Non-perturbative renormalization of quark bilinear operators with Nf=2 (tmQCD) Wilson fermions and the tree-level improved gauge action

TL;DR

This study performs a non-perturbative determination of renormalization constants for bilinear quark operators in Nf=2 twisted mass QCD with a tree-level Symanzik improved gauge action, exploiting maximal twist for automatic O(a) improvement. A novel strategy combines tm and OS valence regularizations to extract scale-independent RCs ZV, ZA and ZP/ZS, with ZV obtained from a Ward identity and ZA/ZPZS from mixed regularizations; RI-MOM is used to obtain Zq, ZS, ZP, ZT, along with careful chiral extrapolations and subtraction of O(g^2 a^2) discretization effects. The paper provides a detailed treatment of Goldstone-pole contamination, momentum selection, finite-size checks, and two complementary analysis paths (M1 and M2), and compares non-perturbative RCs to boosted perturbation theory, finding sizable deviations for ZS and ZP that are crucial for quark-mass determinations. Across three lattice spacings, the results demonstrate good control of discretization and finite-size effects, with RCs at MSbar(2 GeV) furnished for practical phenomenology. The findings offer robust benchmarks for ETMC simulations and guide continuum extrapolation and scheme conversions in lattice QCD with twisted-mass fermions.

Abstract

We present results for the renormalization constants of bilinear quark operators obtained by using the tree-level Symanzik improved gauge action and the Nf=2 twisted mass fermion action at maximal twist, which guarantees automatic O(a)-improvement. Our results are also relevant for the corresponding standard (un-twisted) Wilson fermionic action since the two actions only differ, in the massless limit, by a chiral rotation of the quark fields. The scale-independent renormalization constants ZV, ZA and the ratio ZP/ZS have been computed using the RI-MOM approach, as well as other alternative methods. For ZA and ZP/ZS, the latter are based on both standard twisted mass and Osterwalder-Seiler fermions, while for ZV a Ward Identity has been used. The quark field renormalization constant Zq and the scale dependent renormalization constants ZS, ZP and ZT are determined in the RI-MOM scheme. Leading discretization effects of O(g^2 a^2), evaluated in one-loop perturbation theory, are explicitly subtracted from the RI-MOM estimates.

Figures (9)

• Figure 1: The plateaux quality for $Z_V$ (a), $Z_A$ (b) and $Z_P/Z_S$ (c) for three gauge couplings and masses $a\mu_{sea}^{min}= a\mu_{1}= a\mu_{2}$. From Table \ref{['simuldetails']} we have $a\mu_{sea}^{min}(\beta=3.80) =0.0080$, $a\mu_{sea}^{min}(\beta=3.90)=0.0040$ and $a\mu_{sea}^{min} (\beta=4.05)= 0.0030$. Note that in Fig.(b) the data have been shifted for clarity.
• Figure 2: Left panel: valence chiral limit for the three scale independent RCs at $\beta=3.80$, $a\mu_{sea}=0.0080$ (a), $\beta=3.90$, $a\mu_{sea}=0.0040$ (c) and $\beta=4.05$, $a\mu_{sea}=0.0030$ (e). Right panel: sea chiral limit for the same constants at $\beta=3.80$ (b), $\beta=3.90$ (d) and $\beta=4.05$ (f). Note that the data for $Z_P/Z_S$ have been shifted for clarity.
• Figure 3: The form factors $\Sigma_1$ (top), $\Sigma_2$ (center) and $\Sigma_3$ (bottom), computed at $\beta=3.90$ with $a \mu_{sea}=0.0040$, as a function of $a^2 \tilde{p}^2$.
• Figure 4: The amputated Green functions ${\cal V}_P(p^2,\mu_1,\mu_2)$ and ${\cal V}_S(p^2,\mu_1,\mu_2)$, evaluated at a scale $a^2 \tilde{p}^2\simeq 1$, as a function of the pseudoscalar meson mass squared. The subtracted Green functions are defined in Eq. (\ref{['eq:gpssub']}). Dashed and solid lines illustrate the results of the fit to Eq. (\ref{['eq:gpsfit']}) (with $C=0$ for ${\cal V}_{P(S)}^{\rm sub}$).
• Figure 5: Dependence of $Z_V$, $Z_A$ and $Z_T$ on the valence quark mass, at $a^2 \tilde{p}^2\simeq 1$. The solid lines are linear extrapolations to the (valence) chiral limit.