Higher representations on the lattice: numerical simulations. SU(2) with adjoint fermions

TL;DR

This paper extends lattice gauge theory techniques to fermions in arbitrary representations by detailing a generalized Wilson-Dirac operator, the HMC/RHMC algorithms, and even-odd preconditioning, with a focus on SU(2) and adjoint fermions as a benchmark. It provides a comprehensive validation suite, including Creutz equality, reversibility, operator forces, and spectral gap analyses, and delivers benchmark mesonic observables on small lattices to enable cross-checks across groups. The results lay groundwork for controlled nonperturbative studies of beyond-Standard Model strong dynamics, such as technicolor scenarios, by offering a robust, reproducible computational framework and baseline data. The methodologies—ranging from rational approximations for fractional powers to careful spectral analysis—are broadly applicable to theories with higher representations and different gauge groups, supporting future exploration of phase structure and IR fixed points in BSM candidate theories.

Abstract

We discuss the lattice formulation of gauge theories with fermions in arbitrary representations of the color group, and present in detail the implementation of the HMC/RHMC algorithm for simulating dynamical fermions. We discuss the validation of the implementation through an extensive set of tests, and the stability of simulations by monitoring the distribution of the lowest eigenvalue of the Wilson-Dirac operator. Working with two flavors of Wilson fermions in the adjoint representation, benchmark results for realistic lattice simulations are presented. Runs are performed on different lattice sizes ranging from 4^3x8 to 24^3x64 sites. For the two smallest lattices we also report the measured values of benchmark mesonic observables. These results can be used as a baseline for rapid cross-checks of simulations in higher representations. The results presented here are the first steps towards more extensive investigations with controlled systematic errors, aiming at a detailed understanding of the phase structure of these theories, and of their viability as candidates for strong dynamics beyond the Standard model.

Figures (12)

• Figure 1: Dependence of $\langle \exp\left(\Delta H\right)\rangle$ and $\langle P \rangle$ on the time--step used for the MD integration.
• Figure 2: The expectation value $\langle \Delta H\rangle$ is proportional to $(\Delta\tau)^4$ (upper panel) consistently with the use of a second order integrator (the red curve shows the one parameter fit). The acceptance probability $P_{acc}$ as measured in the test runs (lower panel) is correctly described by the expected large volume behavior $P_{acc}=\mathrm{erfc}(\sqrt{\langle\Delta H\rangle}/2)$ (solid curve, not a fit).
• Figure 3: Reversibility test for several values of the time--step used for the MD integration.
• Figure 4: Time--history for the thermalization of the plaquette in the SU(3) theory with two flavours in the fundamental representation on a $16^4$ lattice at $\beta=5.6$ and $\kappa=0.15750$. The black line shows the evolution of the plaquette value using our HMC algorithm, while the red line represents the same quantity using the DD-HMC algorithm. Trajectory number has been scaled by a factor of 7 for the DD-HMC data.
• Figure 5: Probability distribution of the plaquette. The black (respectively red) curve refers to data from a simulation of the SU(3) gauge theory on a $4^3 \times 8$ lattice at $\beta=5.6$, $\kappa=0.15600$ with two fermions in the fundamental representation (respectively the two--index antisymmetric).