Counting Form Factors of Twist-Two Operators

We present a simple method to count the number of hadronic form factors based on the partial wave formalism and crossing symmetry. In particular, we show that the number of independent nucleon form factors of spin-n, twist2 operators (the vector current and energy-momentum tensor being special examples) is n+1. These generalized form factors define the generalized (offforward) parton distributions that have been studied extensively in the recent literature. In proving this result, we also show how the JPC rules for onium states arise in the helicity formalism. Typeset using REVTEX [email protected] [email protected] The generalized (off-forward) parton distributions of hadrons, the nucleon in particular, have attracted considerable theoretical and experimental interest in the last few years [1]. These distributions generalize the well-known Feynman parton distributions as well as the elastic electromagnetic form factors. Most interestingly, the distributions contain information about the orbital motion of partons in a polarized nucleon. One can, for instance, deduce the amount of the nucleon spin carried by the quark orbital angular momentum once these distributions are known [2]. One way to define the generalized parton distributions is to consider the nucleon matrix elements of the twist-2 operators. An example is the chiral-even spin-independent quark operators Ô1n = ψ(0) i ↔ D (μ1 · · · i ↔ D μn) ψ(0) n = 1, 2, ... , (1) where ↔ D μ1 = ( ← D μ1 − → D μ1 )/2, which form totally symmetric tensor representations of the Lorentz group when the Lorentz indices μ1, · · · , μn are symmetrized and rendered traceless (indicated by parentheses). Clearly, this tower of operators is a generalization of familiar vector current ψ̄γψ. The forward matrix elements of the above operators define the generalized charges, an 〈P |Ô1n |P 〉 = 2an(Q )P μ1 · · ·P μn , (2) where Q is a renormalization scale. The nucleon state is normalized covariantly: 〈P |P 〉 = (2P 0)(2π)3δ3(~ P − ~ P ). Feynman’s unpolarized quark distribution q(x,Q) can be obtained directly from the generalized charges,