Electromagnetic multipole moments of elementary spin-1/2, 1, and 3/2 particles

TL;DR

This paper investigates the electromagnetic multipole moments of elementary particles with spins 1/2, 1, and 3/2, using a covariant projector method, and explores how these moments depend on the particles' Lorentz representations.

Contribution

It introduces a covariant projector approach to calculate multipole moments for various spins and compares these with traditional frameworks, highlighting representation dependence of higher multipoles.

Findings

Charge monopoles and magnetic dipoles are representation independent.

Higher multipoles, like electric quadrupoles, depend on the Lorentz representation.

The bi-vector representation exhibits an electric quadrupole moment opposite to that of the (1/2,1/2) representation.

Abstract

We study multipole decompositions of the electromagnetic currents of spin-1/2, 1, and 3/2 particles described in terms of Lagrangians designed to reproduce representation specific wave equations which are second order in the momenta and which emerge within the recently elaborated Poincar\'e covariant projector method. We calculate the electric multipoles of the above spins for the spinor, the four-vector, and the four-vector--spinor representations, attend to the most general non-Lagrangian spin-3/2 currents which are allowed by Lorentz invariance to be of third order in the momenta and construct the linear current equivalent of identical multipole moments of one of them. We conclude that such non-Lagrangian currents are not necessarily more general than the two-term currents emerging within the covariant projector method. We compare our results with those of the conventional Proca-, and Rarita-Schwinger frameworks. Finally, we test the representation dependence of the multipoles by placing spin-1 and spin-3/2 in the respective (1,0)$\oplus$(0,1), and (3/2,0)$\oplus$(0,3/2) single-spin representations. We observe representation independence of the charge monopoles and the magnetic dipoles, in contrast to the higher multipoles, which turn out to be representation dependent. In particular, we find the bi-vector $(1,0)\oplus (0,1)$ to be characterized by an electric quadrupole moment of opposite sign to the one found in $(1/2,1/2)$, and consequently, to the $W$ boson. Our finding points toward the possibility that the $\rho$ meson could transform as part of an antisymmetric tensor with an $a_{1}$ meson-like state as its representation companion.