# The Ultraviolet and Infrared Behavior of an Abelian Proca Model From the Viewpoint of a One-Parameter Extension of the Covariant Heisenberg Algebra

## TL;DR

This paper explores a modified Abelian Proca model based on a one-parameter extension of the covariant Heisenberg algebra, revealing a new dispersion relation and effective masses, with implications for quantum gravity and electroweak scale physics.

## Contribution

It reformulates the Abelian Proca model using a one-parameter extension of the covariant Heisenberg algebra, deriving a modified dispersion relation and analyzing its physical implications.

## Key findings

- Modified dispersion relation with two massive vector particles
- Maximum length scale near the electroweak length
- Infrared behavior resembles a massive Lee-Wick model

## Abstract

Recently a one-parameter extension of the covariant Heisenberg algebra with the extension parameter $l$ ($l$ is a non-negative constant parameter which has a dimension of $[momentum]^{-1}$) in a $(D+1)$-dimensional Minkowski space-time has been presented [G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior and V. Nikoofard, Effective models of quantum gravity induced by Planck scale modifications in the covariant quantum algebra, Adv. High Energy Phys. 2017 (2017) 4768341]. The Abelian Proca model is reformulated from the viewpoint of the above one-parameter extension of the covariant Heisenberg algebra. It is shown that the free space solutions of the above modified Proca model satisfy the modified dispersion relation $\frac{\textbf{p}^2}{\left(1+\frac{\Lambda^2}{2\hbar^2}\textbf{p}^2\right)^2}=m^2c^2$ where $\Lambda=\hbar l$ is the characteristic length scale in our model. This modified dispersion relation describes two massive vector particles with the effective masses ${\cal M}_{_\pm}(\Lambda)=\frac{2m}{1\mp\sqrt{1-2\left(\frac{mc\Lambda}{\hbar}\right)^2}}$. Numerical estimations show that the maximum value of $\Lambda$ in a four-dimensional space-time is near to the electroweak length scale, i.e., $\Lambda_{_{max}}\sim l_{_{electroweak}}\sim10^{-18}\; m$. We show that in the infrared/large-distance domain the modified Proca model behaves like an Abelian massive Lee-Wick model which has been presented by Accioly and his co-workers [A. Accioly, J. Helayel-Neto, G. Correia, G. Brito, J. de Almeida and W. Herdy, Interparticle potential energy for D-dimensional electromagnetic models from the corresponding scalar ones, Phys. Rev. D 93 (2016) 105042].