Generalized Vector Dominance Theory and Vertex Functions

A simple model formula for the hadronic form factors as a ratio of two $\Gamma$ 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 $〈{\pi }^{+}\left(p_2\right)\left|{V_{\mu }}^0\mathrm{}\right(0\left)\right|{\pi }^{+}\left(p_1\right)〉=F_V\mathrm{}\left(t\right)\mathrm{}{(p_1+p_2)}_{\mu }$ and $〈\sigma \left(p_2\right)\left|{A_{\mu }}^0\mathrm{}\right(0\left)\right|{\pi }^0\left(p_1\right)〉={F_A}^{+}\mathrm{}\left(t\right)\mathrm{}{(p_1+p_2)}_{\mu }+{F_A}^{-}\mathrm{}\left(t\right)\mathrm{}{(p_1+p_2)}_{\mu }$, where $t={(p_1+p_2)}^2$, we find the solutions $F_V\mathrm{}\left(t\right)\mathrm{}\propto \frac{\Gamma (1-{\alpha }_V\mathrm{}(t\left)\right)}{\Gamma (r_V+1-{\alpha }_V\mathrm{}(t\left)\right)}$ and ${F_A}^{+}\mathrm{}\left(t\right)\mathrm{}\propto \frac{\Gamma (1-{\alpha }_A\mathrm{}(t\left)\right)}{\Gamma (r_A+1-{\alpha }_A\mathrm{}(t\left)\right)}$, where ${\alpha }_V\mathrm{}\left(t\right)\mathrm{}$ is the $\rho$ Regge trajectory and ${\alpha }_A\mathrm{}\left(t\right)\mathrm{}$ is the $\pi -A_1$ Regge trajectory. The power behaviors of $F_A\mathrm{}\left(t\right)\mathrm{}$ and $F_V\mathrm{}\left(t\right)\mathrm{}$ for $\mathrm{}\left|t\right|\mathrm{}\to \infty$ are not determined absolutely, but the relative power is found to be $r_V-r_A=\frac{1}{2}$, 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 $P\left(t\right)\mathrm{}$ in $〈\sigma \left(p_2\right)\left|{\partial }_{\mu }{A_{\mu }}^0\mathrm{}\right(0\left)\right|{\pi }^0\left(p_1\right)〉={M_{\pi }}^{2}P\left(t\right)\mathrm{}$ be dominated by the ground-state pion alone, that is, $P\left(t\right)\mathrm{}\propto \frac{\Gamma (-{\alpha }_A\mathrm{}(t\left)\right)}{\Gamma (1-{\alpha }_A\mathrm{}(t\left)\right)}$, in order to ensure absence of non-gauge-invariant terms in the matrix element $〈{\pi }^{+}\left(p_2\right)\left|{V_{\mu }}^0\mathrm{}\right(0\left)\right|{\pi }^{+}\left(p_1\right)〉$. 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 $F_V\mathrm{}\left(t\right)\mathrm{}$ and ${F_A}^{+}\mathrm{}\left(t\right)\mathrm{}$, and make a universality hypothesis that the vector mesons $\rho$ 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 $G_E\mathrm{}\left(t\right)\mathrm{}$ and $G_M\mathrm{}\left(t\right)\mathrm{}$, and the axial-vector form factor $G_A\mathrm{}\left(t\right)\mathrm{}$, which for spacelike values of $t$ agree well with the experimental results.