[Paper Review] Gauge theories with fermions in two-index representations

Thomas DeGrand, Yigal Shamir+1 more|arXiv (Cornell University)|Oct 8, 2013

Quantum Chromodynamics and Particle Interactions6 references3 citations

TL;DR

This paper investigates asymptotically free gauge theories with fermions in two-index representations—specifically SU(3) with two adjoint Dirac fermions and SU(4) with six sextet fermions—using the Schrödinger functional method to compute the nonperturbative beta function and mass anomalous dimension. It finds that the mass anomalous dimension levels off below 0.5 at strong coupling, suggesting a universal bound and casting doubt on these theories as viable extended technicolor candidates.

ABSTRACT

After some introductory comments on the peculiar features of slowly running theories, I will report results obtained using the Schrodinger functional technique for two gauge theories that are believed to lie near the bottom of the conformal window: the SU(3) theory with two adjoint Dirac fermions, and the SU(4) theory with six Dirac fermions in the two-index antisymmetric representation. In both cases we find a small beta function in strong coupling, but we cannot confirm or rule out an infrared fixed point. In both theories the mass anomalous dimension levels off, staying well below 0.5, much like the theories with fermions in the two-index symmetric representation investigated earlier.

Motivation & Objective

• To investigate the nonperturbative behavior of gauge theories with fermions in two-index representations, particularly near the conformal window.
• To determine whether these theories exhibit an infrared fixed point via the nonperturbative beta function and mass anomalous dimension.
• To test the hypothesis that the mass anomalous dimension remains bounded below 0.5 in such theories, which has implications for composite Higgs models.
• To explore large-Nc scaling behavior across SU(2), SU(3), and SU(4) theories with adjoint fermions.

Proposed method

• Uses the Schrödinger functional (SF) scheme to nonperturbatively compute the running coupling and beta function in lattice simulations.
• Employs Wilson-clover fermions with nHYP-smeared gauge links to improve fermion propagator behavior.
• Defines the nonperturbative beta function via the variable $ \tilde{\beta}(u) = d(1/g^2)/d\log L $, with $ u = 1/g^2 $, and fits data to a logarithmic running form $ u(L) = c_0 + c_1 \log(L/(8a)) $.
• Measures the mass anomalous dimension $ \gamma_m $ from the scaling of the pseudoscalar density $ Z_P $, using $ \log Z_P(L) = c_0 + c_1 \log((8a)/L) $, where $ c_1 $ gives $ \gamma_m $.
• Performs continuum extrapolations using linear and quadratic fits to data across multiple lattice volumes $ L = 6,8,10,12,16 $.
• Applies statistical analysis to handle autocorrelations and ensure reliability, especially at strong couplings where data quality degrades.

Experimental results

Research questions

• RQ1 Does the SU(3) gauge theory with two adjoint Dirac fermions exhibit an infrared fixed point, as suggested by perturbative estimates?
• RQ2 Does the SU(4) gauge theory with six sextet fermions show signs of walking behavior, characterized by a near-zero beta function before turning positive?
• RQ3 Is the mass anomalous dimension $ \gamma_m $ bounded below 0.5 in theories with two-index fermions, as observed in previous studies?
• RQ4 How well does large-Nc scaling describe the nonperturbative behavior of these theories, particularly across SU(2), SU(3), and SU(4)?
• RQ5 Can the nonperturbative beta function be reliably extracted in theories with extremely slow running, where lattice perturbation theory fails?

Key findings

• The nonperturbative beta function for both the SU(3)/adjoint and SU(4)/sextet theories shows very small running, with fits to logarithmic running form indicating a nearly constant $ \tilde{\beta}(u) $, though with large statistical errors.
• The mass anomalous dimension $ \gamma_m $ levels off at values well below 0.5 in both theories, particularly for $ g^2 \gtrsim 3 $, indicating a universal bound across two-index fermion representations.
• In the SU(4)/sextet theory, $ \gamma_m $ remains nearly constant at strong coupling, suggesting a possible walking behavior, though no definitive IR fixed point is confirmed.
• The SU(3)/adjoint theory shows similar leveling off of $ \gamma_m $, though with larger statistical noise, and the strongest coupling point is affected by long autocorrelations.
• Large-Nc scaling of the beta function and $ \gamma_m $ shows good consistency across SU(2), SU(3), and SU(4), suggesting robustness of the results down to $ N_c = 2 $.
• The results challenge the viability of these theories as extended technicolor models, since the observed $ \gamma_m < 0.5 $ is too low to generate sufficient fermion masses via technicolor mechanisms.