One-loop renormalization of anisotropic two-scalar quantum field theories

We develop a basis-covariant one-loop renormalization framework for two interacting real scalars in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>D</a:mi> <a:mo>=</a:mo> <a:mn>4</a:mn> <a:mo>−</a:mo> <a:mi>ε</a:mi> </a:math> with the most general parity even two-derivative Lorentz-violating quadratic form, allowing anisotropic spatial gradients and direction-dependent kinetic mixing, together with general cubic and quartic interactions and the relevant lower-dimensional operators forming a renormalization group 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, and linear couplings. 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.