Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Trees to Loops and Back

Andreas Brandhuber, Bill Spence, Gabriele Travaglini

TL;DR

The paper tackles whether generic one-loop amplitudes in supersymmetric Yang–Mills theories can be computed using MHV diagrams in place of conventional Feynman diagrams. It proves covariance of one-loop MHV amplitudes via the Feynman Tree Theorem, shows the correct discontinuities and poles match those from standard methods, and derives universal one-loop collinear splitting functions to all orders in ε. It also analyzes soft limits and provides concrete applications, including a re-derivation of the one-loop integration measure and bubble calculations using the tree theorem. Together, these results provide strong evidence for the equivalence of the MHV-diagram and Feynman-diagram approaches at one loop and suggest avenues for extending the method beyond one loop.

Abstract

We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams. This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. Finally, we present concrete and illustrative applications of Feynman's Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams.

From Trees to Loops and Back

TL;DR

The paper tackles whether generic one-loop amplitudes in supersymmetric Yang–Mills theories can be computed using MHV diagrams in place of conventional Feynman diagrams. It proves covariance of one-loop MHV amplitudes via the Feynman Tree Theorem, shows the correct discontinuities and poles match those from standard methods, and derives universal one-loop collinear splitting functions to all orders in ε. It also analyzes soft limits and provides concrete applications, including a re-derivation of the one-loop integration measure and bubble calculations using the tree theorem. Together, these results provide strong evidence for the equivalence of the MHV-diagram and Feynman-diagram approaches at one loop and suggest avenues for extending the method beyond one loop.

Abstract

We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of the covariance of one-loop non-MHV amplitudes obtained from MHV diagrams. This proof relies only on the local character in Minkowski space of MHV vertices and on an application of the Feynman Tree Theorem. We then show that the discontinuities of one-loop scattering amplitudes computed with MHV diagrams are precisely the same as those computed with standard methods. Furthermore, we analyse collinear limits and soft limits of generic non-MHV amplitudes in supersymmetric Yang-Mills theories with one-loop MHV diagrams. In particular, we find a simple explicit derivation of the universal one-loop splitting functions in supersymmetric Yang-Mills theories to all orders in the dimensional regularisation parameter, which is in complete agreement with known results. Finally, we present concrete and illustrative applications of Feynman's Tree Theorem to one-loop MHV diagrams as well as to one-loop Feynman diagrams.

Paper Structure

This paper contains 24 sections, 121 equations, 17 figures.

Table of Contents

  1. Introduction
  2. The Feynman Tree Theorem
  3. Covariance of the amplitudes at one loop
  4. One-loop MHV amplitude
  5. The NMHV amplitude at one loop
  6. The ${\bf N}^2$MHV amplitude at one loop
  7. General Structure of MHV diagrams at one loop
  8. Proof of universal collinear factorisation in the MHV diagrams method
  9. Collinear limits in the MHV diagrams approach at tree level
  10. "Singular channel" and "non-singular channel" MHV diagrams
  11. $++ \to +$ collinear limit at one loop
  12. $-- \to -$ collinear limit at one loop
  13. Soft limits of amplitudes with MHV diagrams
  14. Tree level
  15. One loop
  16. ...and 9 more sections

Figures (17)

  • Figure 1: The MHV diagrams contributing to a one-loop MHV scattering amplitude. The blobs represent MHV vertices, which should then be dressed with external particles in all possible ways compatible with cyclic ordering, and in such a way that the two vertices have the MHV helicity configuration.
  • Figure 2: One-particle and two-particle MHV diagrams contributing to the one-loop MHV scattering amplitude.
  • Figure 3: In this Figure we represent "missing diagrams", mentioned in the text.
  • Figure 4: These are the MHV diagrams contributing to a one-loop NMHV scattering amplitude.
  • Figure 5: This Figure shows the one-particle cut diagrams generated by cutting open one loop propagator in the diagrams of Figure 4. Notice that the two diagrams on the right hand side are independent and should therefore be included separately.
  • ...and 12 more figures