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Chiral Symmetry Restoration and Scalar-Pseudoscalar partners in QCD

A. Gomez Nicola,1, J. Ruiz de Elvira,1,2, and R. Torres Andres1,

1Departamento de Fsica Teorica II. Univ. Complutense. 28040 Madrid. Spain.2Helmholtz-Institut fur Strahlen- und Kernphysik, Universitat Bonn, D-53115 Bonn, Germany

We describe Scalar-Pseudoscalar partner degeneration at the QCD chiral transition in terms ofthe dominant low-energy physical states for the light quark sector. First, we obtain within model-independent one-loop Chiral Perturbation Theory (ChPT) that the QCD pseudoscalar susceptibilityis proportional to the quark condensate at low T. Next, we show that this chiral-restoring behaviourfor Pis compatible with recent lattice results for screening masses and gives rise to degenerationbetween the scalar and pseudoscalar susceptibilities (S, P) around the transition point, consis-tently with an O(4)-like current restoration pattern. This scenario is clearly confirmed by latticedata when we compare S(T) with the quark condensate, expected to scale as P(T). Finally,we show that saturating S with the /f0(500) broad resonance observed in pion scattering andincluding its finite temperature dependence, allows to describe the peak structure ofS(T) in latticedata and the associated critical temperature. This is carried out within a unitarized ChPT schemewhich generates the resonant state dynamically and is also consistent with partner degeneration.

PACS numbers: 11.10.Wx, 11.30.Rd, 12.39.Fe, 12.38.Gc.

I. INTRODUCTION AND MOTIVATION

Chiral symmetry breaking SUV(Nf) S UA(Nf) SUV(Nf) and its restoration, with Nf light quarkflavours, has been a milestone in our present understand-ing of the Quantum Chromodynamics (QCD) phase di-agram and Hadronic Physics under extreme conditionsof temperatureTand baryon density, as those producedin Heavy-Ion and Nuclear Matter experimental facilitiessuch as RHIC, CERN (ALICE) and FAIR. Lattice simu-lations support that deconfinement and chiral restorationtake place very close to one another in the phase diagram.In the physical case Nf= 2 + 1 (0

=mu = md

mq

ms) and for vanishing baryon chemical potential, theypoint towards a smooth crossover transition at pseudo-critical temperatureTc145-165 MeV[1, 2], the resultsbeing fairly consistent with the O(4) universality class[3], which would hold for two light flavours in the chi-ral limit mq = 0. The crossover nature of the transitionmeans in particular that there is no unique way to iden-tify the transition point, the most efficient one in latticebeing the scalar susceptibility peak position, rather thanthe vanishing point for the quark condensateqqT, the

order parameter, which decreases asymptotically withT

formq= 0.The equivalence with the O(4) O(3) breaking pat-tern for Nf = 2 led to early proposals of me-son degeneration (chiral partners) at chiral restoration[4] which in its simplest linear realization takes placethrough the-component of theO(4) field (, a) acquir-ing a thermal vacuum expectation value and mass bothvanishing at the transition in the chiral limit. Degenera-tion in the vector-axial vector sector (and a1 states) asa signature of chiral restoration has also been thoroughlystudied[5]. Nowadays, we know that the state is wellestablished as a scattering broad resonance for isospinand angular momentumI=J= 0, known as f0(500)[6],which is then difficult to accommodate as an asymptot-ically free state, like in the linear model. Precisely, oneof our main conclusion here will be that this asymptoticdescription is not needed. In fact, in order to study chiralpartner degeneration in the scalar-pseudoscalar sector, itis more appropriate to analyze the corresponding cur-rent correlation functions [7]which can be derived froma chiral effective lagrangian without introducing explic-itly a particle-like degree of freedom. The scalar andpseudoscalar susceptibilities in terms of the correspond-ing QCD S U(2) currents are given by:

S(T) = m

qqT =E

d4xT(qq)(x)(qq)(0)T qq2T=

E

d4x

s(x)

s(0)

Z[s, p]s=mq,pa=0

, (1)

P(T)ab =

E

d4xTPa(x)Pb(0)T abE

d4xKP(x) =

E

d4x

pa(x)

pb(0)Z[s, p]

s=mq,pa=0

, (2)

gomez@fis.ucm.es elvira@hiskp.uni-bonn.de rtandres@fis.ucm.es

arXiv:1304.3

356v2

[hep-ph]17Oct2013

mailto:gomez@fis.ucm.esmailto:elvira@hiskp.uni-bonn.demailto:rtandres@fis.ucm.esmailto:rtandres@fis.ucm.esmailto:elvira@hiskp.uni-bonn.demailto:gomez@fis.ucm.es

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where q = (u, d) is the quark field, Pa(x) = q5aq(x)

and KP(x) are respectively the pseudoscalar currentand its correlator, the Euclidean measure

E

d4x =0

d

d3x with = 1/T andT denotes a thermalaverage. For P, parity invariance of the QCD vacuum(PaT = 0) and isospin symmetry have been used. Inthe above equation, Z[s, p] is the QCD generating func-tional with scalar and pseudoscalar sources (s, pa) cou-

pled to the massless lagrangian in the light sector ass(x)(qq)(x) + ipa(x)Pa(x), so thatZ[mq, 0] is the QCDpartition function.

Thus, should the scalar and pseudoscalar currents be-come degenerate at chiral restoration, P(T) and S(T)would meet at that point. Since S is expected to in-crease, as a measure of the fluctuations of the order pa-rameter, at least up to the transition point, it seems plau-sible that they meet near the transition. In an ideal O(4)pattern, the matching should take place near the maxi-mum ofS.

SincePa has the quantum numbers of the pion fielda,its correlators, likeKP(x), are saturated by the pion state

at low energies. Let us first review the prediction arisingfrom the low-energy theorems of current algebra, equiv-alent to the leading order (LO) in the low-energy expan-sion of chiral lagrangians. At that order, one has Pa 2B0F a (from PCAC theorem) with B0 = M

2/2mq andwhere F and M are the pion decay constant and massrespectively, so that P 4B20F2G(p = 0) +. . . fromEq.(2), being G(p) the pion propagator in momentumspace p (in, p) and n = 2nT the Matsubara fre-quency with integer n. Thus, the pseudoscalar corre-lator, saturated with the dominant pion state, is justproportional to the pion propagator at this order. Inaddition, to LO the Euclidean propagator is just the

free one GLO (p) = 1/(p

2

+M2

) (interactions are sup-pressed at low energies) with p2 = (in)2 |p|2, so that

using also the Gell-Mann-Oakes-Renner (GOR) relationM2F2 =mqqq, valid at this order, we would getP qq/mq, as a first indication of the relation be-tween the pseudoscalar susceptibility and the quark con-densate at the LO given by current algebra.

The latter result can actually be obtained formally as aWard Identity (WI) from the QCD lagrangian[8], in con-nection with the definition of the quark condensate forlattice Wilson fermions [9]. However, both sides of theidentity suffer from QCD renormalization ambiguities, sothat this WI is formally well-defined only for exact chi-ral symmetry [9, 10]. It is therefore interesting to study,and so we will do in the next section, how this identity isrealized within Chiral Perturbation Theory (ChPT)[11],which describes the low-energy chiral symmetry brokenphase of QCD in a model-independent framework wheresymmetry breaking is realized non-linearly and pions arethe only degrees of freedom in the lagrangian. The pre-vious current-algebra results are actually just the LO inthe ChPT expansion in powers of a generic low-energyscalep, denoting pion momenta or temperature, relative,respectively, to 1 GeV and Tc. These are noth-

ing but indicative natural upper limits for the chiral ex-pansion in terms of scattering (typical resonance scale)and thermodynamics (critical phenomena) respectively,although both are treated on the same foot in the chi-ral expansion. In particular, the LO prediction for Pis temperature independent, so it is not obvious that itcan be simply extrapolated as, say,qq qqT. Actu-ally, all the quantities involved change with temperature

due to pion loop corrections, namely M(T), F(T) andqq(T)[12].Similarly, from Eq.(1), one can relate S with the

propagator of a -like state such that it couples lin-early to the external scalar source s(x) in an explicitsymmetry-breaking termLSB = 2B0F s(x)(x). With-out further specification about its nature and its couplingto other physical states such as pions, one already getsS 4B20F2G(p= 0), suggesting a growing behaviourinversely proportional to M2 as the sigma state reducesits mass to become degenerate with the pion.

We also recall that the problem ofS P degenera-tion has been studied in nuclear matter at T= 0 in[13],

to linear order in nuclear density. In that work, currentalgebra is assumed to hold through PCAC in the oper-ator representation and other low-energy theorems suchas GOR, which as discussed in the previous paragraphs,leads to the pseudoscalar correlatorKP(x) being directlyproportional to the pion propagator. The authors in [13]work within low-energy models at finite density for whichthis PCAC realization holds, so that by including theproper finite-density corrections to G, which carries outall the density dependence ofKP through an in-mediummass, and toqq, the relation P qq/mq is foundto hold in the nuclear medium. This result provides an-other supporting argument for the relation between thecondensate and the pseudoscalar susceptibility and rep-resents an additional motivation for our present ChPTanalysis, where we do not need to make any assumptionabout the validity of current algebra.

II. STANDARD CHPT ANALYSIS OF THE

PSEUDOSCALAR CORRELATOR AND

SUSCEPTIBILITY

The Next to Leading Order (NLO) corrections toP can be obtained systematically and in a model-independent way within ChPT, where one can also cal-culate the scalar susceptibility

S to a given order only

in terms of pion degrees of freedom. The price to payis that we expect to reproduce only the behaviour ofS,P(T) for low and moderate temperatures. However,sincePis dominated by pions, whose dynamics are welldescribed through ChPT, we expect to obtain a reason-able qualitative description of its T behaviour, whereasstandard ChPT misses the peak structure ofSnear thetransition. We note in turn that the LO for S vanishes,unlike that ofP. The ChPT NLO result for S(T) canbe found in[14,15].

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For P(T) we consider the effective lagrangianL2 +L4 + . . . , whereL2n =O(p2n), including their depen-dence on the pseudoscalar source pa as given in [11].We follow similar steps as in [15, 16], now for the pseu-doscalar correlator KP(x). The LO comes fromL2 onlyand reproduces the current-algebra prediction. The NLOcorrections to KP(x) are of the following types:

The NLO corrections to the pion propagatorGNLO (x), which come both from L2 one-looptadpole-like contributions GLO (x = 0) and fromtree levelL4 constant terms,

Pion self-interactionsO(pa 3pb) in L2 con-tributing as GLO (x)G

LO (x= 0),

Crossed termsL2 =O(pa) L4 =O(pb) givingGLO (x) multiplied by a NLO contribution,

L4 =O(papb) terms giving rise to a contact contri-bution proportional to(4)(x).

The final result for the pseudoscalar correlator in mo-mentum space for Euclidean four-momentum p can bewritten as:

KP(p) = a +4B20F

2GNLO (p, T)+ c(T)GLO (p)+ O(F2),

(3)where subleading terms are labeled by their F2 depen-dence. The NLO propagator includes wave function andmass renormalization (at this order in ChPT there is noimaginary part for the self-energy):

GNLO (p, T) =

Z(T)

p2

M2

(T)

, (4)

where the LO propagator corresponds to Z = 1, M =Mand is temperature independent. F and M are thelagrangian pion mass and decay constant, related to thevacuum (T = 0) physical values M(0) M 140MeV, F(0) F 93 MeV, byO(F2) corrections[11].

The constant a in Eq.(3) is temperature independentand is a finite combination of low-energy constants (LEC)ofL4 [11]. In the ChPT scheme, the divergent part oftheL4 LEC cancels the loop divergences fromL2 suchthat pion observables are finite and independent of thelow-energy renormalization scale. To NLO in ChPT all

the pion loop contributions in Eq.(3) are proportional tothe tadpole-like contribution GLO (x= 0, T) = G

LO (x=

0, T= 0) + g1(M, T) with the thermal function:

g1(M, T) = T2

22

M/T

dx

x2 (M/T)2

ex 1 , (5)

which is an increasing function ofTfor any mass. Thus,the pion thermal mass in the NLO propagator is givenat this order by M2(T) = M

2(0)

1 + g1(M, T)/2F

2

and is finite and scale-independent. The same holdsfor F2(T) = F

2(0)

1 2g1(M, T)/F2

and forqqT =

qq0

1 3g1(M, T)/2F2

[12]. Note that M2 (T) de-creases with T a factor of 3 slower than the con-densate

qq

T. In addition, to this order it holds

F2(T)M2(T)/qqT = F2 (0)M2(0)/qq0=mq. Thatis, the GOR relation is broken at finite temperature toNLO by the sameT= 0 terms, given in[11]. GOR holdsto NLO only in the chiral limit, including temperatureeffects [17].

The constantc(T) in Eq.(3) includes both LEC contri-butions and loop functions and the same happens withthe wave function renormalization constantZ(T). Bothare divergent, but the combination 4B20F

2Z(T) +c(T)turns out to be finite and scale-independent. Note thatif we replace GLO = G

NLO in the last term in Eq.(3),

which is allowed at this order since c(T) =O(F0) is ofNLO, that combination is precisely the one multiplying

the NLO propagator, i.e, it is the T-dependent residueat the M2(T) pole (when the Euclidean propagator isanalytically continued to the retarded one). That finiteresidue, being finite, has to be then a combination of thefinite observables involved. Actually, it happens to be:

4B20F2Z(T) + c(T) =

F2(T)M4(T)

m2q+ O(F2). (6)

Thus, the expression in Eq.(6) represents the residueof the NLO KP correlator (3) at the thermal pion pole.Note that by showing explicitly that the residue canbe expressed as (6) we obtain that the thermal part

of the pseudoscalar susceptibility P(T) is the same asthat inmqqqT, since F2(T)M2(T)/m2q + O(F2) =(qqT/qq0)(F2M2/m2q) + O(F2), so that

P(T) = KP(p= 0, T) = KP(p= 0, T= 0)

+F2M

2

m2q

qqT qq0qq0 + O(F

2). (7)

Now, sinceqqT qq0 =O(F0), at the NLO or-der we are working, we can replace in Eq.(7) F2M

2 =

mqqq0 +O(F0) so that we get P(T) P(0) =

mq(

qq

T

qq

0) +

O(F2). Furthermore, the con-

stant a appearing in Eq.(3)contains precisely the LECcombination that combines with that in the residue (6)to give the same scaling law, now including the T = 0part. Thus, our final result for the pseudoscalar suscep-tibility in ChPT to NLO (finite and scale-independent)is:

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ChPTP (T) = 4B20

F2

M2+

1

322(4h1 l3) 3

2M2g1(M, T)

+ O(F2) =qq

ChPTT

mq+ O(F2), (8)

where the first term inside brackets is the LO current-algebra O(F2) and l3, h1 are renormalized scale-independent LEC[11].

Therefore, we have obtained the WI connectingqqand Pto NLO in model-independent ChPT, includingfinite-T effects. Furthermore, the mq dependence can-cels in P(T)/P(0) =qqT/qq0, where only mesonparameters show up. To this order, P(T)/P(0) =13g1(M, T)/(2F2) so that the LEC dependence alsodisappears. Recall thath1 comes from a contact term inL4 and is therefore another source of ambiguity in theNLO condensate[11].

Note also that, unlike the approach followed in [13],we have arrived to the result (8) without relying onthe validity of current algebra, which actually holdsonly in ChPT to the lowest order. Actually, as ex-

plained, our result takes into account NLO corrections toF(T), M(T), qqTthrough GOR breaking terms, bothat T= 0 and T= 0, which turn out to be crucial to ob-tain the correct scaling law given in Eq.(8).

III. LATTICE DATA ANALYSIS

We can draw some important conclusions from the pre-vious results. First, P(T) scales like the order parame-terqq, instead of the much softer behaviour 1/M2(T).This scaling suggests a chiral-restoring nature for thepseudoscalar susceptibility, although we cannot draw any

definitive conclusion about chiral restoration just fromour standard ChPT analysis, which makes sense only atlowT. Its behaviour for higher Tapproaching the tran-sition should be considered merely as indicative extrap-olations, pretty much in the same way as the ChPT pre-diction for the vanishing point of the quark condensate is

just a qualitative indication that the restoring behaviourgoes in the right direction. For this reason, in the follow-

ing we will complement our standard ChPT calculationwith a direct lattice data analysis, and later on with aunitarized study which, as we will see, incorporates the

relevant degrees of freedom to achieve a more precise de-scription near the transition point.One can actually observe a clear signal of a criti-

cal chiral-restoring behaviour for P, consistent withour previous ChPT result, in the lattice analysis of Eu-clidean correlators, which determine their large-distancespace-like screening mass Msc in different channels[18].

From Eq.(2) we expect P = KP(p = 0) (MpoleP )2with MpoleP the pole mass associated to KP(p) in a gen-

eral parametrization of the form K1P (, p) =2 +A2(T)|p|2 +MpoleP (T)2 with A(T) = MpoleP (T)/Msc(T)[19]. Here, = in would correspond to the thermalEuclidean propagator and

R+ i to the retarded

Minkowski one setting the dispersion relation. Assuminga soft temperature behaviour forA(T), which is plausiblebelowTc (for instance, A = 1 for the NLO ChPT propa-gator in Eq.(4)) we can then explain the sudden increaseof MscP (T)/M

scP (0) observed in this channel [18] since

we expect that ratio to scale like [P(0)/P(T)]1/2

[qq0/qqT]1/2. We show in Fig.1 (left panel) thesetwo quantities. The correlation between them is notori-ous, given the uncertainties involved, and the mentionedincrease is clearly observed. Data are taken from thesame lattice group and under the same lattice condi-tions[18, 20]. Note that in lattice works (2mq/ms)ss issubtracted from

qq

T in order to avoid renormalization

ambiguities. Estimating the T = 0 condensates fromNLO ChPT 1, this subtraction gives a 6% correctionand, from the lattice values, it is about a 15% correc-tion nearTc. Apart from the screening versus pole massand the strange condensate corrections, one should notforget about the typical lattice uncertainties, like resolu-tion, choice of action, staggered taste breaking and largepion masses[1,20].

Another important conclusion of our analysis is thatthe decrease of P and the increase of S as they ap-proach the critical point, lead to scalar-pseudoscalar sus-ceptibility partner degeneration, which in an ideal O(4)

1 For standard ChPT we use the same LEC values as in [ 14, 15].Their influence is more important in S, due to the vanishing ofthe LO, than in P, qq.

pattern should take place near theSpeak. Once again,this behaviour is observed in lattice data. In Fig.1(rightpanel) we plot the subtracted condensate, expected toscale as P(T)/P(0) according to our previous ChPTand lattice analysis, versus S(T)/P(0), both from thelattice analysis in [1]. The T= 0 values are taken fromChPT. The current degeneration is evident, not only atthe critical point but also above it, where those two quan-tities remain very close to one another.

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MP

scTMPsc0

l,s12 T

0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05

1.0

1.2

1.4

1.6

1.8

2.0

TTc

STP0

l,sT

120 130 140 150 160 170 180 1900.0

0.2

0.4

0.6

0.8

1.0

TMeV

FIG. 1. Left: Comparison between the pseudoscalar screening mass ratio and 1/2l,s , where l,s = r(T)/r(0) with r =

qq (2mq/ms)ss, for the lattice data in[18] (masses) and[20] (condensate) with the same lattice action and resolution andTc 196 MeV. Right: Scalar susceptibility versus l,s P(T)/P(0) from the data in [1] for which Tc 155 MeV.

Recall that both the analysis of the correlation betweenscreening masses and inverse rooted condensate and thatof scalar versus pseudoscalar (condensate) susceptibili-ties, although elaborated from available lattice data, hasnot been presented before, to the best of our knowledge.As commented above, this analysis has been motivatedby our ChPT results in section II and it gives strong sup-port to the scalar-pseudoscalar degeneration pattern atthe transition, as well as providing a natural explanationfor the behaviour of lattice masses in this channel.

IV. UNITARIZED CHPT AND RESULTS

A. Extracting thef0(500) thermal pole fromunitarized ChPT

Since the scalar susceptibility is dominated by the

I = J = 0 lightest state, which does not show up inthe ChPT expansion, let us consider its unitarized ex-tension given by the Inverse Amplitude Method (IAM)[21] which generates dynamically in SU(2) the f0(500)and (770) resonances and has been extended to finitetemperature in [2224]. Thus, before proceeding to de-rive the unitarized susceptibility in section IVB, let usreview briefly here, for the sake of completeness, some ofthe more relevant aspects of the thermal IAM, particu-larly in the scalar channel. We refer to[2225] for a moredetailed analysis.

The IAM scattering amplitude is constructed by de-manding unitarity and matching with the low-energy ex-pansion, for which all the ChPT scattering diagrams atfinite temperature are included up to one-loop [22]. Thedifferent types of those diagrams are represented in Fig-ure 2. The T-dependent corrections to the scatteringamplitude comes from the internal loop Matsubara sumsin the imaginary-time formalism of Thermal Field The-ory. The external pion lines correspond to asymptoticT= 0 states. The thermal amplitude is defined after theapplication of the T= 0 LSZ reduction formula, whichallows to deal just with thermal Green functions. Afterthe Matsubara sums are evaluated, the external lines are

FIG. 2. One-loop diagrams for T-dependent pion scattering.

analytically continued to real frequencies. The full result

for the thermal amplitude to NLO in ChPT is given in[22].

The scattering amplitude can be projected into par-tial waves tIJ(s) in the reference frame p1 =p2 wherethe incoming pions 1,2 are at rest with the thermal bath,so that s = (E1 + E2)

2. The NLO partial waves havethe generic form (we drop in the following the IJ in-dices for brevity)t(s; T) = t2(s) + t4(s; T) wheret2(s) istheO(p2) tree-level T-independent scattering amplitudefromL2 andt4 isO(p4) including the tree level fromL4plus the one-loop from the diagrams in Fig.2. Each par-tial wave satisfies Im t4(s+i; T) = T(s)t2(s)

2 for s >

4M2 with T(s) = 1 4M2/s [1 + 2nB(

s/2; T)] and

nB(x; T) = [exp(x/T) 1]1 the Bose-Einstein distribu-tion. This is the perturbative version of the unitarityrelation for partial waves Im t(s + i; T) = T(s)|t(s; T)|2and T is the two-pion phase space, which at finite Treceives the thermal enhancement proportional to nBwhich has a neat interpretation in terms of emission andabsorption scattering processes allowed in the thermalbath [22, 25]. Precisely imposing that the partial wavessatisfy the above unitarity relation exactly while match-ing the ChPT series at low s and low T, leads to the

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a

Mp2T

Mp20

IJ0

Mp2T 1

4p

2T

Mp20 1

4p

20

IJ0

Mp2T

Mp20

IJ1

0 50 100 150 200

0.2

0.4

0.6

0.8

1.0

TMeV

bpTp0

IJ0

TM2

0M2IJ1

pTp0

IJ1

0 50 100 150 200

0.8

1.0

1.2

1.4

1.6

TMeV

c

Ret4IIsp; T

Ret4IIsp; 0IJ0

Ret4IIsp; T

Ret4IIsp; 0IJ1

0 50 100 150 200

0.2

0.4

0.6

0.8

1.0

TMeV

d

Imt4IIsp; T

Imt4IIsp; 0IJ0

Imt4IIsp; T

Imt4IIsp; 0IJ1

0 50 100 150 200

0.6

0.8

1.0

1.2

1.4

TMeV

FIG. 3. Thermal pole evolution (a)-(b) and contributions from the second sheet amplitude (c)-(d) for the scalar-isoscalar

(I=J= 0) and vector-isovector (I= J= 1) channels.

thermal unitarized IAM amplitude:

tIAM(s; T) = t2(s)

2

t2(s) t4(s; T) (9)

When the IAM amplitude is continued analytically tothe s complex plane [23], it presents poles in the sec-ond Riemann sheettII(s; T) = t2(s)

2/

t2(s) tII4 (s; T)

with tII4 (s; T) =t4(s; T) + 2iTt2(s)2 so that Im tII(s

i) = Im tIAM(s+ i) for s > 4M2 . Those poles cor-respond to the physical resonances, which in the caseof pion scattering are the f0(500) (I = J = 0) and

(770) (I = J = 1). The T-dependent poles can be ex-tracted numerically by searching for zeros of 1/tII(s; T)in the s complex plane. We denote the pole position bysp(T) = [Mp(T) ip(T)/2]2. The LEC for the IAMare chosen so that, within errors, they remain compat-ible with the standard ChPT ones and with the T = 0pole values for the and f0(500) listed in the PDG [6].

Let us comment now on the thermal evolution of theresonance poles, whose main features for this work arerepresented in Figure3. Sincet2(s) = a(s s0) with reala and s

0, the thermal dependence s

p(T) is governed by

that oftII4 at the pole.

In the vector-isovector channel, p 4M2 ) so thebroadening can be explained just by thermal phase spaceincrease up to that temperature. Note that, although this

effect is formallyO(eM/T

), it activates below the tran-sition because of the relative small value of p(0). Abovethat, there is an additional increase of the effective coupling withT [23], to which tadpoles contribute, whichexplains a further increase of the width. This behaviouris represented in Figure3. Observe the softer behaviourof the mass as compared to the width in this channel,and the correlation with the real and imaginary parts ofthe amplitude at the pole position.

In the scalar-isoscalar channel, the one we are inter-

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ested in here, the behavior is remarkably different. Werather talk of a broad resonance pole, since Mp and pare comparable, so that the f0(500) pole is away fromthe real axis. As a consequence, all thermal contribu-tions from diagrams (a)-(e) in Fig.2to tII4 become com-plex at the pole and the real and imaginary parts of thepole equations do not have the simple form of a BW res-onance. In addition, due to the lower value of M2p (0)

as compared to the case, the thermal dependence ismuch more stronger, both for the real and imaginaryparts, and all contributions from those diagrams becomeequally relevant. In particular, the tadpoles in diagrams(d), (e) in Fig.2now come into play. The numerical solu-tion of the pole equations show thatM2p (T) in this chan-nel decreases significatively, while p(T) increases up toT 120 MeV and decreases from that point onwards, asseen in Fig.3. Note that this non-monotonic behaviourfor p cannot be explained now just in terms of phasespace or vertex increasing. On the other hand, a possibleinterpretation of the decreasing M2p is a chiral restor-ing behaviour. Actually, in Fig.3 (a) we also representRe s

p(T) = M2

p(T)

2p

(T)/4

M2S

(T), which wouldcorrespond to the self-energy real part of a scalar parti-cle with energy squareds and p= 0, exchanged betweenthe incoming and outgoing pions. This -like squaredmass not only drops faster but it develops a minimumat a certain temperature, which as we will see below cor-responds to a maximum in the scalar susceptibility. Inthe channel, there is almost no numerical difference be-tweenM2p and Re sp. Thus, a qualitative explanation forthe p(T) change from a increasing to a decreasing be-haviour in the scalar channel would be the influence ofthe strong mass decreasing of the decaying state.

B. Unitarized scalar susceptibility and quarkcondensate

In order to establish a connection between the scalarsusceptibility and the scalar pole, we construct a uni-tarized susceptibility by saturating the scalar propaga-tor with thef0(500) thermal state and assuming that its

p= 0 mass does not vary much with respect to the polemass. Thus, we identify the pole of a scalar state ex-changed in pion scattering with the thermal pole in thescalar channel discussed in the previous section. There-fore, we have:

US(T) = ChPT

S (0)M2

S(0)M2S(T)

, (10)

where we have normalized to the T = 0 ChPT value,since we are demanding that all our T = 0 resultsmatch the model-independent ChPT predictions. Thisnormalization compensates partly the difference betweenthe p = 0 and pole masses. Under this approxima-tion, the self-energy real part is the squared scalar massM2S(T) = M

2p (T) 2p(T)/4, as discussed in the previous

section, and the self-energy imaginary part vanishes at

p= 0.The quark condensate cannot be extracted directly

from the unitarized susceptibility. However, we can ob-tain an approximate description by assuming that therelevant temperature and mass dependence, as far asthe critical behaviour is concerned, comes from pionloop functions as qqU(T, M) = B0T2g(T /M) andS=B

20h(T /M), withf(T) = f(T)f(0). ThisT /M

dependence holds actually to NLO ChPT, as in Eq.(5).Then, from Eq.(1), sinceS=qq/mq, we get

g(x) = g(x0) + xx0

h(y)

y3 dy forx > x0, (11)

withT0 = x0MMa suitable low-Tscale below whichwe use directly NLO ChPT, which has a better analyticbehaviour near T = 0. The h function is obtained fromtheT dependence ofS in Eq.(10).

C. Results

Our theoretical results based on effective theories areplotted in Fig.4. First, ChPT to NLO gives an increas-ing S(T), intersecting P(T) at Td 0.9Tc, whereqqChPTT (Tc) = ChPTP (Tc) = 0. Once again, this re-sult should be considered just as an extrapolation of themodel-independent expressions forS(T) andP(T) be-yond their low-Tapplicability range. With this cautionin mind, standard ChPT supports the idea of partnerdegeneration. Actually, near the chiral limit M T,where critical effects are meant to be enhanced, thedegeneration point Td = Tc 3M/4 +O(M2/Tc),approaching the chiral restoration temperature in thatlimit.

We also plot US(T) in Fig.4. The result agrees withstandard ChPT at low T and improves remarkably thebehaviour near the transition. It actually develops a

maximum at Tc 157 MeV. We show for comparisonthe lattice data of [1]. Furthermore, approaching the chi-ral limit by taking the M = 10 MeV poles from [24]

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PChPT

T

P0

SChPT

T

P0

SUT

P0

q qU T

q q 0

0 50 100 150 200 250 3000.0

0.2

0.4

0.6

0.8

1.0

T MeV

FIG. 4. Scalar versus pseudoscalar susceptibilities in ChPT and in our unitarized description. We show for comparison thelattice data of Fig.1(right).

gives a vanishing MS(T) at Tc 118 MeV and hence adivergent US from Eq.(10) at Tc. Thus, we get, at leastqualitatively, the Tc reduction and stronger S growthnear the chiral limit expected from theoretical [26] andlattice [2, 3] analysis. The clear improvement of the uni-tarized approach with respect to the standard ChPT onefor the scalar susceptibility is essentially due to the in-troduction of the thermal f0(500) state, whose impor-tance in this case is clearly seen from the dependenceS1/M2S, rather than to an enlargement of the appli-cability range in the unitarized approach. Actually, evenwithin the unitarized description, we should be cautiouswhen extrapolating it to nearTc since we may be strictlybeyond the effective theory range.

The resultingqqUT is plotted in Fig.4 with T0 12MeV 2. The critical behaviour is again nicely improvedcompared to the ChPT curves and is in better agreementwith lattice data in that region. In addition, we obtainonce more a scalar-pseudoscalar intersection near theSpeak and hence of chiral restoration. The correspondingqqUTnear the chiral limit (M = 10 MeV) is much moreabrupt, vanishing and meeting Sat Tc as expected.

Recall that in the above unitarized analysis, we arenot performing a fit to lattice points. We just use thesame LEC which generate the T= 0 physical f0(500),states [24] and then provide our results for the suscep-tibility and condensate. The theoretical uncertainties in

the LEC, as well as the lattice errors already discussedshould be taken into account for a more precise com-parison. Besides, our effective theory analysis is not ex-pected to reproduce the chiral restoration pattern aboveTc. In any case, apart from the important consistencyobtained for chiral restoration properties, such as the S

2 We find very small numerical differences changing T0 between10-60 MeV.

peak and the S/P matching, an important point wewant to stress is the crucial role of the f0(500)/state todescribe the scalar susceptibility. SinceS1/M2S, thisobservable is much more sensitive to this broad state, sothat a physically realistic description, including its ther-mal effects, turns out to be essential. This may not be thecase for other thermodynamical observables, for whichother approaches may provide a much better descrip-tion. For instance, the Hadron Resonance Gas (HRG)framework gives very accurate results compared to lat-tice data [27, 28] and particle distributions [29, 30] belowthe transition, by including all known hadronic statesas free particles in the partition function, often includingsmall interaction corrections. Within the HRG approach,

hadron interactions are generically encoded in the reso-nant states. This framework works for most thermody-namic quantities, which are obtained, by construction,as monotonic functions ofT, and generally increase withthe mass of the states considered. For instance, the scalarsusceptibility within that approach would increase withT, as it happens for other quantities such as the traceanomaly. Thus, the effect of a properly included broadT-dependent state arising from pion scattering in order todescribe chiral restoring properties such as the suscepti-bility peak, is once more highlighted. In fact, the stateis just not included in many HRG works [28]or, at most,considered as a BW state [29, 30] with its T = 0 mass

and width, which, as we have commented above, doesnot provide an entirely adequate description. Finally, letus comment that apart from higher mass states, inclu-sion of higher order interactions may also be importantfor certain hadronic observables. For instance, includ-ing interactions in the vector channel are essential todescribe properly the dilepton spectra[5].

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V. CONCLUSIONS

Summarizing, we have analyzed the scalar-pseudoscalar O(4)-like current degeneration patternat chiral symmetry restoration, from lattice simulationsand effective theory analysis. The pseudoscalar suscep-

tibility scales as the quark condensate, which we haveexplicitly shown in ChPT to NLO at low temperature,and becomes degenerate with its chiral partner scalarsusceptibility close to the scalar transition peak. Thelattice data and the unitarized ChPT analysis supportthis picture and are well accounted for by the dominantphysical states: pions and thef0(500) scalar resonancegenerated in pion scattering at finite temperature.

In turn, we have provided a natural explanation forthe sudden growth of lattice masses observed in thepseudoscalar channel. Although we have restricted here,for simplicity, to the Nf = 2 case, the analysis can beextended toNf= 3, where the role of other scalar statessuch as the a0(980) can also be studied [31].

ACKNOWLEDGMENTS

Useful comments from F. Karsch, S. Mukherjee and D.Cabrera are acknowledged. Work partially supported bythe EU FP7 HadronPhysics3 project, the Spanish projectFPA2011-27853-C02-02 and FPI Programme (BES-2009-013672, R.T.A), and by the German DFG (SFB/TR 16,J.R.E.).

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