  We study the long-range force arising between two neutral---of electric
charge---aggregates of matter due to a neutrino-pair exchange, in the limit of
zero neutrino mass. The conceptual basis for the construction of the effective
potential comes from the coherent scattering amplitude at low values of t. This
amplitude is obtained using the methodology of an unsubtracted dispersion
relation in t at threshold for s, where (s, t) are the Lorentz invariant
scattering variables. The ultraviolet behavior is irrelevant for the long-range
force. In turn, the absorptive part in the t-dependence is given by the
corresponding unitarity relation. We show that the potential describing this
force decreases as $r^{-5}$ at large separation distance r. This interaction is
described in terms of its own charge, which we call the weak flavor charge of
the interacting systems, that depends on the flavor of the neutrino as $Q_W^e =
2Z-N$, $Q_W^\mu = Q_W^\tau = -N$. The flavor dependence of the potential
factorizes in the product of the weak charges of the interacting systems, so
that the resulting force is always repulsive. Furthermore, this charge is
proportional to the number of constituent particles, which differs from the
global mass, so this interaction could be disentangled from gravitation through
deviations from the Equivalence Principle.
