Evolution of the coupling constant in SU(2) lattice gauge theory with two adjoint fermions

TL;DR

This work nonperturbatively investigates the running of the SU(2) gauge coupling with two adjoint Dirac fermions using the Schrödinger functional on the lattice. By measuring the finite‑volume coupling $g^2(L)$ across volumes $12^4$–$20^4$ and applying step‑scaling and beta‑function fits, the study finds evidence for an infrared fixed point at $g_*^2 \sim 2$–$3$, well below the perturbative two‑loop estimate. The results indicate conformal behavior in the massless limit and reveal that nonperturbative effects slow the running near the fixed point, underscoring the importance of lattice methods and the need for improved actions to control finite‑a artifacts. These findings have implications for conformal or walking dynamics in beyond‑the‑Standard‑Model scenarios and motivate further nonperturbative and spectrum studies with improved discretisations.

Abstract

We measure the evolution of the coupling constant using the Schroedinger functional method in the lattice formulation of SU(2) gauge theory with two massless Dirac fermions in the adjoint representation. We observe strong evidence for an infrared fixed point, where the theory becomes conformal. We measure the continuum beta-function and the coupling constant as a function of the energy scale.

Figures (5)

• Figure 1: Lattice measurements of $g^2(L/a,\beta_L)$. Continuous lines show $g^2(L)$ integrated from Eq. (\ref{['bfit1']}), constrained to go through lattice points at $L/a=12$.
• Figure 2: Small-statistic measurements of $g^2(L/a,\beta_L)$ for 2 flavours of fundamental representation fermions, to be contrasted with the adjoint representation case in Fig. \ref{['g2_su2a']}. The value of $g^2$ at $\beta_L=2.2$, $L/a=16$ is $26\pm 7$. Continuous lines show $g^2(L)$ integrated using 2-loop $\beta$-function, constrained to go through lattice points at $L/a=16$.
• Figure 3: Step scaling $\Delta(L_1,L_2,\beta_L)$ for different pairs of volumes, see Eq. (\ref{['eq:discrete_beta']}), plotted against $1/g^2(L_1,\beta_L)$. The scaling factor is 2, except for the pair with largest volumes. Also shown is the 2-loop perturbative result for corresponding ratios of volumes.
• Figure 4: The $\beta$-function obtained from Eq. (\ref{['bfit1']}), with $g^2_\ast=2.2$. The shaded area shows the estimated error range of the fit. Shown are also universal perturbative one- and two-loop $\beta$-functions, together with the three- and four-loop results in the MS-scheme vanRitbergen:1997va. Because of the different scheme, these are not directly comparable with the lattice (SF-scheme) results.
• Figure 5: $g^2(\mu=1/L)$ determined from Eq. (\ref{['bfit1']}), together with perturbative $g^2$. Scale $\Lambda_0$ is determined so that $\mu=\Lambda_0$ is at the IR Landau pole of the asymptotically free one-loop coupling and at the UV Landau pole of the measured non-asymptotically free branch ($g^2 > g_\ast^2$). For illustration, the $L/a=16$ lattice points have been placed on the benchmark curve at the corresponding $g^2$-values, indicating the relative scale hierarchy between simulations.