In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneous  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: https://physics.stackexchange.com/q/17944/ , the most highly voted answer, I still can't feel how SSB worked in Higgs mechanism. It seems that the validity of the 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.

To put it simply:

**The spontaneous breaking of what is attributed to Higgs mechanism?**

1. local gauge symmetry

2. global symmetry, since breaking of a "gauge symmetry" should not have any effect on physics. In Higgs mechanism, the really broken symmetry is a global one. Mathematically, it is similar in looking as fixing a gauge, but one should not think of it as a spontaneous breakdown of local gauge symmetry.

3. other.


**Is SSB really indispensable for Higgs mechanism?** 

1. yes, Higgs mechanism is relied on the SSB of some symmetry (above question), the other approaches of description eventually has spontaneously broke some symmetry.

2. 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 vacuum 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.

3. Other.

# **some reference materials**:

1.  https://physics.stackexchange.com/questions/177703/is-elitzurs-theorem-valid-only-in-lattice-field-theory States that SSB of local gauge symmetry is impossible.

2.  [Gauge invariant accounts of the Higgs mechanism](http://arxiv.org/pdf/1102.0468.pdf) in the abstract states that: 

> gauge symmetries merely reflect a redundancy in the state description
> and therefore the spontaneous breaking can not be an essential
> ingredient. Indeed, as already shown by Higgs and Kibble, the
> mechanism can be explained in terms of gauge invariant variables,
> without invoking spontaneous symmetry breaking

3.  [Is electromagnetic gauge invariance spontaneously violated in superconductors?](http://arxiv.org/abs/cond-mat/0503400) In the introduction it says:

>  In particular, we emphasize that global U(1) phase rotation symmetry,
> and not gauge symmetry, is spontaneously violated, and show that the
> BCS wave function is, contrary to claims in the literature, fully
> gauge invariant