Heat kernel approach to the relations between covariant and consistent currents in chiral gauge theories

...................................................................................................................Weapplytheheatkernelmethodtorelationsbetweencovariantandconsistentcurrentsinanoma-lous chiral gauge theories. Banerjee et al. have shown that the relation between these currents is expressed by a “functional curl” of the covariant current. Using the heat kernel method, we evaluate the functional curl explicitly in arbitrary even dimensions. We also apply the heat kernel method to evaluate Osabe and Suzuki’s results of the difference between covariant and consistent currents in two and four dimensions. Applying the arguments of Banerjee et al. to gravitational anomalies, we investigate the relationship between the covariant and consistent energy–momentum tensors. The relation is found to be expressed by a functional curl of the covariant energy–momentum tensor. ...................................................................................................................


Introduction
Chiral gauge anomalies can be viewed in one of two ways, namely, covariant and consistent. Covariant anomalies are defined as covariant divergences of the covariant current, i.e., a covariant divergence of the covariantly regularized expectation value of the current. Consistent anomalies can be considered as gauge transformations of a regularized effective action. From this definition, consistent anomalies satisfy the Wess-Zumino consistency condition (Ref. [1]).
The covariant and consistent anomalies are known to be equivalent in the sense that they lead to the same anomaly-cancelation condition. Bardeen and Zumino (Ref. [2]) have given a general proof for this equivalence of the anomalies using algebraic prescriptions. Their approach does not need any explicit form for the Lagrangians, thus giving model-independent results. Lagrangian-based field-theoretical approaches to the equivalence of the gauge anomalies have been given by various authors (Refs. [3][4][5][6][7][8]). In particular, Banerjee et al. (Ref. [3]) have shown equivalence by introducing a regularized effective action defined through covariant current.
To prove the equivalence of covariant and consistent gauge anomalies, Banerjee et al. (Ref. [3]) gave a relationship between the covariant and consistent currents. The consistent current was derived as a functional derivative of a regularized effective action, which was defined using the covariant current (Ref. [3]). As a result, the relationship between the covariant and consistent currents is PTEP 2017, 033B03 M. Takeuchi and R. Endo expressed by a "functional curl" of the covariant current. 1 The authors of Ref. [3] argued that the functional curl of the covariant current is proportional to the delta function. With the help of the delta-function-type behavior of the functional curl, they have derived the relationship between the covariant and consistent gauge anomalies. Although their result agrees with Bardeen and Zumino (Ref. [2]), the delta-function-type behavior of the functional curl is not clearly explained in their arguments. Thus, it is desirable to prove the behavior of the functional curl more explicitly.
The functional curl of the covariant current has been discussed by various authors (Refs. [6,7,[9][10][11][12]). Fujikawa and Suzuki (Ref. [6]) gave a formal proof of the relationship between the functional curl and the covariant anomaly; this relation was derived by Banerjee et al. (Ref. [3]) using the deltafunction-type behavior of the functional curl. Ohshima et al. (Ref. [7]) evaluated the functional curl of the covariant current in supersymmetric chiral gauge theory. This curl was evaluated explicitly by using the Fourier transformation in four dimensions. Based on their motivation, which differed from that of Banerjee et al., Qiu and Ren (Ref. [12]) evaluated the functional curl explicitly by using the point-splitting method in two and four dimensions. All of these results are consistent with the curl's delta-function-type behavior.
Other studies concerning the relationship between the covariant and consistent currents have been reported in Refs. [4,5,8], where the functional curl does not appear in the arguments. The difference between the covariant and consistent currents has been directly calculated using Pauli-Villars regularization (Ref. [4]) and the point-splitting method (Ref. [5]). Osabe and Suzuki (Ref. [8]) also discussed the difference between covariant and consistent currents, which they defined by invoking different types of exponential regulators. These regulators were then used to obtain a formal expression of the difference between the covariant and consistent currents.
In this paper, by using the heat kernel method (Ref. [13]), we evaluate the functional curl of the covariant current explicitly. The curl that we derive agrees with that of Refs. [3,6]. Our result presents another direct proof of the delta-function-type behavior of the functional curl in arbitrary even dimensions. We also apply the heat kernel method to evaluate Osabe and Suzuki's formal expression of the difference between the covariant and consistent currents (Ref. [8]). This difference, which we calculate in two and four dimensions, agrees with previous results (Refs. [2,3]). The arguments of Banerjee et al. (Ref. [3]) are also applied to gravitational anomalies (Ref. [14]). 2 We investigate the relationship between the covariant and consistent energy-momentum tensors, which is found to be expressed by a functional curl of the covariant energy-momentum tensor.
The rest of this paper is outlined as follows. In Sect. 2, we review the arguments of Banerjee et al. (Ref. [3]) concerning covariant and consistent gauge anomalies. In Sect. 3, we evaluate the functional curl of the covariant current explicitly by using the heat kernel method in arbitrary even dimensions. In Sect. 4, we apply the heat kernel method to Osabe and Suzuki's difference of the covariant and consistent currents (Ref. [8]) in two and four dimensions. In Sect. 5, by applying the arguments of Banerjee et al. (Ref. [3]) to the gravitational anomalies, we investigate the relationship between the covariant and consistent energy-momentum tensors. Section 6 is devoted to a summary and discussion.

Functional curl of the covariant current
We consider a chiral gauge theory given by the following 2n-dimensional Euclidean Lagrangian: where ψ and ψ are the Dirac spinors, and A a μ are the gauge fields. The metric we use is η μν = −δ μν . The Dirac gamma matrices γ μ are antihermitian, and γ 5 = i n γ 1 γ 2 · · · γ 2n is hermitian. The matrices γ μ and the hermitian generators T a satisfy where f abc are the structure constants of the gauge group. The Lagrangian L is invariant under these gauge transformations:

Covariant and consistent currents
Although the Lagrangian (2.1) is invariant under gauge transformations, the effective action is not. The effective action W [A a μ ] transforms as where the gauge anomaly G a (x) is defined by with the vacuum expectation value of the current J μa (x) given by We note that the consistent current J μa (x) cons given by Eq. (2.11) satisfies the integrability condition is gauge invariant, the current J μa (x) cons transforms covariantly under gauge transformation. In the anomalous gauge theory, however, W reg [A b ν ] is not gauge invariant and thus the current J μa (x) cons does not transform covariantly. The covariant current J μa (x) cov is the expectation value of current regularized covariantly with respect to gauge transformation. In contrast with the current J μa (x) cons , the J μa (x) cov transforms covariantly under the gauge transformation (2.7). Consequently, J μa (x) cov cannot be expressed in the form of Eq. (2.11) in the anomalous theory. In particular, the covariant current does not satisfy the integrability condition (2.12). These expectation values are functionals of A b ν . When we need to pay attention to the functional property, we use a symbol such as J μa (x) cov [A b ν ]. Substituting these regularized currents into Eq. (2.9), we obtain the following gauge anomalies: where G a cov (x) and G a cons (x) are called covariant and consistent, respectively. The consistent anomaly G a cons (x) satisfies the Wess-Zumino consistency condition (Ref. [1]), which is ascribed to the integrability condition (2.12).

Relationship between the covariant and consistent currents
We follow Banerjee et al. (Ref. [3]) in deriving the relationship between the covariant and consistent currents. We introduce a parameter g and define Note that the g-dependence arises only through the combination gA a μ ; we obtain where we have used the notation Expression (2.20) has only a formal meaning because the current J μa (x) g is not yet regularized.
The crucial point of the prescription of Ref. [3] is to substitute covariant current J μa (x) We can obtain a consistent current from the regularized effective action (2.22). Taking the functional derivative of Eq. (2.22) with respect to A a μ (x), we obtain the relationship between the covariant and consistent currents (Ref. [3]): . (2.23) Note that the "functional curl" of the covariant current appears in the integrand of the second term on the right-hand side. The functional curl in Eq. (2.23) is obtained by substituting gA a μ into A a μ in the functional curl which does not vanish since the covariant current does not satisfy the integrability condition (2.12) in the anomalous theory. 3 Taking the covariant divergence of Eq. (2.23), we obtain the relationship between the covariant and consistent gauge anomalies:  [3]) have evaluated the functional curl of the covariant current by using the fact that this curl has delta-function-type behavior at x = x : Using expression (2.26), they showed that the functional curl can be expressed by the covariant gauge anomaly, . (2.27) Substituting this equation into Eq. (2.25), they derived an expression for the consistent gauge anomaly that agrees with the result of Ref. [2]. In the arguments of Ref. [3] given above, it is crucial for expression (2.26) to actually hold. In Ref. [3], however, a detailed proof of expression (2.26) is not shown. Considering this point, we evaluate the functional curl explicitly in the next section.

Explicit evaluation of the functional curl of the covariant current
The expectation value of the current can be expressed by To regularize Eq. (3.1), we employ the Gaussian regulator to define a covariant current (Ref. [6]), where s is the cut-off parameter. Because the regulator e −s D 2 is covariant, the current J μa (x) cov transforms covariantly. Taking the functional curl of Eq. (3.2) and using trace properties, we have (Ref. [6]) . Here Fujikawa and Suzuki have shown that the right-hand side of Eq. (3.3) is equal to the functional derivative of the expression for the covariant anomaly (2.15) with respect to the field strength (Ref. [6]), which gives a formal proof of Eq. (2.27) and thus gives the proof of expression (2.26).
In the following, we evaluate the functional curl (3.3) explicitly by using the heat kernel method (Ref. [13]). The functional curl (3.3) can be expressed by where K(x, x ; s) is the heat kernel defined by Substituting the heat-kernel expansion where we have suppressed the symbol lim s→0 . The exponential function appearing on the right-hand side can be understood as the heat kernel of the free theory. That is, if we define where = ∂ μ ∂ μ . A formal solution to Eq. (3.9) can be written as Taking the Taylor expansion of e −s with respect to s, we have 4 With this formula and the integration formula Eq. (3.7) can be written as Considering that the terms higher than zeroth order in s vanish in the limit s → 0, we find that the indices k, l, and m of the surviving terms (3.13) satisfy the condition Here a k (x, x) is given by Eq. (B.7), starting with the term containing 2k factors of γ μ : where the dots on the right-hand side express terms with a lower power of γ μ . Substituting Eq. (B.7) into Eq. (3.18), we obtain the final expression for the functional curl, where the symbol "Str" denotes the symmetrized trace (Ref. [15]) indicating that the factors in the trace are to be totally symmetrized. We notice here that our evaluation gives a direct proof of expression (2.26). Comparing this expression with the final expression of the covariant anomaly (2.16), we again obtain Eq. (2.27).

Explicit evaluation of Osabe and Suzuki's expression for the current difference
Osabe and Suzuki (Ref. [8]) have also discussed the difference between the consistent and covariant currents. Their consistent current J μa (x) cons can be written as in our notation, while the covariant current J μa (x) cov is given by Eq. (3.2), i.e., From these definitions, they derived an expression for the difference between currents. Their derivation can be explained essentially as follows: Introducing / D g = γ μ D g μ = γ μ (∂ μ +igA μ ) and noticing the equality In the third line, we have used the identity Equation (4.4) is equivalent to Osabe and Suzuki's expression for the current difference (Ref. [8]). Now, we calculate current difference (4.4) by applying the heat kernel method. Introducing heat kernels we express Eq. (4.4) as where / A = γ ν A ν (x ) and we have omitted the parity-conserving terms since only parity-violating terms contribute to the anomalies. These kernels K g andK g are not independent of each other. In fact, owing to the relation they satisfyK (4.10) We expand K g (x, x ; s) andK g (x, x ; s) in 2n dimensions as (4.12) Note here that Eq. (4.10) indicatesb where we have suppressed the symbol lim s→0 . With the help of Eqs. (3.11) and (3.12), Eq. (4.14) becomes Eq. (4.14) = i 2 Note that the terms higher than zeroth order in s vanish in the limit s → 0; we find that the indices k, l, and m of the surviving terms on the right-hand side satisfy the condition Below, we work in two and four dimensions.

Relationship between the covariant and consistent energy-momentum tensors
In this section, we apply the arguments of Banerjee et al. (Ref. [3]), as explained in Sect. 2, to gravitational anomalies (Ref. [14]). The vacuum expectation value of the energy-momentum tensor density eT a μ (x) is expressed by the effective action W [e ν b ]: where e μ a is the vielbein field, e = det e a μ , and e a μ is the inverse matrix of e μ a . Gravitational anomalies appear as nonzero values of D μ T a μ (x) (Einstein anomaly) and/or is a divergent quantity. To treat the energymomentum tensor eT a μ (x) meaningfully, we should adopt an appropriate regularization, either consistent or covariant. The consistently regularized energy-momentum tensor eT a μ (x) cons is defined by the regularized effective action W [e ν b ] reg as The covariant energy-momentum tensor eT a μ (x) cov is the expectation value of the energymomentum tensor regularized covariantly with respect to both the general coordinate and local where we have dropped the W t=0 term, since it is an e We can obtain a consistent energy-momentum tensor from the regularized effective action (5.10).
Taking the variation of Eq. (5.10) with respect to e μ a , we obtain the following relationship between the covariant and consistent energy-momentum tensors:  t). (5.12) We emphasize that the "functional curl" of the covariant energy-momentum tensor appears in Eq. (5.11). This curl vanishes only when the theory is not anomalous. In fact, if the theory is anomaly free, the regularized effective action is invariant under the general coordinate and local Lorentz transformations. In this case, the consistent energy-momentum tensor becomes covariant, and thus the covariant energy-momentum tensor satisfies the integrability condition, i.e., the condition of vanishing functional curl. Conversely, if the functional curl of the covariant energy-momentum tensor is zero, the consistent energy-momentum tensor coincides with the covariant one, as seen from Eq. (5.11). In this case, the covariant and consistent gravitational anomalies coincide with each other. The diagrammatic approach to the anomaly, however, tells us that the leading terms of these anomalies differ by the Bose-symmetrization factor 1/(n + 1) in 2n dimensions. This is true only when both anomalies are zero. Thus, the vanishing functional curl indicates an anomaly-free theory.
The relationships between the covariant and consistent gravitational anomalies are derived immediately from Eq.
Taking the covariant divergence of both sides, we obtain a relationship between the covariant and consistent Einstein anomalies: . (5.14) The relationship between the Lorentz anomalies can be similarly obtained.

Summary and discussion
In Sect. 3, we evaluated the functional curl of the covariant current explicitly using the heat kernel method in arbitrary even dimensions. The result gives a direct proof of the delta-function-type behavior of the functional curl. Our explicit form of this curl leads immediately to the relationship between the covariant and consistent currents presented by Bardeen and Zumino (Refs. [2,3]). In Sect. 4, we applied the heat kernel method to evaluate Osabe and Suzuki's results of the difference between the covariant and consistent currents (Ref. [8]) in two and four dimensions. The results are the same as previously reported (Refs. [2,3]). In Sect. 5, applying the arguments of Banerjee et al. (Ref. [3]) to gravitational anomalies, we have investigated the relationship between the covariant and consistent energy-momentum tensors. The relation is found to be expressed by the functional curl of the covariant energy-momentum tensor.
The energy-momentum tensors considered in Sect. 5 have both Einstein and Lorentz anomalies in general. As shown in Ref. [2], these anomalies are not independent of each other. Moreover, using the regularization ambiguity, we can always choose the energy-momentum tensor to have either a vanishing Lorentz anomaly or a vanishing Einstein anomaly. From the covariant regularization viewpoint, this is explained below. 13 Given a covariantly regularized energy-momentum tensor T μν cov , we have in general both the Einstein anomaly D μ T μν cov and the Lorentz anomaly T [μν] cov . Note that these covariant gravitational anomalies are local polynomials of the Riemann curvature (and its derivative for the Einstein anomaly). Because of the regularization ambiguity, we can add a finite, local, and covariant counterterm to T μν cov to obtain another covariantly regularized energy-momentum tensor. Adopting the Lorentz anomaly as a counterterm, we can obtain a Lorentz-anomaly-free energy-momentum tensor, which gives the pure covariant Einstein anomaly D μ T μν pE cov . Since the energy-momentum tensor T μν pE cov , given above, is nothing but the symmetric part of T μν cov , we can say that the pure covariant Einstein anomaly is the covariant divergence of the symmetric part of the covariant energymomentum tensor (Refs. [6,14,17,18]).
We can also define a covariant energy-momentum tensor with a vanishing Einstein anomaly. It is known from Refs. [6,14,[17][18][19][20] that the pure covariant Einstein anomaly has the form D μ T μν pE cov = −D μ L μν , (6.2) where L μν is a local polynomial of Riemann curvature 5 and is antisymmetric with respect to the indices μ and ν. To obtain an Einstein-anomaly-free energy-momentum tensor T μν pL cov , we may adopt L μν as a local counterterm to T μν In Sect. 5, we defined a regularized effective action using the covariant energy-momentum tensor (Eq. (5.10)). Since the covariant energy-momentum tensor retains the ambiguity of adding covariant local curvature and vielbein polynomials, corresponding ambiguity arises in the effective action (5.10). Then, one might wonder what kind of covariant energy-momentum tensor leads to the Lorentz-anomaly-free effective action, which is local Lorentz invariant but which does not have general coordinate invariance. It can be seen that the Lorentz-anomaly-free covariant energy-momentum tensor does not necessarily lead to a Lorentz-anomaly-free effective action. In fact, for spin-1/2 chiral fermions in two-dimensional space-time, an explicit calculation with the use of a symmetric covariant energy-momentum tensor eT μν pE cov [e λ b (t)] to define the effective action (5.10) shows that the second term on the right-hand side of Eq. (5.13) contributes to the consistent Lorentz anomaly. Thus, obtaining a Lorentz-anomaly-free (or Einstein-anomaly-free) consistent energy-momentum tensor is not yet straightforward in the context of Eq. (5.11). Future work will aim to clarify these points.