Yoshio Kikukawa

Weak coupling expansion of massless QCD with a Ginsparg-Wilson fermion and axial U(1) anomaly

Abstract We discuss the weak coupling expansion of massless QCD with the Dirac operator which is derived by Neuberger based on the overlap formalism and satisfies the Ginsparg-Wilson relation. The axial U(1) anomaly associated to the chiral transformation proposed by Luscher is calculated as an application and is shown to have the correct form of the topological charge density for perturbative backgrounds. The coefficient of the anomaly is evaluated as a winding number related to a certain five-dimensional fermion propagator.

Hybrid Monte Carlo on Lefschetz thimbles — A study of the residual sign problem

A bstractWe consider a hybrid Monte Carlo algorithm which is applicable to lattice theories defined on Lefschetz thimbles. In the algorithm, any point (field configuration) on a thimble is parametrized uniquely by the flow-direction and the flow-time defined at a certain asymptotic region close to the critical point, and it is generated by solving the gradient flow equation downward. The associated complete set of tangent vectors is also generated in the same manner. Molecular dynamics is then formulated as a constrained dynamical system, where the equations of motion with Lagrange multipliers are solved by the second-order constraint-preserving symmetric integrator. The algorithm is tested in the λϕ4 model at finite density, by choosing the thimbles associated with the classical vacua for subcritical and supercritical values of chemical potential. For the lattice size L = 4, we find that the residual sign factors average to not less than 0.99 and are safely included by reweighting and that the results of the number density are consistent with those obtained by the complex Langevin simulations.

Nicolai mapping versus exact chiral symmetry on the lattice

The two-dimensional

Overlap in odd dimensions

N=2

Low energy effective action of domain-wall fermion and the Ginsparg-Wilson relation

Wess-Zumino model is constructed on the lattice through Nicolai mapping with a Ginsparg-Wilson fermion. The Nicolai mapping requires a certain would-be surface term in the bosonic action which ensures the vacuum energy cancellation even on the lattice, but inevitably breaks chiral symmetry. With the Ginsparg-Wilson fermion, the holomorphic structure of the would-be surface term is maintained, leaving a discrete subgroup of the exact chiral symmetry intact for a monomial scalar potential. Through this feature both the boson and fermion can be kept massless on the lattice without any fine-tuning.

Axial vector current of exact chiral symmetry on the lattice

Abstract In odd dimensions the lattice overlap formalism is simpler than in even dimensions. Masslessness of fermions can still be preserved without fine tuning and gauge invariance without gauge averaging can be maintained, although, sometimes, only at the expense of parity invariance. When parity invariance is enforced invariance under small gauge transformations can be maintained and continuum global gauge anomalies are reproduced.

Gauge anomaly cancellations in SU(2)L× U(1)Y electroweak theory on the lattice

Abstract We derive the effective action of the light fermion field of the domain-wall fermion, which is referred as q(x) and q (x) by Furman and Shamir. The inverse of the effective Dirac operator turns out to be identical to the inverse of the truncated overlap Dirac operator except a local contact term, which would give the chiral symmetry breaking in the Ginsparg-Wilson relation. We argue that there are direct relations between the low energy observables of the domain-wall QCD and observables of the Ginsparg-Wilson fermion described by the (truncated) overlap Dirac operator.

Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density

Abstract We consider the exact chiral symmetry and its spontaneous breakdown in lattice QCD with the Dirac operators satisfying the Ginsparg-Wilson relation. The axial vector current, which turns out to be related to the vector current simply by the insertion of the operator γ 5 (1 − aD ), is explicitly constructed in the cases of the Neuberger-Dirac operator and its rational approximation. We also discuss a Euclidean proof of the Nambu-Goldstone theorem using the axial Ward-Takahashi identity of this symmetry.

Neutron electric dipole moment with external electric field method in lattice QCD

Abstract We consider the cohomological classification of the (4+2) -dimensional topological field which was proposed by Luscher, for the SU(2) L × U(1) Y electroweak theory. The dependence on the admissible abelian gauge field of U(1) Y is determined through topological argument, with the SU(2) L gauge field fixed as background. We then show the exact cancellation of the local gauge anomaly of the mixed type SU(2) L 2 × U(1) Y at a finite lattice spacing, as well as U(1) Y 3 , using the pseudo reality of SU(2) L and the anomaly cancellation conditions in the electroweak theory given in terms of the hyper-charges of U(1) Y .

Domain wall fermion and chiral gauge theories on the lattice with exact gauge invariance

A bstractWe investigate Lefschetz thimble structure of the complexified path-integration in the one-dimensional lattice massive Thirring model with finite chemical potential. The lattice model is formulated with staggered fermions and a compact auxiliary vector boson (a link field), and the whole set of the critical points (the complex saddle points) are sorted out, where each critical point turns out to be in a one-to-one correspondence with a singular point of the effective action (or a zero point of the fermion determinant). For a subset of critical point solutions in the uniform-field subspace, we examine the upward and downward cycles and the Stokes phenomenon with varying the chemical potential, and we identify the intersection numbers to determine the thimbles contributing to the path-integration of the partition function. We show that the original integration path becomes equivalent to a single Lefschetz thimble at small and large chemical potentials, while in the crossover region multiple thimbles must contribute to the path integration. Finally, reducing the model to a uniform field space, we study the relative importance of multi-thimble contributions and their behavior toward continuum and low-temperature limits quantitatively, and see how the rapid crossover behavior is recovered by adding the multi-thimble contributions at low temperatures. Those findings will be useful for performing Monte-Carlo simulations on the Lefschetz thimbles.