@HyyFly: It would be nice to elaborate a bit on your question and in particular give some motivation. What applications do you have in mind for possible generalizations of the Hubbard-Stratonovich transformation? The latter is just the physics parlance for the formula for the Fourier transform of a Gaussian, *used in reverse*. In any case, if you want to entice people to write useful and thoughtful answers, how you write your question must show some effort.

As to the question itself:

The problem of figuring out when "simple functions" like the exponential of a quadratic has a simple Fourier transform was addressed in the article <a href="https://link.springer.com/article/10.1007%2Fs00029-002-8101-7">"When is the Fourier transform of an elementary function elementary?"</a> by Etingof, Kazhdan and Polishchuk in Selecta Math. 2002.

If all you want is a way of breaking quantum field theory vertices with $p$ legs into trivalent vertices, then one can do that by iterating the Hubbard-Stratonovich transformation (aka intermediate field representation). See, for instance, the article <a href="https://aip.scitation.org/doi/full/10.1063/1.3460320">"Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension"</a> by Rivasseau and Wang in J. Math. Phys. 2010. See in particular Sections III.A and IV.A.