Deep-Inelastic Contribution to Baryon Electromagnetic Mass Differences

Abstract

Unsubtracted dispersion relations for ${t}_{1}({q}^{2}, \ensuremath{\nu})$ and ${t}_{2}({q}^{2}, \ensuremath{\nu})$ are proposed even in the case of the $\ensuremath{\Delta}I=1$ mass difference, with the requirement of the absence of divergences worse than logarithmic ones. By the use of the experimental data on the inelastic nucleon structure functions, the possibility is shown that the deep-inelastic effect leads to the correct sign of the observed $n\ensuremath{-}p$ mass difference, under the condition that $limit of\text{}\ensuremath{\int}{0}^{2M}(1\ensuremath{-}\frac{2{q}^{2}R}{{\ensuremath{\omega}}^{2}}){G}_{2}d\ensuremath{\omega}>0\text{as}{q}^{2}\ensuremath{\rightarrow}\ensuremath{\infty},$ where $\ensuremath{\omega}=\frac{{q}^{2}}{\ensuremath{\nu}}$, $R=\frac{{\ensuremath{\sigma}}_{l}}{{\ensuremath{\sigma}}_{t}}$ is the ratio of virtual-photon cross sections, and ${G}_{2}$ stands for $\ensuremath{\nu}{W}_{2}({q}^{2}, \ensuremath{\nu})$ in the Bjorken limit. The sufficient condition is then found to be either $R\ensuremath{\simeq}k\frac{{\ensuremath{\omega}}^{2}}{{q}^{2}}(k<\frac{1}{2})$ or $R\ensuremath{\propto}\frac{1}{{q}^{2+\ensuremath{\delta}}} (\ensuremath{\delta}>0)$, as ${q}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$. In consideration of the experimental fact that the ratio of structure functions, $\frac{{W}_{2,n}}{{W}_{2,p}}$, in the range where the greater part of the contribution to the relevant integral results is not far away from the threshold value as $\frac{\ensuremath{\omega}}{2M}\ensuremath{\rightarrow}1$ predicted by Bloom and Gilman, the deep-inelastic part of the $n\ensuremath{-}p$ mass difference is effectively written in the form of the magnetic-moment-type self-energy. It is also shown that if this is similarly applicable to other baryons, and if the SU(3) magnetic-moment relations hold, the correct signs and right orders of magnitude of the mass differences ${\ensuremath{\Sigma}}^{\ensuremath{-}}\ensuremath{-}{\ensuremath{\Sigma}}^{+}$, ${\ensuremath{\Xi}}^{\ensuremath{-}}\ensuremath{-}{\ensuremath{\Xi}}^{0}$, as well as $\frac{1}{2}({\ensuremath{\Sigma}}^{+}+{\ensuremath{\Sigma}}^{\ensuremath{-}})\ensuremath{-}{\ensuremath{\Sigma}}^{0}$, are reproduced by the theory with no adjustable parameter except an input of the observed $n\ensuremath{-}p$ mass difference.