Abstract
Scattering amplitudes of any four-dimensional theory with nonabelian gauge group $\mathcal G$ may be recast as two-dimensional correlation functions on the asymptotic two-sphere at null infinity. The soft gluon theorem is shown, for massless theories at the semiclassical level, to be the Ward identity of a holomorphic two-dimensional $\mathcal G$-Kac-Moody symmetry acting on these correlation functions. Holomorphic Kac-Moody current insertions are positive helicity soft gluon insertions. The Kac-Moody transformations are a $CPT$ invariant subgroup of gauge transformations which act nontrivially at null infinity and comprise the four-dimensional asymptotic symmetry group.
Highlights
Gravity is included.1 In this paper, we consider tree-level scattering of massless particles in 4D nonabelian gauge theories with gauge group G
Scattering amplitudes of any four-dimensional theory with nonabelian gauge group G may be recast as two-dimensional correlation functions on the asymptotic twosphere at null infinity
The soft gluon theorem is shown, for massless theories at the semiclassical level, to be the Ward identity of a holomorphic two-dimensional G-Kac-Moody symmetry acting on these correlation functions
Summary
We consider a nonabelian gauge theory with group G and associated Lie algebra g. The adjoint elements of G and generators of g are denoted by g and T a respectively with (T a)bc = −if abc. The four-dimensional matrix valued gauge field is Aμ = AaμT a, where a μ index here and hereafter refers to flat Minkowski coordinates in which the metric is ds2 = −dt2 + dx · dx,. S2 ×R boundary at r = ∞ with coordinates (u, z, z) It has boundaries at u = ±∞, which we denote I±+. I − is the null S2 × R boundary at r = ∞ with coordinates (v, z, z). It has boundaries at v = ±∞, which we denote as I±−.
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