Electromagnetic Scoot for Dyons Revisited

Abstract: In the scattering of two electric charges, the particles acquire a shift in their net boost-like angular momentum, balanced by an opposite field contribution. This electromagnetic scoot effect appears at first order in post-Minkowskian expansion (1PM) order when conservation laws are evaluated on constant-time slices, but disappears at this order on hyperboloidal slices. Here, we extend this analysis to scattering involving both electric and magnetic charges and compare the results with the purely electric case in the context of multiparticle state representations.

• The paper demonstrates that in dyon scattering, the total mechanical angular momentum change, given by 2(e₁g₂ - e₂g₁), is exactly canceled by the electromagnetic field contribution.
• It employs relativistic Lorentz force calculations to derive dyon trajectories and conserved quantities, confirming the persistence of mass moment and angular momentum exchanges across different boundary slices.
• The study underscores the need for new quantum numbers in multiparticle systems and advances our understanding of asymptotic symmetries in gauge and gravitational scattering.

Electromagnetic Scoot Effects in Dyon Scattering: Angular Momentum Exchange and Boundary Dependence

Introduction

The study of scattering processes involving particles with both electric and magnetic charges—dyons—has deep implications for the theoretical structure of the S-matrix, the representation theory of the Poincaré group, and the emergence of topological quantum numbers in asymptotic multiparticle states. This work revisits and substantially extends the phenomenon known as the "electromagnetic scoot": a nontrivial exchange of “boost-like” mass moment and angular momentum between particles and fields during relativistic scattering, previously established for purely electric charges (Gralla et al., 2021), to the general case of dyons. The analysis incorporates both constant time and hyperboloidal boundary slices, examining the persistence or vanishing of the effect under different definitions of conserved charges.

Classical Computation of Dyon Trajectories and Conserved Quantities

The first component of the paper generalizes the post-Minkowskian (1PM) calculation of the electromagnetic scoot in the two-charge case (Gralla et al., 2021) to dyons, accounting for both electric and magnetic charge interactions. Using the relativistic Lorentz force for dyons, the trajectories of two scattering particles were derived in the center-of-momentum frame in the small deflection approximation. The explicit particle worldlines serve as inputs to compute the conserved quantities—energy, momentum, angular momentum, and mass moment—for both the particle sector and the electromagnetic field.

A crucial technical result is that the total change in mechanical angular momentum for two dyons, $\Delta \mathbf{L}_{\mathrm{mech}}$, is shown to be $2(e_1 g_2 - e_2 g_1)\hat{\mathbf{z}}$, and this change is exactly canceled by the electromagnetic field contribution, confirming a robust, non-vanishing transfer of angular momentum between charged particles and their fields during scattering. This establishes that the residual field angular momentum, previously identified in the quantum context by Zwanziger [Phys. Rev. D6, 458 (1972)], manifests at the classical 1PM level and for generic dyonic charges.

Simultaneously, the analogous "boost-like" mass moment exchange persists for dyons, scaling with the quadratic charge combination $e_1 e_2 + g_1 g_2$. The transfer is explicit in the cross-terms of the field components and matches changes in mechanical particle moments, signifying the necessity for new quantum numbers in multiparticle representations—recently tied to pairwise little group constructions (Csáki et al., 2020, Alessio et al., 2024).

The Role of Spacetime Slicings: Constant Time vs Hyperboloidal Boundaries

A systematic reevaluation is performed regarding the boundary hypersurface on which conserved quantities are calculated. Earlier work (Gralla et al., 2024) revealed that the electromagnetic scoot, in the two-charge scattering case, vanishes at 1PM when evaluated on hyperboloidal slices rather than constant-time slices, implying its operational non-observability at null infinity under this slicing.

The current analysis extends this investigation to dyon (specifically charge-monopole) scattering. It is demonstrated via both field and particle sector calculations that, while the field contribution to the mass moment vanishes in the large hyperboloid limit, the total angular momentum change in the $z$-direction, $\Delta L^z = 2 e_1 g_2$, remains robustly nonzero even for the hyperboloidal boundary. Detailed computations in both electromagnetic field theory and classical particle mechanics confirm this persistence. Therefore, the dyonic contribution to the angular momentum scoot effect is immune to the choice of boundary hypersurface, in clear contrast to the charge-only case where the effect disappears on hyperboloids.

Theoretical Implications and Outlook

These results have significant theoretical consequences. First, the explicit calculation of the angular momentum exchange reveals that the residual field-induced quantum number associated with dyon-dyon scattering is not merely a quantum-theoretical artifact but is rooted in classical, physical charge trajectories. The robustness of the dyonic angular momentum exchange under changes in the slicing question the universality of previously held selection rules or nullifications that were observed in purely electric cases.

In the context of asymptotic symmetry and representation theory for multiparticle states, the findings reinforce the necessity to include additional quantum numbers beyond those associated with particle spins and momenta, such as the pairwise helicity and boost charges (Csáki et al., 2020, Alessio et al., 2024, Lippstreu, 2021). Particularly, for systems involving magnetic charges, the natural assignment of field angular momentum is expected to play a central role in classifying states and in the possible emergence of new selection rules for physical processes.

Practically, the semi-classical matching of field and particle angular momenta underscores the importance of careful boundary definitions in perturbative calculations, scattering amplitudes, and classical double copy constructions. Moreover, the observed independence of the dyonic angular momentum transfer on the slicing may have relevance for future explorations of gravitational analogs and their classical limits (Gralla et al., 2021).

Conclusion

This work establishes that the electromagnetic scoot, originally identified in electric charge scattering as a transfer of mass moment, also encompasses a nontrivial, slicing-independent transfer of angular momentum for dyonic systems at leading post-Minkowskian order. While the effect for purely electric systems can be rendered absent by moving to hyperboloidal boundaries, the dyonic component remains, highlighting a critical structural difference in the classical and quantum description of these multiparticle states. The conclusion points to the necessity for further detailed analysis of asymptotic symmetries, quantum numbers, and Hilbert space representations in theories admitting both electric and magnetic charges, a direction that promises new insights for the structure of S-matrix theory and the classical-quantum interface in gauge and gravitational scattering.