Curing the UV/IR mixing for field theories with translation-invariant star products

The ultraviolet/infrared (UV/IR) mixing of noncommutative field theories has been recently shown to be a generic feature of translation-invariant associative products. In this paper we propose to take into account the quantum corrections of the model to modify in this way the noncommutative action. This idea was already used to cure the UV/IR mixing for theories on Moyal space. We show that in the present framework also, this proposal proves successful for curing the mixing. We achieve this task by explicit calculations of one and higher loops Feynman amplitudes. For the sake of completeness, we compute the form of the new action in the matrix base for the Wick-Voros product.

Figure 1

The planar tadpole graph.

Figure 2

The nonplanar tadpole graph.

Figure 3

Planar one-loop four-point graph of incoming momentum $K$.

Figure 4

A nonplanar one-loop four-points graph of total incoming momentum $K$.

Figure 5

A two-loop graph obtained by inserting into the bubble graph of Fig. 3 a nonplanar tadpole of momentum $p$.

Figure 6

A nonplanar graph obtained from the insertion into the bubble graph of Fig. 3 of a chain of nonplanar tadpoles of momenta $p_i$ ($i=1,\dots ,N$).

Figure 7

Insertion of a chain of the bubble graph into some bigger graph.