Ultraviolet modifications of dispersion relations in effective field theory

TL;DR

Planck-scale physics can modify dispersion relations via dimension-5 operators in an EFT with a fixed preferred frame $n^a$ (Lorentz violation). The approach identifies cubic-in-$p$ corrections for scalars, photons, and fermions with explicit dispersions $E^2 \simeq \vec{p}^{\,2}+m^2 + (\kappa / M_{\rm Pl}) |\vec{p}|^3$, $E^2 \simeq p^2 \pm (2\xi / M_{\rm Pl}) p^3$, and $E^2 \simeq p^2 + m^2 + (2|\vec{p}|^3 / M_{\rm Pl})(\eta_1 \pm \eta_2 \gamma_5)$. The results are confronted with terrestrial bounds from clock-comparison and spin-precession measurements, giving $|\eta_{Q,u,d}|,|\xi| \lesssim 10^{-6}$ and $|\eta_L^e| \lesssim 10^{-5}$ after RG running, exceeding some astrophysical limits and challenging certain loop-quantum-gravity predictions. The paper highlights that EFT requires a careful treatment of divergences, motivating the replacement of the background $n^a n^b n^c$ by a traceless $C^{abc}$ to suppress quadratic divergences.

Abstract

The existence of a fundamental ultraviolet scale, such as the Planck scale, may lead to modifications of the dispersion relations for particles at high energies, in some scenarios of quantum gravity. We apply effective field theory to this problem and identify dimension 5 operators that do not mix with dimensions 3 and 4 and lead to cubic modifications of dispersion relations for scalars, fermions, and vector particles. Further we show that, for electrons, photons and light quarks, clock comparison experiments bound these operators at 10^{-5}/Mpl.