Gauged Soft Recursion: On-Shell Construction of Goldstone-Gauge Amplitudes

TL;DR

This work develops a gauged soft recursion framework that extends on-shell bootstrap methods to Nambu-Goldstone bosons coupled to gauge fields. By combining soft photon/gluon theorems with a gauge-invariant amplitude decomposition, it restores Adler's zero for neutral legs and organizes amplitudes into simpler components that can be bootstrapped recursively, including internal gauge-boson effects fixed by angular-momentum considerations. The formalism yields systematic, tree-level constructions for arbitrary numbers of Goldstone and gauge bosons in both Abelian and non-Abelian theories, with explicit examples illustrating 4$+$1 and higher-point configurations and the non-Abelian color-flow generalization. This approach broadens the applicability of on-shell techniques to gauged nonlinear sigma models and sets the stage for potential loop-level generalizations and further explorations of representation-dependent decompositions.

Abstract

We present a new on-shell recursion relation for scattering amplitudes involving Nambu-Goldstone bosons with a gauged unbroken symmetry. A central challenge is that gauge interactions break Adler's zero condition for charged scalars, invalidating the standard soft recursion. To overcome this, we introduce a ``gauged soft recursion'' that leverages the soft theorems of the gauge bosons themselves, combined with a novel decomposition of amplitudes into gauge-invariant components where Adler's zero is partially restored. The formalism, which also incorporates internal gauge bosons via angular momentum constraints, enables the systematic construction of tree-level amplitudes with arbitrary numbers of Goldstone bosons and gauge bosons in both Abelian and non-Abelian theories, as we demonstrate with explicit examples.

Figures (5)

• Figure 1: Here are all possible channels for the recursion of the tree-level amplitude $\mathcal{M}(\pi_1^+,\pi_2^+,\pi_3^0,\pi_4^0,\pi_5^-,\pi_6^-)$.
• Figure 2: These are all the factorization channels in the recursion relation of the tree-level amplitude $\mathcal{M}(\phi_1,\phi_2,\phi_3,\bar{\phi}_4,\bar{\phi}_5,\bar{\phi}_6)$. In computing $\mathcal{A}_6$, which is associated with the flavor structure $\delta_{i_1}^{i_4}\delta_{i_2}^{i_5}\delta_{i_3}^{i_6}$ denoted by the colored lines in the diagrams, 6 of them have a single contraction across the propagator that leads to a valid factorization into $\mathcal{M}_L\times\mathcal{M}_R$ and hence contribute to $\mathcal{A}_6$.
• Figure 3: Here, we use diagrams to represent the group tensors. To make the distinction clearer, we use dashed lines to represent the adjoint representation index $A$ of the Goldstone boson and looped lines to denote the adjoint representation index $B$ of the gauge boson.
• Figure 4: Here are the $\mathcal{M}(\pi_1^+,\pi_2^+,\pi_3^0,\pi_4^0,\pi_5^-,\pi_6^-;\gamma)$ diagrams for the channels $I=(136)$ and $I=(245)$. In the $\mathcal{A}_1$ component, only the particles 1 and 5 carry charges. The effective current is indicated by the light green line with arrow. The current flows through the channel, making the intermediate state effectively charged in this component, although it is a neutral state in the full amplitude.
• Figure 5: Channels for $\phi^6\gamma$ tree amplitude recursion, which carrying the flavor structure $\delta_{i_1}^{i_4}\delta_{i_2}^{i_5}\delta_{i_3}^{i_6}$.