The coherent weak charge of matter

We study the long-range force arising between two aggregates of ordinary matter due to a neutrino-pair exchange, in the limit of zero neutrino mass. Even if matter is neutral of electric charge, it is charged for this weak force. The interaction is described in terms of a coherent charge, which we call the weak flavor charge of aggregated matter. For each one of the interacting aggregates, this charge depends on the neutrino flavor as $Q_W^{\nu_e} = 2Z-N$, $Q_W^{\nu_\mu} = Q_W^{\nu_\tau} = -N$, where $Z$ is the number of protons and $N$ the number of neutrons. $Q_W^{\nu_e}$ depends explicitly on $Z$ because of the charged current contribution to $\nu_e e$ elastic scattering, while the $N$ term in the three charges comes from the universal neutral current contribution. The effective potential describing this force is repulsive and decreases as $r^{-5}$. Due to its specific behavior on $(Z,N)$ and $r$, this interaction is distinguishable from both gravitation and residual electromagnetic forces. As neutrinos are massive and mixed, this potential is valid for $r\lesssim 1/m_\nu$, distances beyond which a Yukawa-like attenuation kicks in.