O(a) errors in 3-D SU(N) Higgs theories

TL;DR

This work advances lattice–continuum matching for 3-D SU($N$) Higgs theories by deriving $O(a)$-level corrections for lattice actions with both adjoint and fundamental scalars, extending prior results beyond SU(2)$\times$U(1). Using an O($a$) Symanzik framework, it provides explicit counterterms ($Z_g$, $Z_\phi$, $Z_\Phi$, $\delta \lambda_i$, $\delta m_i^2$, $Z_m$) and one-loop (and select two-loop) corrections to masses and operator insertions, enabling improved accuracy in nonperturbative lattice studies. The paper also generalizes to SU($N$)$\times$SU($M$) by including an extra scalar singlet and detailing how to augment the operator-renormalization structure, with clear prescriptions for applying these corrections to re-interpret existing data. Although additive $O(a)$ corrections to masses and $\langle \phi^\dagger \phi \rangle$ are not computed (requiring 3-loop work), the results substantially reduce $O(a)$ artifacts in observables related to phase transitions, improving the reliability of lattice investigations of hot gauge theories.

Abstract

We compute the matching conditions between lattice and continuum 3-D SU(N) Higgs theories, with both adjoint and fundamental scalars, at O(a), except for additive corrections to masses and Higgs field operator insertions.

Figures (2)

• Figure 1: The topologically distinct diagrams needed for the Landau gauge calculation. Wavy lines are gauge propagators, solid lines are scalars, and dotted lines are ghosts. A cross is a self energy correction from the measure and a blot is a $\phi^2$ operator insertion.
• Figure 2: Tadpole type diagrams which correct the gauge field self-energy at one and two loops.