[Paper Review] Mass and Axial current renormalization in the Schrödinger functional scheme for the RG-improved gauge and the stout smeared $O(a)$-improved Wilson quark actions

Ken-Ichi Ishikawa, N. Ishizuka+7 more|arXiv (Cornell University)|Nov 27, 2015

Particle physics theoretical and experimental studies3 references5 citations

TL;DR

This paper presents nonperturbative renormalization factors for the axial current ($Z_A = 0.9650(68)(95)$), vector current ($Z_V = 0.95153(76)(1487)$), and quark mass ($Z_m^{\overline{\text{MS}}}(\mu=2\,\text{GeV}) = 0.9950(111)(89)$) in the Schrödinger functional scheme using the RG-improved Iwasaki gauge action and stout-smeared $O(a)$-improved Wilson quarks at $\beta=1.82$. The results enable precise extraction of physical quark masses and decay constants from large-scale $96^4$ lattice simulations.

ABSTRACT

We present the quark mass and axial current renormalization factors for the RG-improved Iwasaki gauge action and three flavors of the stout smeared $O(a)$-improved Wilson quark action. We employ $α=0.1$ and $n_{\mathrm{step}}=6$ for the stout link smearing parameters and all links in the quark action are replaced with the smeared links. Using the Schrödinger functional scheme we evaluate the renormalization factors at $β=1.82$ where large scale simulations are being carried out.

Motivation & Objective

• To determine the quark mass and axial/vector current renormalization factors for the RG-improved Iwasaki gauge action and three-flavor stout-smeared $O(a)$-improved Wilson quarks.
• To enable precise extraction of physical observables such as quark masses and decay constants from large-scale $96^4$ lattice simulations at $\beta=1.82$.
• To provide nonperturbative $Z$-factors in the Schrödinger functional scheme using the SF method with finite-volume and Dirichlet boundary conditions.
• To evaluate the renormalization constants at the physical scale $a^{-1} \sim 2.3$ GeV, relevant for simulations in the HPCI SPIRE Field 5 program.
• To ensure consistency with $O(a)$-improvement by using nonperturbatively determined $c_{\text{SW}}=1.11$ and $O(a)$-mixing corrections assumed negligible.

Proposed method

• The Schrödinger functional scheme is employed with finite temporal and spatial lattice sizes $T = L = 4a$, using Dirichlet boundary conditions in the temporal direction for gauge and quark fields.
• Correlation functions involving boundary operators $O^a$ and $O'^a$ are computed to extract $Z_A$, $Z_V$, and $Z_P$ via the standard SF method using the PCAC relation and normalization conditions.
• The massless limit is taken using the averaged PCAC quark mass $am^{\text{PCAC}}$ for $Z_A$ and $Z_V$, and non-averaged for $Z_P$, to ensure consistency in the renormalization procedure.
• The renormalization constants are extrapolated to the chiral limit using two independent gauge configurations with different valence quark masses, and systematic errors are estimated from the discrepancy between runs.
• The $Z_m^{\overline{\text{MS}}}$ factor is computed using the step-scaling function $\sigma_P(u)$ and the coupling evolution $\sigma(u)$, with perturbative estimates for $\beta(g)$ and $\tau(g)$ in the SF and $\overline{\text{MS}}$ schemes.
• The physical scale is set using a preliminary value $a^{-1} = 2.332(18)$ GeV from Ref. [14], and the final $Z_m^{\overline{\text{MS}}}$ is obtained by combining $Z_A$, $Z_P$, and $u_0$ from independent runs.

Experimental results

Research questions

• RQ1 What are the nonperturbative renormalization factors for the axial current, vector current, and quark mass in the Schrödinger functional scheme for the RG-improved Iwasaki gauge action and stout-smeared $O(a)$-improved Wilson quarks?
• RQ2 How do the $Z_A$, $Z_V$, and $Z_m^{\overline{\text{MS}}}$ factors at $\beta=1.82$ enable precise determination of physical quark masses and decay constants from $96^4$ lattice ensembles?
• RQ3 What is the impact of $O(a)$-mixing on the axial current renormalization, and is it negligible in this setup?
• RQ4 How are the $Z$-factors extracted from correlation functions involving boundary operators and the PCAC relation in the SF scheme?
• RQ5 What is the consistency of the $Z_m^{\overline{\text{MS}}}$ factor when computed via the step-scaling method using $\sigma_P(u)$ and $\sigma(u)$, and how does it compare across independent configurations?

Key findings

• The axial current renormalization factor is determined as $Z_A = 0.9650(68)(95)$ at $\beta=1.82$, with a statistical error of 0.0068 and a systematic error of 0.0095.
• The vector current renormalization factor is $Z_V = 0.95153(76)(1487)$, with a statistical error of 0.00076 and a systematic error of 0.001487, reflecting higher precision in the vector channel.
• The quark mass renormalization constant in the $\overline{\text{MS}}$ scheme at $\mu=2$ GeV is $Z_m^{\overline{\text{MS}}}(\mu=2\,\text{GeV}) = 0.9950(111)(89)$, with a statistical error of 0.0111 and a systematic error of 0.0089.
• The results are obtained using two independent gauge configurations with $\kappa=0.126110$ and $\kappa=0.125120$, and the systematic error is estimated from the discrepancy between the two runs.
• The physical scale is set using $a^{-1} = 2.332(18)$ GeV, and the $Z_m^{\overline{\text{MS}}}$ factor is computed via the step-scaling method using $\sigma_P(u)$ and $\sigma(u)$, with perturbative estimates for $\beta(g)$ and $\tau(g)$.
• The study confirms that $O(a)$-mixing corrections to the axial current are negligible, justifying the use of unimproved current operators in the determination of $Z_A$.