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

Dmitry S. Ageev, Yulia A. Ageeva

TL;DR

This work develops a comprehensive one-loop renormalization framework for a Lorentz-violating, two-scalar quantum field theory with the most general two-derivative kinetic structure. By employing a spectral decomposition of the anisotropic quadratic operator, the authors derive universal master integrals and beta functions expressed in terms of two residues, T and B, weighted by a phase-space kernel, making the impact of anisotropy transparent. They identify how anisotropy and kinetic mixing constrain fixed points, notably restricting the Wilson-Fisher fixed point and revealing a fixed line in the fully quartic sector at one loop. The results provide a clear phase-space interpretation of anisotropy through angular weights and establish a robust framework for extending RG analyses to more fields and sectors with multiple effective metrics.

Abstract

We study a Euclidean quantum field theory of two interacting real scalar fields in $D=4-ε$ dimensions with the most general two--derivative but Lorentz--violating quadratic structure, allowing anisotropic spatial gradients and field--mixing time and cross--gradient terms, together with general cubic and quartic interactions. Although Lorentz violation is introduced only through the kinetic sector, the renormalization structure is nontrivial: interactions generate additional couplings, including masses and linear terms, so a consistent renormalization--group analysis cannot be formulated in terms of modified kinetic terms alone and requires an RG--complete operator basis compatible with the reduced symmetry. We perform a systematic one--loop renormalization and derive the complete set of beta functions for cubic and quartic couplings, masses, and linear terms. We identify and analyze the resulting fixed points and fixed manifolds, showing in particular how anisotropy restricts their existence and stability. In particular we obtain how anisotropy restricts existence of canonical Wilson-Fisher fixed point. Also we provide a transparent physical interpretation of the anisotropy--dependent coefficients appearing in the beta functions and clarify how kinetic mixing reshapes the interaction flow through the available ultraviolet phase space.

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

TL;DR

This work develops a comprehensive one-loop renormalization framework for a Lorentz-violating, two-scalar quantum field theory with the most general two-derivative kinetic structure. By employing a spectral decomposition of the anisotropic quadratic operator, the authors derive universal master integrals and beta functions expressed in terms of two residues, T and B, weighted by a phase-space kernel, making the impact of anisotropy transparent. They identify how anisotropy and kinetic mixing constrain fixed points, notably restricting the Wilson-Fisher fixed point and revealing a fixed line in the fully quartic sector at one loop. The results provide a clear phase-space interpretation of anisotropy through angular weights and establish a robust framework for extending RG analyses to more fields and sectors with multiple effective metrics.

Abstract

We study a Euclidean quantum field theory of two interacting real scalar fields in dimensions with the most general two--derivative but Lorentz--violating quadratic structure, allowing anisotropic spatial gradients and field--mixing time and cross--gradient terms, together with general cubic and quartic interactions. Although Lorentz violation is introduced only through the kinetic sector, the renormalization structure is nontrivial: interactions generate additional couplings, including masses and linear terms, so a consistent renormalization--group analysis cannot be formulated in terms of modified kinetic terms alone and requires an RG--complete operator basis compatible with the reduced symmetry. We perform a systematic one--loop renormalization and derive the complete set of beta functions for cubic and quartic couplings, masses, and linear terms. We identify and analyze the resulting fixed points and fixed manifolds, showing in particular how anisotropy restricts their existence and stability. In particular we obtain how anisotropy restricts existence of canonical Wilson-Fisher fixed point. Also we provide a transparent physical interpretation of the anisotropy--dependent coefficients appearing in the beta functions and clarify how kinetic mixing reshapes the interaction flow through the available ultraviolet phase space.
Paper Structure (17 sections, 290 equations, 1 figure)

This paper contains 17 sections, 290 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup: theory and its decomposition
  3. One-loop diagrams calculation
  4. General expressions for beta functions
  5. Explicit form of beta functions, fixed points and physical interpretation
  6. Cross-gradients turned off $\mathbf{Y}=0$
  7. Turning on cross-gradients: $\mathbf Z_t=\mathbf 1$ and $\mathbf Y\neq 0$
  8. Conclusion
  9. Explicit evaluations of master integrals
  10. Tadpole integral
  11. Bubble integrals
  12. Angular weights $J_{11}$, $J_{22}$ and $J_{12}$ in the limit $\mathbf Z_t=\mathbf 1$ and $\mathbf Y=0$
  13. Weights $J_{11}$, $J_{22}$ and $J_{12}$ in the limit $\mathbf Z_t=\mathbf 1,\mathbf Y=0$
  14. Proof of the inequality $J_{12}^2\le J_{11}J_{22}$
  15. Linear stability of the generalized Wilson--Fisher fixed point
  16. ...and 2 more sections

Figures (1)

  • Figure 1: All possible one loop divergent diagrams for the model with the interaction lagrangian \ref{['eq:L3L4']}. The notations $\lambda$ and $h$ are just schematic and we specify couplings for each concrete diagram in the text. Since there are $\delta j_i \phi_i$ terms in the lagrangian \ref{['eq:L3L4']}, we add the very bottom diagram with one external leg, one propagator and $h_{ijk}$-type vertex as well.