Deconfinement and Chiral Symmetry Restoration in a Strong Magnetic Background

0.29 MB·English

by  Raoul Gatto

YITP-10-96
Deconfinement and Chiral Symmetry Restoration in a Strong Magnetic Background
Raoul Gatto1,∗ and Marco Ruggieri2,†
1Departement de Physique Theorique, Universite de Geneve, CH-1211 Geneve 4, Switzerland
2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Weperformamodelstudyofdeconfinementandchiralsymmetryrestorationinastrongmagnetic
background. We use a Nambu-Jona Lasinio model with the Polyakov loop, taking into account a
possible dependence of the coupling on the Polyakov loop expectation value, as suggested by the
recentliterature. Ourmainresultisthat,withinthismodel,thedeconfinementandchiralcrossovers
ofQCDinstrongmagneticfieldareentangledevenatthelargestvalueofeBconsideredhere,namely
eB=30m2 (that is, B ≈6×1015 Tesla). The amount of split that we measure is, at this value of
π
eB, of the order of 2%. We also study briefly the role of the 8-quark term on the entanglement of
thetwocrossovers. Wethencomparethephasediagramofthismodelwithpreviousresults,aswell
as with available Lattice data.
PACSnumbers: 12.38.Aw,12.38.Mh
Keywords: Hotquarkmatter, Effective modelsofQCD,Deconfinement andchiralsymmetryrestorationin
magneticfields.
1
1
0
I. INTRODUCTION reviews. Inthismodel,theQCDgluon-mediatedinterac-
2
tionsarereplacedbyeffectiveinteractionsamongquarks,

n The study of the Quantum Chromodynamics (QCD) which are built in order to respect the global symme-
a
vacuum, and of its modifications under the influence of tries of QCD. Beside this, the parameters of the model
J
externalfactor like temperature,baryonchemicalpoten- are fixed to reproduce some phenomenological quantity
8
2 tial,externalfields,isoneofthemostattractivetopicsof of the QCD vacuum; therefore, it is reasonable that the
modern physics. One of the best strategies to overcome main characteristics of its phase diagram represent, at
] the difficulty to study chiral symmetry breaking and de- least qualitatively, those of QCD.
h
confinement, which share a non-perturbative origin, is In recent years, the NJL model has been improved
p
- offered by Lattice QCD simulations at zero chemical po- in order to be capable to compute quantities which
p tential [1–5]. At vanishing quark chemical potential, it are related to the confinement-deconfinement transi-
e is established that two crossovers take place in a nar- tion of QCD. It is well known that color confinement
h
row range of temperature; one for quark deconfinement, can be described in terms of the center symmetry of
[
andanotheroneforthe(approximate)restorationofchi- the color gauge group and of the Polyakov loop [16],

2 ral symmetry. Besides, the use of the Schwinger-Dyson which is an order parameter for the center symmetry
v equations for the quark self-energy [6, 7], and the use in the pure gauge theory. In theories with dynami-
1
of functional renormalization group [8] to the Hamilto- cal fermions, the Polyakov loop is still a good indica-
9
2 nianformulationofYang-MillstheoryinCoulombgauge, torfortheconfinement-deconfinementtransition,assug-
1 are very promising [9, 10]. QCD with one quark flavor gested by Lattice data at zero chemical potential [1–
. at finite temperature and quark chemical potential has 5]. Motivated by this property, the Polyakov extended
2
been considered in [11] within the functional renormal- Nambu-Jona Lasinio model (P-NJL model) has been in-
1
0 ization group approach, combined with re-bosonization troduced[17,18],inwhichthe conceptofstatisticalcon-
1 tecniques; in [12], the renormalization group flow of the finement replaces that of the true confinement of QCD,
: Polyakov-Looppotential and the flow of the chiral order and an effective potential describing interaction among
v
parameter have been computed. The results of [12] sug- the chiral condensate and the Polyakov loop is achieved
i
X gest that the chiral and deconfinement phase transition bythecouplingofquarkstoabackgroundtemporalgluon
r temperature agree within a few MeV for vanishing and field, and then integrating over quark fields in the parti-
a
(small) quark chemical potentials. Furthermore, an in- tionfunction. TheP-NJLmodel,aswellasitsrenormal-
teresting possibility to map solutions of the Yang-Mills izable extension, namely the Polyakov extended Quark-
equations of motion with those of a scalar field theory Meson model (P-QM), have been studied extensively in
has been proved in [13]. many contexts [19–36].
An alternative approach to the physics of strong in-
Inaremarkablepaper[37]ithasbeenshownbyKondo
teractions, which is capable to capture some of the
that it is possible to derive the effective 4-quark inter-
non-perturbative properties of the QCD vacuum, is the
action of the NJL model, starting from the QCD la-
Nambu-Jona Lasinio (NJL) model [14], see Refs. [15] for
grangian. In his derivation, Kondo has shown explicitly
that the NJL vertex has a non-local structure, that is, it
is momentum-dependent; besides, the vertex acquires a
∗ [email protected] non-trivialdependenceonthephaseofthePolyakovloop.
† [email protected] ThisideahasbeenimplementedwithintheP-NJLmodel
2
in[31];themodifiedmodelhasthenbeennamedEPNJL, are appropriate for our study, we find a split of the or-
and the Polyakov-loop-dependentvertex has been called der of 2% at the largest value of eB considered, namely
entanglementvertex. Wewillmakeuseofthisnomencla- eB =30m2.
π
ture in the presentarticle. Beforegoing ahead,it worths In [45] a similar computation within a model without
to notice that a low-energy limit of QCD, leading to the entanglement,butwithan8-quarkinteractionadded,has
non-localNambu-JonaLasiniomodelstudiedin[38],has been performed. Following the nomenclature of [31], we
been discussed independently in [39]. In this reference, call the latter model, PNJL model. We find the com-
8
the low-energy limit of the gluon propagator leads to a parison among the EPNJL and the PNJL models very
8
relationamongtheNJLcouplingconstantandthestring instructive. As a matter of fact, the parameters in the
tension. twomodels arechosento reproducethe QCDthermody-
In this article, we report on our study of deconfine- namicsatzeroandimaginarychemicalpotential[28,31].
ment and chiral symmetry restoration at finite temper- Therefore, both of them are capable to describe QCD in
ature in a strong magnetic background. This study is the same regime. It is interesting that, because of the
motivated by several reasons. Firstly, it is extremely in- different interactions content, the two models predict a
teresting to understand how an external field can mod- slight different behavior of hot quark matter in strong
ify the main characteristics of confinement and sponta- magnetic field. In the next future, the comparison with
neouschiralsymmetrybreaking. LatticestudiesonQCD refined Lattice data can enlighten on which of the two
in magnetic (as well as chromo-magnetic) backgrounds models is a more faithful description of QCD.
can be found in [40–42]. Studies of QCD in magnetic Thepaperisorganizedasfollows: inSectionIIwesum-
fields, and of QCD-like theories as well, can be found marize the formalism: we derive the quark propagator
in Refs. [43–50]. Besides, strong magnetic fields might and the equation for the chiral condensates in magnetic
be produced in non-central heavy ion collisions [51, 52]. field, within the EPNJL model, in the one-loop approx-
More concretely, at the center-of-mass energy reachable imation; then, we compute the thermodynamic poten-
at LHC, √sNN 4.5 TeV, the magnetic field can be as tial. in Section III, we collect our results for the chiral
large as 1 eB ≈15m2 according to [52]. It has been ar- condensate and the expectation value of the Polyakov
≈ π
gued that in these conditions, the sphaleron transitions loop,for severalvalues ofthe magnetic fieldstrength. In
of finite temperature QCD, give rise to Chiral Magnetic Section IV, we briefly investigate on the effect of the 8-
Effect (CME) [51, 53]. quarkinteractioninthe EPNJLmodelinmagnetic field.
The novelty of this study is the use of the EPNJL In Section V, we draw the phase diagram of the EPNJL
modelinourcalculations,intheone-loopapproximation. modelinmagneticfield,andmakeacomparisonwithour
In comparisonwith the originalPNJL model of [18], the previous result [45]. Finally, in Section VI we draw our
EPNJL model has two additional parameters. However, conclusions, and briefly comment on possible extensions
they are fixed from the QCD thermodynamics at zero and prosecutions of our study.
magnetic field, as we will discuss in more detail later. We use natural units throughout this paper, ~ = c =
Therefore, the results at a finite value of the magnetic k = 1, and work in Euclidean space-time R4 = βV
B
field strength have to be considered as predictions of , where V is the volume and β = 1/T with T corre-
the model. Our main result is that the entanglement sponding to the temperature of the system. Moreover,
of the NJL coupling constant and the Polyakov loop, we take a non-zero current quark mass. In this case,
might affectcrucially the phase diagramin the tempera- both deconfinement and chiral symmetry breaking are
ture/magneticfieldstrengthplane. Previousmodelstud- crossovers; however, we sometimes will make use of the
ies [43–45] have revealed that both the deconfinement term “phase transition” to describe them, for stylistic
temperature, TL, and the chiral symmetry restoration reasons. It should be clear from the context that our
temperature, Tχ, are enhanced by a magnetic field, in “phase transitions” are meant to be crossovers unless
agreementwith the Lattice data of Ref. [40]. The model stated differently.
resultsareinslightdisagreementwiththeLattice,inthe
sense that the former predict a considerable split of T
L
andT asthe strengthofthe magneticfield isincreased.
χ II. THE MODEL WITH ENTANGLEMENT
Within the EPNJL model, we can anticipate one of the
VERTEX
results,thatis,thesplitamongT andT mightbecon-
L χ
siderably reduced even at large values of the magnetic
We consider a model in which quark interaction is de-
field strength. In particular, using the values of the pa-
scribed by the following lagrangiandensity:
rameters of [31], which arise from a best-fit of Lattice
dataatzeroandimaginarychemicalpotentialandwhich
=ψ¯(iγµD m)ψ+ ; (1)
µ I
L − L
hereψisthequarkDiracspinorinthefundamentalrepre-
1 In this article, we measure eB in units of the vacuum squared sentationofthe flavorSU(2)andthe colorgroup; τ cor-
pion mass m2; then, eB = m2 corresponds to B ≈ 2×1014 respondtothePaulimatricesinflavorspace. Asumover
π π
Tesla. color and flavor is understood. The covariant derivative
3
embeds the QED coupling of the quarks with the exter- study, we follow a more phenomenological approach to
nalmagnetic field, as wellas the QCDcoupling with the the problem, using the ansatz introduced in [31], that is
background gluon field which is related to the Polyakov
loop, see below. Furthermore, we have defined G=g 1 α1LL¯ α2(L3+L¯3) , (5)
− −
(cid:2) (cid:3)
=G ψ¯ψ 2+ iψ¯γ τψ 2 ; (2) and we take L = L¯ from the beginning. The functional
LI h(cid:0) (cid:1) (cid:0) 5 (cid:1) i form in the above equation is constrained by C and ex-
tendedZ symmetry. Wereferto[31]foramoredetailed
3
This interaction term is invariant under SU(2)V ⊗ discussion. Thenumericalvaluesofα1 andα2 havebeen
SU(2)A U(1)V. In the chiral limit, this is the sym- fixed in [31] by a best fit of the available Lattice data at
⊗
metry group of the action as well, if no magnetic field zeroandimaginarychemicalpotentialofRef.[54],which
is applied. However, this group is broken explicitly to have been confirmed recently in [55]. In particular, the
U(1)3V ⊗U(1)3A⊗U(1)V if the magnetic field is coupled fitted data are the critical temperature at zero chemi-
to the quarks, because of the different electric charge of cal potential, and the dependence of the Roberge-Weiss
u and d quarks. Here, the superscript 3 in the V and endpoint on the bare quark mass.
A groups denotes the transformations generated by τ3, The values α1 = α2 α = 0.2 0.05 have been ob-
τ3γ5 respectively. Therefore,thechiralgroupinpresence tained in [31] using a ha≡rd cutoff r±egularization scheme.
eoxfpalimciatlgynebtriockfieenldbyistUhe(1q)u3Va⊗rkUm(1a)s3As.teTrmhistogrUou(1p)i3Vs.then Whaevewvilelrfioficeudstmhaatinilnyoounrthreegcualaserizαat=ion0.2scahsemine,[3o1u].rWree-
We are interested to the interplay among chiral sym- sults are in quantitative agreement with those of [31],
metry restoration and deconfinement in a strong mag- when we take the parameter T = 190 MeV in the
0
netic field. To compute a temperature for the decon- Polyakovloop effective potential in agreement with [31],
finement crossover, we use the expectation value of the see below. Then, we will study how the results change
Polyakov loop, that we denote by L. In order to com- when we vary α.
puteLweintroduceastatic,homogeneousandEuclidean
background temporal gluon field, A = iA = iλ Aa,
0 4 a 4
coupled minimally to the quarks via the QCD covariant A. Quark propagator and chiral condensate in
derivative [18]. Then magnetic field
1
L= Tr exp(iβλ Aa) , (3) We work in the mean field approximation throughout
3 c a 4
this paper, neglecting pseudoscalar condensates; more-
over, we make the assumption that condensation takes
where β = 1/T. In the Polyakov gauge, which is con-
venient for this study, A = iλ φ + iλ φ8; moreover, place only in the flavor channels τ0 and τ3. The mean
0 3 8
field interaction term Eq. (2) can be cast in the form
we work at zero quark chemical potential, therefore we
can take L = L† from the beginning, which implies = 2GΣ u¯u+d¯d GΣ2 , (6)
A8 = 0. This choice is also motivated by the study L − −
4 (cid:0) (cid:1)
of [44], where it is shown that the paramagnetic con- where Σ= u¯u+d¯d .
tribution of the quarks to the thermodynamic potential −h i
In a magnetic field, the chiral condensates of u and d
induces the breaking of the Z symmetry, favoring the
3 quarks have to be different, because the electric charges
configurationswithareal-valuedPolyakovloop(seeSec-
of these quarks are different. Even if the one-loop quark
tionIII.Cof[44]foranexcellentdiscussionofthispoint).
self-energies in Eq. (6) depend on the sum of the two
We have then [18, 19]
condensates, being therefore flavor independent, it is
straightforward to show that the two condensates turn
1+2cos(βφ)
L= . (4) outtobe different,by takingthe traceofthe propagator
3
of the two quarks. The interaction (6) is diagonal in fla-
vor space, therefore we can focus on the propagator of a
As already discussed in the Introduction, it has been
single flavor f.
shown that it is possible to derive the effective 4-quark
To write the quark propagator we use the Ritus
interaction (2) starting from the QCD lagrangian [37].
method [56], which allows to expand the propagator on
In[37]ithasbeenshownthatthe NJL vertexhas anon-
thecompleteandorthonormalsetmadeoftheeigenfunc-
local structure, that is, it is momentum-dependent. An
tions of a charged fermion in a homogeneous and static
analogousconclusionisachievedin[39]. Moreimportant
magneticfield. Thisisawellknownprocedure,discussed
for our study, the NJL vertex acquires a non-trivial de-
many times in the literature, see for example [57–60];
pendence on the phase of the Polyakov loop. Therefore,
therefore it is enough to quote the final result,
in the model we consider here, it is important to keep
into account this dependence. The exact dependence of ∞
dp dp dp 1
G on L has not yet been computed; it is possible that S (x,y)= 0 2 3E (x)Λ E¯ (y) ,
it will be determined in the next future by means of the f kX=0Z (2π)4 P kP ·γ−M P
functional renormalization group approach [37]. In our (7)
4
where E (x) corresponds to the eigenfunction of a To be specific, we consider here the case N = 5, for
P
charged fermion in magnetic field, and E¯ (x) numericalconvenience. AlargervalueofN increasesthe
P
γ (E (x))†γ . In the above equation, ≡ cutoff artifacts, as already discussed in detail in [43, 45,
0 P 0
47], but leaves the qualitative picture unchanged; on the
other hand, a smaller value of N, namely N 3, makes
P =(p +iA ,0, 2k Q eB ,p ) , (8)
0 4 f z ≤
Qq | | notpossible the fitof the pion decayconstantand ofthe
chiral condensate in the vacuum. The (more usual) 3-
where k = 0,1,2,... labels the kth Landau level, and
momentum cutoff regularization scheme is recovered in
sign(Q ), with Q denoting the charge of the fla-
Q ≡ f f the limit N ; we notice that, even if the choice
vor f; Λk is a projector in Dirac space which keeps into N =5mayse→em∞arbitrarytosomeextent,itis notmore
account the degeneracy of the Landau levels; it is given
arbitrary than the choice of the hard cutoff, that is, of a
by
regularizationscheme.
By virtue of Eq. (10) it is easy to argue that the con-
Λ =δ [ δ + δ ]+(1 δ )I , (9)
k k0 + Q,+1 − Q,−1 k0
P P − densatesofuanddquarksaredifferentinmagneticfield.
As a matter of fact, the different value of the charges
where arespinprojectorsandI istheidentitymatrix
±
P makes the r.h.s. of the above equation flavor-dependent,
in Dirac spinor indices. The propagator in Eq. (7) has
even if Σ is flavor-singlet. Once we compute the mean
a non-trivial color structure, due to the coupling to the
field value of Σ by minimization of the thermodynamic
background gauge field, see Eq. (8).
potential (see the next Section), we will use Eq. (10) to
Thetraceofthef-quarkpropagatorisminusthechiral
condensate f¯f , with f = u,d. Taking the trace in evaluate the chiral condensates for u and d quarks in
h i magnetic field.
coordinateandinternalspace,itiseasyto showthatthe
following equation holds:
Q eB ∞ dp M B. Thermodynamic potential
f¯f = N | f | β z f (L,L¯,T p ,k) .
c k z
h i − 2π Xk=0 Z 2π ωf C |
In the one-loop approximation, the thermodynamic
(10)
potential Ω is given by
Here,
1
Ω= (L,L¯,T)+GΣ2 Trlog βS−1 , (16)
(L,L¯,T p ,k)=U 2 (L,L¯,T p ,k) , (11) U − βV
C | z Λ− N | z (cid:0) (cid:1)
and denotes the statistically confining Fermidistribu- where the trace is over color, flavor, Dirac and space-
N
tion function, time indices and the propagator for each flavor is given
byEq.(7). Itisstraightforwardtoderivethefinalresult,
(L,L¯,T p ,k)= 1+2Leβωf +Le2βωf , namely
C | z 1+3Leβωf +3Le2βωf +e3βωf
(12) Ω= (L,L¯,T)+UM
U
Q eB +∞ dp
where | f | βk z (L,L¯,T pz,k) .
− 2π Z 2π G |
fX=u,d Xk −∞
ω2 =p2+2Q eB k+M2 , (13)
f z | f | f (17)
with In the above equation, the first line corresponds to the
classical contribution; we have defined
M =M =m +2GΣ . (14)
u d 0
U =G Σ2 ; (18)
M S
Thefirstandthesecondaddendainther.h.s. ofEq.(10)
correspond to the vacuum and the thermal fluctuations
the second line corresponds to the sum of vacuum and
contribution to the chiral condensate, respectively. The
thermaloneloopcontributions,respectively,andariseaf-
coefficient β = 2 δ keeps into account the degen-
k k0 ter integration over the fermion fluctuations in the func-
−
eracy of the Landau levels. The vacuum contribution is
tional integral. We have defined
ultraviolet divergent. In order to regularize it, we adopt
a smooth regulator UΛ, which is more suitable, from the (L,L¯,T p ,k)=N U (p)ω + 2 log , (19)
numerical point of view, in our model calculation with G | z c Λ f β F
respecttothe hard-cutoffwhichisusedinanalogouscal-
culationswithout magnetic field. Inthis article wechose with
Λ2N (L,L¯,T p ,k)=1+3Le−βωf +3L¯e−2βωf +e−3βωf .
UΛ = Λ2N +(p2+2Q eB k)N . (15) F | z (20)
z | f |
5
The potential term [L,L¯,T] in Eq. (17) is built 1.4
U
by hand in order to reproduce the pure gluonic lat- 1.2
tice data [19, 20]. We adopt the following logarithmic
1.0
form [20],
0.8
0
S
[L,L¯,T]=T4 a(T)L¯L (cid:144)S 0.6 eB=19
U (cid:26)− 2 eB=15
0.4
eB=10
+b(T)ln 1 6L¯L+4(L¯3+L3) 3(L¯L)2 , 0.2 eB=0
− − (cid:27)
(cid:2) (cid:3) 0.0
(21) 160 170 180 190 200 210
THMeVL
with three model parameters (one of four is constrained 0.7
by the Stefan-Boltzmann limit), eB=19
0.6
eB=15
T T 2 0.5 eB=10
0 0
a(T)=a +a +a ,
0 1 2 eB=0
(cid:18)T (cid:19) (cid:18)T (cid:19) 0.4
(22) L
T 3 0.3
0
b(T)=b .
3(cid:18)T (cid:19) 0.2
0.1
The standard choice of the parameters reads a = 3.51,
0
a = 2.47, a = 15.2 and b = 1.75. The parameter 0.0
1 − 2 3 − 160 170 180 190 200 210
T in Eq. (21) sets the deconfinement scale in the pure
0 THMeVL
gauge theory. In absence of dynamical fermions one has
T = 270 MeV. However, dynamical fermions induce a
0
dependence of this parameter on the number of active FIG.1. Upper panel: ChiralcondensateΣ,measuredinunits
flavors[33]. For the case of two light flavorsto which we of the chiral condensate at zero temperature and zero field,
are interested here, we take T0 =212 MeV as in [45]. Σ0, as a function of temperature, for several values of the
As a final remark, it is easy to show that summing magnetic field strength. Lower panel: Expectation value of
Eq. (10) for u and d quarks, one reproduces the equa- thePolyakovloop,L,asafunctionoftemperature,forseveral
tion satisfied by Σ, the latter being obtained from the values of the magnetic field strength. The data are obtained
stationarity condition ∂Ω/∂Σ=0. for α=0.2. In the figures, the magnetic fields are measured
in units of m2.
π
III. DECONFINEMENT AND CHIRAL
SYMMETRY RESTORATION They are computed from the minimization of the ther-
modynamic potential in Eq.(17). In the numericalcom-
InthisSectionwesummarizeournumericalresults. In putation, we have chosen α1 =α2 α=0.2 as in [31].
≡
this study we fix the values of g, Λ and m0 to reproduce The results shown in Fig. 1 are interesting for sev-
the values of fπ and mπ in the vacuum, as well as the eral reasons. Firstly, if we identify the deconfinement
numericalvalueofthelightquarkschiralcondensate. We crossover with the temperature T at which dL/dT is
L
have then Λ = 626.76 MeV, g = 2.02/Λ2. The pion maximum,andthechiralcrossoverwiththetemperature
mass in the vacuum, mπ = 139 MeV, is used to fix the Tχ at which dΣ/dT is maximum, we observe that the
| |
numerical value of the bare quark mass via the GMOR twotemperaturesareveryclosealsoinastrongmagnetic
rwehlaetniownefπt2amke2π =u¯u−2=m(0hu¯2u5i3; MtheiVs )g3i.vesFimna0lly=, w5eMtaekVe fi18el5d.5. MFoerVc.oBncerseidteens,esast, aeBt e=B 1=9m02wewefinfidndTPT ==Tχ19=9
h i − π P
T0 =212MeVinthePolyakovloopeffectivepotentialas MeV and Tχ = 201 MeV. Hence at the strongest value
in [45], unless stated differently. The values of Σ and L of eB considered here, namely eB = 19m2, the entan-
π
arecomputedbytheminimizationofthethermodynamic glement vertex makes the split of the two crossovers
potential. of 1.5%. From the model point of view, it is easy
≈
to understand why deconfinement and chiral symmetry
restoration are entangled also in strong magnetic field.
A. The case α1 =α2 =0.2 As a matter of fact, using the data shown in Fig. 1, it
is possible to show that the NJL coupling constant in
In Fig. 1 we plot our data for the chiral condensate the pseudo-critical region in this model decreases of the
Σ, measured in units of the condensate at T = eB = 0, 15% as a consequence of the deconfinement crossover.
thatisΣ =2 ( 253MeV)3,andtheexpectationvalue Therefore,the strength of the interactionresponsible for
0
× −
of the Polyakov loop as a function of temperature, com- the spontaneous chiral symmetry breaking is strongly
puted for several values of the magnetic field strength. affected by the deconfinement, with the obvious conse-
6
quence that the numericalvalue of the chiralcondensate
2.0 Su
drops down and the chiral crossover takes place. We
Sd eB=30
have verified that the picture remains qualitatively and
01.5
quantitatively unchanged if we perform a calculation at Σ
(cid:144)
d
eB = 30m2. In this case, we find T = 224 MeV and S
π L ,01.0
T =225 MeV. Σ
χ (cid:144)u
Before going ahead, it worths to notice that we have S
0.5
also considered the case T = 190 MeV, which corre-
0 eB=15
sponds to the value considered in [31]. In this case, for
0.0
eB = 0 we find Tχ = TL = 175 MeV, in excellent agree- 180 190 200 210 220 230 240
ment with [31]. This results is comforting, because it THMeVL
shows that even using a different regularization scheme, 0.8
the UV-regulator does not affect the physical predic-
tions at eB = 0. Moreover, at eB/m2 = 30, we find eB=15
π 0.6
Tχ TL =215 MeV. eB=30
≈
This result can be compared with our previous cal-
L0.4
culation [43] in which we did not include the Polyakov
loop dependence of the NJL coupling constant. In [43]
we worked in the chiral limit and we observed that the 0.2
Polyakovloopcrossoverin the PNJLmodelis almostin-
sensitive to the magnetic field; on the other hand, the 0.0
180 190 200 210 220 230 240
chiralphasetransitiontemperaturewasfoundtobe very
THMeVL
sensitive to the strengthof the applied magnetic field, in
agreement with the well known magnetic catalysis sce-
nario [46]. This model prediction has been confirmed
FIG. 2. Upper panel: Chiral condensates of u and d quarks
withinthePolyakovextendedquark-mesonmodelin[44],
as functions of temperatures in the pseudo-critical region, at
when the contributionfromthe vacuumfermion fluctua- eB = 15m2 and eB = 30m2. Condensates are measured in
π π
tionstotheenergydensityiskeptintoaccount2;wethen units of their value at zero magnetic field and zero tempera-
obtainedasimilarresultin[45],inwhichweturnedfrom ture,namelyσ0 =(−253MeV)3. Lowerpanel: Polyakovloop
the chiral to the physical limit at which m =139 MeV, expectationvalueasafunctionoftemperature,ateB=15m2
π π
and introduced the 8-quark term as well (PNJL8 model, and eB=30m2π. Data correspond to α=0.2.
according to the nomenclature of [31]). The comparison
with the results of the PNJL model of [45] is interest-
8
ing because the model considered there, was tuned in symmetry is still broken by a chiral condensate, but de-
order to reproduce the Lattice data at zero and imag- confinement already took place. In the case of the EP-
inary chemical potential [28], like the model we use in NJL model, this window is shrunk to a very small one,
this study. Therefore, they share the property to de- because of the entanglement of the two crossovers at fi-
scribe the QCD thermodynamics at zero and imaginary nite eB. On the other hand, it is worth to stress that
chemical potential; it is therefore instructive to compare the two models share an important qualitative feature:
their predictions at finite eB. both chiral restoration and deconfinement temperatures
For concreteness, in [45] we found TP =185 MeV and are enhanced by a strong magnetic field, in qualitative
Tχ = 208 MeV at eB = 19m2π, corresponding to a split agreement with the existing Lattice data [40].
of 12%. On the other hand, in the present calculation
≈ A further interesting feature of our data is that in the
we measure a split of 1.5% at the largest value of eB
≈ low temperature region, the Polyakovloop is suppressed
considered. Therefore, the results of the two models are
as we increase the strength of the magnetic field; on the
in slight quantitative disagreement; this disagreement is
other hand, in the deconfinement phase, L is enhanced
then reflected in a slightly different phase diagram. We
by the magnetic field. This result was also found in our
will draw the phase diagramof the two models in a next
previous studies [43, 45] with a fixed coupling constant;
Section;however,sincenowitiseasytounderstandwhat
more remarkably, this result is in agreement with the
the main difference consists in: the PNJL model pre-
8 Latticedata[40]. Atthemomentwedonothavephysical
dicts some window in the eB T plane in which chiral
− arguments to explain this result, but we agree with the
authorsofRef.[40]thatthisaspectshouldbeconsidered
in more detail from the theoretical point of view.
For completeness, we plot in Fig 2 the chiral conden-
2 If the vacuum corrections areneglected, the deconfinement and
sates of u and d quarks as a function of temperature,
chiral crossovers are found to be coincident even in very strong
magnetic fields [44], but the critical temperature decreases as a at eB = 15m2π and eB = 30m2π. The condensates are
functionofeB;thisscenarioisveryinterestingtheoretically,but measured in units of their value at zero magnetic field
itseemsitisexcludedfromtherecentLatticesimulations[40]. and zero temperature, namely σ u¯u = d¯d =
0
≡ h i h i
7
240 scheme is used. Therefore, we can expect that a slightly
without8-quarkinteraction different value of α is needed in our case. However, it
220 TΧ,eB=19 is useful to observe that in the limit of zero field, our
regulator is very similar, quantitatively speaking, to the
LV
Me 200 TΧ,eB=0 hard-cutoff of [31], because of the strong power-law de-
H cay of U as p Λ. Therefore, the possible numerical
T Λ | | ≈
TP,eB=19 deviation in the parameters is expected to be very tiny,
180
if any. We have a further corroborationwe are on target
TP,eB=0
with parameters: if we chose T = 190 MeV as in [31],
0
160
0.00 0.05 0.10 0.15 0.20 0.25 0.30 we find Tχ = TL = 175 MeV, in quantitative agreement
with [31]; this convinces that our ultraviolet regulator
Α
0.7 does not lead to quantitative discrepancy with [31] at
zero magnetic field.
0.6
We have investigated the dependence of the pseudo-
0.5 critical temperatures as a function of α, at eB = 0 and
0.4 eB = 19m2. The results of our computations are col-
π
lected in the upper panelof Fig. 3. At α=0, which cor-
0.3 eB=0 eB=19
respondstothe originalPNJLmodel,wemeasureasplit
0.2
ofthetwocriticaltemperaturesof 18%evenateB =0.
≈
0.1 This is wellknownresultinthe PNJLliterature [19,20],
andtoovercomethisproblem,severalsolutionshavebeen
0.0
160 170 180 190 200 210 suggested,liketheuseofthe8-quarkinteraction[24,31].
THMeVL However, the discrepancy of the critical temperatures is
enhancedas the valueofeB is increased. AteB =19m2
π
we find a split of 30%. As we increaseα, the two tem-
FIG. 3. Upper panel: pseudo-critical temperatures as func- ≈
peraturesgetcloserrapidly,bothatzeroandatnon-zero
tions of the parameter α in the NJL coupling constant.
value of the magnetic field strength. At α = 0.2, which
Dashed lines correspond to the Polyakov loop crossover, for
isthe resultthatwe havediscussedpreviouslyandis the
eB = 0 and eB = 19. Dot-dashed lines correspond to the
chiral crossover. The grey squares denote the value of α at value quoted in [31], the two temperatures are almost
which the crossover becomes a first-order phase transition. coincident.
Lower panel: Polyakov loop expectation value (dashed lines) An interesting point that we have found is that in-
andhalfofΣ(eB)/Σ(eB=0)(solidlines)asfunctionsoftem- creasing further the value of α, the crossovers become
perature, for α = 0.275. In both panels the magnetic fields strongerandstronger;thereexistsacriticalvalueofαat
are measured in unitsof m2π. which the crossover is replaced by a sudden jump in the
chiral condensate and in the Polyakov loop expectation
value. This value of α is denoted by a green square in
( 253 MeV)3. They are computed by a two-step pro- Fig. 3, and it is eB dependent. For concreteness,in the
c−edure: firstly we find the values of Σ and L that min- lower panel of Fig.−3 we plot our data for the half of the
imize the thermodynamic potential; then, we make use chiral condensate and the Polyakov loop at α = 0.275,
of Eq. (10) to compute the expectation values of u¯u and whichshould be comparedwith the data in Fig.1 where
d¯d in magnetic field. If we measure the strength of the α=0.2.
crossover by the value of the peak of dΣ/dT , it is ob-
| |
vious from the Figure that the chiral crossover becomes
stronger and stronger as the strength of the magnetic IV. THE EFFECT OF THE 8-QUARK
field is increased, in agreement with [40]. INTERACTION
We havebriefly investigatedonthe roleof other inter-
B. Varying α actions onthe entanglementof deconfinement andchiral
symmetry restorationcrossovers. We report here the re-
sults relatedto the effect of the 8-quarkterm, which has
Since our study is based on a simple model, and we
beenextensivelystudiedin[61]inrelationtothevacuum
can not compute the exact values of the parameter α
stabilityofthe NJL model,to the hadronpropertiesand
in Eq. (5) starting from first principles, we find very in-
to the critical temperature at zero baryon density; be-
structive to measure the robustness of our results as we
sides, it has been studied in the context of the P-NJL
vary the numerical value of this parameter. We empha-
modelin [24, 31,45]. To this end, we addto the interac-
size that in [31] an estimate for the numerical value of
tion lagrangian in Eq. (2), the following term
α is achieved via the best fit of the model data with
tthiael.LHatotwiceevedra,tianathtazterroefearnednciemaagdiinffaerryencthreemgiuclaalripzoatteionn- G8 ψ¯ψ 2+ iψ¯γ5τψ 2 2 . (23)
h(cid:0) (cid:1) (cid:0) (cid:1) i
8
In principle, we expect that the constant G8 acquires a 1.5
dependence on the Polyakov loop as well; however, for
simplicity we neglect this dependence here, since we are
onlyinterestedtounderstandthequalitativeeffectofthe 1.0
0
interaction (23). We remark that the model studied in S
(cid:144)S
this Section is different from the P-NJL model defined
8
0.5
in [31]; in the latter, the dependence of G on L is ne-
glected.
At the one-loop order we have
0.0
160 170 180 190 200 210
= 2u¯u GΣ+2G Σ3 2d¯d GΣ+2G Σ3
L − 8 − 8 THMeVL
3G Σ(cid:2)4 GΣ2 , (cid:3) (cid:2) (cid:3) (24) 0.8
8
− − eB=19
instead of Eq. (6). The thermodynamic potential is still eB=15
0.6 eB=10
given by Eq. (17), with
eB=0
UM =3G8Σ4+GSΣ2 , (25) L 0.4
and
0.2
M =M =m +2(GΣ+2G Σ3) , (26)
u d 0 8
0.0
instead of Eq. (14). The parameters are fixed as in [45], 160 170 180 190 200 210
that is Λ = 589 MeV, g = 5 10−6 MeV−2, G8 = 6 THMeVL
10−22 MeV−8 and m =5.6 M×eV. ×
0
In Fig. 4 we plot our data for the chiral condensate
and the expectation value of the Polyakov loop, as a FIG.4. Upper panel: ChiralcondensateΣ,measuredinunits
function of temperature, for several values of the mag- of the chiral condensate at zero temperature and zero field,
netic field strength and for α = 0.1. We find that even Σ0, as a function of temperature, for several values of the
magnetic field strength. Lower panel: Expectation value of
at this smaller value of α, the two crossovers are entan-
thePolyakovloop P asafunctionoftemperature,forseveral
gledatlargemagneticfields. Onthe otherhand, wefind
valuesofthemagneticfieldstrength. Theresultsareobtained
thattakingarunningNJLcouplingmakesthecrossovers
within the model with 8-quark interaction and α = 0.1. In
sharper,andeventuallywemeasureadiscontinuityinthe the figures, the magnetic fields are measured in units of m2.
π
order parameters for a large magnetic field strength. Lineswiththesamedashingcorrespond tothesamevalueof
InFig.5weplotthecriticaltemperaturesasafunction themagnetic field strength.
of α, for eB = 0 and eB = 19m2. At eB = 0 the two
π
crossoversare entangled already at α=0; therefore, the
effectoftheentanglementvertexistomakethecrossover
sharper. Eventually, at the critical value α = 0.125 the
crossover is replaced by a sudden jump of the expecta-
240
tion values, analogously to the results that we find in with8-quarkinteraction
the modelwithout8-quarkterm. As weincreaseeB,the
220
deconfinement and chiral crossovers are split at α = 0,
in agreement with our previous findings [45]. However, LV TΧ,eB=19
the split in this case is more modest than the split that Me 200
H
we measure in the model with G = 0. In that case, at T
8 TP,eB=19
eB = 19m2 we find a split of 30%, to be compared 180
π ≈ eB=0
withtheonethatwereadfromFig.5forthe modelwith
8-quark interaction, namely 12%. Therefore, our con- 160
≈ 0.00 0.05 0.10 0.15 0.20 0.25 0.30
clusion is that the 8-quark interaction helps to keep the
two crossovers close in a strong magnetic field, even if Α
it is not enough and the entanglement vertex has to be
included, as it is clear from Fig. 5.
FIG. 5. Pseudo-critical temperatures as functions of the pa-
rameter α in the NJL coupling constant. Dashed lines cor-
respond to the Polyakov loop crossover, for eB = 0 and
V. THE PHASE DIAGRAM
eB=19. Dot-dashedlinescorrespondtothechiralcrossover.
Thegreysquaresdenotethevalueofαatwhichthecrossover
InFig.6wecollectourdataonthepseudo-criticaltem- becomes a first-order phasetransition.
peraturesfordeconfinementandchiralsymmetryrestora-
tion,intheformofaphasediagramintheeB T plane.
−
9
In the upper panel we show the results obtained within 1.2
D+ΧSR
theEPNJLmodel;inthelowerpanel,weplottheresults
of the PNJL8 model, that are obtained using the fitting 1.1
functions computed in [45]. In the figure, the magnetic Tc
field is measured in units of m2; temperature is mea- (cid:144)TL
sured in units of the deconfinemπent pseudo-critical tem- T,c 1.0
(cid:144)Χ TΧ
perature at zero magnetic field, namely TB=0 = 185.5 T 0.9 C+ΧSB TL
MeV for the EPNJL model, and T = 175 MeV for
B=0
the PNJL model. For any value of eB, we identify the EPNJL
8
0.8
pseudo-criticaltemperaturewiththepeakoftheeffective
0 5 10 15 20
susceptibility.
eB(cid:144)m 2
Itshouldbekeptinmind,however,thatthe definition Π
1.2
ofapseudo-criticaltemperatureinthiscaseisnotunique,
D+ΧSR
because of the crossover nature of the phenomena that
we describe. Other satisfactory definitions include the 1.1
temperature at which the order parameter reaches one (cid:144)TLc
half of its asymptotic value (which corresponds to the T 1.0
+T →fo0rltimheitPfoolryatkhoevclhoiorapl),caonnddetnhseatpeo,saitniodntooftthheeTpea→k (cid:144)TT,cΧ C+ΧSB TTΧ
0.9 L
∞
in the true susceptibilities. The expectation is that the
critical temperatures computed in these different ways PNJL8
0.8
differ from each other only of few percent. This can be 0 5 10 15 20
confirmed concretely using the data in Fig. 2 at eB =
eB(cid:144)m 2
30m2. Using the peak of the effective susceptibility we Π
π
find T = 225 MeV and T = 224 MeV; on the other
χ L
hand, using the half-value criterion, we find Tχ = 227 FIG. 6. Upper panel: Phase diagram in the eB −T plane
MeV and T = 222 MeV, in very good agreement with
L for the EPNJL model. Temperatures on thevertical axis are
the previous estimate. Therefore, the qualitative picture measured in units of the pseudo-critical temperature for de-
that we derive within our simple calculational scheme, confinementateB=0,namelyTc =185.5MeV.Lowerpanel:
namelytheentanglementofthetwocrossoversinastrong Phase diagram in the eB −T plane for the PNJL8 model.
magnetic field, should not be affected by using different Temperatures on the vertical axis are measured in units of
definitions of the critical temperatures. thepseudo-criticaltemperaturefordeconfinementateB=0,
Firstly we focus on the phase diagram of the EPNJL namely Tc = 175 MeV. In both the phase diagrams, Tχ, TL
correspond to the chiral and deconfinement pseudo-critical
model, which is one of the novelties of our study. In
temperatures, respectively. The grey shaded region denotes
the upper panel of Fig. 6, the dashed and dot-dashed
the portion of phase diagram in which hot quark matter is
lines correspond to the deconfinement and chiral sym-
deconfinedandchiralsymmetryisstillbrokenspontaneously.
metry restoration pseudo-critical temperatures, respec-
tively. Wehavefitourdatausingthefollowingfunctional
form: TABLE I. Coefficients of the fit function defined in Eq. (27)
α for the two models.
T (eB) eB
χ,L
=1+A | | , (27)
Tc (cid:18) Tc2 (cid:19) A α Tc (MeV)
Tχ, EPNJL 2.34×10−3 1.49 185.5
where the subscripts χ,L correspond to the (approx- TL, EPNJL 1.43×10−3 1.68 185.5
imate) chiral restoration and deconfinement tempera-
tures, respectively. The functional form is taken in anal- Tχ,PNJL8 [45] 2.4×10−3 1.85 175
ogytotheoneusedin[40]. Numericalvaluesofthebest- TL, PNJL8 [45] 2.1×10−3 1.41 175
fit coefficients are collected in Table I. As a consequence
oftheentanglement,thetwocrossoversstayclosedalsoin
very strong magnetic field, as we have already discussed zero field in this case is a little smaller than the same
in the previous Sections. The grey region in the Figure temperature for the EPNJL model, because the param-
denotes a phase in which quark matter is (statistically) eters needed to fit the vacuum properties are different.
deconfined, but chiral symmetry is still broken. Accord- However, a comparison among the two phase diagrams
ing to [62, 63], we can call this phase as Constituent is still instructive. The most astonishing feature of the
Quark Phase (CQP). phase diagram of the PNJL model is the entity of the
8
On the lower panel of Fig. 6 we have drawn the splitamongthedeconfinementandthechiralrestoration
phase diagram for the PNJL model; it is obtained us- crossover. The difference with the result of the EPNJL
8
ing Eq. (27) with the coefficients computed in Ref. [45], model is that in the former, the entanglement with the
and collected in Table I. The critical temperature at PolyakovloopisneglectedintheNJLcouplingconstant.
10
As we have already mentioned in the previous Section, try restoration and deconfinement temperatures are en-
the maximum amount of split that we find within the hanced by a strong magnetic field. The models differ
EPNJLmodel,atthelargestvalueofmagneticfieldcon- quantitativelyfortheamountofsplitmeasured(veryfew
sidered here, is of the order of 2%; this number has to percent for the EPNJL model, and of the order of 10%
be compared with the split at eB =20m2 in the PNJL for the other models for eB 20m2).
π 8 ≈ π
model, namely 12%. The larger split causes a consid- Furthermore,wehavecomparedourresultswiththose
≈
erable portion of the phase diagram to be occupied by obtained on the Lattice [40]. In [40], the largestvalue of
the CQP. magnetic field considered is eB 0.75 GeV2, which cor-
Keeping into account the crudeness of the two mod- respondstoeB/m2 38. TheL≈atticedataseemtopoint
π ≈
els, it is fair to say that they share one important fea- towardsthephasediagramoftheEPNJLmodel. Onthe
ture: both deconfinement and chiral symmetry restora- otherhand,theresultsof[40]mightnotbedefinitive: the
tion temperature are increased by a strong magnetic lattice size might be enlarged, the lattice spacing could
field. They disagree quantitatively, in the sense that the be taken smaller (in [40] the lattice spacing is a = 0.3
amount of split in the EPNJL model is more modest fm), and the pion mass could be lowered to its physical
in comparison with that obtained in the PNJL8 model. value in the vacuum. As a consequence, it will be inter-
Before closing, we notice that a similar phase diagram esting to compare our results with more refined data in
calculationhas been performed within the Quark-Meson the future.
(QMinthefollowing)modelin[44]. Inthisreference,the This comparison can be interesting also for another
magneticfieldhasthesameeffectthatwemeasureinthe reason. Indeed,theEPNJLmodelandthePNJL cande-
8
PNJL8 model, that is, to split of deconfinement and chi- scribethesameQCDthermodynamicsatzeroandimag-
ral symmetry restoration crossovers. Besides, both Tχ inary quark chemicalpotential [31], but they differ qual-
and TL are enhanced by a magnetic field, in excellent itatively for the interaction content. As we have shown,
agreementwiththebehaviorthatwefindwithinthesev- they have some quantitative discrepancy for what con-
eral NJL-like models that we have used in this study. cerns the response to a strongmagnetic field. Therefore,
On the Lattice side, the most recent data about this more refined Lattice data in magnetic field might help
kind of study are those of Ref. [40]. In these data, the to discern which of the two models is a more faithful
split among the two crossovers is absent. Then the Lat- description of QCD.
ticedataseemtopointtowardsthephasediagramofthe
From our point of view, it is fair to admit that our
EPNJL model. On the other hand, the results of [40]
studymighthaveaweakpoint,namely,wemissamicro-
might not be definitive, in the sense that both the lat-
scopic computation of the parameters α , α in Eq. (5).
1 2
tice size might be enlarged, and the pion mass could be
Forconcreteness,wehaveusedthebest-fitvaluesquoted
lowered to its physical value in the vacuum. Therefore,
in [31], showing in addition that changing the value of
it will be interesting to compare our results with more
T in the Polyakovloop effective potential as in [31], we
0
refined data.
obtain T =T =175 MeV, in excellent agreementwith
χ L
that reference. This is comforting, since it shows that
ourdifferentUV-regulatordoesnotaffectthequalitative
VI. CONCLUSIONS resultatzerofield,namelythecoincidenceofT andT .
χ L
Besides, we are aware that Eq. (5) is just a particular
In this article, we have studied chiral symmetry choice of the functional dependence of the NJL coupling
restorationanddeconfinementinastrongmagneticback- constantonthe Polyakovloopexpectationvalue. Differ-
ground, using an effective model of QCD. In particular, ent functional forms respecting the C and the extended
wehavereportedourresultsabouttheeffectoftheentan- Z3 symmetryarecertainlypossible,andwithoutarigor-
gledvertexonthephasediagram. Ourmainresultisthat ousderivationofEq.(5)usingfunctionalrenormalization
the entanglement reduces considerably the split among group techniques as suggested in [37], it merits to study
the deconfinement and the chiral symmetry crossovers our problem using different choices for G(L) in the next
studied in [43–45], as expected. future. For these reasons,we prefer to adopt a conserva-
We havealsostudied the effect ofthe 8-quarkterm on tive point of view: it is interesting that a model which
thesplit. Ourresultssuggestthatthe8-quarkinteraction is adjusted in order to reproduce Lattice data at zero
helps the two crossovers to be close. Furthermore, we and imaginary chemical potential, predicts that the two
haveshownthatthecrossoversbecomesharperandthen QCDtransitionsareentangledinastrongmagneticback-
they are replaced by a sudden jump of the expectation ground;however,this conclusionmightnotbe definitive,
values, as the value of α in the entanglement vertex is since there exist other model calculations which share
larger than a critical value. a common basis with ours, and which show a more pro-
Wehavethencomparedourresultswiththoseofother nouncedsplitoftheQCDtransitionsinastrongmagnetic
model calculations, namely the PNJL model [43], the background. MorerefinedLatticedatawillcertainlyhelp
Quark-Mesonmodel[44]andthePNJL model[45]. The to discern which of the two scenarios is the most favor-
8
moststrikingsimilarityamongtheseveralmodelsisthat able.
they all support the scenario in which chiral symme- As a natural continuation of this work, it is worth to