  Neutrino emission caused by Cooper pairing of baryons in neutron stars is
recalculated by accurately taking into account for conservation of the vector
weak current. The vector current contribution to the neutrino emissivity is
found to be several orders of magnitude smaller than that obtained before by
different authors. Therefore, the neutrino energy losses due to singlet-state
pairing of baryons can in practice be neglected in simulations of neutron star
cooling. This makes negligible the neutrino radiation from pairing of protons
or hyperons. The neutrino radiation from triplet pairing takes place through
axial weak currents. For these states, when the total momentum projection is
$m_{j}=0$, the vanishing of the vector weak current contribution results in the
suppression of the neutrino energy losses by about 25%. The neutrino emissivity
due to triplet pairing with $| m_{j}| =2$ is suppressed by about a factor of 3,
caused by the collective contribution of spin-density fluctuations in the
condensate.
