Suppose I study a quantum field theory in which among other fields a shift symmetric scalar field appear: $$\phi\rightarrow\phi+c$$ with $c$ a real constant.
Can this always be interpreted as a Nambu-Goldstone boson for some spontaneous symmetry breaking?
I would say that shift symmetry is a necessary condition to have the non linear realization of the spontaneously broken group: it corresponds to the case in which the parameter of the transformation is taken to be a constant and therefore the gauge fields do not vary, but I am not a lot sure this is exactly how it works, if it works.
Is there any weaker or stronger statement that can be generally made about shift symmetric scalars in order to connect their symmetry to larger symmetries of the theory?