  A study of the Bell-CHSH inequality in gauge field theories is presented. By
using the Kugo-Ojima analysis of the BRST charge cohomology in Fock space, the
Bell-CHSH inequality is formulated in a manifestly BRST invariant way. The
examples of the free four-dimensional Maxwell theory and the Abelian Higgs
model are scrutinized. The inequality is probed by using BRST invariant
squeezed states, allowing for large Bell-CHSH inequality violations, close to
Tsirelson's bound. An illustrative comparison with the entangled state of two
$1/2$ spin particles in Quantum Mechanics is provided.
