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New relations for scattering amplitudes in Yang-Mills theory at loop level

Rutger H. Boels, Reinke Sven Isermann

TL;DR

The paper develops a framework to derive relations among one-loop Yang-Mills amplitudes by analyzing the scaling of permutation and cyclic sums under BCFW-type shifts. It presents concrete decoupling relations for rational terms and for triangle and bubble terms, showing that many coefficients vanish under symmetry sums and reducing the independent coefficient space; it also introduces BCJ-like relations for the one-loop integrand and a loop-momentum convention that reduces the integrand basis from $(n-1)!/2$ to $(n-2)!$. The results hold across helicities and matter content, with numerical checks up to high point counts and potential extensions to QCD and higher loops. These findings imply a surprisingly simple structure for loop amplitudes in Yang-Mills, promising more efficient computations and deeper theoretical control.

Abstract

The calculation of scattering amplitudes in Yang-Mills theory at loop level is important for the analysis of background processes at particle colliders as well as our understanding of perturbation theory at the quantum level. We present tools to derive relations for especially one loop amplitudes, as well as several explicit examples for gauge theory coupled to a wide variety of matter. These tools originate in certain scaling behavior of permutation and cyclic sums of Yang-Mills tree amplitudes and loop integrands. In the latter case evidence exists for relations at all loop orders.

New relations for scattering amplitudes in Yang-Mills theory at loop level

TL;DR

The paper develops a framework to derive relations among one-loop Yang-Mills amplitudes by analyzing the scaling of permutation and cyclic sums under BCFW-type shifts. It presents concrete decoupling relations for rational terms and for triangle and bubble terms, showing that many coefficients vanish under symmetry sums and reducing the independent coefficient space; it also introduces BCJ-like relations for the one-loop integrand and a loop-momentum convention that reduces the integrand basis from to . The results hold across helicities and matter content, with numerical checks up to high point counts and potential extensions to QCD and higher loops. These findings imply a surprisingly simple structure for loop amplitudes in Yang-Mills, promising more efficient computations and deeper theoretical control.

Abstract

The calculation of scattering amplitudes in Yang-Mills theory at loop level is important for the analysis of background processes at particle colliders as well as our understanding of perturbation theory at the quantum level. We present tools to derive relations for especially one loop amplitudes, as well as several explicit examples for gauge theory coupled to a wide variety of matter. These tools originate in certain scaling behavior of permutation and cyclic sums of Yang-Mills tree amplitudes and loop integrands. In the latter case evidence exists for relations at all loop orders.

Paper Structure

This paper contains 6 sections, 22 equations, 4 figures.

Table of Contents

  1. Scaling of permutation and cyclic sums of tree amplitudes
  2. Three photon decoupling relation for rational terms
  3. Five photon decoupling relation for triangle and bubble terms
  4. Relations for massive box contributions to rational terms
  5. Relations for the Yang-Mills integrand at one loop
  6. Conclusion and discussion

Figures (4)

  • Figure 1: Momentum assignment of a massive box coefficient. The $i$-th corner contains gluons with total momentum $K_i$.
  • Figure 2: Massive box coefficient for an amplitude with one photon
  • Figure 3: The Kleiss-Kuijf relation (\ref{['eq:KKrel']}) relates adjacent shifts of massive scalars coupled to photons and gluons to non-adjacent shifts
  • Figure 4: Convention for choosing loop momenta. The bubble represents any color ordered Feynman graph for the planar amplitude. The solid lines indicate the loop legs.