Nonanalyticity of the nonabelian five-dimensional Chern-Simons term
aa r X i v : . [ h e p - t h ] N ov Nonanalyticity of the nonabelian five-dimensional Chern-Simons term
J. F. Assun¸c˜ao,
1, 2
J. Furtado, and T. Mariz Instituto de F´ısica, Universidade Federal de Alagoas, 57072-900, Macei´o, Alagoas, Brazil Departamento de F´ısica, Universidade Regional do Cariri,63180-000, Juazeiro do Norte, Cear´a, Brazil Centro de Ciˆencias e Tecnologia, Universidade Federal do Cariri,63048-080, Juazeiro do Norte, Cear´a, Brazil (Dated: November 26, 2020)
Abstract
In this work, we study the behavior of the nonabelian five-dimensional Chern-Simons term at finitetemperature regime in order the verify the possible nonanalyticity. We employ two methods, a perturbativeand a non-perturbative one. No scheme of regularization is needed, and we verify the nonanalyticity ofthe self-energy of the photon in the origin of momentum space by two conditions that do not commute,namely, the static limit ( k = 0 , ~k →
0) and the long wavelength limit ( k → , ~k = 0), while its tensorialstructure holds in both limits. The large gauge invariance method is used to obtain the effective action forthe Chern-Simons at finite temperature which is invariant under nontrivial gauge transformation. . INTRODUCTION It is well known that in gauge theories with fermions, a topological mass term can be generateddynamically and that, in this case, the effective theory contains a Chern-Simons (CS) term. Thisterm is topological and parity-odd, being introduced and first studied in [1, 2]. It gained attentionin 2 + 1 dimensions due to its relation with planar phenomena, such as superconductivity and thefractional quantum Hall effect [3, 4]. Furthermore, in the presence of such a term, the gauge fieldsbecome massive and for a non-abelian gauge theory, the quantization of the mass parameter appearsas a consequence of the large gauge invariance. While the large gauge invariance mechanism, andconsequently the origin of mass quantization, was well understood at zero temperature, at finitetemperature scenario it only became possible due to the works [5, 6], which showed that onlythe complete effective action is invariant under large gauge transformations, at least for a specificchoice of gauge field background.The CS term can be generalized to any arbitrary odd-dimension d = 2 n + 1 of spacetime. Thefull form (topological structure) of non-abelian CS term, Q n − ( ω, Ω) = n Z dtP n ( ω, Ω n − t ) , (1)can be obtained from the invariant polynomial P n ( ω, Ω n − ) = T r ω Ω n − in one dimension higher,constructed from the connection ω and the curvature two-form Ω = Ω a τ a , being τ a the generatorsof some Lie group and Ω t = t Ω + ( t − t ) ω [7]. The connection ω corresponds to different quantitiesin different theories.Studies in five-dimensional quantum field theories have been attracted much attention re-cently. A correspondence between a six-dimensional field theory compactified on a circle and afive-dimensional field theory, propagating both massive and massless degrees of freedom, is, basi-cally, the origin of the interest of studying five-dimensional CS theories [8]. It is expected that thefive-dimensional CS coupling reveals information on higher dimensional anomalies.Beyond the anomaly cancelation, the CS five-form was used to construct a topological gaugetheory of gravity in five dimensions [9]. In this context of a gauge theory of gravity, the CS form hasalready been used by Witten [10] to construct a theory of gravity that is not only renormalizablebut finite as well.In this paper, we focus on the behavior of the coefficient to verify a possible nonanalyticity,analyze if there is any change in the topological structure of the induced CS term when thephysical system is placed in contact with a thermal bath, and inquire the large gauge invariance in2he effective action of the theory. Since the gauge invariance is an inherent symmetry of the theory,we hope that its effective action remains gauge invariant under nontrivial gauge transformationeven upon consideration of thermal effects.Besides, it is known that at finite temperature regime, the one-loop radiative corrections arenonanalytic functions in the origin of momentum space ( k µ = 0). Thus, as due to the choice of aspecific frame by the thermal bath the radiative corrections, in general, has different dependenciesof k and ~k , the two conditions ( k = 0 , ~k →
0) and ( k → , ~k = 0) do not commute. In fact,this has been shown in the case of Lorentz-violating QED [11], three-dimensional QED [12], hotQCD [13–15], self-interacting scalars [16], and Maxwell-Chern-Simons-Higgs model [17]. The firstcondition ( k = 0 , ~k →
0) is sometimes referred as the “static” limit, while the other condition( k → , ~k = 0) is the “long wavelength” limit. In the physical context, these conditions are relatedto the Debye and plasmon mass responsible for the screening of the gauge field (the damping ofthe gauge field caused by the presence of the thermal virtual pairs), respectively.This paper is structured as follows. In section II, we single out the five-dimensional CS termthrough derivative expansion, and the finite temperature contribution is achieved in the imag-inary time formalism using the Ford expression [18] to compute the sum over the Matsubarafrequencies. In section III, we calculate the triangle, box, and pentagon graphs individually, ina non-perturbative approach, to analyze a possible nonanalytical behavior of the five-dimensionalCS coefficient. In section IV, we induce the large gauge invariant effective action from a particu-lar gauge configuration consistent with the static limit. Finally, in section V, we present a briefsummary of our results.In this paper, we are working with five-dimensional gamma matrices γ α , which are complex-valued 4 × { γ µ , γ ν } = 2 g µν , (2)where g µν = diag(+ , − , − , − , − ) is the Minkowski metric. The trace calculation of such gamma3atrices obey the following rules: tr ( γ µ ) = 0 , (3a) tr ( γ µ γ ν γ λ ) = 0 , (3b) tr ( γ µ γ ν γ λ γ ρ γ α ) = 4 iǫ µνλρα , (3c) tr ( γ µ γ ν γ λ γ ρ γ α γ β γ σ ) = 4 i (cid:16) g βα ǫ λµνρσ + g ρν ǫ αβλµσ − g ρµ ǫ αβλνσ + g ρλ ǫ αβµνσ + g ρβ ǫ αλµνσ + − g ρα ǫ βλµνσ + g νµ ǫ αβλρσ + g νλ ǫ αβµρσ − g νβ ǫ αλµρσ + g να ǫ βλµρσ + − g µλ ǫ αβνρσ + g µβ ǫ αλνρσ − g µα ǫ βλνρσ − g λβ ǫ αµνρσ + g λα ǫ βµνρσ (cid:17) , (3d)and so on. II. THE ONE-LOOP INDUCED FIVE-DIMENSION CHERN-SIMONS TERM
Initially, we are interested in studying the radiative induction of the five-dimensional CS term.For this, the starting point of our model is the usual fermionic sector of the QED, whose Lagrangianis given by L = ¯ ψ ( i /∂ − m − e /A ) ψ. (4)Consequently, the corresponding generating functional is written as Z [ A µ ] = Z D ¯ ψDψe R d x L = e iS eff , (5)and the integration over the fermions gives us the one-loop effective action S eff = − i Tr ln( /p − m − e /A ) . (6)Here, Tr consists in the trace over the Dirac matrices and over the space coordinates.In order to obtain the non-abelian CS term, we must select cubic, quartic, and quintic contri-butions in A µ . To achieve this, we rewrite the expression (6) as S eff = S (0) eff + ∞ X n =1 S ( n ) eff , (7)where S (0) eff = − i Tr ln( /p − m ) and S ( n ) eff = in Tr (cid:2) S ( p ) e /A (cid:3) n , (8)with S ( p ) = ( /p − m ) − . It is important to notice that, as it was first noted in ref [19], there is nononanaliticity present if the loop involves distinct masses in the propagators.4t this point, we have to consider n = 3 , , A µ ( x ) S ( p ) = S ( p − i∂ ) A µ ( x ), and use thecompleteness relation of the momentum space. Then, for n = 3, we get S (3) eff = ie tr Z d x Z d p (2 π ) S ( p ) γ µ S ( p ) γ ν S ( p ) γ λ A µ A ν A λ , (9)where p = p − i∂ , p = p − i∂ − i∂ , and so on. Performing a derivative expansion in the externalmomentum and selecting the terms that will contribute to CS, effective action S (3) eff becomes S (3) CS ,which is written as S (3) CS = − ie tr Z d x Z d p (2 π ) S ( p ) γ µ S ( p ) γ ν S ( p ) γ λ S ( p ) γ ρ S ( p ) γ σ ( ∂ ν A µ )( ∂ ρ A λ ) A σ . (10)Analogously, the contributions for n = 4 and n = 5 yield S (4) CS = − e tr Z d x Z d p (2 π ) S ( p ) γ µ S ( p ) γ ν S ( p ) γ λ S ( p ) γ ρ S ( p ) γ σ ( ∂ ρ A µ ) A ν A λ A σ , (11) S (5) CS = ie tr Z d x Z d p (2 π ) S ( p ) γ µ S ( p ) γ ν S ( p ) γ λ S ( p ) γ ρ S ( p ) γ σ A µ A ν A λ A ρ A σ . (12)In order to obtain the equations (10) and (11), we have used the cyclic property of the trace andthe solution tr Z d p (2 π ) S ( p ) γ µ S ( p ) γ ν S ( p ) γ λ S ( p ) γ ρ S ( p ) γ σ = αǫ µνλρσ , (13)where α is a constant. These expressions (10), (11), and (12) are presented in Fig. 1, respectively. (a) (b) (c) FIG. 1: Feynman graphs
Therefore, we can construct a nonabelian five-dimensional CS structure as follows: S CS = Z d x Z d p (2 π ) tr S ( p ) γ µ S ( p ) γ ν S ( p ) γ λ S ( p ) γ ρ S ( p ) γ σ ×× i − e ( ∂ ν A µ )( ∂ ρ A λ ) A σ + ie
32 ( ∂ ρ A µ ) A ν A λ A σ + e A µ A ν A λ A ρ A σ ] . (14)5ote that, despite the logarithm divergence associated with the coefficient of the five-dimensionalCS term, it is not necessary to use any kind of regularization because the divergent terms do notcontribute to it. This is a general feature of odd dimension theories. Besides, the implementationof temperature effects only affects the coefficient leaving the topological structure unchanged.The non-abelian five-dimensional CS action, as it appears in equation (14), has this form becauseit is written in the fundamental representation, in which A µ ( x ) = A aµ ( x ) τ a and τ a are the hermitiangenerators of the Lie Group SO (1 ,
5) satisfying the commutation relations[ τ a , τ b ] = if abc τ c , (15)where f abc is the structure constant of the group. In the adjoint representation, the CS five-formis given by ω = tr Z d x (cid:18) A ( dA ) + 32 A ( dA ) + 35 A (cid:19) . (16)Due to the common integral appearing in all three terms of the five-dimensional CS action (14),in order to take into account finite temperature effects, we will work uniquely with its tensorialcoefficient S µνλρσ = i Z d p (2 π ) ( /p + m ) γ µ ( /p + m ) γ ν ( /p + m ) γ λ ( /p + m ) γ ρ ( /p + m ) γ σ ( p − m ) = i S ǫ µνλρσ . (17)In calculating the trace of the above numerator, we must consider only the terms that contributeto the CS term, i.e., those with an odd power of mass, since such terms are parity-odd, as required.Then, we have S µνλρσ = i Z d p (2 π ) im − ip m + 4 ip m ( p − m ) ǫ µνλρσ = i Z d p (2 π ) im ( p − m ) ǫ µνλρσ . (18)where the scalar coefficient S takes the form S = Z d p (2 π ) im ( p − m ) . (19)To implement the finite temperature effects, we change from Minkowski to Euclidean spaceand split the internal momentum p µ into its spatial and temporal components. For this, we mustperform the following procedure: g µν → − δ µν , i.e., p → − δ µν p µ p ν = − p , as well as Z d p (2 π ) → Z d ~p (2 π ) i Z dp π . (20)6lso, let us assume from now on that the system is in thermal equilibrium with a temperature T = β − , so that the antiperiodic (periodic) boundary conditions for fermions (bosons) lead todiscrete values p = (2 n + 1) πβ and k = πlβ , with n and l being integers. Thus, with R dp π → β P n ,and considering the adimensional variables ξ p = βω p π and ξ = βm π , where ω p = p ~p + m , we get S = 1 β X n Z d ~p (2 π ) m ( p + m ) = ξ π ) X n (cid:2) ( n + 1 / + ξ (cid:3) − . (21)We complete our calculation by computing the sum over the Matsubara frequencies using the Fordexpression [18], given by X n [( n + b ) + a ] − λ = √ π Γ( λ − / λ )( a ) λ − / + 4 sin( πλ ) f λ ( a, b ) (22)where f λ ( a, b ) = Z ∞| a | dz ( z − a ) λ Re (cid:18) e π ( z + ib ) − (cid:19) , (23)which is valid for Re λ <
1. In our case, as we can see from equation (21), λ = 1, and hence it isout of the range of validity. To deal with this, i.e., to decrease the value of λ , we use the recurrencerelation f λ ( a, b ) = − a λ − λ − f λ − ( a, b ) − a λ − λ − ∂ ∂b f λ − ( a, b ) . (24)Therefore, the CS scalar coefficient is found to be S ( ξ ) = tanh( πξ )16 π . (25)Our result at zero temperature is in agreement, e.g., with [20, 21]. A naive procedure was used atfinite temperature in [22] and the same result was found.Another interesting feature worth commenting on is that, in a perturbative approach, it is notpossible to analyze if there is a nonanalytical behavior of the coefficient. And therefore, in thenext section, we will perform a non-perturbative calculation to verify if there is nonanalyticity inthe five-dimensional CS coefficient. III. NONANALYTICITY OF THE FIVE-DIMENSIONAL CHERN-SIMONS TERMA. Triangle Diagram
Now, let us analyze the nonanalyticity present in contributing to the CS term from the three-point function at the finite temperature regime. As previously noted in equation (18), only linear7erms in the mass contribute to the CS term, so that we get Π µνλ → Π µνλCS , i.e., the convergentintegral Π µνλCS ( k , k ) = ie iβ X n Z d ~p (2 π ) imǫ µνλαβ k α k β ( p + m ) (cid:0) p + m (cid:1) (cid:0) p + m (cid:1) = e S ( k , k ) ǫ µνλαβ k α k β , (26)whose coefficient S ( k , k ) is in Euclidean space, with p µ = p µ − k µ and p µ = p µ − k µ . Sincethe topological structure is independent of the internal momentum and directly obtained from thetrace evaluation, let us take only its coefficient S ( k , k ) to our analysis. Thus, after performingthe frequency sum, we are left with the evaluation of integrals S = β Z d ~p (2 π ) ( ξ tanh ( πξ p ) ξ p (cid:2) ξ p + ( l + iξ p ) (cid:3) (cid:2) ξ p + ( l + iξ p ) (cid:3) + ξ tanh ( πξ p ) ξ p (cid:2) ξ p + ( l − iξ p ) (cid:3) (cid:2) ξ p + ( l − iξ p ) (cid:3) + ξ tanh ( πξ p ) ξ p (cid:2) ξ p + ( l + iξ p ) (cid:3) (cid:2) ξ p + ( l + iξ p ) (cid:3) + ξ tanh ( πξ p ) ξ p (cid:2) ξ p + ( l − iξ p ) (cid:3) (cid:2) ξ p + ( l − iξ p ) (cid:3) + ξ tanh ( πξ p ) ξ p (cid:2) ξ p + ( l + iξ p ) (cid:3) (cid:2) ξ p + ( l − iξ p ) (cid:3) + ξ tanh ( πξ p ) ξ p (cid:2) ξ p + ( l − iξ p ) (cid:3) (cid:2) ξ p + ( l + iξ p ) (cid:3) ) , (27)where we have again used the adimensional variables ξ p = βω p π , with ω p = p ~p + m , and soon, and considered k = πlβ and tan[ π ( l ± iξ )] = ± i tanh( πξ ).At this point, we take the limits on the external momenta in equation (27). There are fourconsiderations, namely, the double static limit, the double long wavelength limit, and two mixedlimits. As previously mentioned, in the static limit, we put k = 0 ( l = 0), and then take thelimit ~k →
0, unlike, in the long wavelength limit, we put ~k = 0, and then take the limit k → l → k ] = ( k = 0 , ~k → { k } = ( k → , ~k = 0), respectively. For the double static limit, we have S ([ k ] , [ k ]) = ξ (cid:18) β π (cid:19) Z d ~p (2 π ) πξ p ) − πξ p [2 πξ p tanh ( πξ p ) + 3] sech ( πξ p )4 ξ p . (28)The mixed limits are given by S ( { k } , [ k ]) = S ([ k ] , { k } ) = ξ (cid:18) β π (cid:19) Z d ~p (2 π ) πξ p ) − πξ p sech ( πξ p )4 ξ p , (29)and the double long wavelength limit is written as S ( { k } , { k } ) = ξ (cid:18) β π (cid:19) Z d ~p (2 π ) πξ p )4 ξ p . (30)In order to evaluate the loop integrals in the above coefficients, we use spherical coordinatesin 4-dimensions and consider the isotropy of the momentum space. The angular integral yields8he solid angle, 2 π , while a change of variable in radial integral from | ~p | to ζ = β π p | ~p | + m isperformed, yielding for the double static limit, S ([ k ] , [ k ]) = Z ∞| ξ | dζ ξ (cid:0) ζ − ξ (cid:1) sech ( πζ ) { πζ ) − πζ [2 πζ tanh( πζ ) + 3] } π ζ = tanh( πξ )16 π ≡ F ( ξ ) , (31)for the mixed limits, S ( { k } , [ k ]) = Z ∞| ξ | dζ ξ (cid:0) ζ − ξ (cid:1) (cid:2) πζ ) − πζ sech ( πζ ) (cid:3) π ζ ≡ G ( ξ ) , (32)and for the double long wavelength limit, S ( { k } , { k } ) = Z ∞| ξ | dζ ξ (cid:0) ζ − ξ (cid:1) tanh( πζ )32 π ζ ≡ H ( ξ ) . (33)The result obtained in equation (31) correspond to that one in equation (25), showing that thederivative expansion method is consistent only to obtain the static limit. This fact is also found inthree- [12, 23] and four-dimensional [11, 24] theories. B. Box Diagram
Analogously, we analyze the nonanalyticity arising in the contribution to the CS term from thefour-point function at finite temperature regime. In this case, in the evaluation of the trace, linearand cubic terms in the mass contribute to the CS term. Thus, considering only these terms, wehave Π µνλρ → Π µνλρCS ( k , k , k ), whereΠ µνλρCS = ie iβ X n Z d ~p (2 π ) ( G + k · k ) k α − ( k · k ) k α + ( G + k · k ) k α ( p + m ) (cid:0) p + m (cid:1) (cid:0) p + m (cid:1) (cid:0) p + m (cid:1) imǫ µνλρα , (34)with p µ = p µ − k µ , p µ = p µ − k µ , p µ = p µ − k µ and G i = p i + m . However, in order tosingle out the nonanalytic contribution, we can use the coefficient-function S ( k i , k j ), defined in theequations (26) and (27), to rewritten the equation (34) as follows:Π µνλρCS = e S ( k , k ) ǫ µνλρα k α + e S ( k , k ) ǫ µνλρα k α + 4 i me β X n Z d ~p (2 π ) ( k · k ) k α − ( k · k ) k α + ( k · k ) k α ( p + m ) (cid:0) p + m (cid:1) (cid:0) p + m (cid:1) (cid:0) p + m (cid:1) ǫ µνλρα . (35)As we can see in the above equation (35), the nonanalytic contributions are presented inside thecoefficients S ( k , k ) and S ( k , k ). Also, there are analytic (remaining) contributions whichare proportional to the external momenta k µi , in a way that if we choose any one of the limits[ k i ] = ( k i = 0 , ~k i →
0) or { k i } = ( k i → , ~k i = 0), they yield a zero result.9s before, once the topological structure of the internal momentum is extracted, it followsthe analysis over the coefficients S ( k , k ) and S ( k , k ). It is possible to remove an externalmomentum, e.g., k µ →
0, so that S ( k , k ) → S ( k , k ) and S ( k , k ) → S ( k , k ). This freedomfollows because the nonanalytic behavior of the function S ( k i , k j ) around the origin was alreadyevaluated in previous subsection for k i = k and k j = k . Hence, we haveΠ µνλρCS ( k , , k ) = e S ( k , k ) ǫ µνλρα k α , (36)where k α = k α + k α . Thus, the limits about external momenta, through four distinct paths,are given by S ([ k ] , [ k ]) equal to F ( ξ ), in double static limit, S ( { k } , [ k ]) = S ([ k ] , { k } ) equal to G ( ξ ), in the mixed limit, and S ( { k } , { k } ) equal to H ( ξ ), in double long wavelength limit. C. Pentagon Diagram
Finally, we proceed with the same analysis for the contribution to the CS term arising from thefive-point function. At this time, in evaluating the trace, linear, cubic, and quintic terms in themass contribute to the CS term, through to the convergent integral Π µνλρσ → Π µνλρσCS ( k , k , k , k ),with Π µνλρσCS = ie iβ X n Z d ~p (2 π ) − imǫ µνλρσ ( p + m ) (cid:0) p + m (cid:1) (cid:0) p + m (cid:1) (cid:0) p + m (cid:1) (cid:0) p + m (cid:1) × [( G + k · k ) ( G + k · k ) + ( G + k · k ) ( k · k ) − ( k · k )( k · k )] , (37)where p µ = p µ − k µ , p µ = p µ − k µ , p µ = p µ − k µ and G i = p i + m . Again, we found analyticcontributions which are proportional to the external momenta k µi and vanish in any one of the limits[ k i ] = ( k i = 0 , ~k i →
0) or { k i } = ( k i → , ~k i = 0), so that only the nonanalytic contributionproportional to G G yields the coefficient-function S ( k , k ). Then, after removing, e.g., the momenta k µ and k µ and taking the limits, we obtainΠ µνλρσCS ( k , , k ,
0) = − e S ( k , k ) ǫ µνλρσ , (38)where newly S ([ k ] , [ k ]) = F ( ξ ), in double static limit, S ( { k } , [ k ]) = S ([ k ] , { k } ) = G ( ξ ), inthe mixed limit, and S ( { k } , { k } ) = H ( ξ ), in double long wavelength limit.10 . The Five-Dimensional Nonabelian Chern-Simons Term at Finite Temperature The one-loop induced nonabelian five-dimensional CS action is obtained by adding the odd-contribution from the triangle, box, and pentagon graphs as follows: S CS = S (0 , Z d xǫ µνλρσ [ − e ( ∂ ν A µ )( ∂ ρ A λ ) A σ + ie
32 ( ∂ σ A µ ) A ν A λ A ρ + e A µ A ν A λ A ρ A σ ] , (39)where, S (0 , F ( ξ ), G ( ξ ), or H ( ξ ) in the double staticlimit, mixed limit, or double long wavelength limit, respectively.As expected, at a finite temperature regime, the one-loop five-dimensional CS action is a non-analytic but covariant function, so that the thermal effects acts only on the coefficient. At thesame time, the topological structure is preserved without any change. This fact was also observedin three-dimensional CS action [12], and it is expected to hold, in general, in odd-dimension oncethe topological structure characteristic of CS term is obtained yet at integrand level.We also discuss the temperature dependence for the coefficient of the CS term. The behaviorof F ( ξ ), G ( ξ ), and H ( ξ ) are numerically plotted in Fig 2. F( (cid:1) )G( (cid:0) )H( (cid:2) ) (cid:3) FIG. 2: Plot of functions F ( ξ ), G ( ξ ) and H ( ξ ). We observe that at zero and infinite temperature, the limits coincide, and then Π µνλCS recoverthe analyticity. When T → ξ → ∞ ) the result obtained by [21] is found. Alternatively, thelimit T → ∞ ( ξ → m →
0) vanishes. This result is in agreement with the fact, well-known inthe literature, that the fermionic mass is a parity-odd quantity in odd-dimension responsible forthe induction of CS term. Between these extremes, as the temperature increases, the fermion mass( ξ = βm/ π ) is attenuated by thermal effects until completely suppressed at T → ∞ .Another interesting feature that is worth to be commented on is the fact that despite the CSaction (39) is not gauge invariant under large gauge transformation, we expected that the effective11ve-dimensional CS action does, as it happens in lower dimensions. Therefore, in the next section,we will verify if there is invariance under large gauge transformation in the five-dimensional CSaction, by performing a non-perturbative calculation in a particular configuration of the gaugefield. IV. LARGE GAUGE INVARIANCE
Since the studies in [5, 6], it is known that the nonabelian CS action is not invariant under largegauge transformation at zero temperature. However, the condition imposed by such nontrivialgauge transformation requires that the change must be proportional to the winding number, whichis an integer, and, therefore, the functional generator remains invariant. As a consequence of thelarge gauge transformation, the CS coefficient becomes quantized.In this section we analyze the large gauge invariance in the five-dimensional CS model at finitetemperature. In our case, the investigation is performed in a particular configuration, in which A = A ( t ) and A i = A i ( x, y, z ) , (40)implying a vanishing electric field and a time independent magnetic field. Then, considering anappropriate gauge transformation, given by A µ → A µ + ∂ µ Ω , with Ω( t ) = (cid:18) − Z t + tβ Z β (cid:19) dtA ( t ′ ) , (41)we obtain that the temporal and spatial components of the gauge field become A → a β = 1 β Z β dtA ( t ) , A i → A i ( x, y, z ) , (42)so that under large gauge transformation, a → a + 2 πn, (43)being n an integer.Now, we will rewrite the effective action presented in equation (6), taking into account theabove considerations, which gives us, in the Euclidian space, the following effective action: S eff = X n Tr ln( ~/p + ˜ ω n γ + m − e~/A ) , (44)with ˜ ω n = (cid:18) n + 12 (cid:19) πβ − e a β . (45)12ccording to the procedure presented in [25], we must derive the effective action with respect to a , ∂S eff ∂a = − eβ X n tr 1 ~/p + ˜ ω n γ + m − e~/A γ (46)and then, after a subsequent expansion, consider only the quadratic contribution in A i , in order toachieve the triangle contribution, so that we get ∂S CSeff ∂a = e β X n tr S ( p ) γ i S ( p − i~∂ ) γ j S ( p − i~∂ ) γ A i A j , (47)where, S ( p ) = ~/p + ˜ ω n γ + m . We can express the above equation in terms of a polarization tensor,as follows: ∂S CSeff ∂a = i Z d x Π ij A i A j , (48)with Π ij = e β X n tr Z d ~p (2 π ) S ( p ) γ i S ( p − i~∂ ) γ j S ( p − i~∂ ) γ = e β X n Z d ~p (2 π ) imǫ ij mn k m k n ( p + m )( p + m )( p + m )= e J ′ ( k , k ) ǫ ij mn k m k n , (49)where J ′ ( k , k ) = ∂J ( k , k ) ∂a = ∂S ( k , k ) ∂a (cid:12)(cid:12)(cid:12) p → ˜ ω n . (50)From now on, the following calculations will be similar to those performed in the previous subsec-tions. The only difference is the substitution of p by ˜ ω n . For simplicity, let us analyze only theCS coefficient in the double static limit, i.e., J ′ ([ k ] , [ k ]) = Z ∞| ξ | dζ ξ (cid:0) ζ − ξ (cid:1) π ζ [cos( a ) + cosh(2 πζ )] (cid:8) πζ )Λ − πζ [1 + cos( a ) cosh(2 πζ )] Λ+4 π ζ sinh(2 πζ ) (cid:2) cos ( a ) − cos( a ) cosh(2 πζ ) − (cid:3)(cid:9) , (51)where Λ = cos( a ) + cosh(2 πζ ), which can be rewrite as J ′ ([ k ] , [ k ]) = ∂∂a Z ∞| ξ | dζ ξ ( ξ − ζ )16 π ζ (cid:26) πζ sin( a ) [2 πζ sinh(2 πζ ) + 3 (cos ( a ) + cosh(2 πζ ))][cos( a ) + cosh(2 πζ )] − − h tan (cid:16) a (cid:17) tanh( πζ ) io = 18 π ∂∂a tan − h tan (cid:16) a (cid:17) tanh( πξ ) i . (52)13herefore, we have J ([ k ] , [ k ]) = 18 π tan − h tan (cid:16) a (cid:17) tanh( πξ ) i . (53)Note that if we expand the above equation around a , we will recover the expressions for F ( ξ )in the double static limit, as expected. This method is consistent only with the static limit due tothe particular gauge choice. Besides, the box and pentagon diagrams, omitted here for simplicity,are computed by the same procedure and yield the same result. This fact is important becausethe topological structure of five-dimensional CS action remains unchanged even in the thermalregime. The result for the five-dimensional CS coefficient, and consequently for the action, foundin this section using large gauge transformation agrees with [26], however, we use a whole differentapproach. V. SUMMARY
As exposed above, we have studied the nonanalytical behavior of the CS coefficient at finite tem-perature. Firstly, we have shown that we can single out from the three graphs, through derivativeexpansion, the same coefficient that generates the nonabelian five-dimensional CS term, namely,the triangle, box, and pentagon ones depicted in Fig 1. We have then calculated its finite temper-ature contribution solving the integral before the sum over the Matsubara frequencies. Our resultis in agreement with the results found in [20, 21].We have verified the nonanalytical behavior of the coefficient through nonperturbative cal-culations of the triangle, box, and pentagon diagrams. These three diagrams were calculatedindividually and they gave rise to the same result for the three limits, namely, the double staticlimit, the mixed limit, and the double wavelength limit.We have obtained the effective action invariant under nontrivial gauge transformation consistentwith the static limit at the finite temperature regime. We also have observed that the topologicalstructure remains unchanged even under thermal effects.14 cknowledgments
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