Renormalization functions for Nf=2 and Nf=4 Twisted Mass fermions

TL;DR

This work advances the precision of renormalization factors in twisted mass lattice QCD by combining non-perturbative RI$'$ determinations with perturbative ${ m O}(g^2 a^ty)$ subtraction of lattice artifacts, followed by conversion to the ${ar{ m MS}}$ scheme at $bc=2$ GeV. Using momentum sources in the RI$'$ framework and an all-orders perturbative treatment, the authors obtain high-precision RFs for quark-field and fermion-bilinear operators, across $N_f=2$ and $N_f=4$ ensembles with Iwasaki gauge action. They perform chiral extrapolations, handle pion-pole contamination for the pseudoscalar channel, and provide final ${ar{ m MS}}$ RFs with quantified statistical and systematic uncertainties, demonstrating the importance of lattice-artifact subtraction for reliable continuum limits. The results enable accurate renormalization of hadron matrix elements in twisted mass QCD and improve consistency with other lattice determinations in the continuum limit.

Abstract

We present results on the renormalization functions of the quark field and fermion bilinears with up to one covariant derivative. For the fermion part of the action we employ the twisted mass formulation with $N_f{=}2$ and $N_f{=}4$ degenerate dynamical quarks, while in the gluon sector we use the Iwasaki improved action. The simulations for $N_f{=}4$ have been performed for pion masses in the range of 390MeV - 760MeV and at three values of the lattice spacing, $a$, corresponding to $β{=}1.90,\,1.95,\,2.10$. The $N_f{=}2$ action includes a clover term with $c_{\rm sw}{=}1.57551$ at $β{=}2.10$, and three ensembles at different values of $m_π$. The evaluation of the renormalization functions is carried out in the RI$'$ scheme using a momentum source. The non-perturbartive evaluation is complemented with a perturbative computation, which is carried out at one-loop level and to all orders in the lattice spacing, $a$. For each renormalization function computed non-perturbatively we subtract the corresponding lattice artifacts to all orders in $a$, so that a large part of the cut-off effects is eliminated. The renormalization functions are converted to the ${\overline{\rm MS}}$ scheme at a reference energy scale of $μ{=}2$ GeV after taking the chiral limit.

Figures (24)

• Figure 1: One-loop diagrams contributing to the fermion propagator. Wavy (solid) lines represent gluons (fermions).
• Figure 2: One-loop diagrams contributing to the computation of the fermion operators. A wavy (solid) line represents gluons (fermions). A cross denotes an insertion of the operator under study. In the case of the ultra-local operators, only the upper right diagram contributes due to the absense of gluons in the vertices.
• Figure 3: $D{\cal Z}_q(a,p)/g^2$ as a function of $(a\,p)^2$ with the Iwasaki gluon action and a $24^3\times48$ lattice for $c_{\rm sw}=0$ (left) and $c_{\rm sw}=1$ (right).
• Figure 4: $D{\cal Z}_S(a,p)/g^2$ as a function of $(a\,p)^2$. The notation is the same as for Fig. \ref{['Zq_pert']}.
• Figure 5: $D{\cal Z}_P(a,p)/g^2$ as a function of $(a\,p)^2$. The notation is the same as for Fig. \ref{['Zq_pert']}.