Collective neutrino-pair emission due to Cooper pairing of protons in superconducting neutron stars

The neutrino emission due to formation and breaking of Cooper pairs of protons in superconducting cores of neutron stars is considered with taking into account the electromagnetic coupling of protons to ambient electrons. It is shown that collective response of electrons to the proton quantum transition contributes coherently to the complete interaction with a neutrino field and enhances the neutrino-pair production. Our calculation shows that the contribution of the vector weak current to the $\nu \bar{\nu}$ emissivity of protons is much larger than that calculated by different authors without taking into account the plasma effects. Partial contribution of the pairing protons to the total neutrino radiation from the neutron star core is very sensitive to the critical temperatures for the proton and neutron pairing. We show domains of these parameters where the neutrino radiation, caused by a singlet-state pairing of protons is dominating