  The recent confirmation by FNAL of the $(g-2)_\mu$ muon anomaly gives strong
evidence for the possible existence of new physics beyond the Standard Model in
the muon sector. Thus it is worthy to revisit the existing experimental
constraints on models suggesting theoretically consistent explanations of the
anomaly. In this work, we point out that accounting for the loss of coherence
between the wave packets(mass states) of solar neutrinos is important for
setting limits on any model with new flavor-sensitive couplings in the neutrino
sector. By taking into account this effect and considering more accurately the
experimental constraints from the BOREXINO measurement of the $^7$Be solar
neutrino interaction rate we corrected the limits previously placed on the
coupling of the light $Z'$ to $L_\mu$-$L_\tau$ current in the parameter space
relevant to the muon $(g-2)_\mu$.
