Tree-level soft emission of a quark pair in association with a gluon

We compute the tree-level current for the emission of a soft quark-antiquark pair in association with a gluon. This soft current is the last missing ingredient to understand the infrared singularities that can arise in next-to-next-to-next-to-leading-order (N$^3$LO) computations in QCD. Its square allows us to understand for the first time the colour correlations induced by the soft emission of a quark pair and a gluon. We find that there are three types of correlations: Besides dipole-type correlations that have already appeared in soft limits of tree-level amplitudes, we uncover for the first time also a three-parton correlation involving a totally symmetric structure constant. We also study the behaviour of collinear splitting amplitudes in the triple-soft limit, and we derive the corresponding factorisation formula.