Covariant Construction of Generalized Form Factors

TL;DR

This paper introduces a systematic, covariant method for constructing hadronic matrix element structures for particles of various spins, providing new form factor bases and correcting previous redundancies.

Contribution

It develops a Lorentz-covariant construction technique using spinor Young tableaux, presenting the first general form factors for spin-3/2 and spin-2 particles.

Findings

Constructed form factor bases for scalar, vector, and tensor operators across multiple spins.

Presented the first general P and T form factors for spin-3/2 and spin-2 particles.

Identified and corrected redundant structures in previous literature for spin-2 particles.

Abstract

We present a systematic technique for constructing the Lorentz-covariant structures of hadronic matrix elements of local operators. The spinor Young tableaux of the Lorentz group is employed to construct all possible structures for the matrix elements of arbitrary operators, using the relativistic wave functions and momenta of the initial and final state particles of arbitrary spin as building blocks. We obtain the form factor bases for the scalar, vector, and rank-2 tensor operators for particles with spin-$\frac{1}{2}$, $1$, $\frac{3}{2}$, and $2$, among which the general $P$ and $T$ form factors for spin-$\frac{3}{2}$ and spin-$2$ particles are presented for the first time. The independent form factor structures are also cross-checked by the non-relativistic counting and Hilbert Series method and we find there is redundant $P$ and $T$ conserved structure for spin-$2$ particles in literature. As an application, the matrix elements of general nonlocal operators can be expanded by towers of the matrix elements of local operators, and thus can be decomposed by the constructed form factor bases above.