Electromagnetic multipole moments of elementary spin-1/2, 1, and3/2particles

Abstract

We study multipole decompositions of the electromagnetic currents of spin-1/2, 1, and 3/2 particles described in terms of representation specific wave equations which are second order in the momenta and which emerge within the recently elaborated Poincaré covariant projector method, where the respective Lagrangians explicitly depend on the Lorentz group generators of the representations of interest. The currents are then the ordinary linear Noether currents related to phase invariance, and present themselves always as two-terms motion-plus spin-magnetization currents. The spin-magnetization currents appear weighted by the gyromagnetic ratio, g, a free parameter in the method which we fix either by unitarity of forward Compton scattering amplitudes in the ultraviolet for spin-1, and spin-3/2, or, in the spin-1/2 case, by their asymptotic vanishing, thus ending up in all three cases with the universal g value of g = 2. Within the method under discussion we calculate the electric multipoles of the above spins for the spinor-, the four-vector, and the fourvector-spinor representations, and find it favorable in some aspects specifically in comparison with the conventional Proca-, and Rarita-Schwinger frameworks. We furthermore attend to the most general non-Lagrangian spin-3/2 currents which are allowed by Lorentz invariance to be up to third order in the momenta and construct the linear-current equivalent of identical multipole moments of one of them. We conclude that non-linear non-Lagrangian spin-3/2 currents are not necessarily more general and more advantageous than the linear spin-3/2 Lagrangian current emerging within the covariant projector formalism. Finally, we test the representation dependence of the multipoles by placing spin-1 and spin-3/2 in the respective (1,0)⊕(0,1), and (3/2,0)⊕(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) ⊕ (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. This observation allows to explain the positive electric quadrupole moment of the ρ meson extracted from recent analyzes of the ρ meson electric form factor. Our finding points toward the possibility that the ρ meson could transform as part of an anti-symmetric tensor with an a1 meson-like state as its representation companion, a possibility consistent with the empirically established ρ-, and a1 vector meson dominance of the hadronic vector-, and axial-vector currents.

Bosonic and fermionic Weinberg-Joos (j,0) ⊕ (0,j) states of arbitrary spins as Lorentz tensors or tensor-spinors and second-order theory

“…In section IV we present the spin- 3 2 Lagrangian and couple it to the electromagnetic field, find the electromagnetic current, and calculate the associated electromagnetic multipole moments. We show that the observables obtained in this fashion reproduce the predictions reported in [16] where the pure spin- 3 2 states have been considered in the standard way as eight-dimensional spinors. Also there we show that the pure spin- 3 2 sector of the antisymmetric tensor spinor describes a particle that propagates causally within an electromagnetic environment.…”

“…The (1, 0)⊕(0, 1) sector is well known and has been frequently elaborated in the literature, listed among others in [23], [16]. We here present it in the momentum space, and denote it by, B…”

“…These states have been constructed for example in [24], [16] and are summarized in the equations (123)- (126) in the Appendix below. In terms of the aforementioned basis vectors, the tensor under discussion expresses as,…”

“…The procedure of extracting the multipole moments from known currents is well established and will not be highlighted here (see for example [25], [16] and references therein for technical details), and amounts to…”

“…In this fashion, the pure spin- "vectors", listed among others in [16], the eight tensor-spinors, w (…”