Quantum noncommutative ABJM field theory: four- and six-point functions

Following our previous paper Quantum noncommutative ABJM theory: first steps, JHEP {\bf 1804} (2018) 070), in this article we investigate one-loop 1PI four-, and six-point functions by using the component formalism in the Landau gauge and show that they are UV finite and have well-defined $(θ^{μν}\rightarrow 0)$ limit. Those results also hold for all one-loop functions which are UV finite by power counting. In summary, taking into account results from previous paper, JHEP {\bf 1804} (2018) 070), and this paper, we conclude that, at least at one-loop order, the NCABJM theory is free from the noncommutative UV and IR instabilities, and that in the limit $θ^{μν}\rightarrow 0$ it flows to the ordinary ABJM theory.

💡 Research Summary

In this paper the authors extend their previous work on the non‑commutative (NC) version of the three‑dimensional ABJM theory by performing a complete one‑loop analysis of the four‑point and six‑point one‑particle‑irreducible (1PI) Green functions. The model under study is the NC U(N)ₖ × U(N)₋ₖ Chern‑Simons‑matter theory defined on a Moyal‑deformed spacetime, where the non‑commutativity is encoded in the antisymmetric matrix θ^{μν} and appears through the usual star‑product phase factors e^{ip∧q}. The authors work in component form rather than superfield formalism and fix the gauge to the Landau gauge (ξ = 0), which simplifies the Chern‑Simons propagator and eliminates spurious infrared (IR) divergences that would otherwise arise from gauge‑fixing terms.

The paper begins with a concise review of the motivation for studying NC gauge theories, emphasizing the notorious UV/IR mixing that plagues many four‑dimensional NC Yang‑Mills models. In three dimensions, however, supersymmetry can dramatically improve the situation. The authors recall that in their earlier work they demonstrated UV finiteness and a smooth commutative limit for the two‑ and three‑point functions. The present work asks whether the same properties persist for higher‑point functions, which involve a richer set of diagrams and more intricate momentum routing.

Section 2 presents the full action, including the Chern‑Simons term, kinetic terms for scalars X^{A} and fermions Ψ^{A}, and the quartic (S₄) and sextic (S₆) interaction terms required by N = 6 supersymmetry. The non‑commutative deformation is introduced by replacing ordinary products with Moyal star products, leading to momentum‑space vertices multiplied by phase factors of the form e^{i p_i∧p_j}. Gauge‑fixing and ghost terms appropriate to the Landau gauge are added, and the complete set of Feynman rules is listed in an appendix.

Section 3 lists the free propagators: the Chern‑Simons gauge propagator is proportional to ε^{μνρ}p_ρ/p², the scalar and fermion propagators are the usual 1/p² and i p·γ/p², respectively, and the ghost propagators are also 1/p². Because ξ = 0, no longitudinal pieces survive, which prevents the appearance of dangerous IR singularities.

The core of the paper is Sections 4 and 5, where the authors compute all one‑loop four‑point scalar Green functions. The diagrams fall into three topological classes: box, triangle, and bubble. Each class contains several diagrams differing by the ordering of external legs and by the placement of the non‑commutative phase factors. The authors employ the symbolic manipulation program FORM to generate the numerators, which are polynomials of at most second order in the loop momentum ℓ. Power‑counting then guarantees that each individual diagram is UV finite. The non‑commutative phases, however, could in principle generate terms that fail to have a well‑defined θ → 0 limit. To address this, the authors invoke Lebesgue’s dominated convergence theorem: they show that the integrands are uniformly bounded by an integrable function independent of θ, allowing the limit to be taken under the integral sign. Explicitly, they demonstrate that the sum over all diagrams leads to cancellations of the θ‑dependent oscillatory pieces, leaving a result that coincides exactly with the ordinary (commutative) ABJM four‑point function.

Section 6 repeats the analysis for the six‑point scalar functions, which arise from the sextic interaction S₆. Although the number of diagrams proliferates, the same strategy applies: the numerators remain at most quadratic in ℓ, the phase factors combine in such a way that the potentially dangerous contributions cancel, and the Lebesgue theorem ensures a smooth commutative limit. The final six‑point amplitude is again UV finite and matches the known result in the undeformed theory.

In the discussion (Section 8), the authors synthesize their findings: at one loop, every UV‑finite 1PI Green function of the NC ABJM theory possesses a well‑defined θ → 0 limit, and that limit reproduces the corresponding function of the ordinary ABJM model. Consequently, the NC deformation does not introduce any new UV divergences nor the characteristic UV/IR mixing that afflicts non‑supersymmetric NC gauge theories. The result strongly suggests that the high degree of supersymmetry (N = 6) protects the theory from the pathological IR behavior, at least perturbatively at one loop. The paper concludes by outlining possible extensions, such as higher‑loop calculations, the inclusion of non‑planar contributions beyond leading order, and the exploration of non‑perturbative aspects of the NC ABJM model. Overall, the work provides a solid perturbative foundation for the consistency of non‑commutative ABJM theory and reinforces its relevance as a laboratory for studying quantum gravity effects, holography, and condensed‑matter applications in a controlled supersymmetric setting.