QCD sum rules and neutron-proton mass difference

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 ($m_u\ne m_d$), and distinguish between $〈0|:\overline{u}u:|0\mathrm{}〉$ and $〈0|:\overline{d}d:|0\mathrm{}〉$, the condensates of the up and down quarks. Using the typical current-quark masses $m_u=5.1\mathrm{}$ MeV and $m_d=8.9\mathrm{}$ 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 $M_n-M_p=(1.35\mathrm{}\pm 0.24) \mathrm{MeV} or (1.42\mathrm{}\pm 0.19) \mathrm{MeV}$, depending on the method of the analysis. Analogously, the difference equation from the unity sum rules yields $M_n-M_p=(1.35\mathrm{}\pm 0.35) \mathrm{MeV} or (0.95\mathrm{}\pm 0.25) \mathrm{MeV}$. These predictions are consistent among themselves and all are in reasonable agreement with the experimental value of 1.29 MeV.