The neutron-proton mass difference

Abstract

We present a lattice calculation of the mass difference between neutron and proton, for which we find $M_n-M_p=1.73(69)\text{MeV}$. This is obtained at 1st order in the $QED$ coupling ${\alpha }_{EM}$ and in the mass difference between $u$ and $d$ quarks $\frac{m_d-m_u}{{\Lambda }_{QCD}}$. We adopt a purely hadronic scheme to renormalize the theory and provide a prescription to separate the $QED$ and strong $IB$ contributions. The simulation is carried out using the ETMC gauge configurations with $N_f=2+1+1$ dynamical quarks. We extrapolate among $3$ values of the lattice spacing and pion masses in the range $M_{\pi }\simeq 200-450$ MeV.