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  • Open Access

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

Dmitry S. Ageev1,2,* and Yulia A. Ageeva2,3,4,†

  • *Contact author: ageev@mi-ras.ru
  • †Contact author: ageeva@inr.ac.ru
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Phys. Rev. D 113, 125010 – Published 8 June, 2026

DOI: https://doi.org/10.1103/3ryd-3hx4

Abstract

We develop a basis-covariant one-loop renormalization framework for two interacting real scalars in D=4−ε 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.

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Phys. Rev. D 113, 125010– Published 8 June, 2026
  • Received 30 January 2026
  • Accepted 12 May 2026
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