When a continuous global symmetry is broken, we lose one degree of freedom.
We also gain a massless spin 0 boson, called a Goldstone boson.
We then say the number of degrees of freedom is the same.
Here is where I have a problem. Doesn't a spin 0 massless particle moving in d+1 dimensions have d degrees of freedom? (I'm fine with an answer in 3+1 dimensions if it makes things simpler, as that's mainly the case I deal with.)