Tau-neutrino charged-current threshold ($E_{\mathrm{thr}}=((m_{\tau }+M_X{)}^2-M_T^2)/(2M_T)$)

For a nearly massless incident tau neutrino and stationary target of mass $M_T$, the initial Mandelstam variables give $s=M_T^2+2M_T{E}_{\nu }$. At threshold the final tau lepton and the lightest allowed recoil system have no relative momentum, so $s=(m_{\tau }+M_X{)}^2$. This proves the formula. A free neutron recoil channel gives approximately $3.45\mathrm{GeV}$; an electron target has a different recoil and a threshold near $3.09\mathrm{TeV}$.