In a 1974 Pati and Salam published the paper "Lepton Number as the Fourth Color", which suggested the gauge group $\rm SU(4)\times SU(2)_L\times SU(2)_R$ could be the fundamental symmetry group, rather than $\rm SU(3)\times SU(2)\times U(1)$.
An important property of any such candidate is that it can reproduce the standard model particle spectrum, and this (as far as I know) occurs through spontaneous symmetry breaking, where you assume a non zero VEV for your potential.
In page 12 of 'A critical study of two Pati-Salam models' (A Barbensi, PhD Thesis, Universita' Di Roma 3, 2019) and in page 7 of arXiv:hep-th/9502222 we are given the most general scalar potential terms, and those include mixing of the two higgs fields.
My goal was the following: I was trying to reproduce the correct particle spectrum for the two spontaneous symmetry breaking patterns $$ \rm SU(4)\times SU(2)_L \times SU(2)_R \mapsto SU(3) \times SU(2)_L \times U(1)_Y \mapsto SU(3) \times U(1)_Q. $$ The initial symmetry breaking pattern can be correctly obtained by looking only at the right-handed Higgs field, responsible for the first breaking, and for that I didn't need the mixing terms or the left handed Higgs field terms in the potentials given above. The result was that, centred around the non zero VEV, only 7 massive Higgs particles remained, which meant we had 9 Goldstone bosons and thus 9 broken generators, as expected.
However, I am struggling to repeat this process now for the second symmetry breaking, as it seems that some mixing terms are necessary to obtain the correct broken potential. I expect three generators to be broken for the second Higgs field, so as to give the correct symmetry breaking pattern.
My questions are the following: What is the simplest potential that could reproduce the second symmetry breaking? Are the mixing terms indeed necessary? If so, wouldn't the initial symmetry breaking pattern be 'disturbed' by these new terms? And finally, how does one read off the scalar particle spectrum of a lagrangian, when one encounters mixing 2nd order terms i.e. terms which involve the product of the left and right Higgs fields?
