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 $D=4-\epsilon$ 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.