Lorentz Invariance and the Kinematic Structure of Vertex Functions

The general kinematic properties of vertex functions which follow from the transformation properties of the initial and final single-particle states, and of the vertex (current) operator under proper and improper Lorentz transformations, are studied for the pseudoscalar (pion) and vector (electromagnetic) vertices. The treatment, which relies strongly on the helicity representation for the states of a relativistic particle introduced by Jacob and Wick, is fully relativistic, and applies to particles of arbitrary spin. The number of independent vertex functions or form factors is determined in each case, and the analogs of the nonrelativistic multipole expansions are obtained. One obtains thereby a complete specification of the dependence of the matrix elements on the spins, helicities, and relative parities of the particles, and some information on the limiting behavior of the form factors for small momentum transfers. A theorem first proved by Ernst, Sachs, and Wali in the special case of spin ½, that the matrix elements of the 4-divergence of the electromagnetic current between states of the same (single) particle vanish independently of the assumption of gauge invariance or current conservation, is extended as a byproduct of the general results to the case of arbitrary spin and to any vector current having the same transformation properties as $j_{\mu }$. Finally, the parametrizations obtained for the pseudoscalar and vector vertex functions are used to calculate the cross sections for the general two-particle scattering process $a+b\to c+d$ in the single-quantum-exchange approximation for relativistic particles of arbitrary spin.