One-loop perturbative coupling of $A$ and $A_\star$ 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~$A_\star$, 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~$A_\star$, 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 constructed a four-dimensional lattice formulation of chiral gauge theories [1], starting from their five-dimensional domain-wall formulation in Ref. [2]. 1 A salient feature of this formulation is that the lattice Dirac operator depends on two gauge fields: one is the original gauge field A, and the other is A , which is given by the gradient flow [4][5][6][7] of A with infinite flow time. In the tree-level continuum limit of the formulation, the left-handed component of the fermion is coupled only to A, while the right-handed one (called the fluffy mirror fermion or "fluff") is coupled only to A . Up to a subtlety associated with the topological charge [1,2,8,9], A basically becomes pure gauge after the infinite-time flow. Then this setup would be regarded as the system of the left-handed Weyl fermion coupled to the gauge field A (in the spirit of Ref. [10]). Since the flow equation preserves the gauge covariance [4][5][6][7], A transforms gauge covariantly under the gauge transformation. Then the fermion determinant is manifestly gauge invariant in this formulation, even if the gauge representation is anomalous. It is crucial to understand, therefore, how this formulation fails when the gauge representation is anomalous. It is conceivable that the locality plays a key role for this, but no definite argument has been given yet.
So far, the explicit form of the four-dimensional lattice Dirac operator in the above formulation has been obtained only when the transition from A to A along the flow is "abrupt" or "sudden"; the resulting Dirac operator is referred to as the chiral overlap operator in the present paper, and is PTEP 2017, 063B08 H. Makino et al. denoted byD χ . As noted above, in the tree-level continuum limit [1], where a is the lattice spacing, m is a parameter of mass dimension one, D μ (A) (D μ (A )) is the covariant derivative defined with A (A ), γ μ is the Dirac matrix, and P ± = (1±γ 5 )/2 are the chirality projection operators. Thus, the lattice Dirac operator does not produce any coupling between two gauge fields, A and A , in the tree-level approximation.
In this paper, we investigate how the above situation is modified under radiative corrections. We thus study the fermion one-loop effective action defined by where the two gauge fields A and A are regarded as independent non-dynamical variables. In the present paper, we assume that the gauge field is perturbative and the Dirac operator has no normalizable zero modes. What we will show in this paper is the gauge invariance implies Then, using Eqs. (1.6) and (1.7) in this relation, we have Now, the gauge field A is given by the gradient flow of A for infinite flow time. Thus, let us assume that A is pure gauge. Although there exists a subtlety as to whether this is actually the case or not for topologically non-trivial gauge field configurations [1,8], this will certainly be the case for topologically trivial configurations. Under this assumption, since the lattice gauge action S G [A] (such as the plaquette action) with which the gauge field A is integrated over will be gauge invariant, we may take a particular gauge in which A ≡ 0. 3 in Eq. (1.6) are constants, from Eqs. (1.6) and (1.11), we have (1.12) We will see that the right-hand side does not vanish even if the gauge representation is anomaly free. Thus, generally, configurations of the gauge field A (in a topologically trivial sector) are integrated with the sum of the gauge-invariant action S G [A] and gauge non-invariant effective action ln Z[A, 0]. 4 For example, we will see that L(A, A = 0; δA, δ ω A | A =0 ) contains a term corresponding to the mass term of the gauge field. Such gauge-breaking effects that are not related to the gauge anomaly should be able to be removed by local counterterms. This expectation is explicitly confirmed in Appendix. Nevertheless, such a necessity for counterterms to restore the gauge symmetry will be undesirable for a possible non-perturbative formulation of chiral gauge theories. This is the implication of our observation (1.3). It appears that the formulation of Ref. [1] with the chiral overlap operatorD χ (i.e., the sudden flow case) should be improved in some possible way. In the rest of this paper, we will explain how Eq. (1.7) is obtained.

Basic formulation
The explicit form of the chiral overlap operatorD χ is given by [1], H. Makino et al. where and are the sign functions [11,12] ≡ H w (A) of the Hermitian Wilson-Dirac operator where ∇ μ and ∇ * μ are forward and backward gauge covariant lattice derivatives, respectively. The parameter m is taken as 0 < am < 2. From Eq. (2.2), depends only on the gauge field A and only on A . By construction, Using these, one can confirm that and, consequently,D χ in Eq. (2.1) satisfies the Ginsparg-Wilson relation [13] γ 5Dχ +D χ γ 5 = aD χ γ 5Dχ .
It is then natural to introduce a modified γ 5 [14,15], which satisfies Note, however, thatγ 5 is not Hermitian in the present formulation,γ † 5 =γ 5 . From the first relation of Eq. (2.8), one can define modified chiral projection operators bŷ The chiral components of the fermion can then be defined aŝ Thanks to the second relation of Eq. (2.8), the action is completely decomposed into the left-handed and right-handed components as

Gauge currents and partial decoupling of the right-handed fluff fermion
In the present paper, we assume that the gauge field is perturbative and the Dirac operator has no normalizable zero modes in infinite volume. Then the change of the effective action (1.2) under the variation of the gauge field δ (1.4), for example, is given by where Tr ≡ x tr, and tr stands for the trace over the spinor and gauge indices. In deriving this, we have used the fermion propagator, (2.14) We will refer to Eq. (2.13) (and a similar expression for the variation δ (1.5)) as the "gauge current." Because of Eq. (2.8), we may decompose the gauge current (2.13) into two parts by inserting chiral projectors: In the right-hand side of this expression, the first term can be regarded as a collection of one-loop diagrams of the physical left-handed fermion containing at least one interaction vertex with A. Similarly, the second term can be regarded as a collection of similar one-loop diagrams but of the right-handed fluff fermion. Interestingly, the last term of Eq. (2.15) identically vanishes even with finite lattice spacings. This might be regarded as a (partial) decoupling of the fluff fermions from the physical gauge field A in the one-loop level; this is certainly a desired property. To see this, we first note that because of 2 = 2 = 1 the chiral overlap operator (2.1) can be written as Then, since δ (1.4) does not change , noting again that 2 = 2 = 1 (and thus δ = − δ ), we have the following sequence of equalities: As relations being dual to these, we also have

Functional curl
The structure of the "gauge current" (2.20) is quite analogous to the covariantly regularized gauge current of the left-handed Weyl fermion [16], which leads to the covariant gauge anomaly [17]. This definition of the gauge current preserves the gauge covariance even for anomalous cases at the expense of the Bose symmetry in fermion one-loop diagrams. The Bose symmetry is restored (in the continuum theory) if the gauge representation of the Weyl fermion is anomaly free. The breaking of the Bose symmetry can be characterized by the "functional curl"; this notion appears in various places in consideration of the anomaly-see for example, Ref. [18] and Sect. 6.6 of Ref. [19]. In our present problem, the analogue of the functional curl (associated with the right-handed fluff fermion) would be We expect that in the continuum limit this combination becomes local because if we neglect the subtlety associated with the definition of the gauge current in quantum theory (such as the covariant versus consistent), then the gauge current would always be given by the derivative of the effective action and then the combination such as Eq. (2.22) would vanish. We will shortly see that this expectation is correct. Note that Eqs. (2.20), (2.21), and (2.13) with δ → δ imply that

Functional curl (2.22) is a local functional
The following argument is almost identical to the one given in Ref. [20], which tries to interpret the lattice formulation of Ref. [21] in terms of the covariant gauge current. Instead of Eq. (2.22) itself, it is convenient to consider 1 Tr 2DχP+ where ≡ δ + δ stands for a general infinitesimal variation of the gauge fields A and A ; in the very final step, we will set 1 = δ and 2 = δ.
Introducing the notation Since no inverse of the Dirac operator is involved in the right-hand side, the functional curl is manifestly a local functional of A and A . We further rewrite Eq. (2.33) as follows: First, we note that (2.34) As we have seen in Eq. (2.17), We decompose this according to the number of γ 5 . Then the parity-odd part of the functional curl (2.22) is given by We see that the parity-odd part is anti-symmetric under the exchange A ↔ A , while the parity-even part is symmetric. We will now present the continuum limit of these expressions.

Continuum limit
The computational strategy for the continuum limit of Eqs. (2.38) and (2.39) is identical to that of Ref. [9]. We thus omit the details of the (very tedious) calculation and show only the results. In what follows, we use the notations and We also define the following lattice integrals. With the abbreviations (2.55) 10    Naturally, this parity-odd part is controlled by the gauge anomaly; it can be confirmed that this combination vanishes when the gauge representation of the fermion is anomaly free. For the parity-even part (2.39), we have Lorentz symmetry-violating terms as well as Lorentzpreserving terms. For the latter, we have (again omitting the symbol tr), L(A, A ; δA, δ A )| parity-even, Lorentz-preserving and δ ω ln Z[A, 0] parity-even, Lorentz-violating The parity-odd breaking term (A.2) is, as expected, the consistent gauge anomaly associated with a single left-handed Weyl fermion. This cannot be written as the gauge variation of a local term, and vanishes if the gauge representation is anomaly free.
Concerning the parity-even breaking terms (A.3) and (A.4), it must be possible to rewrite them as the gauge variation of local terms. In fact, we can see that (again omitting the symbol d 4 x tr) δ ω ln Z[A, 0] parity-even, Lorentz-preserving and δ ω ln Z[A, 0] parity-even, Lorentz-preserving Thus, these breakings can be removed by local counterterms.