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COVARIANT GAUGES AT FINITE TEMPERATURE

arXiv:hep-ph/9205235v2 18 Jun 1993CERN-TH-6491/92ENSLAPP-A-383/92COVARIANT GAUGES AT FINITE TEMPERATUREP V LandshoffCERN, Geneva†A Rebhan‡Laboratoire de Physique Th´eorique ENSLAPP ∗, Annecy-le-VieuxAbstractA prescription is presented for real-time finite-temperature perturbation theory in co-variant gauges, in which only the two physical degrees of freedom of the gauge-fieldpropagator acquire thermal parts. The propagators for the unphysical degrees of free-dom of the gauge field, and for the Faddeev-Popov ghost field, are independent oftemperature.

This prescription is applied to the calculation of the one-loop gluon self-energy and the two-loop interaction pressure, and is found to be simpler to use thanthe conventional one.ENSLAPP-A-383/92CERN-TH-6491/92May 1992 (corrected June 1993)† On leave from DAMTP, University of Cambridge‡ On leave from ITP, Technical University of Vienna; boursier du Minist`ere des AffairesEtrang`eres∗URA 14-36 du CNRS, associ´ee `a l’ENS de Lyon, et au LAPP d’Annecy-le-Vieux

1. INTRODUCTIONIn finite-temperature field theory, one calculates a grand partition functionZ =Xi⟨i |e−βH| i ⟩(1.1a)and thermal averages⟨Q⟩= Z−1 Xi⟨i |e−βHQ| i ⟩.

(1.1b)In both cases, the sum is over a complete orthonormal set of physical states. In scalar field theory, thesestates span the whole Hilbert space, and so one may replace the sums with traces of the operators.But in the case of a gauge theory the Hilbert space contains also unphysical states and so things aremore complicated.In this paper, we reconsider what to do about this for the case of covariant gauges.

The traditionalsolution[1] is, in effect, to sum over all states of the gauge particles, and then cancel the unphysicalcontributions with ghosts. We shall argue that, for calculational purposes, it may be simpler to includein the sums (1.1) only the physical states from the start, and not to introduce unphysical terms thathave to be cancelled.The finite-temperature propagator consists of a vacuum piece, plus a “thermal” part.

In real-timeperturbation theory, which we consider initially, it is a 2 × 2 matrix[2]. Its vacuum piece isiDµν(x) =⟨0|TAµ(x)Aν(0)|0⟩⟨0|Aµ(−iσ, 0)Aν(x)|0⟩⟨0|Aµ(x0 −iσ, x)Aν(0)|0⟩⟨0| ¯T Aµ(x)Aν(0)|0⟩(1.2)with ¯T the anti-time-ordering operator, and 0 ≤σ ≤β.

There is a similar matrix ghost-field propa-gator. The elements of the thermal part of the matrix take account of all the possible states of theheat bath and involve the Bose distribution.

In the traditional approach to covariant gauges, all thecomponents of the gauge-field propagator have a thermal part, as also does the ghost propagator.We shall argue that it is simpler to give only the propagators for the physical degrees of freedom athermal part, so that the unphysical components of the gauge field, and also the ghost field, have onlyvacuum propagators. In the next section, we give a derivation of this from the Gupta-Bleuler methodof quantising the gauge field, though in a sense the result is obvious and can be written down withoutany long derivation.This is because the thermal part of a propagator contains a δ-function that puts the correspondingparticle on shell.

The thermal part represents the absorption of a particle from the heat bath or theemission of one into it. That is, the thermal part is directly associated with the particles in the physicalstates | i ⟩summed in (1.1).

Consequently only the physical degrees of freedom acquire thermal partsfor their propagation.The thermal part of the propagator for the gauge field therefore has the tensor structure2Xλ=1eµλ(k)e∗νλ (k)(1.3a)where eλ(k) are polarisation vectors and k2 = 0. There is some freedom in the choice of these, butsince the heat bath breaks the Lorentz symmetry already, the obvious choice is the one that causes noadditional Lorentz-symmetry breaking.

So we choose polarisation vectors orthogonal to the 4-velocityu of the heat bath, and then (1.3a) isT µν(k, u) = −gµν + kµuν + uµkνu.k−u2 kµkν(u.k)2 . (1.3b)In the frame where the heat bath is at rest this vanishes if either µ or ν is zero, and its other elementsareT ij(k) = δij −kikjk2 .

(1.3c)

To summarise what we have said, part of the finite-temperature propagator of the gauge field is thevacuum piece, which in Feynman gauge reads−gµν1/(k2 + iǫ)−2πi δ(−)(k2)−2πi δ(+)(k2)−1/(k2 −iǫ). (1.4a)We have not written explicitly the colour factor δab.

We have made the particular choice σ = 0 in(1.2) so that the vacuum part of the propagator is temperature-independent; for other choices of σ itdepends on β, but in a simple way. To (1.4a) must be added the thermal piece of the propagator:−iT µν(k) 2πδ(k2) n(|k0|)1111,(1.4b)where n(x) = 1/(eβx −1).There is also the ghost propagator; this has only the vacuum part, which is just equal to the matrixthat appears within (1.4a).

We derive the propagator (1.4) in section 2 and generalise it to othergauges, both covariant and noncovariant.In order to show that this prescription can be simpler than the conventional one, we use it in section 3to calculate the gluon self-energy and the plasma pressure to lowest non-trivial order in the coupling.Section 4 is a discussion of our results. In an Appendix we introduce some alternative versions of theformalism in which the 2 × 2 propagator matrix is diagonal and temperature-independent, and thedependence on the temperature is instead transferred to the interaction vertices.2.

GUPTA-BLEULER THEORY AT FINITE TEMPERATURETo evaluate the grand partition function (1.1a) or a thermal average (1.1b), we need to express theHamiltonian in operator form and to identify the physical states. For this, we use standard Gupta-Bleuler theory[3].

We go through the details only for the case of the Feynman gauge.Start with the standard QCD Lagrangian −14F 2, with the addition of a gauge-fixing term but initiallyno ghost part:L = −14F µνFµν −12(∂.A)2(2.1)The canonical momenta conjugate to the space components Ai of the gauge field areπi = F 0i,(2.2a)while that conjugate to A0 isπ0 = −∂.A . (2.2b)The Hamiltonian density isH = π0 ˙A0 + πi ˙Ai −L = H0 + HINT.

(2.3)Here, H0 is the part of H, expressed as a function of the fields and the canonical momenta with noexplicit time derivatives of the fields, that survives if the coupling g is set equal to zero, andHINT(π, A) = −gπi.A0 ∧Ai + 14(F ijF ij)INT. (2.4)To derive perturbation theory, one introduces interaction-picture operators QI(t) = Λ(t)Q(t)Λ−1(t),where Λ(t) = eiH0I(t−t0)e−iH(t−t0).

These operators evolve with time as free fields: they obey Hamil-ton’s equations of motion with Hamiltonian H0I and the relation between the interaction-picturecanonical momenta and the fields is as in (2.2) but with the coupling g set equal to 0:πiI = ˙AiI −∂iA0I,π0I = −∂.AI. (2.2c)

The interaction-picture field equations are just the zero-mass Klein-Gordon equations. Their solutionsareAµI (x) =Zd3k(2π)312|k|e−ik.xaµ(k) + h c(2.5)where k0 = |k|.

The canonical equal-time commutation relations [AµI (t, x), πνI (t, y)] = iδµνδ(3)(x −y)etc give[aµ(k), aν†(k′)] = −gµν2|k|(2π)3δ(3)(k −k′),[aµ(k), aν(k′)] = [aµ†(k), aν†(k′)] = 0. (2.6)We now have to identify the physical states.

For finite values of t0, the time at which the interactionpicture coincides with the Heisenberg picture, the constraint that they satisfy is non-trivial[4]. How-ever, to derive zero-temperature perturbation theory, or finite-temperature field theory in the real-timeformalism, one may switch offthe gauge coupling adiabatically and work with the limit t0 →−∞.Then the interaction-picture states are just the in-states and the condition that picks out the physicalstates is the same as in abelian gauge theory.

Its most general form is that matrix elements of ∂.AIvanish between physical states, but we may impose the stronger condition[3]a0(k) | phys, in ⟩= 0 = k.a(k) | phys, in ⟩. (2.7)In particular, the vacuum is required to satisfy these conditions.

Together, (2.5), (2.6) and (2.7) makethe vacuum expectation values of T-products of the fields take the usual Feynman-gauge-propagatorforms.The time-development operator for the interaction-picture states away from t = t0 is Λ(t). By con-verting the differential equation it satisfies into an integral equation and iterating, one finds as usualthat the S-matrix is a time-ordered exponential of an integral of HINT(AI, πI).

HINT is given in (2.4).At this stage we may replace the πI with their expressions (2.2c) in terms of AI and ˙AI. This givesHINT = −LINT(AI, ˙AI) + 12g2(A0I ∧AiI)2.

(2.8)The last term here compensates for the presence of g ˙AiI.A0I ∧AiI in the interaction Lagrangian[5].The Hamiltonian formalism of perturbation theory that we have outlined requires that, whenthe˙AiI field propagates between two such neighbouring vertices one should use the propagator⟨0|T ˙AiI(x1) ˙AjI(x2)|0⟩. However, the usual Feynman rules use instead the double time derivative ofthe AI-field propagator, where the two time differentiations are applied also to the θ-functions thatappear in the definition of the T-product.

The last term in (2.8) is then simply omitted[5].From the differential equation satisfied by Λ(t) it is straightforward to show that the S-matrix isunitary. However, this does not imply that probability is conserved, because the completeness relationincludes the unphysical states.In order to achieve probability conservation one must add to theLagrangian (2.1) a ghost part.

This then guarantees that physical initial states do not scatter intounphysical final states. One may show this directly in the operator formalism[4], though we shall becontent here to accept the usual Faddeev-Popov path integral arguments as justification.We now extend the perturbation theory to nonzero temperature.

The same operator Λ enters again,now with complex argument. Simple algebra gives[6]Z =Xi⟨i |e−βH0IΛ(t0 −iβ)Λ−1(t0)| i ⟩⟨Q(t)⟩= Z−1 Xi⟨i |e−βH0IΛ(t0 −iβ)Λ−1(t)QI(t)Λ(t)Λ−1(t0)| i ⟩(2.9)where t0 must be the time at which the interaction picture coincides with the Heisenberg picture andthe sum is still over all physical states, which we now choose to be interaction-picture in-states.

In order now to derive the Feynman rules, it is necessary first to check that Wick’s theorem may beused as usually in thermal field theory. For this, note that the physical-gluon operators commute withthe unphysical gluon and ghost fields.

Also, H0I is a sum of commuting parts containing respectivelyonly physical and unphysical operators, H0I = HPHYS0I+HUNPHYS0I, and HUNPHYS0Ihas zero eigenvaluein the physical states. Hence any unphysical operators in the product of operators in (2.9) may befactorised out: only the vacuum expectation value of their product appears, and the normal zero-temperature Wick’s theorem applies to this.

For the remaining factor, we may replace e−βH0I bye−βHPHYS0I; the normal finite-temperature Wick’s theorem applies to this factor.So the real-time finite-temperature perturbation theory is much as usual[2], except that for the un-physical fields the propagators are just vacuum expectation values (1.2), both for the unphysicalcomponents of A and for the ghost.The gauge-field propagator in Feynman gauge therefore readsDµν = −gµν∆Feσk0θ(−k0) (∆F −∆∗F )e−σk0θ(k0) (∆F −∆∗F )−∆∗F+ Tµν (∆F −∆∗F )1eβ|k0| −11eσk0e−σk01,(2.10)with 0 ≤σ ≤β, and ∆F = 1/(k2 + iǫ). The choice σ = β/2 is usually made on grounds that thismakes the real-time propagator a symmetric 2×2 matrix.

Here this is the case only for the transverseprojection T µρDρσT σν. A more natural choice is σ = 0, for it renders the vacuum piece independentof β.

In the Appendix, we discuss various options for the diagonalisation of (2.10).Although the derivation was done in Feynman gauge, it seems obvious how the resulting propagatorwill read in other gauges. In a general linear gauge with quadratic gauge breaking term12ξ(AµfµfνAν),where fµ is a 4-vector which is either constant or constructed from derivatives ∂/∂x, the vacuum piecegeneralises bygµν →Gµν(k) = gµν −kµfν + fµkνf.k+ (f 2 −ξk2) kµkν(f.k)2 .

(2.11)Here f = f(ik), if the gauge breaking term contains derivatives. With fµ = ikµ, this reproducesthe propagator in covariant gauges, while fµ = (0, ik) corresponds to the Coulomb gauge.

The ghostpropagator is determined by ∆−1ghost = f.k; its 2 × 2 matrix structure is analogous to the vacuumpart of the gauge propagator. In most gauges, other than Feynman or Coulomb, an infinitesimalnonhermitian piece has to be included in Gµν in order to give meaning to the denominators in (2.11).In this case, the vacuum part of Dµν becomes−∆F Gµνeσk0θ(−k0)∆F Gµν −∆∗F G∗µνe−σk0θ(k0)∆F Gµν −∆∗F G∗µν−∆∗F G∗µν.

(2.12)Although, strictly speaking, we have given a derivation only for the Feynman gauge, the results wehave quoted agree with those derived by James for the A0 = 0 gauge[7]. For general covariant gauges,they agree with those in the literature[8] if one sets the temperature to zero in the non-T µν part ofthe propagator.3.

CALCULATIONS3.1. One-loop gluon self-energyIn a high-temperature expansion, the leading-temperature contributions arise from one-loop diagrams,and they determine the physics at the scale gT, where g is the coupling constant.In particular,the one-loop self-energy determines the spectrum of quasiparticles in the plasma, and contains theinformation on screening and Landau damping.

It has been shown by explicit calculations[9] andalso by an analysis of the relevant Ward identities[10] that these “hard thermal loops”, as they have

been called, are independent of the gauge conditions needed to define the gluon propagator. Withthe usual calculational procedure, this gauge independence arises in a rather nontrivial manner, andin particular in covariant gauges the contributions from the thermalised Faddeev-Popov ghosts aredecisive.

With our prescription, the latter are absent and it turns out to be much simpler to establishthe gauge independence.With the usual prescription for covariant gauges, the Feynman diagrams containing the hard-thermal-loop contributions to the gluon self-energy are those given in figure 1, where a slashed line denotesthe thermal part of a propagator, and a bare line the vacuum part. With both vertices of type 1,these diagrams give Re Π11 which coincides with the real part of the proper self-energy correction Πfor timelike momenta.In our formalism, we do not have contributions from the ghost diagrams 1c, 1d, although in diagram 1athe gauge-dependent vacuum gluon propagator does contribute.We shall show that the result isnevertheless gauge independent.With our real-time Feynman rules, the diagram 1b is identical to the corresponding one in Coulombgauge.

Since in Coulomb gauge there are also no contributions from the ghost diagrams, diagram 1ahas to be shown to coincide with the corresponding one in the Coulomb gauge. Evaluating diagram 1ain Coulomb gauge, one finds for its leading-temperature part in SU(N) gauge theoryΠ(a)CGµν(q) = g2NZdDk(2π)(D−1) δ(k2)n(|k0|)−(D −2)4kµkν −2kµqν −2qµkν(k −q)2−k20(k −q)2 Tµν(k),(3.1)where the principal value of the pole of the integrand is understood.

Apart from the piece containingthe Feynman denominator, only the terms with highest degree in k contribute. For this reason, onemay further simplify k20/(k −q)2 →1 in the last term of (3.1).On the other hand, with the Feynman-gauge propagator (2.10) one obtainsΠ(a)µν (q) = g2NZdDk(2π)(D−1) δ(k2)n(|k0|)−(D −2)4kµkν −2kµqν −2qµkν(k −q)2+ (k + q)2(k −q)2 Tµν(k).

(3.2)Because of the presence of the delta-function δ(k2), the numerator in the last term may be rewrittenas (k + q)2 →−(k −q)2 + 2q2, and the q2 dropped because it does not contribute to the hard thermalpart. Thus (3.2) gives the same leading-temperature terms as (3.1) does.Adding the contribution from diagram 1b gives the well-known result [9]Πµν(q) = g2N(D −2)ZdDk(2π)(D−1) δ(k2)n(|k0|)gµν −4kµkν −2kµqν −2qµkν(k −q)2.

(3.3)In gauges other than the Feynman gauge, the vacuum part of the gluon propagator has to be changedaccording to (2.11). Diagram 1b is not affected by this, but diagram 1a receives an additional contri-butiong2NZdDk(2π)(D−1) δ(k2)n(|k0|)T ρσ(k)2f βf.

(k −q) + (ξ(k −q)2 −f 2)(k −q)β[f.(k −q)]2× (k −q)αVαµσ(k −q, q, −k)Vρνβ(k, −q, q −k),(3.4)where in general fµ = fµ(k −q). V is the 3-gluon vertex (without the SU(N) colour factor), whichobeys(k −q)αVαµσ(k −q, q, −k) = gσµ(k2 −q2) −kσkµ + qσqµ.

(3.5)When this is inserted into (3.4), only the two terms quadratic in the loop momentum k contribute tothe leading-temperature part, but both get cancelled, the k2 by the δ-function, and the kσkµ by thetransverse projector Tρσ(k). Therefore the result (3.3) is completely independent of the gauge fixing.For completeness, we also give the result, in our formalism, for the one-loop contribution to thedamping of long-wavelength transverse gluons.

In the conventional formalism, the imaginary part

of the gluon self-energy at one-loop order is given by the diagrams of figure 2 and is strongly gaugedependent. In covariant gauges, the result comes with an unphysical sign, which was even taken to besuggestive of a fundamental instability of the perturbative thermal ground state (see reference 11 for areview).

In our formalism, only diagram 2a contributes, and the result is obviously gauge independent;it readsIm(TµνΠµν11 )(k0, 0) = −16π Ng2T 2. (3.6)By virtue of Im(TµνΠµν) = tanh(k0β/2)Im(TµνΠµν11 ) this gives a contribution to the damping rate oflong-wavelength gluonsγone−loop = g2TN24π .

(3.7)This result coincides with the Coulomb-gauge one of the conventional formalism[11], where also onlytransverse gluons contribute to the imaginary part. It is well-known by now[12] that one-loop resultsfor the gluonic damping rate are incomplete, and the gauge independence of our result (3.7) shouldnot lead one astray.

Still, we find it gratifying that in our formalism the unphysical modes are notable to contribute to (3.7), whereas normally in the bare one-loop calculation they even give rise to anegative sign.3.2. Two-loop gluon-interaction pressureThe thermodynamic pressure of the gluon plasma, P = 1/(βV ) ln Z, is a physical quantity and there-fore gauge independent.

In the following we shall calculate it up to two-loop order in order to performa further check on our Feynman rules. The diagrammatic rules for calculating the partition functionin the real-time formalism have been described in reference 13, to which we refer for more details.At one-loop order, our formalism leads to exactly the same expression for P as the conventional onedoes in Coulomb gauge.

Differences occur starting at two-loop order. The diagrams to be consideredare given in figure 3, where thermal contributions have to be inserted in all possible ways, keeping atleast one of the vertices appearing therein of type 1 [13].Diagram 3b potentially gives rise to integrals involving three powers of the Bose distribution function,but these we find to vanish after performing the momentum algebra because of the three delta functionsassociated with the distribution functions, exactly as in the conventional formalism.Next consider the diagrams where two lines are thermal.

Diagram 3a then identically reproduces theresult of conventional Coulomb gauge,P (a)2= −18N(N 2 −1)g2Z d4q d4k(2π)6 n(|q0|)δ(q2)n(|k0|)δ(k2)23 −z2= −1216N(N 2 −1)g2T 4,(3.8)where we introduced z ≡k.q/|k||q|. Diagram 3b with two thermal and one vacuum gluon propagator isdifferent, though.

The vacuum piece of the gluon propagator contributes potential gauge dependences.However, because of (2.11), all gauge-dependent terms have one 4-momentum contracted with a 3-gluon vertex, and give rise to integrals of the formZd4q d4k n(|q0|)δ(q2)n(|k0|)δ(k2)T µν(k)T σρ(q)(k −q)αVµασ(−k, k −q, q) · · · . (3.9)Because of(k −q)αVµασ(−k, k −q, q) = gσµ(q2 −k2) −qσqµ + kσkµ(3.10)it is easy to see that all the integrals of the form (3.9) vanish — q2 and k2 in (3.10) disappear becausethe δ-functions in (3.9), and the other terms also owing to the presence of the transverse projectors

T µν(k)T σρ(q). Thus the contributions from figure 3b which involve two powers of distribution func-tions are completely gauge independent.

There are actually two such diagrams. The one with both3-vertices of type 1 givesP (b)(11)2= −14N(N 2 −1)g2Z d4q d4k(2π)6 n(|q0|)δ(q2)n(|k0|)δ(k2)× −(1 + z2)(k + q)2 + 8(1 −z2)(k2 + q2)(k −q)2,(3.11a)where the first term can be simplified according to (k + q)2/(k −q)2 →−1 because of the deltafunctions, and the last term vanishes after performing the integrals, making this contribution coincidewith the one of conventional Coulomb gauge.

This yieldsP (b)(11)2= −1432N(N 2 −1)g2T 4. (3.11b)Because the vacuum part of the gluon propagator (2.10) does have a (12)-component, there is a seconddiagram of the form 3b, where one vertex is of type 1, and the other of type 2.

We find after carryingout the momentum algebra that the latter vanishes because of the presence of three δ-functions,δ(q2)δ(k2)δ((k −q)2).Finally, there are the diagrams with only one gluon propagator thermal. Those diagrams, however,contain a zero-temperature one-loop subgraph with its external lines put on-shell by the thermalpropagator, and so are removed by renormalisation.

The additional diagrams containing both a type-1 and a type-2 vertex do not complicate this picture, for they turn out to vanish identically.Discarding the purely zero-temperature contributions, we have therefore that the gluon-interactionpressure at two-loop order is given by the sum of (3.8) and (3.11),P2 = −1144N(N 2 −1)g2T 4,(3.12)reproducing the standard result[13-15]. Notice that the ghost diagram 3c did not contribute to (3.12),although we worked in a general linear gauge.4.

CONCLUSIONWe have demonstrated that in the real-time version of finite-temperature perturbation theory it isquite natural to include a thermal part in the gauge propagator only for the two physical degrees offreedom of the gluons, leaving the unphysical gauge-field components and the Faddeev-Popov ghostsunheated.This allows us to combine the simplicity of covariant gauges for the zero-temperaturepart with the advantages of the noncovariant ghost-free gauges for the thermal contributions. Forthe thermal contributions we single out the rest frame of the heat bath and do not cause additionalviolation of Lorentz symmetry.Our derivation has been carried through for the Feynman gauge.The generalization to arbitrary gauges is immediate, though there may be complications from theprescriptions necessary to define the poles at f.k = 0 in (2.11).We have successfully tested these new Feynman rules in one- and two-loop calculations, and in par-ticular have been able to demonstrate gauge independence of the high-temperature part of the gluonself-energy and for the 2-loop gluon-interaction pressure in a remarkably simple (albeit nontrivial)and general manner.

In the conventional formalism, gauge independence of the hard-thermal gluonself-energy has been explicitly checked only in certain classes of gauges[9], whereas the 2-loop pressurehas been calculated so far only in Feynman[13,14] and axial[15,16] gauges.We have worked throughout in the real-time formalism. This enables us to develop the perturbationtheory in terms of in fields, with the great advantage that not only do the fields obey free-fieldequations, but also we may ignore the interaction in the condition that picks out the physical states.

We have not been able to translate our prescriptions satisfactorily into an imaginary-time formalism.In order to set up such a formalism, one would need to choose an interaction picture that coincideswith the Heisenberg picture at finite time t0. The condition that picks out the physical interaction-picture states is then much more complicated[4].

Even if we were able to implement it, the fact thatthe unphysical-field propagators are temperature-independent and so not periodic in imaginary timewould lead to extra complications[6] and make the formalism difficult to use.An interesting question, to which we hope to return in future work, is how the present approachcould be extended to accommodate a resummation of finite-temperature perturbation theory, as ismandatory for exploring physics at the scale g2T in high-temperature quantum chromodynamics[12](and, incidentially, for handling pinch singularities[17]). An important difference between the bare andthe resummed theory is that, because of the appearance of an additional collective mode at the scalegT, (1.3) no longer covers all of the physical modes in the plasma.

Including the latter in a formalismanalogous to the one presented here might prove to be of conceptual interest for the resummationprogram[12], and might perhaps allow an easier verification of its gauge independence.One of us (PVL) is grateful to Peter van Nieuwenhuizen for patient discussions of the Gupta-Bleulerformalism; AR wishes to thank Patrick Aurenche and Randy Kobes for useful remarks.APPENDIX: Diagonalisation of the matrix propagatorThe real-time version of finite-temperature field theory can be reformulated by diagonalisation of the2×2 matrix propagators, which allows one to associate all thermal contributions with the vertices.A well-known possibility[2, 13] is to diagonalise to a matrix constructed from the ordinary Feynmanpropagator ∆F ,˜D =∆F00−∆∗F. (A.1)In our formalism, the Feynman-gauge propagator may be written asDµν = T µν MT ˜DMT −(gµν + T µν) M0 ˜DM0(A.2a)with (when σ = 0)MT =pn(|k0|)eβ|k0|/2e−βk0/2eβk0/2eβ|k0|/2,(A.2b)and M0 the zero-temperature limit of MT:M0 =1θ(−k0)θ(k0)1.

(A.2c)The ghost propagator is M0 ˜DM0.Recently, a different scheme has been proposed by Aurenche and Becherrawy[18], which diagonalisesto a matrix composed of retarded and advanced propagators,¯D =∆R00∆A,(A.3)instead of Feynman propagators. For this, one has to introduce different matrices to be associatedwith incoming and outgoing lines[18].

In our formalism for the Feynman gaugeDµν = T µν UT ¯DVT −(gµν + T µν) U0 ¯DV0(A.4a)

withUT =1−n(k0)1−(1 + n(k0)),U0 =1θ(−k0)1−θ(k0),VT =1 + n(k0)n(k0)11,V0 =θ(k0)−θ(−k0)11. (A.4b)The ghost propagator is U0 ¯DV0.

The matrices U and V are removed from the propagators andinstead associated with the in- and outgoing lines of the vertices. (Because ofU(k)τ1 = VT (−k),τ1 =0110,more symmetric Feynman rules result if a factor τ1 is combined with U and ¯D, turning the latteroff-diagonal[19].) With (A.4), the analysis of reference 18 carries over to a large extent.

However, inaddition to retarded and advanced lines one has to distinguish between transverse and non-transverseones, and in particular there is no simple causality principle when transverse and non-transverse linescome together at a vertex. For example, a 3-vertex connecting solely retarded or solely advanced linesdoes not vanish, unless the lines are either all transverse or all non-transverse.Such a reformulation in terms of retarded and advanced Green functions leads to well-defined causalproperties, which are not manifest in the usual real-time formalism[20].Causal Green functions,which are relevant for example in linear response theory, are usually obtained in a direct manner inthe imaginary-time formalism.

In our prescription, however, this is not straightforward to implement,and so this diagonalization scheme appears to be particularly useful for our approach.

REFERENCES1 C W Bernard, Phys Rev D9 (1974) 3312; H Hata and T Kugo, Phys Rev D21 (1980) 33332 L V Keldysh, Soviet Physics JETP 20 (1965) 1018; A Niemi and G W Semenoff, Nucl Phys B230(1984) 1813 C Itzykson and J-B Zuber, Quantum field theory (McGraw-Hill, 1980) p 1274 T Kugo and I Ojima, Phys Lett 73B (1978) 459 and Prog Theor Phys 60 (1978) 1869; 61 (1979) 294;61 (1979) 6445 P T Matthews, Phys Rev 76 (1949) 1657; P V Landshoffand J C Taylor, Phys Lett B231 (1989) 1296 K A James and P V Landshoff, Phys Lett B251 (1990) 1677 K A James, Z Phys C48 (1990) 169.8 R L Kobes, G W Semenoffand N Weiss, Z Phys C29 (1985) 371; N P Landsman, Phys Lett B172(1986) 46; K L Kowalski, Z Phys C36 (1987) 6659 O K Kalashnikov and V V Klimov, Sov J Nucl Phys 31 (1980) 699; H A Weldon, Phys Rev D26 (1982)1394; J Frenkel and J C Taylor, Nucl Phys B334 (1990) 199; E Braaten and R D Pisarski, Nucl PhysB337 (1990) 569; B339 (1990) 31010 R Kobes, G Kunstatter and A Rebhan, Nucl Phys B355 (1991) 111 U Heinz, K Kajantie and T Toimela, Ann Phys (NY) 176 (1987) 21812 R D Pisarski, Nucl Phys A525 (1991) 175c13 N P Landsman and Ch G van Weert, Phys Rep 145 (1987) 14114 J I Kapusta, Nucl Phys B148 (1979) 46115 K A James, Z Phys C49 (1991) 11516 H Nachbagauer, PhD thesis, Tech Univ Vienna (1991); Z Phys C56 (1992) 40717 H A Weldon, Phys Rev D45 (1992) 35218 P Aurenche and T Becherrawy, Nucl Phys B379 (1992) 25919 M A van Eijck and Ch G van Weert, Phys Lett B278 (1992) 30520 R L Kobes, Phys Rev D42 (1990) 562; D43 (1991) 1269Figure captions1 Diagrams which contain the hard-thermal-loop contributions to the gluon self energy. Solid lines aregluons and broken lines Faddeev-Popov ghosts.

The slashes denote thermal parts of the propagators;unslashed lines are vacuum propagators. In the formalism described in this paper, diagrams (b) and(c) do not occur.2 The imaginary part of the gluon self-energy, with the same notation as figure 1.

In our formalism, (b)does not occur.3 Diagrams for the 2-loop gluon interaction pressure. The lines represent complete propagators (vacuumplus thermal parts).

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출처: arXiv:9205.235 • 원문 보기