Reparameterization Invariance in Heavy Particle Effective Field Theories

In a low energy effective theory, the fields of heavy particles with mass $m$ and total momentum $p$ depend on both a velocity $v$ and a residual momentum $k$ such that $p=mv+k$. However, there is some arbitrariness in such a description because one may also use a slightly different velocity with a compensating change in the residual momentum. This non-uniqueness is the origin of an invariance under such a reparameterization which imposes non-trivial constraints on the theory. The implications of this invariance for effective theories of heavy spin-0, 1/2, and 1 fields is investigated.