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

TL;DR

This paper uses the heat kernel method to explicitly evaluate the relationship between covariant and consistent currents and energy-momentum tensors in anomalous chiral gauge theories across various dimensions.

Contribution

It provides explicit calculations of the functional curl relating these currents and tensors using the heat kernel approach, extending previous results to arbitrary even dimensions.

Findings

Explicit expression for the functional curl in arbitrary even dimensions.

Evaluation of differences between covariant and consistent currents in 2D and 4D.

Extension of the relation to gravitational anomalies.

Abstract

We apply the heat kernel method to relations between covariant and consistent currents in anomalous 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.

Equations

$$\mathcal{L}=\overline{\psi }{\gamma }^{\mu }({\partial }_{\mu }+i{A}_{\mu }^a{T}^a\frac{1-{\gamma }_5}{2})\psi ,$$

$$\{{\gamma }^{\mu },{\gamma }^{\nu }\}=-2{\delta }^{\mu \nu },$$

$$[T^a,T^b]=i{f}^{abc}{T}^c,$$

$$\text{tr}{T}^a{T}^b=\frac{1}{2}{\delta }^{ab},$$

$${\delta }_{\alpha }\psi (x)=i{\alpha }^a(x)T^a\frac{1-{\gamma }_5}{2}\psi (x),$$

$${\delta }_{\alpha }\overline{\psi }(x)=-i\overline{\psi }(x){\alpha }^a(x)T^a\frac{1+{\gamma }_5}{2},$$

$${\delta }_{\alpha }{A}_{\mu }^a(x)=-D_{\mu }{\alpha }^a(x)=-{\partial }_{\mu }{\alpha }^a(x)+f^{abc}{A}_{\mu }^b(x){\alpha }^c(x).$$

$${\delta }_{\alpha }W[A_{\mu }^a]$$

$$=i\int dx{\alpha }^a(x)G^a(x),$$

$$G^a(x)=D_{\mu }⟨J^{\mu a}(x)⟩$$

$$⟨J^{\mu a}(x)⟩=\frac{1}{i}\frac{\delta }{\delta A_{\mu }^a(x)}W[A_{\nu }^b].$$

$$⟨J^{\mu a}(x){⟩}_{\text{cons}}=\frac{1}{i}\frac{\delta }{\delta A_{\mu }^a(x)}{W}_{\text{reg}}[A_{\nu }^b].$$

$$\frac{\delta }{\delta A_{\mu }^a(x)}⟨J^{\nu b}(x^{\prime }){⟩}_{\text{cons}}-\frac{\delta }{\delta A_{\nu }^b(x^{\prime })}⟨J^{\mu a}(x){⟩}_{\text{cons}}=0.$$

$$G_{\text{cov}}^a(x)=D_{\mu }⟨J^{\mu a}(x){⟩}_{\text{cov}}$$

$$G_{\text{cons}}^a(x)=D_{\mu }⟨J^{\mu a}(x){⟩}_{\text{cons}},$$

$$G_{\text{cov}}^a(x)=\underset{s\to 0}{\lim }\underset{x^{\prime }\to x}{\lim }\text{tr}{T}^a{\gamma }_5{e}^{-s/D^2}\delta (x-x^{\prime }),$$

$$G_{\text{cov}}^a(x)=\frac{(-1{)}^n}{(4\pi {)}^{n}n!}{\epsilon }^{{\mu }_1{\nu }_1\cdots {\mu }_n{\nu }_n}\text{tr}{T}^a{F}_{{\mu }_1{\nu }_1}(x)\cdots F_{{\mu }_n{\nu }_n}(x),$$

$$W_g=W[g{A}_{\mu }^a].$$

$$W[A_{\mu }^a]={\int }_0^{1}dg\frac{\partial W_g}{\partial g}+W_{g=0}.$$

$$W[A_{\mu}^{a}]=\int_{0}^{1}dg\int dx\,\frac{\delta W_{g}}{\delta\bigl{(}gA_{\mu}^{a}(x)\bigr{)}}A_{\mu}^{a}(x),$$

$$W[A_{\mu }^a]=i{\int }_0^{1}dg\int dx{A}_{\mu }^a(x)⟨J^{\mu a}(x){⟩}^g,$$

$$⟨J^{\mu a}(x){⟩}^g=⟨J^{\mu a}(x)⟩[g{A}_{\nu }^b].$$

$$W[A_{\mu }^a{]}_{\text{reg}}=i{\int }_0^{1}dg\int dx{A}_{\mu }^a(x)⟨J^{\mu a}(x){⟩}_{\text{cov}}^g.$$

$$⟨J^{\mu a}(x){⟩}_{\text{cons}}=⟨J^{\mu a}(x){⟩}_{\text{cov}}+{\int }_0^{1}dg\int d{x}^{\prime }g{A}_{\nu }^b(x^{\prime })\left\{\frac{\delta ⟨J^{\nu b}(x^{\prime }){⟩}_{\text{cov}}^g}{\delta \left(g{A}_{\mu }^a(x)\right)}-\frac{\delta ⟨J^{\mu a}(x){⟩}_{\text{cov}}^g}{\delta \left(g{A}_{\nu }^b(x^{\prime })\right)}\right\}.$$

$$\frac{\delta ⟨J^{\nu b}(x^{\prime }){⟩}_{\text{cov}}}{\delta A_{\mu }^a(x)}-\frac{\delta ⟨J^{\mu a}(x){⟩}_{\text{cov}}}{\delta A_{\nu }^b(x^{\prime })},$$

$$G_{\text{cons}}^a(x)=G_{\text{cov}}^a(x)+D_{\mu }^{ac}{\int }_0^{1}dg\int d{x}^{\prime }g{A}_{\nu }^b(x^{\prime })\left\{\frac{\delta ⟨J^{\nu b}(x^{\prime }){⟩}_{\text{cov}}^g}{\delta \left(g{A}_{\mu }^c(x)\right)}-\frac{\delta ⟨J^{\mu c}(x){⟩}_{\text{cov}}^g}{\delta \left(g{A}_{\nu }^b(x^{\prime })\right)}\right\},$$

$$\frac{\delta ⟨J^{\nu b}(x^{\prime }){⟩}_{\text{cov}}}{\delta A_{\mu }^a(x)}-\frac{\delta ⟨J^{\mu a}(x){⟩}_{\text{cov}}}{\delta A_{\nu }^b(x^{\prime })}\propto \delta (x-x^{\prime }).$$

$$\frac{\delta ⟨J^{\nu b}(x^{\prime }){⟩}_{\text{cov}}}{\delta A_{\mu }^a(x)}-\frac{\delta ⟨J^{\mu a}(x){⟩}_{\text{cov}}}{\delta A_{\nu }^b(x^{\prime })}=-2\frac{\delta G_{\text{cov}}^b(x^{\prime })}{\delta F_{\mu \nu }^a(x)}.$$

$$⟨J^{\mu a}(x)⟩$$

$$\displaystyle=\lim_{x^{\prime}\to x}\text{tr}\frac{1+\gamma_{5}}{2}\gamma^{\mu}T^{a}\frac{1}{{{\ooalign{\hfil/\hfil\crcr$D$}}}}\delta(x-x^{\prime}),$$

$$\langle J^{\mu a}(x)\rangle_{\textrm{cov}}=\lim_{s\to 0}\lim_{x^{\prime}\to x}\text{tr}\frac{1+\gamma_{5}}{2}\gamma^{\mu}T^{a}\frac{1}{{{\ooalign{\hfil/\hfil\crcr$D$}}}}e^{-s\not D^{2}}\delta(x-x^{\prime}),$$

$$\frac{\delta ⟨J^{\nu b}(x^{\prime }){⟩}_{\text{cov}}}{\delta A_{\mu }^a(x)}-\frac{\delta ⟨J^{\mu a}(x){⟩}_{\text{cov}}}{\delta A_{\nu }^b(x^{\prime })}$$

$$=-i\underset{s\to 0}{\lim }s\text{tr}{\gamma }_5{\gamma }^{\nu }{T}^b{\int }_0^{1}d\alpha \left(e^{-(1-\alpha )s/D^{\prime 2}}\delta (x^{\prime }-x)\right){\gamma }^{\mu }{T}^a{e}^{-\alpha s/D^2}\delta (x-x^{\prime }),$$

$$\frac{\delta ⟨J^{\nu b}(x^{\prime }){⟩}_{\text{cov}}}{\delta A_{\mu }^a(x)}-\frac{\delta ⟨J^{\mu a}(x){⟩}_{\text{cov}}}{\delta A_{\nu }^b(x^{\prime })}$$

Full text

I 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 the Wess–Zumino consistency condition Wess_Zumino_cons .

The covariant and consistent anomalies are known to be equivalent in the sense that they lead to the same anomaly-cancellation condition. Bardeen and Zumino Bardeen_Zumino_gauge_gravity 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 Banerjee_Banerjee_Mitra ; Banerjee_Banerjee ; Banerjee_Banerjee_point_splitting ; Fujikawa_Suzuki ; Suzuki_SUSY ; Osabe_Suzuki . In particlular, Banerjee et al. Banerjee_Banerjee_Mitra 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. Banerjee_Banerjee_Mitra 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 Banerjee_Banerjee_Mitra . As a result, the relationship between the covariant and consistent currents is expressed by a “functional curl” of the covariant current111The functional curl of the covariant current appears also in the covariant commutator anomaly Dunne_Trugenberger ; Kelnhofer . . The authors of Banerjee_Banerjee_Mitra 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 Bardeen_Zumino_gauge_gravity , 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 Grisaru_Penati ; Dunne_Trugenberger ; Kelnhofer ; Fujikawa_Suzuki ; Suzuki_SUSY ; Z_Qiu_H_Ren . Fujikawa and Suzuki Fujikawa_Suzuki gave a formal proof of the relationship between the functional curl and the covariant anomaly; this relation was derived by Banerjee et al. Banerjee_Banerjee_Mitra using the delta-function-type behavior of the functional curl. Ohshima et al. Suzuki_SUSY 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 Z_Qiu_H_Ren 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. Banerjee_Banerjee ; Banerjee_Banerjee_point_splitting ; Osabe_Suzuki , 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 Banerjee_Banerjee and the point-splitting method Banerjee_Banerjee_point_splitting . Osabe and Suzuki Osabe_Suzuki 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 DeWitt_HK , we evaluate the functional curl of the covariant current explicitly. The curl that we derive agrees with that of Refs. Banerjee_Banerjee_Mitra ; Fujikawa_Suzuki . 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 Osabe_Suzuki . This difference, which we calculate in two and four dimensions, agrees with the previous results Bardeen_Zumino_gauge_gravity ; Banerjee_Banerjee_Mitra . The arguments of Banerjee et al. Banerjee_Banerjee_Mitra are also applied to gravitational anomalies Alvarez-Gaume_Witten 222The equivalence of the covariant and consistent gravitational anomalies is also shown in Ref.Bardeen_Zumino_gauge_gravity by the algebraic approach. We are interested here in the field theoretical approach to the equivalence. . 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 the rest of this paper is outlined as follows. In Sect. II, we review the arguments of Banerjee et al. Banerjee_Banerjee_Mitra concerning covariant and consistent gauge anomalies. In Sect. III, we evaluate the functional curl of the covariant current explicitly by using the heat kernel method in arbitrary even dimensions. In Sect. IV, we apply the heat kernel method to Osabe and Suzuki’s difference of the covariant and consistent currents Osabe_Suzuki in two and four dimensions. In Sect. V, by applying the arguments of Banerjee et al. Banerjee_Banerjee_Mitra to the gravitational anomalies, we investigate the relationship between the covariant and consistent energy–momentum tensors. Section VI is devoted to a summary and discussion.

II Functional curl of the covariant current

We consider a chiral gauge theory given by the following $2n$-dimensional Euclidean Lagrangian

[TABLE]

where $\psi$ and $\overline{\psi}$ are the Dirac spinors, and $A_{\mu}^{a}$ are the gauge fields. The metric we use is $\eta^{\mu\nu}=-\delta^{\mu\nu}$. The Dirac gamma matrices $\gamma^{\mu}$ are anti-hermitian, and $\gamma_{5}=i^{n}\gamma^{1}\gamma^{2}\cdots\gamma^{2n}$ is hermitian. The matrices $\gamma^{\mu}$ and the hermitian generators $T^{a}$ satisfy

[TABLE]

where $f^{abc}$ are the structure constants of the gauge group. The Lagrangian $\mathscr{L}$ is invariant under these gauge transformations:

[TABLE]

II.1 Covariant and consistent currents

Although Lagrangian (1) is invariant under gauge transformations, the effective action is not. The effective action $W[A_{\mu}^{a}]$ transforms as

[TABLE]

where the gauge anomaly $G^{a}(x)$ is defined by

[TABLE]

with the vacuum expectation value of the current $\langle J^{\mu a}(x)\rangle$ given by

[TABLE]

These expressions have only formal meanings, i.e., $\langle J^{\mu a}(x)\rangle$ is divergent since $W[A_{\nu}^{b}]$ is divergent. To treat current (10) meaningfully, we should adopt an appropriate regularization.

We usually adopt either the consistent or covariant regularization. The consistently regularized current $\langle J^{\mu a}(x)\rangle_{\text{cons}}$ is defined through the regularization of the effective action, $W[A_{\nu}^{b}]$. Using the regularized effective action $W_{\text{reg}}[A_{\nu}^{b}]$, we define a regularized current

[TABLE]

We note that the consistent current $\langle J^{\mu a}(x)\rangle_{\text{cons}}$ given by (11) satisfies the integrability condition:

[TABLE]

If $W_{\text{reg}}[A_{\nu}^{b}]$ is gauge invariant, the current $\langle J^{\mu a}(x)\rangle_{\text{cons}}$ transforms covariantly under gauge transformation. In the anomalous gauge theory, however, $W_{\text{reg}}[A_{\nu}^{b}]$ is not gauge invariant and thus the current $\langle J^{\mu a}(x)\rangle_{\text{cons}}$ does not transform covariantly.

The covariant current, $\langle J^{\mu a}(x)\rangle_{\text{cov}}$, is the expectation value of current regularized covariantly with respect to gauge transformation. In contrast with the current $\langle J^{\mu a}(x)\rangle_{\text{cons}}$, $\langle J^{\mu a}(x)\rangle_{\text{cov}}$ transforms covariantly under the gauge transformation (7). Consequently, $\langle J^{\mu a}(x)\rangle_{\text{cov}}$ cannot be expressed in the form of (11) in the anomalous theory. In particular, the covariant current does not satisfy the integrability condition (12). These expectation values are functionals of $A_{\nu}^{b}$. When we need to pay attention to the functional property, we use a symbol such as $\langle J^{\mu a}(x)\rangle_{\text{cov}}[A_{\nu}^{b}]$.

Substituting these regularized currents into equation (9), we obtain the following gauge anomalies:

[TABLE]

and

[TABLE]

where $G^{a}_{\textrm{cov}}(x)$ and $G^{a}_{\textrm{cons}}(x)$ are called covariant and consistent, respectively. The consistent anomaly, $G^{a}_{\textrm{cons}}(x)$, satisfies the Wess–Zumino consistency condition Wess_Zumino_cons , which is ascribed to the integrability condition (12).

The covariant anomaly $G^{a}_{\text{cov}}(x)$ can be expressed as (see, for example, Fujikawa_Suzuki )

[TABLE]

where $s$ is the cut-off parameter and {\ooalign{\hfil/\hfil\crcrD}}=\gamma^{\mu}(\partial_{\mu}+iA_{\mu}^{a}T^{a}). This quantity can be calculated Fujikawa_Suzuki as

[TABLE]

where $\varepsilon^{\mu_{1}\nu_{1}\dotsm\mu_{n}\nu_{n}}$ is the totally antisymmetric tensor with $\varepsilon^{12\dotsm 2n}=1$ and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+i[A_{\mu},A_{\nu}]$ is the field strength of the gauge fields $A_{\mu}=A_{\mu}^{a}T^{a}$. The covariant anomaly $G^{a}_{\textrm{cov}}(x)$ is a finite local polynomial of field strength $F_{\mu\nu}(x)$.

II.2 Relationship between the covariant and consistent currents

We follow Banerjee et al. Banerjee_Banerjee_Mitra in deriving the relationship between the covariant and consistent currents. We introduce a parameter $g$ and define

[TABLE]

If we put $g=1$, $W_{g}$ reduces to the original effective action $W[A_{\mu}^{a}]$. We can express $W[A_{\mu}^{a}]$ using $W_{g}$ as

[TABLE]

Note that the $g$-dependence arises only through the combination $gA_{\mu}^{a}$, we obtain

[TABLE]

where we have dropped the $W_{g=0}$ term since it is an $A_{\mu}^{a}$-independent constant. From definition (10), we rewrite (19) as

[TABLE]

where we have used the notation

[TABLE]

Expression (20) has only a formal meaning because the current $\langle J^{\mu a}(x)\rangle^{g}$ is not yet regularized. The crucial point of the prescription of Ref. Banerjee_Banerjee_Mitra is to substitute covariant current $\langle J^{\mu a}(x)\rangle^{g}_{\text{cov}}=\langle J^{\mu a}(x)\rangle_{\text{cov}}[gA_{\nu}^{b}]$ for $\langle J^{\mu a}(x)\rangle^{g}$ in (20) to construct a regularized effective action, $W[A_{\mu}^{a}]_{\text{reg}}$:

[TABLE]

We can obtain a consistent current from the regularized effective action (22). Taking the functional derivative of (22) with respect to $A_{\mu}^{a}(x)$, we obtain the relationship between the covariant and consistent currents Banerjee_Banerjee_Mitra :

[TABLE]

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 (23) is obtained by substituting $gA_{\mu}^{a}$ into $A_{\mu}^{a}$ in the functional curl

[TABLE]

which does not vanish since the covariant current does not satisfy the integrability condition (12) in the anomalous theory.333It can be seen that the parity-conserving part of the functional curl vanishes. Taking the covariant divergence of (23), we obtain the relationship between the covariant and consistent gauge anomalies:

[TABLE]

where $D_{\mu}^{ac}=\delta^{ac}\partial_{\mu}-f^{adc}A_{\nu}^{d}$.

Banerjee et al. Banerjee_Banerjee_Mitra have evaluated the functional curl of the covariant current by using the fact that this curl has delta-function-type behavior at $x=x^{\prime}$:

[TABLE]

Using (26), they showed that the functional curl can be expressed by the covariant gauge anomaly,

[TABLE]

Substituting this equation into (25), they derived an expression for the consistent gauge anomaly that agrees with the result of Ref. Bardeen_Zumino_gauge_gravity . In the arguments of Ref. Banerjee_Banerjee_Mitra given above, it is crucial for equation (26) to actually hold. In Ref. Banerjee_Banerjee_Mitra , however, a detailed proof of (26) is not shown. Considering this point, we evaluate the functional curl explicitly in the next section.

III Explicit evaluation of the functional curl of the covariant current

The expectation value of the current can be expressed by

[TABLE]

where {\ooalign{\hfil/\hfil\crcrD}}=\gamma^{\mu}(\partial_{\mu}+iA_{\mu}^{a}T^{a}). To regularize (30), we employ the Gaussian regulator to define a covariant current Fujikawa_Suzuki ,

[TABLE]

where $s$ is the cut-off parameter. Because the regulator $e^{-s\not D^{2}}$ is covariant, the current $\langle J^{\mu a}(x)\rangle_{\textrm{cov}}$ transforms covariantly. Taking the functional curl of (31) and using trace properties, we have Fujikawa_Suzuki

[TABLE]

where {\ooalign{\hfil/\hfil\crcrD}}^{\prime}=\gamma^{\mu}(\partial_{\mu}^{\prime}+iA_{\mu}^{a}(x^{\prime})T^{a}). Here Fujikawa and Suzuki have shown that the right-hand side of (32) is equal to the functional derivative of the expression for the covariant anomaly (15) with respect to the field strength Fujikawa_Suzuki , which gives a formal proof of (27) and thus gives the proof of (26).

In the following, we evaluate the functional curl (32) explicitly by using the heat kernel method DeWitt_HK . The functional curl (32) can be expressed by

[TABLE]

where $K(x,x^{\prime};s)$ is the heat kernel defined by

[TABLE]

Substituting the heat-kernel expansion

[TABLE]

into equation (33), we have

[TABLE]

where we have suppressed the symbol $\displaystyle\lim_{s\to 0}$. 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

[TABLE]

then $K_{0}(x,x^{\prime};s)$ satisfies

[TABLE]

where $\square=\partial_{\mu}\partial^{\mu}$. A formal solution to (38) can be written as

[TABLE]

Taking the Taylor expansion of $e^{-s\square}$ with respect to $s$, we have 444A proof of (40) using test functions is given in Appendix A.

[TABLE]

With this formula and the integration formula

[TABLE]

equation (36) can be written as

[TABLE]

Considering that the terms higher than [math]-th order in $s$ vanish in the limit $s\to 0$, we find that the indices $k$, $l$, and $m$ of the surviving terms (42) satisfy the condition

[TABLE]

In addition, the surviving terms must contain at least $2n$ factors of gamma matrices $\gamma^{\mu}$, because of the existence of $\gamma_{5}$ in the trace over spinor indices. As shown in Appendix B, $a_{k}(x,x^{\prime})$ contains at most $2k$ factors of $\gamma^{\mu}$. Consequently, indices $k$ and $l$ of the surviving terms satisfy the condition

[TABLE]

Conditions (43) and (44) lead to

[TABLE]

Then, (42) becomes

[TABLE]

$a_{k}(x,x)$ is given by (162), starting with the term containing $2k$ factors of $\gamma^{\mu}$:

[TABLE]

where the dots on the right-hand side express terms with lower power of $\gamma^{\mu}$. Substituting (162) into (47), we obtain the final expression for the functional curl,

[TABLE]

where the symbol “Str” denotes the symmetrized trace Zumino_Y_S indicating that the factors in the trace are to be totally symmetrized. We notice here that our evaluation gives a direct proof of (26). Comparing this expression with the final expression of the covariant anomaly (16), we again obtain (27).

IV Explicit evaluation of Osabe and Suzuki’s expression for the current difference

Osabe and Suzuki Osabe_Suzuki have also discussed the difference between the consistent and covariant currents. Their consistent current, $\langle J^{\mu a}(x)\rangle_{\textrm{cons}}$, can be written as

[TABLE]

in our notation, while the covariant current, $\langle J^{\mu a}(x)\rangle_{\textrm{cov}}$, is given by (31), i.e.,

[TABLE]

From these definitions, they derived an expression for the difference between currents. Their derivation can be explained essentially as follows: Introducing {\ooalign{\hfil/\hfil\crcrD}}_{g}=\gamma^{\mu}D_{\mu}^{g}=\gamma^{\mu}(\partial_{\mu}+igA_{\mu}) and noticing the equality

[TABLE]

we obtain

[TABLE]

In the third line, we have used the identity:

[TABLE]

Equation (64) is equivalent to Osabe and Suzuki’s expression for the current difference Osabe_Suzuki .

Now, we calculate current difference (64) by applying the heat kernel method. Introducing heat kernels

[TABLE]

we express (64) as

[TABLE]

where {\ooalign{\hfil/\hfil\crcrA}}^{\prime}=\gamma^{\nu}A_{\nu}(x^{\prime}) and we have omitted the parity-conserving terms since only parity-violating terms contribute to the anomalies. These kernels $K_{g}$ and $\tilde{K}_{g}$ are not independent of each other. In fact, owing to the relation

[TABLE]

they satisfy

[TABLE]

We expand $K_{g}(x,x^{\prime};s)$ and $\tilde{K}_{g}(x,x^{\prime};s)$ in $2n$ dimensions as follows:

[TABLE]

Note here that (72) indicates

[TABLE]

Substituting expansions (73) and (74) into (70), we have

[TABLE]

where we have suppressed the symbol $\displaystyle\lim_{s\to 0}$. With the help of (40) and (41), equation (78) becomes

[TABLE]

Note that the terms higher than [math]-th order in $s$ vanish in the limit $s\to 0$, we find that the indices $k$, $l$, and $m$ of the surviving terms on the right-hand side satisfy the condition

[TABLE]

Below, we work in two and four dimensions.

In two dimensions ($n=1$), the condition (82) becomes

[TABLE]

which means that $k=l=m=0$. Thus, equation (81) reads

[TABLE]

where we have used the coincidence limits $b_{0}(x,x)=\tilde{b}_{0}(x,x)=\bm{1}$ ((166) and (75)). This agrees with the previous results Bardeen_Zumino_gauge_gravity ; Banerjee_Banerjee_Mitra .

In four dimensions ($n=2$), the condition (82) becomes

[TABLE]

The solutions of this condition are $(k,l,m)=(0,0,0),(1,0,0),(0,1,0),(0,0,1)$. Calculating the four terms corresponding to these solutions, we obtain

[TABLE]

where we have used Synge’s symbol Synge to denote coincidence limits such as $[\partial_{\alpha}b_{0}]=\displaystyle\lim_{x^{\prime}\to x}\partial_{\alpha}b_{0}(x,x^{\prime})$. Since $[b_{0}]=[\tilde{b}_{0}]=\bm{1}$, the first term in the integrand of (108) vanishes after taking the trace over the spinor indices. With the help of coincidence limits (166), (182), and (75), the second and third terms become

[TABLE]

The last term in (108) can be calculated as follows. Note that

[TABLE]

The coincidence limit of (130) can be evaluated by using (166), (180), (181), and (75); thus, we obtain

[TABLE]

From these results, we finally obtain

[TABLE]

which agrees with the previous results Bardeen_Zumino_gauge_gravity ; Banerjee_Banerjee_Mitra .

V Relationship between the covariant and consistent energy–momentum tensors

In this section, we apply the arguments of Banerjee et al. Banerjee_Banerjee_Mitra , as explained in Sect. II, to gravitational anomalies Alvarez-Gaume_Witten . The vacuum expectation value of the energy–momentum tensor density $\langle eT^{a}_{\ \mu}(x)\rangle$ is expressed by the effective action $W[e_{b}^{\ \nu}]$:

[TABLE]

where $e_{a}^{\ \mu}$ is the vielbein field, $e=\text{det}\,e^{a}_{\ \mu}$ and $e^{a}_{\ \mu}$ is the inverse matrix of $e_{a}^{\ \mu}$. Gravitational anomalies appear as non-zero values of $D^{\mu}\langle T^{a}_{\ \mu}(x)\rangle$ (Einstein anomaly) and/or $\langle T_{[ab]}(x)\rangle=\frac{1}{2}\langle T_{ab}(x)-T_{ba}(x)\rangle$ (Lorentz anomaly).

In equation (133), $\langle eT^{a}_{\ \mu}(x)\rangle$ is ill-defined because $W[e_{b}^{\ \nu}]$ is a divergent quantity. To treat the energy–momentum tensor $\langle eT^{a}_{\ \mu}(x)\rangle$ meaningfully, we should adopt an appropriate regularization, either consistent or covariant. The consistently regularized energy–momentum tensor $\langle eT^{a}_{\ \mu}(x)\rangle_{\text{cons}}$ is defined by the regularized effective action $W[e_{b}^{\ \nu}]_{\text{reg}}$ as

[TABLE]

The covariant energy–momentum tensor $\langle eT^{a}_{\ \mu}(x)\rangle_{\text{cov}}$ is the expectation value of the energy–momentum tensor regularized covariantly with respect to both the general coordinate and local Lorentz transformations. These expectation values are functionals of $e_{a}^{\ \mu}(x)$. When we need to pay attention to the functional property, we use a symbol such as $\langle eT^{a}_{\ \mu}(x)\rangle_{\text{cov}}[e_{b}^{\ \nu}]$.

Now, we introduce a vielbein field, $e_{a}^{\ \mu}(t)=e_{a}^{\ \mu}(x,t)$, with one parameter $t$, which connects the original vielbein $e_{a}^{\ \mu}(x)$ to the flat space–time vielbein $\delta_{a}^{\mu}$ such that

[TABLE]

For example, we may adopt $e_{a}^{\ \mu}(t)=\delta_{a}^{\mu}+t(e_{a}^{\ \mu}(x)-\delta_{a}^{\mu})$ or $e_{a}^{\ \mu}(t)=(e^{tH})_{a}^{\ \mu}$ with the matrix $H_{a}^{\ \mu}=(\ln e)_{a}^{\ \mu}$. We define a $t$-parametrized effective action $W_{t}$ by

[TABLE]

which reduces to the original effective action $W[e_{a}^{\ \mu}]$ if $t=1$. We can express the effective action $W[e_{a}^{\ \mu}]$ by using $W_{t}$ as

[TABLE]

Note that the $t$-dependence of $W_{t}$ arises only through $e_{a}^{\ \mu}(t)$, we obtain

[TABLE]

where we have dropped the $W_{t=0}$ term, since it is an $e_{a}^{\ \mu}$-independent constant. From definition (133), we rewrite this equation as

[TABLE]

where we have used the notation

[TABLE]

To construct a regularized effective action, $W_{\text{reg}}[e_{a}^{\ \mu}]$, we substitute the covariant energy–momentum tensor $\langle eT^{a}_{\ \mu}\rangle^{t}_{\text{cov}}=\langle eT^{a}_{\ \mu}\rangle_{\text{cov}}[e_{b}^{\ \nu}(t)]$ for $\langle eT^{a}_{\ \mu}\rangle^{t}$ on the right-hand side of equation (140):

[TABLE]

We can obtain a consistent energy–momentum tensor from the regularized effective action (142). Taking the variation of (142) with respect to $e_{a}^{\ \mu}$, we obtain the following relationship between the covariant and consistent energy–momentum tensors:

[TABLE]

where we have applied integration by parts to the first term in the second line and used the fact that the $t$-dependence of $\langle eT^{a}_{\ \mu}\rangle^{t}_{\text{cov}}$ arises only through $e_{a}^{\ \mu}(t)$. In (143), primed indices denote those attached at the point $x^{\prime}$ such as

[TABLE]

We emphasize that the “functional curl” of the covariant energy–momentum tensor appears in (143). 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 (143). 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 (143). For example, if we adopt the parametrization $e_{a}^{\ \mu}(t)=\delta_{a}^{\mu}+t(e_{a}^{\ \mu}(x)-\delta_{a}^{\mu})$, equation (143) becomes

[TABLE]

Taking the covariant divergence of both sides, we obtain a relationship between the covariant and consistent Einstein anomalies

[TABLE]

The relationship between the Lorentz anomalies can be similarly obtained.

VI Summary and discussion

In Sect. III, 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 Bardeen_Zumino_gauge_gravity ; Banerjee_Banerjee_Mitra . In Sect. IV, we applied the heat kernel method to evaluate Osabe and Suzuki’s results of the difference between the covariant and consistent currents Osabe_Suzuki in two and four dimensions. The results are the same as previously reported Bardeen_Zumino_gauge_gravity ; Banerjee_Banerjee_Mitra . In Sect. V, applying the arguments of Banerjee et al. Banerjee_Banerjee_Mitra 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. V have both Einstein and Lorentz anomalies in general. As shown in Ref. Bardeen_Zumino_gauge_gravity , 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.

Given a covariantly regularized energy–momentum tensor, $\langle T_{\mu\nu}\rangle_{\text{cov}}$, we have in general both the Einstein anomaly, $D^{\mu}\langle T_{\mu\nu}\rangle_{\text{cov}}$, and the Lorentz anomaly, $\langle T_{[\mu\nu]}\rangle_{\text{cov}}$. 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 $\langle T_{\mu\nu}\rangle_{\text{cov}}$ 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,

[TABLE]

which gives the pure covariant Einstein anomaly $D^{\mu}\langle T_{\mu\nu}\rangle_{\text{cov}}^{\text{pE}}.$ Since the energy–momentum tensor, $\langle T_{\mu\nu}\rangle_{\text{cov}}^{\text{pE}}$, given above is nothing but the symmetric part of $\langle T_{\mu\nu}\rangle_{\text{cov}}$, we can say that the pure covariant Einstein anomaly is the covariant divergence of the symmetric part of the covariant energy–momentum tensor Alvarez-Gaume_Witten ; F-T-Y ; Endo_Takao ; Fujikawa_Suzuki .

We can also define a covariant energy–momentum tensor wiht a vanishing Einstein anomaly. It is known from Alvarez-Gaume_Witten ; F-T-Y ; Matsuki ; Endo_Takao ; Fujikawa_Suzuki ; Peter that the pure covariant Einstein anomaly has the form

[TABLE]

where $L_{\mu\nu}$ is a local polynomial of Riemann curvature555The quantity $L_{\mu\nu}$ is related to axial $U(1)$ anomalies in $d+2$ dimensions Alvarez-Gaume_Witten ; F-T-Y ; Matsuki ; Endo_Takao ; Fujikawa_Suzuki ; Peter .

and is anti-symmetric with respect to the indices $\mu$ and $\nu$. To obtain an Einstein-anomaly-free energy–momentum tensor, $\langle T_{\mu\nu}\rangle_{\text{cov}}^{\text{pL}}$, we may adopt $L_{\mu\nu}$ as a local counterterm to $\langle T_{\mu\nu}\rangle_{\text{cov}}^{\text{pE}}$:

[TABLE]

which has vanishing Einstein anomaly: $D^{\mu}\langle T_{\mu\nu}\rangle_{\text{cov}}^{\text{pL}}=0$. Thus, the pure covariant Lorentz anomaly is given by

[TABLE]

In Sect. V, we have defined a regularized effective action using the covariant energy–momentum tensor (equation (142)). Since the covariant energy–momentum tensor retains the ambiguity of adding covariant local curvature and vielbein polynomials, corresponding ambiguity arises in the effective action (142). 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 $\langle eT_{\mu\nu}\rangle_{\text{cov}}^{\text{pE}}[e_{b}^{\ \lambda}(t)]$ to define the effective action (142) shows that the second term on the right-hand side of equation (145) 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 (143). Future work will aim to clarify these points.

Appendix A Proof of (40) using test functions

In this appendix, we prove (40) using test functions. Namely, we give a proof of equality

[TABLE]

where $f(x)$ is an arbitrary test function. Changing the integration variables from $x^{\mu}$ to $\xi^{\mu}$

[TABLE]

we express the left-hand side of (151) as

[TABLE]

where we have taken the Taylor expansion of $f(\sqrt{4s}\xi)$ with respect to $\xi$ and $f_{,\mu_{1}\mu_{2}\cdots\mu_{l}}={\partial^{l}f}/{\partial x^{\mu_{l}}\cdots\partial x^{\mu_{2}}\partial x^{\mu_{1}}}$. Owing to formula

[TABLE]

(153) becomes

[TABLE]

which is equal to the right-hand side of (151).

Appendix B The largest number of gamma matrices included in $a_{k}(x,x^{\prime})$

The heat kernel $K(x,x^{\prime};s)=e^{-s\not D^{2}}\delta(x-x^{\prime})$ satisfies the differential equation

[TABLE]

and the boundary condition

[TABLE]

We assume the following expansion of $K(x,x^{\prime};s)$,

[TABLE]

From the boundary condition (157), we have

[TABLE]

Equation (156) leads to the following recurrence formulas for $a_{k}$’s:

[TABLE]

From equations (159) and (160), we confirm that $a_{0}(x,x^{\prime})$ is the parallel-displacement matrix of gauge group DeWitt_HK . Then, it is obvious that $a_{0}(x,x^{\prime})$ does not contain any gamma matrices $\gamma^{\mu}$. Equation (161) shows that $a_{k+1}(x,x^{\prime})$ has two more gamma matrices than $a_{k}(x,x^{\prime})$, since $D_{\mu}$ has none. From these observations, we find that the largest number of gamma matrices included in $a_{k}(x,x^{\prime})$ is equal to $2k$.

In the coincidence limit $x^{\prime}\to x$, $a_{k}(x,x^{\prime})$ still has at most $2k$ gamma matrices. In fact, equations (159), (160), and (161) lead us to

[TABLE]

where the dots on the right-hand side express terms with lower power of gamma matrices.

Appendix C Heat kernel appearing in Osabe and Suzuki’s currents

The heat kernel $K_{g}(x,x^{\prime};s)=\langle x|e^{-s\not D\not D_{g}}|x^{\prime}\rangle$ satisfies the differential equation

[TABLE]

and the boundary condition

[TABLE]

We assume the following expansion of $K_{g}(x,x^{\prime};s)$

[TABLE]

From the boundary condition (164), we have

[TABLE]

where we have used Synge’s symbol to denote the coincidence limit $[f]=f(x,x)$. Equation (163) leads to the following recurrence formulas for the $b_{k}$’s:

[TABLE]

where we have used the equation

[TABLE]

with

[TABLE]

and $\gamma^{\mu\nu}=\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]$. From recurrence formulas (167) and (168), together with (166), we obtain the following coincidence limits DeWitt_HK :

[TABLE]

where $\mathscr{F}_{\mu\nu}=\partial_{\mu}\mathscr{A}_{\nu}-\partial_{\nu}\mathscr{A}_{\mu}+[\mathscr{A}_{\mu},\mathscr{A}_{\nu}]$. From these equations, we obtain

[TABLE]