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A new recursion relation for tree-level NLSM amplitudes based on hidden zeros

Xiaodi Li, Kang Zhou

TL;DR

This work addresses the challenge of boundary terms in BCFW-like constructions for tree-level NLSM amplitudes by introducing a new recursion that leverages hidden zeros. The method uses a $z$-shift of Mandelstam variables and two non-overlapping hidden-zero configurations to cancel boundary contributions, enabling reconstruction of higher-point amplitudes strictly from on-shell lower-point ones. It then establishes three key results: (i) Adler zero from soft limits, (ii) a δ-shift construction linking NLSM amplitudes to ${ m Tr}( extphi^3)$ amplitudes via a contour extraction, and (iii) a universal expansion of NLSM amplitudes into bi-adjoint scalar (BAS) amplitudes with fixed coefficients. The approach is dimension-independent and relies only on factorization on physical poles, offering a streamlined route to determine all tree-level NLSM amplitudes and suggesting avenues for future extensions to loop levels and related EFTs.

Abstract

In this note, we propose a novel BCFW-like recursion relation for tree-level non-linear sigma model (NLSM) amplitudes, which circumvents the computation of boundary terms by exploiting the recently discovered hidden zeros. Using this recursion, we reproduce three remarkable features of tree-level NLSM amplitudes: (i) the Adler zero, (ii) the $δ$-shift construction, which generates NLSM amplitudes from ${\rm Tr}(φ^3)$ amplitudes, and (iii) the universal expansion of NLSM amplitudes into bi-adjoint scalar amplitudes. Our results demonstrate that the hidden zeros, combined with standard factorization on physical poles, uniquely determine all tree-level NLSM amplitudes.

A new recursion relation for tree-level NLSM amplitudes based on hidden zeros

TL;DR

This work addresses the challenge of boundary terms in BCFW-like constructions for tree-level NLSM amplitudes by introducing a new recursion that leverages hidden zeros. The method uses a -shift of Mandelstam variables and two non-overlapping hidden-zero configurations to cancel boundary contributions, enabling reconstruction of higher-point amplitudes strictly from on-shell lower-point ones. It then establishes three key results: (i) Adler zero from soft limits, (ii) a δ-shift construction linking NLSM amplitudes to amplitudes via a contour extraction, and (iii) a universal expansion of NLSM amplitudes into bi-adjoint scalar (BAS) amplitudes with fixed coefficients. The approach is dimension-independent and relies only on factorization on physical poles, offering a streamlined route to determine all tree-level NLSM amplitudes and suggesting avenues for future extensions to loop levels and related EFTs.

Abstract

In this note, we propose a novel BCFW-like recursion relation for tree-level non-linear sigma model (NLSM) amplitudes, which circumvents the computation of boundary terms by exploiting the recently discovered hidden zeros. Using this recursion, we reproduce three remarkable features of tree-level NLSM amplitudes: (i) the Adler zero, (ii) the -shift construction, which generates NLSM amplitudes from amplitudes, and (iii) the universal expansion of NLSM amplitudes into bi-adjoint scalar amplitudes. Our results demonstrate that the hidden zeros, combined with standard factorization on physical poles, uniquely determine all tree-level NLSM amplitudes.

Paper Structure

This paper contains 15 sections, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. New BCFW-like recursion relation for tree-level NLSM amplitudes
  3. Design recursion
  4. Example: $6$-point NLSM amplitude
  5. $r_{ij}$ for general $2n$-point amplitude
  6. Applications
  7. Application I: Adler zero
  8. Application II: the $\delta$-shift construction of NLSM amplitude
  9. Constructing NLSM amplitudes from ${\rm Tr}(\phi^3)$ amplitudes via $\delta$-shift
  10. Recursive proof
  11. Application III: the expansion of NLSM amplitudes
  12. Expansion of NLSM amplitudes to BAS amplitudes
  13. Recursive proof
  14. Conclusion and discussion
  15. Proof for Eq. (\ref{['expan-s']})

Figures (3)

  • Figure 1: Illustration of the chosen hidden zeros. The left subfigure corresponds to the hidden zero $\{\tilde{i}_1, \tilde{j}_1\}=\{2n, 2\}$, the right subfigure corresponds to the hidden zero $\{\tilde{i}_2,\tilde{j}_2\}=\{1,3\}$, and the dashed lines in each subfigure correspond to the $s_{ab}$'s in the kinematics of each hidden zero.
  • Figure 2: Illustration of the set of $r_{ij}$'s solved by the momentum-conservation-like conditions in Eq. (\ref{['moment-conser-r']}), and the purple dashed lines corresponds the $r_{ij}$'s.
  • Figure 3: Illustration of the case of $|L|$ being odd. Here the internal line $S_L(z)$ is cut and generated two external lines $I_L$ and $I_R$.