# [particle physics] Higgs mechanism: spontaneous symmetry breaking & gauge invariance

1. May 26, 2012

### nonequilibrium

1. The problem statement, all variables and given/known data
Suppose that there is a gauge group with 24 indepenent symmetries and we find a set of 20 real scalar fields such that the scalar potential has minima that are invariant under only 8 of these symmetries. Using the Brout-Englert-Higss mechanism, how many physical fields are there that are
- massive spin 1
- massless spin 1
- Goldstone scalars
- Higgs scalars

2. Relevant equations
N.A.

3. The attempt at a solution
I'm not sure since I only saw an example worked out where there was a gauge "group" with 1 symmetry and we had 2 real scalar fields and the 1 symmetry was broken, and it ended up giving one massive spin 1, zero massless spin 1, zero Goldstone scalars and 1 Higgs scalar. I'm not sure how to generalize this to a more general case.

But let's give it a try: if we assume that for each broken symmetry, a gauge boson gets mass, we end up with "16 massive spin 1" (since 24 - 8 symmetries are broken). Hence "8 massless spin 1" remain. If I now presume that each gauge boson getting a mass is accompanied with the eating of one Goldstone scalar (which seems sensible from the perspective of the gauge boson gaining one degree of freedom), 16 Goldstone scalars have been eaten, and presuming that no (physical) Goldstone scalars can remain (?) (i.e. "0 Goldstone scalars"), we conclude that from the 20 real scalar fields, "4 Higgs scalars" survive.

Is the answer and/or some of the reasoning correct? Maybe I'm making it too complicated... (for reference we're using Griffiths' Introduction to Elementary Particles, although note that the question is not in the book itself).