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

Carmelo P. Martin, Josip Trampetic, Jiangyang You

Abstract

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.

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

Abstract

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 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 it flows to the ordinary ABJM theory.

Paper Structure

This paper contains 51 sections, 147 equations, 16 figures.

Table of Contents

  1. Introduction
  2. The $\rm U(N)_{\kappa}\times U(N)_{-\kappa}$ quantum field theory action
  3. Free field propagators from the action
  4. Scalar field four-point functions
  5. Box diagram contributions to the 4 scalar field 4-point functions
  6. Triangle diagram contributions to the 4 scalar field 4-point functions
  7. Bubble diagram contributions to the 4 scalar field 4-point functions
  8. Non-planar $(\cal NP)$ bubble + tadpole contributions and $\theta^{\mu\nu}$ limits
  9. $\cal (NP)$ bubble diagram contributions to the 4 scalar field 4-point functions
  10. $\cal (NP)$ tadpole diagram contribution to the 4 scalar field 4-point functions
  11. The $\theta^{\mu\nu}\to 0$ limits for the $\cal (NP)$ part of bubble plus tadpole diagrams
  12. Scalar-guage four-point functions
  13. Scalar-scalar-gauge-hgauge 4-point function: $<XXA\hat{A}>$
  14. Scalar-scalar-(h)gauge-(h)gauge 4-point functions: $<XXAA>$, $<XX\hat{A}\hat{A}>$
  15. Scalar-fermion four-point functions: $<XX\bar{\Psi}\Psi >$,$<XX\Psi\Psi >$,$<XX\bar{\Psi}\bar{\Psi} >$
  16. ...and 36 more sections

Figures (16)

  • Figure 1: Notations and the propagators of the relevant fields. Separate arrows in (h)gauge fields indicate the momentum flow.
  • Figure 2: Scalar field 4-correlator with (h)gauge-scalar box-loop diagrams $S^{box}_{1,2,3,4}$ defined in (\ref{['box1']})-(\ref{['N4']}). Arrows on diagram lines show the flow of charge, while separate arrows indicate the flow of incoming momenta on all four diagrams.
  • Figure 3: Scalar field 4-correlator with (h)guage-scalar box-loop diagrams $S^{box}_{5,6,7,8}$ defined in (\ref{['box5']})-(\ref{['N8']}). Arrows on diagram lines indicate the flow of charge, while separate arrows show the flow of all incoming momenta.
  • Figure 4: Scalar field 4-correlator with triangle-(h)gauge loop diagrams $S^{tri}_{1,2,3,4}$. Arrows on diagram lines show the flow of charge, while separate arrows indicate the flow of incoming momenta.
  • Figure 5: Scalar field 4-correlator with gauge and fermion bubble-loops diagrams: $S^{bub}_{1,2,3}$, and $S^{bub}_{4,5}$, respectively. Here arrows on the diagram lines indicate the flow of charge, while separate arrows indicate the flow of incoming momenta on all five diagrams. In $S_2^{bub},S_3^{bub},S_4^{bub}$ diagrams, momenta are denoted in the same way as in $S_1^{bub}$, while for the $S_5^{bub}$ loop we have the following two propagators momenta: $\ell$, and $-(\ell+p_2+p_3)$, respectively.
  • ...and 11 more figures