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Efficient Calculation of Crossing Symmetric BCJ Tree Numerators

Alex Edison, Fei Teng

TL;DR

This work delivers a crossing-symmetric, RO-free algorithm for constructing BCJ tree numerators directly from CHY integrands, dramatically improving scalability to high multiplicities. By recasting the baseline expansions and dressing rules within a recursive spanning-tree framework, the method recovers crossing symmetry without explicit reference-order averaging, reducing computational complexity. The approach is developed for pure YM, two-fermion, and multi-trace YM-scalar theories, and is extended to include minimally coupled massive states via dimensional uplift, with a Mathematica package provided for practical calculations. These advances enhance the utility of BCJ-compatible numerators for double-copy constructions and open avenues for applications in forward-limit one-loop computations and classical gravity analyses.

Abstract

In this paper, we propose an improved method for directly calculating double-copy-compatible tree numerators in (super-)Yang-Mills and Yang-Mills-scalar theories. Our new scheme gets rid of any explicit dependence on reference orderings, restoring a form of crossing symmetry to the numerators. This in turn improves the computational efficiency of the algorithm, allowing us to go well beyond the number of external particles accessible with the reference order based methods. Motivated by a parallel study of one-loop BCJ numerators from forward limits, we explore the generalization to include a pair of fermions. To improve the accessiblity of the new algorithm, we provide a Mathematica package that implements the numerator construction. The structure of the computation also provides for a straightforward introduction of minimally-coupled massive particles potentially useful for future computations in both classical and quantum gravity.

Efficient Calculation of Crossing Symmetric BCJ Tree Numerators

TL;DR

This work delivers a crossing-symmetric, RO-free algorithm for constructing BCJ tree numerators directly from CHY integrands, dramatically improving scalability to high multiplicities. By recasting the baseline expansions and dressing rules within a recursive spanning-tree framework, the method recovers crossing symmetry without explicit reference-order averaging, reducing computational complexity. The approach is developed for pure YM, two-fermion, and multi-trace YM-scalar theories, and is extended to include minimally coupled massive states via dimensional uplift, with a Mathematica package provided for practical calculations. These advances enhance the utility of BCJ-compatible numerators for double-copy constructions and open avenues for applications in forward-limit one-loop computations and classical gravity analyses.

Abstract

In this paper, we propose an improved method for directly calculating double-copy-compatible tree numerators in (super-)Yang-Mills and Yang-Mills-scalar theories. Our new scheme gets rid of any explicit dependence on reference orderings, restoring a form of crossing symmetry to the numerators. This in turn improves the computational efficiency of the algorithm, allowing us to go well beyond the number of external particles accessible with the reference order based methods. Motivated by a parallel study of one-loop BCJ numerators from forward limits, we explore the generalization to include a pair of fermions. To improve the accessiblity of the new algorithm, we provide a Mathematica package that implements the numerator construction. The structure of the computation also provides for a straightforward introduction of minimally-coupled massive particles potentially useful for future computations in both classical and quantum gravity.

Paper Structure

This paper contains 16 sections, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. CHY Integrands and Baseline Expansion
  3. Super-Yang-Mills (sYM) theories
  4. Yang-Mills-scalar
  5. DDM basis numerators from CHY integrands
  6. Spanning Trees and Reference Orderings
  7. Recursive Constructions without Reference Orderings for Pure Gluon and Two-fermion Numerators
  8. Four-Point YM Example
  9. Recursive Constructions for Multi-trace
  10. Multi-trace Example
  11. Introducing Masses
  12. Conclusion and Outlook
  13. CHY integrands and scattering amplitudes
  14. Mathematica Package
  15. Constituent objects
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Examples of trees with various $\mathop{\mathrm{IT}}\nolimits$ memberships: (a) a tree that only belongs to $\mathop{\mathrm{IT}}\nolimits(1,2,3,4,5)$, (b) a tree that belongs to all $\mathop{\mathrm{IT}}\nolimits(1,(2)\shuffle (3,4),5)$, (c) a tree that belongs to $\mathop{\mathrm{IT}}\nolimits(1,\beta,5)$ for all $\beta\in S_{\{2,3,4\}}$. The baselines are shown in red here and in all future occurances.
  • Figure 2: Merging of branches at the vertex $j$.
  • Figure 3: A typical spanning tree with the baseline shown in red, where $T_a\in M_{\rho_i}$ are the sub-trees merging at $\rho_i$ on the baseline.