Mesonic correlation functions at finite temperature and density in the Nambu-Jona-Lasinio model with a Polyakov loop

0.89 MB·English

by  H. Hansen

Mesonic correlation functions at finite temperature and density in the Nambu – Jona
- Lasinio model with a Polyakov loop
H. Hansen(a),∗ W.M.Alberico(a), A.Beraudo(b), A.Molinari(a), M.Nardi(a), and C.Ratti(c)
(a) INFN, Sezione di Torino and Dipartimento di Fisica Teorica,
University of Torino, via Giuria N.1, 10125 Torino - Italy
(b) Service de Physique Th´eorique, CEA Saclay,
CEA/DSM/SPhT, F-91191, Gif-sur-Yvette - France and
(c) ECT∗, 38050 Villazzano (Trento) - Italy and INFN,
Gruppo Collegato di Trento, via Sommarive, 38050 Povo (Trento) - Italy
We investigate the properties of scalar and pseudo-scalar mesons at finite temperature and
quark chemical potential in the framework of the Nambu–Jona-Lasinio (NJL) model coupled to
the Polyakov loop (PNJL model) with the aim of taking into account features of both chiral sym-
7 metry breaking and deconfinement.
0 The mesonic correlators are obtained by solving the Schwinger–Dyson equation in the RPA ap-
0 proximation with theHartree (mean field) quarkpropagator at finitetemperature and density.
2 Inthephaseofbrokenchiralsymmetryanarrowerwidthfortheσmesonisobtainedwithrespect
tothe NJL case; on theother hand,the pion still behavesas a Goldstone boson.
n
When chiral symmetry is restored, the pion and σ spectral functions tend to merge. The Mott
a
temperaturefor thepion is also computed.
J

9
PACSnumbers: 11.10.Wx, 11.30.Rd,12.38.Aw,12.38.Mh,14.65.Bt,25.75.Nq
2

2
v
6
I. INTRODUCTION
1
1
9 Recently,increasingattentionhasbeendevotedtostudythemodificationofparticlespropagatinginahotordense
0 medium[1,2]. ThepossiblesurvivalofboundstatesinthedeconfinedphaseofQCD[3,4,5,6,7,8,9,10]hasopened
6 interesting scenarios for the identification of the relevant degrees of freedom in the vicinity of the phase transition
0
[11, 12, 13]. At the same time, renewed interest has arisen for the study of the ρ meson spectral function in a hot
/
h medium [14, 15, 16, 17, 18, 19], since precise experimental data have now become available for this observable [20].
p In this paper,we focus on the description of lightscalar and pseudo-scalarmesons at finite temperature and quark
-
chemicalpotential. Besideslattice calculations[21,22, 23,24], hightemperaturecorrelatorsbetweenmesoniccurrent
p
e operatorscanbestudied,startingfromtheQCDlagrangian,withindifferenttheoreticalschemes,likethedimensional
h reduction [25, 26] or the Hard ThermalLoopapproximation[27, 28, 29]. Actually both the above approachesrely on
: a separationofmomentum scaleswhich, strictly speaking,holds only inthe weakcoupling regimeg ≪1. Hence they
v
i cannot tell us anything about what happens in the vicinity of the phase transition.
X
On the other hand a system close to a phase transition is characterizedby large correlationlengths (infinite in the
r caseofasecondorderphasetransition). Itsbehaviourismainlydrivenbythesymmetriesofthelagrangian,ratherthan
a
by the details of the microscopic interactions. In this critical regime of temperatures and densities our investigation
of meson properties is then performed in the framework of an effective model of QCD, namely a modified Nambu
Jona-Lasinio model including Polyakovloop dynamics (referred to as PNJL model) [30, 31, 32, 33, 34, 35, 36, 37] .
Models of the Nambu and Jona-Lasinio (NJL) type [38] have a long history and have been extensively used to
describe the dynamics and thermodynamics of the lightest hadrons [39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. Such
schematicmodelsofferasimpleandpracticalillustrationofthebasicmechanismsthatdrivethespontaneousbreaking
of chiral symmetry, a key feature of QCD in its low-temperature, low-density phase.
In first approximation the behavior of a system ruled by QCD is governed by the symmetry properties of the
Lagrangian, namely the (approximate) global symmetry SU (N ) × SU (N ), which is spontaneously broken to
L f R f
SU (N ) and the (exact) SU (N ) local color symmetry. Indeed in the NJL model the mass of a constituent quark
V f c c
is directly related to the chiral condensate, which is the order parameter of the chiral phase transition and, hence,
is non-vanishing at zero temperature and density. Here the system lives in the phase of spontaneously broken chiral
symmetry: thestronginteraction,bypolarizingthevacuumandturningitintoacondensateofquark-antiquarkpairs,
transforms an initially point-like quark with its small bare mass m into a massive quasiparticle with a finite size.
0
∗Electronicaddress: [email protected]
2
Despite their widespread use, NJL models suffer a major shortcoming: the reduction to global (rather than local)
colour symmetry prevents quark confinement.
On the other hand, in a non-abelian pure gauge theory, the Polyakov loop serves as an order parameter for the
transitionfromthelowtemperature,Z symmetric,confinedphase(theactivedegreesoffreedombeingcolor-singlet
Nc
states,the glueballs),tothe hightemperature,deconfinedphase(the activedegreesoffreedombeingcoloredgluons),
characterized by the spontaneous breaking of the Z (center of SU (N )) symmetry.
Nc c c
With the introduction of dynamical quarks, this symmetry breaking pattern is no longer exact: nevertheless it is
still possible to distinguish a hadronic (confined) phase from a QGP (deconfined) one.
InthePNJLmodelquarksarecoupledsimultaneouslytothechiralcondensateandtothePolyakovloop: themodel
includes features of both chiral and Z symmetry breaking. The model has proven to be successful in reproducing
Nc
lattice data concerning QCD thermodynamics [35]. The coupling to the Polyakovloop, resulting in a suppression of
the unwanted quark contributions to the thermodynamics below the critical temperature, plays a fundamental role
for this purpose.
ItisthereforenaturaltoinvestigatethepredictionsofthePNJLmodelforwhatconcernsmesonicproperties. Since
the “classic” NJL model lacks confinement, the σ meson for example can unphysically decay into a q¯q pair even
in the vacuum: indeed this process is energetically allowed and there is no mechanism which can prevent it. As a
consequence, the σ meson shows, in the NJL model, an unphysical width corresponding to this process. One of our
goals is to check whether the coupling of quarks to the Polyakov loop is able to cure this problem, thus preventing
the decay of the σ meson into a q¯q pair. Accordingly, particular emphasis will be given in our work to the σ spectral
function.
We compute the mesonic correlation functions in ring approximation (i.e. RPA, if one neglects the antisym-
metrisation) with quark propagator evaluated at the Hartree mean field level. The properties of mesons at finite
temperatureandchemicalpotentialarefinallyextractedfromthesecorrelationfunctions. Werestrictourselvestothe
scalar-pseudoscalarsectorsanddiscuss the impact ofthe Polyakovlooponthe mesonicproperties andthe differences
between NJL and PNJL models. Due to the simplicity of the model where dynamical gluonic degrees of freedom are
absent, no true mechanism of confinement is found (we will show that for the σ meson the decay channel σ → qq¯is
still open also below T ).
c
Ourpaperis organizedasfollows: inSectionII webrieflyreviewthe mainfeaturesofthe PNJLmodel, howquarks
arecoupledtothePolyakovloop,ourparameterchoiceandsomeresultsobtainedinRef.[35]whicharerelevanttoour
work. InSectionsIIIandIVweaddressthestudyofcorrelatorsofcurrentoperatorscarryingthequantumnumbersof
physical mesons, and the corresponding mesonic spectral functions and propagators;we obtain the relevant formulas
both in the NJL and in the PNJL cases, and discuss the main differences between the two models. Our numerical
resultsconcerningthe mesonicmassesandspectralfunctions arediscussedinSectionV. Particularattentionisagain
focused on the NJL/PNJL comparison. Final discussions and conclusions are presented in Section VI.
II. THE MODEL
A. Nambu – Jona - Lasinio model
Motivated by the symmetries of QCD, we use the NJL model (see [43, 44, 46, 49] for review papers) for the
descriptionof the coupling between quarks and the chiralcondensate in the scalar-pseudoscalarsector. We will use a
two flavor model, with a degenerate mass matrix for quarks. The associated Lagrangianreads:
L = q¯(iγµ∂ −mˆ)q+G (q¯q)2+(q¯iγ ~τq)2 (1)
NJL µ 1 5
h i
In the above q¯= (u¯,d¯), mˆ = diag(m ,m ), with m = m ≡ m (we keep the isospin symmetry); finally τa (a =
u d u d 0
1,2,3) are SU (2) Pauli matrices acting in flavor space. As it is well known, this Lagrangian is invariant under a
f
global – and not local – color symmetry SU(N = 3) and lacks the confinement feature of QCD. It also satisfies the
c
chiral SU (2)×SU (2) symmetry if mˆ = 0 while mˆ 6= 0 implies an explicit (but small) chiral symmetry breaking
L R
from SU (2)×SU (2) to SU (2) which is still exact, due to the choice m =m ≡m .
L R f u d 0
TheparametersenteringintoEq.(1)areusuallyfixedtoreproducethe massanddecayconstantofthepionaswell
asthe chiralcondensate. The parameterswe usearegiveninTableI, togetherwith the calculatedphysicalquantities
chosen to fix the parameters. The Hartree quark mass (or constituent quark mass) is m = 325 MeV and the pion
decay constant and mass are obtained within a Hartree + RPA calculation.
3
Λ [GeV] G1 [GeV−2] m0 [MeV] |hψ¯uψui|1/3 [MeV] fπ [MeV] mπ [MeV]
0.651 5.04 5.5 251 92.3 139.3
Table I: Parameter set for theNJL Lagrangian given in Eq.(1) and the physicalquantities chosen to fixthe parameters.
B. Pure gauge sector
In this Section, following the arguments given in [50, 51], we discuss how the deconfinement phase transition in a
pure SU(N ) gauge theory can be conveniently described through the introduction of an effective potential for the
c
complex Polyakovloop field, which we define in the following.
Since we want to study the SU(N ) phase structure, first of all an appropriate order parameter has to be defined.
c
For this purpose the Polyakovline
β
L(~x) ≡ Pexp i dτA (~x,τ) (2)
4
" Z0 #
is introduced. In the above, A = iA0 is the temporal component of the Euclidean gauge field (A~,A ), in which the
4 4
strong coupling constant g has been absorbed, P denotes path ordering and the usual notation β = 1/T has been
S
introduced with the Boltzmann constant set to one (k =1).
B
When the theory is regularized on the lattice, the Polyakovloop,
1
l(~x)= TrL(~x), (3)
N
c
is a color singlet under SU(N ), but transforms non-trivially, like a field of charge one, under Z . Its thermal
c Nc
expectationvalue is then chosenasanorderparameterfor the deconfinementphase transition[52, 53, 54]. In fact, in
the usual physical interpretation [55, 56], hl(~x)i is related to the change of free energy occurring when a heavy color
source in the fundamental representation is added to the system. One has:
hl(~x)i=e−β∆FQ(~x). (4)
Inthe Z symmetricphase, hl(~x)i=0,implyingthataninfinite amountoffree energyis requiredto addanisolated
Nc
heavy quark to the system: in this phase color is confined.
Phase transitions are usually characterizedby large correlationlengths, i.e. much larger than the averagedistance
between the elementary degrees of freedom of the system. Effective field theories then turn out to be a useful tool to
describeasystemnearaphasetransition. Inparticular,intheusualLandau-Ginzburgapproach,theorderparameter
is viewed as a field variable and for the latter an effective potential is built, respecting the symmetries of the original
lagrangian. InthecaseoftheSU(3)gaugetheory,thePolyakovlineL(~x)getsreplacedbyitsgaugecovariantaverage
over a finite region of space, denoted as hhL(~x)ii [50]. Note that hhL(~x)ii in general is not a SU(N ) matrix. The
c
Polyakovloop field:
1
Φ(~x)≡hhl(~x)ii= Tr hhL(~x)ii (5)
c
N
c
is then introduced.
Following [35, 50, 51], we define an effective potential for the (complex) Φ field, which is conveniently chosen to
reproduce, at the mean field level, results obtained in lattice calculations. In this approximation one simply sets the
Polyakovloop field Φ(~x) equal to its expectation value Φ=const., which minimizes the potential
U Φ,Φ¯;T =−b2(T)Φ¯Φ− b3 Φ3+Φ¯3 + b4 Φ¯Φ 2 , (6)
T4 2 6 4
(cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1)
where
T T 2 T 3
0 0 0
b (T)=a +a +a +a . (7)
2 0 1 2 3
T T T
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
4
a0 a1 a2 a3 b3 b4
6.75 -1.95 2.625 -7.44 0.75 7.5
Table II:Parameters for theeffective potential in the puregauge sector (Eq.(6)).
A precision fit of the coefficients a , b has been performed in Ref. [35] to reproduce some pure-gauge lattice data.
i i
The results are reported in Table II. These parameters have been fixed to reproduce the lattice data for both the
expectationvalueofthe Polyakovloop[57]andsomethermodynamicquantities[58]. The parameterT isthe critical
0
temperature for the deconfinementphase transition,fixed to 270 MeV accordingto pure gaugelattice findings. With
the present choice of the parameters, Φ and Φ¯ are never larger than one in the pure gauge sector. The lattice data
in Ref. [57] show that for large temperatures the Polyakov loop exceed one, a value which is reached asymptotically
from above. This feature cannot be reproduced in the absence of radiative corrections: therefore, at the mean field
level, it is consistent to have Φ and Φ¯ always smaller than one. In any case, the range of applicability of our model
is limited to temperatures T ≤ 2.5 T (see the discussion at the end of the next section) and for these temperatures
c
there is good agreement between our results and the lattice data for Φ.
The effective potential presents the feature of a phase transition from color confinement (T < T , the minimum
0
of the effective potential being at Φ = 0) to color deconfinement (T > T , the minima of the effective potential
0
occurring at Φ 6= 0) as it can be seen from Fig. 1. The potential possesses the Z symmetry and one can see that,
3
above T , it presents three minima (Z symmetric), showing a spontaneous symmetry breaking.
0 3
C. Coupling between quarks and the gauge sector: the PNJL model
In the presence of dynamical quarks the Z symmetry is explicitly broken. One cannot rigorously talk of a phase
3
transition, but the expectation value of the Polyakov loop still serves as an indicator for the crossover between the
phase where color confinement occurs (Φ−→0) and the one where color is deconfined (Φ−→1).
The PNJL model attempts to describe in a simple way the two characteristic phenomena of QCD, namely decon-
finement and chiral symmetry breaking.
In order to describe the coupling of quarks to the chiral condensate, we start from an NJL description of quarks
(global SU (3) symmetric point-like interaction), coupled in a minimal way to the Polyakov loop, via the following
c
Lagrangian([35])1:
L =q¯(iγ Dµ−mˆ )q+G (q¯q)2+(q¯iγ ~τq)2 −U Φ[A],Φ¯[A];T , (8)
PNJL µ 0 1 5
h i
(cid:0) (cid:1)
where the covariant derivative reads Dµ =∂µ−iAµ and Aµ =δµA0 (Polyakovgauge), with A0 =−iA . The strong
0 4
coupling constant g is absorbed in the definition of Aµ(x) = g Aµ(x)λa where Aµ is the gauge field (SU (3)) and
S S a 2 a c
λ are the Gell–Mann matrices. We notice explicitly that at T =0 the Polyakovloop and the quark sector decouple.
a
In order to address the finite density case,it turns out to be useful to introduce the following effective Lagrangian:
L′ =L +µq¯γ0q , (9)
PNJL PNJL
which leads to the customary grand canonical Hamiltonian. In the above the chemical potential term accounts for
baryon number conservation which, in the grand canonical ensemble, is not imposed exactly, but only through its
expectationvalue. Let us comment here the range ofapplicability ofthe PNJL model. As alreadystatedin Ref. [35],
in the PNJL model the gluon dynamics is reduced to a chiral-point coupling between quarks together with a simple
static background field representing the Polyakov loop. This picture cannot be expected to work outside a limited
range of temperatures. At large temperatures transverse gluons are known to be thermodynamically active degrees
of freedom: they are not taken into account in the PNJL model. Hence based on the conclusions drawn in [60]
according to which transversegluons start to contribute significantly for T >2.5T , we can assume that the range of
c
applicability of the model is limited roughly to T ≤(2−3)T .
c
1 WeuseheretheoriginalLagrangianofRef.[35],withacomplexPolyakovloopeffectivefield,whichimpliesthatatµ6=0theexpectation
valuesofΦandΦ¯ aredifferent. Adifferentchoicecanbemotivated[59]butwehavechecked thatthecalculations ofthepresentwork
arenotsensitivetothisfeature.
5
T =0.26 GeV<T0
“Color confinement”
hΦi=0−→ Nobreaking of Z3
5
U( Φ ) / T4
4
5
3
4
3 2
2
1 1
0
-1 0
-2
1.5
1
0.5
-1.5
-1 0
-0.5 -0.5 Im Φ
0
0.5 -1
Re Φ 1
1.-51.5
T =1 GeV >T0
“Color deconfinement”
hΦi=6 0−→ breakingof Z3
U( Φ ) / T4 0
-0.2
-0.4
0.5
0 -0.6
-0.5 -0.8
-1 -1
-1.5 -1.2
-2
-1.4
-2.5
-3
1.5
1
0.5
-1.5
-1 0
-0.5 -0.5 Im Φ
0
0.5 -1
Re Φ 1
1.-51.5
Figure1: Effectivepotentialinthepuregaugesector(Eq.(6))fortwocharacteristictemperatures,belowandabovethecritical
temperature T0. Onecan see threeminima appearing above T0.
6
D. Field equations
1. Hartree approximation
InthisSectionwederivethe gap equation intheHartreeapproximation,whosesolutionprovidestheself-consistent
PNJL mass of the dressed quark.
WestartfromtheeffectivelagrangiangiveninEq.(9). Theimaginarytimeformalismisemployed. Onedefinesthe
vertices Γ , where M = {S,P}, in the scalar (Γ ≡ I) and pseudo-scalar (Γa ≡ iγ τa) channel. The diagrammatic
M S P 5
Hartree equation reads:
>
= + = + (10)
where the thin line denotes the free propagator in the constant (we work in the mean field) background field A :
4
S (p)= =−(p/−m +γ0(µ−iA ))−1, the thick line the Hartree propagatorS(p)= =−(p/−m+γ0(µ−
0 0 4
iA ))−1,thecross( )thevertexΓ andthedot( )represents2G ,thecouplingconstantinthescalar-pseudoscalar
4 M 1
channel (indeed due to parity invariance only the scalar vertex contributes).
>
Besides, = is the Hartree self-energy and m≡m +Σ. The Hartree equation then reads:
0
+∞ d3p −1
m−m =2G T Tr (11)
0 1 n=−∞ZΛ (2π)3p/−m+γ0(µ−iA4)
X
In allthe aboveformulas,p =iω andω =(2n+1)πT is the Matsubarafrequency for a fermion; the trace is taken
0 n n
over color, Dirac and flavor indices. The symbol denotes the three dimensional momentum regularisation; we use
Λ
an ultraviolet cut-off Λ for both the zero and the finite temperature contributions. Our choice is motivated by our
R
wish to discuss mesonic properties driven by chiral symmetry considerations,a feature not well described if one only
regularizes the T = 0 part (in particular in the vector sector the Weinberg sum rule is not well satisfied). Through
a convenient gauge transformation of the Polyakov line, the background field A in Eq. (11) can always be put in a
4
diagonal form. This allows one to straightforwardly perform the sum over the Matsubara frequencies yielding (see
also section IIIC):
Nc d3p 2m
m−m =2G N [1−f(E −µ+iAii)−f(E +µ−iAii)]. (12)
0 1 f i=1ZΛ (2π)3 Ep p 4 p 4
X
By introducing the modified distribution functions2 f+ and f−, here derived for N = 3 (with the usual notation
Φ Φ c
β =1/T):
Φ+2Φ¯e−β(Ep−µ) e−β(Ep−µ)+e−3β(Ep−µ)
f+(E ) = (13)
Φ p 1+(cid:0) 3 Φ+Φ¯e−β(Ep−(cid:1)µ) e−β(Ep−µ)+e−3β(Ep−µ)
f−(E ) = Φ¯ +(cid:0) 2Φe−β(Ep+µ) e(cid:1)−β(Ep+µ)+e−3β(Ep+µ) , (14)
Φ p 1+(cid:0) 3 Φ+Φ¯e−β(Ep+(cid:1)µ) e−β(Ep+µ)+e−3β(Ep+µ)
the gap equation reads: (cid:0) (cid:1)
d3p 2m
m−m =2G N N [1−f+(E )−f−(E )]. (15)
0 1 f cZΛ (2π)3 Ep Φ p Φ p
2 Wewillexplicitlyderivethesequantities andtheirroleinSec.IIIC.
7
The latter is valid for any N providing one uses the corresponding f+,−. Notice that Eq. (11), after computing
c Φ
the trace on Dirac and isospin indices, can be viewed as a generalization of the corresponding zero temperature and
density NJL gap equation
d4p 1
m−m = 8iG mN N , (16)
0 1 c f (2π)4p2−m2
ZΛ
after adopting the following symbolic replacements:
p=(p ,p~) → (iω +µ−iA ,p~) (17)
0 n 4
d4p 1 d3p
i → −T Tr , (18)
(2π)4 N c (2π)3
ZΛ c n ZΛ
X
2. Grand potential at finite temperature and density in Hartree approximation
The usual techniques [44, 61] can be used to obtain the PNJL grand potential from the Hartree propagator (see
[35]):
(m−m )2 d3p
Ω=Ω(Φ,Φ¯,m;T,µ) = U Φ,Φ¯,T + 0 −2N N E
4G1 c fZΛ (2π)3 p
d3p(cid:0) (cid:1)
−2N T Tr ln 1+L†e−(Ep−µ)/T +Tr ln 1+Le−(Ep+µ)/T .
f ZΛ (2π)3 n c h i c h io
(19)
In the above formula E = p~2+m2 is the Hartree single quasi-particle energy (which includes the constituent
p
quark mass). We then define z+,− and compute them for N =3:
pΦ c
z+ ≡Tr ln 1+L†e−(Ep−µ)/T = ln 1+3 Φ¯ +Φe−(Ep−µ)/T e−(Ep−µ)/T +e−3(Ep−µ)/T (20)
Φ c
h i n (cid:16) (cid:17) o
z− ≡Tr ln 1+Le−(Ep+µ)/T = ln 1+3 Φ+Φ¯e−(Ep+µ)/T e−(Ep+µ)/T +e−3(Ep+µ)/T . (21)
Φ c
h i n (cid:16) (cid:17) o
E. Mean field results
The solutionsofthe meanfieldequationsareobtainedby minimizing the grandpotentialwithrespectto m,Φ and
Φ¯, namely (again below N =3)
c
∂Ω
= 0
∂Φ
T4
= (−b (T)Φ¯ −b Φ2+b ΦΦ¯2)
2 3 4
2
d3p e−2(Ep−µ)/T
−6N T
f ZΛ (2π)3 (1+3 Φ¯ +Φe−(Ep−µ)/T e−(Ep−µ)/T +e−3(Ep−µ)/T
(cid:0) e−(Ep+µ)/T (cid:1)
+ , (22)
1+3 Φ+Φ¯e−(Ep+µ)/T e−(Ep+µ)/T +e−3(Ep+µ)/T)
(cid:0) (cid:1)
∂Ω
= 0
∂Φ¯
T4
= (−b (T)Φ−b Φ¯2+b Φ¯Φ2)
2 3 4
2
d3p e−(Ep−µ)/T
−6N T
f ZΛ (2π)3 (1+3 Φ¯ +Φe−(Ep−µ)/T e−(Ep−µ)/T +e−3(Ep−µ)/T
(cid:0) e−2(Ep+µ)/T(cid:1)
+ (23)
1+3 Φ+Φ¯e−(Ep+µ)/T e−(Ep+µ)/T +e−3(Ep+µ)/T)
(cid:0) (cid:1)
8
and
∂Ω
= 0 (24)
∂m
whichcoincideswith the gapequation(11). A complete discussionofthe results inmeanfieldapproximationis given
in [35]. For the purpose of this article, we only briefly discuss the result obtained in [35] for the net quark number
density, defined by the equation
n (T,µ) 1 ∂Ω(T,µ)
q
=− , (25)
T3 T3 ∂µ
that we display in Fig. 23. Note that an implicit µ-dependence of Ω is also contained in the effective quark mass
m and in the expectation values Φ and Φ¯. Nevertheless, due to stationary equations (22, 23, 24), only the explicit
dependence arising from the statistical factors has to be differentiated.
One can see that the NJL model (corresponding to the Φ→1 limit of PNJL) badly fails in reproducing the lattice
findings, while the PNJL resultsprovidea goodapproximationfor them. One realizesthat, ata givenvalue ofT and
µ, the NJL modelalwaysoverestimatesthe baryondensity, evenif, for largetemperatures,wheninPNJL Φ→1,the
two models merge.
On the other hand in the PNJL model below T (when Φ,Φ¯ → 0) one can see from Eqs. (20) and (21) that
c
contributions coming from one and two (anti-)quarks are frozen, due to their coupling with Φ and Φ¯, while three
(anti-)quarkcontributions are not suppressedevenbelowT . This implies that, atfixed values of T andµ, the PNJL
c
value for n results much lower than in the NJL case. In fact all the possible contributions to the latter turn out to
q
be somehow suppressed: the one- and two-quark contributions because of Φ,Φ¯ → 0, while the thermal excitation of
three quark clusters has a negligible Boltzmann factor.
One would be tempted to identify these clusters of three dressed (anti-)quarks with precursors of (anti-)baryons.
Indeed no binding for these structures is providedby the model. In any case it is encouragingthat coupling the NJL
Lagrangianwith the Polyakovloop field leads to results pointing into the right direction.
In the following Section we explore the PNJL results in the mesonic sector, investigating whether coupling the
(anti-)quarks with the Φ field constrains the dressed qq¯pairs to form stable colorless structures.
III. MESONIC CORRELATORS
In this Section, we addressthe centraltopic ofourpaper, i.e. the study of correlatorsofcurrentoperatorscarrying
thequantumnumbersofphysicalmesons. Wefocusourattentionontwoparticularcases: thepseudoscalariso-vector
current
J a(x) = q¯(x)iγ τaq(x) (pion) (26)
P 5
and the scalar iso-scalar current:
J (x) = q¯(x)q(x)−hq¯(x)q(x)i (sigma). (27)
S
These are in fact the channels of interest to study the chiral symmetry breaking-restoration pattern. In particular
the scalar current represents the fluctuations of the order parameter.
Intermsoftheabovecurrents,thefollowingmesoniccorrelationfunctionsandtheirFouriertransformsaredefined:
CPP(q2)≡i d4xeiq.x 0 T Ja(x)Jb†(0) 0 =CPP(q2)δ (28)
ab P P ab
Z D (cid:12) (cid:16) (cid:17)(cid:12) E
and (cid:12) (cid:12)
(cid:12) (cid:12)
CSS(q2)≡i d4xeiq.x 0 T J (x)J†(0) 0 . (29)
S S
Z D (cid:12) (cid:16) (cid:17)(cid:12) E
In the above equations, the expectation value is taken with(cid:12) respect to the v(cid:12)acuum state and T is the time-ordered
(cid:12) (cid:12)
product.
3 Indeedin[35]adifferentregularizationprocedurewasemployedwithrespecttothechoiceadoptedinthispaper. Namely,nocut-offwas
usedforthe finiteT contribution tothethermodynamical potential. Thischoice was madeinorderto better reproduce latticeresults
uptotemperatures T ∼2Tc. Inany case, forlowertemperature this differenceintheregularizationisunimportant. In particularour
qualitativediscussionoftheroleofthefieldΦinmimickingconfinement isindependent ofthesedetails.
9
1 3
n /T
q
0.8
0.6
0.4
0.2
µ
=0.6 T
c
0
0 0.5 1 1.5 2
T/T
c
Figure2: PNJL (solid line), NJL (dottedline) andlattice results (points)for thenet quarkdensityat µ=0.6T (from [35]).
c
A. Schwinger – Dyson equations at T =µ =0
Here we briefly summarize the usual NJL results for the mesonic correlators [62, 63, 64, 65], which we are going
to generalize in Sec. (IIIC) by including the case in which quarks propagate in the temporal background gauge field
related to the Polyakov loop. The Schwinger – Dyson equation for the meson correlator CMM is solved in the ring
approximation (RPA):
CMM(q2) = ΠMM(q2)+ ΠMM′(2G )CM′M (30)
1
M′
X
where the
ΠMM′ ≡ (2dπ4p)4Tr(ΓMS(p+q)ΓM′S(q)) (31)
ZΛ
are the one loop polarizations and S(p) is the Hartree quark propagator. In terms of diagrams, one defines:
>
ΠMM′ = ΓM ΓM′ (32)
>
10
and
> >
Π Π
CMM = C = + C . (33)
> >
Hence, we need the following (one loop) polarization functions:
d4p
ΠPP(q2) = Tr iγ τaS(p+q)iγ τbS(q) =ΠPP(q2)δ (34)
ab (2π)4 5 5 ab
ZΛ
d4p (cid:0) (cid:1)
ΠSS(q2) = Tr(S(p+q)S(q)). (35)
(2π)4
ZΛ
Thus, for example, for the pion channel:
d4p m2−p2+q2/4
ΠPP(q2) = −4iN N (36)
c f (2π)4[(p+q/2)2−m2][(p−q/2)2−m2]
ZΛ
= 4iN N I −2iN N q2I (q2)
c f 1 c f 2
the loop integrals being:
d4p 1
I = (37)
1 (2π)4p2−m2
ZΛ
d4p 1
I (q2) = . (38)
2 (2π)4[(p+q)2−m2][p2−m2]
ZΛ
By defining4:
f2(q2) = −4iN m2I (q2) (39)
P c 2
and owing to the fact that the Hartree equation (16) implies
m−m
0
I = , (40)
1
8iG mN N
1 c f
one shows that [63]
m−m q2
ΠPP(q2) = 0 +f2(q2) (41)
2G m P m2
1
m−m q2−4m2
ΠSS(q2) = 0 +f2(q2) . (42)
2G m P m2
1
The explicit solutions of the Schwinger–Dysonequations in ring approximation then read:
• Scalar iso-scalar sector
CSS(q2) = ΠSS(q2)+ΠSS(q2)(2G )CSS(q2) (43)
1
ΠSS(q2)
⇒CSS = . (44)
1−2G ΠSS(q2)
1
• Pseudo-scalariso-vector sector
CPP(q2) = ΠPP(q2)+ΠPP(q2)(2G )CPP(q2) (45)
1
ΠPP(q2)
⇒CPP = . (46)
1−2G ΠPP(q2)
1
4 f2(q2=0)isthepiondecayconstant f2 inthechirallimit[44].
P π