Calculation of $<p|\overline{u}u-\overline{d}d|p>$ from QCD sum rule and the neutron-proton mass difference

Abstract

The proton matrix element of the isovector-scalar density, $<p|\overline{u}u-\overline{d}d|p>/2M_p$, is calculated by evaluating the nucleon current correlation function in an external isovector-scalar field using the QCD sum-rule method. In addition to the usual chiral and gluon condensates of the QCD vacuum, the response of the chiral condensates to the external isovector field enters the calculation. The latter is determined by two independent methods. One relates it to the difference between the up and down quark chiral condensates and the other uses the chiral perturbation theory. To first order in the quark mass difference $δm=m_d-m_u$, the non-electromagnetic part of the neutron-proton mass difference is given by the product $δm <p|\overline{u}u-\overline{d}d|p>/2M_p$; the resulting value is in reasonable agreement with the experimental value.