Abstract

1. Substantiating the Four-Neutrino Hypothesis from the Gösgen ExperimentA.C. Foster and D.J. ErnstDepartment of Physics and Astronomy, Vanderbilt University Nashville, TN Abstract Neutrino Oscillations – Probability The Simulation Recent evidence from nuclear reactor experiments indicate unique, non-zero neutrino masses per flavor state; furthermore, data also reveals the existence of neutrino oscillations as a result of the superposition of their mass and flavor eigenstates. Despite this extension beyond the Standard Model of particle physics, not all of the world's data may be ultimately corroborated in terms of the three-neutrino model. As a result, a fourth, light, “sterile” neutrino uncoupled from the Z boson [See FIG. I] has been posited, limited by a lower mass-squared difference of Δm2 sterile > 1 eV2, following from experiment.1 Here, a FORTRAN 95 program simulation was created in order to obtain the data reported from an electron antineutrino vanishing experiment conducted in Gösgen, Switzerland, using a 2800-MW thermal nuclear reactor at varying distances from a proton-rich liquid scintillator detector, and a two-neutrino mixing model. The output was formatted specifically for easy importing of the data into Gnuplot. After careful debugging procedures, we will analyze the data with a three-neutrino model, and then finally a four-neutrino model, having generated 2x2 contour plots with χ2 magnitudes for the mixing parameters Δm2 and sin2 2θ. Based on similar analyses of other experimental data, we expect that the four-neutrino hypothesis will be a more cogent explanation of the reactor data recorded worldwide. which necessarily implies that the energies of the different mass states cannot be equal. After an arbitrary time t, the time evolution of the initial beam therefore becomes Zacek et al. define a χ2 expression to test the cogency of a particular oscillation hypothesis5: (EQ.9) (EQ.3) where Yjexpt(Ek) are the empirical positron yields at distances Lj with σj,k as the associated uncertainties per measurement per energy bin. The detector energy bins are represented via k = 1, …, 16. The bins are evenly spaced at 0.305 MeV with lower and upper limits at 0.877 and 5.452 MeV. The parameters Njare the appropriate normalizations with associated uncertainties σj. For j = 1, 2, σj = ± 0.015; for j = 3, σj = ± 0.030. The expected yields, which are supplied in the literature and follow from Monte Carlo simulation, are thus compared to the experimental yields through EQ. 9. For 51 variances, minus 6 controlled parameters (3 of which are, unfortunately, not defined in the literature), 45 degrees of freedom exist. The simulation is conceptually straightforward and currently departs very scantly from the described theory. Prior to running the program, the operator is to set the number of integration points for computation, along with oscillation parameter bounds, in order to specify a plot resolution. The program in essence runs through various oscillation parameter values based on the programmer's specifications; the idea is that one should make a 2x2 exclusion contour plot with associated χ2 for magnitudes of 4.61, 5.99, and 9.21 for two degrees of freedom and confidence levels of 90%, 95%, and 99%, respectively. To do so, the program must implement the empirical and expected spectra computations, the sigmas, individual bin energies, and the three detector distances from the neutrino source. To determine the appropriate χ2 magnitudes, an integration loop was defined, and a dummy variable was used to integrate over the detector energy bins for each experimental propagation distance – a total of 16 * 3 = 48 loops each time the function is called and passed parameter values. Although not listed here, the code for the program as a whole is partially available upon request. Finally, the computed data are stored in space-delimited text files arranged in three columns reserved each for mass, mixing angle, and associated χ2. The data are then read into the Gnuplot software, plotted, and then saved as a PNG image file for analysis. Within the Gnuplot console, the contour parameter levels are set manually at 4.61, 5.99, and 9.21. Also, the standard Gnuplot contour is a 3D rendering; for our purposes, one must use a command to cut away the 3D component of the contour, and then make Gnuplot view the base of the contour like a topographical map, with only the indicated magnitudes displayed. If one wished to find the amplitude of discovering a neutrino in state l' at time t, one invokes the following: (EQ.4&5) Introduction From the above developments, suffice it to say for the sake of brevity that the probability of finding this neutrino at a distance (m) L ≈ ct in a state l' is Within the past decade, the existence of neutrino oscillations (and, therefore, neutrino mixing) has been experimentally verified. The resulting data from experiments conducted worldwide bear striking implications not only for the contemporary particle physics zeitgeist, but also modern cosmology and astrophysics. Furthermore, although the existence of three neutrino flavors has also been confirmed, not all of the experimental data can be corroborated by the current three neutrino model. Consequently, a fourth neutrino, sometimes referred to as the “sterile” neutrino, has been postulated in order to explain apparent flux deficits in existing reactor data. The sterile neutrino is hypothesized to be a neutral lepton with no ordinary weak interactions (thus the name) except those induced by mixing.2 In addition, it turns out that light (Δm2 ≈ 1 eV2) sterile neutrinos happen to be good candidate constituent particles of dark matter. Here the concern is with neutrinos produced during nuclear fission in a nuclear power reactor located in Gösgen, Switzerland. The researchers at Gösgen, Zacek et al., analyzed their recorded data (positron energy spectra) in terms of a two-neutrino mixing model, for the oscillation parameters Δm2 and sin2 2θ. The ultimate purpose of this endeavor is to extend the analysis via computational means to three- and four-neutrino models for those same mixing parameters, which will perhaps further support the four-neutrino hypothesis when combined with similar analytical extensions of data obtained from other reactors around the world. (EQ.6) where E is neutrino energy in MeV, and Δm2 b,a = | mb2 - ma2 | in eV2. The probability equation defined above can apply to both appearance and disappearance experiments. Based on the mixing parameters for a two-dimensional unitary mixing matrix parameterized by a mixing angle θ as defined below (EQ.7) and considering the fact that the Gösgen experiment is of the disappearance type, the probability of finding the original weak-interaction eigenstate at some distance L from the source and for some energy E of the neutrino may be shown to be equal to3 Neutrino Oscillations – Mixing (EQ.8) Fortunately, the germane theory is conceptually simple and elegant. Given the fact that unique, non-zero masses exist per neutrino flavor state, neutrino mass eigenstates are constrained to propagate at different speeds, leading to linear superpositions of the physical mass eigenstates. Consider a neutrino beam created by the charged current weak interaction. A neutrino, νl, of state l in the beam is not a physical quantum state, but rather a linear superposition of the physical fields νa, each with their own masses ma, where the subscript a = e, μ, τ, … indicates weak interaction eigenstates. Thus, neutrino mixing may be defined by Conclusion Experimental Detection Method in Brief The program, some 900 lines of code, is still under development. Although capable of generating exclusion plots (see the image to the right for an example of a contour), minor problematic artifacts crop up in the end result which contradict expectation, calling at least for careful debugging procedures. Upon weeding out these bugs, we expect that extending our results to three- and four-neutrino analyses will contribute to the data in support of a light, sterile neutrino. The FORTRAN 95 simulation is being designed in order to re-obtain the data recorded by Zacek et al. in the literature. Essentially, the energy spectra of electron antineutrinos, νe, emerging from a thermal nuclear reactor operating at 2800 MW at three reactor-detector distances Lj, j = 1, 2, 3 (corresponding to measurements of 37.9 m, 45.9 m, and 64.7 m respectively) were measured. The detector recorded positron energy spectra which result from the well-known β- decay reaction, νe + p →e+ + n via 30 liquid scintillator-filled cells arranged in five planes.4 (EQ.1) U being a unitary mixing matrix signifying neutrino mixing. Given their relativistic nature, the energy for component va is given by Acknowledgments Dr. David Ernst, the Vanderbilt Dept. of Physics and Astronomy, and Vanderbilt University, along with Alyce Dobyns and the VU 2014 REU students. UNCP RISE. This work was supported by grant  #5R25GM077634-04 from the NIGMS (National Institute of General Medical Sciences) supporting the UNCP RISE Program. FIG. I: Neutrinos and their antiparticle counterparts interact with matter via coupling to the Z gauge boson, according to the Standard Model. (EQ.2) References • 1. 1 K. Abazajiana et al. (2012). “Light Sterile Neutrinos: A White Paper”. 1-2. • 2. Ibid., 3-4. • 3. G. Zacek et al. "Neutrino-oscillation Experiments at the Gösgen Nuclear Power Reactor." Physical Review D, 1986, 2622. • 4. Ibid., 2625. • 5. Ibid., 2632.