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Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis (A toy model with angular momentum mixing)) Model Hamiltonian (solvable)Case I: " # Case II:    QCD case4The one-gluon exchange potential (equal helicity) .51( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B !The partial wave expansion (m=0):""C 5The angular dependence with non-zero azimuthal number66% HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! / 0` Ot{h______` M <ff33̙3` +ffO=ff̙H7` fff3f̙` Tff33ff` 0Ky{kOz` )R{f` GiIfff̙fR` ̙|̙3f` 3ff~>?" dd@'?lFd@  nK'o`P( n?" dd@   @@``PT     o (`0p>> ''^(  ZB  c $D"`   6 "M }  LUSQdkYkHrh7h_     0L "; `  8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    0 "` `  X*   0 "`   Z*   0` "` `  Z* T     "`h2   s *"`h2   s *"h2   s *"@h2   s *"0`h2   s *"0h2  s *"0@h2  s *"`0h2  s *"`h2  s *"h2  s *"@h2  s *"`n2  0" h2  s *"``h2  s *"`h2  s *"@`n2  0"``h2  s *"`h2  s *"n2  0"@n2  0"`h2  s *" h2  s *"`@n2  0"@n2   0"@@h2 ! s *"`@n2 " 0"``n2 # 0"`h2 $ s *"`@h2 % s *"``h2 & s *" h2 ' s *"` H  0޽h ? 3ff~___PPT10i. s+D=' U= @B + Network 0 (((  ZB  c $D"   6U "&wf U LUSQdkYkHrh7h_    0ȎU "wQ  U NUSQdkYkHroRh7h_   00U "` ` U X*   0ؗU "`  U Z*   0U "` ` U Z* T `]   "]p h2   s *"`]h2   s *"]h2   s *"]Eh2   s *"`h2   s *"h2  s *"Eh2  s *"yh2  s *"`B h2  s *"B h2  s *"EB h2  s *"yB n2  0",B h2  s *"`u  h2  s *"u  h2  s *"u E n2  0"yu  h2  s *"`(  h2  s *"(  n2  0"( E n2  0"y(  h2  s *",(  h2  s *"` Z n2  0" Z n2   0" EZ h2 ! s *"y Z n2 " 0"`  n2 # 0"  h2 $ s *" E h2 % s *"y  h2 & s *"A  h2 ' s *"yA  `B ( s *D"H  0޽h ? 3ff~___PPT10i. s+D=' U= @B + 0` Ot{h______` M <ff33̙3` +ffO=ff̙H7` fff3f̙` Tff33ff` 0Ky{kOz` )R{f` GiIfff̙fR` ̙|̙3f` 3ff~>?" dd@'?lFd@  nK'o`P( n?" dd@   @@``PR    o (` p>> p''^(  ZB  c $D"`   6U "M } U LUSQdkYkHrh7h_     0U "; ` U 8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    0U "` ` U X*   0U "`  U Z*   0U "` ` U Z* T     "`h2   s *"`h2   s *"h2   s *"@h2   s *"0`h2   s *"0h2  s *"0@h2  s *"`0h2  s *"`h2  s *"h2  s *"@h2  s *"`n2  0" h2  s *"``h2  s *"`h2  s *"@`n2  0"``h2  s *"`h2  s *"n2  0"@n2  0"`h2  s *" h2  s *"`@n2  0"@n2   0"@@h2 ! s *"`@n2 " 0"``n2 # 0"`h2 $ s *"`@h2 % s *"``h2 & s *" h2 ' s *"` H  0޽h ? 3ff~___PPT10i. s+D=' U= @B + 1_Network 0 (((  ZB  c $D"   6V "&wf V LUSQdkYkHrh7h_    0V "wQ  V NUSQdkYkHroRh7h_   0@ V "` ` V X*   0tV "`  V Z*   0V "` ` V Z* T `]   "]p h2   s *"`]h2   s *"]h2   s *"]Eh2   s *"`h2   s *"h2  s *"Eh2  s *"yh2  s *"`B h2  s *"B h2  s *"EB h2  s *"yB n2  0",B h2  s *"`u  h2  s *"u  h2  s *"u E n2  0"yu  h2  s *"`(  h2  s *"(  n2  0"( E n2  0"y(  h2  s *",(  h2  s *"` Z n2  0" Z n2   0" EZ h2 ! s *"y Z n2 " 0"`  n2 # 0"  h2 $ s *" E h2 % s *"y  h2 & s *"A  h2 ' s *"yA  `B ( s *D"H  0޽h ? 3ff~___PPT10i. s+D=' U= @B +0 ~v(    0,V P   V P*    00V    V \* d  c $ ?  V  0<5V  0 V 8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    6;V _P  V Z*   6(AV _  V \* H  0޽h ? 3380___PPT10.L`>0  0(  x  c $QV00 V x  c $LRV7_  V H  0޽h ? 33___PPT10i.o?+D=' U= @B + 0 0T0(  Tx" T c $X_VM }  V x T c $`LV; ` V H T 0޽h ? 3fffff3̙3f̙___PPT10i.~7++D=' U= @B +  0 ! (  r  S \qVP   V r  S 4rV ` V   <sV"`@h  1Noninteracting quarks (s=1/2): (Pauli principle)202$  <A ?"6@ NNN?N   <pyV"`@T  KInteracting quarks:04  # A    <}V3"`P &= BCooper theorem0l  68c"` 0   <܀V"`p b  QColor superconductivity03  HA ? ?"6@ NNN?N  H  0޽h ? 3ff~___PPT10i. A+D=' U= @B + 0   (  r  S lVM }  V   HA ?  V  <V"` n TAt ultra-high density: CFL03q  <TV; X  At moderate density: Beta equilibrium Charge neutrality Non-vanishing s quark mass Give rise to a mismatch \delta\mu 03  <ЗVfF: DM. Alford, K. Rajagopal, T. Schaefer, A. Schmitt: arXiv:0709.4635v2E0E .H  0޽h ? 3ff~___PPT10i.l5+D=' U= @B + 0 P=(  r  S pV+0  V X  0A? f X  0A?R `   <|VW D  _)A. Schmitt Phys. Rev D 71, 054016 (2005)*0*H  0޽h ? 3ff~___PPT10i.vq+D=' U= @B +5  0 L D ` (  r  S xVP   V r  S (V` ` V   <V"` QTransition temperature:0j  S A ??"?jZGY   <V"`  TGap equation (non-linear):0a8     xgZB   s *D30 @0    <V  C No mixing 0 3X  0A? 0   <`V `   21) Spherical solutions: no angular momentum mixing30338XBF '[   `r  S A  ??"?'0 r  S A  ??"?t[ 8  WJ> ZB  s *D3   <VWJ> rAngular momentum mixng03  <\V` `,G  ,2) Non-spherical solutions (equal helicity):-0-3"H  0޽h ? 3ff~___PPT10i.`XZ+D=' U= @B +|  0   p'# (  r  S `VM }  V    <8V"`;  @ V j  S A   ??"?0    <V'"   VIn mean field approximation:03  <V B  G Gap equation:03  <hV 4   F Free energy: 0 3X  0A ? 0 | X  0A? p h X  0A? k  pp  # #" ` ` @ V  <V?pp  P o(`B  0o ?pp`B  0o ?p p `B  0o ?pp `B  0o ?pp p pp  "3 #" ` P@ # <D ?pp  P o(fB $ 6o ?ppfB % 6o ?p p fB & 6o ?pp fB ' 6o ?pp H  0޽h ? 3ff~___PPT10i.H㺬+D=' U= @B + 0 tl(     6V"`p   X  0A?ppX  0A ?   <(P7 o-The interaction potential can be expanded as &.0,3  <P 7  TThe gap function: &03X  0A? p X  0A?^    <4 t  8with0  <|g SThe free energy: &03X  0A?\ 2l      ,$@0ZB  s *D3  TB  c $D3    <%g & N,$0 TNo angular momentum mixing03H  0޽h ? 3ff~___PPT10.Q*"Q+YyBDa' U= @B D' = @BA?%,( < +O%,( < +DS' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+8+0+0 +" 0 QI (     61"` p   X  0A?  <3| SThe free energy: &03X  0A?   <8g TThe gap function: &03  <=I 0 |0 ,$0 Z Angular momentum mixing occurred!0!3@  <;"` 0p,$D 0 XThe free energy in case II with angular momentum mixing is lower than that of the case IY0YX  0AF?@r 8    `B  0D3 TB  c $D3 H  0޽h ? 3ff~y q ___PPT10Q .Vg+5D' H= @B Dp' = @BA?%,( < +O%,( < +D$' =%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* +p+0+ 0 ++0+ 0 + 0 &;(  x  c $@UM }      <U"` 0@ U 4  # Ap=j  S A ??"?_   <P   J The pairing strength are equal in leading order, similar to the case II"Ku3K3  <UI 0  2 The subleading terms fall off with increasing J"3u333 "  <d   G Forward singularity of QCD brings about the energy dependence of gap"Hu3H38  "ZB  s *DZB  s *DTB   c $D0ZB ! s *Dj % S Aa `??"?-  `H  0޽h ? 3ff~___PPT10i.YP5+D=' U= @B +I 0   @< (  ~  s *$M }   ~  s *; `  `  c $A ? T X  0A?@ b   <p ['The two-loop approximation to \gamma_2(0(^r  6 &6   <\6 S  ? Powers of T 0   <t=)  DOrder of g^2mu^40  <䗠o FV  EStationary points0X  0A?s   f2  0"` 7f2  0"` JX  0A?I0X  0A?v;B  c $Dz ,$@0B  c $Dz9w,$D0  0A?V FS ,$@0B  s *Dm ,$@0B  s *D] ,$D0  <؟P S 3(D. Rischke Prog. Part. Nucl. Phys. 52 197 (2004))404>H  0޽h ? 3ff~E=___PPT10.\d+2D ' U= @B D ' = @BA?%,( < +O%,( < +D' =%(D=' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+ 0  - p(  8  ,x # c &A "??"?@  "x $ c &A  ??"?   x % c &A$ ??"?   & BP7 lIn Nambu-Gorkov space:0   ' <  IFor one flavor paring, the simplest choice of \Phi reads (equal helicity)J0J@X ( 0A ?0  ) <@6'  KThe NG blocks of S read0X * 0A!? p H  0޽h ? 3ff~___PPT10i.g0(+D=' U= @B +" 0 _ W A (    <Š P9  j   S A# &??"?`B &B  s *D,$D0  <TȠ,$0 REnergy density of normal phase0 l   = ,$@0TB  c $DTB  c $D TB  c $D  TB B c $D  ZB  s *D   <Π   GFree energy density0  S A" (??"?Q 8 ($D0l  $  @ $ ,$D0TB . c $D  3 <A' 0??0 $  0TB : c $D ZB ; s *D @ l  C ? C,$D0TB  c $D ZB   s *D ` r $ S A% ,??"? e H  , 7 <A( 2?? C 2ZB < s *DP  l    A  ,$D0 , <A& .??0   .ZB - s *D   TB > c $D p H  0޽h ? 3ff~80___PPT10.k+ ,jD' U= @B Dg' = @BA?%,( < +O%,( < +DS' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-6B'blinds(horizontal)*<3<*=DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =-6B'blinds(horizontal)*<3<*?D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B'blinds(horizontal)*<3<*@D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =-6B'blinds(horizontal)*<3<*A+8+0+ 0 + 0   P! d (  r  S @0    8      BA* 5??M 5t  6"`   B n F Gap equation     <8 p EMinimization of F0~  BA- J??V J@l ^  ^ ,$D0f  6A+?^ `B  0DO O   BA. K?? S K`B  0DO O  l   Y !  Y,$D0@ (   ( f  6A,?( `B  0Dvv`B  0Dk   < s Y 3We have assumed the gap depends on x=cos\theta only404% H  0޽h ? 3ff~XP___PPT100.z@wq+L[D' U= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-6B'blinds(horizontal)*<3<*!+x- 0   p$X (  $r $ S   %  x  $ <A) R??p Rx  $ <A/ S??v  Sx  $ <A0 T??  T  $ 0A1?W 7 ,$D0X $ 0A2?@- $ 0A3?} T ,$D0B $ s *DT ,$@0 $ 0q ,$@0 $ 0A4? P ,$@0F @ $# D,$@0TB $ c $D@@ZB $ s *D@ $ 0A5? }C ,$D 0 $ <\ }E,$0 ; 1st order    $ <s  ,$ 0 ; 2nd order   @z   $   ,$@ 0TB $ c $D ZB $ s *D   $ <A6 U?? w 8 U$@ 0 $ 0A7? # U ,$D 0 $ <7 LThe gap equation becomes0H $ 0޽h ? 3ff~X P ___PPT100 .z+JD' U= @B DO' = @BA?%,( < +O%,( < +DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$D' =%(DJ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-6B'blinds(horizontal)*<3<* $D' =%(D=' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$D ' =%(D ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$+p+0+$0 ++0+$0 +_ 0 vn,(  ,x , <A6 W??!  W , <T0P k7We introduce a parameter b, which satisfy the condition808X , 0A8?0   , <l5L ITo the leading order:0X  , 0A9?0 @t f  , 0"` j , S A: q??"?0 @   q , <:3"` P   V"The angular dependence is defined:#0# ,0 \A; r??"? @ @P  r  F # ) `  , P 0 ` TB , c $D` ) ` ZB ,B s *D# `  , <@ p  GS-wave gap function0 , <hD'PL A Gap equation 0H , 0޽h ? 3ff~___PPT10i.|0}H+D=' U= @B + 0 <R(  <x < c $tIM }   j < S A< t??"?  t < <VGH@8___PPT9 ^The spherical gap, f(x)=1 is a trivial solution and there is no angular momentum mixing.*H10_I8X < <4Z H@8___PPT9 gThe spin-1 gap, carried over from the polar phase of He_3, f(x)~x, fails to satisfy this equation*C1%0h;'8X < <f ,  L3) Conserve the parity0 < <,h {  ]The solution to this integral equation can be obtained from a variational principle as below.^0^> H < 0޽h ? 3ff~___PPT10i.P^eP+D=' U= @B +S  0 j b &H(  Hr H S  s   X H 0A>?@ Q H <u~ 9where0X H 0A??@ H <8zJ >0  H <h} QConsidering a trial function:0j H S AC v??"?   vj H S A= ??"? @p  8  @  $H @  H <3"`   KThe minimization yields0n H 0"`@ @ @  H <w F ^  Z&The corresponding free energy density:'0'j #H S A@ ??"? p   %H BTC p(  `*A. Schmitt Phys. Rev. D 71, 054016 (2005)+0+  &H B pb0  a+T. Schaefer Phys. Rev. D 62, 094007 (2000),0, H H 0޽h ? 3ff~___PPT10i.8+D=' U= @B + 0  L^(  Lr L S 8k00  X L 0AA?@S  L < D  g3Free energy density correspond to red-dashed line: 404j  L S AB ??"?     L <؞3"` "  RLower than that without mixing0  L <t4 @H q;Red-dashed line: with mixing Black-solid line: p-wave only<0<  L <l D h4Free energy density correspond to black-solid line: 505j L S A@ ??"?0 H L 0޽h ? 3ff~___PPT10i.+D=' U= @B + 0 @`:(  `r ` S [ `   X ` 0AD?0` X ` 0AE?  ` <p>P u7The first three expansion coefficients of gap function "80lFZ8 ` <3"`w 6<d  e1No surprise the component J=1 is the biggest one202H ` 0޽h ? 3ff~___PPT10i.'ܢ+D=' U= @B +g 0 ~vPd(  dr d S  `  j d S Af ??"?[H  d <6  tThe integral equation for f(x)0j  d S AG ??"?    d <$ ,  LThe free energy density:0^ d 688j d S AK ??"?] Z  H d 0޽h ? 3ff~___PPT10i.+D=' U= @B + 0   p1 (  p p <Fr EA trial function:0^ p 6j p S AI ??"?  F  @  p c@ P p <3"`   KThe minimization yields0n  p 0"`@ @ @ j  p S AJ ??"?    p < d `0  m9The free energy density corresponding to trial function::0:  p0 \AH ??"?    X p 0AL?  p <\\ v B  k7The free energy density corresponding to final results:808j p S AM ??"? p0 8 p b  p M p BpC (  `*A. Schmitt Phys. Rev. D 71, 054016 (2005)+0+  p B4p b0  a+T. Schaefer Phys. Rev. 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Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis (A toy model with angular momentum mixing)) Model Hamiltonian (solvable)Case I: " # Case II:    QCD case4The one-gluon exchange potential (equal helicity) .51( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B !The partial wave expansion (m=0):""C 5The angular dependence with non-zero azimuthal number66% HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /  0 'Q(  x  c $@UM }      <U"` 0@ U 4  # Ap=j  S A ??"?_   <P   J The pairing strength are equal in leading order, similar to the case II"Ku3K3  <UI 0  2 The subleading terms fall off with increasing J"3u333 "  <d   G Forward singularity of QCD brings about the energy dependence of gap"Hu3H38  "ZB  s *DZB  s *DTB   c $D0ZB ! s *Dj % S Aa `??"?-  ` ' <U 2 6W.Brown, J. 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Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis (A toy model with angular momentum mixing)) Model Hamiltonian (solvable)Case I: " # Case II:    QCD case4The one-gluon exchange potential (equal helicity) .51( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B !The partial wave expansion (m=0):""C 5The angular dependence with non-zero azimuthal number66% HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! / 0  - m(  8  ,x # c &A "??"?@  "x $ c &A  ??"?   x % c &A$ ??"?   & B"`P" 7 rIn Nambu-Gorkov space:0  ' < @  &For one flavor paring (equal helicity)'0'3X ( 0A ?0  ) <@ '  QThe NG blocks of S read03X * 0A!? p H  0޽h ? 3ff~___PPT10i.g0(+D=' A= @B +2xkLGpv;;|VO!XPE4MP,!1iciƤFۨmiB?Rk[kM65kLJg^ ̷{ffg; ƫ!*>j{MM:(m4f;/? wR18'=b(K~vZFhZ$I^jELp9Q)>?;hX3sZ-XҋU5Ջ˝U%N0*ɱ knoN0VZ귔T1guteΏOo{!vtk%.qiYA_'`=X.kuqLB*Zie\e;a`Zْ2 wq^``ذk >ZҨրa#R8j Ll}Zj:u/ xÛ OJ0ph#H@: 荫7H\0n* .4IAk~妕sS}%4a-Fe-OJHBVb(Yr>Y2yWߎc5>埄MچQN$BǀIx̆ƷJ *46h4z7[WW )f|yt8n&kJ {7-^޺7öómP3 6/V]}}%*C>Pmtu15N)s{)w0d64aTŚ_t.vTRMQxp<D9)\Y<< TB5JiT[ *_pӲ7Fvc6_cm462;*ַ ,RAj{p".iN9b!pLVd2"ޑBXTr|P(Cc=*?b>y'A$=g:k6ZIjm4uԶUMz,IG,Ek^JxwyB }~řINFcK 3+$7F'"d X2ӫkr*jJ~e-;'>.A?wQ[;P{?ڱ#/Qr;Q5xxnLQn qGbis?[t?~VnL{K0x?$Cڋm@D/4 $O -4\p} IV:[ E80>܇>-5߷Y.rOL4=?sxY54FV<ƃfՆF?sreIB TQx1E(    Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLL0|Ԗ0Ԗ"D[SOalngsRomanLL0|Ԗ0Ԗ DTimes New RomanLL0|Ԗ0Ԗ0DWingdingsRomanLL0|Ԗ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'()*-/0(47(8)9.:B E"GILO'PTUX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@ph:m ʚ;ʚ;g4DdDd0ppp@  <4ddddc20Lme___PPT10E .$ fengbo.6? +O  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis (A toy model with angular momentum mixing)) Model Hamiltonian (solvable)Case I: " # Case II:    QCD case4The one-gluon exchange potential (equal helicity) .51( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B !The partial wave expansion (m=0):""C 5The angular dependence with non-zero azimuthal number66% HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /# 0   A (    NŠ "`P@  j   S A# &??"?`B &B  s *D,$D0  <TȠ ,$0 XEnergy density of normal phase03l x =x,$@0TB  c $DTB  c $D TB  c $D  TB B c $D  ZB  s *D   <Π x MFree energy density03  S A" (??"?Q 8 ($D0l  $  @ $ ,$D0TB . c $D  3 <A' 0??0 $  0TB : c $D ZB ; s *D @ l  C ? C,$D0TB  c $D ZB   s *D ` r $ S A% ,??"? e H  , 7 <A( 2?? C 2ZB < s *DP  l    A  ,$D0 , <A& .??0   .ZB - s *D   TB > c $D p H  0޽h ? 3ff~80___PPT10.k+ ,jD' U= @B Dg' = @BA?%,( < +O%,( < +DS' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-6B'blinds(horizontal)*<3<*=DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =-6B'blinds(horizontal)*<3<*?D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B'blinds(horizontal)*<3<*@D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =-6B'blinds(horizontal)*<3<*A+8+0+ 0 +2xkLGpv;;|VO!XPE4MP,!1iciƤFۨmiB?Rk[kM65kLJg^ ̷{ffg; ƫ!*>j{MM:(m4f;/? wR18'=b(K~vZFhZ$I^jELp9Q)>?;hX3sZ-XҋU5Ջ˝U%N0*ɱ knoN0VZ귔T1guteΏOo{!vtk%.qiYA_'`=X.kuqLB*Zie\e;a`Zْ2 wq^``ذk >ZҨրa#R8j Ll}Zj:u/ xÛ OJ0ph#H@: 荫7H\0n* .4IAk~妕sS}%4a-Fe-OJHBVb(Yr>Y2yWߎc5>埄MچQN$BǀIx̆ƷJ *46h4z7[WW )f|yt8n&kJ {7-^޺7öómP3 6/V]}}%*C>Pmtu15N)s{)w0d64aTŚ_t.vTRMQxp<D9)\Y<< TB5JiT[ *_pӲ7Fvc6_cm462;*ַ ,RAj{p".iN9b!pLVd2"ޑBXTr|P(Cc=*?b>y'A$=g:k6ZIjm4uԶUMz,IG,Ek^JxwyB }~řINFcK 3+$7F'"d X2ӫkr*jJ~e-;'>.A?wQ[;P{?ڱ#/Qr;Q5xxnLQn qGbis?[t?~VnL{K0x?$Cڋm@D/4 $O -4\p} IV:[ E80>܇>-5߷Y.rOL4=?sxY54FV<ƃfՆF?srQ i tQ{x1E(    Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLL0|Ԗ0Ԗ"D[SOalngsRomanLL0|Ԗ0Ԗ DTimes New RomanLL0|Ԗ0Ԗ0DWingdingsRomanLL0|Ԗ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'()*-/0(47(8)9.:B E"GILO'PTUX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@ph:m ʚ;ʚ;g4DdDd0ppp@  <4ddddc20Lme___PPT10E .$ fengbo.6? +O  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis (A toy model with angular momentum mixing)) Model Hamiltonian (solvable)Case I: " # Case II:    QCD case4The one-gluon exchange potential (equal helicity) .51( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B !The partial wave expansion (m=0):""C 5The angular dependence with non-zero azimuthal number66% HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! / 0   P! z (  r  S @0    8      BA* 5??M 5t  6"`   B n P Gap equation( 3 3   <8 ,p KMinimization of F03~  BA- J??V J@l ^  ^ ,$D0f  6A+?^ `B  0DO O   BA. K?? S K`B  0DO O l   Y !  Y,$D0@ (   ( f  6A,?( `B  0Dvv`B  0Dk   < " Y 3We have assumed the gap depends on x=cos\theta only 4043% H  0޽h ? 3ff~XP___PPT100.z@wq+L[D' U= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-6B'blinds(horizontal)*<3<*!+- 0   p$ (  $ $  6"`  %  x  $ <A) R??p Rx  $ <A/ S??v  Sx  $ <A0 T??  T  $ 0A1?W 7 ,$D0X $ 0A2?@- $ 0A3?} T ,$D0B $ s *DT ,$@0 $ 0q ,$@0 $ 0A4? P ,$@0F @ $# D,$@0TB $ c $D@@ZB $ s *D@ $ 0A5? }C ,$D 0 $ <\ }E,$0 ; 1st order    $ <s  ,$ 0 ; 2nd order   @z   $   ,$@ 0TB $ c $D ZB $ s *D   $ <A6 U?? w 8 U$@ 0 $ 0A7? # U ,$D 0 $ <7p RThe gap equation becomes03H $ 0޽h ? 3ff~X P ___PPT100 .z+JD' U= @B DO' = @BA?%,( < +O%,( < +DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$D' =%(DJ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-6B'blinds(horizontal)*<3<* $D' =%(D=' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$D ' =%(D ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$+p+0+$0 ++0+$0 + 0 ,,(  ,x , <A6 W??!  W , <T0P q7We introduce a parameter b, which satisfy the condition8083X , 0A8?0   , <l5LX OTo the leading order:03X  , 0A9?0 @t f  , 0"` j , S A: q??"?0 @   q , <:3"` P\  \"The angular dependence is defined:#0# ,0 \A; r??"? P  r  F # ) `  ,  @TB , c $D` ) ` ZB ,B s *D# `  , <@   MS-wave gap function03 , <hD'P O Gap equation &0 3H , 0޽h ? 3ff~___PPT10i.|0}H+D=' U= @B +2xkLGpv;;|VO!XPE4MP,!1iciƤFۨmiB?Rk[kM65kLJg^ ̷{ffg; ƫ!*>j{MM:(m4f;/? wR18'=b(K~vZFhZ$I^jELp9Q)>?;hX3sZ-XҋU5Ջ˝U%N0*ɱ knoN0VZ귔T1guteΏOo{!vtk%.qiYA_'`=X.kuqLB*Zie\e;a`Zْ2 wq^``ذk >ZҨրa#R8j Ll}Zj:u/ xÛ OJ0ph#H@: 荫7H\0n* .4IAk~妕sS}%4a-Fe-OJHBVb(Yr>Y2yWߎc5>埄MچQN$BǀIx̆ƷJ *46h4z7[WW )f|yt8n&kJ {7-^޺7öómP3 6/V]}}%*C>Pmtu15N)s{)w0d64aTŚ_t.vTRMQxp<D9)\Y<< TB5JiT[ *_pӲ7Fvc6_cm462;*ַ ,RAj{p".iN9b!pLVd2"ޑBXTr|P(Cc=*?b>y'A$=g:k6ZIjm4uԶUMz,IG,Ek^JxwyB }~řINFcK 3+$7F'"d X2ӫkr*jJ~e-;'>.A?wQ[;P{?ڱ#/Qr;Q5xxnLQn qGbis?[t?~VnL{K0x?$Cڋm@D/4 $O -4\p} IV:[ E80>܇>-5߷Y.rOL4=?sxY54FV<ƃfՆF?sr$Q. J6GrYx1E(    Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLL0|Ԗ0Ԗ"D[SOalngsRomanLL0|Ԗ0Ԗ DTimes New RomanLL0|Ԗ0Ԗ0DWingdingsRomanLL0|Ԗ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'()*-/0(47(8)9.:B E"GILO'PTUX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@ph:m ʚ;ʚ;g4DdDd0ppp@  <4ddddc20Lme___PPT10E .$ fengbo.6? +O  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis (A toy model with angular momentum mixing)) Model Hamiltonian (solvable)Case I: " # Case II:    QCD case4The one-gluon exchange potential (equal helicity) .51( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B !The partial wave expansion (m=0):""C 5The angular dependence with non-zero azimuthal number66% HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! / 0 <r(  <x < c $tIM }   j < S A< t??"?  t < <VG.H@8___PPT9 ^The spherical gap, f(x)=1 is a trivial solution and there is no angular momentum mixing.0H10_3I8X < <4Z H@8___PPT9 gThe spin-1 gap, carried over from the polar phase of He_3, f(x)~x, fails to satisfy this equation0C1%0h3;'8X < <f   R3) Conserve the parity03- < <,h B{  ]The solution to this integral equation can be obtained from a variational principle as below.&^0\> H < 0޽h ? 3ff~___PPT10i.P^eP+D=' U= @B +2xkLGpv;;|VO!XPE4MP,!1iciƤFۨmiB?Rk[kM65kLJg^ ̷{ffg; ƫ!*>j{MM:(m4f;/? wR18'=b(K~vZFhZ$I^jELp9Q)>?;hX3sZ-XҋU5Ջ˝U%N0*ɱ knoN0VZ귔T1guteΏOo{!vtk%.qiYA_'`=X.kuqLB*Zie\e;a`Zْ2 wq^``ذk >ZҨրa#R8j Ll}Zj:u/ xÛ OJ0ph#H@: 荫7H\0n* .4IAk~妕sS}%4a-Fe-OJHBVb(Yr>Y2yWߎc5>埄MچQN$BǀIx̆ƷJ *46h4z7[WW )f|yt8n&kJ {7-^޺7öómP3 6/V]}}%*C>Pmtu15N)s{)w0d64aTŚ_t.vTRMQxp<D9)\Y<< TB5JiT[ *_pӲ7Fvc6_cm462;*ַ ,RAj{p".iN9b!pLVd2"ޑBXTr|P(Cc=*?b>y'A$=g:k6ZIjm4uԶUMz,IG,Ek^JxwyB }~řINFcK 3+$7F'"d X2ӫkr*jJ~e-;'>.A?wQ[;P{?ڱ#/Qr;Q5xxnLQn qGbis?[t?~VnL{K0x?$Cڋm@D/4 $O -4\p} IV:[ E80>܇>-5߷Y.rOL4=?sxY54FV<ƃfՆF?srYn:Y]x1E(    Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*Math      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxz{|}~Type 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLL0|Ԗ0Ԗ"D[SOalngsRomanLL0|Ԗ0Ԗ DTimes New RomanLL0|Ԗ0Ԗ0DWingdingsRomanLL0|Ԗ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'()*-/0(47(8)9.:B E"GILO'PTUX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@ph:m ʚ;ʚ;g4DdDd0ppp@  <4ddddc20Lme___PPT10E .$ fengbo.6? +O  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis (A toy model with angular momentum mixing)) Model Hamiltonian (solvable)Case I: " # Case II:    QCD case4The one-gluon exchange potential (equal helicity) .51( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B !The partial wave expansion (m=0):""C 5The angular dependence with non-zero azimuthal number66% HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /K  0 b Z &H(  H H  6 s"`  p  X H 0A>?@ Q H <u~ ?where03X H 0A??@ H <8zJ >0  H <h}  WConsidering a trial function:03j H S AC v??"?   vj H S A= ??"? @p   H B"` L  QThe minimization yields0l H 6"`  @  H <w F ^  Z&The corresponding free energy density:'0'j #H S A@ ??"? p   %H BTC p(  `*A. Schmitt Phys. Rev. D 71, 054016 (2005)+0+  &H B pb0  a+T. Schaefer Phys. Rev. D 62, 094007 (2000),0, H H 0޽h ? 3ff~___PPT10i.8+D=' U= @B +2xkLGpv;;|VO!XPE4MP,!1iciƤFۨmiB?Rk[kM65kLJg^ ̷{ffg; ƫ!*>j{MM:(m4f;/? wR18'=b(K~vZFhZ$I^jELp9Q)>?;hX3sZ-XҋU5Ջ˝U%N0*ɱ knoN0VZ귔T1guteΏOo{!vtk%.qiYA_'`=X.kuqLB*Zie\e;a`Zْ2 wq^``ذk >ZҨրa#R8j Ll}Zj:u/ xÛ OJ0ph#H@: 荫7H\0n* .4IAk~妕sS}%4a-Fe-OJHBVb(Yr>Y2yWߎc5>埄MچQN$BǀIx̆ƷJ *46h4z7[WW )f|yt8n&kJ {7-^޺7öómP3 6/V]}}%*C>Pmtu15N)s{)w0d64aTŚ_t.vTRMQxp<D9)\Y<< TB5JiT[ *_pӲ7Fvc6_cm462;*ַ ,RAj{p".iN9b!pLVd2"ޑBXTr|P(Cc=*?b>y'A$=g:k6ZIjm4uԶUMz,IG,Ek^JxwyB }~řINFcK 3+$7F'"d X2ӫkr*jJ~e-;'>.A?wQ[;P{?ڱ#/Qr;Q5xxnLQn qGbis?[t?~VnL{K0x?$Cڋm@D/4 $O -4\p} IV:[ E80>܇>-5߷Y.rOL4=?sxY54FV<ƃfՆF?sr<~}x1`F(    Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLL0|Ԗ0Ԗ"D[SOalngsRomanLL0|Ԗ0Ԗ DTimes New RomanLL0|Ԗ0Ԗ0DWingdingsRomanLL0|Ԗ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'()*-/0(47(8)9.:B E"GILO'PT UX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@p:m ʚ;ʚ;g4DdDd0ppp@  <4ddddc20Le___PPT10E .$ fengbo.6r___PPT9TL ? +O  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: ) HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /|  0   p'# (  r  S `V   V    <8V"`;  @ V j  S A   ??"?0    <V'"   VIn mean field approximation:03  <V B  G Gap equation:03  <hV 4   F Free energy: 0 3X  0A ? 0 | X  0A? p h X  0A? k  pp  # #" ` ` @ V  <V?pp  P o(`B  0o ?pp`B  0o ?p p `B  0o ?pp `B  0o ?pp p pp  "3 #" ` P@ # <D ?pp  P o(fB $ 6o ?ppfB % 6o ?p p fB & 6o ?pp fB ' 6o ?pp H  0޽h ? 3ff~___PPT10i.H㺬+D=' U= @B +  0 'Q(  x  c $@UM }      <U"` 0@ U 4  # Ap=j  S A ??"?_   <P   J The pairing strength are equal in leading order, similar to the case II"Ku3K3  <UI 0  2 The subleading terms fall off with increasing J"3u333 "  <d   G Forward singularity of QCD brings about the energy dependence of gap"Hu3H38  "ZB  s *DZB  s *DTB   c $D0ZB ! s *Dj % S Aa `??"?-  ` ' <U 2 6W.Brown, J. Liu, H-c Ren Phys. Rev. D 62 054016 (2000)707 .H  0޽h ? 3ff~___PPT10i.YP5+D=' U= @B +W  0 n f &H(  H H  6 s"`  p  X H 0A>?@ Q H <u~ ?where03X H 0A??@ H <8zJ >0  H <h}  WConsidering a trial function:03j H S AC v??"?   vj H S A= ??"? @p   H B"` L  QThe minimization yields03r H <g "`  @  H <w  ^  `&The corresponding free energy density:'0'3j #H S A@ ??"? p   %H BTC p(  `*A. Schmitt Phys. Rev. D 71, 054016 (2005)+0+  &H B pb0  a+T. Schaefer Phys. Rev. D 62, 094007 (2000),0, H H 0޽h ? 3ff~___PPT10i.8+D=' U= @B + 0   L(  L L  68"`   X L 0AA?0  L <   u3Free energy density correspond to red-dashed line: &4033j  L S AB ??"?     L <؞3"` D XLower than that without mixing0  L <t4 @H q;Red-dashed line: with mixing Black-solid line: p-wave only<0<  L 0l  v4Free energy density correspond to black-solid line: &5043j L S A@ ??"?0 H L 0޽h ? 3ff~___PPT10i.+D=' U= @B + 0 0(@ `(  `X ` 0AD?p X ` 0AE?  ` <p> 7The first three expansion coefficients of gap function 080lFZ73 ` <3"` 6.  k1No surprise the component J=1 is the biggest one202  ` <W> > xPartial wave expasion (m=0):03 H ` 0޽h ? 3ff~___PPT10i.'ܢ+D=' U= @B + 0 Pd@(  d d  6"``  j d S Af ??"?   d <6J  zThe integral equation for f(x)03j  d S AG ??"?    d <$ ,  RThe free energy density:03^ d 688j d S AK ??"?] Z  H d 0޽h ? 3ff~___PPT10i.+D=' U= @B +2xkLGpv;;|VO!XPE4MP,!1iciƤFۨmiB?Rk[kM65kLJg^ ̷{ffg; ƫ!*>j{MM:(m4f;/? wR18'=b(K~vZFhZ$I^jELp9Q)>?;hX3sZ-XҋU5Ջ˝U%N0*ɱ knoN0VZ귔T1guteΏOo{!vtk%.qiYA_'`=X.kuqLB*Zie\e;a`Zْ2 wq^``ذk >ZҨրa#R8j Ll}Zj:u/ xÛ OJ0ph#H@: 荫7H\0n* .4IAk~妕sS}%4a-Fe-OJHBVb(Yr>Y2yWߎc5>埄MچQN$BǀIx̆ƷJ *46h4z7[WW )f|yt8n&kJ {7-^޺7öómP3 6/V]}}%*C>Pmtu15N)s{)w0d64aTŚ_t.vTRMQxp<D9)\Y<< TB5JiT[ *_pӲ7Fvc6_cm462;*ַ ,RAj{p".iN9b!pLVd2"ޑBXTr|P(Cc=*?b>y'A$=g:k6ZIjm4uԶUMz,IG,Ek^JxwyB }~řINFcK 3+$7F'"d X2ӫkr*jJ~e-;'>.A?wQ[;P{?ڱ#/Qr;Q5xxnLQn qGbis?[t?~VnL{K0x?$Cڋm@D/4 $O -4\p} IV:[ E80>܇>-5߷Y.rOL4=?sxY54FV<ƃfՆF?sr4 lN ʀZ< d oB v){x1F(    Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLLԖ0Ԗ"D[SOalngsRomanLLԖ0Ԗ DTimes New RomanLLԖ0Ԗ0DWingdingsRomanLLԖ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'()*-/0(47(8)9.:B E"GILO'PT UX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@p:m ʚ;ʚ;g4DdDddY0ppp@  <4ddddc20Le___PPT10E .$ fengbo.6r___PPT9TL ?&-2008-4-25O  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! / 0` Ot{h______` M <ff33̙3` +ffO=ff̙H7` fff3f̙` Tff33ff` 0Ky{kOz` )R{f` GiIfff̙fR` ̙|̙3f` 3ff~>?" dd@'?lFd@  nK'o`P( n?" dd@   @@``PT     o (`0p>> ''^(  ZB  c $D"`   6t' "M }  LUSQdkYkHrh7h_     08* "; `  8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    00 "` `  X*   0p7 "`   Z*   09 "` `  Z* T     "`h2   s *"`h2   s *"h2   s *"@h2   s *"0`h2   s *"0h2  s *"0@h2  s *"`0h2  s *"`h2  s *"h2  s *"@h2  s *"`n2  0" h2  s *"``h2  s *"`h2  s *"@`n2  0"``h2  s *"`h2  s *"n2  0"@n2  0"`h2  s *" h2  s *"`@n2  0"@n2   0"@@h2 ! s *"`@n2 " 0"``n2 # 0"`h2 $ s *"`@h2 % s *"``h2 & s *" h2 ' s *"` H  0޽h ? 3ff~___PPT10i. s+D='  = @B + Network 0 (((  ZB  c $D"   6  "&wf   LUSQdkYkHrh7h_    0̤  "wQ    NUSQdkYkHroRh7h_   0&g  "` `   X*   0&g  "`    Z*   0<*g  "` `   Z* T `]   "]p h2   s *"`]h2   s *"]h2   s *"]Eh2   s *"`h2   s *"h2  s *"Eh2  s *"yh2  s *"`B h2  s *"B h2  s *"EB h2  s *"yB n2  0",B h2  s *"`u  h2  s *"u  h2  s *"u E n2  0"yu  h2  s *"`(  h2  s *"(  n2  0"( E n2  0"y(  h2  s *",(  h2  s *"` Z n2  0" Z n2   0" EZ h2 ! s *"y Z n2 " 0"`  n2 # 0"  h2 $ s *" E h2 % s *"y  h2 & s *"A  h2 ' s *"yA  `B ( s *D"H  0޽h ? 3ff~___PPT10i. s+D='  = @B + 0` Ot{h______` M <ff33̙3` +ffO=ff̙H7` fff3f̙` Tff33ff` 0Ky{kOz` )R{f` GiIfff̙fR` ̙|̙3f` 3ff~>?" dd@'?lFd@  nK'o`P( n?" dd@   @@``PR    o (` p>> p''^(  ZB  c $D"`   6le  "M }   LUSQdkYkHrh7h_     0$h  "; `   8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    0n  "` `   X*   0t  "`    Z*   0x  "` `   Z* T     "`h2   s *"`h2   s *"h2   s *"@h2   s *"0`h2   s *"0h2  s *"0@h2  s *"`0h2  s *"`h2  s *"h2  s *"@h2  s *"`n2  0" h2  s *"``h2  s *"`h2  s *"@`n2  0"``h2  s *"`h2  s *"n2  0"@n2  0"`h2  s *" h2  s *"`@n2  0"@n2   0"@@h2 ! s *"`@n2 " 0"``n2 # 0"`h2 $ s *"`@h2 % s *"``h2 & s *" h2 ' s *"` H  0޽h ? 3ff~___PPT10i. s+D='  = @B + 1_Network 0 (((  ZB  c $D"   6T<  "&wf   LUSQdkYkHrh7h_    0Y "wQ    NUSQdkYkHroRh7h_   0X] "` `   X*   0c "`   Z*   0g "` `  Z* T `]   "]p h2   s *"`]h2   s *"]h2   s *"]Eh2   s *"`h2   s *"h2  s *"Eh2  s *"yh2  s *"`B h2  s *"B h2  s *"EB h2  s *"yB n2  0",B h2  s *"`u  h2  s *"u  h2  s *"u E n2  0"yu  h2  s *"`(  h2  s *"(  n2  0"( E n2  0"y(  h2  s *",(  h2  s *"` Z n2  0" Z n2   0" EZ h2 ! s *"y Z n2 " 0"`  n2 # 0"  h2 $ s *" E h2 % s *"y  h2 & s *"A  h2 ' s *"yA  `B ( s *D"H  0޽h ? 3ff~___PPT10i. s+D='  = @B + 0 0T0(  Tx" T c $܀g M }  g  x T c $g ; ` g  H T 0޽h ? 3fffff3̙3f̙___PPT10i.~7++D='  = @B +]  0 t l &H(  H H  6 "`  p  X H 0A>?@ Q H <|P~ ?where03X H 0A??@ H <UJ >0  H <X  WConsidering a trial function:03j H S AC v??"?   vj H S A= ??"? @p   H HL]3"` R  QThe minimization yields03r H <g "`  @  H <aw  ^  `&The corresponding free energy density:'0'3j #H S A@ ??"? p   %H BgC p(  `*A. Schmitt Phys. Rev. D 71, 054016 (2005)+0+  &H Bk pb0  a+T. Schaefer Phys. Rev. D 62, 094007 (2000),0, H H 0޽h ? 3ff~___PPT10i.8+D='  = @B +Z 0 q i p(  p p <F KA trial function:03^ p 6j p S AI ??"?   p Hd3"`cRP QThe minimization yields03l  p 6"` j  p S AJ ??"?    p <d 80  s9The free energy density corresponding to trial function::0:3  p0 \AH ??"?    X p 0AL?  p <\ v   q7The free energy density corresponding to final results:8083j p S AM ??"? p0 8 p b  p M p B(pC (  `*A. Schmitt Phys. Rev. D 71, 054016 (2005)+0+  p B?p b0  a+T. Schaefer Phys. Rev. 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Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! / 0` Ot{h______` M <ff33̙3` +ffO=ff̙H7` fff3f̙` Tff33ff` 0Ky{kOz` )R{f` GiIfff̙fR` ̙|̙3f` 3ff~>?" dd@'?lFd@  nK'o`P( n?" dd@   @@``PT     o (`0p>> ''^(  ZB  c $D"`   6<-  "M } -  LUSQdkYkHrh7h_     0-  "; ` -  8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    0|-  "` ` -  X*   0x-  "`  -  Z*   0-  "` ` -  Z* T     "`h2   s *"`h2   s *"h2   s *"@h2   s *"0`h2   s *"0h2  s *"0@h2  s *"`0h2  s *"`h2  s *"h2  s *"@h2  s *"`n2  0" h2  s *"``h2  s *"`h2  s *"@`n2  0"``h2  s *"`h2  s *"n2  0"@n2  0"`h2  s *" h2  s *"`@n2  0"@n2   0"@@h2 ! s *"`@n2 " 0"``n2 # 0"`h2 $ s *"`@h2 % s *"``h2 & s *" h2 ' s *"` H  0޽h ? 3ff~___PPT10i. s+D='  = @B + Network 0 (((  ZB  c $D"   6U  "&wf -  LUSQdkYkHrh7h_    0`X  "wQ    NUSQdkYkHroRh7h_   0<]  "` `   X*   0b  "`    Z*   0  "` `   Z* T `]   "]p h2   s *"`]h2   s *"]h2   s *"]Eh2   s *"`h2   s *"h2  s *"Eh2  s *"yh2  s *"`B h2  s *"B h2  s *"EB h2  s *"yB n2  0",B h2  s *"`u  h2  s *"u  h2  s *"u E n2  0"yu  h2  s *"`(  h2  s *"(  n2  0"( E n2  0"y(  h2  s *",(  h2  s *"` Z n2  0" Z n2   0" EZ h2 ! s *"y Z n2 " 0"`  n2 # 0"  h2 $ s *" E h2 % s *"y  h2 & s *"A  h2 ' s *"yA  `B ( s *D"H  0޽h ? 3ff~___PPT10i. s+D='  = @B + 0` Ot{h______` M <ff33̙3` +ffO=ff̙H7` fff3f̙` Tff33ff` 0Ky{kOz` )R{f` GiIfff̙fR` ̙|̙3f` 3ff~>?" dd@'?lFd@  nK'o`P( n?" dd@   @@``PR    o (` p>> p''^(  ZB  c $D"`   6\  "M }   LUSQdkYkHrh7h_     0D!  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Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /0  5(  x  c $4 00   x  c $ _      <  E  9Institute of Particle physics, Huazhong Normal University:0:H  0޽h ? 33___PPT10i.o?+D='  = @B +j  0  y p) (  r  S X%         <0& "`;  @   j  S A   ??"?0    <' '"   VIn mean field approximation:03  <1  B  G Gap equation:03  <6  4   F Free energy: 0 3X  0A ? 0 | X  0A? p h X  0A? k | ` ` @ ( #" ` ` @    <| ?` ` @ P o(`B  031 ?` ` ` `B  031 ?@` @`B  031 ?` @`B  031 ?` ` ` @d  ` P@ ) #" ` P@ # <<<  ? ` P@ P o(fB $ 631 ? ` P` fB % 631 ? @P@fB & 631 ? ` @fB ' 631 ?P` P@H  0޽h ? 3ff~___PPT10i.H㺬+D='  = @B +r+ a/;x1F(   - Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation ""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLL- Ԗ0Ԗ"D[SOalngsRomanLL- Ԗ0Ԗ DTimes New RomanLL- Ԗ0Ԗ0DWingdingsRomanLL- Ԗ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'*)*-/0(47(8)9.:B E"GILO'PT UX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@p:m ʚ;ʚ;g4DdDd- 0ppp@  <4ddddc20Le___PPT10E .$ fengbo.6r___PPT9TL ?Z-2008.4.26-4.29 ",{ASJ\hQV|P[irtf[/gO WSNO  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /r<;݂x1F(   - Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLL- Ԗ0Ԗ"D[SOalngsRomanLL- Ԗ0Ԗ DTimes New RomanLL- Ԗ0Ԗ0DWingdingsRomanLL- Ԗ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'*)*-/0(47(8)9.:B E"GILO'PT UX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@p:m ʚ;ʚ;g4DdDd- 0ppp@  <4ddddc20Le___PPT10E .$ fengbo.6r___PPT9TL ?Z-2008.4.26-4.29 ",{ASJ\hQV|P[irtf[/gO WSNO  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /} 0 $(  r  S ` M }   r  S 8  `k   H  0޽h ? 3ff~___PPT10i.:+D='  = @B +ri\x1F(   - Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLL- Ԗ0Ԗ"D[SOalngsRomanLL- Ԗ0Ԗ DTimes New RomanLL- Ԗ0Ԗ0DWingdingsRomanLL- Ԗ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'*)*-/0(47(8)9.:B E"GILO'PT UX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@p:m ʚ;ʚ;g4DdDd- 0ppp@  <4ddddc20Le___PPT10E .$ fengbo.6r___PPT9TL ?Z-2008.4.26-4.29 ",{ASJ\hQV|P[irtf[/gO WSNO  =>Angular Momentum Mixing in Non-spherical Color superconductors??  Feng bo, Hou de-fu, Ren hai-cang!!ZoutlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /rt^x1vF(   U Equation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLLGԖ0ԖD[SOalngsRomanLLGԖ0Ԗ DTimes New RomanLLGԖ0Ԗ0DWingdingsRomanLLGԖ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'*)*-/0(47(8)9.:B E"GILO'PT UX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@p:m ʚ;ʚ;g4DdDd0ppp@  <4ddddc20Le___PPT10E .$ fengbo.6r___PPT9TL ?Z-2008.4.26-4.29 ",{ASJ\hQV|P[irtf[/gO WSNO  =>Angular Momentum Mixing in Non-spherical Color superconductors?? &Q l O_[ Nwml outlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /a0 xp (  x  c $L00  x  c $M_     <DO` 0Z  dNS-N^'Yf[|P[irtxvz@b0 H  0޽h ? 33___PPT10i.o?+D=' V= @B +rZny]x1E(  V }&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLLGԖ0ԖD[SOalngsRomanLLGԖ0Ԗ DTimes New RomanLLGԖ0Ԗ0DWingdingsRomanLLGԖ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'*)*-/0(47(8)9.:B E"GILO'PT UX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@p:m ʚ;ʚ;g4DdDd0ppp@  <4ddddc20Le___PPT10E .$ fengbo.6r___PPT9TL ?Z-2008.4.26-4.29 ",{ASJ\hQV|P[irtf[/gO WSNO  =>Angular Momentum Mixing in Non-spherical Color superconductors?? &Q l O_[ Nwml outlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! / 0 ` .(  r  S P    r  S `` `    <"`@ ' TGap equation (non-linear):0X  0A?0   <$I ` 0  21) Spherical solutions: no angular momentum mixing30338XBF '[   0 r  S A  ??"?'0 r  S A  ??"?t[ 8  WJ> @'ZB  s *D3   <DWJ> rAngular momentum mixng03  < `,  ,2) Non-spherical solutions (equal helicity):-0-3"H  0޽h ? 3ff~___PPT10i.`XZ+D=' d= @B + 0 tl(     6/ "`p   X  0A?ppX  0A ?p  < 0 PP7 o-The interaction potential can be expanded as &.0,3  < 6 P 7  TThe gap function: &03X  0A? p X  0A?^    <,;  t  8with0  <>   SThe free energy: &03X  0A? 2l      ,$@0ZB  s *D3  TB  c $D3    <D g & N,$0 TNo angular momentum mixing03H  0޽h ? 3ff~___PPT10.Q*"Q+YyBDa' d= @B D' = @BA?%,( < +O%,( < +DS' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+8+0+0 +r] ]ݻx1E(  V  Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEqua      tion Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation""Equation Equation.DSMT40*MathType 5.0 Equation #Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation&&Equation Equation.DSMT40*MathType 5.0 Equation('Equation Equation.DSMT40*MathType 5.0 Equation,(Equation Equation.DSMT40*MathType 5.0 Equation.)Equation Equation.DSMT40*MathType 5.0 Equation0*Equation Equation.DSMT40*MathType 5.0 Equation2+Equation Equation.DSMT40*MathType 5.0 Equation5,Equation Equation.DSMT40*MathType 5.0 EquationJ0Equation Equation.DSMT40*MathType 5.0 EquationK1Equation Equation.DSMT40*MathType 5.0 EquationR2Equation Equation.DSMT40*MathType 5.0 EquationS3Equation Equation.DSMT40*MathType 5.0 EquationT4Equation Equation.DSMT40*MathType 5.0 EquationU5Equation Equation.DSMT40*MathType 5.0 EquationW7Equation Equation.DSMT40*MathType 5.0 Equationq8Equation Equation.DSMT40*MathType 5.0 Equationr9Equation Equation.DSMT40*MathType 5.0 Equationt;Equation Equation.DSMT40*MathType 5.0 Equationv>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 Equation`jEquation Equation.DSMT40*MathType 5.0 EquationqEquation Equation.DSMT40*MathType 5.0 Equation/ 00DArialngsRomanLLGԖ0ԖD[SOalngsRomanLLGԖ0Ԗ DTimes New RomanLLGԖ0Ԗ0DWingdingsRomanLLGԖ0Ԗ@0.  @n?" dd@  @@`` xppm5   5()!"%&'*)*-/0(47(8)9.:B E"GILO'PT UX&0` co2$RDv/-: "$_ܒ>aNISo}G 2$RD:4!S2$/:;<2BDX2$G_eCe 8s2]2$3csj<(R4`2$ z(٭s]b2$R@|%o|e2$jӦ@1qʯ#ɡ:k2$ ѭhU!$$$$$$$$$$$$$$$$$$$2$ <`{ $$$$2$-ǏQ38$@rZ~fb 0AA@33@p:m ʚ;ʚ;g4DdDd̤0ppp@  <4ddddc20Le___PPT10E .$ fengbo.6r___PPT9TL ?Z-2008.4.26-4.29 ",{ASJ\hQV|P[irtf[/gO WSNO  =>Angular Momentum Mixing in Non-spherical Color superconductors?? &Q l O_[ Nwml outlineIntroduction Angular momentum mixing (zero azimuthal number) (i) general analysis (ii) a toy model (iii) QCD case Angular momentum mixing with nonzero azimuthal number Conclusion and Outlook =>M%M4+ l  Introduction'Ground state of cold dense quark matter((QCD phase diagram   7One flavor CSC is free from the Fermi momentum mismatch8Z8 Angular momentum mixing General analysis *A toy model with angular momentum mixing+l+ Model Hamiltonian (solvable)Case I: " # Case II:    QCD case l 4The one-gluon exchange potential (equal helicity) .51"( CJT effective action          The CJT effective action.  Gap equation   / JSon s approximation and new variables&&6:The integral equation of f(x) < The free energy density= The numerical results of f(x)B  C 9 The angular dependence with non- zero azimuthal number:: "& HiConclusion and outlookSBecause of the nonlinearity of the gap equation, non-spherical gap function can not be restricted to one angular momentum channel, the mixing is bound to show up The free energy of the polar phase had been lowered by angular momentum mixing, but still higher than that of CSL The systematic survey of the angular momentum mixing in cross helicity pairing is under construction, since the calculations suggest that is the preferred pairing pattern Another place where the angular momentum mixing show up is the CSC-LOFF state in the presence of momentum mismatch.CZiZZyZAZ;Z|er Thank you! /r!  x1Root EntrydO){ (%PicturesYCurrent User2SummaryInformation(           > !"#$%&'()*+,-./0123456789:;<=?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz|}~(       !"#&()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz|}~ õƬD 4<Version_  jlpingjlping