Non-perturbative Renormalisation of Domain Wall Fermions: Quark Bilinears

TL;DR

This work presents a first-principles study of non-perturbative renormalisation of the quark field and flavour non-singlet bilinears for domain wall fermions in the RI scheme on quenched lattices. The authors combine off-shell vertex functions, exact Ward-Takahashi identities, and hadronic matrix elements to determine $Z_q$, $Z_A$, $Z_V$, $Z_S$, $Z_P$, and $Z_T$, while carefully treating explicit and spontaneous chiral symmetry breaking, pole terms, and zero-mode effects. They demonstrate that explicit chiral symmetry breaking is negligible, extract RG-invariant renormalisation factors, and show that the scale dependence is small after accounting for lattice artifacts. When compared with perturbative predictions, the non-perturbative results align reasonably well after appropriate $M_5$-dependent corrections and mean-field improvements, validating domain wall fermions as a robust framework for non-perturbative operator renormalisation.

Abstract

We find the renormalisation coefficients of the quark field and the flavour non-singlet fermion bilinear operators for the domain wall fermion action, in the regularisation independent (RI) renormalisation scheme. Our results are from a quenched simulation, on a 16^3x32 lattice, with beta=6.0 and an extent in the fifth dimension of 16. We also discuss the expected effects of the residual chiral symmetry breaking inherent in a domain wall fermion simulation with a finite fifth dimension, and study the evidence for both explicit and spontaneous chiral symmetry breaking effects in our numerical results. We find that the relations between different renormalisation factors predicted by chiral symmetry are, to a good approximation, satisfied by our results and that systematic effects due to the (low energy) spontaneous chiral symmetry breaking and zero-modes can be controlled. Our results are compared against the perturbative predictions for both their absolute value and renormalisation scale dependence.

Figures (21)

• Figure 1: A plot of $\frac{1}{12}{\rm Tr}\left( S^{-1}_{latt} \right)$ versus $(ap)^2$ showing that for moderate values of $(ap)^2$ the effects of explicit chiral symmetry breaking are small.
• Figure 2: The value of $\frac{1}{12}{\rm Tr}\left( S^{-1} \right)$ extrapolated to $m_f=0$ vs $(ap)^2$. For moderate $(ap)^2$ the extrapolated value is zero within errors, showing that the residual mass is small.
• Figure 3: $Z_m Z_q$ calculated from the slope of $\frac{1}{12}{\rm Tr}\left( S^{-1} \right)$ versus $m_f$ plotted as a function of $(ap)^2$.
• Figure 4: A plot of $\Lambda_A - \Lambda_V$ versus $(ap)^2$, showing that there is no significant difference between $Z_A$ and $Z_V$, even for moderate values of $(ap)^2$.
• Figure 5: A graph of $\frac{1}{2}\left\{\Lambda_A + \Lambda_V\right\}$ versus $(ap)^2$, which up to lattice artifacts, gives $Z_A/Z_q$ and $Z_V/Z_q$.