Massive vectors from projective-invariance breaking

TL;DR

This paper explores how breaking projective invariance in metric-affine gravity can generate massive vector fields, providing a potential mechanism similar to electroweak symmetry breaking.

Contribution

It demonstrates that incorporating the tensor of homothetic curvature and specific torsion constraints can produce massive vectors within the metric-affine gravity framework.

Findings

Tensor of homothetic curvature replaces unphysical constraints.

Breaking projective invariance yields massive vectors.

Mechanism analogous to electroweak symmetry breaking.

Abstract

A general affine connection has enough degrees of freedom to describe the classical gravitational and electromagnetic fields in the metric-affine formulation of gravity. The gravitational field is represented in the Lagrangian by the symmetric part of the Ricci tensor and the classical electromagnetic field can be represented by the tensor of homothetic curvature. The simplest metric-affine Lagrangian that depends on the tensor of homothetic curvature generates the Einstein-Maxwell equations for a massless vector. Metric-affine Lagrangians with matter fields depending on the connection are subject to an unphysical constraint because the symmetrized Ricci tensor is projectively invariant while matter fields are not. We show that the appearance of the tensor of homothetic curvature, which is not projectively invariant, in the Lagrangian replaces this constraint with the Maxwell equations and restores projective invariance of the total action. We also examine several constraints on the torsion tensor to show that algebraic constraints on the torsion that break projective invariance of the connection and impose projective invariance on the tensor of homothetic curvature replace the massless vector with a massive vector. We conclude that the metric-affine formulation of gravity allows for a mechanism that generates masses of vectors, as it happens for electroweak gauge bosons via spontaneous symmetry breaking.