Domain-wall fermions with U(1) dynamical gauge fields in (4+1)-dimensions

TL;DR

This paper presents a numerical simulation of a (4+1)-dimensional domain-wall model with a quenched U(1) gauge field, revealing challenges in realizing lattice chiral gauge theories due to the absence of zero modes in the symmetric phase.

Contribution

It introduces a numerical approach to study domain-wall fermions with U(1) gauge fields in higher dimensions, highlighting difficulties in achieving chiral symmetry on the lattice.

Findings

Zero mode appears absent in the symmetric phase

Challenges in constructing lattice chiral gauge theories

Simulation results suggest limitations in continuum limit

Abstract

We carry out a numerical simulation of a domain-wall model in (4+1) dimensions, in the presence of a quenched U(1) dynamical gauge field only in an extra dimension, corresponding to the weak coupling limit of a (4-dimensional) physical gauge coupling. Our numerical data suggest that the zero mode seems absent in the symmetric phase, so that it is difficult to construct a lattice chiral gauge theory in the continuum limit.

Equations

$$G(p{)}_{s,t}=[\left(-i\sum _{\mu }{\gamma }_{\mu }{\bar{p}}_{\mu }+M(z)\right)G_R(p)P_L$$

$$+{\left(-i\sum _{\mu }{\gamma }_{\mu }{\bar{p}}_{\mu }+M^{†}(z)\right)G_L(p)P_R]}_{s,t},$$

$$S_G={\beta }_s\sum _{s,x,\hat{\mu }}\mathrm{Re}\mathrm{Tr}\left[U_D(x,s)U_D^{†}(x+\hat{\mu },s)\right],$$

Full text

1 Introduction

Construction of chiral gauge theory is one of the long standing problems of lattice field theories. Because of the fermion doubling problem, the lattice field theory discretized naively becomes non-chiral. Several approaches on lattice have been proposed to overcome this difficulties, but none of them have been proven to be successful.

Recently Kaplan has proposed a domain-wall model in order to construct lattice chiral gauge theories[1]. The model consists of Wilson fermion action in (2$n$+1) dimensions with a fermion mass term being the shape of a domain wall in the (extra) (2$n$+1)th dimension. In the case of free fermions it is shown for $0<m_{0}<1$, where $m_{0}$ denotes the domain wall mass height, that a massless chiral state arises as a zero mode bound to the $2n$-dimensional domain wall while all doublers have large masses of the lattice cutoff scale.

In a way of introducing dynamical gauge fields two variants of this model have been proposed: the waveguide model[2] and the overlap formula[3]. However, it has been reported that the chiral zero mode disappears for these two variants even in the weak coupling limit, due to the roughness of gauge field[2]. In the original model the roughness of the gauge field is replaced with the dynamical gauge field in the extra dimension, and in the weak coupling limit of the $2n$ dimensional coupling, $2n$ dimensional links $U_{\mu}(x,s)$ = 1 ($\mu=1,\cdots,2n$) while only extra dimensional links $U_{2n+1}(x,s)$ become dynamical. We would like to know whether the fermionic zero modes exist on the domain wall or not in this limit. This question has already been investigated in (2+1) dimensions[4], but the result, in particular, in the symmetric phase is not conclusive, due to peculiarity of the phase transition in the 2 dimensional U(1) spin model. Therefore we numerically investigate this model with U(1) gauge field in (4+1) dimensions and report the result here.

2 Mean-field analysis

Before numerical investigation we estimate the effect of the dynamical gauge field in the extra dimension using the mean-field analysis. In our mean-field analysis all link variables in the extra dimension are replaced with $(x,s)$-independent constant $z$ $(0<z<1)$, so that the fermion propagator is easily obtained[4] 111Note that $G_{L}$ $(G_{R})$ here was denoted $G_{R}$ $(G_{L})$ in Ref.[4]

[TABLE]

where $\bar{p}_{\mu}\equiv\sin(p_{\mu})$ and $P_{R/L}=(1\pm\gamma_{5})/2$. Corresponding fermion masses are obtained from $G_{R}$ and $G_{L}$ in ${\bar{p}}\rightarrow 0$ limit, and we found that fermionic zero modes exist only for $1-z<m_{0}<1$. Thus the critical value of domain wall mass is $m_{0}^{c}=1-z$ and therefore no zero mode survives if $z=0$.

3 Numerical analysis

We investigated the (4+1) dimensional U(1) model numerically, using a quenched approximation. At the zero physical gauge coupling the gauge field action of the model is reduced to the $2n$ dimensional spin model with many copies:

[TABLE]

where $D=2n+1$. Therefore there exists a phase transition which separates a broken phase from a symmetric phase. We calculated the order parameter $v$ using a rotational technique[4] and found $\beta_{s}=0.29$ corresponds to the symmetric phase and $\beta_{s}=0.5$ to the broken phase. We calculated the fermion propagator over 50 configurations at $\beta_{s}=0.5$ and 0.29 on $L^{3}\times 32\times 16$ lattices with $L=$4, 6, 8. For the fermion field we take periodic boundary conditions except the anti-periodic boundary condition in the 4th direction. The errors are estimated by the single elimination jack-knife method.

At $s=0$ we obtained the inverse propagator $G_{R}^{-1}$ and $G_{L}^{-1}$ for $p_{1},p_{2},p_{3}=0$ as a function of $p_{4}$. We obtain fermion mass squared, $m_{f}^{2}$, by extrapolating $G_{R}^{-1}$ and $G_{L}^{-1}$ linearly to $p_{4}=0$.

3.1 Broken phase

In Fig.1 we plotted $m_{f}$ in the broken phase ($\beta_{s}=0.5$) as a function of $m_{0}$.

As seen from this figure, the finite size effect is small, and the left-handed modes are always massive, while the right-handed modes are massless if $m_{0}$ is larger than about $0.6$. Therefore, we conclude that chiral zero modes can exist in the broken phase.

3.2 Symmetric phase

Let us show the fermion mass in the symmetric phase ($\beta_{s}=0.29$) in Fig.2. In the smallest lattice size the chiral zero modes seem to exist.

However, for large lattices, the mass difference between the left- and right-handed modes becomes smaller. This suggest that the fermion spectrum becomes vector-like in the infinite volume limit. However, since the fermion mass near $m_{0}=1.0$ is so small, from this data alone, we cannot exclude a possibility that the critical mass $m_{0}^{c}$ is very close to $1.0$.

To make a definite conclusion on the absence of chiral zero modes in the symmetric phase, we try to fit the fermion propagator using the form of the mean-field propagator with the fitting parameter $z$. We show the quality of the fit in Fig.3.

This figure shows that the fermion propagator is well described by the mean-field propagator.

In Fig.4 and Fig.5 we plotted the parameter $z$ obtained the above fit as a function of $1/L$

The parameter $z$’s are almost independent of $m_{0}$ at the each $1/L$ except for the right-handed ones at $m_{0}=0.99$. The solid circles represent the order parameter $v$. The behaviors of $z$ at different $m_{0}$ are almost identical each other and are very similar to that of $v$ except the right-handed ones at $m_{0}=0.99$. This suggest that $z$ can be identified with $v$ and therefore $z$ becomes zero as the lattice size goes to the infinity. If this is the case the fermion spectrum of this model becomes vector-like in the symmetric phase.

4 Conclusions

We have carried out the numerical simulations of the U(1) original domain-wall model in (4+1) dimensions in the weak coupling limit of the 4-dimensional coupling.

In the broken phase, there exist chiral zero modes on the domain wall for $m_{0}>m_{0}^{c}$. The existence of the critical mass $m_{0}^{c}$ is predicted by the mean-field analysis. On the other hand, in the symmetric phase, the analysis using the mean-field propagator suggests that this model becomes vector-like. We should note, however, that the right-handed modes at $m_{0}=0.99$ behaves differently and the similar behavior was also found in the (2+1) dimensional model[4]. Therefore in the future, we must investigate this point in detail, for example, increasing the statistics.

Besides this point the results from both phases suggest that this model becomes vector-like in the continuum limit, which should be taken at the critical point (: the point between two phases). Therefore, it seems also difficult to construct a chiral gauge theory on lattice via the original domain-wall model.