  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.
