One-loop perturbative coupling of A and A ? through the chiral overlap operator

Recently, Grabowska and Kaplan constructed a four-dimensional lattice formulation of chiral gauge theories on the basis of the chiral overlap operator. At least in the tree-level approximation, the left-handed fermion is coupled only to the original gauge field A, while the right-handed one is coupled only to the gauge field A?, a deformation of A by the gradient flow with infinite flow time. In this paper, we study the fermion one-loop effective action in their formulation. We show that the continuum limit of this effective action contains local interaction terms between A and A?, even if the anomaly cancellation condition is met. These non-vanishing terms would lead an undesired perturbative spectrum in the formulation.


Introduction and discussion
Recently, Grabowska and Kaplan proposed a four-dimensional lattice formulation of chiral gauge theories [1].This formulation is based on the so-called overlap operator, which can be obtained from their five-dimensional domain-wall formulation [2] 1 by the traditional way [4][5][6].In this formulation, along the fifth dimension, the original gauge field A is deformed by the gradient flow [7][8][9][10] for infinite flow time.Since the gradient flow preserves the gauge covariance, this formulation is manifestly gauge invariant, even if the anomaly cancellation condition is not met.Although there is a subtlety associated with the topological charge [1,2,[11][12][13], the smeared gauge field after the infinite-flow time, A , only to which the right-handed (invisible) fermion would be coupled, can be basically considered as pure gauge (see Appendix A).Then one would regard their setup as the system of the left-handed fermion interacting with the gauge field A; 2 this picture was however confirmed only in the tree-level approximation [1].It is thus a crucial problem whether radiative corrections induce the physical coupling of the right-handed fermion or not.
First, let us see the tree-level decoupling between the physical and invisible sectors.So far, only when the transition of the flowed gauge field along the fifth dimension is abrupt, the four-dimensional lattice Dirac operator has been obtained as an explicit form; this is referred to as the chiral overlap Speaker, e-mail: o-morikawa@phys.kyushu-u.ac.jpAcknowledges partial support by JSPS Grants-in-Aid for Scientific Research Grant Number JP16H03982. 1 As a closely related six-dimensional domain-wall formulation, see Ref. [3]. 2 Grabowska and Kaplan's formulation is a modification of that of Álvarez-Gaumé and Ginsparg [14].The latter takes A = 0 identically without the gradient flow and it breaks the gauge invariance.arXiv:1710.00536v1[hep-lat] 2 Oct 2017 operator Dχ .The operator Dχ is given by [1] where a is the lattice spacing, and ( ) is the sign function [15,16] ≡ H w (A) of the Hermitian Wilson Dirac operator where m is the parameter of the domain-wall height, and γ µ is the Dirac matrix.In this expression, ∇ µ is the forward gauge covariant lattice derivative and ∇ * µ is the backward one.With the assumption of abruptness, this Dirac operator depends on the two gauge fields, A and A .In the classical continuum limit [1], where is the covariant derivative defined with respect to A (A ), and P ± = (1 ± γ 5 )/2 are the chirality projection operators.Therefore, the coupling between the gauge fields, A and A , is not produced in the tree-level approximation.
Let us study how the decoupling between A and A is modified under radiative corrections.The fermion one-loop effective action is defined by where A and A are regarded as independent non-dynamical variables.To investigate the (de)coupling, two infinitesimal variations δ and δ are introduced such that δ acts only on A but not on A , and δ acts in an opposite way, Then, we will find that in the continuum limit a double variation of the effective action is given as where L(A, A ; δA, δ A ) is a local polynomial of its arguments and their spacetime derivatives.
To find a possible implication of Eq. ( 8), we take gauge variations as δ and δ : Since, as a property of the gradient flow, the two gauge fields A and A transform in the same way under the gauge transformation, the gauge invariance of the effective action implies Therefore, using Eq. ( 8), we can obtain Now, let us assume that A becomes pure gauge under the gradient flow with infinite flow time (see Appendix A): Then the gauge transformation A g −1 makes A = 0, where That is, we can impose the A = 0 gauge on Eq. ( 12) We will see below that the right-hand side does not vanish even if the anomaly cancellation condition is met.
It will be shown in the next section that ln Z[A, 0] has the term thus the mass term tr A µ A µ is produced in the one-loop level.The propagator of the gauge potential in this A = 0 thus has the structure, where we have defined the mass parameter m A as Therefore, the perturbative spectrum is modified in a weird way; this would not be what we want to obtain for chiral gauge theories.Since these effects in the one-loop effective action ( 16) should be removed by local counterterms, the formulation of Grabowska and Kaplan will be undesirable as a non-perturbative formulation of chiral gauge theories.Then their formulation with the abrupt transition should be improved in some possible way.
2 Explicit forms of L and δ ω ln Z In this section, we show the results of the continuum limit of L(A, A ; δA, δ A ). 3 In what follows, we use the variables and the field strength We also define the following lattice integrals: where f i (am) (i = 0, . . ., 5) as the function of am are plotted in Figs.1-6.The local functional L has three parts, according to the parity and Lorentz symmetry: (i) the parity-odd and Lorentz-preserving part, (ii) the parity-even and Lorentz-preserving part, and (iii) the parity-even and Lorentz-violating part.First, the parity-odd part of L is given by where and in what follows the symbol tr is assumed to be omitted.This part is proportional to the gauge anomaly coefficient; thus this vanishes if the anomaly cancellation condition is met.Second, we have the parity-even and Lorentz-preserving part of L, L(A, A ; δA, δ A )| parity-even, Lorentz-preserving and finally the parity-even and Lorentz-violating part is given by (34) By using the above form of L(A, A ; δA, δ A ), one can deduce the gauge variation of ln Z[A, 0], δ ω ln Z[A, 0] (see Appendix A of Ref. [17] for details).The parity-odd part of L gives rise to (leaving out the symbol d 4 x tr) which is the consistent gauge anomaly associated with a left-handed fermion.It is impossible to rewrite this expression as the gauge variation of a local term.On the other hand, the parity-even part of δ ω ln Z can be written as the gauge variation of local terms: