  We consider the Lagrangian of gravity covariantly amended by the mass and
polynomial interaction terms with arbitrary coefficients, and reinvestigate the
consistency of such a theory in the decoupling limit, up to the fifth order in
the nonlinearities. We calculate explicitly the self-interactions of the
helicity-0 mode, as well as the nonlinear mixing between the helicity-0 and -2
modes. We show that ghost-like pathologies in these interactions disappear for
special choices of the polynomial interactions, and argue that this result
remains true to all orders in the decoupling limit. Moreover, we show that the
linear, and some of the nonlinear mixing terms between the helicity-0 and -2
modes can be absorbed by a local change of variables, which then naturally
generates the cubic, quartic, and quintic Galileon interactions, introduced in
a different context. We also point out that the mixing between the helicity-0
and 2 modes can be at most quartic in the decoupling limit. Finally, we discuss
the implications of our findings for the consistency of the effective field
theory away from the decoupling limit, and for the Boulware-Deser problem.
