Axial U(1) current in Grabowska and Kaplan's formulation

TL;DR

This paper examines the axial-vector current in a non-perturbative chiral gauge theory formulation by Grabowska and Kaplan, revealing the current's non-local bulk contribution and proposing a local current definition.

Contribution

It clarifies the nature of the axial-vector current in Grabowska and Kaplan's formulation, highlighting the bulk contribution and proposing a local current definition.

Findings

The conserved current includes both domain-wall and bulk contributions.

The local axial current is obtained by subtracting the bulk contribution.

The current does not reproduce the correct anomaly without this subtraction.

Abstract

Recently, Grabowska and Kaplan suggested a non-perturbative formulation of a chiral gauge theory, which consists of the conventional domain-wall fermion and a gauge field that evolves by the gradient flow from one domain wall to the other. In this paper, we discuss the U(1) axial-vector current in 4 dimensions using this formulation. We introduce two sets of domain-wall fermions belonging to complex conjugate representations so that the effective theory is a 4-dimensional vector-like gauge theory. Then, as a natural definition of the axial-vector current, we consider a current that generates the simultaneous phase transformations for the massless modes in 4 dimensions. However, this current is exactly conserved and does not reproduce the correct anomaly. In order to investigate this point precisely, we consider the mechanism of the conservation. We find that this current includes not only the axial current on the domain wall but also a contribution from the bulk, which is non-local in the sense of 4-dimensional fields. Therefore, the local current is obtained by subtracting the bulk contribution from it.

Figures

Equations

$$\mathcal{L}=\bar{\psi }[/{\partial }_{2n+1}-Mϵ(s)]\psi .$$

$${\partial }_s{\bar{A}}_{\nu }(x,s)=\frac{ϵ(s)}{M^{\prime }}{D}_{\mu }{\bar{F}}_{\mu \nu },$$

$$\Delta (A)\equiv \frac{\det \left(/D_{2n+1}^{(R)}-Mϵ(s)\right)}{\det \left(/D_{2n+1}^{(R)}+M\right)},$$

$$S_{2n+1}^{(CS)}=c_{2n+1}\frac{M}{|M|}\int [ϵ(s)+1]{\omega }_{2n+1}.$$

$$c_{2n+1}=\frac{i^n}{2^{n+1}{\pi }^n(n+1)!},$$

$$\log \Delta (A)=\mathrm{Tr}\log \left(/D_{2n+1}-Mϵ(s)\right)-\mathrm{Tr}\log \left(/D_{2n+1}+M\right).$$

$$\mathrm{Tr}\log \left(/D_{2n+1}-Mϵ(s)\right)\to \mathrm{Tr}\log \left(/D_{2n+1}-Mϵ(s)\right)+\sum _i{C}_i\mathrm{Tr}\log \left(/D_{2n+1}-M_iϵ(s)\right),$$

$$\mathrm{Tr}\log \left(/D_{2n+1}+M\right)\to \mathrm{Tr}\log \left(/D_{2n+1}+M\right)+\sum _i{C}_i^{\prime }\mathrm{Tr}\log \left(/D_{2n+1}+M_i^{\prime }\right).$$

$$\log \Delta (A{)}_{reg.}$$

$$\mathrm{Tr}\log \left(/D_{2n+1}-Mϵ(s)\right)-\mathrm{Tr}\log \left(/D_{2n+1}+M\right)$$

$$+\sum _i{C}_i\left[\mathrm{Tr}\log \left(/D_{2n+1}-M_iϵ(s)\right)-\mathrm{Tr}\log \left(/D_{2n+1}+M_i\right)\right]$$

$$\stackrel{\mathrm{around}s=0}{\to }\left[\mathrm{Tr}\log (/D_{2n}{P}_{-}+/{\partial }_{2n}{P}_{+})-\mathrm{Tr}\log (/D_{2n}{P}_{-}+/{\partial }_{2n}{P}_{+}-M)\right]$$

$$+\sum _i{C}_i\left[\mathrm{Tr}\log (/D_{2n}{P}_{-}+/{\partial }_{2n}{P}_{+})-\mathrm{Tr}\log (/D_{2n}{P}_{-}+/{\partial }_{2n}{P}_{+}-M_i)\right],$$

$$\begin{array}{rlr}M+\sum _i{C}_i{M}_i& =& 0,\\ M^2+\sum _i{C}_i(M_i{)}^2& =& 0,\\ M^3+\sum _i{C}_i(M_i{)}^3& =& 0,\\ ⋮& & .\end{array}$$

$$\mathrm{Tr}\log \left(/D_{2n+1}-M\right)-\mathrm{Tr}\log \left(/D_{2n+1}+M\right)$$

$$+\sum _i{C}_i\left[\mathrm{Tr}\log \left(/D_{2n+1}-M_i\right)-\mathrm{Tr}\log \left(/D_{2n+1}+M_i\right)\right].$$

$$\begin{array}{rlr}M+\sum _i{C}_i{M}_i& =& 0,\\ M^3+\sum _i{C}_i(M_i{)}^3& =& 0,\\ ⋮& & ,\end{array}$$

$$\sum _i{C}_i=0,$$

$$\mathrm{Tr}\log (/D_{2n+1}-Mϵ(s))-\mathrm{Tr}\log (/D_{2n+1}+M).$$

$$\left[i/p+{\gamma }^5{\partial }_s-ϵ(s)M\right]G(p,s;s^{\prime })=\delta (s-s^{\prime }),$$

$$G(x,s;x^{\prime },s^{\prime })=\int \frac{d^{2n}p}{(2\pi {)}^{2n}}{e}^{-ip\cdot (x-x^{\prime })}G(p,s;s^{\prime }).$$

$$G(p,s;s^{\prime })=\{\begin{array}{ll}{S}^{(+)}(p,s-s^{\prime })+{\mathcal{D}}^{(+)}(p)e^{-(s^{\prime }+s)\sqrt{p^2+M^2}}& (0<s,s^{\prime })\\ S^{(-)}(p,s-s^{\prime })+{\mathcal{D}}^{(-)}(p)e^{(s^{\prime }+s)\sqrt{p^2+M^2}}& (s,s^{\prime }<0)\\ {\mathcal{D}}^{(-+)}(p)e^{(s-s^{\prime })\sqrt{p^2+M^2}}& (s<0<s^{\prime })\\ {\mathcal{D}}^{(+-)}(p)e^{(s^{\prime }-s)\sqrt{p^2+M^2}}& (s^{\prime }<0<s),\end{array}$$

$$S^{(+)}(p,s-s^{\prime })$$

$${\mathcal{D}}^{(+)}(p)=-\frac{i/pM(M+\sqrt{p^2+M^2}{\gamma }^5)}{2p^2\sqrt{p^2+M^2}}+\frac{M}{2\sqrt{p^2+M^2}},$$

$${\mathcal{D}}^{(-+)}(p)=-\frac{i/p(\sqrt{p^2+M^2}+M{\gamma }^5)}{2p^2}-\frac{1}{2}{\gamma }^5,$$

$$\mathrm{Tr}\log (/D_{2n+1}-ϵ(s)M)-\mathrm{Tr}\log (/D_{2n+1}+M)$$

$$\equiv$$

$$\underset{L\to \infty }{\lim }{I}_m^{(L)}\equiv I_m.$$

$$I_2=\int \frac{d^{d}k}{(2\pi {)}^d}{\int }_{-\infty }^{\infty }ds{\int }_{-\infty }^{\infty }d{s}^{\prime }\mathrm{tr}\left[{\bar{A}}_{\mu }(-k,s^{\prime }){\bar{A}}_{\nu }(k,s)\right]{\Gamma }^{\mu \nu }(k,s,s^{\prime }),$$

$${\Gamma }^{\mu \nu }(k,s,s^{\prime })=\int \frac{d^{d}p}{(2\pi {)}^d}\left[\mathrm{tr}\left[{\gamma }^{\mu }G(p,s^{\prime };s){\gamma }^{\nu }G(p^{\prime },s;s^{\prime })\right]-\mathrm{tr}\left[{\gamma }^{\mu }{S}^{(-)}(p,s^{\prime };s){\gamma }^{\nu }{S}^{(-)}(p^{\prime },s;s^{\prime })\right]\right].$$

$$\mathrm{I}:\{s^{\prime }<s<0\}\oplus \mathrm{II}:\{s<0<s^{\prime }\}\oplus \mathrm{III}:\{0<s^{\prime }<s\}\oplus {\mathrm{I}}^{\prime }\oplus \mathrm{I}{\mathrm{I}}^{\prime }\oplus \mathrm{II}{\mathrm{I}}^{\prime },$$

$$\mathrm{I}\equiv {\iint }_{(\mathrm{I})}\mathrm{tr}\left[{\bar{A}}_{\mu }(-k,s^{\prime }){\bar{A}}_{\nu }(k,s)\right]{\Gamma }^{\mu \nu }(k,s,s^{\prime }).$$

$$G(p^{\prime },s;s^{\prime })=S^{(-)}(p^{\prime },s-s^{\prime })+D^{(-)}(p^{\prime },s+s^{\prime }),$$

Full text

In this paper, we discuss the U(1) axial-vector current in 4 dimensions using this formulation. We introduce two sets of domain-wall fermions belonging to complex conjugate representations so that the effective theory is a 4-dimensional vector-like gauge theory. Then, as a natural definition of the axial-vector current, we consider a current that generates the simultaneous phase transformations for the massless modes in 4 dimensions. However, this current is exactly conserved and does not reproduce the correct anomaly. In order to investigate this point precisely, we consider the mechanism of the conservation. We find that this current includes not only the axial current on the domain wall but also a contribution from the bulk, which is non-local in the sense of 4-dimensional fields. Therefore, the local current is obtained by subtracting the bulk contribution from it.

1 Introduction

Formulating a chiral gauge theory non-perturbatively has been a long-standing problem[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Recently[11, 12], Grabowska and Kaplan suggested a formulation that consists of the domain-wall fermion in $2n$+1 dimensions and a gauge field that evolves by the gradient flow from one domain wall to the other. A long-distance flow makes the gauge field pure gauge, and thus one of the massless modes (“fluffy mirror fermion” or “fluff”) does not couple with the gauge field. Therefore, we obtain a chiral gauge theory including only the other massless mode that couples with the gauge field. However, the heavy modes in the bulk induce some terms, which can not be renormalized to the 4-dimensional Lagrangian. To cancel the bulk terms, Grabowska and Kaplan introduced a subtracting field, which has a loop factor +1 and a constant mass. It is known that the cancellation is not complete, but there remains a Chern-Simons-like term[13, 14]. However, if the anomaly-free condition $d^{abc}=0$ is satisfied, the Chern-Simons-like term vanishes and then we obtain the 4-dimensional local theory.

In order to investigate the consistency of this formulation, we consider a vector-like theory by introducing two sets of domain-wall fermions belonging to complex conjugate representations[12, 15, 16, 17]. Each of the fermions induces one left-handed physical fermion and one right-handed fluff fermion. Therefore, if the fluffs are decoupled correctly, we have the 4-dimensional vector-like gauge theory with one right-handed and one left-handed chiral fermions after we apply the charge conjugation to one of the physical fermions. In this paper, we consider the U(1) axial-vector current and discuss how the anomaly arises. We first define a current that generates the simultaneous phase transformations for the left-handed physical fermions in 4 dimensions. From the viewpoint of the effective theory, this current looks like the U(1) axial-vector current. However, it is pointed out in Ref.[15] that this current is exactly conserved and does not reproduce the correct anomaly. In order to solve this paradox, we investigate the mechanism of the conservation. We find that this current contains a bulk contribution in addition to the axial-vector current on the domain wall. Therefore, the proper local current is obtained by subtracting the bulk part.

This paper is organized as follows. In Sec. 2, we review the formulation of Grabowska and Kaplan in the lattice space. In Sec. 3, we consider a regularization of this formulation in the continuum space in order to simplify the calculations in the subsequent sections. We find that one needs to introduce Pauli-Villars fields for both of the domain-wall fermion and the subtracting field. In Sec. 4, we calculate the 1-loop effective action to the quadratic order in the gauge fields, and confirm that the effective action consists of three parts. One is equal to the effective action of a chiral fermion with Pauli-Villars-like regularization. The second is the Chern-Simons term in the bulk. The third are various divergent terms, which will be cancelled by our regularization. In Sec. 5, we discuss the axial-vector current in 4-dimensions. In Sec. 6, we give summary and conclusions.

2 Review of Grabowska and Kaplan’s method

We review the formulation of Grabowska and Kaplan. There are recent studies[16, 17, 18] based on this formulation. In this section, we consider the lattice space although we use the symbols in the continuum space. We will discuss the continuum regularization in the next section.

We start with a domain-wall fermion in $2n$+1 dimensions:

[TABLE]

Here $\psi$ is the Dirac field with $2^{n}$ components. $s$ is the $2n$+1th coordinate, $s\in[-L,L]$ with periodic boundary condition, and $\epsilon(s)=\mathrm{sgn}(s)$. In the limit of $L\rightarrow\infty$, two massless modes are localized on the $2n$-dimensional wall $s=0$ and $s=L$, which have the chirality $-1$ and $+1$ respectively. The heavy modes that live in the bulk will be decoupled classically in the limit of $M\rightarrow\infty$. In order to obtain a chiral gauge theory, in which only the left-handed mode couples with the gauge field, the $2n$+1-dimensional gauge field $\bar{A}_{\mu}$ is constructed by the gradient flow[19, 20, 21] from $s=0$ to $s=\pm L$:

[TABLE]

with $\bar{A}_{\mu}(x,0)=A_{\mu}(x)$, $\mu,\nu=1,\cdots,2n$, and $\bar{A}_{2n+1}=0$. $\bar{F}_{\mu\nu}$ is the field strength of the gauge field $\bar{A}_{\mu}$. We assume $M^{\prime}\gg M$ so that $\bar{A}_{\mu}(x,s)$ is close to $A_{\mu}(x)$ near the domain wall $|s|\lesssim 1/M$. Since the gradient flow damps the physical degrees of freedom, the gauge field $\bar{A}_{\mu}$ becomes to pure gauge111More precisely, it is also possible for the gauge field to become an instanton configuration. We don’t consider this case in this paper. at $s=L$ in the limit of $L\rightarrow\infty$. Thus the right-handed mode on $s=L$ is decoupled and we obtain the $2n$-dimensional chiral gauge theory if the bulk degrees of freedom are decoupled.

In order to cancel the bulk degrees of freedom222For the case that the gauge field is constant in the $s$ direction, $\bar{A}_{\mu}(x,s)=A_{\mu}(x)$, we don’t have to cancel the bulk terms because they can be absorbed in the 4-dimensional Lagrangian. However, it is not possible when we consider the gradient flow., we introduce a “subtracting field”333In Ref.[11, 12], this field is called ”Pauli-Villars field”. But we distinguish this field from a conventional Pauli-Villars field, whose role is a regularization., which has a loop factor +1 and a constant mass $-M$. This setting is equivalent to defining the fermion determinant as follows:

[TABLE]

where $\not{D}^{(R)}_{2n+1}$ is the $2n$+1-dimensional Dirac operator belonging to the representation $R$. Indeed the terms that are even functions of $M$ in the bulk are cancelled. On the other hand, for the odd terms, the parity anomaly survives and the effective action contains a bulk term[13, 14]:

[TABLE]

Here

[TABLE]

and $\omega_{2n+1}$ is the $2n$+1-dimensional Chern-Simons form. $S_{2n+1}^{(CS)}$ vanishes if the representation satisfies the condition for the anomaly cancellation in $2n$ dimensions.

In order to perform the calculation easily, we consider a continuum version of this formulation in the following sections.

3 Regularization in the continuum formulation

In this section, we regularize the formulation given in Sec. 2 in the continuum space. The bare effective action corresponding to Eq. (2.3) is given by444We drop the superscript ”$(R)$” in $\not{D}_{2n+1}$.

[TABLE]

Here, we adopt the Pauli-Villars regularization555The dimensional regularization can not be used for the $2^{n}$-component Dirac field in $2n+1$ dimensions..

We regularize the domain-wall fermion and the subtracting field respectively as follows:

[TABLE]

[TABLE]

Note that while the subtracting field is regularized as usual, the domain-wall fermion is regularized by additional domain-wall fermions with mass $M_{i}\epsilon(s)$. The parameters $C_{i},M_{i},C_{i}^{\prime},M_{i}^{\prime}$ will be determined later so that the regularized effective action converges as usual. Here, we choose $C_{i}^{\prime}=C_{i}$ and $M_{i}^{\prime}=M_{i}$ so that the Pauli-Villars fields do not generate extra bulk effective action. In other words, we introduce pairs of Pauli-Villars fields consisting of a domain-wall fermion and a subtracting field, which we call Pauli-Villars pairs. Thus the regularized effective action is

[TABLE]

Let us write down the condition for the effective action to converge. For a necessary condition, divergences arising on the walls should be cancelled. As we will see in Eq. (4.76), a pair of a domain-wall fermion and a subtracting field behaves like a chiral fermion with a Pauli-Villars-like field666This $2n$-dimensional Pauli-Villars-like field is not the Pauli-Villars field that we have introduced in Eq. (3.2) and Eq. (3.3). around $s=0$. Therefore, all pairs including the Pauli-Villars pairs give the following contribution to the effective action from the near-wall region:

[TABLE]

where $P_{-}$ and $P_{+}$ are the chirality projection operators. $\mathrm{Tr}\log(\not{D}_{2n}P_{-}+\not{\partial}_{2n}P_{+})$ and $\mathrm{Tr}\log(\not{D}_{2n}P_{-}+\not{\partial}_{2n}P_{+}-M)$ are the effective action of the left-handed chiral fermion and the Pauli-Villars-like field, respectively. We will derive Eq. (3.5) in Sec. 4. The conditions to cancel the divergences in Eq. (3.5) are777Generally, we need $d$ conditions in $d$ dimensions.

[TABLE]

Note that the leading divergences in Eq. (3.5), which are independent of $M$ and $M_{i}$, are cancelled in each pair.

Eq. (3.6) are also sufficient to cancel the divergences from the bulk. In the bulk region $-L<s<0$, the cancellation is trivial because the domain-wall fermions and the subtracting fields have the same mass in each pair. In the bulk region $0<s<L$, Eq. (3.4) reduces to

[TABLE]

In Eq. (3.7), terms that are even functions of $M$ and $M_{i}$ are trivially cancelled. On the other hand, the odd terms are cancelled if the following conditions are satisfied:

[TABLE]

which are part of Eq. (3.6). Therefore, Eq. (3.6) is the necessary and sufficient condition for the effective action to converge.

However, we need to prevent the Pauli-Villars fields from changing the physical degrees of freedom. In fact, each of the Pauli-Villars pairs induces a massless mode on the wall and a Chern-Simons term in the bulk, which will not be decoupled even if we take the limit $M_{i}\rightarrow\infty$. Thus one observes $\sum_{i}C_{i}$ additional massless modes and Chern-Simons terms. These extra contributions vanish by imposing an additional condition:

[TABLE]

which we will confirm in Eq. (4.79).

Thus we conclude that a continuum version of the regularized effective action is given by Eq. (3.4) with Eq. (3.6) and Eq. (3.9).

4 Calculation of the effective action

In this section, we calculate the regularized effective action, Eq. (3.4), by expanding with respect to the gauge field $\bar{A}_{\mu}$. In order to do it, it is sufficient to calculate the one pair of a domain-wall fermion and a subtracting field:

[TABLE]

The other pairs are obtained by replacing the mass and loop factor. As we will see later, Eq. (4.1) consists of three parts. One is the effective action of the $2n$-dimensional chiral fermions with a Pauli-Villars-like regularization. This confirms that the massless modes localized on the walls behave as chiral fermions even at the quantum level. The second is the Chern-Simons term in $2n$+1 dimensions. The third are various divergent terms, which will be cancelled after summing up with the Pauli-Villars pairs.

4.1 Propagator of domain-wall fermion

We begin with deriving the propagator of the domain-wall fermion in the continuum888The propagator in the lattice theory is derived in Ref[7, 8] .. As we will see below, this propagator can be regarded as a sum of two processes. One is the bulk propagation with a constant mass $\pm M$. The other is the massless propagation along the domain walls. Thus this propagator includes both of the heavy bulk modes and the massless domain-wall modes.

The propagator is a solution of the following equation:

[TABLE]

where $G(p,s;s^{\prime})$ is the Fourier transform of the propagator in the $2n$ directions,

[TABLE]

We use the symbol $\gamma^{5}$ as the chirality matrix even in $2n$+1 dimensions, i,e, $\gamma^{5}\equiv\gamma^{1}\cdots\gamma^{2n}$.

In order to concentrate on the modes around $s=0$, we take the limit $L\rightarrow\infty$, and obtain the following expression (See Appendix A):

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and $\theta(s-s^{\prime})$ is the step function. $S^{(-)}$ is obtained from replacing $M\rightarrow-M$ in $S^{(+)}$. $\mathcal{D}^{(-)},\mathcal{D}^{(+-)}$ are obtained from $\mathcal{D}^{(+)},\mathcal{D}^{(-+)}$, respectively, by replacing $M\rightarrow-M$ and $\gamma^{5}\rightarrow-\gamma^{5}$. Note that $S^{(+)}$ and $S^{(-)}$ are the conventional propagators in $2n$+1 dimensions with constant mass $M$ and $-M$, respectively, and represent the heavy modes. The other terms in Eq. (4.4) represent the massless modes localized on the wall $s=0$.

These results are consistent with the physical intuition that the propagator $G(p,s;s^{\prime})$ reduces to the conventional one with constant mass $\pm M$ in the region far from the domain wall, $s,s^{\prime}\ll-1/M$ or $1/M\ll s,s^{\prime}$.

4.2 Calculation of effective action –vacuum polarization–

Let us expand Eq. (4.1) as follows:

[TABLE]

where $d=2n$ and $\Gamma^{\mu_{1}\cdots\mu_{m}}$ is the sum of the fermion loops with $m$ vertices for the domain-wall fermion and the subtracting field. Note that $k_{i}(i=1,\cdots,m)$ are the $2n$-dimensional momenta, and $s_{i}$ are the $2n$+1th coordinates. As in the previous section, we take the limit $L\rightarrow\infty$ and consider

[TABLE]

In the following, we give an explicit calculation for $I_{2}$, which is nothing but the vacuum polarization loop:

[TABLE]

where

[TABLE]

Here $p^{\prime}$ stands for $p+k$ so that we must substitute $p^{\prime}=p+k$ in Eq. (4.13) before integrating with respect to $p$. The second term in Eq. (4.13) comes from the subtracting field.

It is convenient to divide the range of $s$ and $s^{\prime}$ into six regions:

[TABLE]

where the regions I, II, III correspond to diagrams in Fig. 1. $\mathrm{I^{\prime}},~{}\mathrm{II^{\prime}},~{}\mathrm{III^{\prime}}$ are obtained by interchanging $s\leftrightarrow s^{\prime}$ in $\mathrm{I},~{}\mathrm{II},~{}\mathrm{III}$, respectively.

We denote the contribution from the region I by I:

[TABLE]

In this region, the propagator $G$ can be written as (See Eq. (4.4)):

[TABLE]

and

[TABLE]

where

[TABLE]

Thus $\Gamma^{\mu\nu}(k,s,s^{\prime})$ is 999We drop the arguments $p,p^{\prime},s,s^{\prime}$ for simplicity. The symbols without prime such as $G,S^{(-)},D^{(-)}$ mean that their arguments are $(p,s;s^{\prime})$. On the other hand, the symbols with prime stand for the arguments $(p^{\prime},s;s^{\prime})$.

[TABLE]

where $T_{local}^{(-)}(p,p^{\prime},s,s^{\prime})$ depends on $s,s^{\prime}$ as follows :

[TABLE]

Here, $\alpha,\beta,\gamma$ are functions of $p,p^{\prime}$ and obtained from $\mathrm{tr}\left[\gamma^{\mu}D^{(-)}\gamma^{\nu}D^{\prime(-)}\right]$, $\mathrm{tr}\left[\gamma^{\mu}D^{(-)}\gamma^{\nu}S^{\prime(-)}\right]$, $\mathrm{tr}\left[\gamma^{\mu}S^{(-)}\gamma^{\nu}D^{\prime(-)}\right]$, respectively (See Appendix. B). Note that the bulk term from the domain-wall fermion, $\mathrm{tr}\left[\gamma^{\mu}S^{(-)}\gamma^{\nu}S^{(-)}\right]$, has been cancelled by the subtracting field, and there remains only $T_{local}^{(-)}$ , which damps exponentially in $s,s^{\prime}$. Therefore, $\Gamma^{\mu\nu}(k,s,s^{\prime})$ in the region I has values only in $-1/M\lesssim s^{\prime},s<0$ .

Using this fact, we evaluate the integral with respect to $s,s^{\prime}$ in Eq. (4.15) as follows. First, we approximate that $\bar{A}(x,s)$, which evolves by the gradient flow, is constant in the region $-1/M\lesssim s^{\prime},s<0$. Thus we can write

[TABLE]

The approximation “$\sim$” will be exact if we take the limit $M^{\prime}\rightarrow\infty$. Then, for example, for the first term in Eq. (4.24), we have

[TABLE]

The other exponentials in Eq. (4.24) can be integrated in the same way and we obtain the expression Eq. (B.11).

We denote the contribution from the region II by II:

[TABLE]

The propagator $G$ in this region is given by (See Eq. (4.4))

[TABLE]

[TABLE]

Thus $\Gamma^{\mu\nu}(k,s,s^{\prime})$ is

[TABLE]

Note that $\Gamma^{\mu\nu}$ has values only when $-1/M\lesssim s<0<s^{\prime}\lesssim 1/M$ because the integrand in Eq. (4.40) is proportional to $\exp[(s-s^{\prime})(\sqrt{p^{2}+M^{2}}+\sqrt{p^{\prime 2}+M^{2}})]$. Therefore, the calculation can be performed similarly to the region I, and we obtain the resulting expression Eq. (B.13).

We denote the contribution from the region III by III:

[TABLE]

In this region, the propagator $G$ can be written as

[TABLE]

and

[TABLE]

where

[TABLE]

Thus $\Gamma^{\mu\nu}(k,s,s^{\prime})$ is calculated as

[TABLE]

where

[TABLE]

[TABLE]

Here, $\alpha,\beta,\gamma$ in Eq. (4.52) are the same as those in Eq. (4.24)(See Appendix. B). Note that $T_{local}^{(+)}$ , which is localized on $s=0$, gives the same contribution as $T_{local}^{(-)}$ after integrating over $s$ and $s^{\prime}$. On the other hand, $T_{bulk}$ is the bulk term, which does not vanish unlike the region I because of the opposite signs of the masses. We will discuss this point in the next subsection.

Next, we consider the other regions $\mathrm{I}^{\prime},\mathrm{II}^{\prime},\mathrm{III}^{\prime}$. The net effect of interchanging $s\leftrightarrow s^{\prime}$ is to change the signs of $\gamma^{5}$ in $S^{(+)}$ and $S^{(-)}$ (See Eq. (4.5)). Therefore the contributions from $\mathrm{I}^{\prime},\mathrm{II}^{\prime},\mathrm{III}^{\prime}$ are obtained from those of $\mathrm{I},\mathrm{II},\mathrm{III}$, respectively, by changing the signs of $\gamma^{5}$ in $S^{(+)}$ and $S^{(-)}$. (See the discussions below Eq. (B.11) and Eq. (B.13))

All contributions (Eq. (B.11), (B.13), (4.49)) sum up to

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Here, $N^{\mu\nu}\equiv p\cdot p^{\prime}\delta^{\mu\nu}-p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu}$. The first term in Eq. (4.54) represents the contribution from the localized terms. $\Pi^{\mu\nu}_{{(anomalous)}}$ and $\Pi^{\mu\nu}_{{(non-anomalous)}}$ are the parts with and without $\gamma^{5}$, respectively. As we will see in the next subsection,

[TABLE]

is equal to the vacuum polarization of a left-handed chiral fermion with a Pauli-Villars-like regulator of mass $M$. $I_{2}^{bulk}$ represents the contribution from the bulk region $0<s,s^{\prime}<\infty$.

Note that there are no leading UV divergences, terms that have degree of divergence $d-2$, in Eq. (4.55)-(4.57). Therefore, all UV divergences are cancelled by the Pauli-Villars pairs under the conditions, Eq. (3.6).

4.3 Comparison with chiral fermion in $2n$ dimensions

We first consider the vacuum polarization of a left-handed chiral fermion:

[TABLE]

where $P_{-}=\frac{1-\gamma^{5}}{2}$ . By introducing one Pauli-Villars field, we have

[TABLE]

where

[TABLE]

[TABLE]

The non-anomalous part $V^{\mu\nu}_{(non-anomalous)}$ is precisely equal to Eq. (4.55).

We evaluate the difference of Eq. (4.56) and Eq. (4.63), and show it is zero in the limit of $M\rightarrow\infty$. This is trivial for $d>2$ since both of them vanish. Thus we consider the case $d=2$. The difference is calculated as:

[TABLE]

Because this integral is finite at $k=0$, we can expand it around $k=0$:

[TABLE]

where,

[TABLE]

Thus we obtain

[TABLE]

Therefore, Eq. (4.58) is equal to Eq. (4.61) in the limit of $M\rightarrow\infty$.

We extend the above result to general cases $m>2$. It is expected that $I_{m}$ given by Eq. (4.11) is also written as

[TABLE]

where $I_{m}^{s=0}$ is the $m$-vertex loop of the left-handed chiral fermion with the Pauli-Villars field on the domain wall $s=0$. $I_{m}^{bulk}$ is the $m$-vertex loop of the heavy mode and the subtracting field in the bulk region $0<s<\infty$. Note that the divergent terms that are included in $I_{m}^{s=0}$ and $I_{m}^{bulk}$ will be cancelled by adding the Pauli-Villars pairs. Then, from Eq. (4.70), we have

[TABLE]

The first term in Eq. (4.72) can be regarded as

[TABLE]

Here, the first term and the second term in Eq. (4.73) are the effective actions of the left-handed chiral fermion and the Pauli-Villars field, respectively. On the other hand, the second term in Eq. (4.72) can be written as[13, 14]:

[TABLE]

where $S_{2n+1}^{(CS)}$ is the Chern-Simons term given by Eq. (2.4). The ultra-violet (UV) divergence in $\delta S_{2n+1}(M)$ is cancelled after combining with the Pauli-Villars pairs.

So far, we have neglected the domain wall $s=L$ by taking the limit $L\rightarrow\infty$. There, the similar result for the right-handed chiral fermion to Eq. (4.73) should be obtained. Therefore the effective action Eq. (4.9) is

[TABLE]

Here, $()_{s=0}$ and $()_{s=L}$ stand for substituting the gauge field $\bar{A}(x,s=0)$ and $\bar{A}(x,s=L)$ into the covariant derivative, respectively.

By adding the Pauli-Villars pairs, the regularized effective action is obtained as follows. For the sake of simplicity, we write Eq. (3.4) as

[TABLE]

where $C_{0}=1$ and $M_{0}=M$. By applying Eq. (4.76) to each pair, we have

[TABLE]

The last term in Eq. (4.79) is UV-finite and vanishes in the limit of $M,M_{i}\rightarrow\infty$, which we drop in the following expressions. As argued in Sec.3, the extra massless modes and Chern-Simons terms have vanished by the condition $\sum_{i=1}C_{i}=0$. Thus there are no artificial degrees of freedom. In addition, the regularized effective action Eq. (4.79) converges under the condition, Eq. (3.6).

Note that Eq. (4.79) is gauge invariant because gauge anomalies from the three lines are cancelled. For example, for $n=2$, the gauge variation of the Chern-Simons term is

[TABLE]

where $\chi$ is the gauge function. On the other hand, the first and second lines in Eq. (4.79) give the anomaly of the left- and right-handed chiral fermions in 4 dimensions, respectively, which cancel with Eq. (4.80). This cancellation agrees with the manifestly gauge invariant construction, Eq. (3.4).

5 Axial-vector current in vector-like gauge theory

We investigate the consistency of this formulation by introducing two sets of domain-wall fermions belonging to complex conjugate representations. As a simple example, we consider a 5-dimensional U(1) gauge theory. We assume that each set of fermions consists of a domain-wall fermion, a subtracting field and Pauli-Villars pairs. The two domain-wall fermions $\psi$ and $\psi^{\prime}$ have U(1) charge $\pm 1$, respectively. While left-handed physical fermions are localized on $s=0$, right-handed fluff fermions are localized on $s=L$. We denote the former coming from $\psi$ and $\psi^{\prime}$ by $q_{L}$ and $q^{\prime}_{L}$, respectively. Because the gradient flow makes the fluff fermions decouple, we obtain a 4-dimensional effective theory consisting of the left-handed physical fermions $q_{L},q^{\prime}_{L}$. Here, the Chern-Simons term vanishes due to the representations, and the effective theory is equivalent to the vector-like theory after applying the charge conjugation: $q_{R}\equiv q_{L}^{\prime C}$. In the following, we will show that the axial-vector current that is defined naturally does not reproduce the correct anomaly[15].

One natural way to define such current is to introduce a fictitious U(1) gauge field $B_{\mu}$ that couples to $q_{L}$ and $q_{L}^{\prime}$ with charge +1. Then the current is defined by the variation with respect to the gauge field $B_{\mu}(x)$. In order to realize it, we consider the bulk U(1) gauge field $\bar{B}_{\mu}(x,s)$ that couples to $\psi$ and $\psi^{\prime}$ with charge +1. We assume that $\bar{B}_{\mu}$ also evolves by the gradient flow from $s=0$ to $s=\pm L$:

[TABLE]

with $\mu,\nu=1,\cdots,4$, $\bar{B}_{\mu}(x,0)=B_{\mu}(x)$ and $\bar{B}_{5}=0$. $\bar{F}^{(B)}_{\mu\nu}$ denotes the field strength of $\bar{B}_{\mu}$, and $M^{\prime\prime}\gg M$ as $M^{\prime}$ in Eq. (2.2). Then, we define $J_{\mu}^{B}(x)$ by

[TABLE]

where $\mu=1,\cdots,4$. $S_{eff}[\bar{A},\bar{B}]$ is the effective action obtained by integrating out $\psi$ and $\psi^{\prime}$. The symbol $\langle\rangle_{A}$ stands for the expectation value in the presence of the background gauge field $A_{\mu}$, which we drop in the expressions below. $J_{\mu}^{B}$ seems to be the U(1) axial-vector current:

[TABLE]

However, it does not reproduce the correct axial anomaly. Indeed, as we will see below, $J_{\mu}^{B}$ is exactly conserved[15]:

[TABLE]

On the other hand, from the viewpoint of the 5-dimensional theory, this conservation is natural because this current is a Noether current of this system. In order to solve this paradox, we investigate the mechanism of this conservation.

First we discuss how the effective action changes under the gauge transformation of $B_{\mu}(x)$,

[TABLE]

Because $\bar{B}_{\mu}(x,s)$ is changed as

[TABLE]

the variation of the effective action $S_{eff}[\bar{A},\bar{B}]$ can be written in the following two ways:

[TABLE]

Thus we obtain

[TABLE]

where

[TABLE]

Note that the region $-L<s<0$ has no contribution to Eq. (5.10) because no terms are induced there as we have seen in Sec. 4.2. The above expression indicates that there is a contribution from the bulk to the divergence of the current as well as that from the domain wall, Eq. (5.3).

As we have seen in the previous section, $S_{eff}[\bar{A},\bar{B}]$ consists of the effective action of the chiral fermions on $s=0,L$ and the Chern-Simons term in the bulk, in the limit of $M\rightarrow\infty$. Thus we can write

[TABLE]

where $J_{\mu}^{(q_{L},q_{R})},~{}J_{\mu}^{(\mathrm{fluff})}$ are currents of the chiral fermions on each boundary, and

[TABLE]

$j_{\mu}^{(CS)}(x,s)$ is the Chern-Simons current:

[TABLE]

In the presence of the gauge fields $\bar{A}$ and $\bar{B}$, the Chern-Simons form $\omega_{5}$ is

[TABLE]

Note that the $\bar{B}$ dependent part does not vanish although the anomaly-free condition for $\bar{A}$ is satisfied. By substituting Eq. (5.18) into Eq. (5.15), we obtain

[TABLE]

where $\mu=1,\cdots,4$ because $\bar{B}_{5}=0$, and $a,b,c,d=1,\cdots,5$. Then, the divergence of $J_{\mu}^{(CS)}$ is calculated as follows:

[TABLE]

with $\mu,\nu,\lambda,\rho=1,\cdots,4$. We have used in the last line that $\bar{A}_{\mu}(x,s=L)$ is pure gauge.

On the other hand, the anomaly of $J^{(q_{L},q_{R})}_{\mu}$ is the same as the conventional axial anomaly of the vector-like fermion[22, 23]:

[TABLE]

$\partial_{\mu}J_{\mu}^{(\mathrm{fluff})}$ is similar, but vanishes because $\bar{A}(x,s=L)$ is pure gauge101010The fluff fermions are indeed decoupled even for the anomaly..

Thus the 4-dimensional current $J_{\mu}^{B}$ is conserved as mentioned above:

[TABLE]

In addition, the current is non-local in the sense of the 4-dimensional field theory because it includes the bulk contribution. Therefore we can not regard $J_{\mu}^{B}$ as the local U(1) axial current in the effective theory.

In order to obtain the local and correctly anomalous current, we subtract the bulk contribution from $J_{\mu}^{B}$:

[TABLE]

Indeed, Eq. (5.29) can be written as

[TABLE]

which is manifestly local and reproduces the correct anomaly:

[TABLE]

Note that the Chern-Simons current $j_{\mu}^{(CS)}(x,s)$ and $j_{\mu}^{B}(x,s)$ are gauge invariant (See Eq. (5.19) and Eq. (5.11)). Therefore $J_{\mu}^{axial}(x)$ is also gauge invariant. This is true also when the gauge group of the gauge field $\bar{A}_{\mu}$ is non-Abelian. In such case, indeed, the Chern-Simons form is

[TABLE]

where $r$ and $\bar{r}$ are the representations of the two fermions $\psi$ and $\psi^{\prime}$, respectively. Thus the Chern-Simons current is written as

[TABLE]

which is manifestly gauge invariant.

6 Summary and conclusions

In this paper, we have studied the formulation in Ref.[11, 12] in the continuum. In Sec. 3, we have given the regularization by Eq. (3.4) with Eq. (3.6) and Eq. (3.9). The Pauli-Villars pairs could generate extra massless modes on the walls and Chern-Simons terms in the bulk. However, the condition, Eq. (3.9), eliminates these extra contributions.

In Sec. 4, we have calculated the effective action to the quadratic order in the gauge field, and we have found that the effective action consists of three parts. One is the effective action of the chiral fermions on the domain walls with Pauli-Villars-like regularization. The second is the Chern-Simons term in the bulk. The third are divergent terms, which are cancelled by the Pauli-Villars pairs.

In Sec. 5, we have argued the axial-vector current in 4 dimensions. We have introduced two sets of domain-wall fermions belonging to complex conjugate representations so that the effective theory is the vector-like gauge theory. Then we have considered the axial-vector current that generates the simultaneous phase transformations for the fermions. This current is exactly conserved, but it contains the contribution from the bulk, which is non-local from the viewpoint of the 4-dimensional theory. Therefore the local gauge invariant axial-vector current is obtained by subtracting the bulk part.

Acknowledgment

We would like to thank Hiroshi Suzuki, Okuto Morikawa and Ryuichiro Kitano for useful discussion at KEK Theory Workshop 2016.

Appendix A Propagator of domain-wall fermion

The propagator of the domain-wall fermion is a solution of the following equation:

[TABLE]

where $G(p,s;s^{\prime})$ is the Fourier transform of the propagator in the $2n$ directions,

[TABLE]

We first consider the region $s^{\prime}>0$. Then we have three cases for $s$:

[TABLE]

We denote the propagators for (i),(ii),(iii) by $G^{(1)}$, $G^{(2)}$, $G^{(3)}$, respectively. From Eq. (A.1), we have

[TABLE]

[TABLE]

[TABLE]

where $C_{1},C_{2},C_{3}$ are $s$ independent matrices. Note that the sign of the mass in $G^{(3)}$ is different from the others. We impose the following boundary conditions:

[TABLE]

The first equation is obtained from Eq. (A.1) by integrating for $s$ around $s^{\prime}$. The second is to connect $G$ continuously at $s=0$. The third is the periodic boundary condition. Thus matrices $C_{1},C_{2},C_{3}$ are all determined. Then, by using the identity

[TABLE]

we obtain

[TABLE]

In the following, we consider the limit of $L\rightarrow\infty$. Then, $G^{(1)}$ becomes

[TABLE]

$G^{(2)}$ and $G^{(3)}$ can be calculated similarly. The result can be summarized as

[TABLE]

where

[TABLE]

The propagator for $s^{\prime}<0$ is obtained by replacing $M\rightarrow-M$ and $\gamma^{5}\rightarrow-\gamma^{5}$ in the above expressions (A.12)-(A.15):

[TABLE]

where

[TABLE]

Appendix B Vacuum polarization

We give the concrete expressions of I, II, III that are defined in Eq. (4.15), (4.33), (4.41), respectively. I is given by

[TABLE]

where

[TABLE]

Here, $\mathrm{tr}\left[\gamma^{\mu}D^{(-)}\gamma^{\nu}D^{\prime(-)}\right]$ is calculated as follows:

[TABLE]

where

[TABLE]

and $N^{\mu\nu}=p\cdot p^{\prime}\delta^{\mu\nu}-p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu}$ .

Similarly, $\mathrm{tr}\left[\gamma^{\mu}D^{(-)}\gamma^{\nu}S^{\prime(-)}\right]$ is given by

[TABLE]

where

[TABLE]

$\mathrm{tr}\left[\gamma^{\mu}S^{(-)}\gamma^{\nu}D^{\prime(-)}\right]$ is given by

[TABLE]

where

[TABLE]

Consequently, I is obtained as follows:

[TABLE]

The last term that includes $\mathrm{tr}\left[\gamma^{\mu}\gamma^{\nu}\gamma^{5}\right]$ will be cancelled with the contribution from the region $\mathrm{I}^{\prime}$ because the net effect of interchanging $s\leftrightarrow s^{\prime}$ changes the sign of $\gamma^{5}$ in $S^{(-)}$.

Similarly, II is given by

[TABLE]

Again the last term will be cancelled with the contribution from the region $\mathrm{II}^{\prime}$.

III is given by

[TABLE]

where

[TABLE]

[TABLE]

Here, $\mathrm{tr}\left[\gamma^{\mu}D^{(+)}\gamma^{\nu}D^{\prime(+)}\right]$ is calculated similarly to Eq. (B.5):

[TABLE]

Note that $\alpha(p,p^{\prime})$ in Eq. (B.19) is equal to Eq. (B.6). We obtain the similar results for $\mathrm{tr}\left[\gamma^{\mu}D^{(+)}\gamma^{\nu}S^{\prime(+)}\right]$ and $\mathrm{tr}\left[\gamma^{\mu}S^{(+)}\gamma^{\nu}D^{\prime(+)}\right]$, and $T_{local}^{(+)}$ is written as Eq. (4.52).