Worldline formalism in Snyder spaces

We study the ${\varphi }_{\star }^4$ model for a scalar field in a linearization of the Snyder model, using the methods of the worldline formalism. Our main result is a master equation for the 1-loop n-point function. From this we derive the renormalization of the coupling parameters of the theory and observe the appearance of a ${\varphi }^6$ divergent contribution that opens the question of whether this theory is renormalizable or not. Additionally, we observe that some terms in the renormalized action can be interpreted as coming from an effective metric proportional to the square of the field.