# Nonassociative Snyder phi4 Quantum Field Theory

## TL;DR

This paper develops a truncated nonassociative Snyder phi^4 quantum field theory, analyzing its one-loop correlation functions and revealing UV divergences and renormalizability issues at linear order in the deformation parameter.

## Contribution

It introduces a linear-order approximation of nonassociative Snyder phi^4 theory and studies its quantum properties using the functional method in momentum space.

## Key findings

- Two- and four-point functions exhibit UV divergences faster than in commutative theories.
- Six-point function shows logarithmic UV divergence, indicating nonrenormalizability at beta^1 order.
- Nonassociativity effects cancel out at this order due to permutation averaging.

## Abstract

In this article we define and quantize a truncated form of the nonassociative and noncommutative Snyder phi^4 field theory using the functional method in momentum space. More precisely, the action is approximated by expanding up to the linear order in the Snyder deformation parameter beta, producing an effective model on commutative spacetime for the computation of the two-, four- and six-point functions. The two- and four-point functions at one loop have the same structure as at the tree level, with UV divergences faster than in the commutative theory. The same behavior appears in the six-point function, with a logarithmic UV divergence and renders the theory unrenormalizable at beta^1 order except for the special choice of free parameters s_1=-s_2. We expect effects from nonassociativity on the correlation functions at beta^1 order, but these are cancelled due to the average over permutations.