On single and double soft behaviors in NLSM

TL;DR

This work develops a comprehensive diagrammatic and algebraic treatment of single- and double-soft limits in the nonlinear sigma model (NLSM). By deploying Berends-Giele recursion with the Cayley parametrization and exploiting Kleiss-Kuijf relations, the authors prove the leading single-soft Adler zero for currents, establish the leading and subleading double-soft behavior for currents with adjacent soft particles, and demonstrate that these current-level soft theorems consistently reduce to the known on-shell soft theorems for amplitudes. They further extend the results to nonadjacent soft emissions via KK relations, providing explicit expressions for both leading and subleading nonadjoint soft limits and revealing cancellations that constrain the amplitude structure. The analysis offers a transparent, diagrammatic understanding of soft theorems in NLSM and suggests deeper symmetry or current-algebra origins for the subleading double-soft terms, with potential extensions to loop level and broader effective field theories.

Abstract

In this paper, we study the single and double soft behaviors of tree level off-shell currents and on-shell amplitudes in nonlinear sigma model(NLSM). We first propose and prove the leading soft behavior of the tree level currents with a single soft particle. In the on-shell limit, this single soft emission becomes the Adler's zero. Then we establish the leading and sub-leading soft behaviors of tree level currents with two adjacent soft particles. With a careful analysis of the on-shell limit, we obtain the double soft behaviors of on-shell amplitudes where the two soft particles are adjacent to each other. By applying Kleiss-Kuijf (KK) relation, we further obtain the leading and sub-leading behaviors of amplitudes with two nonadjacent soft particles.

Figures (9)

• Figure 1: Diagrams which contribute to the six-point current $J(2,3,4,5,6)$.
• Figure 2: Diagrams for the current $J(\widetilde{2},\dots,2n)$ with $2$ soft.
• Figure 3: Diagrams for the current $J(2,\dots,i-1,\widetilde{i},i+1,\dots,2n)$ with $i$ soft.
• Figure 4: A typical diagram with the soft particles $2$ and $3$ attached to the off-shell line is given by (A). (B) is the boundary case with the off-shell line connected to a four-point vertex.
• Figure 5: A typical diagram with the soft particles $2$ attached to the off-shell line and the soft particle $3$ in a subcurrent is given by (A). (B) and (C) are two boundary cases with the off-shell line connected to a four-point vertex.