Electroproduction Structure Functions, Integral Representations, and Light-Cone Commutators

Abstract

The electroproduction inelastic structure functions ${W}_{i}(\ensuremath{\kappa}, \ensuremath{\nu})$ ($i=1, 2$; $\ensuremath{\kappa}={q}^{2}=\mathrm{momentum}\mathrm{transfer}\mathrm{squared}$; $\ensuremath{\nu}=q\ifmmode\cdot\else\textperiodcentered\fi{}p=\mathrm{energy}\mathrm{transfer}$; ${p}^{2}=1$) are studied in the Bjorken ($A$) limit ($\ensuremath{\nu}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\rho}\ensuremath{\equiv}\ensuremath{-}\frac{\ensuremath{\nu}}{\ensuremath{\kappa}}$ fixed) and in the Regge ($R$) limit ($\ensuremath{\nu}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\kappa}$ fixed). Finite $A$ limits [${F}_{2}(\ensuremath{\rho})=\mathrm{lim}\ensuremath{\nu}{W}_{2}(\ensuremath{\kappa}, \ensuremath{\nu})$, ${F}_{1}(\ensuremath{\rho})=\mathrm{lim}{W}_{1}(\ensuremath{\kappa}, \ensuremath{\nu})$] and Pomeranchuk-dominated $R$ limits [${w}^{2}(\ensuremath{\kappa}){\ensuremath{\nu}}^{\ensuremath{-}1}=\mathrm{lim}{W}_{2}$, ${w}_{1}(\ensuremath{\kappa})\ensuremath{\nu}=\mathrm{lim}{W}_{1}$] are assumed. These two limits are first related by use of Deser-Gilbert-Sudarshan (DGS) representations for causal functions ${V}_{i}(\ensuremath{\kappa}, \ensuremath{\nu})$ related to the ${W}_{i}$ by ${W}_{2}=\ensuremath{\kappa}{V}_{2}$ and ${W}_{1}=\ensuremath{\kappa}{V}_{1}\ensuremath{-}{\ensuremath{\nu}}^{2}{V}_{2}$. The above $A$- and $R$-limit assumptions, together with a smoothness assumption on the DGS spectral functions motivated by the existence of some equal-time commutators, are shown to imply that ${F}_{2}(\ensuremath{\infty})={w}_{2}(\ensuremath{-}\ensuremath{\infty})\ensuremath{\equiv}{w}_{2}=\mathrm{const}$ and ${\mathrm{lim}}_{\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}}\frac{{F}_{1}(\ensuremath{\rho})}{\ensuremath{\rho}}=\ensuremath{-}{\mathrm{lim}}_{\ensuremath{\kappa}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}}{w}_{2}(\ensuremath{\kappa})\ensuremath{\kappa}\ensuremath{\equiv}{w}_{1}=\mathrm{const}$. These results agree with experiment if ${w}_{i}\ensuremath{\ne}0$. The properties of the spectral functions that are obtained are used to discuss some equal-time commutators and properties of the photon structure function and the cross section ${\ensuremath{\sigma}}_{\ensuremath{\gamma}}(\ensuremath{\nu})$. The empirical value of ${\ensuremath{\sigma}}_{\ensuremath{\gamma}}(\ensuremath{\infty})$ is used to roughly calculate ${w}_{2}$. It is stressed, however, that the above assumptions do not preclude the possibility that ${w}_{2}=0$. The Fourier transforms ${\stackrel{^}{W}}_{i}({x}^{2}, p\ifmmode\cdot\else\textperiodcentered\fi{}x)$ of the ${W}_{i}(\ensuremath{\kappa}, \ensuremath{\nu})$ are next studied and used to again relate the $A$ and $R$ limits. Restrictions on the ${W}_{i}$ imposed by the requirements of finite $A$ limits and correct $R$ limits are determined and results equivalent to the above ones are obtained. This analysis determines the configuration-space behavior corresponding to the $A$ and $R$ limits for large $\ensuremath{\rho}$ and $\ensuremath{\kappa}$, respectively. These limits are shown to determine the behavior of the ${\stackrel{^}{W}}_{i}$ near the light cone ${x}^{2}=0$, and the results are that $\ensuremath{\pi}{\stackrel{^}{W}}_{2}\ensuremath{\sim}\ensuremath{\delta}({x}^{2})\frac{1}{2}\ensuremath{\epsilon}({x}_{0}){\stackrel{^}{f}}_{2}({x}_{0})$ and $\ensuremath{\pi}{W}_{1}\ensuremath{\sim}{\ensuremath{\delta}}^{\ensuremath{'}}({x}^{2})2|{x}_{0}|{\stackrel{^}{f}}_{1}({x}_{0})$ for ${x}^{2}\ensuremath{\sim}0$, where ${w}_{2}=\frac{1}{2}i\ensuremath{\int}d{x}_{0}{\stackrel{^}{f}}_{2}({x}_{0})$ and ${w}_{1}=2i[{f}_{1}(\ensuremath{\infty})\ensuremath{-}{\stackrel{^}{f}}_{1}(\ensuremath{-}\ensuremath{\infty})]$. Corresponding properties of the ${V}_{i}$ are derived, and the equivalence of this approach with the one using integral representations is established. The light-cone behavior of each component of $〈p|[{J}_{\ensuremath{\mu}}(x), {J}_{\ensuremath{\nu}}(0)]|p〉$ can be determined, and it is shown in particular that $〈p|[{J}_{0}\ensuremath{-}{J}_{3}, {J}_{0}\ensuremath{-}{J}_{3}]|p〉\ensuremath{\sim}\ensuremath{\delta}({x}^{2})\ensuremath{\epsilon}({x}_{0}){\stackrel{^}{f}}_{2}({x}_{0})$, apart from total derivatives with respect to ${x}_{0}\ensuremath{-}{x}_{3}$. The light-cone behavior equivalent to the Fubini-Dashen-Gell-Mann sum rule is then put in a form which can accommodate this result so well that we are led to propose a highly symmetric universal structure for the light-cone commutator of two $\mathrm{SU}(3)$ currents. The universality is made precise by use of the ${[SU(3)\ensuremath{\bigotimes}\mathrm{SU}(3)]}_{\ensuremath{\beta}}$ equal-time commutation relations. The equal-time implications of the proposal are considered and are shown to be consistent with, and suggested by, the gluon mode. In the context of this model, ${w}_{2}$ is numerically estimated and found to agree with the experimental value.