Abstract
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.
Highlights
A first working definition of a p-adic quantum field theory is a theory defined through a functional integral over maps φ∶ Qp → R, where Qp denotes the p-adic numbers and R denotes the reals.1 We may expand our definition of p-adic quantum field theories by replacing Qp with a field extension of Qp, and by allowing φ to be valued in some vector space over R
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
We instead add a relevant deformation to the free bilocal theory, and under appropriate conditions we find that the two-point function shows universal infrared behavior
Summary
A first working definition of a p-adic quantum field theory is a theory defined through a functional integral over maps φ∶ Qp → R, where Qp denotes the p-adic numbers and R denotes the reals. We may expand our definition of p-adic quantum field theories by replacing Qp with a field extension of Qp, and by allowing φ to be valued in some vector space over R. Progress has been made in understanding a p-adic version of the anti-de Sitter/conformal field theory correspondence (AdS=CFT) [5,6]. (iii) For some sign characters, in order to get a theory with well-defined scaling behavior both in the ultraviolet and in the infrared, we are obliged to alter the global symmetry group of the theory from OðNÞ3 to SpðNÞ3, where SpðNÞ is the noncompact group of real-valued N × N matrices preserving a symplectic structure. Readers wishing to see a tabulation of our main results can consult Table I, in which we indicate for each field (R or Qp) and each sign character (parametrized by τ) what sort of theory we must consider in order to have a renormalization group flow from a free ultraviolet theory to the universal infrared scaling behavior of the two-point function characteristic of SYK-type theories
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