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We extend a known mass-gap equation for pure gluodynamics in global colourmodels (formulated in equal time quantization in Coulomb gauge) to one in whichgluons 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 (assuggested by Grand Unification arguments at large scales) it is immediate tosee that different masses are generated for each subgroup. This global symmetryis not broken, but the split masses erase accidental symmetries that might bepresent 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 oflow-dimensional examples of simple Lie groups, but in spite of the systemhaving a form that would seem to allow spontaneous symmetry breaking, it is nottriggered for these groups whose algebra has no ideal, and the dispersionrelations for the various gluons converge to the same form.

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