Remark on neutrino oscillations

Abstract

The oscillations of ultra-relativistic neutrinos are realized by the propagation of assumed zero-mass on-shell neutrinos with the speed of light in vacuum combined with the phase modulation by the small mass term exp[-i(mνk2/2|p→|)τ]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp [-i(m^{2}_{\ u _{k}}/2|\\vec {p}|)\ au ]$$\\end{document} with a time parameter τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. This picture is realized in the first quantization by the mass expansion and in field theory by the use of δ(x0-y0-τ)⟨0|T⋆νLk(x)νLk(y)¯|0⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta (x^{0}-y^{0}-\ au ) \\langle 0|T^{\\star }\ u _{L k}(x)\\overline{\ u _{L k}(y)}|0\\rangle $$\\end{document} with the neutrino mass eigenstates νLk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _{L k}$$\\end{document} and a finite positive τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} after the contour integral of the propagating neutrino energies. By noting that the conventional detectors are insensitive to neutrino masses, the measured energy-momenta of the initial and final states with assumed zero-mass neutrinos are conserved. The propagating neutrinos preserve the three-momentum in this sense but the energies of the massive neutrinos are conserved up to uncertainty relations and thus leading to oscillations. Conceptual complications in the case of Majorana neutrinos due to the charge conjugation in d=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=4$$\\end{document} are also discussed.