  Melonic field theories are defined over the $p$-adic numbers with the help of
a sign character. Our construction works over the reals as well as the
$p$-adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky
models as special cases; depending on the sign character, the symmetry group of
the field theory can be either orthogonal or symplectic. Analysis of the
Schwinger-Dyson equation for the two-point function in the leading melonic
limit shows that power law scaling behavior in the infrared arises for
fermionic theories when the sign character is non-trivial, and for bosonic
theories when the sign character is trivial. In certain cases, the
Schwinger-Dyson equation can be solved exactly using a quartic polynomial
equation, and the solution interpolates between the ultraviolet scaling
controlled by the spectral parameter and the universal infrared scaling. As a
by-product of our analysis, we see that melonic field theories defined over the
real numbers can be modified by replacing the time derivative by a bilocal
kinetic term with a continuously variable spectral parameter. The infrared
scaling of the resulting two-point function is universal, independent of the
spectral parameter of the ultraviolet theory.

High Energy Physics - Theory