# Einstein-Cartan-Dirac gravity with $U(1)$ symmetry breaking

## TL;DR

This paper extends Einstein-Cartan-Dirac gravity by adding a Maxwell field that breaks $U(1)$ symmetry, exploring implications for high-density astrophysical objects and early Universe phase transitions.

## Contribution

It introduces a novel Einstein-Cartan-Dirac-Maxwell theory with $U(1)$ symmetry breaking, deriving field equations and discussing potential astrophysical and cosmological implications.

## Key findings

- Derived generalized field equations for the extended theory.
- Estimated the magnitude of torsion-induced corrections.
- Discussed possible effects in neutron stars and early Universe.

## Abstract

Einstein-Cartan theory is an extension of the standard formulation of General Relativity where torsion (the antisymmetric part of the affine connection) is non-vanishing. Just as the space-time metric is sourced by the stress-energy tensor of the matter fields, torsion is sourced via the spin density tensor, whose physical effects become relevant at very high spin densities. In this work we introduce an extension of the Einstein-Cartan-Dirac theory with an electromagnetic (Maxwell) contribution minimally coupled to torsion. This contribution breaks the $U(1)$ gauge symmetry, which is suggested by the possibility of a torsion-induced phase transition in the early Universe, yielding new physics in extreme (spin) density regimes. We obtain the generalized gravitational, electromagnetic and fermionic field equations for this theory, estimate the strength of the corrections, and discuss the corresponding phenomenology. In particular, we briefly address some astrophysical considerations regarding the relevance of the effects which might take place inside ultra-dense neutron stars with strong magnetic fields (magnetars).