QCD sum rules and neutron-proton mass difference.

Abstract

We use the method of QCD sum rules to investigate the neutron-proton mass difference. We include diagrams consistently up to dimension 9, assuming different up and down current-quark masses (${m}_{u}\ensuremath{\ne}{m}_{d}$), and distinguish between $〈0|:\overline{u}u:|0〉$ and $〈0|:\overline{d}d:|0〉$, the condensates of the up and down quarks. Using the typical current-quark masses ${m}_{u}=5.1$ MeV and ${m}_{d}=8.9$ MeV and the standard condensate values for average current-quark masses, we perform numerical analyses of the resultant QCD $p$ and unity sum rules. In particular, numerical analyses of the difference equation from the $p$ sum rules yield ${M}_{n}\ensuremath{-}{M}_{p}=(1.35\ifmmode\pm\else\textpm\fi{}0.24) \mathrm{MeV} or (1.42\ifmmode\pm\else\textpm\fi{}0.19) \mathrm{MeV}$, depending on the method of the analysis. Analogously, the difference equation from the unity sum rules yields ${M}_{n}\ensuremath{-}{M}_{p}=(1.35\ifmmode\pm\else\textpm\fi{}0.35) \mathrm{MeV} or (0.95\ifmmode\pm\else\textpm\fi{}0.25) \mathrm{MeV}$. These predictions are consistent among themselves and all are in reasonable agreement with the experimental value of 1.29 MeV.