Perturbative analysis for Kaplan’s lattice chiral fermions

S. Aoki and H. Hirose

Institute of Physics, University of Tsukuba, Tsukuba Ibaraki-305, Japan (July 17, 1994)

Abstract

Perturbation theory for lattice fermions with domain wall mass terms is developed and is applied to investigate the chiral Schwinger model formulated on the lattice by Kaplan’s method. We calculate the effective action for gauge fields to one loop, and find that it contains a longitudinal component even for anomaly-free cases. From the effective action we obtain gauge anomalies and Chern-Simons current without ambiguity. We also show that the current corresponding to the fermion number has a non-zero divergence and it flows off the wall into the extra dimension. Similar results are obtained for a proposal by Shamir, who used a constant mass term with free boundaries instead of domain walls.

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Contents

I  Introduction  3

II Action, Fermion Propagator and Chiral Zero Modes 5 A Lattice Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 B Chiral Zero Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 C Fermion Propagator and Zero Modes . . . . . . . . . . . . . . . . . . . . . 8 D Fermion Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

III Perturbative Calculations for the Chiral Schwinger Model 11 A Effective Action at Fermion One-Loop . . . . . . . . . . . . . . . . . . . . 12 B Evaluation of Zero Mode Contributions . . . . . . . . . . . . . . . . . . . 12 C Evaluation of Remaining Contributions . . . . . . . . . . . . . . . . . . . 14 D Total Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

IV Anomalies in the Chiral Schwinger Model 17 A Currents and their Divergence . . . . . . . . . . . . . . . . . . . . . . . . 17 B Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 C Gauge Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 D Chern-Simons Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 E Pythagorean Chiral Schwinger Model and Anomaly in Fermion Number

Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

V  conclusions  23

I. INTRODUCTION

Construction of chiral gauge theories is one of the long-standing problems of lattice field theories. Because of the fermion doubling phenomenon, a naively discretized lattice fermion field yields 2^{d} fermion modes, half of one chirality and half of the other, so that the theory is non-chiral. Several lattice approaches have been proposed to define chiral gauge theories, but so far none of them have been proven to work successfully.

Kaplan has proposed a new approach [1] to this problem. He suggested that it may be possible to simulate the behavior of massless chiral fermions in 2k dimensions by a lattice theory of massive fermions in 2k+1 dimensions if the fermion mass has a shape of a domain wall in the 2k+1-th dimension. He showed for the weak gauge coupling limit that massless chiral states arise as zero-modes bound to the 2k-dimensional domain wall while all doublers can be given large gauge invariant masses. If the chiral fermion content that appears on the domain wall is anomalous the 2k-dimensional gauge current flows off the wall into the extra dimension so that the theory can not be 2k-dimensional. Therefore he argued that this approach possibly simulates the 2k-dimensional chiral fermions only for anomaly-free cases.

His idea was tested for smooth external gauge fields. Jansen [2] showed numerically that in the case of the chiral Schwinger model in 2 dimensions with three fermions of charge 3, 4, and 5 the anomalies in the gauge currents cancel on the wall. The Chern-Simons current far away from the domain wall was calculated in Ref. [3]. It is shown that the 2k+1-th component of the current is non-zero in the positive mass region and zero in the negative mass region such that the derivative of the current cancels the 2k-dimensional gauge anomaly on the wall, as was argued in Ref. [1].

In the continuum perturbation theory Frolov and Slavnov [4] proposed a gauge invariant regularization of the standard model through an infinite tower of regulator fields. Some similarity between their proposal and the Kaplan’s approach was pointed out by Narayanan and Neuberger [5]. It has been also shown that the chiral fermion determinant can be nonperturbatively defined as an overlap of two vacua [6], which can be extended to lattice theories. Using Narayanan and Neuberger’s point of view, we observed in Ref. [7] that non-gauge (chiral) anomalies are correctly reproduced within the Frolov and Slavnov’s regularization method.

The results above provide positive indications that Kaplan’s method for chiral fermion may work. There exists, however, several potential problems in his approach. Since the original 2k+1-dimensional model is vector-like, there always exists an anti-chiral mode, localized on an anti-domain wall formed by periodicity of the extra dimension. If the chiral mode and the anti-chiral mode are paired into a Dirac mode, this approach fails to simulate chiral gauge theories. Without dynamical gauge fields, the overlap between the chiral mode and the anti-chiral mode is suppressed as O(e^{−L}) where L is the size of the extra dimension. If gauge fields become dynamical, the overlap depends on the gauge coupling. In the original paper [1] the strong coupling limit of the gauge coupling in the extra dimension was proposed to suppress the overlap. However, a mean-field calculation [8] in this limit indicated that the chiral mode disappears and the model becomes vector-like.

More recently Distler and Rey [9] pointed out that the Kaplan’s method may have a problem in reproducing fermion number non-conservation expected in the standard model. Using the 2-dimensional chiral Schwinger model they argued that either the 2-dimensional fermion number current is exactly conserved or the light degree of freedom flows off the wall into the extra dimension so that the model can not be 2-dimensional.

In this paper we carry out a detailed perturbative analysis of the Kaplan’s proposal for smooth background gauge fields on a finite lattice taking the chiral Schwinger model in 2-dimensions as a concrete example. In sect. II, we formulate the lattice perturbation theory for the Kaplan’s method with the periodic boundary condition. Since translational invariance is violated by domain wall mass terms, usual Feynman rules in the momentum space can not be used except in the regions far away from the domain wall [3]. To perform perturbative calculations near or on the wall, we use the Feynman rules in real space of the extra dimension, as proposed in ref. [5]. We calculate the fermion propagator for the periodic boundary condition, which reproduces the fermion propagator in ref. [5] near the origin of the extra dimension. A similar calculation is also made for the constant fermion mass with free boundaries in the 2k+1-th dimension. As shown by Shamir [10] the constant mass term with this boundary condition can also produce the chiral zero mode on the 2k-dimensional boundary. The results are similar but simpler than those by the Kaplan’s method. In sect. III, using the Feynman rules of sect. II we calculate a fermion one-loop effective action for the U(1) gauge field of the chiral Schwinger model simulated by the Kaplan’s method. We find that the effective action contains the longitudinal component as well as parity-odd terms, and that this longitudinal component, which breaks gauge invariance, remains nonzero even for anomaly-free cases. This result is compared with those of the conventional Wilson fermion formulation of this model [11]. In sect. IV we derive gauge anomalies as well as the Chern-Simons current from the effective action without ambiguity. Then we show that the current corresponding to the fermion number has a non-zero divergence and the fermion number current flows off the walls into the extra dimension. In sect. V, we give our conclusions and discussions.

II. ACTION, FERMION PROPAGATOR AND CHIRAL ZERO MODES

A. Lattice Action

We consider a vector gauge theory in D=2k+1 dimensions with a domain wall mass term. For later convenience we use the notation of ref. [5], where the fermionic action is written in terms of a d=2k dimensional theory with infinitely many flavors. Our action is denoted as

S = S_{G} + S_{F} .  (1)

The action for gauge field S_{G} is given by

S_{G} = β ∑_{n,μ>ν} ∑_{s} Re{Tr[U_{μν}(n, s)]}
+ β_{D} ∑_{n,μ} ∑_{s} Re{Tr[U_{μD}(n, s)]}  (2)

where μ, ν run from 1 to d, n is a point on a d-dimensional lattice and s a coordinate in the extra dimension, β is the inverse gauge coupling for plaquettes U_{μν} and β_{D} that for plaquettes U_{μD}. In general we can take β ≠ β_{D}. The fermionic part of the action S_{F} is given by

S_{F} = \frac{1}{2} ∑_{n,μ} ∑_{s} ψ̄_{s}(n)γ_{μ}[U_{s,μ}(n)ψ_{s}(n + μ) − U_{s,μ}^{†}(n − μ)ψ_{s}(n − μ)]
+ ∑_{n} ∑_{s,t} ψ̄_{s}(n)[M_{0}P_{R} + M_{0}^{†}P_{L}]_{st}ψ_{t}(n)
+ \frac{1}{2} ∑_{n,μ} ∑_{s} ψ̄_{s}(n)[U_{s,μ}(n)ψ_{s}(n + μ) + U_{s,μ}^{†}(n − μ)ψ_{s}(n − μ) − 2ψ_{s}(n)]  (3)

where s, t are considered as flavor indices, P_{R/L} = (1 ± γ_{2k+1})/2,

(M_{0})_{st} = U_{s,D}(n)δ_{s+1,t} − a(s)δ_{st}  (4a)
(M_{0}^{†})_{st} = U_{s−1,D}^{†}(n)δ_{s−1,t} − a(s)δ_{st},  (4b)

and U_{s,μ}(n), U_{s,D}(n) are link variables for gauge fields. We consider the above model with periodic boundaries in the extra dimension, so that s, t run from −L to L − 1, and we take

a(s) = 1 − m_{0}[sign(s + \frac{1}{2}) · sign(L − s − \frac{1}{2})] = \{ \begin{matrix}1 − m_{0}, & −\frac{1}{2} < s < L − \frac{1}{2} \\ 1 + m_{0}, & −L − \frac{1}{2} < s < −\frac{1}{2} \end{matrix}  (5)

for −L ≤ s < L. It is easy to see [5] that S_{F} above is identical to the Kaplan’s action in D=2k+1 dimensions [1] with the Wilson parameter r = 1. In fact the second term in eq.

(3) can be rewritten as

\frac{1}{2} ψ̄_{s}γ_{D}[U_{s,D}ψ_{s+1} − U_{s−1,D}ψ_{s−1}]
+ \frac{1}{2}ψ̄_{s}[U_{s,D}ψ_{s+1} + U_{s−1,D}ψ_{s−1} − 2ψ_{s}] + M(s)ψ̄_{s}ψ_{s}  (6)

with M(s) = m_{0}[sign(s + 1/2) · sign(L − s − 1/2)]. Note that our action is slightly different from that of ref. [5]: we have the D-th component of the link variable U_{s,D}(n) and all link variables have s dependence. With the gauge fixing condition U_{s,D}(n) = 1 for all s and n [9], our action becomes almost identical to that of ref. [5], but still the s dependence exists in our link variables in d dimensions. The model in ref. [5] corresponds to our model at β_{D} = ∞, where s dependences of gauge fields are completely suppressed, and the model at β_{D} = 0 was investigated by the mean field method [8].

B. Chiral Zero Modes

We now consider chiral zero modes of the action S_{F} in the weak coupling limit , i.e. ∀U_{s,μ} = ∀U_{s,D} = 1. According to ref. [5], the right-handed zero modes are given by zero modes of the operator M and the left-handed zero modes by those of the operator M^{†}, where

(M)_{st} = (M_{0})_{st} + \frac{∇(p)}{2}δ_{st}, (M^{†})_{st} = (M_{0}^{†})_{st} + \frac{∇(p)}{2}δ_{st}  (7)

with ∇(p) ≡ ∑_{μ=1}^{d} 2[cos(p_{μ}a) − 1] in momentum space of d dimensions. It is noted that 0 ≤ −∇(p) ≤ 4d and zero modes exist if and only if −∇(p) ≤ 2m_{0} [3]. Hereafter we only consider the case that 0 < m_{0} < 2. In this range of m_{0}, there is only one right-handed zero mode u_{R} satisfying M · u_{R} = 0, which is given by

u_{R}(s) = \{ \begin{matrix}(1 − ∇(p)/2 − m_{0})^{s}C_{0}^{−1} & for s ≥ 0 \\ (1 − ∇(p)/2 + m_{0})^{s}C_{0}^{−1} & for s < 0 \end{matrix}  (8)

where the d-dimensional momentum p has to be restricted to 0 ≤ m_{0} + ∇(p)/2 so that (1 − ∇/2 − m_{0}) ≤ 1. The normalization constant C_{0} takes the value

\frac{1 − (1 − ∇(p)/2 − m_{0})^{L}}{m_{0} + ∇(p)/2} + \frac{1 − (1 − ∇(p)/2 + m_{0})^{−L}}{m_{0} − ∇(p)/2} .  (9)

This zero mode is localized around s = 0. On a finite lattice (i.e. L ≠ ∞) with the periodic boundary condition, there exists another zero mode u_{L} with the opposite chirality satisfying M · u_{L} = 0, which is given by u_{L}(s) = u_{R}(L − t − 1) and is localized around s = L. The overlap between the two zero modes vanishes exponentially as L → ∞;

∑_{s=−L}^{L−1} u_{R}(s)u_{L}(s) = C_{0}^{−2}L × (1 − \frac{∇(p)}{2} − m_{0})^{L}
× (1 − \frac{∇(p)}{2} + m_{0})^{−L} ⟶ 0,  (10)

We illustrate the shape of the two zero modes u_{R} and u_{L} at m_{0} = 0.1 and 0.5 for p = 0 in Fig. 1.

C. Fermion Propagator and Zero Modes

The fermion propagator in d-dimensional momentum space and in real D-th space has been obtained in ref. [5] for the infinite D-th space(i.e. L = ∞ ). It is not difficult to obtain the fermion propagator for a finite lattice with periodic boundaries. We have

S_{F} (p)_{st} = − [[(i ∑_{μ} γ_{μ}p̄_{μ} + M)G_{L}(p)]_{st}P_{L}
+ [(i ∑_{μ} γ_{μ}p̄_{μ} + M^{†})G_{R}(p)]_{st}P_{R}]  (11)

where

G_{L}(p) = \frac{1}{p̄^{2} + M^{†}M} , G_{R}(p) = \frac{1}{p̄^{2} + MM^{†}}  (12)

with p̄_{μ} = sin(p_{μ}a) . Explicit expressions for G_{L} and G_{R} are complicated in general, but they become simple for large L where we neglect terms of order O(e^{−cL}) with c > 0. We obtain

G_{L}(p)_{st} = \{ \begin{matrix}Be^{−α_{+}|s−t|} + (A_{L} − B)e^{−α_{+}(s+t)} + (A_{R} − B)e^{−α_{+}(2L−s−t)}, & (s, t ≥ 0) \\ A_{L}e^{−α_{+}s+α_{−}t} + A_{R}e^{−α_{+}(L−s)−α_{−}(L+t)}, & (s ≥ 0, t ≤ 0) \\ A_{L}e^{α_{−}s−α_{+}t} + A_{R}e^{−α_{−}(L+s)−α_{+}(L−t)}, & (s ≤ 0, t ≥ 0) \\ Ce^{−α_{−}|s−t|} + (A_{L} − C)e^{α_{−}(s+t)} + (A_{R} − C)e^{−α_{−}(2L+s+t)}, & (s, t ≤ 0) \end{matrix} (13)

G_{R}(p)_{st} = \{ \begin{matrix}Be^{−α_{+}|s−t|} + (A_{R} − B)e^{−α_{+}(s+t+2)} + (A_{L} − B)e^{−α_{+}(2L−s−t−2)}, & (s, t ≥ −1) \\ A_{R}e^{−α_{+}(s+1)+α_{−}(t+1)} + A_{L}e^{−α_{+}(L−s−1)−α_{−}(L+t+1)}, & (s ≥ −1, t ≤ −1) \\ A_{R}e^{α_{−}(s+1)−α_{+}(t+1)} + A_{L}e^{−α_{−}(L+s+1)−α_{+}(L−t−1)}, & (s ≤ −1, t ≥ −1) \\ Ce^{−α_{−}|s−t|} + (A_{R} − C)e^{α_{−}(s+t+2)} + (A_{L} − C)e^{−α_{−}(2L+s+t+2)}, & (s, t ≤ −1) \end{matrix} (14)

where

a_{±} = 1 − \frac{∇(p)}{2} ∓ m_{0}  (15a)
α_{±} = arccosh[\frac{1}{2}(a_{±} + \frac{1 + p̄^{2}}{a_{±}})] ≥ 0,  (15b)
A_{L} = \frac{1}{a_{+}e^{α_{+}} − a_{−}e^{−α_{−}}} ,  A_{R} = \frac{1}{a_{−}e^{α_{−}} − a_{+}e^{−α_{+}}}  (15c)
B = \frac{1}{2a_{+}sinhα_{+}} ,  C = \frac{1}{2a_{−}sinhα_{−}} .  (15d)

For |s|, |t| ≪ L the propagator above coincides with that of ref. [5]. From the form of A_{L}, A_{R}, B and C, it is easy to see [5] that singularities occur only in A_{L} at p = 0;

A_{L} ⟶ \frac{m_{0}(4 − m_{0}^{2})}{4p^{2}a^{2}},  p → 0.  (16)

Therefore the propagator G_{L} describes a massless right-handed fermion around s, t = 0 and G_{R} a massless left-handed fermion around |s|, |t| = L, which correspond to the two zero modes in the previous subsection. Later we use the above forms of the fermion propagator to calculate fermion one-loop diagrams. It is also noted that the fermion propagator away from the two domain walls approaches the Wilson fermion propagator with a constant mass term, i.e. ,

S_{F} (p) → ∫ \frac{dp_{D}}{2π} \frac{e^{iap_{D}(s−t)}}{iγ · p + iγ_{D}p_{D} ± m_{0} − ∇(p) + 1 − cos(p_{D}a)}  (17)

for 1 ≪ |s|, |t|, |L − s|, |L − t| with s · t > 0, where +m_{0} is taken for s, t > 0 and −m_{0} for s, t < 0. Therefore the calculation in ref. [3] is valid in this region of s and t.

Before closing this subsection, we give the form of propagator for the Shamir’s free boundary fermions [10]. This is again given by eq.(11) with M_{st} = δ_{s+1,t} − a_{+}δ_{s,t} and M_{st}^{†} = δ_{s−1,t} − a_{+}δ_{s,t}. For large L, it becomes [10]

G_{L}(p)_{st} = Be^{−α_{+}|s−t|} + A′_{L}e^{−α_{+}(s+t)} + A′_{R}e^{α_{+}(s+t−2L)}  (18)
G_{R}(p)_{st} = Be^{−α_{+}|s−t|} + A′_{R}e^{−α_{+}(s+t)} + A′_{L}e^{α_{+}(s+t−2L)},  (19)

where

A′_{L} = B \frac{1 − a_{+}e^{−α_{+}}}{a_{+}e^{α_{+}} − 1} ,  A′_{R} = −Be^{−2α_{+}}  (20)

and now 0 ≤ s, t ≤ L. Singularities occur only in A′_{L} at p = 0 such that

A′_{L} → \frac{m_{0}(2 − m_{0})}{p^{2}a^{2}}.  (21)

Thus, the propagator describes a right-handed massless fermion around one boundary at s, t = 0 and a left-handed massless around the other boundary at s, t = L.

D. Fermion Feynman Rules

In this subsection, we write down the lattice Feynman rules for fermions relevant for fermion one-loop calculations, which will be performed in the next section. We first choose the axial gauge fixing U_{s,D} = 1. Although the full gauge symmetries in D dimensions are lost, the theory is still invariant under gauge transformations independent of s [9]. We consider the limit of small d-dimensional gauge coupling, and take

U_{s,μ}(n) = exp[iagA_{μ}(s, n + μ/2)]  (22)

where a is the lattice spacing, and g ∝ 1/\sqrt{β} is the gauge coupling constant whose mass dimension is 2 − D/2 (mass dimension of the gauge fields A_{μ} is D/2 − 1). It is noted that the other gauge coupling g_{s} ∝ 1/\sqrt{β_{s}} is not necessarily small and can be made arbitrary large. We consider Feynman rules in momentum space for the physical d dimensions but in real space for the extra dimension.

• The fermion propagator is given by

< ψ_{s}(−p)ψ̄_{t}(p) >= S_{F} (p)_{st}.  (23)

where S_{F} has been given in eq.(11) with eqs.(13,14) for the Kaplan’s fermions or with eqs.(18,19) for the Shamir’s fermions.

• The fermion vertex coupled to a single gauge field is given by

agψ̄_{s}(q)∂_{μ}[S_{F}^{−1}(\frac{q + p}{2})]_{ss}A_{μ}(s, p − q)ψ_{s}(−p).  (24)

Here ∂_{μ}S_{F}^{−1}(q) = \frac{∂S_{F}^{−1}(q)}{∂(q_{μ}a)} = iC_{μ}(q)γ_{μ} + S_{μ}(q) with C_{μ}(q) = cos(q_{μ}a) and S_{μ}(q) =

sin(q_{μ}a). From this form of the vertex it is easy to see that the fermion tadpole diagram for an external gauge field vanishes identically.

• The fermion vertex with two gauge fields is given by

−a^{2} \frac{g^{2}}{2} ψ̄_{s}(q)[∂_{μ}^{2}S_{F}^{−1}(\frac{q + p}{2})]_{ss}A_{μ}^{2}(s, p − q)ψ_{s}(−p)  (25)

where A_{μ}^{2}(s, p) = A_{μ}(s, p − p_{1})A_{μ}(s, p_{1}) with p and p_{1} fixed.

III. PERTURBATIVE CALCULATIONS FOR THE CHIRAL SCHWINGER MODEL

In the following two sections we analyze the chiral Schwinger model formulated via the Kaplan’s method for lattice chiral fermions. Using the Feynman rules of the previous section for D = 3, we calculate the effective action for external gauge fields, from which we derive gauge anomalies, Chern-Simons current, and anomaly of the fermion number current. We perform the calculations for the Shamir’s method in parallel with those for the Kaplan’s method.

A. Effective Action at Fermion One-Loop

Since [A_{μ}, A_{ν}] = 0 for U(1) gauge fields, all diagrams with odd number of external gauge fields vanishes identically. Furthermore diagrams with four or more external gauge fields are all convergent. Therefore only the diagrams with two external gauge fields are potentially divergent. The effective action for two external gauge fields is denoted by

S_{eff}^{(2)} ≡ −\frac{g^{2}}{2} ∑_{p,s,t} A_{μ}(s, p)A_{ν}(t, −p)I^{μν}(p)_{st}
= −\frac{g^{2}}{2} ∑_{p,s,t} A_{μ}(s, p)A_{ν}(t, −p)[I_{a}^{(2)} + I_{b}^{(2)}]_{st}^{μν},  (26)

where

[I_{a}^{(2)}]_{st}^{μν} = ∫_{−π/a}^{π/a} \frac{d^{2}q}{(2π)^{2}} tr \{[∂_{μ}S_{F}^{−1}(q + \frac{p}{2}) · S_{F} (q + p)]_{st}
× [∂_{ν}S_{F}^{−1}(q + \frac{p}{2}) · S_{F} (q)]_{ts}\} × a^{2}  (27)

and

[I_{b}^{(2)}]_{st}^{μν} = −δ_{st}δ_{μν} ∫_{−π/a}^{π/a} \frac{d^{2}q}{(2π)^{2}} tr[∂_{μ}^{2}S_{F}^{−1}(q) · S_{F} (q)]_{ss} × a^{2}  (28)

with tr meaning trace over spinor indices.

B. Evaluation of Zero Mode Contributions

To evaluate I^{μν}(p) we decompose it into two parts as

I^{μν}(p) = I_{0}^{μν}(p) + [I^{μν}(p) − I_{0}^{μν}(p)]  (29)

where I_{0}^{μν} is the contribution of zero modes and I^{μν} −I_{0}^{μν} is the remaining contribution. For I_{0}^{μν} we replace the integrand of I^{μν} with that in the a → 0 limit, and we obtain

I_{0}^{μν}(p)_{st} = ∑_{X} ∫_{−∇(q)≤2m_{0}} \frac{d^{2}q}{(2π)^{2}}
× tr [iγ_{μ}(−iγ_{α}(q + p)_{α}a)G_{X}^{0}(q + p)_{st}P_{X}
× iγ_{ν}(−iγ_{β}(q + p)_{β}a)G_{X}^{0}(q + p)_{ts}P_{X}]] × a^{2},  (30)

with X = L for |s|, |t| ≈ 0, or X = R for |s|, |t| ≈ L. The zero mode propagators G_{X}^{0} are given by

G_{X}^{0}(q)_{st} = lim_{a→0} G_{X}(q)_{st} = \frac{1}{q^{2}a^{2}} F_{X}(s, t)  (31)

where F_{X}(s, t) = F_{X}(t, s) and

F_{L}(s, t) = \frac{m_{0}(4 − m_{0}^{2})}{4} × \{ \begin{matrix}(1 − m_{0})^{s+t} & for s, t ≥ 0 \\ (1 − m_{0})^{s}(1 + m_{0})^{t} & for s ≥ 0 and t < 0 \\ (1 + m_{0})^{s+t} & for s, t ≤ 0 \end{matrix}  (32)

F_{R}(s, t) = \frac{m_{0}(4 − m_{0}^{2})}{4} × \{ \begin{matrix}(1 − m_{0})^{2L−s−t−2} & for s, t ≥ 0 \\ (1 − m_{0})^{L−s−1}(1 + m_{0})^{−L−t−1} & for s ≥ 0 and t < 0 \\ (1 + m_{0})^{−2L−s−t−2} & for s, t ≤ 0 \end{matrix}  (33)

for the Kaplan’s fermion with the domain wall mass terms, and

F_{L}(s, t) = m_{0}(2 − m_{0})(1 − m_{0})^{s+t}  (34a)
F_{R}(s, t) = m_{0}(2 − m_{0})(1 − m_{0})^{2L−s−t}  (34b)

for the Shamir’s fermion with the constant mass terms and free boundaries. We evaluate I_{0}^{μν}(p) in the a → 0 limit. In this limit

lim_{a→0} ∫_{−∇(q)≤2m_{0}} \frac{d^{2}q}{(2π)^{2}} tr[P_{X}γ_{μ}γ_{α}γ_{ν}γ_{β}]\frac{(q + p)_{α}q_{β}}{(q + p)^{2}q^{2}}
= ∫_{−∞}^{∞} \frac{d^{2}q}{(2π)^{2}} tr[P_{X}γ_{μ}γ_{α}γ_{ν}γ_{β}]\frac{(q + p)_{α}q_{β}}{(q + p)^{2}q^{2}}
= \frac{1}{2π} [δ_{X}iϵ^{μα} \frac{p_{ν}p_{μ}}{p^{2}} + (δ^{μν} − \frac{p_{μ}p_{ν}}{p^{2}} ) − \frac{δ^{μν}}{2} ] ,  (35)

therefore we obtain

lim_{a→0} I_{0}^{μν}(p) = ∑_{X} \frac{1}{2π} [δ_{X}iϵ^{μα} \frac{p_{ν}p_{μ}}{p^{2}} + (δ^{μν} − \frac{p_{μ}p_{ν}}{p^{2}} ) − \frac{δ^{μν}}{2} ]
× F_{X}(s, t)^{2}  (36)

where δ_{L} = 1 and δ_{R} = −1. It is noted that F_{X} satisfies

∑_{t} F_{X}(s, t)^{2} = F_{X}(s, s),  ∑_{s,t} F_{X}(s, t)^{2} = 1.  (37)

C. Evaluation of Remaining Contributions

We consider the remaining terms in I^{μν}. Since the combination I^{μν}(p) − I_{0}^{μν}(p) is infrared finite, we can change the integration variable from q to qa and take the a → 0 limit in the integrand. Thus we obtain

lim_{a→0} I^{μν}(p) − I_{0}^{μν}(p) = I^{μν}(0) = \frac{1}{2π} · [iϵ^{μν}Γ_{CS} + δ^{μν}K]  (38)

where

Γ_{CS}(s, t) = \frac{ϵ^{μν}}{i} 2π ∫ \frac{d^{2}q}{(2π)^{2}} tr \{[∂_{μ}S_{F}^{−1}(q)S_{F} (q)]_{st}
× [∂_{ν}S_{F}^{−1}(q)S_{F} (q)]_{ts}}  (39)

and

K(s, t) = 2π ∫ \frac{d^{2}q}{(2π)^{2}} [tr \{[∂_{μ}S_{F}^{−1}(q)S_{F} (q)]_{st}

× [∂_{μ}S_{F}^{−1}(q)S_{F} (q)]_{ts}} − δ_{st}tr[∂_{μ}^{2}S_{F}^{−1}(q)S_{F} (q)]_{ss}] .  (40)

Here no summation over μ, ν is implied.

The parity-odd term Γ_{CS} is the coefficient function of a 3-dimensional Chern-Simons term in the axial gauge [9], which satisfies Γ_{CS}(s, t) = −Γ_{CS}(t, s). It is easy to show that

∑_{t} Γ_{CS}(s, t) = −\frac{ϵ^{μν}}{i} ∫ \frac{d^{2}q}{2π} tr \{∂_{μ}S_{F}^{−1}(q)∂_{ν}S_{F} (q)\}_{ss}

= \frac{ϵ^{μν}}{i} [∫ \frac{dq_{μ}}{2π} tr \{∂_{μ}S_{F}^{−1}(q) · S_{F} (q)\}_{ss}]_{q_{ν}=ϵ}^{π}

This would be zero if there were no infra-red singularities in S_{F}. However, because of the contribution from zero modes, S_{F} is singular at q = 0. Therefore

∑_{t} Γ_{CS}(s, t) = − ∑_{t} Γ_{CS}(t, s) = 4 lim_{ϵ→0} [∫_{ϵ}^{π/2} \frac{dq_{μ}}{2π} ∑_{X} δ_{X}[S_{ν}(q)C_{μ}(q)G_{X}^{0}(q)]_{ss}]_{q_{ν}=ϵ}^{π/2}
= −4 ∑_{X} δ_{X}F_{X}(s, s) lim_{ϵ→0} ∫_{ϵ}^{π/2} \frac{dq_{μ}}{2π} \frac{ϵ}{q_{μ}^{2} + ϵ^{2}}
= − ∑_{X} 2δ_{X} \frac{F_{X}(s, s)}{π} [tan^{−1} \frac{q}{ϵ} ]_{ϵ}^{π/2}
= − ∑_{X} \frac{δ_{X}}{2} F_{X}(s, s),  (41)

where F_{X}(s, t) is given in the previous subsection.

Since S_{F} becomes the Wilson fermion propagator with constant mass term for 1 ≪ |s|, |t|, |L − s|, |L − t| with s · t > 0 [see eq.(17) ], it becomes

∑_{s,t} Γ_{CS}^{μν}(s, t)A_{μ}(s, p)A_{μ}(t, −p) ⟶

− ϵ^{μν} ∫ dp_{3}A_{μ}(p_{3}, p)p_{3}A_{μ}(−p_{3}, −p) ∫ \frac{d^{3}q}{(2π)^{2}}

× tr {[∂_{μ}S_{F}^{−1} · S_{F}][∂_{3}S_{F}^{−1} · S_{F}][∂_{ν}S_{F}^{−1} · S_{F}]} ,  (42)

which coincides with the result of ref. [3]. This is a good check of our calculation. From Ref.

[3] we obtain

= ϵ^{μν} ∫ dp_{3}A_{μ}(p_{3}, p)p_{3}A_{ν}(−p_{3}, −p) × \{ \begin{matrix}1 & for +m_{0} \\ 0 & for −m_{0} \end{matrix} .  (43)

The parity-even term K satisfies K(s, t) = K(t, s) and

∑t K(s, t) = ∑_{t} K(t, s)
= − ∫ \frac{d^{2}q}{(2π)^{2}} tr \{[∂_{μ}S_{F}^{−1} · ∂_{μ}S_{F}]_{ss} + [∂_{μ}^{2}S_{F}^{−1} · S_{F}]_{ss}\}
= ∑_{X} \frac{1}{2} · F_{X}(s, s)  (44)

The derivation of the last equality is similar to that of eq.(41).

D. Total Contributions

Combining the above contributions we finally obtain

S_{eff}^{(2)} = −\frac{g^{2}}{4π} ∑_{s,t} ∫ d^{2}x \{∑_{X} F_{X}(s, t)^{2}[A_{μ}(s, x)(δ^{μν} − \frac{∂_{μ}∂_{ν}}{□})A_{ν}(t, x)]
+ [K(s, t) − ∑_{X} \frac{1}{2} · F_{X}(s, t)^{2}]A_{μ}(s, x)A_{μ}(t, x)
+ ∑_{X} iδ_{X}F_{X}(s, t)^{2}[\frac{∂_{μ}}{□} A_{μ}(s, x)ϵ^{αν}∂_{α}A_{ν}(t, x)] + iΓ_{CS}(s, t)ϵ^{μν}A_{μ}(s, x)A_{ν}(t, x)\} . (45)

This is the main result of this paper. It is noted that the above formula is valid for both the Kaplan’s and the Shamir’s methods. The following consequences can be drawn from eq.

(45) above.

The Parity-odd terms, which are proportional to ϵ^{αν}, are unambiguously defined, contrary to the case of the continuum regularization for anomaly free chiral gauge theories [4] which only regulates the parity even terms [5,7]. These parity-odd terms break gauge invariance in the 2-dimensional sense.

For anomalous chiral Schwinger model, the parity-odd term with X = R is localized around s = 0 and that for X = L is localized around s = L. The effective action above for anomalous chiral Schwinger model via the Kaplan’s method or the Shamir’s variation is different from the one via the usual Wilson fermion in 2 dimensions [11]: The term proportional to Γ_{CS}, which can not be evaluated analytically for s dependent gauge fields, is special for chiral fermions from 3-dimensional theories, and the presence of this term prevents us from concluding whether the anomalous chiral Schwinger model can be consistently defined via the Kaplan’s (Shamir’s) method.

For anomaly free cases such that ∑_{R} g_{R}^{2} = ∑_{L} g_{L}^{2}, the parity-odd terms are exactly cancelled locally in s space. Here g_{R(L)} is the coupling constant of a fermion with positive (negative) m_{0} which generate a right-handed (left-handed) zero mode around s = 0. The simplest but non-trivial example is a Pythagorean case, g_{R} = 3, 4 and g_{L} = 5 [9]. Even for these anomaly free cases, the longitudinal term, whose coefficient is K − F^{2}/2, remains nonzero in the effective action, so that gauge invariance in the 2-dimensional sense is violated. In this regard the form of the effective action via the Kaplan’s (Shamir’s) method is similar to the one via the usual Wilson fermion [11].

Let us consider the effective action for s independent gauge fields as in ref. [5]. Since ∑_{μ,ν} ϵ^{μν}A_{μ}(x)A_{ν}(x) = 0 for s independent gauge fields, the Chern-Simons term vanishes. The other parity-odd term is cancelled between the two zero modes since ∑_{X,s,t} δ_{X}F_{X}(s, t)^{2} = 0. The longitudinal term also vanishes due to the identity

∑_{t}[K(s, t) − \frac{F_{X}(s, t)^{2}}{2} ] = ∑_{t}[\frac{F_{X}(s, t)^{2}}{2} − \frac{F_{X}(s, t)^{2}}{2} ] = 0  (46)

[see eqs. (37) and (44)]. Therefore the effective action becomes

S_{eff}^{(2)} = −2 × \frac{g^{2}}{4π} ∫ d^{2}x[A_{μ}(x)(δ^{μν} − \frac{∂_{μ}∂_{ν}}{□})A_{ν}(x)].  (47)

This effective action is transverse and thus gauge invariant in the 2-dimensional sense. Both zero modes around s = 0 and s = L equally contribute so that a factor 2 appears in the above result. This is consistent with the general formula derived in ref. [6]. The anomalous chiral Schwinger model can not be simulated by the Kaplan’s (Shamir’s) method with the s-independent gauge fields, since the gauge fields see both of the zero modes so that it fails to reproduce the parity-odd term, expected to exist [12]

IV. ANOMALIES IN THE CHIRAL SCHWINGER MODEL

A. Currents and their Divergence

From the effective action obtained in the previous section, we can calculate the vacuum expectation values of various currents in the presence of back-ground gauge fields. Let us define the fermion number current as

⟨J_{μ}^{g}(s, x)⟩ = \frac{δS_{eff}^{(2)}}{gδA_{μ}(s, x)}  (48)

where the index g in the current explicitly shows the charge of the fermion. From eq. (45) we obtain

J_{μ}^{g}(s, x) = i\frac{g}{4π} ∑_{t}[∑_{X} δ_{X}F_{X}(s, t)^{2}(ϵ^{αν}∂_{μ} + ϵ^{αμ}∂_{ν})\frac{∂_{α}}{□} A_{ν}(t, x) − 2Γ_{CS}ϵ^{μν}A_{ν}(t, x)]
− \frac{g}{4π} ∑_{t}[2 ∑_{X} F_{X}(s, t)^{2}(δ^{μν} − \frac{∂_{μ}∂_{ν}}{□})A_{ν}(t, x) + (2K(s, t) − ∑_{X} F_{X}(s, t)^{2})A_{μ}(t, x)]
≡ J_{μ}^{g,odd} + J_{μ}^{g,even}  (49)

where J_{μ}^{g,odd} is a parity-odd current (the first two terms) and J_{μ}^{g,even} is a parity-even current (the last two terms). Hereafter all J_{μ} should be understood as vacuum expectation values, though ⟨ ⟩ is suppressed. From the fermion number current the gauge current for a fermion with charge g is easily constructed as J_{μ}^{G}(s, x) ≡ gJ_{μ}^{g}(s, x).

Divergences of the parity-odd and parity-even currents become

∂_{μ}J_{μ}^{g,odd}(s, x) = i\frac{g}{4π} ∑_{t}[∑_{X} δ_{X}F_{X}(s, t)^{2} − 2Γ_{CS}(s, t)]
× ϵ^{μν}∂_{μ}A_{ν}(t, x),  (50)
∂_{μ}J_{g,even}^{μ}(s, x) = \frac{g}{4π} ∑_{t}[∑_{X} F_{X}(s, t)^{2} − 2K(s, t)]
× ∂_{μ}A_{μ}(t, x).  (51)

B. Gauge invariance

As mentioned in Sect. II, the action in A_{3} = 0 gauge is invariant under s independent gauge transformation. This invariance implies ∑_{s} ∂_{μ}J_{μ}^{G}(s, x) = 0. This identity is satisfied in our calculation of the effective action, since

∑_{s} [∑_{X} δ_{X}F_{X}(s, t)^{2} − 2Γ_{CS}(s, t)]
= ∑_{X} δ_{X}[F_{X}(t, t) − F_{X}(t, t)] = 0  (52)

∑_{s} [∑_{X} F_{X}(s, t)^{2} − 2K(s, t)]
= ∑_{X}[F_{X}(t, t) − F_{X}(t, t)] = 0  (53)

from eqs.(37,41,44).

C. Gauge Anomalies

The gauge anomaly for a fermion with a charge g, denoted T_{g}, is defined by T_{g} = g∂_{μ}J_{μ}^{g,odd}, and it becomes

T_{g}(s, x) = ∑_{t} g^{2}C(s, t) × T^{0}(t, x)  (54)

where

T^{0}(t, x) = i\frac{1}{4π} ϵ^{μν}∂_{μ}A_{ν}(t, x)  (55)

is the gauge anomaly of a 2-dimensional theory, and

C(s, t) = ∑_{X} δ_{X}F_{X}(s, t)^{2} − 2Γ_{CS}(s, t)  (56)

represents the spread of the gauge anomaly over the 3rd direction due to the spread of zero modes. This spread of the anomaly has been observed in a numerical computation [2]. It is noted that the divergence of the gauge current J_{μ}^{G} also contains parity-even contributions, given by

∑_{t} D(s, t) × \frac{g^{2}}{4π} ∂_{μ}A_{μ}(t, x)  (57)

where D(s, t) = ∑_{X} F_{X}(s, t)^{2} − 2K(s, t).

The one-loop integral (39) defining Γ_{CS} is too complicated to calculate analytically. For t-independent gauge fields A_{μ}(t, x) = A_{μ}(x) there is a considerable simplification and we obtain

T_{g}(s, x) = g^{2}C(s) × T^{0}(x)  (58)

where

T^{0}(x) = i\frac{1}{4π} ϵ^{μν}∂_{μ}A_{ν}(x)  (59)

and

C(s) = ∑_{t} C(s, t) = 2 ∑_{X} δ_{X}F_{X}(s, s)
= \frac{m_{0}(4 − m_{0}^{2})}{2} × \{ \begin{matrix}[(1 − m_{0})^{2s} − (1 − m_{0})^{2(L−s−1)}] & for s ≥ 0 \\ [(1 + m_{0})^{2s} − (1 + m_{0})^{−2(L+s+1)}] & for s ≤ 0 \end{matrix} .  (60)

for the Kaplan’s method. We plot C(s) as a function of s at m_{0} = 0.1 and 0.5 in Fig.2. For the Shamir’s method, we obtain

C(s) = 2m_{0}(2 − m_{0}) · [(1 − m_{0})^{2s} − (1 − m_{0})^{2(L−s)}]  (61)

and plot this in Fig.3. It is noted that there is no parity-even contribution in ∂_{μ}J_{μ}^{G} since ∑_{t} D(s, t) = 0 in this case.

D. Chern-Simons Current

From the 3-dimensional point of view, the gauge anomaly should be cancelled in such a way that T_{g} +∂_{3}gJ_{3}^{CS}(s, x) = 0 [1,3], where J_{3}^{CS} is the third component of the Chern-Simons current for the 3-dimensional vector gauge theory. With our gauge fixing the effective action does not depends on A_{3}, and it is difficult to calculate J_{3}^{CS} analytically except in the region away from domain-walls [3]. However, for t-independent gauge fields, we can obtain the Chern-Simons current everywhere via the relation ∂_{3}gJ_{g,odd}^{3}(s, x) = −T_{g}, which becomes

J_{3}^{CS}(s + \frac{1}{2}, x) − J_{3}^{CS}(s − \frac{1}{2}, x) = g^{2}C(s) × T^{0}(x).  (62)

Taking J_{3}^{CS}(s + \frac{1}{2}, x) = g^{2}I(s) · T^{0}(x), we obtain

I(s + \frac{1}{2}) − I(s − \frac{1}{2}) = C(s).  (63)

We have to solve this equation with the boundary condition I(s) → −2 as s → +∞ [3]. For a finite s space s → +∞ means 1 ≪ s ≪ L.

For the Kaplan’s method we obtain

I(s − \frac{1}{2}) = \{ \begin{matrix}\frac{2 + m_{0}}{2} · [(1 − m_{0})^{2s} + (1 − m_{0})^{2(L−s)}] − 2 & 0 ≤ s ≤ L \\ −\frac{2 − m_{0}}{2} · [(1 + m_{0})^{2s} + (1 + m_{0})^{−2(L+s)}] & −L ≤ s ≤ 0 \end{matrix} .  (64)

This solution automatically satisfies the other boundary condition that I(s) → 0 as → −∞ [3]. Again → −∞ means 1 ≪ −s ≪ L for a finite s space. We plot I(s) as a function of s at m_{0} = 0.1 and 0.5 in Fig. 4. For the Shamir’s method we obtain

I(s − \frac{1}{2}) = 2[(1 − m_{0})^{2s} + (1 − m_{0})^{2(L−s+1)}] − 2
for 0 ≤ s ≤ L − 1,  (65)

which is plotted in Fig. 5.

E. Pythagorean Chiral Schwinger Model and Anomaly in Fermion Number Current

Let us consider the Pythagorean chiral Schwinger model [9]. In this model there are two right-handed fermions with charges g_{1} and g_{2}, and one left-handed fermion with charge g_{3}. Formulation of this model via the Kaplan’s method has already been discussed in ref. [1,2,9] ( an extension to the Shamir’s method is straightforward ). We assign +m_{0} for fermions with charge g_{1} and g_{2}, and −m_{0} for a fermion with charge g_{3}. The value of |m_{0}| should be equal for all fermions, as will be seen below.

The theory has a U(1)^{3} symmetry [9] corresponding to independent phase rotations of three fermions. The corresponding currents are

\{ \begin{matrix}J_{μ}^{G} & = & g_{1}J_{μ}^{g_{1}} + g_{2}J_{μ}^{g_{2}} + g_{3}J_{μ}^{g_{3}} \\ J_{μ}^{R} & = & g_{2}J_{μ}^{g_{1}} − g_{1}J_{μ}^{g_{2}} \\ J_{μ}^{F} & = & J_{μ}^{g_{1}} + J_{μ}^{g_{2}} + J_{μ}^{g_{3}} \end{matrix}  (66)

The first one is the gauge current, whose divergence becomes

∂_{μ}J_{μ}^{G}(s, x) = (g_{1}^{2} + g_{2}^{2} − g_{3}^{2}) × ∑_{t} C(s, t) · T^{0}(t, x).  (67)

Therefore, if g_{1}^{2} + g_{2}^{2} = g_{3}^{2} (Pythagorean relation) is satisfied and if all fermions have the same value of |m_{0}| to give the same C(s, t), this current is conserved and there is no gauge anomaly for any background gauge fields. The second current is non-anomalous, since

∂_{μ}J_{μ}^{R}(s, x) = (g_{2} · g_{1} − g_{1} · g_{2}) ∑_{t} C(s, t) · T^{0}(t, x) = 0.  (68)

The third current, which corresponds to the fermion number of the theory, is anomalous, since

∂_{μ}J_{μ}^{F} (s, x) = (g_{1} + g_{2} − g_{3}) × ∑_{t} C(s, t) · T^{0}(t, x).  (69)

The Kaplan’s method as well as the Shamir’s one successfully give a non-zero divergence for the fermion number current, though the coefficient C(s, t) has a finite width. For tindependent gauge fields, this anomaly becomes

(g_{1} + g_{2} − g_{3}) · C(s) · T^{0}(x)  (70)

where C(s) is almost localized at s = 0 and at s = L as seen in Fig.2 and Fig.3. Since the fermion number is conserved in the 3-dimensional theory, the third component of the fermion number current should satisfy ∂_{3}J_{3}^{F} + (g_{1} + g_{2} − g_{3}) · C(s) · T^{0}(x) = 0 [9]. Therefore we obtain

J_{3}^{F} (s, x) = (g_{1} + g_{2} − g_{3}) · I(s) · T^{0}(x).  (71)

V. CONCLUSIONS

In this paper we have formulated a lattice perturbative expansion for the Kaplan’s chiral fermion theories, extending the suggestion by Narayanan and Neuberger [5]. Applying our perturbative technique to the chiral Schwinger model formulated via the Kaplan’s or the Shamir’s method, we have calculates the fermion one-loop effective action for gauge fields. The effective action contains parity-odd terms and longitudinal terms, both of which break 2-dimensional gauge invariance, and the anomaly of the gauge current is obtained from the effective action. The gauge anomaly is calculable in the Kaplan’s (Shamir’s) method if the perturbative expansion is carefully formulated. For the anomaly-free Pythagorean chiral Schwinger model, the fermion number current is anomalous. To obtain this anomaly the fermion number current should not be summed over s, in contrast to the case of the continuum calculation [7], where the anomaly comes from an infinite summation over s.

The main conclusions drawn from the results are as follows.

1. Anomaly of the fermion number current is shown to be non-zero in this method, though the current flows off walls into the extra dimension. Since the current is external we feel that this does not affect the dynamics of the model and therefore does not spoil the 2-dimensional nature of the chiral zero mode, contrary to the suggestion of Ref. [9]. The 3-dimensional nature of the Kaplan’s (Shamir’s) formulation manifests itself only in the non-conservation of the fermion number, which is expected to occur in Nature.

2. Two-dimensional gauge invariance at low energy can not be assured by the Kaplan’s (Shamir’s) method, except for s-independent gauge fields, even for anomaly-free cases. This is similar to the situation with lattice chiral gauge theories formulated with the ordinary Wilson mass term [11]. In this point the Kaplan’s (Shamir’s) method does not seem better than the conventional approaches. At this moment it is not clear whether this violation of gauge invariance spoils the whole program of this method. In particular the effects of the longitudinal component of gauge fields has to be analyzed

further.

3. If the theory is anomaly free and gauge fields are s-independent [5], the gauge invariance as a 2-dimensional theory can be maintained. However, the gauge fields feel both of the zero modes even in the L → ∞ limit, and the fermion loop contribution to the effective action is twice as large as the one expected from a single chiral fermion. Therefore we have to take a square-root of the fermion determinant to obtain the correct contribution. For fermion quantities such as the fermion number current, however, it seems possible to separate the contribution of the chiral zero mode at s = 0 from that of the anti-chiral zero mode at s = L, as seen in the previous section.

Perturbative calculations performed in this paper can be extended to 4+1 dimensional theories. Of course actual calculations become much more complicated and difficult because of severe ultra-violet divergences in 4+1 dimensions than in 2+1 dimensions. Work in this direction is in progress.

ACKNOWLEDGMENTS

We would like to think Prof. Ukawa for discussions and the careful reading of the manuscript.

After finishing this work, a new paper by Narayanan and Neuberger [13] appeared. In the paper the gauge anomaly for the chiral Schwinger model was calculated semi-analytically via the overlap formula of ref. [6].

REFERENCES

1 D. B. Kaplan, Phys. Lett. B288, 342 (1992). 2 K. Jansen, Phys. Lett. B288, 348 (1992). 3 M. F. L. Golterman, K. Jansen, and D. B. Kaplan, Phys. Lett. B301, 219 (1993). 4 S. A. Frolov and A. A. Slavnov, Phys. Lett. B309, 344 (1993). 5 R. Narayanan and H. Neuberger, Phys. Lett. B302, 62 (1993). 6 R. Narayanan and H. Neuberger, RU-93-25, Rutgers University preprint, July 1993. 7 S. Aoki and Y. Kikukawa, UTHEP-258/KUNS-1204, University of Tsukuba preprint,

June 1993. 8 C. P. Korthals-Altes, S. Nicolis and J. Prades, CPT-93/P.2920, Center de Physique

Théorique preprint, June 1993. 9 J. Distler and S.-J. Rey, PUPT-1386/NSF-ITP-93-66/SNUTP 93-27, Princeton University

preprint, May 1993. 10 Y. Shamir, WIS-93/20/FEB-PH, Weizmann Institute preprint, February 1993. 11 S. Aoki, Phys. Rev Lett. 60 2109 (1988); K. Funakubo and T. Kashiwa,ibid 60 2113

(1988); T. D. Kieu, D. Sen, S.-S. Xue, ibid 60 2117 (1988); S. Aoki, Phys. Rev. D38 618 (1988).

12 R. Jackiw and R. Rajaraman, Phys. Rev Lett. 54 1219 (1985). 13 R. Narayanan and H. Neuberger, RU-93-34, Rutgers University preprint, August 1993.

FIGURES

FIG. 1. Two zero modes u_{L} and u_{R} as a function of s at m_{0} = 0.1 and 0.5 for p_{1} = p_{2} = 0. We take L = 100.

FIG. 2. The coefficient of the anomaly C(s) for the Kaplan’s method as a function of s at m_{0} = 0.1 and 0.5 for L = 100.

FIG. 3. The coefficient of the anomaly C(s) for the Shamir’s method as a function of s at m_{0} = 0.1 and 0.5 for L = 100.

FIG. 4. The coefficient of the Chern-Simons current I(s) for the Kaplan’s method as a function of s at m_{0} = 0.1 and 0.5 for L = 100.

FIG. 5. The coefficient of the Chern-Simons current I(s) for the Shamir’s method as a function of s at m_{0} = 0.1 and 0.5 for L = 100.