## On neutrino mixing in Quantum Field Theory

##### Abstract

Neutrino physics is one of the most important areas in research on the fundamental interactions, both theoretically and experimentally. A central issue in the study of these particles is related to the properties of mixing and ﬂavor oscillations. In particular, a necessary condition for the mixing is that neutrinos have mass, a feature which is not considered by the Standard Model, and which poses the problem of the origin and nature of these masses. The theoretical basis of neutrino mixing has been studied in many details and in the late ’90 a quantum ﬁeld theory (QFT) formalism for mixed ﬁelds has been developed thanks to which it was found that mixing transformations induce a condensed structure in the ﬂavour vacum with possible phenomenological consequences. In particular, the usual formulas of Pontecorvo oscillation are arising as the relativistic limit of exact formulas in the context of ﬁeld theory. Within this framework it is inserted our work, ﬁnalized to the study of algebraic properties of the mixing transformations generator in QFT and its components. In quantum mechanics (QM) the mixing transformation looks like a rotation operating on massive neutrino states. We show explicitly that such a rotation is not suﬃcient for implementing the mixing transformation at level of ﬁelds. It is necessary, in fact, also the action of a Bogoliubov transformation which operates a suitable mass shift. Such a property of Bogoliubov transformations has been already known and used since long time, e.g. in renormalization theory or in the dynamical generation of mass. We then analyze the condensate nature of the ﬂavor vacuum and the roˆle played by the non commutativity between the rotation and the Bogoliubov transformation. This structure of the vacuum also suggests a thermodynamical interpretation which we investigate, showing peculiarities in the thermal behavior due to
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the character of the particle-antiparticle condensate involved in the ﬂavor vacuum. The key point in our analysis is the non commutativity between rotation and Bogoliubov transformations, a feature which turns out to be at the origin of the inequivalence among mass and ﬂavor vacua. From another point of view, the Bogoliubov transformation are shown to naturally arise when studying the neutrino mixing in the contest of the Noncommutative Spectral Geometry in Alain Connes’ construction and the algebra doubling he introduces. Given the algebraic nature of our arguments, we have good reasons to believe that the results we have obtained are general and also extend to the mixing phenomenon of any particle, even if our analysis is limited to the case of two Dirac neutrinos.
Thethesisisstructuredasfollows: afteranIntroduction tothearguments of our work is presented; in Chapter 1 we brieﬂy summarise the main aspects of the Standard Model and of neutrino mixing. Chapter 2 is dedicated to neutrino mixing and oscillations, considering both the quantum mechanics approach and the quantum ﬁeld theory formalism. In Chapter 3 we analyze the mixing generator, decomposing it into components, and the ﬂavor vacuum structure, studying its thermodynamical properties. A noncommutative structure will arise. Chapter 4 is focused non commutativity, giving some known examples of noncommutative systems, and brieﬂy presenting noncommutative geometry and noncommutative spectral geometry (NCSG) elements. In Chapter 5 we introduce further notions on Alain Connes’ construction; we summarize how neutrinos appear within this construction and werelatethealgebradoubling,whichisacrucialelementoftheNCSGmodel, totheHopfnon-commutativealgebraandBogoliubovtransformations,which play a key role in the neutrino mixing. In Chapter 6, in order to better understand the mixing phenomenon, we study a classical system analogue for it. We then close with our Conclusions and Outlook.
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The results obtained and presented in this thesis have been published in the following international journals: • M.V. Gargiulo, M. Sakellariadou, G.Vitiello, Doubling of the Algebra and Neutrino Mixing within Noncommutative Spectral Geometry, EPJ C (2014) 74 2695 ; • M.V. Gargiulo, M. Sakellariadou, G.Vitiello, Noncommutativespectralgeometry, Bogoliubovtransformationsandneutrino oscillations, Journal of Physics: Conference Series (2015) 626 012014 ; • M.Blasone, M.V.Gargiulo and G.Vitiello, Disentangling mass and angle dependence in neutrino mixing, Journal of Physics: Conference Series (2015) 626 012026 ; • M. Blasone, M.V. Gargiulo, G. Vitiello, On the role of rotations and Bogoliubov transformations in neutrino mixing, Phys. Lett. B (2016) 761 104; • M.Blasone, M.V. Gargiulo, G.Vitiello, Semiclassical aspect of neutrino mixing Journal of Physics: Conference Series (2017), in press.
Work in progress • M.Blasone, M.V. Gargiulo, G.Vitiello, Noncommutative aspects of neutrino mixing, work in progress. [edited by author]