Scattering amplitudes for monopoles: pairwise little group and pairwise helicity

Abstract

On-shell methods are particularly suited for exploring the scattering of electrically and magnetically charged objects, for which there is no local and Lorentz invariant Lagrangian description. In this paper we show how to construct a Lorentz-invariant S-matrix for the scattering of electrically and magnetically charged particles, without ever having to refer to a Dirac string. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. The corresponding “pairwise helicity” is identified with the quantized “cross product” of charges, e1g2− e2g1, for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind of pairwise spinor-helicity variable, which serves as an additional building block for electric-magnetic scattering amplitudes. We then construct the most general 3-point S-matrix elements, as well as the full partial wave decomposition for the 2 → 2 fermion-monopole S-matrix. In particular, we derive the famous helicity flip in the lowest partial wave as a simple consequence of a generalized spin-helicity selection rule, as well as the full angular dependence for the higher partial waves. Our construction provides a significant new achievement for the on-shell program, succeeding where the Lagrangian description has so far failed.

Zwanziger’s pairwise little group on the celestial sphere

“…In this paper we propose a partial 1 answer to this question for massless matter particles. Using the insights made in the recent analyses [6,7], which fully clarified the representation 1 We expect that a complete description of the Poincaré transformation properties of the asymptotic states will require one to also specify the in/out vacuum states, similar to those given in [4,5]. Herein we limit our discussion to the non-vacuum sector.…”

“…We review Zwanziger's construction in more detail in section 3.1. More complete details can be found in the original work [8], as well as the recent studies [6,7] which resurfaced Zwanziger's work by recasting the construction in terms of the massive spinor-helicity formalism of [50] by introducing a new pairwise spinor-helicity variable, thereby streamlining the study of the three point and partial wave amplitudes of electric-magnetic particle scattering. One of the reasons why Zwanziger's work has remained dormant for so long may be due to the equivocal representation theory of the additional U(1) phase.…”

“…One of the reasons why Zwanziger's work has remained dormant for so long may be due to the equivocal representation theory of the additional U(1) phase. The recent works [6,7] clarified the crucial point that the additional phase should be thought of as arising from the Poincaré group acting an additional pairwise state, over and above the tensor product of two particle states that Zwanziger wrote down. The useful nomenclature of a pairwise state and a pairwise helicity, used throughout in this paper, also originates from these recent works.…”

“…Let us now emphasize a conceptual point which was first developed in [6,7]. For simplicity let us return to the proposed transformation law for a two particle state…”

“…(3.28) 7 In some works, for example [9], the (1 + |z| 2 ) iηλ+1 prefactor is not explicitly indicated. In those works the energy variable with respect to which one is taking the Mellin transform is understood to be ω = E 1+|z| 2 .…”