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"%&'() *+ ,# .#$/0123455789:;<==?@A BCEGCFFIJKKMMOOR SUTV2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$i]pf.(`zc-+s2$Q;5/t2$,4\P3!f2$zTXYTkΠ5'2$;[žr5b$l 'Mg}6_Qܱ[W#s2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$I 2$bLqxݰ97$9+l 2$f:emV5Bjve) 2$C 84AkV $2$6GA}0  b 2$~j݊ֈaƘz^Wu <AA@8VT~ʚ;o89ʚ;g4BdBdHw 0Zppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0` f3f3` ff̙` ___` ff` 3ff3f>?" dd@,?lPd@   Z l<@ d`"  n?" dd@   @@``PR    @ ` ` p>> nf (   ^T y   "yZ   S B7 C DEHFPv @?q 6 6 n(dx  G b q %(@"B y2   ZG1?"  T(gֳgֳ ?"P  LUSQdkYkHrh7h_    Tgֳgֳ ?"``  Z*  T\gֳgֳ ?"`   \*  Tgֳgֳ ?"`   \*   0ȶ "  8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~  `  Hd޽h? ?  f3f3 Soaring0   0  @(  T t  "tZ  S BsC5 DEHFPv @?3 r4 rv$[I_l3HxF 3 %(@" 2  3 CRENGCJRQ1? `TO`TR`TO`TRR`TRR"t  TTgֳgֳ ?"/O  LUSQdkYkHrh7h_    Tgֳgֳ ?"pp   NUSQdkYkHroRh7h_   Tgֳgֳ ?"``  Z*  TLgֳgֳ ?"`   \*   T,gֳgֳ ?"`   \*`  Hd޽h? ? f3f3 0 ~v@(    0y P    P*    0}     \* d  c $ ?    0聍  @  8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    6ԇ `P   Z*   6 `   \* H  0޽h ? ̙3380___PPT10.Cxx 0X(  X X 0 P    P*   X 0     Z*  X 6t  `P   X*  X 6 `   Z* H X 0޽h ? ̙3380___PPT10.{P{  0  :(  r  S 4  )     S h>  0  "p`PpH  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +   0 P|(  |r | S 0 P    | c 0e0e    "p`PpH | 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +v  0 p-(  r  S      r  S  2    0H 6  W  ]  0    Two nonets above or below/near 1 GeV : 1. f_0(600), f_0(980), K_0^{*} (800), a_0(980) 2. f_0(1370), f_0(1500)/ f_0(1710), K_0^{*} (1430), a_0(1450)   0 c '' }5 It has been suggested that the scalar mesons above 1 GeV can be identified as conventional two-quark states, even with some possible gluon component. The light scalar mesons below or near 1GeV are dominated by multi-quark states. F. Close and N.A. Tornqvist, J. Phys. G 28, 249 (2002) ,6P9H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 ,(  ,r , S 06    , c 7gֳgֳ ?    H , 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +O  0 vn 4(  4r 4 S \K    r 4 S 4L P   F 4 S A Ό  4 0$!  `  ^" lP j 4 BA *?  4 0O 0 P  /The decay constants \bar{f}_s is defined as: 0 lP0j  4 BA *? 0   4 0In general, the above two twist-3 DAs have the following form: ? lP?j  < BA |?I7H < 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 DT(  Dr D S     D c |gֳgֳ ? ` `0   j D BA ?  P D 0l   The correlation functions can be calculated by virtue of OPE in deep Euclidean region (-q^2>>0). The results can be written as >0lPZ0lPZH D 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +m  0 0L$(  L L c @gֳgֳ ? `0    L ZA   ? P  j L BA ? Q@H L 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +m  0 PT$(  T T c ͇gֳgֳ ? `0   j T BA  ?S C  T ZA   ? `  H T 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +   0 p \R(  \r \ S ч `0   \ c ҇gֳgֳ ? `00   j \ BA   ?@j \ BA  |?@ j \ BA |? @ j  \ BA أ? @ H \ 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +   0 +#d(  dr d S  'G   d c lgֳgֳ ?    j d BA ?@PE d 0   In order to suppress the contribution from excited states and continuum states, we apply Borel transformations to both sides of above equations$0lPZj d BA Ό?  H d 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +   0 D<l(  lr l S  2   l c gֳgֳ ? 0   ^ l 6A?@`  j l BA ? ` pj l BA ?@`  H l 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +=   0 d\t(  t t c  [gֳgֳ ? `0  [ j t BA ? P0 j t BA ?0 P H t 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 n(  r  S l     c `gֳgֳ ? P   j  BA ?p` j  BA ? IH  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +m  0 $(  r  S p[ `  [ r  S H[   [ H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +m  0 0$(  r  S  [ P  [ r  S [ 00  [ H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B + ( 0 P\(  \r \ S -[ p [  \ c -[gֳgֳ ? `0  [ j \ BA+ Ό?G  \ 0/[  ~ < H \ 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 Pz(  r  S d&     r  S <' 0   j  BA |?%j  BA |? I j  BA  |? @ H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 p e(    Tgֳgֳ ?P 8the mass of a_0 within Borel window at s_0 = 4.5 GeV^299C  NA ? @   c gֳgֳ ?      TDgֳgֳ ? eMoments for twist-3 DA \phi_{a_0}^{s} of a_0 The mass within Borel window can be plotted as below. Te lP.G 7G "?!H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0  ^(  r  S + jT: + j  BA |?   T+gֳgֳ ?  ^    0X+ P XSimilarly, we can display the decay constant within Borel window [1.3,1.6] GeV^2 below. Y lPY^  6A?p  c 8+gֳgֳ ?   + H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0  ;(  r  S $+ 0@ + j  BA |?0 PP  0l+  aMaking use of the above mass and decay constant, we can plot the first two moments of a_0 below. b lPb^  6A?  `0  0+ @P` \  lP   0h+ w >the second moment of a_0 (\xi^{s}_{2,a_0}) within Borel window"? lP?  c $+gֳgֳ ?  0  + H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0  (    NA ? 0   +  0+  ``  =the forth moment of a_0 (\xi^{s}_{2,a_0}) within Borel window"> lP>#  0+   q The values of the first two moments for a_0 from sum rules (16) are [0.30,0.35] and [0.18,0.22] respectively. r lPr  T+gֳgֳ ?  Y  G   c `+gֳgֳ ?    + H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 (  r  S \u+ `  + r  S 4v+   + j  BA! |?@``  s *T~+ i  /The first two moments for \phi_{a_0}^{\sigma} of a_0 within Borel window [1.2,1.5]GeV^2 and [1.1,1.4]GeV^2 are showed below. From the figures, we can obtain the number of these two moments: \xi_{\sigma}^{2,a_0} =[0.21,0.23], \xi_{\sigma}^{4,a_0} =[0.099,0.107] . 8 lPq lP0H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 (    c (e+gֳgֳ ? `0  +   NA" ? p  +  0`f+ P  F the second moment of a_0 (\xi^{\sigma}_{2,a_0}) within Borel window"G lPGH  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 0(    c  4+gֳgֳ ? `0  +   NA# ? z  +  0^+  @ E the forth moment of a_0 (\xi^{\sigma}_{4,a_0}) within Borel window"F lPFH  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +9  0 `XP(    c G+gֳgֳ ? P  +   c H+0e0e  P `p + H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +L  0 skp(  r  S @*+ 0  + r  S ++ `0  + j  BA$ |?a e  0t.+  P   Here the threshold value is chosen as (5.4 \pm 0.3) GeV^2. The first moment of \phi_{k^{*}_0}^{s} is not zero due to SU(3) symmetry breaking effect. 8? lP] lPH  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +Y  0 x(  r  S 4[ `  [   c [gֳgֳ ? 0  [ j  BA% |?   0+ m F the first moment of k^{*}_0 ( \xi^{s}_{1,k_0}) within Borel window4G lP'H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 5-(    c P[gֳgֳ ? `  [   ZA& | ?   [   0+  0, E second moment of k^{*}_0 ( \xi^{s}_{2, k_0}) within Borel window4F lP)H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 (  r  S X[  [   c 0[gֳgֳ ? `0  [ H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  0 %(  r  S [ `0  [ r  S T[  [ p  HA( | ?  0[  K the first moment of k^{*}_0 ( \xi^{\sigma}_{1,k_0}) within Borel window4L lP,H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B +.  0 UM (      c [gֳgֳ ? `  [    6[: n Z The number of these two moments are in the range of [0.02,0.036] and [0.13,0.17] .89 lP" lP[   ZA' * ? 0  [   0[  0  I second moment of k^{*}_0 ( \xi^{\sigma}_{2,k_0}) within Borel window 4J lP-H   0޽h ? f3f3y___PPT10Y+D=' 2 = @B +9 ! 0 `X0(    c [gֳgֳ ? P  [   c X[0e0e   [ H  0޽h ? f3f3y___PPT10Y+D=' 2 = @B + " 0 P (   r   S  0     c gֳgֳ ? `   H   0޽h ? f3f3y___PPT10Y+D=' 2 = @B +  # 0 3+p$(  $ $ c \[gֳgֳ ? `0  [  $ ZA) * ?   [  $ 0|[ P @ C the second moment of f_0 ( \xi^{s}_{2,f_0}) within Borel window4D lP'H $ 0޽h ? f3f3y___PPT10Y+D=' 2 = @B + $ 0 "4(  4r 4 S }[ `  [ r 4 S <~[   [ p 4 HA/ Ό ?k 4 0[ < H the second moment of f_0 ( \xi^{\sigma}_{2,f_0}) within Borel window4I lP,H 4 0޽h ? f3f3y___PPT10Y+D=' 2 = @B +$  0 ;3`d(  d d s o[gֳgֳ ? #" 0e  [  d 6p[o 3 The value of this moment is located at the range of [0.13,0.18]. The forth moment of f_0 (\xi^{\sigma}_{4,f_0}) could not be obtained due to unstable platform.h7 lP lP> lP* lPH d 0޽h ? f3f3___PPT10i.+D=' 2 = @B +Q % 0 xp<(  <r < S a[ `  [ r < S b[ z [ j < BA. |?`  j < BA* |? 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R4sE"~1-z>Lx6Nւ7RKE5GeGT:1Vޡo7;vrjwK 0qynuuq&?߿}7fjgNWh\ @-UrGGS0qdոq .W0͉cSO nB2$PxkL$5I-ac2$ZN㱅!2$I 2$bLqxݰ97$9+l 2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 2$6GA}0  b 2$~j݊ֈaƘz^Wu <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 p(    Tgֳgֳ ?P 8the mass of a_0 within Borel window at s_0 = 4.5 GeV^299C  c gֳgֳ ?      T@gֳgֳ ? eMoments for twist-3 DA \phi_{a_0}^{s} of a_0 The mass within Borel window can be plotted as below. Te lP.G 7G "?!  # lA- Ό ? k H  0޽h ? f3f3y___PPT10Y+D=' = @B +  0  j(  r  S @˘ jT:   j  BA |?   T( gֳgֳ ?  ^    0D3  P XSimilarly, we can display the decay constant within Borel window [1.3,1.6] GeV^2 below. Y lPY  c - gֳgֳ ?    j  BA ?u H  0޽h ? f3f3y___PPT10Y+D=' = @B +rV($*V[1lEn amplitudes of scalar mesons from QCD sum rulesOutline Introduction Introduction Introduction+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesons+Sum rules for the moments of scalar mesonsNumerical calculationsMoments for twist-3 DAs of a_0/Moments for twist-3 DA \phi_{a_0}^{s} of a_0 /Moments for twist-3 DA \phi_{a_0}^{s} of a_0  /Moments for twist-3 DA \phi_{a_0}^{s} of a_0 /Moments for twist-3 DA \phi_{a_0}^{s} of a_0 /Moments for twist-3 DA \phi_{a_0}^{s} of a_0 5Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0 5Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0 5Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0 #Moments for twist-3 DAs of k^{*}_05Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_05Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_05Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_05Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0<Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 Moments for twist-3 DAs of f_0-Moments for twist-3 DA \phi_{f_0}^{s} of f_0-Moments for twist-3 DA \phi_{f_0}^{s} of f_04Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_04Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_0Summary and OutlookSummary and OutlookThanks!  õʾĸģǶ OLE  õƬ)\˼sg{ւ!:'[}YZ'Y R܋M垵fޚ imw=kuzOx}V}mgݳvYCCY詰lam(@,kSGo OZБ}ׯ}ۏÚ=YpY}E56o{oU!~osskZ9i-瀣xطاr߶OrN~9`YT0Ǿ]^5 OZZrν~uf ߯ikZ>լoߒ)^vLzYI6^:3 |hOSIւ%g>ʜr|ؿ-ϗ2 e{SY QWm:3ʽmv߫5T蓬6{gF4{jVڬeݫcZ@mk3 ?@ǰѳǙת8+f詰`N~>ν_ } d]v7EU=nH\T)zcC:XOZ@gmf/3484+͜rזs嵶0ne׸> kǵ嵶Zq=N{%}Ff7+`ݠsʮeݳM׶: IXUsr֦y`׶>kQDŽ 9Wtmv8cjoaGyp-^7@d-xV`G~kYkm~k쩖/UryԶ_󚢯yb ϋr󑛮q)l&'Y m?GF痣?G//yNpY}Zi9L~?{$<)[-ajs{aa˳{f=_j<&Ua5aW=??Q76a{a-詬rdmf?ە3i}ZQ4z.ܧ'lV@ gw]'a-=j{wa@ܹfϦ1}=У_}#'a-%f g.詰@5ÉQƬqa@ͺGk{q}l\'bjVsSa- =q!5IU.{7zeG|18gcXY`x=ׯ_G߿h7oތFOOO?jTO|<<Z@N޳F=lFo=~V1}R_=kx4?jnճpUc5f5c=FzX=,Y.f=i$޷hy??G 1|a]г$QC۷gث]s]k]kacVY[ 쵛YY-Fgcw@9Ezed{Yb=+lO~u=g-~ױ:+,usg7Uvv}wB/;imɺf7\yLyXѷ{o=FؓD͞5bqX=+>`ׯvlVY.1@ 5MtqTUyjoaq F@Eszz{uc_v{Ya>8fAYm}xx-y뼾5zO>Y ذ.31 wE"{Y'_,{ŖgԳFZOju68f2 v=]6;83+)Tu}I=Kj,,.V9oX}:]^wгb-Hk68lpfA @fv{V@Oe-`[{#~6ssF@d-W{droW|7=,$kёYC ]7w8Ow83cZ]] L_){ظmG >Z&R'-^jY}b^詬,~mc^ؾ+ؗ~5_{F_@d-`kٷ]ݳ3+@O`b8}WK={O)]]^pæ~oViVSa-ܾk쳖'vЫ@֢ w=Y0zZ |6+>k ,&eOZС,O<9a2Tp1@}RYk[vuky{6b, :@`k^c>8f5 >E蓰;?Wl5Y竳6zsTX fWk^va>E'Y kZv8f~x\qTpZ0`843Ev->ZЫX84+9gJy|Tւk^ -Ή'Y ڧw+?]G3y;vNԊf=`He8yls:' @d-Чй-sb8zYDAf 詰=)_ G`f[@@ @[@DYn}Vu~~WVϔr-ZSؖD'cק{~VйƶO-k zZ5W[@f8ЁYkklc>5 k>\c[kl}b3TZ X5q?[z*^qyc@d-h8'ى>vtl>uf<+֬詬Ő{~zZ51ckl$kO9.u_جagTBO lj>}}}R4yMqn kx}}<>>Z,=+Y=--'ǵ3<;^S3Z~WcгZ }*\Ov Sa-V9=i}}3{3ǠOZ=ev9:ʎ&ѻZW Y0+OYɠZic5]c5a{k l|&ru$[Ƶ:8+٬KD=EyS'ɤǍ^׺]96s $_jvzFS\&Ϫ kxfvEw^kpqfAD֢&׾)r渼/m9s쾴>լY>M98:'k-6zFS: E u9z`ɠ: `3?ߗ6~(гZb=84+ؓYMuF蓬iV`߬@Oe-gD>Z tV`/YyR_IZ5MgDޭ3'a-qQz|:4+0_+Z8Q蓰a~FT_(If~y7 uVψ:9)kk@ĞlQ> kAgf+OӺz*Yj<&lr~O{7@gf_tマ k@~m~mk'Y lqVYXi<>詬tIGZ'ziעO |Vp,ZT_׺$k]=ظO=xǨGSY ؆ؗמدEd-~-$f܃詰`}FNqq1}h^slxV` kS垞&=մ׍os߼y3 kۯZz֡Qo=ȗ/_^_Nun-lV/?kC5n;[}ϳН؛u}ʽצYc8,2{ѿZ sf!=YYuX ۽~=kY'f[ 8笀Zmy{٘U^׭'٘.;gѹAZm-_oZOϷ0a~>i_蓬,9+gV`b_6kON~ƽꩬ%'G!~o_O=/f~mϖ}ؿMyVOe-nƵG|IXqVsfb_Oƹȱ_;{詬/ѧ~m_['iOZ]ROC>iY X7`1_=O0 ~>$kf܃ӇTZj<&&kCPz#luVYkxJ PեIzVHrVЬ&fwyJ V|͞OҳБY} {,)=k|د'Y2+ןY='a-Ib ?pZ蓬tg詬> kԬYSa-IHjV̬Z蓰$5+֬z֢ 7y`X}RYC[}'yER =@d-f 詬> k@RfV=@d-RlVSY }6+Z蓬>)Y X7}'a-[ Z蓘xLA Q^k'y$5+oV@O'Y 3+z*kOZI Wz*kOZ蓬Ŋ߹ kOҹfSY } Y3Z蓬Ԭ[ kOZԬYT)E5@o7pVf$k>Z$uYSa-IHjV`gV@Oe-IX RlV=@d-6+詬> kI Sׯ_G߿h7oތFOOOW?)կ02[kg+z!5~o{g r||-n^@Z=Y׺׵OZ3^˹ȪYY\C ¼qo\úhjVmc:KHgy]d]Ǟx7+lZﴉ=uϞ}߸N5dٳ;k8bqX=+laV`ϬkT{U3zEg`}Ҭs^5z}hr8no{Y f{f@O|pOY69ѫVU˯7cxob5zr}-q5qӪzV37+0ڧ6]7Ҿne^:8zO@j-I k-Z \;WSa-I:W̬z*kOZ$5+֬詰$k$5+phVSy> k@RfSY IZj<&hިլY5蓬$kԹ5+z*>Z) |6+$I \@O'Y f z*kOZ@Ro 詰$kOJjV @d-IX 4+\SY }m֢ Y=>k 蓬$5+0 詰$k@J Z蓰@RSY }@d-8sZ蓬$u[詬> kԬYSa-IHjV`߬Z蓰$5+gV=@ԽcFfլI}H\f@O'Y kI \SY }f Z蓬@Ro z*kOZ@RfTX }'%5+ߝYO$[s詬> k$5+0+f=UkQ鍼~ |6+Z蓬HjVڬ詬> k$5+pY='a-IbYTX }84+Zatzz:nnn}ӷ}iTqZ  m7<==Mzګi}jƯ#7oF_~z֕^lշžj˗/K}OX `'{fRbZFQ8г[ƌp}1[1KZ`V`42+vbulq\̾j\3dYN˳Wݲ׍Z f>gV`^;}u{DogHM*={13_/bW' O*vׯ.sS|\̺NvD_;gE=⌦Yׯy zY=+$:+pfV@G-M~[^_t'-ӷ23WWWύ~axkV-@d-84+詬> kI Sa-I@Ԭ^f>Z蓰lxV`;='a-f>TxL0Ykz*> kԬYSY }HjVmgT@'Yg+詰$kI+oV='a-3+z*>ZI ۬Z蓰4+٬z*ka-I`(}RYk^j<&HodO@gf@O'a- YfT@ YC詰$kԬY@Oe-IX HjV`Ϭ kOfƿչY->Z蓰lpVo9WT@R6+詰nE5XìY}^k'a-xkVT@8ZZ|8NN~8?e?:zVf DZ_?|}7VU_7Qw~Yc׳ Y.ľ=n}yM\= @k-ڻjoۦ?7__ $Efcw{>+0YSa-'b ; 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Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. 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"%&'() *+ ,# .#$/0123455789:;<==?@A BCEGCFFIJKKMMOOR SUTV2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$i]pf.(`zc-+s2$Q;5/t2$,4\P3!f2$zTXYTkΠ5'2$;[žr5b$l 'Mg}6_Qܱ[W#s2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$I 2$bLqxݰ97$9+l 2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 2$6GA}0  b 2$~j݊ֈaƘz^Wu <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang 5  Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 =5 (    0xp   `9  =the forth moment of a_0 (\xi^{s}_{2,a_0}) within Borel window"> lP>#  0    q The values of the first two moments for a_0 from sum rules (16) are [0.30,0.35] and [0.18,0.22] respectively. r lPr  T$ gֳgֳ ?  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Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 6.(    c  gֳgֳ ? `0     0  P  F the second moment of a_0 (\xi^{\sigma}_{2,a_0}) within Borel window"G lPG  # lA Ό ? sQg  H  0޽h ? f3f3y___PPT10Y+D=' = @B +r @4?[1lE(                           ! 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"%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJKKMMOOR SUTV2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$i]pf.(`zc-+s2$Q;5/t2$,4\P3!f2$zTXYTkΠ5'2$;[žr5b$l 'Mg}6_Qܱ[W#s2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}$2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$P|P*A~DBw 2$#8D~g>I 2$bLqxݰ97$9+l 2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 2$6GA}0  b 2$~j݊ֈaƘz^Wu <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 5-0(    c  gֳgֳ ? `0     0   @ E the forth moment of a_0 (\xi^{\sigma}_{4,a_0}) within Borel window"F lPF  # lA" Ό ? 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"%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJKKMMOOR SUTV2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$i]pf.(`zc-+s2$Q;5/t2$,4\P3!f2$zTXYTkΠ5'2$;[žr5b$l 'Mg}6_Qܱ[W#s2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 $2$#8D~g>I 2$bLqxݰ97$9+l 2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 2$6GA}0  b 2$~j݊ֈaƘz^Wu <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0|ppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. 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Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(Q % 0 xp <(  <r < S Q  `  Q  r < S $Q  z Q  j < BA( Ό?3s j  < BA1 Ό?s H < 0޽h ? f3f3y___PPT10Y+D=' = @B +rJPc&&of[1E( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh5 0D[SOes New Romanhh5 0 DArialNew Romanhh5 00DWingdingsRomanhh5 0@DўSOgdingsRomanhh5 0@ .  @n?" dd@  @@``   dZSS Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$i]pf.(`zc-+s2$Q;5/t2$,4\P3!f2$zTXYTkΠ5'2$;[žr5b$l 'Mg}6_Qܱ[W#s2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. 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"%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(                          ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | }  aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(E   0 ldp \(  \r \ S  `0   \ c dgֳgֳ ? `00   j  \ BA  Ό?H \ 0޽h ? f3f3y___PPT10Y+D=' d= @B +r  %[1E( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh5 0D[SOes New Romanhh5 0 DArialNew Romanhh5 00DWingdingsRomanhh5 0@DўSOgdingsRomanhh5 0@ .  @n?" dd@  @@``   dZPS Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0 D<l(  lr l S  2   l c gֳgֳ ? 0   ^ l 6A?@`  j l BA ? |<pj l BA ?@|< H l 0޽h ? f3f3y___PPT10Y+D=' d= @B +ra-? =RC[1E( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh5 0D[SOes New Romanhh5 0 DArialNew Romanhh5 00DWingdingsRomanhh5 0@DўSOgdingsRomanhh5 0@ .  @n?" dd@  @@``   dZPS Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0 D<l(  lr l S  2   l c gֳgֳ ? 0   ^ l 6A?@`  j l BA ? Ej l BA ?@D H l 0޽h ? f3f3y___PPT10Y+D=' d= @B +rCZ jC[1E( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh5 0D[SOes New Romanhh5 0 DArialNew Romanhh5 00DWingdingsRomanhh5 0@DўSOgdingsRomanhh5 0@ .  @n?" dd@  @@``   dZPS Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(aY.M Wang Collaborated with C.D Lu and H. Zou Institute of High Energy Physics, CAS"bZS OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the ma ss and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0 D<l(  lr l S  2   l c gֳgֳ ? 0   ^ l 6A?@`  j l BA ? q<j l BA ?@< H l 0޽h ? f3f3y___PPT10Y+D=' d= @B +r [1E( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh5 0D[SOes New Romanhh5 0 DArialNew Romanhh5 00DWingdingsRomanhh5 0@DўSOgdingsRomanhh5 0@ .  @n?" dd@  @@``   dZPS Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =0CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(eY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS"fZW OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/ *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  :(  r  S ,+  )     S 0  0  "p`PpH  0޽h ? f3f3y___PPT10Y+D=' v= @B +rC [1~I( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh5 0D[SOes New Romanhh5 0 DArialNew Romanhh5 0"0DWingdingsRomanhh5 0@DўSOgdingsRomanhh5 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0|ppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =3CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(eY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS"fZW OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * iGt . 5 F *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  ph(  hr h S v i v  h s X4vgֳgֳ ? #" 0e  v H h 0޽h ? f3f380___PPT10.#r \j+[ l\1qI( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh5 0D[SOes New Romanhh5 0 DArialNew Romanhh5 0"0DWingdingsRomanhh5 0@DўSOgdingsRomanhh5 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd\q 0jppp@ <4!d!d4w 0h5<4dddd4w 0h5 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =3CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(eY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS"fZW OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $ZStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. Production of scalar mesons in B decays can be very useful to discover the characters of these mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions.b Z!Zg Z2!Z/IP/\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jKb . 5 Q *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  ph(  hr h S v i v  h s X4vgֳgֳ ? #" 0e  v H h 0޽h ? f3f380___PPT10.#rl\f+l\1I( X  W Microsoft lQ_ 3.0                           ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0dppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =h3CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(eY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS"fZW OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP#PD\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jKj . 5 Q *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 ,(  ,r , S & <   , c 'gֳgֳ ?    H , 0޽h ? f3f3y___PPT10Y+D=' = @B +rN s*N\1H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0jppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =2CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(eY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS"fZW OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP#PD\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  ph(  hr h S ( <   h s Hgֳgֳ ? #" 0e   H h 0޽h ? f3f380___PPT10.#r\=N+fP\1H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0jppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =2CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(fY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "gZX OutlinecIntroduction Sum rules for the moments of scalar mesons Numerical calculations Summary and outlook 2c$%c$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP#PD\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  :(  r  S @C  )     S Nv n 0  "p`PpH  0޽h ? f3f3y___PPT10Y+D=' = @B +r%QٙQd\1H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0jppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =3CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(fY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "gZX OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP#PD\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.  B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  :(  r  S @C  )     S Nv  0{  "p`PpH  0޽h ? f3f3y___PPT10Y+D=' )= @B +   0 P|(  |r | S ( P    | c 0e0e    "p`PpH | 0޽h ? f3f3y___PPT10Y+D=' )= @B +  # 0 3+p$(  $ $ c  gֳgֳ ? `0    $ ZA) * ?     $ 0 P @ C the second moment of f_0 ( \xi^{s}_{2,f_0}) within Borel window4D lP'H $ 0޽h ? f3f3y___PPT10Y+D=' )= @B +r w J$|\1H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0jppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =3CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(fY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "gZX OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP#PD\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  # 0 3+p$(  $ $ c  gֳgֳ ? `0    $ ZA) * ?     $ 0 P @ C the second moment of f_0 ( \xi^{s}_{2,f_0}) within Borel window4D lP'H $ 0޽h ? f3f3y___PPT10Y+D=' = @B +rSJ*8$/><\1H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0jppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =3CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(fY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "gZX OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP#PD\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(.  0 UM (      c вQ gֳgֳ ? `  Q     6L : n Z The number of these two moments are in the range of [0.02,0.036] and [0.13,0.17] .89 lP" lP[   ZA' * ? HDt      0\L  0  I second moment of k^{*}_0 ( \xi^{\sigma}_{2,k_0}) within Borel window 4J lP-H   0޽h ? f3f3y___PPT10Y+D=' = @B +rz<DQ V<\1H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0jppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =-3CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(yY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, C "zZk OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate that effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP#PD\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7                        ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  "Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  :(  r  S @C  )     S Nv  0Y  "p`PpH  0޽h ? f3f3y___PPT10Y+D=' = @B +rÊ8\1H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanh.-h, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0dppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =,3CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(yY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, C "zZk OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP"PC\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 ,(  ,r , S & <   , c 'gֳgֳ ?    H , 0޽h ? f3f3y___PPT10Y+D=' = @B +rt ]P8"\1H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0dppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =2CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(yY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, C "zZk OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $Study on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Hadronic distribution amplitudes are the important ingredients when applying factorizable theorem to analyze exclusive processes. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. f$Z3%Z4$Z %ZP"\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 ,(  ,r , S & <   , c 'gֳgֳ ?    H , 0޽h ? f3f3y___PPT10Y+D=' = @B +rt" kP"m\1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhh, ) 0D[SOes New Romanhh, ) 0 DArialNew Romanhh, ) 0"0DWingdingsRomanhh, ) 0@DўSOgdingsRomanhh, ) 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0jppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =82CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(yY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, C "zZk OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. T$Z3%Z$ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 ,(  ,r , S & ,i   , c 'gֳgֳ ?    H ,$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$ nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd ) 0jppp@ <4!d!d4w 0h )<4dddd4w 0h ) <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =82CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(yY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, C "zZk OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. T$Z3%Z$ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 ,(  ,r , S & ,   , c 'gֳgֳ ?    H , 0޽h ? f3f3y___PPT10Y+D=' )= @B +  0   <(  < < c 8Wgֳgֳ ? 0    < ZA أ ?   < 0f p  ISetting y=-x=z and expanding the above definitions around z^2=0, we have *J lPHj < BA ?    < 0Pj  >In general, the above two twist-3 DAs have the following form: ? lP?j  < BA |?I7H < 0޽h ? f3f3y___PPT10Y+D=' )= @B +r  u a \1G( X  Root EntrydO)0{ PicturesCurrent User8SummaryInformation(Z ed       !"#$%&'()*+,-./0123456789:;<=>?@ABC FGHIJKNOPQRSbcifgh jklmnopqrstuvwxyz{|}~  !"#$%2$q}]Ro|a"&Ied:l2$0ǴhGK2$:$oU4!x\52$3;vSm)cL 2$U¦D)ӟْ8Ila: 2$Wk~W`|;AUTmb$q*ŤcqJq 2$STĒP%/2$p s e 12$ n n~>=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =52CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  :(  r  S 5  )     S l;  0Y  "p`PpH  0޽h ? f3f3y___PPT10Y+D=' ' = @B +r U  "X \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0hvppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =52CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$3RG evolution of decay constant, mass and condensate4 4#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 ,(  ,r , S p  IY   , c 4gֳgֳ ?    H , 0޽h ? f3f3y___PPT10Y+D=' = @B +r^X P :X + \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4Bd      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{}~Bd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =;2CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 DT(  Dr D S z    D c {gֳgֳ ? ` `0   j D BA ? P D 0l~   The correlation functions can be calculated by virtue of OPE in deep Euclidean region (-q^2>>0). The results can be written as >0lPZ0lPZH D 0޽h ? f3f3y___PPT10Y+D=' = @B +  0 n(  r  S t     c Lgֳgֳ ? P   j  BA ?p` j  BA ? IH  0޽h ? f3f3y___PPT10Y+D=' = @B +rg _ " C \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =;2CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks! /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 n(  r  S t     c Lgֳgֳ ? P   j  BA ?p j  BA ? vH  0޽h ? f3f3y___PPT10Y+D=' = @B +r ";  ? \1 H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =W2CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z( ' 0 L(  Lr L S p'  p@  '  H L 0޽h ? f3f3y___PPT10Y+D=' = @B +r? T (? \1 H( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =W2CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  (  r  S  nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =W2CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$uMaking use of quark-hadron duality, we can match two different forms of correlation functions by dispersion relation,v Zv *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z( $ 0 "4(  4r 4 S U `  U r 4 S U   U p 4 HA/ Ό ? 4 0 '  < H the second moment of f_0 ( \xi^{\sigma}_{2,f_0}) within Borel window4I lP,H 4 0޽h ? f3f3y___PPT10Y+D=' = @B +r N %_ " \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =32CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$QWe can match two different forms of correlation functions by dispersion relation,R$R *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0 +#d(  dr d S ͇ 'G   d c ·gֳgֳ ?    j d BA ?}@PE d 0   In order to suppress the contribution from excited states and continuum states, we apply Borel transformations to both sides of above equations$0lPZj d BA Ό? cH d 0޽h ? f3f3y___PPT10Y+D=' = @B +r" j  " o \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =32CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$QWe can match two different forms of correlation functions by dispersion relation,R$R *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to e _shaopengshaopeng՜.+,0     Ļʾihep)' 0Times New RomanArial WingdingsSoaringMicrosoft ʽ 3.0DTwist-3 distributioxplore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0 +#d(  dr d S ͇ 'G   d c ·gֳgֳ ?    j d BA ?}@PE d 0 S   In order to suppress the contribution from excited states and continuum states, we apply Borel transformations to both sides of above equations$0lPZj d BA Ό? H d 0޽h ? f3f3y___PPT10Y+D=' = @B +rp   o \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =32CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$QWe can match two different forms of correlation functions by dispersion relation,R$R *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0 D<l(  lr l S ܇ 2   l c ݇gֳgֳ ? 0   ^ l 6A?@`  j l BA ? q<j l BA ?X< H l 0޽h ? f3f3y___PPT10Y+D=' = @B +rF 6  " [ \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKdr 0h|ppp@ <4!d!d4w 0hP3<4dddd4w 0hP3 <4BdBd4x 0h6 j0___PPT10 pp*___PPT9 z? %O  =32CTwist-3 distribution amplitudes of scalar mesons from QCD sum rulesDD#(zY.M Wang In collaboration with C.D Lu and H. Zou Institute of High Energy Physics, CAS "{Zl OutlineIntroduction Sum rules for the moments of twist-3 distribution amplitudes of scalar mesons Numerical calculations Summary and outlook 2$%$ Introduction $Structures of scalar mesons are still not clear. two-quark state, multi-quark state, meson-meson states, glueball& :1 M!1I}  Introduction $oStudy on production of scalar mesons in B meson decay can provide much useful information about their inner structure of scalar mesons. Wei Wang, Yue-Long Shen, Ying Li and Cai-Dian Lu, hep-ph/0609082 However, only twist-2 distribution amplitudes (DAs) of scalar mesons are available now. So it is necessary to calculate the moments of twist-3 DAs in order to give more accuracy predictions. Twist-3 DAs of pion are very important for B decays to pion pion, so it is necessary to investigate the effects of twist-3 DAs of scalar mesons in B decays to scalar mesons. P Z3!Z ZP\+ Introduction The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). * jK *Sum rules for the moments of scalar mesons++$,Definition of twist-3 DAs for scalar mesons:- -*Sum rules for the moments of scalar mesons++$ *Sum rules for the moments of scalar mesons++$rIn order to calculate the moments of distribution amplitudes, we consider the following two correlation functions ,r Z!Zs*Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$  *Sum rules for the moments of scalar mesons++$AThe correlation functions can also be calculated in hadron level.B B *Sum rules for the moments of scalar mesons++$QWe can match two different forms of correlation functions by dispersion relation,R$R *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0 D<l(  lr l S ܇ 2   l c ݇gֳgֳ ? 0   ^ l 6A?@`  j l BA ? q<wj l BA ?X< H l 0޽h ? f3f3y___PPT10Y+D=' = @B +r Q  s U \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(   0 P|(  |r | S H P    | c |0e0e    "p`PpH | 0޽h ? f3f3y___PPT10Y+D=' = @B +rU  U \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxz{|}~DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 5The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.6 6D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(Y  0 x(  r  S P6 `     c (7gֳgֳ ? 0   j  BA% |?TI   0lB m F the first moment of k^{*}_0 ( \xi^{s}_{1,k_0}) within Borel window4G lP'H  0޽h ? f3f3y___PPT10Y+D=' = @B +  0 5-(    c Qgֳgֳ ? `     ZA& | ? 9     0^  0, E second moment of k^{*}_0 ( \xi^{s}_{2, k_0}) within Borel window4F lP)H  0޽h ? f3f3y___PPT10Y+D=' = @B +r ; >* ˠ @ \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 4The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.5 5D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  (  r  S  r `0   r  S r q    0d}  K the first moment of k^{*}_0 ( \xi^{\sigma}_{1,k_0}) within Borel window"L lPLj  BA# Ό?C*tH  0޽h ? f3f3y___PPT10Y+D=' = @B +r A]9 ` K= \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 4The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.5 5D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  3T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(r= &c= ` \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz|}~B B *Sum rules for the moments of scalar mesons++$QWe can match two different forms of correlation functions by dispersion relation,R R *Sum rules for the moments of scalar mesons++$RThen, we can find the sum rules of moments for twist-3 DAs of scalar mesons below.S ZS *Sum rules for the moments of scalar mesons++$ "*Sum rules for the moments of scalar mesons++$9RG evolution of decay constant, quark mass and condensate: :#Numerical calculationsbMoments for twist-3 DAs of a_0 Moments for twist-3 DAs of K^{*}_0 Moments for twist-3 DAs of f_04C ! c0Moments for twist-3 DAs of a_0B$bMoments for twist-3 DA \phi_{a_0}^{s} of a_0 Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_06. !4 cY).Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B GThe numerical parameters at 1 GeV scale used in this paper are taken asH H$.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B U Here a_0 indicate that the scalar meson is composed of u \bar{d}. In order to obtain the value of its moments from Eq. (16), we should calculate the mass and decay constant for a_0 before. The mass can be obtained by taking logarithm of both sides of Eq. (16), and then applying differential operator to both sides of Eq. (16), while the decay constant can be immediately calculated from Eq. (16) once the mass is known. For the sum rules of mass, the threshold value and Borel parameter are taken as . (Q !V4tW( B  *.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B KFrom the above figure, we can find that the mass of a_0 is in the range of  L J,.Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B dThe value of decay constant is within Borel window.,J Z!Ze..Moments for twist-3 DA \phi_{a_0}^{s} of a_0 //B  24Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B (MThe mass and decay constant can also be calculated from (18). The results areN N44Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 54Moments for twist-3 DA \phi_{a_0}^{\sigma} of a_0  53B ( 7"Moments for twist-3 DAs of k^{*}_0##B$sMoments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_0 Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_067 !< t94Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B Here k^{*}_0 indicates that the flavor content of scalar meson is s \bar{u}. Following the same procedure as a_0, we can derive the mass and decay constant within Borel window [1.9,2.1]GeV^2 and [1.3, 1.7] GeV^2 from (16) as  ;4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B SThe first two moments for \phi_{k^{*}_0}^{s} within Borel window are showed below.T T>4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 4The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.5 5D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(E   0 ldp \(  \r \ S P `0   \ c (gֳgֳ ? `00   j  \ BA  Ό?H \ 0޽h ? f3f3y___PPT10Y+D=' = @B +.  0 UM (      c gֳgֳ ? `      6  Z The number of these two moments are in the range of [0.02,0.036] and [0.13,0.17] .89 lP" lP[   ZA' * ? HI     0    I second moment of k^{*}_0 ( \xi^{\sigma}_{2,k_0}) within Borel window 4J lP-H   0޽h ? f3f3y___PPT10Y+D=' = @B + & 0 D(  Dr D S 43_ `  _  D c  4_gֳgֳ ? `0  _ H D 0޽h ? f3f3y___PPT10Y+D=' = @B +r  K D Q  p \1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 4The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.5 5D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(.  0 UM (      c gֳgֳ ? `      6  Z The number of these two moments are in the range of [0.02,0.036] and [0.13,0.17] .89 lP" lP[   ZA' * ? Ht     0    I second moment of k^{*}_0 ( \xi^{\sigma}_{2,k_0}) within Borel window 4J lP-H   0޽h ? f3f3y___PPT10Y+D=' = @B +r D   %\1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 4The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.5 5D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0  ph(  hr h S 0. <   h s /gֳgֳ ? #" 0e   H h 0޽h ? f3f380___PPT10.#r&\m+%p\1G( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 4The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.5 5D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(rp+pt\1D( X  W Microsoft lQ_ 3.0 Equation.30 Microsoft lQ_ 3.0D/ 0|DTimes New Romanhht3 0D[SOes New Romanhht3 0 DArialNew Romanhht3 00DWingdingsRomanhht3 0@DўSOgdingsRomanhht3 0@ .  @n?" dd@  @@``   h[ST Q          ! "%&'() *+ ,# .#$/01 23 455789:;<==?@A BCEGCFFIJK KMMOOR SUTVWl2$.5*U+b$Jj#-B!c^J9IU+2$J K/ԁZzB t2$*[/~:b$K+;%Bfv65X2$xšGh2$,Lu${'ZsOF9b$\?0'Jm(ȭ&ab${.-C-2$~FHimV'EyF]2$Q;5/t2$,4\P3!f$$$2$^Uz޴.-6^MD> nB2$PxkL$5I-ac2$ZN㱅!2$=g}2$EI1_vj3g_N2$S1Gµ= 8 % 2$?^Z!\l +=۹ b$ -ݰM0N? b$K%9W̷~Yrk@B7 2$7~;zv Yв#ۊg2$#8D~g>I $2$f:emV5Bjve) 2$C 84AkV 2$uO`"ϿyksVa5 $2$~j݊ֈaƘz^Wu $b$$ga5~q$B9 <AA@8VT~ʚ;o89ʚ;g4KdKd4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B  ?4Moments for twist-3 DA \phi_{k^{*}_0}^{s} of k^{*}_055B From the above figures, we can find that the value of these two moments are in the range of [0.0017,0.0023] and [0.21,0.29] respectively. It is obvious that the first moment corresponding to SU(3) symmetry breaking effect is tiny.  A;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B 4The mass and decay constant of k^{*}_0 from (18) are in the range of [1449,1543] MeV and [350,376] MeV corresponding to Borel window [2.1,2.3] GeV^2 and [1.3,1.6]] GeV^2 respectively. The first two moments for \phi_{k^{*}_0}^{\sigma} within Borel window [1.8,2.6] GeV^2 and [1.0,1.2] GeV^2 are showed below.5 5D ;Moments for twist-3 DA \phi_{k^{*}_0}^{\sigma} of k^{*}_0 <<B  G"Moments for twist-3 DAs of f_0B$bMoments for twist-3 DA \phi_{f_0}^{s} of f_0 Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_06. !4 cH#,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B Here f_0 refers to the scalar meson which is constitute of s \bar{s} quarks. The mass and decay constant of f_0 from sum rules (16) are [1629,1713] MeV and [371,393] MeV corresponding to threshold value s_0=(6.5 \pm 0.3) GeV^2 with Borel window [2.5, 2.7]GeV^2 and [1.7, 2.0]GeV^2. The second moment of f_0 (\xi^{s}_{2,f_0}) within Borel window [1.3, 1.6]GeV^2 is showed below. The value of this moment is located at the range of [0.18,0.26]. The forth moment of f_0 (\xi^{s}_{4,f_0}) could not be obtained due to unstable platform. @ 54J$,Moments for twist-3 DA \phi_{f_0}^{s} of f_0--B  N%3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B The mass and decay constant of f_0 from sum rules (18) are [1616,1703] MeV and [380,427] MeV within Borel window [2.5, 2.7]GeV^2 and [1.2, 1.6]GeV^2. The second moment of f_0 (\xi^{\sigma}_{2,f_0}) within Borel window [1.5, 1.8]GeV^2 is showed below.  [*3Moments for twist-3 DA \phi_{f_0}^{\sigma} of f_044B  P&Summary and OutlookIn this work, we have calculated the moments of twist-3 DAs for scalar mesons. For convenience, we collect the results of mass, decay constant and Gegenbauer moments at 1 GeV scale as follows. ZQ'Summary and OutlookOur results indicates that the Gegenbauer moments of twist-3 DAs for scalar mesons are small. The twist-3 DAs of scalar mesons can be applied to various approaches, such as PQCD, QCDF, LCSR, to explore the inner structure of scalar mesons in the exclusive process of scalar meson production in heavy flavor hadron decay. We can also investigate the DAs of glueball component for scalar mesons to help us discover the mystery of scalar meson.  T(Thanks!6 /D     ! %&')+-/1368:<=@BCEK!L"M#O$R%S&U'Z(  0 aYph(  h h s /gֳgֳ ? #" 0e    h 0$ q< _ The idea of the QCDSR formulism is to approach the bound state problem in QCD from the asymptotic freedom side, i.e., to start at short distances and move to larger distances where confinement effects become important, asymptotic freedom starts to break down and resonances emerge as a reflection of the fact that quarks and gluons are permanently confined within hadrons. The breakdown of asymptotic freedom is signalled by emergence of power corrections due to non-perturbative effects in the QCD vacuum. These are introduced via nonvanishing vacuum expectation values of quark and gluon condensates operators. L.J. Reinders, H.R. Rubinstein, S. Yazaki, Phys. Rep. 127, 1 (1985). P %iE6uH h 0޽h ? f3f380___PPT10.#r\+{\1Root EntrydO)ی PicturesCurrent User8SummaryInformation(Zd      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz|}~