UV-IR mixing in nonassociative Snyder ϕ4 theory

Abstract

Using a quantization of the nonassociative and noncommutative Snyder ϕ 4 scalar field theory in a Hermitian realization, we present in this article analytical formulas for the momentum-conserving part of the one-loop two-point function of this theory in D-, 4-, and 3-dimensional Euclidean spaces, which are exact with respect to the noncommutative deformation parameter β. We prove that these integrals are regularized by the Snyder deformation. These results indicate that the Snyder deformation does partially regularize the UV divergences of the undeformed theory, as it was proposed decades ago. Furthermore, it is observed that different nonassociative ϕ 4 products can generate different momentum-conserving integrals. Finally, most importantly, a logarithmic infrared divergence emerges in one of these interaction terms. We then analyze sample momentum nonconserving integral qualitatively and show that it could exhibit IR divergence too. Therefore, infrared divergences should exist, in general, in the Snyder ϕ 4 theory. We consider infrared divergences at the limit p → 0 as UV/IR mixings induced by nonassociativity, since they are associated to the matching UV divergence in the zero-momentum limit and appear in specific types of nonassociative ϕ 4 products. We also discuss the extrapolation of the Snyder deformation parameter β to negative values as well as certain general properties of one-loop quantum corrections in Snyder ϕ 4 theory at the zero-momentum limit.

Twist for Snyder space

“…In this way, the free scalar field theory in the Hermitian realization (3) has been developed in [43], where it was shown that it is equivalent to the standard commutative theory. The interacting φ 4 theory has been investigated in a linear approximation in β in [44] and in the full β dependence in [63]. Of course, the interacting theory is no longer equivalent to its commutative counterpart.…”

“…This fact makes the computation of these diagrams particularly difficult. A partial evaluation has been performed in [63], where it has been shown that the Snyder star product can introduce a natural regularization on at least some of the one-loop diagrams of the two-point functions. However, in the limit of vanishing incoming momenta, a variant of the UV/IR mixing of noncommutative field theory [7,8,64] appears.…”

The Snyder Model and Quantum Field Theory

“…However, in spite of several attempts using various methods [14][15][16][17][18][19][20][21], the issue of finiteness of the Snyder field theory has not been established up to now. Here, we review an attempt to investigate this topic using the formalism of noncommutative QFT [22][23][24].…”

“…Attempting instead a calculation at all orders in , not all diagrams can be explicitly computed [24]. It can be shown, however, that the divergences are suppressed with respect to the noncommutative theory, and there are even indications that the integrals might be finite, at least for the interaction (34).…”

Inferring the covariant Θ-exact noncommutative coupling in the top quark pair production at linear colliders

“…The role of supersymmetric Yang-Mills theory in the noncommutative UV/IR mixing [17,18,[22][23][24] were extensively scrutinized to overcome the divergences which appeared in the field non-local interaction. Other class of NC theories namely, non-geometric theories like κ-Minkowski spacetime and Snyder spacetime also exhibits the UV/IR mixing at one-loop level [25,27,28]. In general, the UV/IR mixing is the universal property of all NCQFT.…”