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Hard thermal effective actions in the Schwinger formulation Ashok Dasa,b and J. Frenkelc a Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171, USA b Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700064, INDIA and c Instituto de F´ısica, Universidade de São Paulo, São Paulo, SP 05315-970, BRAZIL We derive the properties of hard thermal effective actions in gauge theories from the point of view of Schwinger’s proper time formulation. This analysis is simplified by introducing a set of generalizedenergyandmomentawhichareconservedandarenon-localingeneral. Theseconstants of motion, which embody energy-momentum exchanges between the fields and the particles along their trajectories, can be related to a class of gauge invariant or covariant potentials in the hard thermal regime. We show that in this regime the generalized energy, which is non-local in general, generates the characteristic non-local behaviorof thehard thermal effective actions. 7 PACSnumbers: 11.10.Wx 0 0 2 I. INTRODUCTION termine the set of associated gauge invariant potentials [19]inthehardthermalregime. InsectionIII,weextend n thisanalysistonon-Abeliangaugetheoriesandderivethe a The high temperature properties of the quark-gluon J plasma in QCD are of great interest not only in their corresponding generalized momenta and the related set 7 own right, but also as a starting point for the resum- ofgaugecovariantpotentialsinthehardthermalregime. 1 mation of perturbation theory [1, 2]. These physical These potentials, which generate correctly the relevant hard thermal contributions to all orders, are in general properties are encoded in the hard thermal effective ac- 2 non-local. We conclude this note with a brief summary tion whose coefficient (the overall multiplicative factor) v is proportionalto T2 whereT representsthe equilibrium in section IV. 5 2 temperature. Such leading contributions to the effective 0 action arise from one loop diagrams where the internal 2 momentum is of order T which is much larger than any II. EFFECTIVE ACTION FOR QED 1 externalmomentum. Thecorrespondinghardthermalef- 6 fective actions have been derived from the point of view Let us consider scalar QED in 3+1 dimensions de- 0 of thermal field theory [3, 4, 5, 6, 7, 8, 9] as well as from scribed by the Lagrangian density (our metric has the / h the point of view of semi-classical transport equations signature (+,−,−,−)) -t [10,11,12,13,14,15,16]. Itisknownfromthesestudies ep that the hard thermal effective actions are gauge invari- L=((∂µ−ieAµ)φ)†(∂µ−ieAµ)φ−m2φ†φ. (1) ant and, in general, are non-local except in the static h : limit where they become local. The Lagrangian density is quadratic in the scalar fields v The one-loopeffective actions at zero temperature, on which can be integrated out in the path integral leading i X the other hand, are commonly derived using the proper to the generating functional time formulation of Schwinger [17] which is manifestly r a gauge invariant. However, this formulation is not as Z[Aµ] = N det (∂µ−ieAµ)(∂µ−ieAµ)+m2 −1 much developed at finite temperature [18] and the main (cid:2) (cid:0) (cid:1)(cid:3) 1 purpose of this short paper is to derive the hard ther- = det 1 (Π −eA )(Πµ−eAµ)−m2 −,(2) µ µ mal effective actions as well as their properties from (cid:20) 2m (cid:21) (cid:0) (cid:1) Schwinger’s proper time approach. We find that, in this wherewehaveidentifiedthecanonicalmomentumconju- approach, all the information about the hard thermal gateto thecoordinatexµ asΠ =−i∂ andhavechosen effective action is contained in a set of conserved gener- µ µ a particular form for the normalization constant N. It alized momenta, which are in general non-local. In the followsnowthattheone-loopeffectiveactionforthisthe- hard thermal regime,these momenta can be related to a ory can be written as class of gauge invariant or covariant potentials [19]. In this regime, the generalized energy, which is in general 1 non-local, contains all the information about the non- Γ = iTr ln (Π −eA )(Πµ−eAµ)−m2 µ µ 2m localbehavioroftheeffectiveactionswhichbecomelocal (cid:0) (cid:1) = iTr lnH, (3) only in the static limit. While our analysis holds for all theories,forbrevitywediscussonlygaugetheoriesinthis where “Tr”stands for trace overa complete set of states paper. In section II, we recapitulate briefly Schwinger’s and we have identified proper time approach and derive the hard thermal effective action resulting from scalar QED. In this case, 1 weconstructtheconservedgeneralizedmomentaandde- H = (Πµ−eAµ)(Πµ−eAµ)−m2 . (4) 2m (cid:0) (cid:1) 2 The expression (3) can lead to the effective action at obvious. However, the extension of the propagator to fi- zero as well as at finite temperature depending on the nite temperature can be carried out as follows. periodicity condition and the basis states chosen. Letusnotefrom(7)thatwheninteractionsarepresent, In Schwinger’s approach, the effective action (3) can the momentum p (or Π ) is not conserved. However, µ µ be written (in a regularized manner) as even in such a case, we can define a generalized momentum that is a constant of motion in the following way. Γ=−iTr ∞ dτ e−τH, (5) We note from (7) that the time evolution of functions in Z τ the phase space is given by the operator 0 where H (defined in (4)) can be thought of as the evolu- d 1 ∂ = (p·∂)+eF pν . (9) tion operator for the proper time parameter τ. Defining dτ m(cid:18) µν ∂p (cid:19) µ the kinematic momentum as Letusnextdefinethe (non-local)operator(wenotehere pµ =Πµ−eAµ, (6) parenthetically that this is, in fact, the operator that arisesintheconventionalcalculationsofthehardthermal we can determine the proper time evolution of the coor- effective action for QED) dinates and momenta from the canonical commutation relations to be e ∂ K = F pν . (10) dxµ pµ p·∂ µν ∂pµ = −i[xµ,H]= , dτ m Then, it can be easily checked with the help of (9) that dp e µ = −i[p ,H]= F pν, (7) µ µν dτ m 1 P = p +Y =p − Kp µ µ µ µ µ 1+K where F denotes the Abelian field strength tensor. (It µν 1 isworthpointingoutherethatwehavenotworriedabout = p , (11) µ the order of factors in the second equation, which is not 1+K relevantinthehardthermalregimethatweareinterested is conserved, namely, in.) Ifthedynamicalequations(7)forsuchaparticlecan be solvedin a closedform, one can construct a complete dP set of states and evaluate the effective action (5) in a µ =0. (12) dτ closedform. In general,this is notpossible for a particle interacting with an arbitrary external field and in such Thecoordinate,canonicallyconjugatetothisgeneralized a case, one studies the effective action in a perturbative momentum, can also be derived and has the form manner[17]. However,as wewillshow,inthe hardther- 1 1 malregimewhereΠ≫eA,thereisagreatsimplification Xµ =xµ+ Yµ, (13) yielding the leading result for the effective action in a 1+K p·∂ straight forward manner. and it satisfies the equation Tocalculatethe hardthermaleffectiveaction,wenote thatthecurrentofthetheoryatzerotemperaturefollows dXµ Pµ from (3) to be = . (14) dτ m jµ(0)(x)= δΓ(0) Thus, we see that in these generalized variables, the dy- δAµ(x) namicalequationsreducetothoseoffreeparticlemotion. d4Π i We note that this generalized momentum, Pµ, is in =−2e (Πµ−eAµ) .(8) generalnon-local. However,it can be easily verifiedthat Z (2π)4 (Π−eA)2−m2 P Pµ =p pµ, (15) If we know the current, it can, of course, be function- µ µ allyintegrated(inprinciple)to yieldthe effective action. so that the zero temperature propagator can be written We note that the denominator in (8) can be thought in these variables as a free propagator of as an effective scalar propagator in a space-time dependent background field A (x). In order to calculate µ i i the hard thermal (retarded) effective action, we need to i∆(0) = = , (16) (Π−eA)2−m2 P2−m2 define the current at finite temperature by generalizing thispropagatortotheappropriatefinitetemperatureone where we have used both (6) as well as (15). The gener- through a suitable analytic continuation [1, 2, 20] (in ei- alization to finite temperature is now immediate ther imaginary time or real time formalism). Because of thepresenceofthebackgroundfieldinthezerotempera- i i∆(T) = +2πn(|P |)δ(P2−m2), (17) turepropagator,suchageneralizationisnotimmediately P2−m2 0 3 where n(|P |) denotes the bosonic distribution function. will also satisfy such a relation. 0 We note here in passing that since the thermal propaga- Using (17) we can determine the temperature depen- tor is related to the zero temperature one through the dentpartofthe current(see (8)) (we areusing the nota- thermal operator [21], the currents at finite temperature tion jµ(T) =jµ(0)+jµ(β)) d4Π jµ(β) = −2e (Πµ−eAµ)2πn(|P |)δ(P2−m2) Z (2π)4 0 d4P e e = −2e 1− P ∂ Fρσ+··· Pµ+ P Fµα+··· 2πn(|P |)δ(P2−m2), (18) Z (2π)4 (cid:18) (P ·∂)2 σ ρ (cid:19)(cid:16) P ·∂ α (cid:17) 0 where the first factor arises from the Jacobian for the understand the non-local behavior of P better, let us 0 change of variables. In the hard thermal limit where write (11) as we can neglect masses and assume P ≫ ∂, the leading contribution to the current occurs only at order e2 and Pµ =pµ+eAµ, (24) takes the form where in the hard thermal regime, we can identify d4P jµ(β) = −2e2 δ(P2)n(|P |) 1 1 HTL Z (2π)3 0 Aµ(x,pˆ)= pˆνFνµ =Aµ− ∂µpˆ·A. (25) pˆ·∂ pˆ·∂ Pµ∂ν Pρ × ηµν − F . (19) νρ Inthisregime,werecognizeA tocorrespondtotheclass (cid:18) P ·∂ (cid:19)P ·∂ µ of gauge invariant potentials [19] which are, in general, Using the standard integral path dependent and non-local. For example, using the retarded path integral form for the operator 1 [22], we pˆ∂ ∞ π2T2 can explicitly express these potentials as · dx xn(x)= , (20) Z 6 0 t Aµ(x,pˆ) = − dt′ Fµ(t′,x−pˆ(t−t′)) and integrating the current (19), we obtain the temper- Z −∞ ature dependent hard thermal effective action to be t = − dt′ Fµ(x′(t′)), (26) Z e2T2 dΩ pˆ pˆσ Γ(β) = d4x Fµν ν F , (21) −∞ HTL 12 Z Z 4π (pˆ·∂)2 σµ where eF =epˆνF denotes the Lorentz force four vec- µ µν tor where we have labeled the variable of integrationin (19) as p and have defined eFµ =e(pˆ·F,F), F=E+pˆ×B, (27) pˆµ =(1,pˆ), (22) with E,B representing the electric and the magnetic fields respectively. It follows, therefore, that and dΩ denotes the angular integration over the unit t vectoRr pˆ. eA (x,pˆ)=−e dt pˆ·E(x(t)), 0 ′ ′ ′ The integrand of the effective action in (21) is mani- Z −∞ festlygaugeinvariantandLorentzinvariant,butappears t to be manifestly non-local as well. The locality/non- eA(x,pˆ)=−e dt′(E(x′(t′))+pˆ×B(x′(t′))),(28) Z locality of the hard thermal effective action can be best −∞ understoodintermsofthekinematicmomentum. Inthis canbeinterpretedinthiscaseastheenergyandmomen- variable, the hard thermal current (19) takes the form tumexchangedbetweentheparticleandthefieldalonga trajectoryparallelto p. As is clear,in generalthe gauge jµ(β) =−2e d4p pµδ(p2)n(|P |), (23) invariantpotentialsarenon-local. However,wenotefrom HTL Z (2π)3 0 (25)thatinthestaticlimit(whenthebackgroundfieldis static)A and,therefore,P islocal. Itthenfollowsfrom 0 0 and we see that all the non-locality of the current (and, (23) that although the hard thermal current as well as therefore, the effective action) is contained in the gener- theeffectiveactionarenon-localingeneral,theybecome alized energy P and manifests through the dependence local only in the static limit. It is interesting to remark 0 of the integrand on the distribution function n(|P |). To here that integrating by parts (21) and using (25), the 0 4 hard thermal effective action may be expressed in terms and we have identified the kinematic momentum to be of the gauge invariant potentials in the simple form p =Π −gAaTa. (36) µ µ µ m2 dΩ Γ(HβT)L = 2ph Z d4xZ 4π Aµ(x,pˆ)Aµ(x,pˆ), (29) We note that since the generators Ta do not commute (see(31)),inthiscase,inadditiontotheusualequations wherem = eT representsthethermalmassofthepho- of motion (34) for the coordinates and momenta of the ph √6 particle, we will also have ton. The form (29) is reminiscent of the gauge invariant mass generatedin the Schwinger model (at zero temper- dTa g =−i[Ta,H]=− fabc(p·Ab)Tc. (37) ature) [23]. dτ m Together with (37), therefore, the equations in (34) de- III. EFFECTIVE ACTION FOR YANG-MILLS scribea“spinning”particleintheinternalspaceandthis THEORY “spin” becomes an additional degree of freedom in this case. (We mention here again that we have disregarded The derivation of the hard thermal effective action the ordering of the factors which are not relevant in the (and its properties) for the Yang-Mills theory follows in hardthermalregime.) Furthermore,weremarkherethat acompletelyparallelmannertothediscussioninthelast the expectation value of the generators in the semiclas- section. Therefore,weonlygiveabriefdescriptionofthe sicallimit (for largequantumnumbers)canbe identified essential steps involved in such a derivation. As in sec- with the color charge of the classical particle [24, 25]. tion II, let us consider a complex scalar field in a given As in the last section, we note that neither the canon- representation of SU(N) interacting with a background ical momentum nor the kinematic momentum is con- non-Abeliangaugefield. TheLagrangiandensityhasthe served. However, we can determine a generalized mo- form (compare with (1)) mentum (see (11)) which is conserved in the following way. Let us define a derivative operator L= (∂µ−igAaµTa)φ †(∂µ−igAµbTb)φ−m2φ†φ. (30) ∂ Here t(cid:0)he scalar field is(cid:1)a matrix of SU(N) and Ta,a = D˜µ =∂µ−gfabcAbµTc∂Ta. (38) 1,2,··· ,N2 − 1 represent the generators of the group It is easy to see that acting on the space of functions of (in the representation to which the scalar fields belong) the kind fa(x,p)Ta, this gives satisfying the commutation relations D˜ (fa(x,p)Ta)=(D f(x,p))aTa, (39) [Ta,Tb]=ifabcTc, (31) µ µ where the covariant derivative is defined to be with fabc denoting the structure constants. Following the discussion in section II, we can write the one-loop (D f(x,p))a =∂ fa+gfabcAbfc. (40) µ µ µ effective action as (see (3)) Withtheseand(34)aswellas(37),itcanbeshownthat Γ=iTr lnH, (32) the conserved generalized momentum takes the form with the Hamiltonian given by 1 P = p +Y =p − Kp µ µ µ µ µ 1+K 1 H = (Π −gAaTa)(Πµ−gAµbTb)−m2 . (33) 1 2m µ µ = pµ, (41) (cid:0) (cid:1) 1+K We note that “Tr” in (32) denotes summing over a com- where in the present case pletesetofstatesaswellasasumoverthematrixindices of the group. g ∂ K = Fa Tapν . (42) The effective action in (32) can now be given a proper p·D˜ (cid:18) µν ∂pµ(cid:19) time representation as in (5) and the dynamical proper time evolutions for the coordinates and momenta can be Although these generalized conserved momenta are, in determined to be general, non-local, as in (15) it can be verified that dxµ = −i[xµ,H]= pµ, PµPµ =pµpµ. (43) dτ m dp g Therefore, following the analysis of the previous section, dτµ = −i[pµ,H]= mFµaνTapν, (34) wecanwrite the temperature dependent partofthe current in the hard thermal regime as (see (23)) where Fa denotes the non-Abelian field strength tensor µν d4pdT jµa(β) =−2g Tapµδ(p2)n(|P |). (44) Fµaν =∂µAaν −∂νAaµ+gfabcAbµAcν, (35) HTL Z (2π)3 0 5 Here dT denotestheintegrationoverthe“spin”degrees where m = NgT is the thermal gluon mass, in com- gl 6 offreeRdom(colorcharge). Onceagainweemphasizehere q plete analogy with (29). Alternatively, one could have that the hard thermal current and, therefore, the action started from a manifestly gauge invariantform like (50), will be non-local in general simply because P (defined 0 whichrepresentsanaturalgeneralisationoftheQEDac- in (41)) has this behavior. tion (29). Then, noticing that the overall multiplicative To understand this non-local behavior better, we note factor in (50) can be uniquely determined by an explicit that in the hard thermal regime, we can write (see (41)) evaluation of the gluon self-energy, one would be readily P =p +gAaTa, (45) led to the well knownhard thermal effective action(49). µ µ µ where with the help of (39) we find a a 1 1 Aa(x,pˆ)= pˆνF =Aa − ∂ pˆ·A . µ (cid:18)pˆ·D νµ(cid:19) µ (cid:18)pˆ·D µ (cid:19) IV. CONCLUSION (46) Here D represents the covariant derivative defined in µ (40). The gauge covariant potentials in (46) represent Inthiswork,wehavediscussedthehardthermaleffec- a natural extension of the class of invariant potentials tiveactionsandtheirpropertiesfromthepointofviewof in (25). These non-Abelian potentials, which are in gen- Schwinger’s proper time approach. We have shown that eralnon-localandpathdependent,generatecorrectlythe the non-localities of the current as well as the effective leadinghardthermalcontributionstoallordersing. Fur- action can be understood from the behavior of a set of thermore, as in the case of QED, we note that when the generalized momenta, Pµ, which are conserved (even in background field is static, the generalized energy in (45) the presence of interactions) and include the exchange has the simple local form of energy andmomentum between the backgroundfields and the particle along its trajectory. These conserved P0 =p0+gAa0Ta, (47) momenta are in general non-local, but satisfy the condition P Pµ = p pµ, where p denotes the kinematic µ µ µ so that only in this limit, the hard thermal current as momentum. As a result, the Lorentz invariant effective well as the effective action become local. In fact, in this actions at zero temperature, which are functions of P2, limit only the gluon self-energy is leading at high tem- will be local as expected. On the other hand, at finite perature. In general, since P0 is conserved, n(|P0|) is temperature, Lorentz invariance is broken because the also constant under τ evolution. Using this fact, it can rest frame of the heat bath defines a preferred reference be readily shown that the hard thermal current (44) is frame. Consequently, the thermal effective actions will covariantly conserved, depend onthe non-localenergy P throughthe distribu- 0 tion function. It is precisely this feature that generates DµjHµaT(Lβ) =∂µjHµaT(Lβ)+gfabcAbµjHµcT(βL) =0. (48) the non-localbehavior of the effective action in the hard thermal loop regime. An exception occurs only when Asaresult,thehardthermaleffectiveaction,whichisob- the background is static, in which case the exchanged tained by functionally integrating the hard thermal cur- energy is local, and this is reflected in the local forms rent is gauge invariant. The explicit form of the hard of the corresponding hard thermal effective actions. Al- thermal effective action for the Yang-Mills fields can be though our discussion has been completely within the written as [3, 4] context of gauge field backgrounds, this analysis holds as well when the background involves scalar fields. The Γ(β) = N(gT)2 d4x dΩ Fµνa pˆνpˆσ abFb . approach outlined in this paper may also be useful to HTL 12 Z Z 4π (cid:18)(pˆ·D)2(cid:19) σµ study the properties of effective actions describing non- (49) equilibrium systems at high temperature [26]. Thesimilarityoftheaboveactionwith(21)isworthnot- ing. It is interesting to observe that integrating this ex- pressionby parts,we can rewritethe hardthermal effec- tiveaction(49)intermsofthegaugecovariantpotentials Acknowledgment: (46) in the simple form This work was supported in part by US DOE Grant m2 dΩ number DE-FG 02-91ER40685,by CNPq and FAPESP, Γ(β) = gl d4x Aa(x,pˆ)Aµa(x,pˆ), (50) Brazil. HTL 2 Z Z 4π µ [1] J. I. Kapusta, Finite Temperature Field Theory (Cam- [2] M.LeBellac,Thermal FieldTheory(CambridgeUniver- bridge UniversityPress, Cambridge, England, 1989). 6 sity Press, Cambridge, England, 1996). [15] R. Pisarski, ”Non-abelian Debye screening, tsunami [3] E. Braaten and R. Pisarski, Nucl. Phys. B337, 569 waves and worldline fermions,” hep-ph/9710370. (1990);ibidB339,310(1990);ibidPhys.Rev.D45,1827 [16] D. Litim and C. Manuel, Phys. Rep. 364, 451 (2002). (1992). [17] J. Schwinger, Phys. Rev. 82, 664 (1951). [4] J. Frenkel and J. C. Taylor, Nucl. Phys. B334, 199 [18] J. Jalilian-Marian, S. Jeon, R. Venugopalan and J. (1990); ibidB374, 156 (1992); ibid B685, 393 (2004). Wirstam, Phys. Rev. D62, 045020 (2000). [5] J. C. Taylor and S. M. Wong, Nucl. Phys. B346, 115 [19] P. A. M. Dirac, Can. J. Phys. 33, 650 (1955); B. De (1990). Witt,Phys.Rev.125,2189(1962);S.Mandelstam,Ann. [6] E. Efraty and V. P.Nair, Phys.Rev. D47, 5601 (1993). Phys. 19, 1 (1962); A. Sisakyan, O. Shevchenko and I. [7] R.JackiwandV.P.Nair,Phys.Rev.D48,4991 (1993). Solovstov, Sov. J. Part. Nucl. 21, 285 (1990); P. Gaete, [8] D. McKeon and A. Rebhan, Phys. Rev. D47, 5487 Z. Phys. C76, 355 (1997). (1993); ibidD49, 1047 (1994). [20] A. Das, Finite Temperature Field Theory (World Scien- [9] F. T. Brandt and D. McKeon, Phys. Rev. D54, 6435 tific, Singapore, 1997). (1996). [21] F. T. Brandt, A. Das, O. Espinosa, J. Frenkel and S. [10] H.Elze and U.Heinz, Phys. Rep. 183, 81 (1989). Perez,Phys.Rev.D72,085006(2005);ibidD73,065010 [11] J.P.BlaizotandE.Iancu,Nucl.Phys.B390,589(1993); (2006); ibidD73, 067702 (2006). ibidB417,608(1994);ibidB434,662(1994);ibidPhys. [22] E.LifshitzandL.Pitaevski,Physical Kinetics,chapter3 Rep.359, 355 (2002). (Pergamon Press, Oxford, England, 1981). [12] V.P.Nair,Phys.Rev.D48,3432(1993);ibidD50,4201 [23] J. Schwinger, Phys. Rev. 128, 2425 (1962). (1994). [24] S. K. Wong, NuovoCim. 65A, 689 (1970). [13] R.Jackiw,Q.LiuandC.Lucchesi,Phys.Rev.D49,6787 [25] L. S. Brown and W. I. Weisberger, Nucl. Phys. B157, (1994). 285 (1979). [14] P. F. Kelly, Q. Liu, C. Lucchesi and C. Manuel, Phys. [26] S. Mrowczynski, A. Rebhan and M. Strickland, Phys. Rev. Lett. 72, 3461 (1994); ibid Phys. Rev. D50, 4209 Rev. D70, 025004 (2004). (1994).