On $p$-form theories with gauge invariant second order field equations

We explore field theories of a single $p$-form with equations of motions of order strictly equal to 2 and gauge invariance. We give a general method for the classification of such theories which are extensions to the $p$-forms of the Galileon models for scalars. Our classification scheme allows us to compute an upper bound on the number of different such theories depending on $p$ and on the space-time dimension. We are also able to build a nontrivial Galileon-like theory for a 3-form with gauge invariance and an action which is polynomial into the derivatives of the form. This theory has gauge invariant field equations but an action which is not, like a Chern-Simons theory. Hence the recently discovered no-go theorem stating that there are no nontrivial gauge invariant vector Galileons (which we are also able here to confirm with our method) does not extend to other odd-$p$ cases.