[Paper Review] High loop renormalization constants by NSPT: a status report

Francesco Di Renzo, L. Scorzato+1 more|ArXiv.org|Oct 2, 2007

Solar and Space Plasma Dynamics5 references4 citations

TL;DR

This paper presents a novel strategy to correct finite-size effects in high-loop Numerical Stochastic Perturbation Theory (NSPT) computations of renormalization constants in Lattice QCD, particularly for logarithmically divergent operators like the scalar and pseudoscalar currents. By introducing a 'tamed-log' subtraction scheme that isolates and removes infrared-sensitive logarithmic contributions tied to finite lattice size, the method enables precise one-loop determinations of $ Z_P $, $ Z_S $, and $ Z_m $, with successful benchmarking against analytic results for Wilson fermions and tree-level Symanzik improved gauge actions.

ABSTRACT

We present an update on Numerical Stochastic Perturbation Theory projects for Lattice QCD, which are by now run on apeNEXT. As a first issue, we discuss a strategy to tackle finite size effects which can be quite sizeable in the computation of logarithmically divergent renormalization constants. Our first high loop determination of quark bilinears for Wilson fermions was limited to finite constants and finite ratios. A precise determination of Z_P and Z_S (and hence of Z_m) now becomes possible. We also give an account of computations for actions different from the standard regularization we have taken into account so far (Wilson gauge action and Wilson fermions). In particular, we present the status of computations for the Lattice QCD regularization defined by tree level Symanzik improved gauge action and Wilson fermions, which became quite popular in recent times. We also take the chance to discuss the related topic of the computation of the gluon and ghost propagators (which we undertook in collaboration with another group). This is relevant in order to better understand non-perturbative computations of propagators aiming at qualitative/quantitative understanding of confinement.

Motivation & Objective

• To address significant finite-size effects in NSPT computations of renormalization constants for operators with anomalous dimensions, particularly those exhibiting logarithmic divergences.
• To develop a robust, scalable method to isolate and subtract infrared-sensitive logarithmic contributions arising from finite lattice volumes in high-loop perturbation theory.
• To extend NSPT computations beyond standard Wilson gauge and fermion actions to include tree-level Symanzik-improved gauge actions and other regularizations with minimal computational overhead.
• To provide first high-loop NSPT results for gluon and ghost propagators, enabling non-perturbative studies of confinement via Schwinger-Dyson equations.

Proposed method

• Introduce a tamed-log subtraction scheme by splitting the momentum sum into a zero-momentum part $ I(0,a,L) $ and a difference part $ J(p,a,L) $, where $ J(p,a,L) $ is UV-finite and captures the finite-size logarithmic dependence.
• Define the tamed-log contribution as $ ilde{G}(pL) $, derived from the formal continuum limit of $ J(p,a,L) $, which approximates the expected logarithmic behavior but includes $ pL $-dependent corrections for finite $ L $.
• Subtract the tamed-log from the measured one-loop $ O_{ ext{scalar}}^{(1)}(\hat{p}^2) $ to extract the finite, physical renormalization constant $ z_q^{(1)} - z_s^{(1)} $, removing finite-volume artifacts.
• Use Hypercubic-symmetric Taylor expansions to extrapolate the subtracted quantity to zero momentum, ensuring the removal of residual $ \hat{p} $-dependence.
• Implement the same NSPT framework for different gauge and fermion actions (e.g., Iwasaki, Symanzik-improved, Wilson, Clover) without deriving new Feynman rules, leveraging the computational efficiency of apeNEXT.
• Measure gluon and ghost propagators via momentum-space correlators and Faddeev-Popov matrix inversion, respectively, using Landau gauge and FFT acceleration.

Experimental results

Research questions

• RQ1 How can finite-size effects be effectively corrected in NSPT computations of logarithmically divergent renormalization constants?
• RQ2 Can the tamed-log subtraction scheme reliably recover analytic results for $ Z_S $, $ Z_P $, and $ Z_m $ at one-loop order in finite volumes?
• RQ3 To what extent can NSPT be extended to different Lattice QCD regularizations (e.g., tree-level Symanzik-improved gauge action) with minimal computational cost?
• RQ4 How do statistical fluctuations affect the measurement of the gluon and ghost propagators in NSPT, and can they be mitigated?

Key findings

• The tamed-log subtraction method successfully recovers the analytic one-loop result for $ Z_S $, correcting for finite-volume effects that previously caused large deviations in the unsubtracted $ O_s^{(1)}(\hat{p}^2) $.
• After applying the tamed-log subtraction, the computed $ Z_P $ and $ Z_S $ values agree with analytic expectations, enabling a precise determination of $ Z_m $.
• The method is robust and scalable, allowing high-loop computations of renormalization constants for operators with anomalous dimensions in finite volumes.
• NSPT computations for the tree-level Symanzik-improved gauge action with Wilson fermions show good agreement with analytic benchmarks after normalization correction, validating the framework for new regularizations.
• The gluon propagator is successfully computed at tree level and one-loop order in NSPT, with results matching known analytic constants in the continuum limit after logarithmic subtraction.
• Despite higher statistical noise, the ghost propagator is measured reliably via Faddeev-Popov matrix inversion, with results consistent across different lattice sizes and configurations.