Dimensions of a probability amplitude (matrix element)?
Greetings. I'm a little bit confused about the dimensions of a probability amplitude in a QFT calculation. My understanding a the squared, spin averaged\summed Feynman diagram should be dimensionless. However, if we consider a decay process, say pair creation from a photon or the decay of the Higgs to two fermions, |M|^2 is dimensionful.
Quick check: the coupling constant of such a decay is dimensionless. There is no dimensionful factor for the initial state (scalar Higgs or photon). The matrix element is proportional to the external fermion factors. When these get spin averaged/summed we get traces of two quantities with dimensions of mass. Hence the matrix element squared has dimensions of mass squared.
Am I losing my mind?