Higher melonic theories

We classify a large set of melonic theories with arbitrary q-fold interactions, demonstrating that the interaction vertices exhibit a range of symmetries, always of the form ℤ₂ ⁿ for some n, which may be 0. The number of different theories proliferates quickly as q increases above 8 and is related to the problem of counting one-factorizations of complete graphs. The symmetries of the interaction vertex lead to an effective interaction strength that enters into the Schwinger-Dyson equation for the two-point function as well as the kernel used for constructing higher-point functions.