Renormalization constants for 2-twist operators in twisted mass QCD

TL;DR

This work provides a comprehensive non-perturbative determination of renormalization constants for fermion fields and one-derivative twist-2 operators in twisted mass QCD with two degenerate quarks. Using RI$'$-MOM conditions, the authors compute $Z_q$, $Z_{ m DV}$, $Z_{ m DA}$, and $Z_{ m DT}$ non-perturbatively, systematically subtract ${ m O}(a^2)$ lattice artifacts with perturbative input, and extrapolate to $a^2p^2=0$ to control discretization effects. The results are evolved to a reference scale and converted to the MSbar scheme at 2 GeV, yielding precise renormalization factors (e.g., $Z_{ m DV1}$, $Z_{ m DV2}$, $Z_{ m DA1}$, $Z_{ m DA2}$) across several lattice spacings, which are essential for comparing lattice-derived twist-2 matrix elements to experiment. The methodology provides a robust framework for connecting lattice QCD results on hadron structure to phenomenology, with quantified systematic uncertainties from momentum range selection and finite-volume effects.

Abstract

Perturbative and non-perturbative results on the renormalization constants of the fermion field and the twist-2 fermion bilinears are presented with emphasis on the non-perturbative evaluation of the one-derivative twist-2 vector and axial vector operators. 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 have been performed for pion masses in the range of about 450-260 MeV and at three values of the lattice spacing $a$ corresponding to $β=3.9, 4.05, 4.20$. Subtraction of ${\cal O}(a^2)$ terms is carried out by performing the perturbative evaluation of these operators at 1-loop and up to ${\cal O}(a^2)$. The renormalization conditions are defined in the RI$'$-MOM scheme, for both perturbative and non-perturbative results. The renormalization factors, obtained for different values of the renormalization scale, are evolved perturbatively to a reference scale set by the inverse of the lattice spacing. In addition, they are translated to ${\bar{\rm MS}}$ at 2 GeV using 3-loop perturbative results for the conversion factors.

Figures (15)

• Figure 1: One-loop diagrams contributing to the fermion propagator. Wavy lines represent gluons and solid lines fermions.
• Figure 2: One-loop diagrams contributing to the computation of the twist-2 operators. A wavy (solid) line represents gluons (fermions). A cross denotes an insertion of the operator under study.
• Figure 3: $Z^{0\nu}_{\rm DA}$ for $\beta=3.9$ ($a^{-1}$=2.217 GeV) and $m_\pi=0.430$ GeV for method 1 (open symbols) and method 2 (filled symbols). The upper plot corresponds to non-perturbative results, where the index A, B represents the set of momenta with spatial components $2\,\pi/L\,(3,3,3)$ and $2\,\pi/L\,(2,2,2)$, respectively. The lower plot shows the non-perturbative results after subtracting the perturbative ${\cal O}(g^2\,a^2)$-terms, where the two methods give almost identical results. Moreover, in method 1, the jump between the two sets of momenta disappears.
• Figure 4: In the left panel we show $Z^{0\nu}_{\rm DV}$ using the same notation as in Fig. \ref{['fig3']}. In the right panel, upper graph we show $Z^{\mu\nu}_{\rm DV}$ again using the notation of Fig.\ref{['fig3']} whereas in the lower graph we show a comparison between method 1 after subtracting the perturbative ${\cal O}(a^2)$-terms (diamonds) and method 2 without any subtractions (filled squares).
• Figure 5: $Z_{\rm DV1}$ at $\beta=3.9$, as a function of the pion mass: $m_\pi=0.302$ GeV ($a\mu_0=0.004$), $m_\pi=0.375$ GeV ($a\mu_0=0.0064$) and $m_\pi=0.429$ GeV ($a\mu_0=0.0085$). The left plot regards the unsubtracted non-perturbative results and the right one corresponds to the subtracted data.