Main Reference Zee (Quantum Mechanics in a Nutshell).

1) Global symmetry

A global symmetry means that the Lagrangian is invariant by a transformation whose parameters are constant.

For a continuous global symmetry, if the symmetry of the Lagrangian is the group $G$, and if the symmetry of the vacuum is the group $H$, a subgroup of $G$, you have ($dim G-dim H$) Goldstone bosons.

For instance, take a complex scalar field $\Phi$ with a Mexican hat potential , so that the total Lagrangian density is $L = \partial \phi^\dagger \partial \phi + \mu^2 \phi^\dagger \phi - \lambda (\phi^\dagger \phi)^2$.

The group symmetry is here $G=O(2)$

Define $\phi = \rho e^{i\theta}$

Breaking the symmetry means choosing for the vacuum the minima for the potential, and a particular angle, that is :

$\rho_V = v, \theta_V = \theta_0$

The group $H$ is trivial here.

Define :
$\rho = v + \chi$, where $v = \sqrt{\frac{\mu^2}{2\lambda}}$

Developping the Lagrangian, you get a term $v^2(\partial \theta)^2$, which is the dynamical part for a massless field $\theta$, so $\theta$ is our Goldstone boson (There is one because $dim G - dim H = 1 - 0 = 1 $).

So, we see, that spontaneous symmetry breaking could arise in a global continuous symmetry.

2) Local symmetry

A local symmetry means that the Lagrangian is invariant by a transformation whose parameters are functions of space-time.

"Gauging" means (continous) local symmetry.
So you don't need "gauging" to have a spontaneous symmetry breaking.

With a local symmetry, some of the Goldstone Bosons are "eaten" by the Gauge field ($A_\mu$), so that these gauge fields (which are massless) become massive. In a 4d space-time dimension, a massless Gauge field has $2$ degrees of freedom, while a massive gauge field has $3$ degrees of freedom. To do that, the Gauge field has to "eat" one degree of freedom (one Goldstone boson)

3) Global symmetry as a special case of Local Symmetry

In the set of local symmetry, global symmetry is a very special case (a very special subset), where transformation parameters are constant. So, if you want, you can consider that global symmetry are "included" into local symmetry.

This post imported from StackExchange Physics at 2014-05-21 15:21 (UCT), posted by SE-user Trimok