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Octet Baryons at Finite Temperature:

arXiv:hep-ph/9306231v1 5 Jun 1993MSUCL-870January, 1993Octet Baryons at Finite Temperature:QCD Sum Rules vs. Chiral SymmetryYuji KOIKENational Superconducting Cyclotron Laboratory, Michigan State UniversityEast Lansing, MI 48824-1321, USAAbstractCorrelators of the octet baryons in the hot pion gas are studied in the framework of theQCD sum rule. The condensates appearing in the OPE side of the correlators become T-dependent through the interaction with thermal pions.

We present an explicit demonstrationthat the O(T 2)-dependence of the condensates is completely compensated by the change ofthe pole residue and the π + B →B′ scattering effect in the spectral functions. Thereforethe baryon masses are constant to this order, although ⟨¯uu⟩T ≃⟨¯uu⟩0(1 −T 2/8f 2π), which isconsistent with the chiral symmetry constraint by Leutwyler and Smilga.

1IntroductionModification of the hadronic properties at finite temperature has been receiving greaterattention in connection with the relativistic heavy ion collisions such as RHIC and LHC.Although the finite-T properties of various mesons have been extensively studied in theliterature using QCD Sum Rules (QSR) [1, 2, 3] and effective theories for QCD [4, 5, 6, 7],baryon properties have not been seriously investigated except for the nucleon [8, 9, 10]. Inprinciple, the change of baryon masses should manifest itself in the yield modification andthe threshold behaviour of lepton pairs coming from the baryon-antibaryon annihilation inthe relativistic heavy ion collisions.

Therefore it is a pressing issue to have a QCD predictionfor the finite-T behavior of the baryon correlators.Since the work by Shifman, Vainstein and Zakharov [11], the QCD sum rule method hasbeen extensively used as a systematic tool to study various resonance properties based onQCD (see [12] for a review).Especially, the masses of octet and decuplet baryons havebeen well reproduced in terms of the vacuum condensates [13, 14, 15, 16]. Hence it is worthwhile to extend this strongly QCD motivated phenomenology to study finite-T behavior ofbaryon correlators.

In this paper we shall analyze the correlators of the octet baryons (N,Λ, Σ and Ξ) at finite temperature. The source currents we will investigate are the followinginterpolating fields which have been used in QSR:ηN(x)=ǫabcua(x)Cγµub(x)γ5γµdc(x),(1.1)ηΛ(x)=s23ǫabc hua(x)Cγµsb(x)γ5γµdc(x) −da(x)Cγµsb(x)γ5γµuc(x)i,(1.2)ηΣ(x)=ǫabcua(x)Cγµub(x)γ5γµsc(x),(1.3)ηΞ(x)=−ǫabcsa(x)Cγµsb(x)γ5γµuc(x),(1.4)where a, b, c are the color indices and C is the charge conjugation matrix.

Although onecan choose other sets of source currents for the octet baryons, it has been known that theabove combination gives the best description of the octet baryon masses.A useful quantity for studying the finite-T behavior of baryons is the retarded correlationfunction for the above fields [17]:ΠRαβ(ω, q, T) = iZd4x eiqxθ(x0)⟨ηα(x)¯ηβ(0) + ¯ηβ(0)ηα(x)⟩T ,(1.5)where q = (ω, q) is the external four momentum, ⟨O⟩= TrOe−H/T /Z (Z = Tr(e−H/T))is the thermal average for the operator O with H being the QCD hamiltonian, and α and βare the spinor indices of the interpolating fields (1.1)–(1.4). The retarded correlator satisfiesthe dispersion relation with an appropriate spectral function ρ(ω, q, T):ΠRαβ(ω, q, T) =Z ∞−∞du ραβ(u, q, T)u −ω −iǫ .

(1.6)In the QCD sum rule method, one applies the operator product expansion (OPE) to the lefthand side of the correlator in the deep Euclidean region Q2 = −q2 →∞and adjusts the1

resonance parameters (resonance masses, pole residues, width of the resonances, continuumthreshold, etc) in the spectral function ρ so as to reproduce the OPE side of the correlator.This procedure provides us with an expression for the resonance parameters in terms of thecondensates. Therefore, in the QSR approach, the change of the condensates at finite-Tnaturally causes a response of the baryons as a change of their properties at finite-T. Forexample, one might naively expect that the nucleon mass will drop at finite temperature asthe chiral order parameter decreases [10].On the other hand, Leutwyler and Smilga [8] considered the nucleon correlator in a ther-mal pion gas and showed that the nucleon mass does not have O(T 2)-dependence: In thethermal pion gas, the T-dependence of the nucleon correlator is associated with the pion-nucleon forward scattering amplitude.

It becomes zero in the soft pion limit (Adler’s zero)and thus the real part of the self-energy becomes zero at the pole position of T = 0. Thesame statement is also true for all the octet baryons.The purpose of this paper is to demonstrate how this seeming contradiction can bereconciled in the framework of the finite-T QCD sum rules.

We organize the QSR at T ̸= 0for the nucleon in the dilute pion gas [2]. This approximation should be valid at relativelylow temperature (T ≤150 MeV) well inside the confined phase.

In this framework, thecondensates appearing in the OPE side of the correlators receive T-dependence through thepion matrix elements of the same operators. We pay particular attention to the effect ofthe π + N →N scattering in the phenomenological side of the QSR as was suggested byEletsky [9].

Then it can be shown that the O(T 2)-dependence of the condensates is exactlycompensated by this scattering and the mass does not have O(T 2)-dependence as was foundby Leutwyler and Smilga. Thus the naive expectation motivated by the Ioffe’s mass formulafor the nucleon, MN(T) ≃(−2(2π)2⟨¯uu⟩T)1/3, does not work at finite-T.

The situation isanalogous for the other octet baryons.The construction of this paper is the following: In section 2, after briefly summarizingour finite-T QSR for the case of the nucleon along the line of [2], we present an explicitdemonstration that the nucleon mass does not have a O(T 2)-dependence in the QSR. Thediscussion for the hyperons is similar.

So we will not repeat it, but present some of theformulas in the Appendix. In section 3, we shall analyze the baryon correlators in the piongas in terms of PCAC without using QCD sum rules.

We shall see that the octet baryoncorrelators at finite-T can be written in terms of the correlators at T = 0 with T-dependentcoefficients and our OPE expression derived in section 2 is consistent with those relations.This explains why the pole positions of the octet baryon correlators do not have O(T 2)-dependence. Section 4 is devoted to a brief summary and outlook.2QCD Sum Rules for Nucleon at T ̸= 02.1OPE at T ̸= 0We shall first discuss the QCD side of the correlator.

For simplicity, we study the nucleonat rest, i.e. q = 0.

In the region, Q2 = −q2 = −ω2 > 0, the retarded correlator (1.5) isidentical to the causal correlator:Παβ(ω, q, T) = iZd4x eiqx⟨T (ηα(x)¯ηβ(0))⟩T. (2.1)2

At T ̸= 0, Παβ(q, T) can be decomposed into the three scalar components:Παβ(q, T) = Π1(q, T)δαβ + Π2(q, T)/qαβ + Π3(q, T)/uαβ,(2.2)where uµ is the average four-flow velocity of the medium equal to uµ = (1, 0, 0, 0) in the restframe. In the deep Euclidean region Q2 →∞, one can apply OPE to Π(q, T), which, for Π2as an example, can be schematically written as,Π2(q, T)=XiCµ1···µsi(q, µ)Oi(d,s)µ1···µs(µ)=−164π4q4ln(Q2) + ⟨O(4,0)⟩T + qµqνq2 ⟨O(4,2)µν ⟩T!ln(Q2)+1q2 ⟨O(6,0)⟩T + qµqνq2 ⟨O(6,2)µν ⟩T + qµqνqλqσq4⟨O(6,4)µνλσ⟩T!+ · · ·,(2.3)where the i-th local operator Oi(d,s)µ1···µs(µ) renormalized at the scale µ has dimension-d andspin-s with s Lorentz indices and Cµ1···µsi(q, µ) is the corresponding Wilson coefficient.

As acomplete set of the local operators in (2.3), one can always choose symmetric and tracelessoperators with respect to all Lorentz indices. We will hereafter assume this symmetry con-dition for all nonscalar operators.

In the above equation the following two features peculiarto T ̸= 0 are implemented [2]:(i) T-dependence of the correlators appears only as a thermal average of the local op-erators in the OPE as a consequence of the QCD factorization. This is indeed natural ifwe note that such a soft effect should be ascribed to the condensates ⟨Oi⟩T as long as thetemperature is low enough compared to the separation scale µ, i.e.

T ≪µ ≪Q. (ii) At T ̸= 0, there is no Lorentz invariance due to the presence of the thermal factore−H/T, and hence nonscalar operators survive as condensates:⟨Oiµ1···µs(µ)⟩T = (uµ1 · · · uµs −traces)ai(µ, T).

(2.4)At relatively low temperature in the confined phase, the system can be regarded as a non-interacting gas of Goldstone bosons (pions). In this approximation, T-dependence of thecondensates can be written as⟨Oi⟩T ≃⟨Oi⟩+3Xa=1Zd3p2ε(2π)3⟨πa(p)|Oi|πa(p)⟩nB(ε/T),(2.5)where ε =qp2 + m2π, a denotes the isospin index, nB(x) = [ex −1]−1 is the Bose-Einsteindistribution function, and ⟨·⟩is the usual vacuum average.

Here we have used the covariantnormalization for the pion state: ⟨πa(p)|πb(p′)⟩= 2ε(2π)3δabδ3(p −p′). Thus we need pionmatrix elements of the local operators appearing in the OPE to carry out the finite-T sumrule.

For the scalar operators O(d,0), we can apply the soft pion theorem⟨πa(p)|O|πb(p)⟩= −1f 2π⟨0|hF a5 ,hF b5, Oii|0⟩+ O m2πΛ2HAD!,(2.6)3

where ΛHAD is a typical hadronic scale of O(1 GeV) and F a5 is the isovector axial chargedefined byF a5 =Zd3x ¯q(x)γ0γ5τ a2 q(x). (2.7)Pion matrix elements of the nonscalar operators are associated with the pion structure func-tions measurable in the hard processes (deep inelastic scattering, Drell-Yan, direct photonproduction etc.).

Experimentally, however, only twist-2 part of these matrix elements areknown to some extent, and therefore we have to await future precise measurements of thetwist-4 pion structure functions to carry out the satisfactory QSR at finite-T. Pion matrixelements of these operators read⟨π(p)|Oiµ1···µs|π(p)⟩∼(pµ1 · · · pµs −traces)Ai(µ),(2.8)and therefore the contribution of nonscalar condensates becomes the effect of O(T 4) or higherin T (mπ ∼T is assumed) as is easily seen by inserting (2.8) into (2.5). In [2] we foundthat these effects can be neglected below T = 160 MeV.

We shall consistently ignore thesecondensates in this work. For the same reason, Π3(q, T) also becomes higher order withrespect to temperature.

The effect of the heavier resonances (K, η etc) was also found to benegligible at T ≤160 MeV because of the suppression coming from the distribution function∼e−mK/T [2].Applying the soft pion theorem to scalar operators appearing in the OPE of the nucleoncorrelators, we getΠN1 (q, T)=14π2⟨¯uu⟩ 1 −ζ8!q2ln(Q2),(2.9)ΠN2 (q, T)=−164π4q4ln(Q2) −132π2⟨αsπ G2⟩ln(Q2) −2⟨¯uu⟩23q2,(2.10)where ζ = (T 2/f 2π)B1(mπ/T) withB1(z) = 6π2Z ∞zdyqy2 −z21ey −1. (2.11)In (2.9) and (2.10) we note the following points:(i) Although we discarded the terms proportional to mu and md because of their smallness,we included the pion mass correction coming from the Bose-Einstein distribution, since itappears in the form of mπ/T in B1.B1 approaches 1 at mπ ≪T, while it is stronglysuppressed as B1 ∼e−mπ/T at mπ ≫T.

(ii) The chiral condensate ⟨¯uu⟩T changes as ⟨¯uu⟩0(1 −ζ/8). Its T-dependence was cal-culated by the chiral perturbation theory including up to O(T 6) effects [18], which gives thesame coefficient as above for the O(T 2) effect.

In the temperature range T ≤160 MeV, bothcalculations agree well within 5 %. (iii) By using the QCD trace anomaly, T-dependence of ⟨αsπ G2⟩T can be estimated [2].It gives a negligible change of the condensate at finite T (0.5% at T = 200 MeV).

We thusignored its T-dependence in (2.10).4

(iv) The four-quark condensate in the nucleon channel turned out to be T-independent,which is quite different from the behavior of the square of the chiral order parameter. Herewe used the vacuum saturation assumption, i.e., ⟨(¯uΓu)2⟩→⟨¯uu⟩2 after applying the softpion theorem, as is usually adopted in the vacuum QSR.

The calculation of the four-quarkcondensates is somewhat tedious, so we shall present a nonfactorized form of the four-quarkcondensates for the octet baryons in the Appendix. There one sees that the T-dependenceof the four-quark condensates is different in different channels.

For example, the four-quarkcondensate in the nucleon channel is T-independent, while it is proportional to 1 −ζ/6 inthe Π1 structure in the Σ-channel (See also (A.11)–(A.15) in the Appendix).2.2Spectral functionIn the pion gas, the spectral function for the nucleon current acquires a contribution fromthe scattering with the thermal pions: π + N →N, ∆etc. The effect of these scatteringshas to be taken into account in the spectral function before the change of the condensates isascribed to the shift of the pole position.

For the purpose of identifying these contributions,we introduce the expression for the spectral function[17]:ραβ(ω, q, T)=1πImΠRαβ(ω, q, T)=1Z (2π)3 Xn,m⟨n|ηα(0)|m⟩⟨m|¯ηβ(0)|n⟩×(e−εn/T + e−εm/T)δ(ω −ωmn)δ(3)(q −pmn),(2.12)where the states |m⟩and |n⟩have the four momentum (εm, pm) and (εn, pn), respectively,and ωmn = εm −εn, pmn = pm −pn. If we put |n⟩= |0⟩, |m⟩= |N(p)⟩( |m⟩= |0⟩,|n⟩= | ¯N(p)⟩) in (2.12), this is the contribution from the nucleon (anti-nucleon) at T = 0.By introducing the nucleon and the anti-nucleon spinor by the relation⟨0|ηα(0)|N(p)⟩= λNuα(p),⟨0|¯ηα(0)| ¯N(p)⟩= λN ¯vα(p)(2.13)with the normalization ¯u(p)u(p) = 2MN and ¯v(p)v(p) = −2MN, we get for this contributionρ(ω, q) = λ2N2p0(/q + MN)δ(ω −p0) −δ(ω + p0)(2.14)with p0 =qq2 + M2N.

Using the Borel sum rule method, we can study the T-dependence ofthe mass MN(T), of the pole residue λN(T), and of the continuum threshold S0(T). However,it is difficult to incorporate the effects of the width and the scattering contribution inducedin the pion gas.

Thus we shall first list up the effects which should not be associated with thechange of the above three resonance parameters. Then we put these additional structuresat T ̸= 0 in the spectral function when we carry out the Borel sum rule.

(i) π + N →N (π + ¯N →¯N) contribution; |n⟩= |π(k)⟩, |m⟩= |N(p)⟩(|n⟩= | ¯N(p)⟩,|m⟩= |π(k)⟩): By taking into account the Bose symmetrization among pions which equallyfill both |n⟩and |m⟩, but do not interact with the nucleon current, we arrive atρπ+N→N(q, T)=(2π)3Zd3k(2π)32k0nB(k0/T)Zd3p(2π)32p05

×Xspin=±1/2Xa⟨πa(k)|ηα(0)|N(p)⟩⟨N(p)|¯ηβ(0)|πa(k)⟩×δ(ω −p0 + k0)δ(3)(q −p + k) + δ(ω −p0 −k0)δ(3)(q −p −k)+Xspin=±1/2Xa⟨¯N(p)|ηα(0)|πa(k)⟩⟨πa(k)|¯ηβ(0)| ¯N(p)⟩×δ(ω + p0 + k0)δ(3)(q + p + k) + δ(ω + p0 −k0)δ(3)(q + p −k)i. (2.15)Here we discarded the thermal factor for the nucleon (Fermi distribution function), since1/(eMN/T −1) ≃0 at T ≤200 MeV.

We have to include two kinds of contribution to thematrix element ⟨πa(k)|ηα(0)|N(p)⟩[19]. (a) Direct coupling of ηN to the pion (Fig.

1): This contribution can be calculated byapplying the soft pion theorem:⟨πa(k)|ηpα(0)|N(p)⟩=−ifπ⟨0| [F a5 , ηpα(0)] |N(p)⟩,(2.16)[F a5 , ηpα(0)]=τ a222 γ5ηp −τ a212 γ5ηn,(2.17)where ηp is the proton current defined in (1.1) and ηn = −ǫabcda(x)Cγµdb(x)γ5γµuc(x) is theneutron current. Using (2.16) and (2.17) in (2.15) and putting q = 0, we arrive at 1ρπ+N→N(0)(q, T)= −3λ2N4f 2π!

Zd3k(2π)32k02p0nB(k0/T) ×hγ5(/p + MN)γ5δ(ω −p0 + k0) + δ(ω −p0 −k0)+ γ5(/p −MN)γ5δ(ω + p0 + k0) + δ(ω + p0 −k0)i. (2.18)Since we used the soft pion theorem in calculating the matrix element in (2.16), it sufficesto consider the soft pion limit k = (k0, k) →0 in (2.18).

Then (2.18) becomesρπ+N→N(0)(q, T) = λ2Nζ32MN! (δ(ω −MN) −δ(ω + MN)) (/q −MN) .

(2.19)Equation (2.18) was derived by Eletsky [9] taking the imaginary part of the correspondingretarded correlator.2 There it was shown that (2.18) has a localized structure around thenucleon and anti-nucleon poles, which can be well approximated by (2.19).1Here p has k or −k corresponding to two δ(3)-functions in (2.15).2The result given in eq. (11) of [9] is two times larger than the one given in (2.18) by mistake.

I thankV. L. Eletsky for correspondence to clarify this point.6

(b) Coupling of ηN to the nucleon which interacts with π (Fig. 2): This contribution canbe calculated by using the vertex⟨π(k)|η(0)|N(p)⟩= λN⟨π(k)|ψN(0)|N(p)⟩= λNgπNN2MN1/p −/k −MN/kγ5u(p),(2.20)where we assumed the π −N −N interaction lagrangian asLπNN = gπNN2MN¯Nγ5γµ⃗τN∂µ⃗π.

(2.21)Inserting (2.20) into (2.15), one obtains in the q = 0 limit:ρπ+N→N(1)(q, T)=Zd3k(2π)32k02p0nB(k0/T) 3 λNgπNN2MN!21(q2 −MN)2× [(/q + MN) /kγ5 (/p + MN) /kγ5 (/q + MN)×δ(ω −p0 + k0) + δ(ω −p0 −k0)+ (−/q + MN) /kγ5 (/p −MN) /kγ5 (−/q + MN)×δ(ω + p0 + k0) + δ(ω + p0 −k0)i(2.22)= gA2λ2Nζ32MN! (δ(ω −MN) −δ(ω + MN)) (/q + MN) .

(2.23)In (2.23), we used the Goldberger-Treiman relation gπNN/MN = gA/fπ. Note that in (2.19)the two structures proportional to 1 and /q have opposite signs while they have the same signin (2.23).One can easily check that the crossing term between (a) and (b) disappears.In the above (a) and (b), we have obtained the π + N →N scattering contribution tothe spectral function at q = 0 in the form:ρπ+N→N(ω, T)=ρπ+N→N(0)(q, T) + ρπ+N→N(1)(q, T)= λ2Nζ32MN!

(δ(ω −MN) −δ(ω + MN))n1 + g2A/q −1 −g2AMNo. (2.24)We note that up to O(T 2) the use of the nucleon mass and the pole residue of T = 0 in(2.24) is consistent with our treatment of the OPE side because of the presence of the factorζ in (2.24).

(From a physical ground, one might wish to replace them by those at T ̸= 0.But these two procedures cause only an O(T 4) difference in the final result. )In (2.24), the π + N →N scattering term eventually becomes an effective delta functionat the pole position of T = 0.

However, it contributes differently to the 1 and /q structuresof the correlators. Therefore its effect must be taken into account in the spectral function7

when we make use of the Borel sum rule method. Otherwise, an erroneous mass shift of thenucleon would occur.

(ii) π+N →∆contribution: Using the lowest order piece of the chiral invariant π−N−∆interaction lagrangian LπN∆∼gπN∆¯∆µγ5N∂µπ [20], we can calculate the vertex:⟨π(k)|η(0)|∆(p)⟩∼λNgπN∆/p −/k −MNkµ∆µ(p),(2.25)where ∆µ(p) is the Rarita-Shwinger spinor for ∆. The sum over spin for ∆gives the projec-tion operator [20]Xspin∆µ(p)∆ν(p) = Pµν ="gµν −23M2∆pµpν −13γµγν −13M∆(pµγν −pνγµ)#(/p + M∆).

(2.26)In the soft pion limit, the π +N →∆scattering term appears at ω ∼M∆where the nucleonpropagator in (2.25) becomes proportional to 1/(M2∆−M2N) and the vertex contribution iskµkνPµν ∼k2. Therefore the π + N →∆scattering contribution becomes O(T 4) and weshall discard it.One can easily repeat the same steps as above (i) (a) (b) and (ii) for the other octetbaryons.

By the same reason as (ii), the effect of the transition to the decuplet baryons(π + Σ →Σ∗(1385), π + Ξ →Ξ∗(1530)) is O(T 4). Even among octet baryons, (i)(b) typescattering between Λ and Σ is O(T 4) because of their mass difference.2.3Borel sum ruleWe are now ready to perform the Borel sum rule analysis for the octet baryons.

We assumedthat the finite-T medium is the dilute pion gas which has zero chemical potential, and thusthe charge conjugation symmetry is preserved, i.e., there appears no splitting between thebaryon and anti-baryon poles.3Therefore the spectral function for the nucleon reads atq = 0ρN(u, T)=λ2N(T)δ(u2 −M2N(T))(/q + MN(T))sign(u) + ρπ+N→N(u, T)+(continuum by step function),(2.27)where ρπ+N→N is defined in (2.24).Corresponding to (2.2), we decompose the spectralfunction as ρN(q) = ρ1(q)+/qρ2(q). Then both ρ1 and ρ2 satisfy the relation ρi(−ω, q = 0) =−ρi(ω, q = 0) (i = 1, 2) as was obtained in the previous subsection.

In the deep Euclideanregion, the dispersion relation (1.6) can be written asΠRi (ω2 = −Q2, q = 0, T) = Πi(Q2, T) =Z ∞0du2uρi(u, q = 0)u2 + Q2. (i = 1, 2)(2.28)3 This is in contrast to the system with a finite baryon number such as the nuclear matter.

In order toorganize sum rules for baryons in such a system, the dispersion relation needs some modification. See [21]for the detail.8

Applying the Borel transformΠi(M2, T)≡ˆBMΠi(Q2, T)≡lim Q2, n →∞Q2/n = M2 : fixed!1(n −1)! (Q2)n −ddQ2!nΠi(Q2, T)(2.29)=Z ∞02udu e−u2/M2ρi(u, T)(2.30)to (2.9) and (2.10) with the spectral function (2.27), we get the following relation:2a 1 −ζ8!M4 = ˜λ2N(T)MN(T)e−M2N(T)/M2 −˜λ2N(0)(1 −gA2)MN(0)ζ16e−M2N(0)/M2, (2.31)M6 + M2b + 43a2 = ˜λ2N(T)e−M2N(T)/M2 +˜λ2N(0)(1 + gA2)ζ16e−M2N(0)/M2, (2.32)wherea=−(2π)2⟨¯uu⟩,(2.33)b=π2⟨αsπ G2⟩,(2.34)˜λ2N(T)=2(2π)4λ2N(T).

(2.35)From (2.31) and (2.32) one gets the expression for the nucleon mass:MN(T) =2a1 −ζ8M4 +˜λ2N(0)(1 −gA2)MN(0)ζ/16e−M2N(0)/M2M6 + M2b + 43a2 −˜λ2N(0)(1 + gA2)ζ/16e−M2N(0)/M2. (2.36)In (2.36), we omitted the correction due to the continuum contribution for brevity.

Theformula which include the correction is given by the following replacement:M6→M6(1 − 1 + S0M2 + S202M4!e−S0/M2),M4→M41 −1 + S0M2e−S0/M2,M2→M2 1 −e−S0/M2. (2.37)An important consequence from (2.36) is that MN(T) is completely T-independent.

In fact,if we replace ˜λ2N(0) in the numerator of (2.36) by the one obtained from (2.31) by setting9

T = 0, and replace ˜λ2N(0) in the denominator of (2.36) by the one obtained from (2.32),we can easily see that the T-dependence disappears from MN(T). If we did not include thescattering term, the nucleon mass would behave as MN(T) = MN(0)(1 −ζ/8) as can beseen from (2.36).

This means that the T-dependence of the condensates caused through theinteraction with the thermal pions is completely compensated by the π −N scattering termand the pole position does not move at least to order O(T 2). This is consistent with thestatement of Leutwyler and Smilga based on the chiral lagrangian [8]: If we calculate the selfenergy of the nucleon using the πNN effective lagrangian (2.21), we can easily check that thereal part of the self energy is zero at the T = 0 pole position.

This is a direct consequence ofthe Adler’s consistency condition required for the use of PCAC [19] (in another word, (2.21)has a derivative coupling).The pole residue changes as ˜λ2N(T) = ˜λ2N(0)(1 −(1 + g2A)ζ/16) as is seen from (2.31) and(2.32); the same result obtained by [8] in the chiral lagrangian approach.From the above demonstration, it is clear that we have to take into account the newstructure in the spectral function consistently with the T-dependence in the OPE side of thecorrelator. Otherwise the usual procedure in the sum rule leads to an artificial change of theresonance parameters.In [10], the nucleon mass was calculated by a finite-T QCD sum rule method.

The authorsfound a dropping nucleon mass even in the low temperature region as ⟨¯uu⟩T decreased atfinite-T. They did not take into account the π + N →N scattering effect and assumedthat the T-dependence of the four-quark condensate is the same as (⟨¯uu⟩T)2.

Although theircalculation was not based on the pion gas approximation, the present consideration showsit is crucial to treat both the OPE side and the phenomenological side consistently. Correcttreatment of the T-dependence of all the condensates is also required.3Analysis of Correlators using PCACIn this section we shall examine the octet baryon correlators starting from the pion gasapproximation without using OPE:Π(q, T) ≃Π(q, 0) + iZd4x eiqxZd3k(2π)32k0nB(k0/T)⟨πa(k)|T (η(x)¯η(0)) |πa(k)⟩.

(3.1)Applying the LSZ reduction formula for the pion, then using the PCAC relation ∂µAaµ(x) =fπm2πφa(x) for the pion field φa(x) and taking the soft pion limit, one arrives atΠ(q, T)≃Π(q, 0) −iδabζ24Zd4x eiqx ×n⟨ThF a5 (x0),hF b5(x0), η(x)ii¯η(0)⟩+ ⟨Tη(x)hF a5 (0),hF b5(0), ¯η(0)ii⟩+⟨ThF a5 (x0), η(x)i hF b5(0), ¯η(0)i⟩+ ⟨ThF b5(x0), η(x)i[F a5 (0), ¯η(0)]⟩o+ · ··,(3.2)where + · ·· denotes the terms associated with the axial charges carried by the octet baryons(terms with gA in the previous section), which becomes O(T 4) or higher except at the pole10

position of T = 0.4 Utilizing the formula (2.17) together withδab hF a5 ,hF b5, ηNii= 34ηN,(3.3)we obtain the following expression for the nucleon correlator at T ̸= 0:ΠN(q, T)= 1 −ζ16!ΠN(q, 0) −ζ16γ5ΠN(q, 0)γ5 + · · ·. (3.4)The second term of the r.h.s.

of (3.4) corresponds to the π + N →N scattering term inthe sum rule approach in the previous section and the first term of the r.h.s. of (3.4) is themodification of the residue of the nucleon current.

The above analysis of the current simplytells us that the nucleon correlator at T ̸= 0 can be written as a superposition of the samecorrelator at T = 0 with T-dependent coefficients and there is no O(T 2) shift of the poleposition, which is consistent with the observation of [8]. (As was shown in [8] using (2.21),the contribution to the real part of the self-energy from + · ·· in (3.4) becomes zero at thepole position but induces an O(T 2) wave function renormalization.

)Equation (3.4) reads ΠN1 (q, T) = ΠN1 (q, 0)(1−ζ/8)+··· and ΠN2 (q, T) = ΠN2 (q, 0)+···. TheT-dependence of these two equations is the same as the OPE given in (2.9) and (2.10).

This isthe reason we observed no mass shift in the Borel sum rule. We remind the readers once againthat the factorization of the four-quark condensate in the medium level ⟨(¯qΓq)2⟩T →⟨¯qq⟩2Tis not justified.

If we adopted this procedure in section 2, the T-dependence of the OPE sideof the correlator would be different from (3.4).It is easy to extend the above analysis to other octet baryon correlators. For this purposewe need the following commutators:hF a5 , ηΛi=s23γ5"−τ a221 ηΣ+ +τ a212 ηΣ−+ 1√2τ a211 −τ a222ηΣ0#,(3.5)hF a5 , ηΣ+i=τ a12√2γ5ηΛ′,(3.6)hF a5 , ηΞ0i=τ a211 γ5ηΞ0 +τ a212 γ5ηΞ−,(3.7)δab hF a5 ,hF b5, ηΛii=−√3ηΛ′,(3.8)δab hF a5 ,hF b5, ηΣii=ηΣ,(3.9)δab hF a5 ,hF b5, ηΞii=34ηΞ,(3.10)where we introduced a new current ηΛ′ defined byηΛ′ =√2ǫabcuaCγ5γµdbγµsc,(3.11)4In the sum rule, we needed to integrate over the spectral functions, which is why the O(T 2) contributionappeared.11

in (3.6) and (3.8). By using these relations in (3.2), we obtain the following relations amongthe correlators:ΠΛ(q, T)=ΠΛ(q, 0) +√3ζ24 ΠΛΛ′(q, 0) −ζ12γ5ΠΣ(q, 0)γ5 + · · ·,(3.12)ΠΣ(q, T)= 1 −ζ12!ΠΣ(q, 0) −ζ12γ5ΠΛ′(q, 0)γ5 + · · ·,(3.13)ΠΞ(q, T)= 1 −ζ16!ΠΞ(q, 0) −ζ16γ5ΠΞ(q, 0)γ5 + · · ·,(3.14)withΠΛΛ′αβ (q, 0) = iZd4x eiqx⟨TηΛα(x)¯ηΛ′β (0) + ηΛ′α (x)¯ηΛβ (0)⟩.

(3.15)The current ηΛ′ is anti-symmetric under the exchange between u and d quarks and thus ηΛ′(or γ5ηΛ′) should have some overlap with isoscalar strangeness=–1 baryons such as Λ(1115),Λ∗(1405) (JP = 1/2−) as well as the π−Σ continuum contribution. As is shown in [16], thereare 5 independent interpolating fields without derivatives for Λ.

One can see by the Fierzrearrangement between s and u (or d) that ηΛ′ consists of ηΛ and those others. We tried toidentify its structure by the vacuum QSR including the operators up to dimension-6, but itdoes not seem to have a dominant pole contribution.

In any case, (3.13) tells us that thefinite-T Σ-correlator can be written as the modification of the residue and the mixing withΛ′ correlator. In principle, the second term of (3.12) also describes the modification of theresidue.

As for Ξ, the situation is completely parallel with the nucleon. The T-dependenceof the OPE expressions (A.15) and (A.16) is consistent with (3.14) as they should.4Summary and OutlookIn this paper we have presented an explicit demonstration that the O(T 2) dependence ofthe condensates which appear in the OPE of the octet baryon correlators is totally absorbedby the scattering terms π + B →B′ and the modifications of the pole residues, in theframework of the QCD sum rules.

This result is consistent with the statement by Leutwylerand Smilga [8]. The result stems from the fact that the baryon correlators in the thermal piongas can be written as a superposition of the correlators of T = 0 with T-dependent coefficientsup to O(T 2).

The procedure for achieving consistency with this relation is somewhat intricatein the QSR. Therefore one has to pay particular attention to the consistency between theassumption made to estimate the T-dependence of the condensates and the new structureappearing in the phenomenological spectral function.In the chiral lagrangian approach, the O(T 2) mass shift of baryons is caused by tadpoleinteractions such as msKK ¯BB/f 2π (K is the kaon field) [22].

In the mu = md = 0, ms ̸= 0limit studied in this work, the kaon or η field always accompanies the tadpole contributionas in the case of the above interaction. Thus without including kaons or η in the heat bath,baryons do not receive a O(T 2) mass shift.

However, the presence of those massive excitations12

is suppressed as ∼e−mK/T. For example, the above term in the effective lagrangian causesthe mass shift of the order of msT 2/(24f 2π)B1(mK/T) ∼1 MeV at T = 150 MeV.

If weincluded the kaons and η’s in the heat bath together with the nonzero strange-quark mass,the T-dependence of the OPE for the hyperons shown in (A.11)–(A.16) would be differentfrom (3.12)–(3.14). This would lead to the O(T 2) mass shift of the octet baryons, althoughit should be tiny because of B1(mK/T) (∼0.06 at T = 150 MeV).To go beyond O(T 2), one needs more information on the pion structure functions (twist-4 effects), O(T 4) or higher T-dependence of the scalar condensates and the analysis of thestructure Π3 in (2.2) as well as more involved treatment for the phenomenological side (suchas octet →decuplet transitions).

These issues are beyond the scope of the present study. Ihope the lesson we learned through the demonstration in this work will be useful for moreadvanced studies on these higher order effects.AcknowledgementI would like to thank T. Hatsuda for illuminating discussion and useful comments onthe manuscript.

I’m also grateful to P. Danielewicz for useful discussions and reading themanuscript. Conversations with S. H. Lee, T. Matsui, S. Nagamiya and K. Yazaki are alsoacknowledged.

This work is supported in part by the US National Science Foundation undergrant PHY-9017077.13

AppendixHere we will present the nonfactorized form of the four-quark operators which appearin the OPE for the octet baryon currents. Nonscalar four-quark operators are associatedwith the twist-4 contribution in the pion structure function.

We ignore them since theybecome O(T 4) effect in the QCD sum rules at T ̸= 0. We decompose the contribution ofthe four-quark condensate into the two pieces corresponding to the two structures of thecorrelators:ΠN,Λ,Σ,Ξ4−quark(q) = ˜ΠN,Λ,Σ,Ξ1+ ˜ΠN,Λ,Σ,Ξ2/q.

(A.1)(i) N:˜ΠN1 (q)=0,(A.2)˜ΠN2 (q)=−1q21 −1Nc 52⟨¯uγµu ¯dγµd⟩+ 32⟨¯uγµγ5u ¯dγµγ5d⟩−1252⟨¯uγµλau ¯dγµλad⟩+ 32⟨¯uγµγ5λau ¯dγµγ5λad⟩+1 −1Nc 12⟨(¯uu)2⟩−12⟨(¯uγ5u)2⟩−14⟨(¯uγµγ5u)2⟩+ 14⟨(¯uγµu)2⟩−1212⟨(¯uλau)2⟩−12⟨(¯uγ5λau)2⟩−14⟨(¯uγµγ5λau)2⟩+ 14⟨(¯uγµλau)2⟩,(A.3)where λa is the SU(3) color matrix and we explicitly kept the Nc(= 3) dependence. As isseen from (A.3), the four-quark operators always appear in the form of1 −1Nc{¯qΓq¯q′Γq′ + ··} −12{¯qΓλaq¯q′Γλaq′ + ··}.We will henceforth use the abbreviation −12{with λa} to denote the second contribution.

(ii) Λ:˜ΠΛ1 (q)=−ms3q21 −1Nc n4⟨¯uu ¯dd⟩−⟨¯uσµνu ¯dσµνd⟩−4⟨¯ud ¯du⟩+ ⟨¯uσµνd ¯dσµνu⟩o−12 {with λa}+ ms3q21 −1Nc ⟨¯uu¯ss⟩+ ⟨¯dd¯ss⟩−12⟨¯uγµu¯sγµs⟩−12⟨¯dγµd¯sγµs⟩−12⟨¯uγµγ5u¯sγµγ5s⟩−12⟨¯dγµγ5d¯sγµγ5s⟩14

−12 {with λa},(A.4)˜ΠΛ2 (q)=−23q2121 −1Nc 52⟨¯uγµu ¯dγµd⟩+ 32⟨¯uγµγ5u ¯dγµγ5d⟩+ 54⟨¯uγµu¯sγµs⟩+34⟨¯uγµγ5u¯sγµγ5s⟩+ 54⟨¯dγµd¯sγµs⟩+ 34⟨¯dγµγ5d¯sγµγ5s⟩−14 {with λa}+1 −1Nc ⟨¯uu¯ss⟩−⟨¯uγ5u¯sγ5s⟩+ 12⟨¯uγµu¯sγµs⟩−12⟨¯uγµγ5u¯sγµγ5s⟩+⟨¯dd¯ss⟩−⟨¯dγ5d¯sγ5s⟩+ 12⟨¯dγµd¯sγµs⟩−12⟨¯dγµγ5d¯sγµγ5s⟩−12 {with λa}−121 −1Nc 52⟨¯uγµd ¯dγµu⟩+ 32⟨¯uγµγ5d ¯dγµγ5u⟩+ 14 {with λa}. (A.5)(iii) Σ:˜ΠΣ1 (q)=msq21 −1Nc −⟨(¯uu)2⟩+ ⟨(¯uγ5u)2⟩−12⟨(¯uγµu)2⟩+ 12⟨(¯uγµγ5u)2⟩−12 {with λa}+1 −1Nc{⟨¯uγµu¯sγµs⟩−⟨¯uγµγ5u¯sγµγ5s⟩} −12 {with λa},(A.6)˜ΠΣ2 (q)=−1q21 −1Nc 52⟨¯uγµu¯sγµs⟩+ 32⟨¯uγµγ5u¯sγµγ5s⟩−12 {with λa}+1 −1Nc 12⟨(¯uu)2⟩−12⟨(¯uγ5u)2⟩−14⟨(¯uγµγ5u)2⟩+ 14⟨(¯uγµu)2⟩−12 {with λa}.

(A.7)(iv) Ξ:˜ΠΞ1 (q)=msq21 −1Nc −3⟨¯uu¯ss⟩+ ⟨¯uσµνu¯sσµνs⟩+ 12⟨¯uσµνγ5u¯sσµνγ5s⟩−12 {with λa},(A.8)15

˜ΠΞ2 (q)=−1q21 −1Nc 52⟨¯uγµu¯sγµs⟩+ 32⟨¯uγµγ5u¯sγµγ5s⟩−12 {with λa}+1 −1Nc 12⟨(¯ss)2⟩−12⟨(¯sγ5s)2⟩−14⟨(¯sγµγ5s)2⟩+ 14⟨(¯sγµs)2⟩−12 {with λa}. (A.9)Factorizing these four-quark operators into the square of the chiral order parameter, one caneasily get the form of (2.9), (2.10) and the following (A.11)–(A.16) at T = 0.

To get the T-dependence of the four-quark condensates, we need to calculate the double commutators of(A.2)–(A.9) with the isovector axial charge. The calculation is tedious but straightforward.The following formulas are useful in carrying out the calculation (q = (u, d)):[F a5 , ¯qq]=−¯qγ5τ aq,[F a5 , ¯qγ5q]=−¯qτ aq,[F a5 , ¯qσµνq]=−¯qσµνγ5τ aq,[F a5 , ¯qσµνγ5q]=−¯qσµντ aq,[F a5 , ¯qγµq]=[F a5 , ¯qγµγ5q] = 0,hF a5 , ¯qτ bqi=−δab¯qγ5q,hF a5 , ¯qγ5τ bqi=−δab¯qq,hF a5 , ¯qσµντ bqi=−δab¯qσµνγ5q,hF a5 , ¯qσµνγ5τ bqi=−δab¯qσµνq,hF a5 , ¯qγµτ bqi=iǫabc¯qγµγ5τ cq,hF a5 , ¯qγµγ5τ bqi=iǫabc¯qγµτ cq.

(A.10)After calculating the double commutators, we end up with other four-quark operators. Ap-plying the factorization to these four-quark operators, we eventually got (2.9), (2.10) andthe following (A.11)–(A.16):ΠΛ1 (q, T)=112π2 4⟨¯uu⟩ 1 −ζ8!−⟨¯ss⟩!q2ln(Q2) + ms96π4q4ln(Q2)−4ms3q2 ⟨¯uu⟩2 1 −ζ6!+ 4ms9q2 ⟨¯uu⟩⟨¯ss⟩ 1 −ζ8!,(A.11)ΠΛ2 (q, T)=−164π4q4ln(Q2) + ms12π2 4⟨¯uu⟩ 1 −ζ8!−3⟨¯ss⟩!ln(Q2)16

−132π2⟨αsπ G2⟩ln(Q2) + 29q2⟨¯uu⟩2 1 −ζ2!−89q2⟨¯uu⟩⟨¯ss⟩ 1 −ζ8!,(A.12)ΠΣ1 (q, T)=14π2⟨¯ss⟩q2ln(Q2) −ms32π2q4ln(Q2) −4ms3q2 ⟨¯uu⟩2 1 −ζ6!,(A.13)ΠΣ2 (q, T)=−164π4q4ln(Q2) −ms4π2⟨¯ss⟩ln(Q2)−132π2⟨αsπ G2⟩ln(Q2) −2⟨¯uu⟩23q2 1 −ζ8!,(A.14)ΠΞ1 (q, T)=14π2⟨¯uu⟩ 1 −ζ8!−2msq2 ⟨¯uu⟩⟨¯ss⟩ 1 −ζ8!,(A.15)ΠΞ2 (q, T)=−164π4q4ln(Q2) −132π2⟨αsπ G2⟩ln(Q2) −2⟨¯ss⟩23q2 . (A.16)The T-dependence of (A.15) and (A.16) is the same as (3.14).

We also note the T-dependenceof the four-quark operators is different in the different channels.17

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Figure CaptionsFig. 1 π + N →N scattering term in which ηN couples to π directly.Fig.

2 π + N →N scattering term in which ηN couples to the nucleon that interacts withπ.19

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