Multi-soft theorems in Gauge Theory from MHV Diagrams

TL;DR

This paper demonstrates that scattering amplitudes in gauge theories exhibit universal multi-soft behavior when an arbitrary number of adjacent particles become soft simultaneously, using the CSW (MHV) framework to identify leading contributions. It provides compact rules to construct the leading multi-soft factors $S_m$ for $m$ gluons with $k$ negative helicities and applies these to explicit cases ($k=1,2$). In ${\cal N}=4$ SYM, the work leverages analytic supervertices to derive multi-soft factors for two scalars or two fermions in a singlet plus $m-2$ gluons, revealing a stronger $1/\delta^2$ leading divergence in certain double-soft singlet limits. The results yield simple expressions for triple- and quadri-soft limits in singlet configurations and suggest avenues for extending the framework to subleading terms and inverse soft-limit reconstructions, contributing to a more efficient understanding of amplitudes in gauge theories.

Abstract

In this work we employ the MHV technique to show that scattering amplitudes with any number of consecutive soft particles behave universally in the multi-soft limit in which all particles go soft simultaneously. After identifying the diagrams which give the leading contribution we give the general rules for writing down compact expressions for the multi-soft factor of m gluons, k of which have negative helicity. We explicitly consider the cases where k equals 1 and 2. In N =4 SYM, the multi-soft factors of 2 scalars or 2 fermions forming a singlet of SU(4) R-symmetry, and m-2 positive helicity gluons are derived. The special case of the double-soft limit gives an amplitude whose leading divergence is 1/δ^2 and not 1/δas in the case of 2 scalars or 2 fermions that do not form a singlet under SU(4). The construction based on the analytic supervertices allows us to obtain simple expressions for the triple-soft limit of 1 scalar and 2 positive helicity fermions, as well as for the quadrapole-soft limit of 4 positive helicity fermions, in a singlet configuration.

Figures (7)

• Figure 1: On the left tree diagrams with MHV vertices contributing to the leading multi-soft factor $S_m^{(0)}(1^+,2^+,...,i^-,...m^+)$. On the right tree diagrams which give subleading contributions to the multi-soft factor. In the case of the factorisation of a generic "hard" amplitude the right vertex should be "dressed" with additional MHV vertices. Apparently, this will not change the soft factor \ref{['1soft']}, \ref{['12-fin']}.
• Figure 2: Diagrams contributing to the multi-soft factor involving 2 negative helicity gluons $S_m^{(0)}(1^+,...,i^-,...,j^-,...,m^+)$. The negative helicity gluons are on the left MHV vertex. The red dashed line denotes the additional gluons and possibly vertices of the hard amplitude that factorises in the soft limit.
• Figure 3: Diagrams contributing to the multi-soft factor involving 2 negative helicity gluons $S_m^{(0)}(1^+,...,i^-,...,j^-,...,m^+)$. Here the negative helicity gluons are sitting one on the left MHV vertex and one on the middle MHV vertex. The red dashed line denotes the additional gluons and possibly vertices of the hard amplitude that factorises in the soft limit.
• Figure 4: Diagrams contributing to the multi-soft factor involving 2 negative helicity gluons $S_m^{(0)}(1^+,...,i^-,...,j^-,...,m^+)$. Here the negative helicity gluons are sitting one on the left MHV vertex and one on the right MHV vertex. The red dashed line denotes the additional gluons and possibly vertices of the hard amplitude that factorises in the soft limit.
• Figure 5: Tree diagrams with 2 analytic supervertices contributing to the degree-12 superamplitude of Eq. \ref{['finrest']}. $n_1$ and $n_2$ are the number of legs in the right and left vertex respectively.