Parametrized relativistic dynamical framework for neutrino oscillations

Mass state transitions are a key feature of parametrized relativistic dynamics (PRD). PRD is a manifestly covariant quantum theory with invariant evolution parameter. The theory has been applied to neutrino flavor oscillations between two mass states. It is generalized to transitions between three mass states and applied to the survival of electron neutrinos. The analysis shows that significant differences exist between theoretical results of the conventional model and the PRD model.


Introduction
Experiments with solar neutrinos, atmospheric neutrinos, reactor neutrinos, and accelerator neutrinos have demonstrated that flavor mixing can occur between two or three neutrino flavors composed of up to three neutrino mass states [1,2]. Mass state transitions are a key feature of parametrized relativistic dynamics (PRD). Introductions to PRD are presented by Fanchi [3,4], Pavsic [5], and Horwitz [6]. A review of the highlights of the history of neutrinos is presented to establish a framework for analysing neutrino oscillations in the context of PRD. We then extend the PRD formalism for two-state flavor mixing [7] to three-state flavor mixing. The formalism is illustrated by applying it to the neutrino transition P Q Q o e from an electron neutrino e Q to a muon neutrino P Q in vacuum. In this application we assume the transition W Q Q o e from an electron neutrino e Q to a tau neutrino W Q is negligible and that only two flavor states are involved. Calculation of electron neutrino survival probabilities using both the conventional analysis and PRD shows that there are significant differences between the two theories.

The Neutrino Hypothesis
We begin our review of some highlights of the history of neutrinos with the observation that particles can be emitted by radioactive elements. Wilhelm Roentgen [8] observed a new kind of radiation called x-rays. Roentgen's x-rays could pass through a human body and create a shadowy image of bone. Henri Becquerel learned about x-rays and began to look for a connection between x-rays and the phosphorescence of a uranium salt. Becquerel [9] published a series of articles about his discovery ofa The beta decay process was evidence that the neutron was not a stable particle. Fermi [20] adopted Pauli's "neutron" concept, which he renamed "neutrino," and hypothesized a 4-point beta decay in which the neutron decayed into a proton, an electron, and a neutrino. The study of cosmic rays led Sakata and Inoue [23] to suggest that there may be two neutrinos: the electron neutrino ߥ associated with electrons, and the muon neutrino ߥ ఓ associated with muons discovered in cosmic rays.

Detecting the Neutrino
It was reasonable to infer from Fermi's 4-point theory that the neutrino could interact with other particles in such a way that it could be detectable. One interaction is beta decay in which the neutron decays into a proton, an electron, and an electron antineutrino: ݊ ՜ ‫‬ ା ݁ ି ߥ ഥ . Another possible interaction is inverse beta decay in which an electron antineutrino interacts with a proton to form a neutrino and a positron: ߥ ഥ ‫‬ ା ՜ ݊ ݁ ା .The electron capture process occurs when an electron in a lower energy level of an atom is captured by a proton in the nucleus to produce a neutron and a neutrino: ݁ ି ‫‬ ା ՜ ݊ ߥ . It was not clear at the time if the neutrino and the antineutrino behaved differently when interacting with matter. Two possible beta emission processes were positive beta emission ‫‬ ା ՜ ݊ ݁ ା ߥ in which a proton transformed into a neutron, a positron and a neutrino, and negative beta ഥ which may be viewed as the transformation of a neutron inside the nucleus to a proton: Bruno Pontecorvo [24] proposed detecting neutrinos by allowing the neutrino to interact with a proton or neutron of a nucleus heavier than hydrogen so that Z > 1. The interaction would change a proton into a neutron and yield an isotope of a new element. Neutrino detection would be achieved by starting with a large amount of a pure isotope and look for the formation of a neutrino-generated element as a function of time. One reaction proposed by Pontecorvo was the reaction in which a neutrino interacts with a proton in a chlorine nucleus to produce an electron and an argon nucleus: . Ray Davis performed the experiment using radiation emitted by different reactors and detectors containing carbon tetrachloride.
Nuclear reactors emit antineutrinos from negative beta decays of fission products.
assumed a neutrino would interact with a proton. Davis was aware that his experiment was testing the relative performance of neutrinos and antineutrinos: "If neutrinos and antineutrinos areidentical in their interactions with nucleons one shouldbe able to observe the process upon carrying the experimentto the required sensitivity. However, if neutrinosand antineutrinos differ in their interactions withnucleons one would not expect to induce the reaction ‫݈ܥ‬ ." [25, pg. 766] Assuming the experiment had adequate sensitivity, Davis argued that a null result using chlorine in the detector implied that the interaction between neutrinos and nucleons is not the same as the interaction between antineutrinos and nucleons.His early results were not conclusive [25,26]. The first tentative observation of neutrinos was made by Reines and Cowan in 1953 at the Hanford nuclear reactor [27] on the Columbia River in the state of Washington. Electron neutrinos from nuclear fission reactors have since been identified as electron antineutrinos. Reines and Cowan observed the inverse beta decay process in which an electron antineutrino interacted with a proton to from a neutrino and a positron ߥ ഥ ‫‬ ା ՜ ݊ ݁ ା .The positron produced by the reaction interacted with an electron in a pair-annihilation process to yield two gamma rays in the reaction ݁ ା ݁ ି ՜ ʹߛ. The Hanford work identified gamma rays from pair annihilation, but this was not considered sufficiently conclusive because the gamma ray signal could not be distinguished from cosmic ray signals. The experiment was moved to the Savannah River Nuclear Reactor facility near Augusta, Georgia where it could be shielded from cosmic rays. The experiment was also modified to include a neutron absorber (cadmium chloride) in the scintillation tank. The neutron produced in the inverse beta decay process was absorbed by a cadmium nucleus in the neutron capture process ݊ ‫݀ܥ‬ ସ଼ ଵ଼ ՜ ‫݀ܥ‬ ସ଼ ଵଽ ߛ. Photomultiplier tubes in the scintillation tank should be able to detect gamma rays from the pair-annihilation process approximately 5 microseconds before gamma ray signals from the neutron capture process.The neutrino signature was the combination of the detection of gamma rays from pair-annihilation followed by the detection of gamma rays from the neutron capture process. Cowan, Reines and collaborators [28] reported the direct detection of the electron neutrino in 1956.

Neutrino Oscillations
Ray Davis continued to improve his experimental techniques. Together with his colleagues, Davis increased the volume of ‫݈ܥ‬ ଵ ଷ in his detector and moved the detector underground to the Homestake gold mine in South Dakota. His goal was to minimize extraneous signals from cosmic rays as he sought to measure neutrino production by fusion reactions in the interior of the sun. Davis [29] found that solar neutrino flux was much lower than predicted by John Bahcall [30]. Bahcall and colleagues [31,32] refined their theoretical predictions, but additional measurements by Davis and colleagues [32] showed that observed solar neutrino flux wasstill lower than predicted. Where were the missing solar neutrinos?
Work with kaons had shown that kaons could oscillate between neutral kaon ‫ܭ‬ and neutral anti-kaon ‫ܭ‬ തതതത flavor states [33]. The observable kaons were the long-lived kaon state (symmetric superposition ‫ܭ‬ ൌ ଵ ξଶ ൫‫ܭ‬ ‫ܭ‬ തതതത ൯) and short-lived kaon state (anti-symmetric superposition ‫ܭ‬ ௌ ൌ ଵ ξଶ ൫‫ܭ‬ െ ‫ܭ‬ തതതത ൯). By analogy, Pontecorvo suggested that the electron neutrino ߥ , and the muon neutrino ߥ ఓ could oscillate between states [34,35]. Danby, et al. [36] showedthat there is experimental evidence that there are at least two different types of neutrinos.Maki, Nakagawa and Sakata [37] presented a theory of 2-neutrino mixing between the muon neutrino and electron neutrino. A third type of neutrino was proposed by Perl, et al. [38]. They were the first to report evidence of the tau lepton and suggested that a third neutrino, the tau neutrino, may be associated with the tau lepton. Thetau neutrino was observed in 2000 by the DONUT(Direct Observation of Nu Tau) collaboration at Fermilab in the USA and Super-Kamiokande in Japan [39]. Super-Kamiokandebegan operation in 1996. The collaboration detected oscillations in atmospheric neutrinos by 1998 [40], which implies that neutrinos have mass.

Transitions between Flavor States: Electron Neutrino Disappearance
The formalism for three mass states presented in the previous section is illustrated by applying the formalism to the disappearance of electron neutrinos. We begin with a pure beam of electron neutrinos

Application to Neutrino Oscillation Experiments
The evolution equation in PRD for a state may be written in terms of the evolution parameter s as where j K is the eigenvalue of the mass operator for mass state j . The evolution parameter dependent solution of Eq. (5.1) in the mass basis for two mass states is » Flavor oscillations may be described by quantifying the behavior of two particles. One particle propagates without interaction or oscillation from the source to the detector and serves as a "clock" for the scalar evolution parameter s. The other particle is the oscillating particle. In this application, the source and detector are separated by a distance L . The most probable trajectory of the non-interacting s-clock particle is The distance x G traveled by the s-clock particle in the interval t G is L , so we obtain > @