Renormalization constants of local operators for Wilson type improved fermions

TL;DR

This paper delivers a comprehensive determination of renormalization constants for the quark field and local bilinear currents in Wilson-type lattice QCD with twisted-mass fermions. It combines perturbative one-loop results (including ${\cal O}(a^2)$ terms) with non-perturbative RI$'$-MOM calculations, performing a careful subtraction of lattice artifacts and a pion-pole subtraction for $Z_P$, then converts to ${\overline{\rm MS}}$ at $\mu=2$ GeV. The study spans three lattice spacings (three $\beta$ values) and uses two degenerate quark flavors, providing a consistent, cross-checked set of $Z$-factors and their scale evolution, along with a Mathematica input file containing the perturbative expressions. The results offer precise renormalization inputs for phenomenology and help quantify discretization effects in ${\cal O}(a^2)$-improved Wilson-type actions, enabling more accurate lattice QCD predictions. Overall, the work advances non-perturbative renormalization accuracy in twisted-mmass Wilson lattice QCD and supplies practical data for converting lattice results to the widely used ${\overline{\rm MS}}$ scheme at 2 GeV.

Abstract

Perturbative and non-perturbative results are presented on the renormalization constants of the quark field and the vector, axial-vector, pseudoscalar, scalar and tensor currents. The perturbative computation, carried out at one-loop level and up to second order in the lattice spacing, is performed for a fermion action, which includes the clover term and the twisted mass parameter yielding results that are applicable for unimproved Wilson fermions, as well as for improved clover and twisted mass fermions. We consider ten variants of the Symanzik improved gauge action corresponding to ten different values of the plaquette coefficients. Non-perturbative results are obtained using the twisted mass Wilson fermion formulation employing two degenerate dynamical quarks and the tree-level Symanzik improved gluon action. The simulations are performed for pion masses in the range of 480 MeV to 260 MeV and at three values of the lattice spacing, a, corresponding to beta=3.9, 4.05, 4.20. For each renormalization factor computed non-perturbatively we subtract its perturbative O(a^2) terms so that we eliminate part of the cut-off artifacts. The renormalization constants are converted to MS-bar at a scale of mu=2 GeV. The perturbative results depend on a large number of parameters and are made easily accessible to the reader by including them in the distribution package of this paper, as a Mathematica input file.

Figures (11)

• Figure 1: One-loop diagrams contributing to the fermion propagator. Wavy (solid) lines represent gluons (fermions).
• Figure 2: One-loop diagram contributing to the bilinear operators. A wavy (solid) line represents gluons (fermions). A cross denotes the Dirac matrix $\Gamma$.
• Figure 3: $Z_{\rm q},\,Z_{\rm V},\,Z_{\rm A},\,Z_{\rm T}$ at $\beta=3.9$, as a function of the pion mass. Computations were performed at pion masses of $m_\pi=0.302$ GeV ($a\mu_0=0.004$), $m_\pi=0.375$ GeV ($a\mu_0=0.0064$), $m_\pi=0.429$ GeV ($a\mu_0=0.0085$) and $m_\pi=0.468$ GeV ($a\mu_0=0.01$).
• Figure 4: $Z_{\rm P}$ at $\beta=3.9$ (left panel) and $\beta=4.05$ (right panel) for various masses. The upper plot shows the results before the pion pole subtraction as described by Eq. (\ref{['PionPole']}), while the lower figure the results upon the appropriate subtraction given in Eq. (\ref{['PionPoleSub']}).
• Figure 5: $Z_{\rm P}$ for $\beta=4.20$ for the two pion masses. Results are shown upon the pion pole subtraction as described in Eq. (\ref{['PionPoleSub']}).