[Paper Review] Non-perturbative evaluation of cSW for smeared link clover fermion and Iwasaki gauge action

Yusuke Taniguchi|arXiv (Cornell University)|Mar 1, 2013

Quantum Chromodynamics and Particle Interactions5 references3 citations

TL;DR

This study non-perturbatively evaluates the $c_{\rm SW}$ coefficient for APE stout-smeared link clover fermions with the Iwasaki gauge action at a lattice spacing of approximately 0.1 fm. Using the Schrödinger functional method, it finds that $c_{\rm SW}$ decreases monotonically with increasing smearing steps ($N_{\rm smear} = 1$ to $6$), remaining about 10% below the tree-level value even at $N_{\rm smear} = 6$, indicating significant non-perturbative corrections.

ABSTRACT

We performed a rough estimate of the non-perturbative value of the clover term coefficient cSW for the APE stout link Wilson fermion. We varied the number of smearings from Nsmear=1 to 6 and adopted beta values roughly corresponding to the lattice spacing of 0.1 fm. We used the Schroedinger functional technique for an evaluation of cSW and found that cSW decreases monotonically as we increase Nsmear but has a 10% order of deviation from the tree level value for Nsmear=6.

Motivation & Objective

• To non-perturbatively determine the $c_{\rm SW}$ coefficient for APE stout-smeared link clover fermions with the Iwasaki gauge action.
• To investigate how $c_{\rm SW}$ evolves with increasing smearing steps ($N_{\rm smear}$) while keeping the lattice spacing near 0.1 fm.
• To assess the consistency of the tree-level $c_{\rm SW} = 1$ value with non-perturbative results for highly smeared fermions.
• To evaluate the improvement coefficient $c_A$ of the axial current as a byproduct of the $c_{\rm SW}$ determination.

Proposed method

• The Schrödinger functional technique is employed to compute $c_{\rm SW}$ using boundary conditions in the temporal direction on an $8^3 \times 16$ lattice.
• Two-point correlation functions of the axial current $A_0$ and pseudo-scalar density $P$ are computed at bulk and boundary points to extract the PCAC quark mass.
• The improvement condition $\Delta M = M(3T/4, T/4) - M'(3T/4, T/4) \to 0$ is used to determine the non-perturbative $c_{\rm SW}$ value at the critical hopping parameter $\kappa_c$.
• The critical hopping parameter $\kappa_c$ is tuned so that the PCAC mass $M(T/2, T/4) \to 0$, and $\kappa_c$ is fitted linearly as a function of $c_{\rm SW}$ to locate the $c_{\rm SW}$ value where $\Delta M = 0$.
• The axial current improvement coefficient $c_A$ is extracted from the $x_0$-dependence of $c_A' = -(r - r')/(s - s')$, with $c_A = c_A'(T/4)$, and evaluated at the non-perturbative $c_{\rm SW}$.
• Simulations use $N_f = 3$ degenerate flavors, $\rho = 0.1$ for smearing, and $\beta$ values tuned to achieve $a \approx 0.1$ fm using the Sommer scale $r_0$ and $\Omega$ baryon mass.

Experimental results

Research questions

• RQ1 How does the non-perturbative $c_{\rm SW}$ coefficient change with increasing APE stout smearing steps ($N_{\rm smear}$) at fixed lattice spacing?
• RQ2 Is the tree-level value $c_{\rm SW} = 1$ consistent with the non-perturbative result for highly smeared fermions?
• RQ3 What is the behavior of the critical hopping parameter $\kappa_c$ as a function of $N_{\rm smear}$, and how close is it to the tree-level value $1/8$?
• RQ4 What is the non-perturbative value of the axial current improvement coefficient $c_A$, and is it consistent with zero?
• RQ5 How well does the linear fit of $\Delta M$ vs. $c_{\rm SW}$ determine the $c_{\rm SW}$ value at which $\Delta M = 0$?

Key findings

• The non-perturbative $c_{\rm SW}$ value decreases monotonically from 1.342(21) at $N_{\rm smear} = 1$ to 1.057(20) at $N_{\rm smear} = 6$, indicating a significant reduction from the tree-level value of 1.
• Even at $N_{\rm smear} = 6$, $c_{\rm SW}$ remains approximately 10% below the tree-level value, with a value of 1.057(20), suggesting non-perturbative corrections are substantial.
• The critical hopping parameter $\kappa_c$ decreases smoothly with increasing $N_{\rm smear}$, approaching the tree-level value of $1/8 = 0.125$ at $N_{\rm smear} = 6$, where $\kappa_c = 0.12634(22)$, indicating improved chiral symmetry.
• The axial current improvement coefficient $c_A$ is found to be consistent with zero within statistical errors for all values of $N_{\rm smear}$, confirming the absence of $O(a)$ errors in the axial current at the non-perturbative $c_{\rm SW}$.
• The linear fit of $\Delta M$ vs. $c_{\rm SW}$ yields a reliable determination of $c_{\rm SW}$, with the zero-crossing point well-determined across all $N_{\rm smear}$ values.
• The $\beta$ values are tuned to achieve $a \approx 0.1$ fm, with $\beta = 1.82$ at $N_{\rm smear} = 6$, and the lattice spacing is confirmed using the Sommer scale $r_0$ and $\Omega$ baryon mass.