Eikonal Amplitudes and Nonglobal Logarithms from the BMS Equation

The Banfi–Marchesini–Smye (BMS) equation accounts for resummation of nonglobal logarithms to all orders in perturbation theory in the large- $${{{\text{N}}}_{{\text{c}}}}$$ approximation. We show that the squared amplitudes for the emission of soft energy-ordered gluons are correctly embedded in this equation, and explicitly verify that they coincide with those derived in our previous work in the large- $${{{\text{N}}}_{{\text{c}}}}$$ limit up to sixth order in the strong coupling. We perform analytical calculations for the nonglobal logarithms up to fourth order for the specific hemisphere mass distribution in $${{e}^{ + }}{{e}^{ - }}$$ collisions, thus confirming our previous semi-numerical results. We show that the solution to the BMS equation may be cast into a product of an infinite number of exponentials each of whichresums a class of Feynman diagrams that manifest a symmetry pattern, and explicitly carry out the computation of the first of these exponentials.