In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously symmetry breaking(SSB), some people saying that Higgs mechanism is the results of SSB of local gauge symmetry, some people says that we can formulate Higgs mechanism in a gauge invariant way, some people also says that we need only a non-zero vaccum expectation value... I am confused about this different or maybe same point of views.
In this post:How does the Higgs mechanism work? , the most highly voted answer, I still can't feel how SSB worked in Higgs mechanism. It seem that the validity of last part, the appearance of a mass term for $A$, is guaranteed if we have a non-zero equilibrium value $\phi_0$ to expand around. I do not see that the requirement that the phase of the field $\phi$ need to be fixed at some particular value to generate mass term. Thus it seems to me it is not true that SSB is really indispensable for Higgs mechanism.
Put it straight forwardly:
The spontanously breaking of what is attributed to Higgs mechanism?
local gauge symmetry
global symmetry, since breaking of a "gauge symmetry" should not have any effect on physics.
other.
Is SSB really indispensable for Higgs mechanism?
yes, Higgs mechanism is relied on the SSB of some symmetry(above question), the other approches of description eventually has spontanously broke some symmetry.
No, the SSB is just one way to describe Higgs mechanism(or even not a complete way), what is really need is the non-zero vaccum expectation value, for example in the linked post the requirement for the mass term to occur is to have some non-zero expectation value of $\phi$ to expand around, we do not need the phase of the field to be fixed, thus the symmetry is not broken.
Ohter.