In this paper we provide an updated analysis of the neutrino magnetic moments (NMMs), discussing both the constraints on the magnitudes of the three transition moments Lambda_i as well as the role of the CP violating phases present both in the mixing matrix and in the NMM matrix. The scattering of solar neutrinos off electrons in Borexino provides the most stringent restrictions, due to its robust statistics and the low energies observed, below 1 MeV. Our new limit on the effective neutrino magnetic moment which follows from the most recent Borexino data is 3.1 x 10^-11 mu_B at 90% C.L. This corresponds to the individual transition magnetic moment constraints: |Lambda_1| < 5.6 x10^-11 mu_B, |Lambda_2| < 4.0 x 10^-11 mu_B, and |Lambda_3| < 3.1 x 10^-11 mu_B (90% C.L.), irrespective of any complex phase. Indeed, the incoherent admixture of neutrino mass eigenstates present in the solar flux makes Borexino insensitive to the Majorana phases present in the NMM matrix. For this reason we also provide a global analysis including the case of reactor and accelerator neutrino sources, and presenting the resulting constraints for different values of the relevant CP phases. Improved reactor and accelerator neutrino experiments will be needed in order to underpin the full profile of the neutrino electromagnetic properties.
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