  We propose a system of coupled, real-valued, effective Langevin equations for
the nonequilibrium phase transition exhibited by the pair contact process with
diffusion (and similar triplet and quadruplet, n-uplet, processes). A
combination of analytical and numerical results demonstrate that these
equations account for all known phenomenology in all physical dimensions,
including estimates of critical exponents in agreement with those reported for
the best-behaved microscopic models. We show in particular that the upper
critical dimension of these n-uplet transitions is 4/n, and 4/n-1 for their
anisotropic (biased) versions.
