Fractional Soft Limits of Scattering Amplitudes

It is a common lore that the amplitude for a scattering process involving one soft Nambu-Goldstone boson should scale like an integer power of the soft momentum. We revisit this expectation by considering the 2→2 scattering of phonons in solids. We show that, depending on the helicities of the phonons involved in the scattering process, the scattering amplitude may in fact vanish like a fractional power of the soft momentum. This is a peculiarity of the 4-point amplitude, which can be traced back to (1) the (spontaneous or explicit) breaking of Lorentz invariance, and (2) the approximately collinear kinematics arising when one of the phonons becomes soft. Our results extend to the general class of nonrelativistic shift-invariant theories of a vector field.