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All One-loop Maximally Helicity Violating Gluonic Amplitudes in QCD

Carola F. Berger, Zvi Bern, Lance J. Dixon, Darren Forde, David A. Kosower

TL;DR

This work computes the rational parts of all-n gluon one-loop MHV amplitudes in QCD using an on-shell recursion bootstrap, building on known cut-containing terms to yield complete expressions for arbitrary multiplicity. By decomposing amplitudes into completed-cut and rational components and employing carefully chosen complex-momentum shifts, the authors derive all-n closed-form rational terms for the ${ m N}=0$ scalar-loop contribution and verify results against established six-, seven-, and eight-point data. The method demonstrates modest growth in complexity with increasing $n$ and provides numerically checkable results and symmetries, contributing to robust NLO predictions for high-multiplicity jet processes and offering insights into the twistor-space structure of QCD amplitudes. The all-n expressions enable faster, more stable numerical implementations and lay groundwork for extending on-shell bootstrap techniques to broader classes of amplitudes in collider phenomenology.

Abstract

We use on-shell recursion relations to compute analytically the one-loop corrections to maximally-helicity-violating n-gluon amplitudes in QCD. The cut-containing parts have been computed previously; our work supplies the remaining rational parts for these amplitudes, which contain two gluons of negative helicity and the rest positive, in an arbitrary color ordering. We also present formulae specific to the six-gluon cases, with helicities (- + - + + +) and (- + + - + +), as well as numerical results for six, seven, and eight gluons. Our construction of the n-gluon amplitudes illustrates the relatively modest growth in complexity of the on-shell-recursive calculation as the number of external legs increases. These amplitudes add to the growing body of one-loop amplitudes known for all n, which are useful for studies of general properties of amplitudes, including their twistor-space structure.

All One-loop Maximally Helicity Violating Gluonic Amplitudes in QCD

TL;DR

This work computes the rational parts of all-n gluon one-loop MHV amplitudes in QCD using an on-shell recursion bootstrap, building on known cut-containing terms to yield complete expressions for arbitrary multiplicity. By decomposing amplitudes into completed-cut and rational components and employing carefully chosen complex-momentum shifts, the authors derive all-n closed-form rational terms for the scalar-loop contribution and verify results against established six-, seven-, and eight-point data. The method demonstrates modest growth in complexity with increasing and provides numerically checkable results and symmetries, contributing to robust NLO predictions for high-multiplicity jet processes and offering insights into the twistor-space structure of QCD amplitudes. The all-n expressions enable faster, more stable numerical implementations and lay groundwork for extending on-shell bootstrap techniques to broader classes of amplitudes in collider phenomenology.

Abstract

We use on-shell recursion relations to compute analytically the one-loop corrections to maximally-helicity-violating n-gluon amplitudes in QCD. The cut-containing parts have been computed previously; our work supplies the remaining rational parts for these amplitudes, which contain two gluons of negative helicity and the rest positive, in an arbitrary color ordering. We also present formulae specific to the six-gluon cases, with helicities (- + - + + +) and (- + + - + +), as well as numerical results for six, seven, and eight gluons. Our construction of the n-gluon amplitudes illustrates the relatively modest growth in complexity of the on-shell-recursive calculation as the number of external legs increases. These amplitudes add to the growing body of one-loop amplitudes known for all n, which are useful for studies of general properties of amplitudes, including their twistor-space structure.

Paper Structure

This paper contains 13 sections, 130 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Review of Recursive Bootstrap Approach
  4. Five-, Six-, Seven-, and Eight-Point Amplitudes
  5. Recomputation of the five-point amplitude $A_{5;1}^{{\cal N}=0}(1^-,2^+,3^-,4^+,5^+)$
  6. Six-point amplitudes with two negative helicities
  7. Numerical results for various six-, seven- and eight-point amplitudes
  8. One-Loop $n$-Gluon ${\cal N}=0$ Amplitudes with Two Negative Helicities
  9. Solving the recursion relation
  10. The final result
  11. Conclusions
  12. Completed-Cut Terms of ${\cal N}=0$ MHV Amplitudes
  13. Rational Parts of Six-Gluon ${\cal N}=0$ MHV Amplitudes

Figures (11)

  • Figure 1: Schematic representation of tree-level recursive contributions. The labels '$T$' refer to tree vertices which are on-shell amplitudes. The momenta $\hat{\jmath}$ and $\hat{l}$ are shifted, on-shell momenta, evaluated according to eqs. (\ref{['eq:shifted_mom_over_1']}) and (\ref{['eq:shifted_mom_over_2']}).
  • Figure 2: Schematic representation of one-loop recursive contributions. The labels '$T$' and '$L$' refer to tree and loop vertices. The factorization-function contribution (c) does not appear for MHV amplitudes.
  • Figure 3: Diagrammatic representation of overlap contributions. Each overlap diagram corresponds to a physical channel.
  • Figure 4: Non-vanishing recursive diagrams for the amplitude $A^{{\cal N}=0}_{5;1}(1^-,2^+,3^-,4^+,5^+)$, using a ${[1,3\rangle}$ shift, as given in eqs. (\ref{['rec513a']})-(\ref{['rec513d']}).
  • Figure 5: Channels giving overlap contributions to $A^{{\cal N}=0}_{5;1}(1^-,2^+,3^-,4^+,5^+)$ with a ${[1,3\rangle}$ shift, at the values of $z$ given in eq. (\ref{['olap513']}).
  • ...and 6 more figures