[Paper Review] Determination of the mass anomalous dimension for $N_f=12$ and $N_f=9$ SU($3$) gauge theories

Etsuko Itou, Akio Tomiya|arXiv (Cornell University)|Nov 5, 2014

Particle physics theoretical and experimental studies6 references5 citations

TL;DR

This paper presents a lattice QCD study of the mass anomalous dimension in SU(3) gauge theories with $N_f=12$ and $N_f=9$ fundamental fermions, using two independent methods—step scaling and hyperscaling of the Dirac mode number—on the continuum limit with fixed renormalized coupling. The key result is a consistent determination of $\gamma_m^* \approx 0.081 \pm 0.018$ (stat.) for $N_f=12$, with the $N_f=9$ case initiated to probe the conformal window boundary.

ABSTRACT

We show the numerical simulation result for the mass anomalous dimension of the SU($3$) gauge theory coupled to $N_f = 12$ fundamental fermions. We use two independent methods, namely the step scaling method and the hyperscaling method of the Dirac mode number, to determine the anomalous dimension in the vicinity of the infrared fixed point of the theory. We show the continuum extrapolations keeping the renormalized coupling constant as a reference in both analyses. Furthermore, some recent works seems to suggest the lower boundary of the conformal window of the SU($3$) gauge theory exists between $N_f=8$ and $10$. We also briefly report our new project, in which the numerical simulation of the SU($3$) gauge theory coupled to $N_f=9$ fundamental fermions has been performed.

Motivation & Objective

• To determine the mass anomalous dimension $\gamma_m^*$ at the infrared fixed point (IRFP) in SU(3) gauge theories with $N_f=12$ fundamental fermions using independent lattice methods.
• To investigate the consistency of results from the step scaling method and the hyperscaling method of the Dirac mode number across lattice volumes and continuum extrapolation.
• To initiate numerical simulations for $N_f=9$ to probe the lower boundary of the conformal window of SU(3) gauge theories.
• To assess the impact of lattice artifacts and fermion discretization schemes on the determination of universal quantities like $\gamma_m^*$.

Proposed method

• The step scaling method is applied using the pseudo-scalar operator renormalization factor $Z_P$, defined via the ratio of nonperturbative to tree-level pseudo-scalar correlators at fixed propagation time $t$ and lattice temporal extent $T$.
• The mass step scaling function $\Sigma_P(\beta, a/L; s)$ is computed for $s=2$, and its continuum limit $\sigma_P(u,s)$ is taken while keeping the renormalized coupling $u = g_R^2(1/L)$ constant.
• The anomalous dimension at the IRFP is extracted via $\gamma_m^*(u^*) = -\log|\sigma_P(u^*,s)| / \log|s|$.
• The hyperscaling method analyzes the Dirac mode number $\nu(\lambda)$ in the $100 < \nu(\lambda) < 2000$ range with fine resolution $\Delta(a\lambda) = 0.01$ or $0.02$, using rescaling to test scale invariance.
• Lattice configurations are generated using the Hybrid Monte Carlo (HMC) algorithm with twisted boundary conditions in $x,y$ directions and periodic in $z,t$, enabling exact massless simulations.
• Continuum extrapolations are performed using three-point linear fits in $(a/L)^2$, with results kept consistent across different lattice volumes and $\beta$ values tuned to the IRFP in the TPL scheme.

Experimental results

Research questions

• RQ1 What is the value of the mass anomalous dimension $\gamma_m^*$ at the infrared fixed point for $N_f=12$ SU(3) gauge theory?
• RQ2 Do the step scaling and hyperscaling methods yield consistent results for $\gamma_m^*$ in the continuum limit?
• RQ3 What is the behavior of the Dirac mode number and its scaling properties in the $N_f=12$ theory, indicating conformal invariance?
• RQ4 Does the $N_f=9$ theory exhibit signs of conformal behavior, and where does it lie relative to the conformal window boundary?

Key findings

• The mass anomalous dimension for $N_f=12$ is determined as $\gamma_m^* = 0.081 \pm 0.018$ (stat.) $^{+0.025}_{-0}$ (syst.) using the step scaling method.
• The hyperscaling method of the Dirac mode number shows scale invariance in the continuum limit, with $\gamma_m$ values in the range $0.05 \leq \gamma_m \leq 0.08$ for the finest lattices, consistent with the step scaling result within $1\sigma$.
• The $10^3 \times 20$ and $20^3 \times 40$ lattices show a signal of scale invariance at $\gamma_m \approx 0.02$, though the $8^3 \times 16$ and $16^3 \times 32$ lattices do not, likely due to coarse spacing and small volume.
• The $N_f=9$ case is initiated with HMC simulations using Wilson fermions and Iwasaki gauge action, with $\beta$-$\kappa$ scans to locate the line of constant renormalized mass.
• The results are consistent with a $2\sigma$-compatible discrepancy with previous results using improved staggered fermions ($\gamma_m^* \approx 0.24$), possibly due to zero-mode effects from twisted boundary conditions.
• The study confirms the existence of an IRFP at $g_{\text{TPL}}^{*2} = 2.69 \pm 0.14$ (stat.) $^{+0}_{-0.16}$ (syst.) for $N_f=12$, supporting the existence of a conformal window.