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Theoretical Physics Institute

arXiv:hep-ph/9302300v1 24 Feb 1993Theoretical Physics InstituteUniversity of MinnesotaTPI-MINN-92/37-TTemperature Dependence of Electric and MagneticGluon CondensatesV.L. Eletsky†,Theoretical Physics Institute, University of MinnesotaMinneapolis, MN 55455P.J.

Ellis and J.I. KapustaSchool of Physics and Astronomy, University of MinnesotaMinneapolis, MN 55455AbstractThe contribution of Lorentz non-scalar operators to finite temperature correlation func-tions is discussed.

Using the local duality approach for the one-pion matrix element of aproduct of two vector currents, the temperature dependence of the average gluonic stresstensor is estimated in the chiral limit to be ⟨E2 + B2⟩T = π210bT 4. At a normalization pointµ = 0.5 GeV we obtain b ≈1.1.Together with the known temperature dependence ofthe Lorentz scalar gluon condensate we are able to infer ⟨E2⟩T and ⟨B2⟩T separately in thelow-temperature hadronic phase.†Permanent address: Institute of Theoretical and Experimental Physics, Moscow 117259,Russia.

Correlators of currents with the quantum numbers of hadrons are known to be useful toobtain information about the masses and couplings of hadrons; they are employed in theQCD sum rule approach and in lattice calculations. In both approaches the correlators areconsidered at large Euclidean distances or imaginary times where the dominant contributioncomes from the lowest state with the corresponding quantum numbers.

QCD sum rules givepredictions also for form factors and structure functions of hadrons. (For a recent review ofapplications of QCD correlation functions see ref.

[1]. )In recent years there has been increasing interest in finite temperature QCD and hadronicphysics due to the expectation that at high enough temperatures the QCD vacuum, spec-ified by nonperturbative condensates of quark and gluon fields, will “melt” and undergo atransition to a quark-gluon plasma.

Melting is usually understood in the sense that chiralsymmetry restoration and deconfinement take place. The former means that with increasingtemperature quark condensates evaporate, while the latter means that hadrons do not rep-resent stable degrees of freedom.

It was shown by Leutwyler and his collaborators [2] usingthe chiral Lagrangian approach that the quark condensate indeed decreases with rising tem-perature. From the usual QCD sum rules at T = 0 it is well known that the properties ofhadrons are, to a large extent, determined by nonperturbative quark and gluon condensates[3].

Naturally, a large number of papers were devoted to the generalization of QCD sumrules to finite temperature in attempts to relate the temperature dependence of the hadronicspectrum to the temperature dependence of the condensates (see, e.g. [4, 5, 6]).

In this casethe vacuum average of the product of currents becomes the Gibbs average over the thermalensemble. To calculate the Gibbs average one must choose a basis for the states.

As arguedin refs. [5, 7] at temperatures which are much less than the energy scale of confinement theappropriate basis is that of hadronic states, rather than the quark-gluon basis used in earlypapers on the subject (see, e.g.

ref. [4]).

Using this basis it was also shown [5] that at lowT the thermal correlators are expressed as a mixture of zero-temperature correlators withdifferent parity. It is also clear that if the operator product expansion (OPE) is applied to athermal correlator then the temperature dependence appears only in the matrix elements ofthe operators (condensates), the coefficient functions being obtained through a perturbativecalculation at T = 0.

QCD sum rules at low temperature were recently reexamined alongthese lines in ref. [8].1 At high temperatures, corresponding to the quark-gluon plasma, thecalculation of thermal correlators should be performed in a basis consisting of quark andgluon states.

In this case the perturbative temperature-dependent parts of the condensatesdue to quarks and gluons from the thermal ensemble may be included in the coefficient1We thank T. Hatsuda for drawing our attention to this paper.1

functions [6].Thus the QCD sum rule method, understood as a tool to get information about theimaginary parts of correlators via analyticity, seems to be tractable both at very low and veryhigh temperatures, but not in the region of a phase transition where a drastic rearrangementof the spectrum takes place.An additional feature of finite temperature sum rules is the appearence of new conden-sates due to Lorentz non-scalar operators; these were, of course, present in the OPE, butgave zero contribution when averaged over the vacuum. At finite temperatures Lorentz in-variance is broken and these operators should contribute [1, 6].

The same applies to the caseof finite density [9]. However, each of these new condensates is an unknown nonperturbativeparameter.

In principle they may be fixed from the physical spectral densities of the corre-lators, just as in the zero temperature case the now well-established condensates were fixedby the hadronic spectrum.Consider the correlator of two isovector vector currents at finite temperature T andeuclidean momentum q, where T 2 ≪Q2 = −q2 and Q2 >∼1 GeV2:iZd4xeiqx Xn⟨n|T jµ(x)jν(0)e(Ω−H)/T |n⟩= (gµνq2 −qµqν)C1(q, T) + utµutνC2(q, T) ,(1)where T denotes a time-ordered product, jµ = 12(¯uγµu −¯dγµd), utµ = uµ −(u · q)qµ/q2 is thetransverse part of the heat bath four-velocity uµ and Ω= −T log(Pn⟨n|e−H/T |n⟩). Eq.

(1) isthe most general expression compatible with conservation of the vector current. The Lorentzinvariance breaking term proportional to utµutν must be absent at T = 0.

This means thatC2(q, T) goes to zero as T →0, while C1(q, T) becomes the usual zero temperature correlator.Notice that eq. (1) may be considered to be the amplitude for forward scattering of a virtualphoton by the heat bath.

Then the imaginary parts of C1 and C2 are the structure functionsof deep inelastic scattering of leptons by the heat bath (uTµ is similar to the transversecomponent of the target momentum, pµ −(p · q)qµ/q2).At low T, when the contributions from all particles except pions are exponentially sup-pressed in the Gibbs average, the functions C1 and C2 may be estimated by expanding inthe density of thermal pions. In the first order of this expansion only matrix elements overone-pion states are taken into account.

This approximation was made in ref. [5] for C1.The one-pion matrix elements were estimated via PCAC and current algebra.

It was shownthat C1 and its counterpart from the axial channel are given by T-dependent mixtures oftheir zero temperature values and, as a result, the corresponding screening lengths tend toconverge with increasing temperature.The purpose of the present letter is to estimate C2. Let us start from the one-pion matrix2

element in the chiral limitiZd4xeiqx⟨π(p)|T jµ(x)jν(0)|π(p)⟩,(2)where we assume p ∼T ≪Q, since eq. (2) is to be integrated over p with Bose occupa-tion probabilities.

If ˆOµ1µ2...µn is an operator of Lorentz spin n, then the matrix element⟨π(p)| ˆOµ1µ2...µn|π(p)⟩∝pµ1pµ2 . .

. pµn, and cannot be reduced via PCAC to a vacuum matrixelement.

It is clear that at low temperatures, T ≪Q, the main contribution to C2 comesfrom operators of lowest spin, namely spin 2. In leading twist there are two spin 2 operatorswhich are related to the energy-momentum tensor:θqµ1µ2 = 12i(¯qγµ1Dµ2q + ¯qγµ2Dµ1q),q = u, d, s .

. .θGµ1µ2 = Gaµ1αGaαµ2 −14gµ1µ2GaβαGaαβ ,(3)where Dµ is the covariant derivative.

Graphs which correspond to the contributions of theseoperators to the matrix element in eq. (2) are shown in fig.

1. If the normalization pointfor the operators is taken to be µ2 = Q2, then the operator θGµ1µ2 does not contribute to theOPE in the leading log approximation, and the contribution of twist 2, spin 2 operators toeq.

(2) involves1Q2⟨π(p)|θuµν + θdµν|π(p)⟩= 1Q2⟨π(p)|θtotµν −θGµν|π(p)⟩. (4)Here we neglected the contributions of heavy quarks.The matrix element of the totalenergy-momentum tensor is ⟨π(p)|θtotµν |π(p)⟩= 2pµpν (the states are normalized such that⟨π(p)|π(p′)⟩= (2π)32Eδ(3)(p−p′)), while the matrix element of the gluon energy-momentumtensor⟨π(p)|θGµν|π(p′)⟩= bpµpν(5)contains an unknown constant b.

This constant is related to the matrix element of the energydensity of the gluon fieldb =12p2⟨π(p)|E2 + B2|π(p)⟩µ=Q . (6)Note that b depends on the normalization point, µ, in the operator product expansion.

Thisdependence will be discussed later.Let us try to estimate b within a quark-hadron duality approach, saturating the amplitudeof eq. (2) by hadrons, ⟨π|T jµ(x)jν(0)|π⟩= Pn⟨π|jµ(x)|n⟩⟨n|jν(0)|π⟩.

Focussing on spin 2contributions to C2, we then have2 −bQ2+ cQ4 + . .

. = 1πZ ∞0dsρ(s)F 2n(Q2)s + Q2,(7)3

where Fn(Q2) is the part of the form factor ⟨π(p)|jµ|n(p + q)⟩proportional to pµ and ρ(s) isthe spectral density in the s-channel. The states |n⟩are normalized as in eq.

(5), the n-statecontribution to ρ(s) being πδ(s −m2n). On the l.h.s.

of eq. (7) the term c/Q4 denotes thecontribution of three different spin 2, twist 4 operators [10] whose individual contributionscannot be separated.

The constants b and c are considered as parameters to be fitted. Theellipsis in eq.

(7) corresponds to spin 2 terms of higher twist. Note that eq.

(7) is justthe sum rule for the second moment of the deep inelastic structure function,R 10 F2(x, Q2)dx,divided by Q2. It is valid in the asymptotic region, Q2 →∞, with all higher states in thes-channel equally important in this region.

Our goal here is to see whether eq. (7) can besatisfied in a region of intermediate Q2 ∼1 GeV2 where the r.h.s.

may be approximated bythe contribution of a few low-lying states2.First consider the case of charged pions. The lowest states in the s-channel are the π anda1(1260) mesons.

Assuming ρ-dominance for the form factors (which is known to be a goodapproximation for the pion form factor up to Q2 ≃2 GeV2), ⟨π|jµ|n⟩= −m2ρgρ ερµ⟨πρ|n⟩(Q2 +m2ρ)−1, where ερµ is the ρ-meson polarization vector and g2ρ/4π ≃2.9. We obtain for the r.h.s.of eq.

(7)8m4ρ(Q2 + m2ρ)2 1Q2 +14m2a1g2ρ(Q2 + m2a1)ng2a1ρπ + ga1ρπha1ρπ(Q2 −m2a1)+ 14h2a1ρπ(Q2 + m2a1)2o. (8)Here we used⟨π+(p)ρ0(q)|π+(p + q)⟩= gρππ ερ∗· (2p + q)(9)andi⟨π+(p)ρ0(q)|a+1 (p + q)⟩= ga1ρπ ερ∗· εa1 + ha1ρπ ερ∗· (p + q) εa1 · p .

(10)The notation corresponds to that of ref. [13].

Note that gρππ = gρ within the ρ-dominanceapproach. The couplings ga1ρπ and ha1ρπ cannot, of course, be determined from the a1 widthalone.

To this end we use an effective chiral Lagrangian with spin 1 mesons [12, 13, 14].In this approach the constants in question are expressed in terms of parameters of this2In ref. [11] the transverse photon structure function in the region of intermediate x was calculatedstarting from the V V V V four-point correlation function, using the OPE in the photon virtuality p2 andextrapolating to p2 = 0.

One could think of doing the same thing for the pion structure function, startingfrom the AV V A correlator. It can be shown, however, that just as in the case of the longitudinal photonstructure function, there are difficulties in the extrapolation to on-shell pions.

The AV V A box diagram alsocannot be used, via a triple dispersion relation, to model the continuum contribution to the real part of theforward scattering amplitude in the usual manner because of the zero momentum transfer in the t-channel.4

Lagrangian which are fitted to reproduce masses and widths. The Lagrangian in questioncontains a massive Yang-Mills part and two higher derivative termsLAV φ=−12Tr(F LµνF Lµν + F RµνF Rµν) + m20Tr(ALµALµ + ARµ ARµ)−iξTr(DµUDνU†F Lµν + DµU†DνUF Rµν) + σTrF LµνUF RµνU†,(11)where U = exp(2iφ/Fπ), φ = φaτ a/√2, ALµ =12(Vµ + Aµ), ARµ =12(Vµ −Aµ), F L,Rµν=∂µAL,Rν−∂νAL,Rµ−ighAL,Rµ, AL,Rνiand the covariant derivative DµU = ∂µU −igALµU +igUARµ .The quadratic piece of this Lagrangian is non-diagonal in ∂µφ and Aµ.

After diagonalizationthe physical masses are given bym2V = m2ρ =m201 −σ;m2A = m2a1 =11 + σ m20 + g2F 2π4!,(12)and Fπ is related to the physical coupling ˜Fπ = 135 MeV through˜Fπ = ZFπ ,Z2 = 1 −g2 ˜F 2π4m20= 1 −σ1 + σm2Vm2A. (13)The couplings ga1ρπ, ha1ρπ and gρππ are expressed through g, ξ, σ and the meson masses byha1ρπ = −2Z2˜Fπ21 −σ212 (σ + gξ) ,(14)ga1ρπ = 12(m2V + m2A −m2π)ha1ρπ + m2V˜Fπ21 −σ212 h(1 −σ)(1 −Z2) + 2gξZ2i,(15)gρππ =gq2(1 −σ)"1 −12(1 −Z2) +gξ(1 −σ)Z4(1 −Z2)#.

(16)Here we retain a non-zero mass for the pion for the purposes of fitting the coupling constants.The widths are expressed through these couplings in the following way3Γρ→ππ =16πm2ρ|qπ|3g2ρππ ,(17)andΓa1→ρπ =|qπ|12πm2a12g2a1ρπ + Eρmρga1ρπ −ma1mρ|qπ|2ha1ρπ!2. (18)With the four available parameters g, σ, ξ and m0 it is possible to fit both the masses andthe widths of the ρ and the a1 [14].

We have refitted these parameters using a recent value3We note that the minus sign in eq. (18) is correct in contrast to refs.

[13, 14] which are written with anincorrect plus sign.5

of the width, Γa1 = 400 MeV [15, 16]. There are two possible solutions:(A)σ = 0.340,ξ = 0.446,g = 8.37 ,(B)σ = −0.291,ξ = 0.0585,g = 7.95 .

(19)which correspond to(A)ga1ρπ = −5.42 GeV,ha1ρπ = −16.7 GeV−1,γ = 0.52 ,(B)ga1ρπ = 4.25 GeV,ha1ρπ = −2.05 GeV−1,γ = 0.33 . (20)Here the quantity γ is the ratio of polar- and axial-vector contributions to radiative piondecay.Both solutions are reasonably consistent with the positive experimental value of∼0.4 discussed by Holstein [13].

However it can be shown that the opposite-sign solution,(B), is excluded by the QCD sum rule estimates of Ioffe and Smilga [17] for the two formfactors entering the non-diagonal matrix element ⟨a1|jµ|π⟩. They use couplings g1 and g2 toparameterize these form factors in a ρ-dominance approach, and the relation to ga1ρπ andha1ρπ is given byga1ρπ = g1ma1,ha1ρπ =2ma1"g1 + m2ρm2a1g2#.

(21)While the absolute values of g1 and g2 obtained in ref. [17] contain large uncertainties, theyare definitely of the same sign, thus ruling out solution (B).

Therefore we choose the like-signsolution (A).We shall display our results for the r.h.s. of eq.

(7) multiplied by Q4, which accordingto the l.h.s should give the linear relation (2 −b)Q2 + c. The results from eq. (8) are givenby the dashed line for the charged pion case in fig.

2. It is seen that there is a good lineardependence for Q2 ≥0.9 GeV2 .

We cannot use values of Q2 larger than plotted in thefigure since higher states, which are not accounted for, become important and ρ-dominanceis not applicable either. There is an excited pion state π∗(1300) which may contribute forthe values of Q2 in question.

Its coupling to ρπ defined through ⟨π∗|ρ(q)π(p)⟩= g∗ερ ·p maybe roughly estimated using the rather uncertain data [15] on the width, Γtotπ∗= 200 −600MeV and Γπ∗→πρ = 13Γtotπ∗. This gives g∗≈5.

The contribution of the π∗to the r.h.s. of eq.

(7) is then2g∗2m4ρg2ρ(Q2 + m2ρ)2(Q2 + m2π∗) . (22)The result of taking into account the π∗is shown in fig.

2 by the full line. It is clear thatthe effect of the π∗is quite small.

By fitting a straight line to the curve we estimate b = 1.14and c = 1.14 GeV2.6

The matrix element of the gluon field energy density, eq. (6), must be the same forcharged and neutral pions.This may be used to check our calculation.So, let us nowconsider the case of neutral pions.

Isotopic spin invariance forbids the π0 and a01 mesons inthe s-channel so the lowest allowed state is the ω meson. The ωρπ vertex has the the formi⟨π(p)ρ(q)|ω(p + q)⟩= gωρπǫαβστ εωαερβpσqτ ,(23)where εωα and ερβ are the polarization vectors of the ω and ρ mesons.

Then the contributionof the ω to the r.h.s. of eq.

(7) is2m4ρQ2(Q2 + m2ρ)2(Q2 + m2ω) gωρπgρ!2. (24)To be consistent we should use the value of the coupling constant gωρπ obtained from the de-cay ω →πγ using ρ-dominance [18], gωρπ ≃14.9 GeV−1.

The corresponding Q2 dependenceof eq. (24) (multiplied by Q4) is shown in fig.

2 by the dashed curve for the neutral case.There is, however, an excited state, ω∗(1390), which can contribute. The dominant decaymode is to the ρπ channel and taking this to account for the full width of 230 ± 40 MeV[15], we deduce a coupling constant gω∗ρπ = 5.29 GeV−1.

The result of including both theω and ω∗is shown by the full line in fig. 2.

There is a noticable curvature and a linear fitin this case results in larger uncertainties: b = 1 −1.2 and c = 0 −0.3 GeV2. While thevalue for b agrees with the one obtained from charged pions, it is clear that the interceptc is different.

This should have been expected since c involves the contributions of quarkoperators and their averages over charged and neutral pions need not be the same.Thus, we adopt the value b ≃1.14, corresponding to a normalization point µ ∼Q ∼1GeV. This is in good agreement with the value b = 1.03 deduced from the analysis of ref.

[8] in which the matrix element ⟨π|θu+dµν |π⟩was extracted from a fit [19] to the quark andgluon distribution functions in the pion. In the leading log approximation the dependenceon the normalization point is determined by the renormalization group.

However, as is wellknown [20], operators of the same twist get mixed under renormalization due to radiativegluon corrections. The diagonal combinations in the case of two quark flavors areθtotµν = θuµν + θdµν + θGµν[0] ,Rµν = θuµν + θdµν −38θGµνh−4487i,∆µν = θuµν −θdµνh−3287i.

(25)The numbers in parentheses are the anomalous dimensions γ of the corresponding diagonaloperators which are renormalized multiplicatively,ˆOQ = κγ ˆOµ,κ = αs(µ2)αs(Q2) = log(Q/ΛQCD)log(µ/ΛQCD) ,(26)7

where ΛQCD ≈150 MeV. Then the evolution of b, defined by eq.

(5), under a change of thenormalization point is given byb(µ) = 16111 −κ44/87+ b(Q)κ44/87 . (27)It can be seen that according to eq.

(27) b decreases with µ and becomes zero at µ = 1.1ΛQCD.At the standard normalization point used in QCD sum rules, µ = 0.5 GeV, we get b = 1.06.Note that the small value of the normalization point for which b = 0 (meaning that there isno gluon component in the pion) agrees with the results of ref. [21] where it was shown thata quark model description of deep inelastic scattering of leptons on nucleons is consistentwith experimental data provided µ ≈mπ.Coming back to finite temperatures, the temperature dependence of the condensate ⟨E2+B2⟩is determined by the integral over the thermal pion phase space⟨E2 + B2⟩T = 3bZd3p(2π)3|p|exp(|p|/T) −1 = bπ210 T 4 ,(28)where the factor of 3 in front of the integral accounts for the three charged states of pions.The structure function C2 in eq.

(1) is obtained in the same wayC2(Q, T) = π2T 410Q2 2 −b + ¯cQ2 + . .

. !+ O T 6Q4!.

(29)where ¯c = 23ccharged + 13cneutral ≈23 is the charge averaged value of the constant c.Let us now briefly summarize what is known about behavior of condensates at low tem-peratures in the chiral limit. The temperature dependence of the usual (Lorentz scalar)condensates at low T was considered on the basis of chiral perturbation theory up to three-loop order [2].

The low T expansion of the quark condensate begins with a term of orderT 2/F 2π, because for pions with zero momentum the matrix element ⟨π|¯qq|π⟩is non-zero andproportional to ⟨0|¯qq|0⟩/F 2π. In the case of the operator GaµνGaµν, which is a chiral singlet,the one-pion matrix elements vanish.

The T dependence of the gluon condensate is relatedthrough the trace anomaly to ⟨θµµ⟩T. The first non-zero contribution to this matrix elementappears only at the three-loop level.

As a result, the T dependence of the gluon condensatebegins at order T 8/F 4π,⟨αsπ G2µν⟩T = ⟨αsπ G2µν⟩0 −4π23645N2f (N2f −1)T 8F 4πlog ΛpT −14+ . .

.,(30)where Λp ≃275 MeV is a scale encountered in the three-loop calculation of the pressure of ahot pion gas within chiral perturbation theory [2]. The sign of this contribution corresponds8

to the melting of the gluon condenate with rising temperature. However, this melting ismuch slower than in the case of the quark condensate, and ⟨G2⟩T is practically constant upto T ∼150 MeV, that is, in the region of applicability of the approximation of a hadronicgas.One-pion matrix elements of Lorentz non-scalar operators cannot be estimated using thesoft pion approach, because they are proportional to the pion momentum p. Since p ∼T, thecorresponding condensates naturally vanish as T →0.

Since ⟨B2−E2⟩T ≃⟨B2−E2⟩0, we getfrom eq. (28) the T dependence of the condensates of chromomagnetic and chromoelectricfields,⟨B2⟩T = ⟨B2⟩0 + bπ220 T 4 ,⟨E2⟩T = ⟨E2⟩0 + bπ220 T 4 ,(31)where ⟨B2⟩0 = −⟨E2⟩0 ≃2 × 10−2 GeV4, using a renormalization scale µ = 0.5 GeV.

Weindicate the predicted ratios ⟨B2⟩T/⟨B2⟩0 and ⟨E2⟩T/⟨E2⟩0 by the dashed curves in fig. 3.

Itis seen that the T dependence is rather weak at low T and, at T ∼150 MeV, the condensatesare changed from their T = 0 value by only about 1%. The fact that the change is small isqualitatively consistent with the results extracted from the lattice data [6], however we donot agree with the lattice predictions for the sign.

We suggest that the lattice calculationsare probably not sufficiently accurate to predict such small effects.We notice that keeping mπ finite would not affect the values of b and c within the ac-curacy of our approach. The only differences would appear in the integral over the thermaldistribution function, eq.

(28), and in a lower order contribution to eq. (30).

It is straight-forward to perform the calculation numerically and this results in the full curves shown infig. 3.

We observe that eq. (31) is a good approximation, indeed for the electric field theresults are indistinguishable.

We remark that at very low T, T ≪mπ, we have for µ = 0.5GeV⟨B2⟩T = ⟨B2⟩0 −0.033m5/2π T 3/2e−mπ/T ,⟨E2⟩T = ⟨E2⟩0 + 0.20m5/2π T 3/2e−mπ/T(32)The numerical effect is exceedingly small, but it is interesting to observe that the magneticcondensate ⟨B2⟩T slightly decreases at very low T before increasing. The behavior of ⟨E2⟩Tis, however, monotonic.Finally we briefly comment on the effects of higher spin and twist operators.

The averagesof Lorentz non-singlet operators of spin larger than 2 are necessarily proportional to higherpowers of T and their contribution to thermal correlators will be suppressed by powersof T 2/Q2. The operators of spin 2, but of higher twist, are suppressed by µ2h/Q2, whereµh is some hadronic mass scale ∼ΛQCD.In the case of vector currents three twist 4,spin 2 operators [10] contribute to the constant c in eq.

(7) and to disentangle individual9

contributions some extra information must be used. In our opinion, this problem deservesfurther consideration.We acknowledge useful discussions with A. Gorski, M. Shifman, E. Shuryak, C.S.

Songand A. Vainshtein. V.E.

would like to thank the staffof the Nuclear Theory Group andTheoretical Physics Institute, especially Larry McLerran, for the warm hospitality extendedto him during his stay at the University of Minnesota. This work was supported in part bythe Department of Energy under contract DE-FG02-87ER40328.References[1] E.V.

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Figure Captions:• Fig. 1: Diagrams contributing to (a) ⟨π|θqµ1µ2|π⟩and (b) ⟨π|θGµ1µ2|π⟩.

The dashed linescorrespond to gluons.• Fig. 2: The r.h.s.

of eq. (7) multiplied by Q4 shown as a function of Q2.

In the case ofcharged pions, the dashed curve is obtained with π- and a1-meson intermediate statesand the full curve also includes the π∗meson. In the case of neutral pions, the dashedcurve is obtained with an ω-meson intermediate state and the full curve also includesthe ω∗meson.• Fig.

3: The curves labelled B and E give, repectively, ⟨B2⟩T/⟨B2⟩0 and ⟨E2⟩T/⟨E2⟩0as a function of temperature. Normalization point is µ = 0.5 GeV.

The dashed curvesgive the results for zero pion mass and the full curves correspond to non-zero pionmass. In case E these two curves are indistinguishable.12

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출처: arXiv:9302.300 • 원문 보기