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Massive Loop Amplitudes from Unitarity

Z. Bern, A. G. Morgan

TL;DR

This paper extends unitarity-based methods to massive one-loop gauge-theory amplitudes, showing that complete amplitudes can be constructed from tree-level cuts supplemented by known ultraviolet and infrared mass singularities. By performing cuts to all orders in ε within the FDH scheme and using dimensional regularization, the authors fix cut-free rational terms and derive explicit four-gluon amplitudes with massive fermion or scalar loops, including new results for the five-gluon case with identical helicities. The work demonstrates a SUSY-like decomposition between fermion and scalar contributions, confirms consistency with massless limits, and outlines how these techniques generalize to higher-point amplitudes and potentially two-loop calculations. Overall, the approach provides a compact, on-shell framework for massive-loop amplitudes with practical implications for heavy-quark thresholds and precision QCD predictions.

Abstract

We show, for previously uncalculated examples containing a uniform mass in the loop, that it is possible to obtain complete massive one-loop gauge theory amplitudes solely from unitarity and known ultraviolet or infrared mass singularities. In particular, we calculate four-gluon scattering via massive quark loops in QCD. The contribution of a heavy quark to five-gluon scattering with identical helicities is also presented.

Massive Loop Amplitudes from Unitarity

TL;DR

This paper extends unitarity-based methods to massive one-loop gauge-theory amplitudes, showing that complete amplitudes can be constructed from tree-level cuts supplemented by known ultraviolet and infrared mass singularities. By performing cuts to all orders in ε within the FDH scheme and using dimensional regularization, the authors fix cut-free rational terms and derive explicit four-gluon amplitudes with massive fermion or scalar loops, including new results for the five-gluon case with identical helicities. The work demonstrates a SUSY-like decomposition between fermion and scalar contributions, confirms consistency with massless limits, and outlines how these techniques generalize to higher-point amplitudes and potentially two-loop calculations. Overall, the approach provides a compact, on-shell framework for massive-loop amplitudes with practical implications for heavy-quark thresholds and precision QCD predictions.

Abstract

We show, for previously uncalculated examples containing a uniform mass in the loop, that it is possible to obtain complete massive one-loop gauge theory amplitudes solely from unitarity and known ultraviolet or infrared mass singularities. In particular, we calculate four-gluon scattering via massive quark loops in QCD. The contribution of a heavy quark to five-gluon scattering with identical helicities is also presented.

Paper Structure

This paper contains 18 sections, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of previous techniques
  3. Naive application of Cutkosky rules
  4. Fixing the ambiguity
  5. Cuts with dimensional regularization
  6. Fixing the remaining ambiguity
  7. Fermion loop amplitudes
  8. The $A_4(1^+, 2^+, 3^+, 4^+)$ amplitude
  9. The $A_4(1^-,2^-,3^+, 4^+)$ amplitude
  10. Remaining amplitudes
  11. Summary and discussion
  12. Integrals
  13. The $D=4-2\epsilon$ scalar integral functions
  14. The higher dimension integrals
  15. Feynman parameter reduction
  16. ...and 3 more sections

Figures (2)

  • Figure 1: The $s$- and $t$-channel cuts of a one-loop four-gluon amplitude. The cut lines represent either a fermion or scalar.
  • Figure 2: Examples of labels on integral functions with four-point kinematics. The triangle $I_3^{(2)}$ in (b) is obtained by removing the internal propagator between external legs 1 and 2 in (a). Similarly, the bubble $I_2^{(2,4)}$ in (c) is obtained by removing the internal propagator between legs 3 and 4 in (b).