Split gluon masses in $SU(N)\times SU(M)$ theories

TL;DR

This paper extends gluodynamics models to include different gluon masses in $SU(N)\times SU(M)$ theories, showing that mass splitting occurs without symmetry breaking and analyzing specific low-dimensional examples.

Contribution

It introduces a mass-gap equation for split gluon masses in $SU(N)\times SU(M)$ theories and explores their implications and numerical behavior.

Findings

Different gluon masses are generated for each subgroup.

Mass splitting erases accidental symmetries without breaking the global symmetry.

Numerical analysis shows no spontaneous symmetry breaking in studied examples.

Abstract

We extend a known mass-gap equation for pure gluodynamics in global colour models (formulated in equal time quantization in Coulomb gauge) to one in which gluons split into two sets which may have different masses. If the theory is $SU(N)\times SU(M)$ with gluons in both groups having identical couplings (as suggested by Grand Unification arguments at large scales) it is immediate to see that different masses are generated for each subgroup. This global symmetry is not broken, but the split masses erase accidental symmetries that might be present due to the two couplings being the same at the large scale, such as $SU(N\times M)$ or similar. We also numerically explore a couple of low-dimensional examples of simple Lie groups, but in spite of the system having a form that would seem to allow spontaneous symmetry breaking, it is not triggered for these groups whose algebra has no ideal, and the dispersion relations for the various gluons converge to the same form.