Second order theory of $(j,0)\oplus (0,j)$ single high spins as Lorentz tensors

Abstract

We show that higher order differential equations and matrix spinor calculus are completely avoidable in the description of pure high spin-j Weinberg-Joos states, (j, 0)⊕(0, j). The case is made on the example of 3 2 , 0 ⊕ 0, 3 2 , for the sake of concreteness and without loss of generality. Namely, we use as a vehicle for the aforementioned covariant single spin-3 2 description the direct sum of 3 2 , 0 ⊕ 0, 3 2 with the Dirac field, ψ ≃ 1 2 , 0 ⊕ 0, 1 2 , on the one side, and 1 2 , 1 ⊕ 1, 1 2 , on the other, which amounts to the antisymmetric tensor of second rank with Dirac spinor components, Ψ2 sector of interest is then tracked down in two steps. First we search for spin-3 2 by means of a covariant spin projector constructed from the Casimir invariants of the Poincaré algebra, the squared four momentum, P 2 , and the squared Pauli-Lubanski vector, W 2 . This projector is second order in the momenta. Afterwords we identify the wanted irreducible representation space by means of a momentum independent (static) projector designed on the basis of the Casimir invariants of the Lorentz algebra. The latter projectors have the property to unambiguously identify any irreducible so(1, 3) subspace of any Lorentz tensor and without rising the order of the differential equation. In this fashion, a Lagrangian that is second order in the momenta is furnished. The method proposed correctly reproduces the electromagnetic multipole moments earlier calculated for 3 2 , 0 ⊕ 0, 3 2 in treating it in the standard way as eight dimensional spinor. We furthermore calculate Compton scattering off the pure spin-3 2 under discussion, and show that the differential cross section satisfies unitarity in forward direction for a gyromagnetic ratio of g = 2 3 . This result hints on the possible validity of Belinfante's conjecture for pure spin-states, while the natural value of g = 2 seems more likely to characterize the highest spins in the Rarita-Schwinger representation spaces. The scheme straightforwardly extends to any (j, 0) ⊕ (0, j) Weinberg-Joos state and brings the advantage of avoiding rectangular matrix couplings between states of different spins, replacing them by simple Lorentz contractions.

Higher-spin particles at high-energy colliders

“…This fact has been often ignored in nuclear physics when computing the pion-nucleon interaction mediated by a spin-3/2 ∆ resonance [19][20][21][22][23][24], for instance, and in collider studies of generic spin-3/2 particles. There have been attempts to cure the problem by rewriting the interactions of the spin-3/2 fields [25][26][27][28][29][30][31][32][33][34][35][36][37]. Difficulties in formulating ∆-resonance interactions in a Lagrangian description have been discussed in Ref.…”