**... but theory is still under [discuss]**~~{construct}~~ion

Neutrino physics is one of the most interesting and vividly discussed topics in high-energy physics today. Especiallythe question whether the neutrinos can oscillate or not (i.e. different neutrinos can change into each other) gave rise to a huge number of experiments to actually observe these oscillations. At least since the results from the Super Kamiokande ... and the SNO experiment ... are published, it is widely believed that neutrino oscillations are an experimentally verified fact. However, the first hint has already been found in 1964 when the Homestake experiment ... discovered the solar neutrino problem. That is, the number of measured electron neutrinos from the sun is by a factor of 2-3 less than the number of neutrinos predicted by the standard solar model (SSM).

Since within the standard model (SM) of particle physics the neutrinos are massless, and consequently cannot oscillate, their measurement shows that new physics beyond the SM exists. And indeed nowadays the experiments on neutrino oscillations are important to measure the unknown parameters of the SM and its minimal extensions. In particular, these unknown parameters are the neutrino masses and the entries in the neutrino mixing matrix.

From all the measurements made to discoverneutrino oscillationsone should think that the theory behind [them] is well established and understood. But surprisingly this is not the case. The first who mentioned the idea ofneutrino oscillations, though he assumed neutrino-antineutrino oscillations, was Pontecorvo in 1957 [Pon57, Pon58]. A few years later Maki, Nakagawa and Saka were the first to consider oscillations between the electron and the muon neutrino [MNS62]. Then it took around 20 years before Kayser in 1981 showed that the up to that point used plane-wave approximation cannot hold for oscillating neutrinos and he proposed a wave packet treatment [Kay81], which then has again not been discussed for around 10 years. In the early 90s the discussion on the theoretical description of neutrino oscillations finally started with several seminal papers. First, Giunti, Kim and Lee explicitly calculated the oscillation probability for the neutrinos in a wave packet model [GKL91] and then showed that the state vectors used for the quantum mechanical description are, in general, ill-defined [GKL92]. In 1993 they published together with Lee a calculation of the probability in a quantum field theoretical framework without using state vectors for the neutrinos [GKLL93]. And finally, in 1995 Blasone and Vitiello showed that the description of mixed particles in quantum field theory (QFT) yields unexpected problems for the interpretation of neutrinos as particles. By only using exact—without perturbation—QFT methods they calculated an oscillation probability which differs significantly from the other results [BV95].All these different approaches are even today still under discussion, but however under the assumption of relativistic neutrinos which have tiny mass squared differences, all approaches give the same result. Thus, the theoretical discussion on the right description of the neutrinos does not spoil the experimental results, because today we are only able to measure ultra-relativistic neutrinos whose energy is at least a few orders of magnitude higher than their mass.

Diploma ThesisOn Theories of Neutrino Oscillations (Summary and Characterisation of the Problematic Aspects)Daniel Kruppke September 2007

**A quantum field theory for flavor states ...**

The study of mixing of fields of different masses in the context of Quantum Field Theory (QFT) has produced recently very interesting and in some sense unexpected results ...The story begins in 1995 when in Ref.[1], it was proved the unitary inequivalence of the Hilbert spaces for (fermion) fields with definite flavor on one side and those (free fields) with definite mass, on the other. The proof was then generalized to any number of fermion generations [7] and to bosonic fields [2, 5].This result strikes with the common sense of Quantum Mechanics (QM), where one has only one Hilbert space at hand: the inconsistencies that arise there have generated much controversy and it was also claimed that it is impossible to construct an Hilbert space for flavor states [16] (see however Ref.[6] for a criticism of that argument). In fact, not only the flavor Hilbert space can be consistently defined [1], but it also provide a tool for the calculation of flavor oscillation formulas in QFT ..., which exhibit corrections with respect to the usual QM ones [20, 21]. From a more general point of view, the above results show that mixing is an “example of non-perturbative physics which can be exactly solved”, as stated in Ref.[13]. Indeed, the flavor Hilbert space is a space for particles which are not on-shell and this situation is analogous to that one encounters when quantizing fields at finite temperature [22] or in a curved background [23]. In the derivation of the oscillation formulas by use of the flavor Hilbert space, both for bosons and for fermions, a central role is played by the flavor charges [9] and indeed it was found that these operators satisfy very specific physical requirements [6, 8].

(Submitted on 23 May 2003 (v1), last revised 10 Jun 2003 (this version, v2))

**... with an unfinished taste**

Blasone and Vitiello (BV) have attempted to construct a Fock space for neutrino flavor states [4]... Giunti conclude that “the Fock spaces of flavor neutrinos are ingenuous mathematical constructs without physical relevance” [3].

...there is another issue that plagues the scheme in [5]. The problem is that the neutrino flavor vacuum defined in [5] is time-dependent and hence Lorentz invariance is manifestly broken. Recently, BV and collaborators attempted to tackle this issue by proposing neutrino mixing as a consequence of neutrino interactions with an external non-abelian gauge field [7]. Under this framework, the Lorentz violation of the neutrino flavor vacuum can be attributed to the presence of a fixed external field which specifies a preferred direction in spacetime. However, at the moment, there is not a single sign of such a non-abelian gauge field in neutrino experiments. They proposed that this scheme can be tested in the tritium decay, but again the indefinite mass mνα becomes an observable quantity. Also, given the current stringent bounds on Lorentz violations [8], it is unclear whether this scheme will survive...

In this article,we first gave a detailed review on the current status of the understanding about the neutrino flavor states. At the end of the review, we were led to conclude that it is currently unclear how to construct a consistent and physically relevant Fock space of neutrino flavor states. We proceeded to prove that if one insists on second-quantizing the neutrino flavor fields and thereby constructing the flavor states, then they are approximately well-defined only when neutrinos are ultra-relativistic or the mass differences are negligible compared to energy...

However,we showed that one can consistently describe weak interactions by only neutrino mass eigenstates. At the same time, we argued that the second quantization of neutrino flavor fields generally lacks physical relevance because their masses are indefinite. Thus, neutrino flavor states lose their physical significance and they should simply be interpreted as definitions to denote specific linear combinations of mass eigenstates involved in weak interactions.Under this interpretation, there is no physical motivation to construct the Fock space of neutrino flavor states from the first principles of quantum field theory.

(Submitted on 16 Sep 2012 (v1), last revised 26 Nov 2012 (this version, v2))

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