We use lattice simulations and the continuous renormalization-group method, based on the gradient flow, to calculate the 𝜷 function and anomalous dimensions of the SU(3) gauge theory with N_f = 10 flavors of fermions in the fundamental representation. We employ several improvements to extend the range of available renormalized couplings, including the addition of heavy Pauli-Villars bosons to reduce cutoff effects and the combination of a range of gradient flow transformations. While in the weak coupling regime our result is consistent with those of earlier studies, our techniques allow us to study the system at much stronger couplings than previously possible. We find that the renormalization group 𝜷 function develops a zero, corresponding to an infrared-stable fixed point, at gradient-flow coupling g² = 15.0(5). We also determine the mass and tensor anomalous dimensions: At the fixed point we find 𝜸_m ≃ 0.6, suggesting that this system might be deep inside the conformal window.

We use lattice simulations and the continuous renormalization-group method, based on the gradient flow, to study a candidate theory of composite Higgs and a partially composite top. The model is an SU(4) gauge theory with four Dirac fermions in each of the fundamental and two-index antisymmetric representations. We find that the theory has an infrared fixed point at g² ≈ 15.5 in the gradient flow scheme. The mass anomalous dimension of each representation is large at the fixed point. On the other hand, the anomalous dimensions of top-partner operators do not exceed 0.5 at the fixed point. This may not be large enough for a phenomenologically successful model of partial compositeness.

As fermions are added to a lattice gauge theory, one is driven to stronger bare coupling in order to maintain the same renormalized coupling. Stronger bare couplings are usually associated with larger gauge fluctuations, leading to larger cutoff effects and more expensive simulations. In theories with many light fermions, sometimes the desired physical region cannot be reached before encountering a phase boundary. We show that these undesired effects can be reduced by adding Pauli–Villars fields. We reach significantly larger renormalized couplings while at the same time damping short-distance fluctuations of the gauge field. This may allow for controlled continuum extrapolations from large lattice spacings.

Models for what may lie behind the Standard Model often require non-perturbative calculations in strongly coupled field theory. This creates opportunities for lattice methods, to obtain quantities of phenomenological interest as well as to address fundamental dynamical questions. I survey recent work in this area.