Electron-positron annihilation and hadronic sources

The canonical leading light-cone contribution ${\ensuremath{\partial}}_{\ensuremath{\mu}}{\ensuremath{\Delta}}_{+}(x){p}_{\ensuremath{\nu}}f(x\ifmmode\cdot\else\textperiodcentered\fi{}p)+\ensuremath{\cdots}$ to the electroproduction [$\ensuremath{\gamma}(q)+h(p)\ensuremath{\rightarrow}\mathrm{anything};\ensuremath{\gamma}=\mathrm{photon}, h=\mathrm{hadron}$] matrix element $〈h(p)|{J}_{\ensuremath{\mu}}(x){J}_{\ensuremath{\nu}}(0)|h(p)〉$ is such that $f(\ensuremath{\lambda})$, by virtue of the ${J}_{\ensuremath{\mu}}{J}_{\ensuremath{\nu}}$ operator-product expansion (OPE), is analytic for small $\ensuremath{\lambda}$. The corresponding contribution ${\ensuremath{\partial}}_{\ensuremath{\mu}}{\ensuremath{\Delta}}_{+}(x){p}_{\ensuremath{\nu}}\overline{f}(x\ifmmode\cdot\else\textperiodcentered\fi{}p)+\ensuremath{\cdots}$ to the annihilation [${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\gamma}(q)\ensuremath{\rightarrow}h(p)+\mathrm{anything}$] matrix element $〈0|R({J}_{\ensuremath{\mu}}(x)\stackrel{^}{S}(p))R({J}_{\ensuremath{\nu}}(x)\stackrel{^}{S}\ifmmode\dagger\else\textdagger\fi{}(\ensuremath{-}p))|0〉$, where $\stackrel{^}{S}(p)$ is the Fourier transform of a source operator $S(y)$ for $h(p)$, is such that $\overline{f}(\ensuremath{\lambda})$ can be singular for $\ensuremath{\lambda}\ensuremath{\rightarrow}0$; $\overline{f}(\ensuremath{\lambda})\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\sigma}}$. We show how the multiple OPE's among the operators involved can determine the degree $\ensuremath{\sigma}$ of this short-distance singularity provided long-distance effects are not important. We refer to $\ensuremath{\sigma}$ as the "slant" of the matrix element and we explicitly calculate $\ensuremath{\sigma}$ in terms of the minimal dimension $d$ of the source $S(y)$. In the canonical case, we find $\ensuremath{\sigma}=d\ensuremath{-}\frac{1}{2}$ if $h$ is a pseudoscalar particle and $\ensuremath{\sigma}=d\ensuremath{-}1$ if $h$ is a spinor particle. These singularities imply that the scaling functions behave like ${\ensuremath{\omega}}^{\ensuremath{\sigma}}$ for $\ensuremath{\omega}\ensuremath{\equiv}\frac{{q}^{2}}{2q\ifmmode\cdot\else\textperiodcentered\fi{}p}\ensuremath{\rightarrow}\ensuremath{\infty}$ and that the multiplicities behave like ${(\sqrt{{q}^{2}})}^{\ensuremath{\sigma}\ensuremath{-}3}$ for ${q}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$. These results provide handles on the heretofore elusive source dimensions $d$. For example, if logarithmic or greater multiplicities are observed ($\ensuremath{\sigma}\ensuremath{\ge}3$) along with canonical scaling, it can be concluded that $d$ cannot have its canonical elementary value ($d=\frac{5}{2}$ for spinors, $d=3$ for scalars) but rather $d\ensuremath{\ge}\frac{7}{2}$ (spinors) or $d\ensuremath{\ge}4$ (scalars). These results can be readily generalized to noncanonical cases. For example, they lead to simple explanations of known results in renormalized perturbation theory (${\ensuremath{\varphi}}^{3}$ theory, ${\ensuremath{\varphi}}^{4}$ theory, pseudoscalar-meson theory, quantum electrodynamics). Since $\ensuremath{\lambda}$ (and $\ensuremath{\omega}$) is dimensionless, the dimensional analysis involved in our treatment is on a different footing from that which determines the usual short-distance and light-cone singularities of current products. The slants are nevertheless simple functions of the field and source dimensions. Phenomenologically, the operatorial nature of our approach makes it easily extendable to the treatment of other inclusive lepton hadron processes with one or more particles observed in the final state.