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New Recursions for the Canonical Scalar-Scaffolded Yang-Mills Amplitude

Jeffrey V. Backus

Abstract

The recently-developed "scalar-scaffolding" formulation of gluon amplitudes casts the Yang-Mills (YM) amplitude as a well-defined Laurent series expansion in scalar variables, valid for any spacetime dimension and helicity configuration. In this letter, we exploit this new perspective to develop conceptually novel methods of computing YM tree amplitudes. First, using standard gluon factorization to determine all terms with poles, we show how gauge invariance uniquely fixes the piece with no poles (the "contact term") from only terms that have a single pole. This allows us to write a YM recursion not only for the full amplitude but also for the amplitude up to any order in the Laurent series. Next, by imposing gauge invariance for terms with poles, we write down relations which compute numerators recursively in the amplitude's Laurent series expansion. Starting from an initial set of cuts depending only on the $(n-1)$-point amplitude, these formulae allow us to determine the remaining terms in the $n$-point amplitude. Finally, we use this "Laurent series recursion" to derive a recursion solely for the contact term. We speculate on the possibility that this and analogous recursions for any term in the amplitude may be solved. In attached Mathematica notebooks, we give implementations of these three recursions.

New Recursions for the Canonical Scalar-Scaffolded Yang-Mills Amplitude

Abstract

The recently-developed "scalar-scaffolding" formulation of gluon amplitudes casts the Yang-Mills (YM) amplitude as a well-defined Laurent series expansion in scalar variables, valid for any spacetime dimension and helicity configuration. In this letter, we exploit this new perspective to develop conceptually novel methods of computing YM tree amplitudes. First, using standard gluon factorization to determine all terms with poles, we show how gauge invariance uniquely fixes the piece with no poles (the "contact term") from only terms that have a single pole. This allows us to write a YM recursion not only for the full amplitude but also for the amplitude up to any order in the Laurent series. Next, by imposing gauge invariance for terms with poles, we write down relations which compute numerators recursively in the amplitude's Laurent series expansion. Starting from an initial set of cuts depending only on the -point amplitude, these formulae allow us to determine the remaining terms in the -point amplitude. Finally, we use this "Laurent series recursion" to derive a recursion solely for the contact term. We speculate on the possibility that this and analogous recursions for any term in the amplitude may be solved. In attached Mathematica notebooks, we give implementations of these three recursions.
Paper Structure (6 sections, 54 equations, 5 figures)

This paper contains 6 sections, 54 equations, 5 figures.

Table of Contents

  1. Appendix A: $\Delta T_{2n}$ is linear in $\delta_j$
  2. Appendix B: $a$ tensors from gauge invariance in adjacent gluons
  3. Appendix C: Linearity proofs for the $N_\mathcal{I}$
  4. Appendix D: Simplifying the non-zero $a$ matrix elements
  5. Appendix E: \ref{['eq:app-D-gen']} when $k = 3$
  6. Appendix F: Contact team recursion relations

Figures (5)

  • Figure 1: (Left) The five-point momentum polygon. Overlaid is the dual Feynman graph, where dotted lines represent scalars and solid black lines gluons. (Right) Gluon factorization on pole $X_{a,b} = 0$.
  • Figure 2: Consider computing $N_{\{ (5,11), (7,11) \} }$ (in green) by cutting on $X_{5,11}$ for the six-point amplitude using \ref{['eq:N-fact']}. Then, for the left surface, we have $\mathcal{I}_L = 0$ and $L_0 = \{ 1 \}$, while, for the right surface, $\mathcal{I}_R = \{ (7,11) \}$ and thus $R_0 = 0$. Therefore, we need to know the four-point numerators $N^{(L)}_0$, $N^{(L)}_{(1,5)}$ (in blue) and $N^{(R)}_{(7,11)}$.
  • Figure 3: At five-points, any two chords drawn on the momentum polygon always leave at least two neighboring points open: $e.g.$, points $3,4$ on the left, and points $5,6$ on the right.
  • Figure 4: (Left) The set $\mathcal{I}$ contains the chord $X_{k,m}$ for $k,m \neq 1, 2n-1$ as well as any collection of non-intersecting chords in the shaded green region "behind" it. (Right) To determine $N_{(5,11),(5,9)}$ of the eight-point amplitude, we need to know $T_{1}^{(16)}$ (purple and green), $T_{15}^{(16)}$ (blue and green), as well as $a_{i,j}$ (purple chord $X_{1,5}$ and green).
  • Figure 5: The factorization of $N_{\mathcal{J}\cup(j,1)}^{(R)}$ in \ref{['eq:for-fig-6']} on cut $X_{j,1} = 0$.