Signs of the time: Melonic theories over diverse number systems
2017
We define melonic field theories 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.
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