[Paper Review] Non-perturbative renormalization in lattice QCD

Yasumichi Aoki|arXiv (Cornell University)|May 13, 2010

Quantum Chromodynamics and Particle Interactions66 references4 citations

TL;DR

This paper reviews non-perturbative renormalization in lattice QCD, focusing on the RI/MOM and RI/SMOM schemes to address the 'window problem' in momentum-space renormalization. It presents recent advances in handling systematic errors, compares results for the strong coupling constant and strange quark mass across methods, and confirms consistency in the continuum limit despite differing actions and schemes.

ABSTRACT

Recent developments in non-perturbative renormalization for lattice QCD are reviewed with a particular emphasis on RI/MOM scheme and its variants, RI/SMOM schemes. Summary of recent developments in Schroedinger functional scheme, as well as the summary of related topics are presented. Comparison of strong coupling constant and the strange quark mass from various methods are made.

Motivation & Objective

• To address the 'window problem' in the RI/MOM scheme, where the renormalization scale μ must satisfy ΛQCD ≪ μ ≪ 1/a, leading to systematic errors in typical lattice simulations.
• To improve the reliability of non-perturbative renormalization by introducing and applying the RI/SMOM scheme, which reduces non-perturbative contaminations.
• To compare results for the strong coupling constant αs and strange quark mass m_s across different lattice actions and renormalization schemes, ensuring consistency in the continuum limit.
• To summarize recent developments in the Schrödinger functional scheme and its application to light and heavy quark systems, including HQET extensions.
• To support precision lattice QCD calculations by reviewing lattice perturbation theory results relevant to renormalization, including two-loop and O(α²a²) corrections.

Proposed method

• Uses the RI/MOM scheme to impose renormalization conditions on the amputated vertex function ΠΓ(p₁,p₂) with off-shell quark states at momentum p, enforcing ZΓ/Zq · Tr[ΠΓ · PΓ] = 1.
• Applies the RI/SMOM scheme as a variant of RI/MOM that reduces non-perturbative effects by modifying the momentum-space projection and momentum configuration.
• Employs the Schrödinger functional scheme with step scaling to compute renormalization constants and match to continuum schemes like ¯MS.
• Combines non-perturbative renormalization with continuum perturbation theory (cPT) for matching to ¯MS, avoiding reliance on lattice perturbation theory (LPT) with poor convergence.
• Utilizes lattice perturbation theory (LPT) and numerical stochastic perturbation theory (NSPS) to compute higher-order corrections, such as two-loop results for Zq and improvement coefficients.
• Applies the window-optimized RI/MOM and RI/SMOM schemes to reduce systematic errors in quark mass and αs determinations at physical lattice spacings.

Experimental results

Research questions

• RQ1 How can the 'window problem' in the RI/MOM scheme be mitigated to reduce systematic errors in lattice QCD simulations?
• RQ2 To what extent does the RI/SMOM scheme improve the reliability of non-perturbative renormalization compared to RI/MOM by suppressing non-perturbative effects?
• RQ3 How consistent are determinations of the strong coupling constant αs and strange quark mass m_s across different lattice actions and renormalization schemes?
• RQ4 What is the role of the Schrödinger functional scheme in non-perturbative renormalization, especially for light and heavy quarks?
• RQ5 How do higher-order lattice perturbation theory results (e.g., two-loop, O(α²a²)) support and improve non-perturbative renormalization procedures?

Key findings

• The RI/SMOM scheme effectively reduces non-perturbative contaminations in the renormalization condition by modifying the momentum-space setup, offering a more robust alternative to RI/MOM.
• Despite differences in lattice actions and renormalization schemes, determinations of the strong coupling constant αs and strange quark mass m_s in the continuum limit show good consistency across methods.
• The Schrödinger functional scheme enables precise non-perturbative renormalization for light quarks and heavy quarks via HQET, with applications to b-quark systems and step-scaling procedures.
• Two-loop results in lattice perturbation theory for Zq and improvement coefficients (e.g., cSW) are available and used to correct for O(α²a²) effects, improving accuracy.
• Numerical stochastic perturbation theory (NSPS) has been developed to compute three-loop corrections to the ghost propagator and quark self-energy, supporting future precision calculations.
• The combination of non-perturbative renormalization with continuum perturbation theory for matching to ¯MS ensures that physical matrix elements remain scheme- and scale-independent.