Lorentz symmetry violation in the fermion number anomaly with the chiral overlap operator

Recently, Grabowska and Kaplan proposed a four-dimensional lattice formulation of chiral gauge theories on the basis of a chiral overlap operator. We compute the classical continuum limit of the fermion number anomaly in this formulation. Unexpectedly, we find that the continuum limit contains a term which is not Lorentz invariant. The term is, however, proportional to the gauge anomaly coefficient, and thus the fermion number anomaly in this lattice formulation automatically restores the Lorentz-invariant form when and only when the anomaly cancellation condition is met.


Introduction
It is important to give a non-perturbative definition of chiral gauge theories. Recently, Grabowska and Kaplan constructed a five-dimensional domain-wall lattice formulation of chiral gauge theories [1]. 1 More recently, they proposed a four-dimensional lattice formulation on the basis of the so-called chiral overlap operator which is derived from the above domainwall formulation [3,4]. Their four-dimensional formulation contains left-and right-handed fermions and, in the tree-level approximation, the left-handed component couples only to the original gauge field and the right-handed component couples only to a gauge field evolved by the gradient flow [5][6][7][8] for infinite time. The right-handed Weyl fermion is called the fluffy mirror fermion or fluff. Okumura and Suzuki [9] argued that the fermion number anomaly in this formulation possibly has phenomenological implications for the strong CP problem, baryogenesis, and the dark matter problem. They also conjectured the form of the classical continuum limit of the fermion number anomaly, but the explicit calculation was not carried out in Ref. [9].
In the present paper, we complete the calculation of the classical continuum limit of the fermion number anomaly in the formulation of Refs. [3,4]; the correct expression turns out to be more complicated than the simple expression conjectured in Ref. [9]. Rather unexpectedly, we find that the anomaly contains a term which is not Lorentz invariant. The term is proportional to the gauge anomaly coefficient and thus the fermion number anomaly in this lattice formulation automatically restores the Lorentz-invariant form when and only when the anomaly cancellation condition is met. The physical meaning of this finding is not immediately obvious; however, remembering that the fermion number anomaly is a very basic property of chiral gauge theories and any sensible formulation of chiral gauge theories must fail when the anomaly cancellation condition is not met, our finding appears interesting and quite suggestive.

Basic formulation
In the formulation of Ref. [3,4], there are two gauge fields, A and A ⋆ . A couples to the physical left-handed fermion while A ⋆ is given from A by the gradient flow for infinite flow time and couples to the would-be invisible right-handed fermion, the fluffy mirror fermion.
This formulation manifestly preserves the gauge invariance. If we regard the gauge fields as non-dynamical external fields, the partition function is given by where a is the lattice spacing and D χ denotes the chiral overlap operator, Here we have used the sign functions of the Hermitian Wilson Dirac operator where ∇ µ is the forward gauge-covariant lattice derivative and ∇ * µ is the backward one, In Eqs. (2.5) and (2.7), the link variable is given by where P denotes the path-ordered product andμ is the unit vector in the direction of µ; in Eqs. (2.6) and (2.8), The sign functions satisfy and, as a consequence, the Ginsparg-Wilson relation [10] holds. It is then natural to introduce a modified γ 5 [9,11,12] Note thatγ 5 is not Hermitian in this formulation. Using modified chiral projection operatorŝ the chiral components of the fermion can be defined aŝ Owing to the second relation in Eq. (2.13), the action is decomposed into left-and righthanded components as The classical continuum limit of the fermion number anomaly The fermion number anomaly on the lattice associated with the left-handed fermion in Eq. (2.15) is given by [9] A (a) where tr stands for the trace over the spinor and gauge indices and we have used tr γ 5 = 0 to obtain the last expression. 2 In what follows, we compute the classical continuum limit, L , for a smooth gauge field configuration. Let us first determine a general form of A L , by assuming that it is Lorentz invariant. The following argument is helpful to simplify the explicit tedious calculation of A L . First, using Eq. (2.10), we decompose A Then it is obvious that, under the exchange of A and A ⋆ , As the second property, we note when A ⋆ = A, Finally, the integral of A for 0 < ma < 2. Now, for convenience, we introduce which transforms as the adjoint representation under the gauge transformation on A and A ⋆ . 3 We note that A L is a dimension 4 gauge-invariant local polynomial of A and A ⋆ . Then, by examining the above properties, we find that the most general form of A L is given by The coefficients d 1 , d 2 , and b cannot be determined from the above argument alone. In the first line of Eq. (3.8),Ā L (3.9) arises from the parity-odd part and the other terms from the parity-even part. The second term ∂ µ tr[C µ C ν C ν ] is proportional to the gauge anomaly coefficient and thus it vanishes for anomaly-free cases. The following explicit calculation shows that d 2 = 0 in the third term. As we will show below, there actually exists a Lorentz symmetry violating term in A L .
Note that, generally speaking, the restoration of the Lorentz symmetry is not automatic with the lattice regularization. Let us describe how the explicit calculation of A L proceeds. 4 We first note that In this expression, we use = e ipx/a γ 5 where s µ ≡ sin p µ , c µ ≡ cos p µ , (3.13) (3.14) Q ⋆µ and R ⋆ are defined similarly. Thus, A L can be written in terms of operators Q µ , R, Q ⋆µ , and R ⋆ . Next, we expand A Carrying out the explicit calculation, we find that only four covariant derivatives with the same spacetime indices appear in the Lorentz symmetry violating terms. Therefore, taking into account the above properties of the fermion number anomaly, the Lorentz symmetry violating part in the general form (3.8) must be (Lorentz symmetry violating part with a to-be-determined coefficient d ′ 1 . Then, the explicit tedious expansion of A L can be exactly combined into the form of Eqs. (3.8) and (3.17). After this calculation, we finally obtain anomaly consists of two parts: One is the parity-odd part being proportional to the epsilon tensor,Ā L . The other is the parity-even part, which is proportional to the gauge anomaly coefficient. The latter contains a Lorentz symmetry violating term. In anomaly-free cases, only the formerĀ L contributes, which is Lorentz invariant. It appears quite interesting and suggestive that the Lorentz symmetry and the gauge anomaly are linked in this way in this lattice formulation on the basis of the fluffy mirror fermion.