Covariant Spinor Formalism for Multipole Expanded Form Factor
We present a systematic technique for constructing Lorentz covariant orbital-spin ($LS$) bases for matrix elements of local operators and the associated form factors, thereby extending the traditional multipole expansion to a Lorentz covariant formalism. In the spinor-helicity formalism, matrix elements of local operators for spin-$j$ particles can be treated as several massive 3-point scattering amplitudes, each of which can be further decomposed into different $LS$ partial wave amplitudes. We obtain explicit complete and linearly independent $LS$ amplitude bases for scalar, vector, and rank 2 tensor form factor of particles with spin-$\frac{1}{2}$, $1$, and $\frac{3}{2}$. In the Breit frame, it recovers the traditional multipole expansion expression, and we show the explicit equivalence among the traditional multipole expansion, canonical $LS$ expansion, and the $\mathrm{SO}(3)$ Zemach tensor expansion. Finally noting covariant structures built from the relativistic external wave functions and momenta of the initial and final state particles, we give a universal construction formula for form factor of arbitrary Lorentz tensor operators for arbitrary external spin particles.