Generalized Vector Dominance Theory and Vertex Functions

Abstract

A simple model formula for the hadronic form factors as a ratio of two $\ensuremath{\Gamma}$ functions is examined from the point of view of its consistency with the beta-function model and with the chiral properties of the vector and axial-vector currents. It is shown that similar models may hold simultaneously for the simplest vector and axial-vector form factors while maintaining consistency with the Veneziano model of the four-point function, and with the requirements of a conserved vector current, partially conserved axial-vector current, and the Gell-Mann algebra of currents. An infinite class of such solutions is found: Defining form factors by $〈{\ensuremath{\pi}}^{+}({p}_{2})|{{V}_{\ensuremath{\mu}}}^{0}(0)|{\ensuremath{\pi}}^{+}({p}_{1})〉={F}_{V}(t){({p}_{1}+{p}_{2})}_{\ensuremath{\mu}}$ and $〈\ensuremath{\sigma}({p}_{2})|{{A}_{\ensuremath{\mu}}}^{0}(0)|{\ensuremath{\pi}}^{0}({p}_{1})〉={{F}_{A}}^{+}(t){({p}_{1}+{p}_{2})}_{\ensuremath{\mu}}+{{F}_{A}}^{\ensuremath{-}}(t){({p}_{1}+{p}_{2})}_{\ensuremath{\mu}}$, where $t={({p}_{1}+{p}_{2})}^{2}$, we find the solutions ${F}_{V}(t)\ensuremath{\propto}\frac{\ensuremath{\Gamma}(1\ensuremath{-}{\ensuremath{\alpha}}_{V}(t))}{\ensuremath{\Gamma}({r}_{V}+1\ensuremath{-}{\ensuremath{\alpha}}_{V}(t))}$ and ${{F}_{A}}^{+}(t)\ensuremath{\propto}\frac{\ensuremath{\Gamma}(1\ensuremath{-}{\ensuremath{\alpha}}_{A}(t))}{\ensuremath{\Gamma}({r}_{A}+1\ensuremath{-}{\ensuremath{\alpha}}_{A}(t))}$, where ${\ensuremath{\alpha}}_{V}(t)$ is the $\ensuremath{\rho}$ Regge trajectory and ${\ensuremath{\alpha}}_{A}(t)$ is the $\ensuremath{\pi}\ensuremath{-}{A}_{1}$ Regge trajectory. The power behaviors of ${F}_{A}(t)$ and ${F}_{V}(t)$ for $|t|\ensuremath{\rightarrow}\ensuremath{\infty}$ are not determined absolutely, but the relative power is found to be ${r}_{V}\ensuremath{-}{r}_{A}=\frac{1}{2}$, as a consequence of a quantization rule for the trajectory intercepts. Scalar and pseudoscalar form factors are treated also, with similar results; in the approximation used here, however, it is required that the pseudoscalar form factor $P(t)$ in $〈\ensuremath{\sigma}({p}_{2})|{\ensuremath{\partial}}_{\ensuremath{\mu}}{{A}_{\ensuremath{\mu}}}^{0}(0)|{\ensuremath{\pi}}^{0}({p}_{1})〉={{M}_{\ensuremath{\pi}}}^{2}P(t)$ be dominated by the ground-state pion alone, that is, $P(t)\ensuremath{\propto}\frac{\ensuremath{\Gamma}(\ensuremath{-}{\ensuremath{\alpha}}_{A}(t))}{\ensuremath{\Gamma}(1\ensuremath{-}{\ensuremath{\alpha}}_{A}(t))}$, in order to ensure absence of non-gauge-invariant terms in the matrix element $〈{\ensuremath{\pi}}^{+}({p}_{2})|{{V}_{\ensuremath{\mu}}}^{0}(0)|{\ensuremath{\pi}}^{+}({p}_{1})〉$. In order to consider matrix elements of the vector- and axial-vector-current operators between other hadron states, we write generalized field-current identities, using the forms of ${F}_{V}(t)$ and ${{F}_{A}}^{+}(t)$, and make a universality hypothesis that the vector mesons $\ensuremath{\rho}$ and ${A}_{1}$ and their higher recurrences couple universally to the independent helicity amplitudes of the Breit frame. As an example, a treatment of the nucleon electromagnetic and axial-vector form factors is given, leading to simple model formulas for the Sachs form factors ${G}_{E}(t)$ and ${G}_{M}(t)$, and the axial-vector form factor ${G}_{A}(t)$, which for spacelike values of $t$ agree well with the experimental results.