- Email: [email protected]
Chiral symmetry restoration in the instanton liquid at finite density
NUCLEAR PHYSICS A Nuclear Physics A642 ( 1998) 7 l c-77c
ELSEVIER
Chiral
Symmetry
Restoration
in the Instanton
Ralf Rapp * Department of Physics and Astronomy, Stony Brook, NY 11794-3800, U.S.A.
State University
Liquid
at Finite
Density
of New York at Stony Brook,
The properties of the QCD partition function at finite chemical potential are studied within the instanton liquid model. It is shown that the density dependence of the quarkinduced instanton-antiinstanton (I-A) interaction leads to the formation of topologically neutral I-A pairs (‘molecules’), resulting in a first order chiral phase transition at a critical chemical potential p: 2 310 MeV. At somewhat higher densities (,up > 360 MeV), the quark Fermi surface becomes instable with respect to diquark condensation (Cooper pairs) generating BCS-type energy gaps of order 50 MeV. 1. INTRODUCTION The investigation of the phase diagram of QCD is one of the central issues in understanding the properties of strong interactions. While the finite temperature axis has been theoretically explored in quite some detail, much less is known about the finite density &-) axis, which is partly due to the fact that first principle QCD lattice calculations at finite pq encounter the problem of a complex fermionic determinant when integrating the QCD partition function. The phase structure of QCD at finite density, however, might be very rich, including new forms of condensates (other than the ordinary chiral condensate present in the QCD vacuum), different orders of phase transitions in the ,!+T-plane (entailing a tricritical point [I]), etc.. In this contribution, however, we will focus on the T=O-, pLg>O- axis and try to examine the nature of the chiral phase transition in this regime. We will do so within the framework of the instanton liquid model (ILM) of &CD, which, although lacking explicit confinement, yields a very successful phenomenology of the QCD vacuum structure, and the low-lying hadron spectrum. Recently it also provided an interesting mechanism for chiral symmetry restoration at finite temperature, based on the rearrangement of the (anti-) instantons within the liquid, rather than on a mere disappearance of them as suggested earlier. Our objective here is to investigate whether a similar mechanism could be responsible for the restoration of chiral symmetry at finite density as well. We start by briefly recalling some features of the instanton liquid model at zero density (sect. 2) and then turn to the finite density case (sect. 3), where we first calculate the pL4dependence of the instanton-antiinstanton interaction and then assess its impact on the chiral phase transition. Under certain approximations we will be able to avoid a complex ‘supported by the A.-v.-Humboldt foundation Energy under grant no. DEFGOW38ER40388.
as a Feodor-Lynen
0375-9474/98/$19.00 0 1998 Elsevier Science B.V. PI1 SO375-9474(98)00501-6
All rights reserved
fellow and by the US-Department
of
BuylcJsgss suognIos aporu o~az sec[ II!$s cp!q~ ‘(aaeds wapyma u!) uoyenba aw!a 113!suap al!uy ay$ u10.13~re$s a~ ‘Iz?gua$od Imituay~ ymnb acguy 1~ suoys~a$u! ~!aql 13nJ$suo2 0% amy lmj a~ aIqurasua uo~u~~su~ aq? 30 suo~y2~y~pow mupam dpnls 0A tori a7pud 712 uoqeJa*uI y-1 -1-g VIJ pm sapon 0Jaz y.nx@ ~.;CI~N~~O'~~[~Z-NON;L~~~~~~~I~~I~N~;CN~~~NI
3m
es
spsal ~usu!wa~ap 3!uoyurla3 aq$ ‘(0 +- /u1) sassmu ymnb quamw ~Iwus Buy3aI%au put ‘V‘Oa pm fo& ‘sapour olaz asay? Lq pauusds syeq ay? UI ‘uo!$nIos B.wua olaz papuey-q3al (-c@~II ) e sassassod pIay uo~we~su~ (-ye) ayl u! uoyenba ml!a aqL wa!wLs ayl30 say.xadoJd yInq aq? %!ssasss ~03 aI%?quossal aq pInoys qq~ ‘sapour BupCl IsaMoI ay? 1cIuo Bu!daay Lq pawInaIs3 LIa~am!xo~dds s! ‘spIay ymnb ayq JaAo uoyeel8 -a?u! ay$ urog %!s~E ‘lolmado X?Jra ayJ30 ~uwI!urJa~ap ayL ‘IUFs_a (d)u dp J z = A/N aq 0% suouitma3 30 aauasqI2 aql u! &suap uo~ue~su~ Is?07 ay? auyIla%ap zUFs uoyma~ -u! uo$us~su! au[$ 30 qmd a!uonIS aq? pus (suor$aalJo3 um~usnb BmpnIm!) (W)u apny!Id -um uoygsu!-aISu!s ayL Y3UO~UE$SU!JUS -N PUS SUO~US?SU! +N 30 (UO!$V~Ua!.IO 10103 puv az!s ‘uog!sod) { In ‘Id ‘12) = 1~ sapurploo3 aar$3aIIo3 paIIw-os aql .IaAo uogslSa$u! UB olu! pa$laauoa uaaq ssq suoyem8yuo~ pIay aIqysod 118 .IaAo IwSaqu!-ypd aqy aJayM
(1) -de
uo!pqsu!
‘
ay$
(~UO~U’T?!JSU~,)
U!
pX+Svp!UIas
pa!yeu!wop s! umnam
uogmn3
uoyysd
a!Jnl!$suo3
sauroaaq uaq? uoyur!xold aab aqA .suoysnba sII!II\I-%~A aql30 suoyqos SuoyWn~guo3 play a%ea aayeq.mJad-uou icq aqq uo paseq s! aa@ u! lapour uo~uv+su~ ay;L
q3q~
ay$ ~sqv uoydumssa
iC;CIsN3ao~3z;Cv?3ao~aaInbI?
NO&N~~;LSNI~H&
-z
.uoydpmap play-ueaur mo u! sam%y [g‘z] d+g~mpuo3~adns ~0103 30 cus!uey~auI passnmp AI’$uaDal ay!$ MOM MOT+ pm aX3ms !urJad aq$qe syrenb uaaM?aq suo!qviaJlo3 awes apnIm! aJomaylm3 aM .yasoldde ad4 play-warn K!uyp~ Ieyuayod Im!tuaya I’ee3y~maq$ apurysa PUT?uoymn3 uoyqmd
13c
R. Rapp/Nuclear Physics A642 (1998) 71c-77c
Constraining ourselves to zero temperature, the gluonic (instanton-) fields entering the covariant derivative are not affected by 1-1~.The explicit form of the quark zero modes has been determined in ref. [5],
with the function II(Z) = (1-t p2/s2) and a spin-color spinor x. Note that the solution the adjoint Dirac equation, @,,(?
-I.Lq) (i PI -
iPqY4) = 0 >
of
(6)
carries the chemical potential argument with opposite sign. This is necessary for a consistent definition of expectation values at finite Jo,, and in particular renders a finite norm
With the properly constructed quark wave functions we can now evaluate the fermionic overlap matrix element representing the zero mode part of the full Dirac operator. Choosing for simplicity the sum ansatz for the gauge-field configurations, A, = AE+A$, entering the covariant derivative, allows us to replace the latter by an ordinary one, yielding TIA(Z,%pq)
.I
=
-
=
i u4 fl(',r;pq)
d4x '$,(z
- ZI; -PLq) (i $ - &'Y4)
QO,A(x
- zA;&)
+ i
The breaking of Lorentz invariance in the medium implies the existence of two independent (real-valued) functions fi, fz which are shown in fig. 1. Similar to what has been found at finite temperature [6], TEA is strongly enhanced in temporal direction with increasing pLq,but oscillates as N sin(2p,r) in spatial direction [7]. The latter effectively suppresses the interaction once integrating over r. 3.2. Thermodynamics of the ILM at Finite ps To investigate the finite density properties of the thermodynamic potential (or free energy) we here resort to the so-called cocktail model introduced in ref. [8]. It amounts to a mean-field type description including three major components in the system: essentially random (anti-) instantons (the ‘atomic’ component), strongly correlated I-A-pairs (the ‘molecular’ component) and a Fermi sphere of constituent quarks (‘quasiparticles’). Thus
where the constituent
quark contribution
is zero for Fermi energies ,u~ < mq, with m, denoting instanton part of the free energy, Glst(%,
%Tl; Pq)=
-
ln[Zinst(~a, hi
v,
A)1
=
-71, I”[:]
the constituent
-n,
ln[z]
quark mass.
,
The
(11)
142
R. Rapp/Nuclear
-
Physics A642 (1998) 71c-77c
----_--
&.=O 0.3GeV 0.6 GeV
-.--
0.9 Ge”
t
i
3.0 ’
---.
f,(r) 2.0
--------
pq=o 0.3 Ge” 0.6 GeV 0.9G.a”
1
I
-1.0;.-.. 0.0
1.0
”
3.0
2.0
r
4.0
-I
5.0
[PI
Figure 1. Quark-induced I-A-interaction at finite density for the most attractive color orientation zlq=l, G=O as well as r=O (left panel) and for ud=O, lu’l=l and r=O (right panel).
is related to the atomic and molecular Gl = = &L
‘activities’
2 C pbP4 eWsi”” (TIA(P~)TA&))~~‘~ C2 #b-4) e-z5’i,t
([TIA(CL,)TAr(~U,)I”NI)
(14
The minimization of 52 with respect to the corresponding (4-dimensional) densities na and n, determines the equilibrium state of the system at fixed pLp. In particular, Rinsl encodes all the features of the T = pq = 0 instanton ensemble, as e.g. the quark condensate and the constituent quark mass, which in mean-field approximation are given and m4 c( -p”(qq). Therefore the normalization constant by (rlq) = -li(~)(3/2nP C IX (AQCn)b can be fixed to give N/V=1.4 fmP4, being realized for n,=1.34 fmP4 and n,=O.O3 fmP4, which is not unreasonable. The gluonic interaction has been approximated by an average repulsion Sint = -rcp4(n, + 27~~). At finite pq the Dirac operator is not hermitian any more, i.e. TAI&) # TjA(pq), resulting, in general, in a complex fermionic determinant, entailing the well-known ‘sign’ problem. However, assuming an average gluonic interaction that does not depend on density allows us to perform the color averages implied in eqs. (12) analytically, i.e.
ic h ensures the pressure to remain real. Fig. 2 shows our (Nf = 2, .f,’= .A(*&), w h’ results as function of the quark chemical potential (left panel) for N, = 3, Nf = 2. At small pclsessentially nothing happens until, at a critical value & N 310 MeV, the system
R. RappINuclear
1%
Physics A642 (1998) 71c-77c
1
-
Fq=O --0.16 GeV ----- 0.24GeV
or-
0.0
i
I
:
: :
/
/
H d
’
x10! “ #’
: Quark 0.2 :
I
BCS !’ 1 Gap!
0.4 r
7 o,3 1 Constituent
*
Mass
7
,i’
: 0.1 T
, ,/
8'
-0.6
/ ”
”
”
0.1
”
"
”
0.2
"
0.3
"
f
’ 0.0
” 0.4
”
0.5
/’
,/'
-0.7
#'
Eb
0.0 0.0
,
/
0.1
0.2 mq
0.3
0.4
WV1
r-l,[GW
Figure 2. Our results for the instanton (‘atomic’) and molecule densities (upper left panel), the pressure p = -0 (middle left panel) and the constituent quark mass (lower left panel) after minimizing the free energy, eq. (9)) with respect to n, and n,; additionally including scalar diquark correlations (as discussed in sect. 3.3) adds the dashed-dotted line to the lower left panel, representing the BCS energy gap at the quark Fermi surface. The right panel shows the free energy as a function of constituent quark mass m, K nAf2, indicating a first order transition from the minimum at finite mp to the one at m4 = 0.
jumps into the chirally restored phase, the latter being characterized by n, = 0. The transition is of first order, as can be seen by inspection of the m,-dependence of the free energy (right panel of fig. 2). Below pi, the pressure actually decreases slightly with increasing ~1~indicating a mixed phase-type instability, similar to what has been discussed in refs. [2,9]. A significant difference, however, is given by the fact that in our approach the total instanton density at the transition (residing in I-A-molecules) is still appreciable, N/V = 2n, 21 1.1 fmW4, providing the major part of the pressure at this point; in other words: a substantial part of the nonperturbative vacuum pressure persists in the chirally restored phase (of course, eventually it will be suppressed due to the Debye screening of the instanton fields, which we have not included here).
76c
R. Rapp/Nuclear Physics A642 (1998) 71c-77c
Color Superconductivity As has recently been pointed out in refs. [2,3], the quark Fermi surface in the plasma phase might be unstable with respect to the formation of quark-Cooper pairs, once an attractive q-q interaction is present. Using the instanton-induced interaction in the scalar, color-antitriplett diquark channel (which is essentially the Fierzed-transformed interaction leading to a deeply bound pion and is also phenomenolgically well-supported by baryonic spectroscopy), BCS-type energy gaps of As = 50-100 MeV have been predicted. In the mean-field approach employed here, this interaction modifies the quark quasipartitle contribution according to 3.3.
$&I = tr log[D(m,,Ao)] - tr Mm,,
AoMAo)]
+ Gtr Mm,,
Ao)]
tr [F(m,, A,)] (14)
with the quasiparticle quark-propagator D and the annomalous (Gorkov) propagator F corresponding to creation/annihilation of a bound Cooper pair. G is the effective coupling constant derived from the instanton q-q-vertex [3]. From “t one can obtain the standard gap equation for As. However, here we also have to account for the fact that a finite A0 will damp the quark zero-mode propagators in the instanton part of the free energy, which results in a suppression of the quark induced I-A interaction TIA of eq. (8). Thus, Rinst disfavors finite A,%. The resulting expression for the free energy is of the form
(the last term accounting for unpaired quarks), which now has to be minimized W.T.t. n,, n, and A,. We find that color superconductivity does not appear before puQN 360 MeV (see dashed-dotted curve in the lower left panel of fig. 2): although the quark-part of the free energy by itself, eq. (14), always favors a finite value for A,, the suppression caused by A0 in Rinst (i.e. the damping of quark-propagation in TIA) prevents the formation of a diquark condensate below p,r c~ 360 MeV. This again reflects the fact that the instanton contribution to the pressure is still dominant in the region somewhat above pi, so that the gain due to a finite As in fit cannot overcome the ‘penalty’ in Rinst. Above pq N 360 MeV energy gaps of order N 50 MeV arise, in line with the findings of ref. [3]. As far as the thermodynamic properties of the system are concerned, no changes as compared to the previous section occur below pq = 360 MeV, and even above the results would be hardly distinguishable from the curves for n&J, nm(& p(PJ and m,&) displayed in fig. 2. In summary, we have studied the QCD partition function at finite density using the instanton model. Employing a simple mean-field approach, chiral symmetry restoration emerges at pg =310 MeV as a first order transition from an essentially random (anti-/) instanton liquid into a phase of strongly correlated I-A molecules (and massless quarks), driven by the density dependent increase of the quark-induced I-A-interaction. Neglecting possible medium modifications in the gluonic interaction, which are expected to be small, a complex pressure could be avoided. Accounting for correlations in the scalar diquark channel, color superconductivity sets in for chemical potentials p,r > 360 MeV, associated with BCS gaps of up to N 50 MeV.
R. Rapp/Nuclear
Physics A642 (1998) 71c-77c
IIC
Acknowledgement
I thank my collaborators
E. Shuryak, T. Schafer and M. Velkovsky for fruitful discussions.
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
M.A. Halasz et al., hep-ph/9804290; M. Stephanov, theses proceedings. M. Alford, K. Rajagopal and F. Wilczek, hep-ph/9711395 and Phys. Lett. B (1998), in press. R. Rapp, T. Schafer, E. Shuryak and M. Velkovsky, hep-ph/9711396, and Phys. Rev. Lett. (1998), in press. T. Schafer and E. Shuryak, hep-ph/9610451, Rev. Mod. Phys. , April 1998. A.A. Abrikosov Jr., Sov. J. of Nucl. Phys. 37 (1983) 459. E. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. B364 (1991) 255. T. Sch;ifer, hep-ph/9708256, Phys. Rev. D, in print. E.-M. Ilgenfritz and E. Shuryak, Phys. Lett. B325 (1994) 263. M. Buballa, Nucl. Phys. A611 (1996) 393.
ELSEVIER
Chiral
Symmetry
Restoration
in the Instanton
Ralf Rapp * Department of Physics and Astronomy, Stony Brook, NY 11794-3800, U.S.A.
State University
Liquid
at Finite
Density
of New York at Stony Brook,
The properties of the QCD partition function at finite chemical potential are studied within the instanton liquid model. It is shown that the density dependence of the quarkinduced instanton-antiinstanton (I-A) interaction leads to the formation of topologically neutral I-A pairs (‘molecules’), resulting in a first order chiral phase transition at a critical chemical potential p: 2 310 MeV. At somewhat higher densities (,up > 360 MeV), the quark Fermi surface becomes instable with respect to diquark condensation (Cooper pairs) generating BCS-type energy gaps of order 50 MeV. 1. INTRODUCTION The investigation of the phase diagram of QCD is one of the central issues in understanding the properties of strong interactions. While the finite temperature axis has been theoretically explored in quite some detail, much less is known about the finite density &-) axis, which is partly due to the fact that first principle QCD lattice calculations at finite pq encounter the problem of a complex fermionic determinant when integrating the QCD partition function. The phase structure of QCD at finite density, however, might be very rich, including new forms of condensates (other than the ordinary chiral condensate present in the QCD vacuum), different orders of phase transitions in the ,!+T-plane (entailing a tricritical point [I]), etc.. In this contribution, however, we will focus on the T=O-, pLg>O- axis and try to examine the nature of the chiral phase transition in this regime. We will do so within the framework of the instanton liquid model (ILM) of &CD, which, although lacking explicit confinement, yields a very successful phenomenology of the QCD vacuum structure, and the low-lying hadron spectrum. Recently it also provided an interesting mechanism for chiral symmetry restoration at finite temperature, based on the rearrangement of the (anti-) instantons within the liquid, rather than on a mere disappearance of them as suggested earlier. Our objective here is to investigate whether a similar mechanism could be responsible for the restoration of chiral symmetry at finite density as well. We start by briefly recalling some features of the instanton liquid model at zero density (sect. 2) and then turn to the finite density case (sect. 3), where we first calculate the pL4dependence of the instanton-antiinstanton interaction and then assess its impact on the chiral phase transition. Under certain approximations we will be able to avoid a complex ‘supported by the A.-v.-Humboldt foundation Energy under grant no. DEFGOW38ER40388.
as a Feodor-Lynen
0375-9474/98/$19.00 0 1998 Elsevier Science B.V. PI1 SO375-9474(98)00501-6
All rights reserved
fellow and by the US-Department
of
BuylcJsgss suognIos aporu o~az sec[ II!$s cp!q~ ‘(aaeds wapyma u!) uoyenba aw!a 113!suap al!uy ay$ u10.13~re$s a~ ‘Iz?gua$od Imituay~ ymnb acguy 1~ suoys~a$u! ~!aql 13nJ$suo2 0% amy lmj a~ aIqurasua uo~u~~su~ aq? 30 suo~y2~y~pow mupam dpnls 0A tori a7pud 712 uoqeJa*uI y-1 -1-g VIJ pm sapon 0Jaz y.nx@ ~.;CI~N~~O'~~[~Z-NON;L~~~~~~~I~~I~N~;CN~~~NI
3m
es
spsal ~usu!wa~ap 3!uoyurla3 aq$ ‘(0 +- /u1) sassmu ymnb quamw ~Iwus Buy3aI%au put ‘V‘Oa pm fo& ‘sapour olaz asay? Lq pauusds syeq ay? UI ‘uo!$nIos B.wua olaz papuey-q3al (-c@~II ) e sassassod pIay uo~we~su~ (-ye) ayl u! uoyenba ml!a aqL wa!wLs ayl30 say.xadoJd yInq aq? %!ssasss ~03 aI%?quossal aq pInoys qq~ ‘sapour BupCl IsaMoI ay? 1cIuo Bu!daay Lq pawInaIs3 LIa~am!xo~dds s! ‘spIay ymnb ayq JaAo uoyeel8 -a?u! ay$ urog %!s~E ‘lolmado X?Jra ayJ30 ~uwI!urJa~ap ayL ‘IUFs_a (d)u dp J z = A/N aq 0% suouitma3 30 aauasqI2 aql u! &suap uo~ue~su~ Is?07 ay? auyIla%ap zUFs uoyma~ -u! uo$us~su! au[$ 30 qmd a!uonIS aq? pus (suor$aalJo3 um~usnb BmpnIm!) (W)u apny!Id -um uoygsu!-aISu!s ayL Y3UO~UE$SU!JUS -N PUS SUO~US?SU! +N 30 (UO!$V~Ua!.IO 10103 puv az!s ‘uog!sod) { In ‘Id ‘12) = 1~ sapurploo3 aar$3aIIo3 paIIw-os aql .IaAo uogslSa$u! UB olu! pa$laauoa uaaq ssq suoyem8yuo~ pIay aIqysod 118 .IaAo IwSaqu!-ypd aqy aJayM
(1) -de
uo!pqsu!
‘
ay$
(~UO~U’T?!JSU~,)
U!
pX+Svp!UIas
pa!yeu!wop s! umnam
uogmn3
uoyysd
a!Jnl!$suo3
sauroaaq uaq? uoyur!xold aab aqA .suoysnba sII!II\I-%~A aql30 suoyqos SuoyWn~guo3 play a%ea aayeq.mJad-uou icq aqq uo paseq s! aa@ u! lapour uo~uv+su~ ay;L
q3q~
ay$ ~sqv uoydumssa
iC;CIsN3ao~3z;Cv?3ao~aaInbI?
NO&N~~;LSNI~H&
-z
.uoydpmap play-ueaur mo u! sam%y [g‘z] d+g~mpuo3~adns ~0103 30 cus!uey~auI passnmp AI’$uaDal ay!$ MOM MOT+ pm aX3ms !urJad aq$qe syrenb uaaM?aq suo!qviaJlo3 awes apnIm! aJomaylm3 aM .yasoldde ad4 play-warn K!uyp~ Ieyuayod Im!tuaya I’ee3y~maq$ apurysa PUT?uoymn3 uoyqmd
13c
R. Rapp/Nuclear Physics A642 (1998) 71c-77c
Constraining ourselves to zero temperature, the gluonic (instanton-) fields entering the covariant derivative are not affected by 1-1~.The explicit form of the quark zero modes has been determined in ref. [5],
with the function II(Z) = (1-t p2/s2) and a spin-color spinor x. Note that the solution the adjoint Dirac equation, @,,(?
-I.Lq) (i PI -
iPqY4) = 0 >
of
(6)
carries the chemical potential argument with opposite sign. This is necessary for a consistent definition of expectation values at finite Jo,, and in particular renders a finite norm
With the properly constructed quark wave functions we can now evaluate the fermionic overlap matrix element representing the zero mode part of the full Dirac operator. Choosing for simplicity the sum ansatz for the gauge-field configurations, A, = AE+A$, entering the covariant derivative, allows us to replace the latter by an ordinary one, yielding TIA(Z,%pq)
.I
=
-
=
i u4 fl(',r;pq)
d4x '$,(z
- ZI; -PLq) (i $ - &'Y4)
QO,A(x
- zA;&)
+ i
The breaking of Lorentz invariance in the medium implies the existence of two independent (real-valued) functions fi, fz which are shown in fig. 1. Similar to what has been found at finite temperature [6], TEA is strongly enhanced in temporal direction with increasing pLq,but oscillates as N sin(2p,r) in spatial direction [7]. The latter effectively suppresses the interaction once integrating over r. 3.2. Thermodynamics of the ILM at Finite ps To investigate the finite density properties of the thermodynamic potential (or free energy) we here resort to the so-called cocktail model introduced in ref. [8]. It amounts to a mean-field type description including three major components in the system: essentially random (anti-) instantons (the ‘atomic’ component), strongly correlated I-A-pairs (the ‘molecular’ component) and a Fermi sphere of constituent quarks (‘quasiparticles’). Thus
where the constituent
quark contribution
is zero for Fermi energies ,u~ < mq, with m, denoting instanton part of the free energy, Glst(%,
%Tl; Pq)=
-
ln[Zinst(~a, hi
v,
A)1
=
-71, I”[:]
the constituent
-n,
ln[z]
quark mass.
,
The
(11)
142
R. Rapp/Nuclear
-
Physics A642 (1998) 71c-77c
----_--
&.=O 0.3GeV 0.6 GeV
-.--
0.9 Ge”
t
i
3.0 ’
---.
f,(r) 2.0
--------
pq=o 0.3 Ge” 0.6 GeV 0.9G.a”
1
I
-1.0;.-.. 0.0
1.0
”
3.0
2.0
r
4.0
-I
5.0
[PI
Figure 1. Quark-induced I-A-interaction at finite density for the most attractive color orientation zlq=l, G=O as well as r=O (left panel) and for ud=O, lu’l=l and r=O (right panel).
is related to the atomic and molecular Gl = = &L
‘activities’
2 C pbP4 eWsi”” (TIA(P~)TA&))~~‘~ C2 #b-4) e-z5’i,t
([TIA(CL,)TAr(~U,)I”NI)
(14
The minimization of 52 with respect to the corresponding (4-dimensional) densities na and n, determines the equilibrium state of the system at fixed pLp. In particular, Rinsl encodes all the features of the T = pq = 0 instanton ensemble, as e.g. the quark condensate and the constituent quark mass, which in mean-field approximation are given and m4 c( -p”(qq). Therefore the normalization constant by (rlq) = -li(~)(3/2nP C IX (AQCn)b can be fixed to give N/V=1.4 fmP4, being realized for n,=1.34 fmP4 and n,=O.O3 fmP4, which is not unreasonable. The gluonic interaction has been approximated by an average repulsion Sint = -rcp4(n, + 27~~). At finite pq the Dirac operator is not hermitian any more, i.e. TAI&) # TjA(pq), resulting, in general, in a complex fermionic determinant, entailing the well-known ‘sign’ problem. However, assuming an average gluonic interaction that does not depend on density allows us to perform the color averages implied in eqs. (12) analytically, i.e.
ic h ensures the pressure to remain real. Fig. 2 shows our (Nf = 2, .f,’= .A(*&), w h’ results as function of the quark chemical potential (left panel) for N, = 3, Nf = 2. At small pclsessentially nothing happens until, at a critical value & N 310 MeV, the system
R. RappINuclear
1%
Physics A642 (1998) 71c-77c
1
-
Fq=O --0.16 GeV ----- 0.24GeV
or-
0.0
i
I
:
: :
/
/
H d
’
x10! “ #’
: Quark 0.2 :
I
BCS !’ 1 Gap!
0.4 r
7 o,3 1 Constituent
*
Mass
7
,i’
: 0.1 T
, ,/
8'
-0.6
/ ”
”
”
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0.3
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Figure 2. Our results for the instanton (‘atomic’) and molecule densities (upper left panel), the pressure p = -0 (middle left panel) and the constituent quark mass (lower left panel) after minimizing the free energy, eq. (9)) with respect to n, and n,; additionally including scalar diquark correlations (as discussed in sect. 3.3) adds the dashed-dotted line to the lower left panel, representing the BCS energy gap at the quark Fermi surface. The right panel shows the free energy as a function of constituent quark mass m, K nAf2, indicating a first order transition from the minimum at finite mp to the one at m4 = 0.
jumps into the chirally restored phase, the latter being characterized by n, = 0. The transition is of first order, as can be seen by inspection of the m,-dependence of the free energy (right panel of fig. 2). Below pi, the pressure actually decreases slightly with increasing ~1~indicating a mixed phase-type instability, similar to what has been discussed in refs. [2,9]. A significant difference, however, is given by the fact that in our approach the total instanton density at the transition (residing in I-A-molecules) is still appreciable, N/V = 2n, 21 1.1 fmW4, providing the major part of the pressure at this point; in other words: a substantial part of the nonperturbative vacuum pressure persists in the chirally restored phase (of course, eventually it will be suppressed due to the Debye screening of the instanton fields, which we have not included here).
76c
R. Rapp/Nuclear Physics A642 (1998) 71c-77c
Color Superconductivity As has recently been pointed out in refs. [2,3], the quark Fermi surface in the plasma phase might be unstable with respect to the formation of quark-Cooper pairs, once an attractive q-q interaction is present. Using the instanton-induced interaction in the scalar, color-antitriplett diquark channel (which is essentially the Fierzed-transformed interaction leading to a deeply bound pion and is also phenomenolgically well-supported by baryonic spectroscopy), BCS-type energy gaps of As = 50-100 MeV have been predicted. In the mean-field approach employed here, this interaction modifies the quark quasipartitle contribution according to 3.3.
$&I = tr log[D(m,,Ao)] - tr Mm,,
AoMAo)]
+ Gtr Mm,,
Ao)]
tr [F(m,, A,)] (14)
with the quasiparticle quark-propagator D and the annomalous (Gorkov) propagator F corresponding to creation/annihilation of a bound Cooper pair. G is the effective coupling constant derived from the instanton q-q-vertex [3]. From “t one can obtain the standard gap equation for As. However, here we also have to account for the fact that a finite A0 will damp the quark zero-mode propagators in the instanton part of the free energy, which results in a suppression of the quark induced I-A interaction TIA of eq. (8). Thus, Rinst disfavors finite A,%. The resulting expression for the free energy is of the form
(the last term accounting for unpaired quarks), which now has to be minimized W.T.t. n,, n, and A,. We find that color superconductivity does not appear before puQN 360 MeV (see dashed-dotted curve in the lower left panel of fig. 2): although the quark-part of the free energy by itself, eq. (14), always favors a finite value for A,, the suppression caused by A0 in Rinst (i.e. the damping of quark-propagation in TIA) prevents the formation of a diquark condensate below p,r c~ 360 MeV. This again reflects the fact that the instanton contribution to the pressure is still dominant in the region somewhat above pi, so that the gain due to a finite As in fit cannot overcome the ‘penalty’ in Rinst. Above pq N 360 MeV energy gaps of order N 50 MeV arise, in line with the findings of ref. [3]. As far as the thermodynamic properties of the system are concerned, no changes as compared to the previous section occur below pq = 360 MeV, and even above the results would be hardly distinguishable from the curves for n&J, nm(& p(PJ and m,&) displayed in fig. 2. In summary, we have studied the QCD partition function at finite density using the instanton model. Employing a simple mean-field approach, chiral symmetry restoration emerges at pg =310 MeV as a first order transition from an essentially random (anti-/) instanton liquid into a phase of strongly correlated I-A molecules (and massless quarks), driven by the density dependent increase of the quark-induced I-A-interaction. Neglecting possible medium modifications in the gluonic interaction, which are expected to be small, a complex pressure could be avoided. Accounting for correlations in the scalar diquark channel, color superconductivity sets in for chemical potentials p,r > 360 MeV, associated with BCS gaps of up to N 50 MeV.
R. Rapp/Nuclear
Physics A642 (1998) 71c-77c
IIC
Acknowledgement
I thank my collaborators
E. Shuryak, T. Schafer and M. Velkovsky for fruitful discussions.
REFERENCES
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