Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A unitarity compatible approach to one-loop amplitudes with massive fermions

Simon Badger, Christian Brønnum-Hansen, Francesco Buciuni, Donal O'Connell

TL;DR

This work develops a unitarity-compatible method to compute one-loop amplitudes with massive fermions by embedding mass in six dimensions and performing $d$-dimensional generalized cuts with the six-dimensional spinor-helicity formalism. Divergences from wavefunction renormalization are circumvented, and remaining ambiguities are fixed by matching universal infrared poles in $4-2\epsilon$ and ultraviolet poles in $6-2\epsilon$ using an effective six-dimensional QCD Lagrangian with a minimal set of higher-dimension counterterms. The method yields explicit procedures for extracting integral coefficients, includes a state-sum reduction to control extra polarizations, and provides concrete results for $gg\to t\bar t$ with a detailed Mathematica notebook. Overall, the approach offers a gauge-invariant, on-shell framework that avoids regulator-induced gauge issues and has potential for extension to more complex multi-fermion processes and higher loops.

Abstract

We explain how one-loop amplitudes with massive fermions can be computed using only on-shell information. We first use the spinor-helicity formalism in six dimensions to perform generalised unitarity cuts in $d$ dimensions. We then show that divergent wavefunction cuts can be avoided, and the remaining ambiguities in the renormalised amplitudes can be fixed, by matching to universal infrared poles in $4-2ε$ dimensions and ultraviolet poles in $6-2ε$ dimensions. In the latter case we construct an effective Lagrangian in six dimensions and reduce the additional constraint to an on-shell tree-level computation.

A unitarity compatible approach to one-loop amplitudes with massive fermions

TL;DR

This work develops a unitarity-compatible method to compute one-loop amplitudes with massive fermions by embedding mass in six dimensions and performing -dimensional generalized cuts with the six-dimensional spinor-helicity formalism. Divergences from wavefunction renormalization are circumvented, and remaining ambiguities are fixed by matching universal infrared poles in and ultraviolet poles in using an effective six-dimensional QCD Lagrangian with a minimal set of higher-dimension counterterms. The method yields explicit procedures for extracting integral coefficients, includes a state-sum reduction to control extra polarizations, and provides concrete results for with a detailed Mathematica notebook. Overall, the approach offers a gauge-invariant, on-shell framework that avoids regulator-induced gauge issues and has potential for extension to more complex multi-fermion processes and higher loops.

Abstract

We explain how one-loop amplitudes with massive fermions can be computed using only on-shell information. We first use the spinor-helicity formalism in six dimensions to perform generalised unitarity cuts in dimensions. We then show that divergent wavefunction cuts can be avoided, and the remaining ambiguities in the renormalised amplitudes can be fixed, by matching to universal infrared poles in dimensions and ultraviolet poles in dimensions. In the latter case we construct an effective Lagrangian in six dimensions and reduce the additional constraint to an on-shell tree-level computation.

Paper Structure

This paper contains 20 sections, 102 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of one-loop amplitudes and integrands
  3. Massive fermion spinors
  4. Massive fermions from massless six dimensional spinors
  5. Interactions and state-sum reduction
  6. A one-loop example calculation
  7. Generalised unitarity cuts in six dimensions
  8. Determining the remaining integral coefficients
  9. Fixing $c_{2,m^2}$ by matching the poles in $4-2\epsilon$ dimensions
  10. Counterterms for QCD in six dimensions
  11. Fixing $c_1$ by matching poles in $6-2\epsilon$ dimensions
  12. Conclusions
  13. Conventions and spinor construction
  14. Tree-level amplitudes in six dimensions
  15. Three-point amplitudes
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Decomposing the tree amplitude appearing on the left hand side of a wavefunction cut reveals a divergent graph.
  • Figure 2: Feynman diagram for one-loop contribution to the coupling between a massive fermion pair and an off-shell scalar. All external momenta are outgoing.
  • Figure 3: Configurations for left- and right-moving primitive amplitudes contributing to $gg\to t{\bar{t}}$ scattering.
  • Figure 4: The complete set of cuts for $B^{[L]}\left( 1_{t},2,3,4_{{\bar{t}}}\right)$. Double lines represent massive fermions.
  • Figure 5: The complete set of cuts for $B^{[R]}\left( 1_{t},2,3,4_{{\bar{t}}}\right)$. Double lines represent massive fermions.
  • ...and 2 more figures