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On-shell methods are particularly suited for exploring the scattering ofelectrically and magnetically charged objects, for which there is no local andLorentz invariant Lagrangian description. In this paper we show how toconstruct a Lorentz-invariant S-matrix for the scattering of electrically andmagnetically charged particles, without ever having to refer to a Dirac string.A key ingredient is a revision of our fundamental understanding ofmulti-particle representations of the Poincar\'e group. Surprisingly, theasymptotic states for electric-magnetic scattering transform with an additionallittle group phase, associated with pairs of electrically and magneticallycharged particles. The corresponding "pairwise helicity" is identified with thequantized "cross product" of charges, $e_1 g_2 - e_2 g_1$, for everycharge-monopole pair, and represents the extra angular momentum stored in theasymptotic electromagnetic field. We define a new kind of pairwisespinor-helicity variable, which serves as an additional building block forelectric-magnetic scattering amplitudes. We then construct the most general3-point S-matrix elements, as well as the full partial wave decomposition forthe $2\to 2$ fermion-monopole S-matrix. In particular, we derive the famoushelicity flip in the lowest partial wave as a simple consequence of ageneralized spin-helicity selection rule, as well as the full angulardependence for the higher partial waves. Our construction provides asignificant new achievement for the on-shell program, succeeding where theLagrangian description has so far failed.

Scattering Amplitudes for Monopoles: Pairwise Little Group and Pairwise Helicity