Phase structure, critical points and susceptibilities in Nambu-Jona-Lasinio type models

0.2 MB·English

by  C. A de Sousa

FrascatiPhysicsSeriesVol.XLVI(2007), pp. 000-000
HADRON07: XII Int. Conf. on Hadron Spectroscopy*Frascati,October8-13,2007
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PHASE STRUCTURE, CRITICAL POINTS,

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h AND SUSCEPTIBILITIES IN NAMBU-JONA-LASINIO TYPE
p MODELS
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C. A de Sousa, Pedro Costa and M. C. Ruivo
e
Departamento de F´ısica, Universidade de Coimbra, P - 3004 - 516
h
[ Coimbra, Portugal

1
v
6
2
4 Abstract
3
. We investigate the chiral phase transition at finite temperature and chemical
1
0 potential within SU(2) and SU(3) Nambu-Jona-Lasinio type models. The be-
8 haviorofthebaryonnumbersusceptibilityandthespecificheat,inthevicinity
0 of the critical end point, is studied. The class of the critical points is analyzed
:
v by calculating critical exponents.
i
X
r
a 1 Introduction
Strongly interacting matter at non-zero temperature and chemical potential is
an exciting topic for physicists coming from different areas, either theoretical
or experimental. One of the main goals in the heavy-ion physics program
nowadays is to study the effects of several macroscopic phenomena occurring
under extreme conditions. The discussion about the existence of a tricritical
point (TCP) or a critical end point (CEP) is also a topic of recent interest.
As is well known, a TCP separates the first order transition at high chemical
potentialsfromthesecondordertransitionathightemperatures. Ifthesecond
order transition is replaced by a smooth crossover, a CEP which separates
the two lines is found. At the CEP the phase transition is of second order
and probably falls into the same universality class of the three-dimensional
Ising model. The existence of the CEP in QCD was suggested at the end
1)
of the eighties , and its properties in the context of several models have
2, 3, 4)
been studied since then . The most recent lattice results with N =
f
2+1 staggered quarks of physical masses indicate the location of the CEP at
TCEP =162±2MeVandµCEP =360±40MeV5),howeveritsexactlocation
is not yet known.
This pointofthe phasediagramisthe specialfocus ofthe presentcontri-
bution. Nambu-Jona-Lasinio (NJL) type models are used and the main goal
is to locate the critical end point and confront the results with universality
arguments.
We remark that most of the work done in this area has been performed
with non strange quarks only. We will discuss the class of the critical points
by including the analyzes in the chiral limit of both SU(2) and SU(3) versions
of the NJL model.
6, 7)
The Lagrangianof the SU(3) NJL model is given by:
8
g
L = q¯(i∂·γ−mˆ)q+ S (q¯λaq)2+(q¯(iγ )λaq)2
5
2
Xa=0h i
+ g det q¯(1+γ )q +det q¯(1−γ )q . (1)
D 5 5
h (cid:2) (cid:3) (cid:2) (cid:3)i
8, 9)
Using a standardhadronizationprocedure , the baryonicthermody-
namicpotential,Ω(T,V,µ ),isobtaineddirectlyfromthe effectiveaction. The
i
baryonnumbersusceptibilityχ andthespecificheatC describe,respectively,
B
the response of the baryon density ρ and the entropy S with respect to the
B
chemical potential µ and the temperature T:
i
1 ∂ρ T ∂S
i
χ = and C = . (2)
B
3 (cid:18)∂µ (cid:19) V (cid:18)∂T(cid:19)
i=Xu,d,s i T Ni
These physical quantities are relevant observables to be studied in the
context of possible signatures for chiral symmetry restoration.
Ourmodelofstronginteracting mattercansimulate regionsofahot and
densefireballcreatedinaheavy-ioncollision. Sinceelectronsandpositronsare
not involvedin the strong interaction, we impose the condition µ =0. So, we
e
naturally get the chemical equilibrium condition µ =µ =µ =µ that will
u d s B
be used.
After this presentation of the model, we discuss the phase diagrams in
sec. 2. The behavior of the baryon number susceptibility and the specific
heat in the T −µ plane around the CEP is studied in sec. 3, as well as the
B
corresponding critical exponents. Finally, we conclude in sec. 4 with a brief
summary of our results.
2 Phase diagrams in SU(2) and SU(3) NJL models
In this section we analyze the phase diagrams in different conditions in the
T−µ plane. DependingonthenumberofquarkflavorsN =2orN =3,and
B f f
on the masses of the quarks, different situations can occur and the transition
fromhadronicmattertoQGPmaybeoffirstorder,secondorder,oracrossover
transition.
Westartbyanalyzingthedifferencesbetweenthethree-flavorNJLmodel
anditssimplerversionintheSU(2)sector. Thephasediagramsforbothmodels
are presented in fig. 1 as a function of µ and T.
B
Concerning the SU(2) model, and using physical values of the quark
masses, m = m = 6 MeV, together with Λ = 590 MeV and g Λ2 = 2.435,
u d S
we find that the CEP is localized at TCEP = 79.9 MeV and µCEP = 331.72
B
MeV. We verified that, in the chiral limit, the transition is of second order at
µ =0 and, as µ increases,the line of secondorder phase transitionwill end
B B
inafirstorderlineattheTCP.TheTCPislocatedatµTCP =286.1MeVand
B
TTCP =112.1 MeV.
240 200
220 crossover 180
200 160 ms= 140.7 MeV
180
V) 160 second order V) 140 crossover second order
T (Me 111024000 TCP T (Me 11802000 TCP
80 CEP 60 CEP
60
40 Chiral Limit 40 Chiral Limit
20 20
0 0
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400
B (MeV) B (MeV)
Figure 1: Phase diagram in the SU(2) (left) and SU(3) (right) NJL models.
The solid line represents the first order transition, the dashed the second order
and the dotted the crossover transition.
For the SU(3) NJL model, also in the chiral limit (m =m =m =0),
u d s
we verify that the phase diagram does not exhibit a TCP: chiral symmetry
is restored via a first order transition for all baryonic chemical potentials and
temperatures (see right panel of fig. 1). This pattern of chiral symmetry
restoration remains for m =m =0 and m <mcrit. In our model we found
u d s s
mcrit = 18.3 MeV for m = m = 0 4). When m ≥ mcrit, at µ = 0, the
s u d s s B
transition is second order and, as µ increases, the second order line will end
B
in a first order line at the TCP. The TCP for m = 140.7 MeV is located
s
at µTCP = 265.9 MeV and TTCP = 100.5 MeV. If we choose m = m 6= 0,
B u d
insteadofsecondordertransitionwehaveasmoothcrossoverwhichcriticalline
7, 9)
will end in the first order line at the CEP. Using the set of parameters :
m = m = 5.5 MeV, m = 140.7 MeV, g Λ2 = 3.67, g Λ5 = −12.36 and
u d s S D
Λ=602.3MeV,thispointislocalizedatTCEP =67.7MeVandµCEP =318.4
B
MeV.
Wepointoutthatbothsituationsareinagreementwithwhatisexpected
10)
at µ = 0 : the phase transition in the chiral limit is of second order for
B
N =2 and first order for N ≥3.
f f
3 Susceptibilities and critical exponents in the vicinity of the CEP
ThephenomenologicalrelevanceoffluctuationsaroundtheCEP/TCPofQCD
has been recognized by several authors.
20 40
µ µCEP
T = T CEP B = B
15 30
χ -2 (fm)B 10 T > T CEP -3 C (fm) 20 µ µCEP µ B < µ BCEP
5 T < T CEP 10 B > B
0 0
290 300 310µ 320 330 340 350 40 50 60 70 80 90 100
B (MeV) T (MeV)
Figure 2: Response functions in the SU(3) NJL model. Left panel: Baryon
number susceptibility as a function of µ for different T around the CEP
B
(TCEP = 67.7 MeV andT = TCEP ± 10 MeV). Right panel: Specific heat
as a function of T for different µ around the CEP (µCEP =318.4 MeV and
B B
µ =µCEP ±10 MeV).
B B
In the left panel of fig. 2, the baryonnumber susceptibility is plotted for
three different temperatures around the CEP. For temperatures below TCEP
the phase transition is first order and, consequently, χ has a discontinuity.
B
For T = TCEP the susceptibility diverges at µ = µCEP (the slope of the
B B
baryon number density tends to infinity). For temperatures above TCEP, in
the crossover region, the discontinuity of χ disappears at the transition line.
B
A similar behavior is found for the specific heat for three different chemical
potentials around the CEP, as we can observe from the right panel of fig. 2.
ThesecalculationshavebeenperformedintheSU(3)NJLmodel,butthesame
4)
qualitative behavior can be found in the SU(2) NJL version .
Summarizing, the baryon number susceptibility and the specific heat di-
verge at T = TCEP and µ = µCEP, respectively. 2, 3, 4) In order to make
this statement more precise, we will focus on the values of a set of indices, the
so-calledcriticalexponents,whichdescribethe behaviornearthe criticalpoint
of various quantities of interest (in our case ǫ and α are the critical exponents
of χ and C, respectively). If the critical region of the CEP is small, it is
B
expectedthatmostofthefluctuationsassociatedwiththeCEPwillcomefrom
2)
the mean field region around the CEP .
To a better understanding of the critical behavior of the system, we also
analyze what happens in the SU(2) NJL model.
102 e = 0.66 m 0.01 SU(2) 102 e = 0.67 m 0.01 SU(3)
-2 m) e’ = 0.66 m 0.01 -2 m) m
c (fB 101 c (fB 101 e’ = 0.68 0.01
m B < m BCEP m B < m BCEP
100 10-2 m B > m BCE1P0-1 100 101 100 10-2 m B > m BCEP10-1 100 101
CEP CEP
|m B- m B | (MeV) |m B- m B | (MeV)
Figure 3: Baryon number susceptibility as a function of |µ −µCEP| at fixed
B B
temperature T = TCEP in SU(2) (left panel) and SU(3) (right panel) NJL
models.
′
Toobtainthecriticalexponentǫ(ǫ)forthebaryonnumbersusceptibility,
we willconsidera path parallelto the µ -axisin the T −µ plane, fromlower
B B
(higher)µ towardsthecriticalµCEP,atfixedtemperatureT =TCEP. Tothis
B B
purpose we consider a linear logarithmic fit of the type lnχ = −ǫ(′)ln|µ −
B B
µCEP|+c(′) , where the term c (c′) is independent of µ .
B 1 1 1 B
The values presented in fig. 3 for these critical exponents, calculated in
both SU(2) and SU(3) NJL models, are consistent with the mean field theory
predictionǫ=2/3. Thismeansthatthesizeoftheregionisapproximatelythe
same independently of the direction of the path parallel to the µ -axis.
B
Paying now attention to the specific heat around the CEP, we have used
a path parallelto the T-axisin the T −µ plane from lower/higherT towards
B
TCEP at fixed µ =µCEP. In fig. 4 we plot C as a function of T close to the
B B
CEPinalogarithmicscaleforbothSU(2)andSU(3)calculations. Inthis case
we use a linear logarithmic fit, lnC = −αln|T −TCEP|+c , where the term
2
c is independent of T.
2
Starting with the SU(2) case, we observe (see left panel of fig. 4), for
T <TCEP, that the slope of the fitting of data points changes for |T −TCEP|
around0.3MeV.Sowehaveachangefromthecriticalexponentα=0.59±0.01
2)
to α =0.45±0.01. As pointed out in , this change of the exponent can be
1
interpreted as a crossoverof different universality classes, with the CEP being
affected by the TCP. It seems that the effect of the hidden TCP on the CEP
is relevant for the specific heat contrarily to what happens to χ .
B
m m
102 a = 0.59 0.01 SU(2) 102 a = 0.61 0.01 SU(3)
-3 m) m -3 m) a’ = 0.67 m 0.01
C (f 101 a’ = 0.69 m 0.01 a1 = 0.45 0.01 C (f 101
10-2 TT <> TT CCEE1PP0-1 100 101 100 10-2 TT <> TT CCEEPP10-1 100 101
CEP CEP
|T -T | (MeV) |T- T | (MeV)
Figure 4: Specific heat as a function of T for different values of µ around
B
µ=µCEP in SU(2) (left panel) and SU(3) (right panel) NJL models.
B
We also observe that there is no clear evidence of change of the slope of
thefittingofdatapointsinthethree-flavorNJLmodel(seefig. 4,rightpanel).
Infact,nowweonlyobtainacriticalexponentα=0.61±0.01whenthecritical
point is approached from below. When the critical point is approached from
′
above the trivial exponent α =0.67±0.01 is obtained.
TojustifythepossibleeffectofthehiddenTCPontheCEP,assuggested
2, 3)
in , we analyze the behavior of the specific heat around the TCP. We
find nontrivial critical exponents α = 0.40±0.01 and α = 0.45±0.01, for
SU(2) and SU(3) cases, respectively. This result, in spite of being close, is not
in agreement with the respective mean field value (α = 1/2). However, they
can justify the crossing effect observed. We notice that the closest distance
betweenthe TCPandthe CEPin the phasediagramoccursinthe T-direction
((TTCP −TCEP)<(µCEP −µTCP)), and is more clear in the SU(2) case.
B B
4 Summary
We verified that our model calculation reproduces qualitative features of the
QCDphasediagramatµ =0: form =0thechiraltransitionissecond-order
B i
for N = 2 and first-order for N ≥ 3. Using realistic values for the current
f f
quark masses we find the location of the CEP in both SU(2) and SU(3) NJL
models.
It wasshownthat the baryonnumber susceptibility and the specific heat
diverge at the CEP. The critical exponents for χ around the CEP, in both
B
N = 2 and N = 3 NJL models, are consistent with the mean field values
f f
′
ǫ=ǫ =2/3. For the specific heat, the nontrivial values of α (1/2<α<2/3)
aroundthe CEP can be interpreted as a crossoverfrom a mean field tricritical
exponent (α=1/2) to an Ising-like critical exponent (α=2/3).
Abetterinsighttothedifficulttaskoftheanalysisofthephasediagramof
QCDcanbe providedbyanextensionoftheNJL modelwherequarksinteract
with the temporal gluon field represented by the Polyakovloop dynamics.
Acknowledgments
WorksupportedbygrantSFRH/BPD/23252/2005fromF.C.T.(P.Costa),
Centro de F´ısica Teo´rica and FCT under project POCI 2010/FP/63945/2005.
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