One-Loop Renormalization of Anisotropic Two-Scalar Quantum Field Theories

TL;DR

This paper develops a covariant one-loop renormalization framework for anisotropic two-scalar quantum field theories, deriving beta functions and classifying fixed points with anisotropy effects included.

Contribution

It introduces a universal spectral representation for divergences and classifies fixed points in anisotropic scalar theories, extending previous isotropic analyses.

Findings

Derived closed-form beta functions for anisotropic couplings.

Classified fixed points and showed anisotropy restricts certain fixed points.

Provided a phase-space interpretation of UV mode interactions.

Abstract

We develop a basis--covariant one--loop renormalization framework for two interacting real scalars in $D=4-\epsilon$ with the most general two--derivative Lorentz--violating quadratic form, allowing anisotropic spatial gradients and direction--dependent kinetic mixing, together with general cubic and quartic interactions forming RG complete set of operators at one-loop. In dimensional regularization with minimal subtraction we compute the full set of one--loop UV divergences and obtain closed beta functions for quartic and cubic couplings, masses. The pole coefficients admit a universal spectral representation as angular averages over the direction--dependent eigenvalues and projectors of the UV kinetic matrix; all anisotropy dependence enters through a single universal kernel admitting two--particle phase--space interpretation. We classify fixed points and fixed manifolds and show, in particular, that anisotropy restricts the existence of the coupled Wilson--Fisher--type fixed point. When the cross--gradients are turned on the coefficients in beta functions are governed by six phase--space weights admitting interpretation in terms of encoding ``populations'' and ``coherences'' of the UV normal modes.