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Gravitational effective action and entanglement entropy in UV modified theories with and without Lorentz symmetry

Dmitry Nesterov, Sergey N. Solodukhin

TL;DR

This work analyzes how UV modifications to field propagators, including Lorentz-invariant higher-derivative and non-Lorentz-invariant Lifshitz-type operators, influence low-energy gravitational couplings and entanglement entropy. Employing generalized heat-kernel methods and the replica trick on conical spaces, it derives how the cosmological constant and Newton's constant are induced and how entanglement entropy scales with the entangling surface area. The results show that UV modifications can alter the divergence structure (e.g., quadratic, logarithmic) but do not universally eliminate UV divergences; notably, Lifshitz-type UV terms can render extrinsic-curvature couplings UV-finite under certain conditions, while entropy remains area-law controlled and largely insensitive to Lorentz symmetry. Together, these findings illuminate the interplay between UV physics and gravitational effective actions, with implications for quantum gravity and condensed-matter analogs.

Abstract

We calculate parameters in the low energy gravitational effective action and the entanglement entropy in a wide class of theories characterized by improved ultraviolet (UV) behavior. These include i) local and non-local Lorentz invariant theories in which inverse propagator is modified by higher-derivative terms and ii) theories described by non-Lorentz invariant Lifshitz type field operators. We demonstrate that the induced cosmological constant, gravitational couplings and the entropy are sensitive to the way the theory is modified in UV. For non-Lorentz invariant theories the induced gravitational effective action is of the Horava-Lifshitz type. We show that under certain conditions imposed on the dimension of the Lifshitz operator the couplings of the extrinsic curvature terms in the effective action are UV finite. Throughout the paper we systematically exploit the heat kernel method appropriately generalized for the class of theories under consideration.

Gravitational effective action and entanglement entropy in UV modified theories with and without Lorentz symmetry

TL;DR

This work analyzes how UV modifications to field propagators, including Lorentz-invariant higher-derivative and non-Lorentz-invariant Lifshitz-type operators, influence low-energy gravitational couplings and entanglement entropy. Employing generalized heat-kernel methods and the replica trick on conical spaces, it derives how the cosmological constant and Newton's constant are induced and how entanglement entropy scales with the entangling surface area. The results show that UV modifications can alter the divergence structure (e.g., quadratic, logarithmic) but do not universally eliminate UV divergences; notably, Lifshitz-type UV terms can render extrinsic-curvature couplings UV-finite under certain conditions, while entropy remains area-law controlled and largely insensitive to Lorentz symmetry. Together, these findings illuminate the interplay between UV physics and gravitational effective actions, with implications for quantum gravity and condensed-matter analogs.

Abstract

We calculate parameters in the low energy gravitational effective action and the entanglement entropy in a wide class of theories characterized by improved ultraviolet (UV) behavior. These include i) local and non-local Lorentz invariant theories in which inverse propagator is modified by higher-derivative terms and ii) theories described by non-Lorentz invariant Lifshitz type field operators. We demonstrate that the induced cosmological constant, gravitational couplings and the entropy are sensitive to the way the theory is modified in UV. For non-Lorentz invariant theories the induced gravitational effective action is of the Horava-Lifshitz type. We show that under certain conditions imposed on the dimension of the Lifshitz operator the couplings of the extrinsic curvature terms in the effective action are UV finite. Throughout the paper we systematically exploit the heat kernel method appropriately generalized for the class of theories under consideration.

Paper Structure

This paper contains 15 sections, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vacuum Energy/Cosmological Constant
  3. Entanglement Entropy
  4. The replica method
  5. The Lorentz invariant theories
  6. Newton's constant
  7. Complete covariant decomposition
  8. Non-Lorentz invariant UV modifications
  9. Scaling arguments
  10. Entanglement entropy: non-Lorentz invariant theories
  11. Gravitational couplings in curved spacetime
  12. Conclusions
  13. Heat kernel of operator $F(\Box_g)$ in momentum representation
  14. Some Integrals in Momentum Space
  15. Heat kernel of operator $\Delta_\tau+F(\Delta_x)$

Figures (1)

  • Figure 1: Contour of $\sigma$-integration starts at negative imaginary infinity, ends at positive imaginary infinity and passes zero from the positive real side.