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The Noncommutative S-Matrix

Suvrat Raju

Abstract

As a simple example of how recently developed on-shell techniques apply to nonlocal theories, we study the S-matrix of noncommutative gauge theories. In the complex plane, this S-matrix has essential singularities that signal the nonlocal behavior of the theory. In spite of this, we show that tree-level amplitudes may be obtained by BCFW type recursion relations. At one loop we find a complete basis of master integrals (this basis is larger than the corresponding basis in the ordinary theory). Any one-loop noncommutative amplitude may be written as a linear combination of these integrals with coefficients that we relate to products of tree amplitudes. We show that the noncommutative N=4 SYM theory has a structurally simple S-matrix, just like the ordinary N=4 SYM theory.

The Noncommutative S-Matrix

Abstract

As a simple example of how recently developed on-shell techniques apply to nonlocal theories, we study the S-matrix of noncommutative gauge theories. In the complex plane, this S-matrix has essential singularities that signal the nonlocal behavior of the theory. In spite of this, we show that tree-level amplitudes may be obtained by BCFW type recursion relations. At one loop we find a complete basis of master integrals (this basis is larger than the corresponding basis in the ordinary theory). Any one-loop noncommutative amplitude may be written as a linear combination of these integrals with coefficients that we relate to products of tree amplitudes. We show that the noncommutative N=4 SYM theory has a structurally simple S-matrix, just like the ordinary N=4 SYM theory.

Paper Structure

This paper contains 39 sections, 106 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. On-Shell Methods for Noncommutative Tree Amplitudes
  3. Review
  4. BCFW extension
  5. BCFW Analysis for Noncommutative Amplitudes
  6. One-Loop Analysis
  7. Review
  8. One Loop Analysis in Ordinary Theories
  9. Noncommutative Non-Planar Amplitudes
  10. A Basis of Master Integrals
  11. Obtaining the Coefficients
  12. Box Coefficients
  13. Triangle Coefficients
  14. Bubble Coefficients
  15. The Remainder
  16. ...and 24 more sections

Figures (6)

  • Figure 1: Three Point Vertex
  • Figure 2: Non-Planar Diagram
  • Figure 3: Box: 4-pt YM amplitude
  • Figure 4: Triangle 1: 4-pt YM amplitude
  • Figure 5: Triangle 2: 4-pt YM amplitude
  • ...and 1 more figures