By considering the characteristic curves of shift-symmetric scalar field theories, caustics for SO(p) waves (on a flat spacetime background) are identified. Only theories which have a global galilean symmetry are free of such caustics.

Caustics are intersections of characteristic curves - physically this corresponds to two light rays crossing. In a relativistic field theory, at these points the first derivative of the field is not well-defined, and its second derivative diverges. This corresponds to a breakdown of the effective field theory, and is a clear indication that some `new physics' must enter in order to resolve these points. However, it is interesting to note that some effective theories (here the Galileon and DBI) are automatically free of caustics for all SO(p) waves on a flat spacetime background. (for arbitrary waves, 1602.00735 suggests that only the canonical free scalar is free of caustics)