On space-time noncommutative U(1) model at high temperature1 Alexei Strelchenko Dnepropetrovsk National University, 8 49050 Dnepropetrovsk, Ukraine 0 E-mail: [email protected] 0 2 n a J Abstract: WeextendtheresultsofRef. [1]tononcommutativegaugetheoriesatfinitetemperature. 0 In particular, by making use of thebackground field method, we analyze renormalization issues and 3 thehigh-temperatureasymptoticsoftheone-loopEuclideanfreeenergyofthenoncommutativeU(1) ] gauge model within imaginary time formalism. As a by-product, the heat trace of the non-minimal h photonkineticoperatoronnoncommutativeS1×R3manifoldtakeninanarbitrarybackgroundgauge -t isinvestigated. AllpossibletypesofnoncommutativityonS1×R3areconsidered. Itisdemonstrated p that thenon-planarsector of themodel does not contributetothefree energy of thesystem at high e h temperature. The obtained results are discussed. [ 2 v 3 PACS numbers: 11.15.-q, 11.10.Wx, 11.10.Nx, 11.15.Kc 5 6 2 1 Introduction . 2 1 Understanding fundamental properties of hot plasma in noncommutative gauge theories, especially in 7 NC QED, remains one of the most challenging problems in high-energy physics. Indeed, because of the 0 noncommutative nature of space-time, even the simplest thermal U(1) model exhibits such odd features : v asgenerationofthemagneticmass(associatedwithnoncommutativetransversemodes),appearanceofa i tachyoninthespectrumofquasi-particleexcitationsetc. [2,3,4,5,6,7,9,10]. Theseobservationsconcern X mainly space/space noncommutative theories where there are no notorious difficulties with causality r a and unitarity [11, 12]. At the same time, it was realized that a space/space NC QFT may have non- renormalizable divergences as a consequence of UV/IR mixing phenomenon [13] (see also [14] for recent discussion). ThepurposeofthepresentworkistogainsomebetterinsightintobasicaspectsoftheEuclidean-time formalisminthermalgaugetheoriesonNCS1 R3,includingrenormalizationandthehigh-temperature × asymptotic of the (Euclidean) free energy (FE). For the sake of completeness, three different types of noncommutative space-time will be worked out: namely, space/space, full-rank and pure space/time noncommutativities. Webeginouranalysiswiththeinvestigationofone-loopdivergencesintheEuclidean NC U(1)gaugemodelonS1 R3 to makesurethat the theorydoes existatleastatthe leadingorderof × the loopexpansion. Then we will turnto the evaluationof the high-temperatureasymptotics ofthe one- loop FE. The main attention will be paid to the non-planar sector of the perturbative expansion. Thus, it was discovered in Ref. [4, 5] that there is a drastic reduction of the degrees of freedom in non-planar part of FE. Here we will arrive at the same qualitative picture for all types of noncommutativity. 1Talkgivenatthe8thWorkshop”QuantumFieldTheoryUndertheInfluenceofExternalConditions”,Leipzig, Germany,17-21 September2007. 2 The model Consider U(1) gauge model on NC S1 R3. Its action reads2 × 1 S = d4x G ⋆G , (1) −4g2 µν µν ZM where the integration is carried out over = S1 R3 manifold and G denotes the curvature tensor µν M × of U(1) gauge connection. To investigate quantum corrections to (1) we employ the background field method. To this aim we split the field A into a classical backgroundfield B and quantum fluctuations Q , i.e. A =B +Q . µ µ µ µ µ µ Then, substituting this decomposition into (1), we extract the part of the action (1) that is quadratic in quantum fluctuations. In a covariant background gauge it is written in the form (we use notations of Ref. [27]): 1 S [B,Q,C,C]= d4x Q (x)D(ξ)Q (x)+C(x) DC(x) , (2) 2 −2g2 µ µν ν ZM (cid:18) (cid:19) where 1 Dξ = δ 2+( 1) +2(L (F ) R (F )) (3) µν − µν∇ ξ − ∇µ∇ν Θ µν − Θ µν h i isthephotonkineticoperatorandD = istheinversepropagatorofghostparticles. Here and µ µ µ −∇ ∇ ∇ F stand for the covariantderivative and the curvature tensor of the backgroundfield B , respectively. µν µ Functionalintegrationofthepartitionfunctionw.r.t. quantumfieldsgivesthefollowingformalexpression for the 1-loop effective action (EA), 1 Γ(1)[B]=Γ [B]+Γ [B]= lndet Dξ lndet(D). (4) gauge ghost 2 − (cid:0) (cid:1) As well-known this quantity is divergent and must be regularized. This will be done by zeta-function regularizationin what follows. For the study of thermal QFT one needs to introduce another important object – the free energy of the system. Recall, that there are two definitions of this quantity. One of them presents the canonical FE, FC(β)=β−1 ln 1 e−βω , (5) − ω X (cid:0) (cid:1) which has clear physical meaning of ”summation over modes”. The other one expresses FE in terms of the Euclidean EA, FE(β)=β−1ΓE(β), (6) and is much more convenient from practical point of view. These two definitions are related by FE(β)=FC(β)+E , 0 where E is the energyofvacuum fluctuations. Itshouldbe noted, however,that a rigorousproofofthis 0 relation even in conventional field theories may be a highly non-trivial task (e.g. for thermal systems in curved spaces, see for instance Refs. [15, 16]). The equivalence of the canonical and Euclidean FE in QFT with space-time noncommutativity (althoughwith some heuristic assumptions)was discussedin Ref. [1]. 3 Zeta-function regularization. In the zeta regularizationscheme, the regularized EA (4) is represented by [17, 18, 19] 1 Γ(1)[B]= µ2sΓ(s) ζ s,Dξ 2ζ(s,D) , (7) s −2 − 2As usual, we will work in the rest frameof the heat b(cid:0)ath(cid:0)with(cid:1)u=(0,0,0,1)(cid:1), where u is the heat bath four velocity. Allfieldsobeyperiodicboundaryconditions. 2 where ζ s,Dξ and ζ(s,D) are zeta-functions of each operator in (4), s is a renormalization parameter and µ is introduced to render the mass dimension correct. The regularization is removed in the limit (cid:0) (cid:1) s 0 giving → 1 1 1 Γ(1) [B]= γ +lnµ2 ζ (0) ζ′ (s), (8) s→0 −2 s − E tot − 2 tot (cid:18) (cid:19) where γ is the Euler constant and ζ (s)=ζ s,Dξ 2ζ(s,D). E tot − To deal with the zeta-functions we need to introduce the heat traces for the operators Dξ and D, (cid:0) (cid:1) respectively. Recall that for a star-differential operator it is define as D K(t, )=TrL2(exp( t ) volume term), (9) D − D − where t is the spectral (or ”proper time”) parameter. Symbol TrL2 denotes L2-trace taken on the space of square integrable functions ( on S1 R3 with periodic boundary conditions in our case)and may also × involve the trace over vector, spinor etc. indices. The main technical result here is that on a (flat) NC manifold the heat trace (9) can be expanded in power series in small t as: ∞ K(t, )= t(n−4)/2a ( ). (10) n D D n=1 X Forfurtherdetails,werefertheinterestedreadertoRefs. [20,21,22,23,24,1,25]. Now,thezeta-function ζ (s) has the following integral representation, tot 1 ∞ dt ζ (s)= Kξ t,Dξ 2K(t,D) , (11) tot Γ(s) t1−s − Z0 (cid:0) (cid:0) (cid:1) (cid:1) and to analyze the structure of (8) one should actually evaluate the heat trace coefficients for each operator entering (4). For instance, taking into account the relation a ( ) = Res Γ(s)ζ(s, ), k s=(4−k)/2 D D the pole part of (8) can be re-expressed through the heat trace coefficients as 1 1 Γ(1) [B]= γ +lnµ2 a (Dξ) 2a (D) . (12) pole −2 s − E 4 − 4 (cid:18) (cid:19) (cid:0) (cid:1) That is, on a 4-dimensional manifold it is determined by the 4th heat trace coefficients. 4 Evaluation of the heat trace coefficients Toobtaintheheattraceasymptoticsofthenon-minimaloperator(9)itisconvenienttousethecalculating methodbyEndo[26]generalizedonaNCcase[27]. Namely,ifthebackgroundfieldsatisfiestheequation of motion, the following relation holds3: Kξ t,D(ξ) =Kξ=1 t,D(ξ=1) (13) (cid:16) (cid:17)t (cid:16) (cid:17) ξ − dτ d4x ∇µ∇′µK(x,x′;τ|β)−volume term |x=x′, Zt ZM (cid:0) (cid:1) whereK(x,x′;τ β) isthethermalheatoperatorofthe inverseghostpropagator. NoticethatRHSofthis | relationconsistsofthe heattracesofminimal star-differentialoperators. Calculatingprocedureforsuch objects is standard and described, for instance, in Ref. [23]. In particular, it was found that the heat trace expansion for a generalized star-Laplacian4 contains coefficients of two types: so-called planar and mixed heat trace coefficients. In our example, the first planar heat trace coefficient is given by 1 11 apl.tot. :=a (D(ξ)) 2a (D)= d4xF ⋆F . (14) 4 4 − 4 16π2 − 3 µν µν (cid:18) (cid:19)ZM 3Noticethatonehastoeliminatevolumedivergences byaddingappropriateterms,cf. expr. (9). 4Thatis,whichincludesbothleftandrightMoyalmultiplications. 3 Evaluationofthemixedheattracecoefficients,however,ismoreinvolved. Hereweinspectthreedifferent cases. (i)Full-ranknoncommutativity. Tosimplifycomputationsweassumethatthe deformationmatrixΘhas the canonical form: θS 0 0 1 Θ= , S = . (15) 0 ϑS 1 0 (cid:18) (cid:19) (cid:18) − (cid:19) However, the reader should be warned that, in general, a reference frame where the matrix Θ has the block off-diagonalform(15) does notnecessarilycoincide with the reference frame ofthe heatbath. The first nontrivial mixed coefficient can be now easily evaluated and has the form (see also [1] for some technical details) ξ−1/2 amix.tot. = dx dx4 dy dy4 dx3 (16) 5 −2βθ2π5/2 n∈ZZR2×S1 ⊥ ZR2×S1 ⊥ ZR × X π ϑn π ϑn B x1,x2,x3+ | | ;x4 B y1,y2,x3 | | ;y4 . µ µ × β − β µ,µ=6 3 (cid:18) (cid:19) (cid:18) (cid:19) X This coefficient is divergent as θ 0 and/or ϑ 0 that is a manifestation of the well-known UV/IR → → phenomenon [28, 29, 30]. (ii) Pure space/time noncommutativity (Θij =0 and Θi4 is directed along x axis). In this case the first k mixed heat trace coefficient is presented by 1 amix.tot. = 2 ξ dx4dy4 dx dx (17) 3 −2βπ3/2 (cid:16) −p (cid:17)nX∈ZZS1×S1 ZR3 ⊥ k × π ϑn π ϑn B x ,x + | | ;x4 B x ,x | | ;y4 . × 4 ⊥ k β 4 ⊥ k− β (cid:18) (cid:19) (cid:18) (cid:19) (iii) Space/space noncommutativity (Θij =0, Θ4i =0). One finds 6 (lnξ 2) amix.tot. = − dx3dx4 dx dy 4 8θ2π3 ZS1×R ZR2×R2 ⊥ ⊥× B (x ,x3;x4) B (y ,x3;x4). (18) i ⊥ i ⊥ × i=1,2 X From (12) we see that this coefficient does contribute to the pole term of the one-loop EA and, hence, affects renormalization of the model that will be explained in a moment. 5 Renormalization and high-temperature asymptotics Let us now looka little moreclosely atthe divergentpartofEA(12). Clearly,in the caseofnoncommu- tative compact dimension it is defined solely by the planar heat trace coefficient (14). That is, the pole part of the one-loop EA has the form 1 22 Γ(1) [B]= d4x (4π)−2 F ⋆F , (19) pole −2s − 6 µν µν ZM (cid:18) (cid:19) leading thus to the standard renormalizationgroup. We see that the source of the UV divergence in (8) is associated with the original four-dimensional field theory and this divergence is removed by ordinary renormalization at zero temperature. However, the situation changes drastically when the compact coordinateis commutative: in this particularcasethe expression(19) containsanadditionaltermdue to the mixed heat trace coefficient (18). Although this new term is also temperature independent, it brings 4 into EA a non-local and, moreover, gauge-fixing dependent divergence which cannot be eliminated by any renormalizationprescription. To obtain high-temperature asymptotics of the one-loop EA we rewrite (7) as ∞ dt 1 Γ(s1)[B]=µ2s t3−stk2 −2ak(Dξ)+ak(D) + k=2Z0 (cid:18)(cid:18) (cid:19) X +2 e−β24nt2 1aplanar(Dξ)+aplanar(D) , (20) n=1 (cid:18)−2 k k (cid:19)! X where we retained all exponentially small terms in the planar sector as well. (They must be taken into account when the parameter β is small). The evaluation of the planar part proceeds exactly as in the conventional thermal SU(2) gluodynamics giving 1 S [B]+Γ(1) [B] d4xF ⋆F (21) tree planar ≃ −4g2(T) µν µν (cid:18) R ZM 2k−4 β + aplanar(Dξ) 2aplanar(D) ζ(2k 4)Γ(k 2) , 2 k − k − − ! Xk=6(cid:18) (cid:19) (cid:16) (cid:17) from which one deduces high temperature behaviour of NC U(1) effective coupling: g2 11 −1 g2(T)=g2 1+ R ln(T/T ) . (22) R R 4π2 3 0 (cid:18) (cid:19) It should be emphasized, however, that the formula (22) makes sense unless a compact dimension is commutative: as we have already seen, within space/space NC U(1) model one cannot renormalize the charge because of the non-planar contribution (18). Nowconsiderthenon-planarpartofEA.Forthesakeofdefinitenessletusfocusonthepurespace/time noncommutativity. Firstofall,wenotethattheexpression(17)isvalidwheneverthecondition ϑ/β =0 | | 6 holds. Hence, it is interesting to explore high temperature regime when ϑ/β C , C R . We 0 0 + | | ≫ ∈ assumedthatthe backgroundfieldB C∞(S1 R3) and,therefore,itshouldvanishexponentially fast µ ∈ × at large distances. For n=0 one estimates 6 π ϑn π ϑn ϑ ϑ B x ,x ,z+ | | ;x B y ,y ,z | | ;y C exp C | | , | | C , µ 1 2 4 µ 1 2 4 2 1 0 β − β ∼ − β β ≫ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) where C is a positive constant which characterizes the fall-off of the gauge potential at large distances. 1 Uptoaninessentialoverallconstantthecontributionofthefirstmixedcoefficienttotheeffectivepotential can be estimated as 1+√ξ atot = dx4dy4 dxB x¯;x4 B x¯;y4 . (23) 3 (cid:0)2β(π)3/2(cid:1)ZS1×S1 ZR3 4 4 (cid:0) (cid:1) (cid:0) (cid:1) Notice that this expression is insensitive to the value of the deformation parameter5. Moreover,since in the limit β 0 the main contribution to (23) comes from the zero bosonic modes, the mixed heat trace → coefficients behave as βC, where C is some temperature-independent quantity. From the definition ∼ (6) it follows that, at least on the one-loop level, the non-planar part of EA provides the temperature- independent contribution to the Euclidean FE and therefore can be neglected in the high temperature limit. 5Of course, this does not mean that the expression (4) possesses a smooth commutative limit: in obtaining high- temperatureasymptotics for(23)weassumed|ϑ|6=0. 5 6 Conclusion In this paper we have investigated the one-loop quantum corrections to EA (resp. Euclidean FE) in NC thermal U(1) theory within the imaginary time formalism. Let us summarize the obtained results. First, in the space/space noncommutative QED, the renormalizability of the theory is ruined by the non-planar sector of the perturbative expansion. This phenomenon was already observed, for instance, in Ref. [13] (see also [24, 27]). At the same time, in the case of a noncommutative compact dimension the theory can be renormalized,at least on one-looplevel, by the standardrenormalizationprescription. Second,wecalculatedthe heattraceasymptoticsfor the non-minimalphotonkinetic operatoronNC S1 R3. We saw, in particular, that the noncommutativity of the compact coordinate results in arising × of additional odd-numbered coefficients in the heat trace expansion. Furthermore, in the case of pure space/timenoncommutativitythe firstnontrivialmixedcontributiontothe heattraceappearsinamixed. 3 Although this coefficient does not affect counterterms in the zeta function regularization, it can lead to certain troubles in different regularization schemes, see Ref.[1] for further discussion. Third, we obtain the high-temperature asymptotics of the one-loop Euclidean FE (6). It is rather remarkable that the non-planar sector does not contribute at high temperature for any type of noncom- mutativity. This seems to be in accordance with observations made in earlier works where a drastic reduction of the degrees of freedom in non-planar part of FE was discovered [4, 5]. There is a subtlety, however, that one should keep in mind. Namely, if noncommutativity does not involve time, there are no difficulties in developing the Hamiltonianformalismfor a NC theory and equivalence of the canonical and Euclidean free energies is proved by standard arguments [31]. Contrary to this, in the space/time NC theories there is no good definition of the canonicalHamiltonian and, consequently, of the canonical FE (5) although some progress in this direction has been made recently in Ref.[1]. Finally,anextensionofourresultstomoregeneralcaseofU(N)gaugesymmetrycanbedonestraight- forwardly. Indeed,onecanshowthatthemixedheattracecoefficientsarecompletelydeterminedbyU(1) partofthemodel[27]. 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