We propose a regularization of four-dimensional chiral gauge theories using six-dimensional Dirac fermions. In our formulation, we consider two different mass terms having domain-wall profiles in the fifth and the sixth directions, respectively. A Weyl fermion appears as a localized mode at the junction of two different domain walls. One domain wall naturally exhibits the Stora–Zumino chain of the anomaly descent equations, starting from the axial anomaly in six dimensions to the gauge anomaly in four dimensions. Another domain wall implies a similar inflow of the global anomalies. The anomaly-free condition is equivalent to requiring that the axial anomaly and the parity anomaly are canceled among the six-dimensional Dirac fermions. Since our formulation is based on a massive vector-like fermion determinant, a nonperturbative regularization will be possible on a lattice. Putting the gauge field at the four-dimensional junction and extending it to the bulk using the Yang–Mills gradient flow, as recently proposed by Grabowska and Kaplan, we define the four-dimensional path integral of the target chiral gauge theory.

Japan

Hidenori Fukaya

Tetsuya Onogi

Shota Yamamoto

Ryo Yamamura

Six-dimensional regularization of chiral gauge theories

Recently, the Yang–Mills gradient flow has been found to be a useful concept not only in lattice simulations but also in continuous field theories. Since its smearing property is similar to the Wilsonian “block spin transformation”, there might be a deeper connection between them. In this work, we define the “effective action”, which generates configurations at a finite flow time, and derive the exact differential equation to investigate the flow-time dependence of the action. Then the Yang–Mills gradient flow can be regarded as the flow of the effective action. We also propose a flow-time-dependent gradient, where the differential equation becomes similar to the renormalization group equation. We discuss the possibility of regarding the time evolution of the effective action as the Wilsonian renormalization group flow.

The Yang—Mills gradient flow and lattice effective action

Extended QCD (XQCD), proposed by Kaplan [D. B. Kaplan, arXiv:1306.5818], is an interesting reformulation of QCD with additional bosonic auxiliary fields. While its partition function is kept exactly the same as that of original QCD, XQCD naturally contains properties of low-energy hadronic models. We analyze the renormalization group flow of 2D (X)QCD, which is solvable in the limit of a large number of colors , to understand what kind of roles the auxiliary degrees of freedom play and how the hadronic picture emerges in the low-energy region.