The formalism for neutrino flavor change induced by lepton family number
violating interactions is developed for the three-neutrino case, and used to
derive the corresponding flavor change probabilities in matter. Applied to the
solar and atmospheric neutrino fluxes, it is argued that the observed
anomalies, including the zenith dependence for the atmospheric case, could be
due to such interactions.Comment: 6 pages, 1 figure Modifications: include limit from tau -> rho mu
  (leads to somewhat different conclusions) and a change to the figur

Brooijmans, G.

English

arXiv

The formalism for neutrino flavor change induced by lepton family number
violating interactions is developed for the three-neutrino case, and used to
derive the corresponding flavor change probabilities in matter. Applied to the
solar and atmospheric neutrino fluxes, it is argued that the observed
anomalies, including the zenith dependence for the atmospheric case, could be
due to such interactions.
 Comment: 6 pages, 1 figure Modifications: include limit from tau -> rho mu
  (leads to somewhat different conclusions) and a change to the figur

DIAL UCLouvain

arXiv:hep-ph/9808498v2  23 Nov 1998A Supersymmetric Solution to the Solar and Atmospheric Neutrino AnomaliesG. BrooijmansInstitut de Physique Nucle´aire, Universite´ catholique de Louvain, 1348 Louvain-la-Neuve, BelgiumInstitut Interuniversitaire des Sciences Nucle´aires, Belgium(Now at Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510)(February 1, 2008)The formalism for neutrino flavor change induced by lepton family number violating interac-tions is developed for the three-neutrino case, and used to derive the corresponding flavor changeprobabilities in matter. Applied to the solar and atmospheric neutrino fluxes, it is argued that theobserved anomalies, including the zenith dependence for the atmospheric case, could be due to suchinteractions.PACS Numbers: 13.15.+g, 11.30.Pb, 26.65.+tI. INTRODUCTIONThe deficit in the observed solar neutrino fluxes (see for example reference [1]), and the deviation of the atmospheric“ratio of ratios” from 1.0 [2] are two of the outstanding problems in today’s particle physics. These are usuallyinterpreted as being due to neutrino oscillations, for which, however, there is as yet no unambiguous evidence.In this paper, the possibility that both the solar and atmospheric neutrino anomalies could be due to the existence oflepton number violating interactions, and in particular supersymmetric R-parity violating interactions, is investigated.Whereas various authors [3–6] have recently suggested these interactions could lead to neutrino masses and mixingsleading to the “standard” vacuum oscillation solution, here possible direct, matter-induced flavor change is studied.II. FLAVOR CHANGING INTERACTIONS AND THE MSW EFFECTThe time dependency of the flavor composition of neutrinos propagating in matter is governed by the equationiddt νeνµντ = c11 +√2 GFNe b12 b13b12 c22 b23b13 b23 c33 νeνµντ , (1)where the term√2 GFNe – Ne is the electron density – corresponds to the usual νe− e− charged-current interactions(this is the term at the origin of the classic MSW effect [7,8]), and the cii and bij coefficients represent flavor-diagonal(FD) and flavor-changing (FC) interactions, respectively. Note that we have assumed CP conservation here (i.e. thepropagation matrix is symmetric). If such interactions exists, they would lead to additional electroweak potentialenergy for the propagating neutrino eigenstates, thereby opening the possibility for resonant enhancement of flavorchange. In the Standard Model, electroweak neutral currents are FD interactions, while FC interactions do not existin the leptonic sector.In supersymmetric theories with R-parity violation however, we find both new FD and FC interactions may exist.The two types of interactions that are relevant for neutrinos give rise to ν − d-type quark and ν − e scattering viad-type squark and selectron exchange, respectively. They correspond to the following two terms in the lepton numberviolating part of the superpotential:W = λijkLiLLjLEkR + λ′ijkLiLQjLDkR. (2)Here i, j and k are the generation indices, LL and QL are the left-handed lepton and quark doublet superfields, andER and DR are the right-handed lepton and quark singlets. Note that the λ couplings are antisymmetric in the firsttwo indices.For example, for neutrino energies small compared to the squark mass (mq˜), the termλ′131λ′3312m2b˜(νeLγµντLdRγµdR + ντLγµνeLdRγµdR)(3)1corresponds to flavor-changing neutrino scattering on a d-quark where the b-squark acts as the mediator of theinteraction. We can now rewrite the coefficients of the propagation matrix in equation (1) in terms of the R-parityviolating couplings. Taking only into account scattering on d quarks (matter is essentially first-generation) we findfor example:b12 =[λ′111λ′2114m2d˜+λ′121λ′2214m2s˜+λ′131λ′2314m2b˜+λ′112λ′2124m2s˜+λ′113λ′2134m2b˜+λ′121λ′2124m2s˜+λ′131λ′2134m2b˜+λ′112λ′2214m2s˜+λ′113λ′2314m2b˜]Nd, (4)where Nd is the down-quark density in matter. All the other matrix coefficients may be expressed likewise.Various authors have suggested [9,10] or established [11] that the solar neutrino problem could be explained by thepresence of non-standard FD and FC interactions. The solutions have recently been analyzed again by P.I. Krastevand J.N.Bahcall [12] taking into account the newly available solar neutrino data.However, all these studies only considered the solar neutrino problem, and used a simplified two-family model(involving the first and third generations). In this work, a 3-family analysis is developed, in the attempt to explainboth the solar and atmospheric neutrino anomalies.The procedure to derive the flavor change probabilities in the 3-family case is as follows. Let |να > be the flavoreigenstates (α = e, µ or τ) and |νi > the propagation eigenstates (i=1,2 or 3). We then have|να(t) >=3∑i=1Uαie−iHt|νi >=3∑i=1Uαie−ieit|νi >=3∑f=1(3∑i=1Uαie−ieit)U†fi|νf >, (5)where Uαi is a unitary rotation and the ei are the propagation energy eigenlevels, eigenvalues of the propagationmatrix given in equation (1). Since the elements of the propagation matrix, hence also its eigenvalues, depend onmatter density it is advisable to replace the time t by the distance traveled x with the approximation t = x. Theflavor changing amplitude is then given byA (να(x = 0)→ νβ(x = L)) =3∑i=1Uαie−ieiLU†iβ . (6)However, since the columns of the rotation matrix U which diagonalizes the propagation matrix are precisely theeigenvectors of this matrix,one finally obtainsP (να(x = 0)→ νβ(x = L)) =∣∣∣∣∣∣3∑j=1Vjαe−iejtV ∗jβ∣∣∣∣∣∣2, (7)where V is the matrix whose columns are the eigenvectors of the propagation matrix.III. DATACombining solar and atmospheric neutrino data yields 13 constraints: there are three different types of solar neutrinoexperiments, each sensitive to a different energy range, and the atmospheric “ratio of ratios” is measured in 5 zenithangle bins for two different energy ranges (sub- and multi-GeV).The three types of solar neutrino experiments (Gallium: Sage and Gallex, Chlorine: Homestake, and waterCˇerenkov: Super-Kamiokande) have different lower thresholds for the neutrino energy. This has allowed to showthat there seems to be an energy-dependent suppression of the solar neutrino flux (see for example N. Hata and P.Langacker [1]), which could be a strong hint for neutrino oscillations, were it not that neutrinos of different energiesare produced at different radii in the Sun and therefore “see” different matter densities on their way out.Table III summarizes the results from the various experiments (the results from the Gallium experiments havebeen combined into one), and gives the ratio of observed rates over rates predicted by the Bahcall-Pinsonneault 95Standard Solar Model [13]. The experimental results can be found in Refs. [14] (Homestake), [15] (Sage), [16] (Gallex)and [17] (Super-Kamiokande).2For the atmospheric neutrino data, the results published recently by the Super-Kamiokande Collaboration [2] arebeing used. Fogli et al. [18] have graphically extracted the numerical values from the Super-Kamiokande plots, yieldingthe observed number of data events and the expected number of events both for data and Monte-Carlo, such that allthe detection efficiencies can be taken into account. Table III gives the observed “ratio of ratios” (ratio of muon-likeover electron-like events for data divided by the same ratio for Monte-carlo) for each of the five zenith bins, both forthe sub- and multi-GeV samples. The first error is statistical [18] and the second is systematic [2].We have extracted upper limits on the values of the coefficients cii and bij from the present limits [19,20] on R-parity violating couplings. For reasons of simplicity, only the λ′ijk couplings (corresponding to scattering by squarkexchange) and scattering on d quarks are included, so that the limits are given by equations analogous to equation(4). These limits have then been transformed to limits on 6 variables ǫ11, ǫ12, ǫ13, ǫ22, ǫ23 and ǫ33 defined by:ǫii =cii√2 GFNd, ǫij =bij√2 GFNd, i 6= j, (8)which give the relative strength of the interaction w.r.t. the weak interaction.Furthermore, we can subtract c11× 1 from the right-hand side of equation (1). This does not affect our result sinceit is equivalent to a redefinition of the zero of the neutrino electroweak potential energy.IV. ANALYSIS AND RESULTSA. Analysis ProcedureTo find the values of the couplings which would produce effects similar to those observed in the data, a χ2-likefunction with 13 terms corresponding to the three types of solar neutrino experiments and the atmospheric datais constructed. The minimum of this χ2 is then searched for (using Minuit [21]), as a function of the 4 variablesǫ13, ǫ23, ǫ22 and ǫ33. ǫ12 has been excluded from the fit since the strong limits arising from the non-observation ofµ T i→ e T i [19] make its contribution to any of the two anomalies negligible.The Solar model used is the Bahcall-Pinsonneault Standard Solar Model [13], which “cuts” the Sun into layers,and for each of these layers, gives the average electron and neutron densities (the Sun is assumed neutral so that theproton density equals the electron density) as well as the fraction of neutrinos produced for each of the major neutrinosources (pp, 8B, 13N, 15O,7 Be and pep). Our program steps through each layer, adds the newly produced neutrinosand changes the flavor of a fraction of the entering and produced neutrinos in the manner determined by the flavorchange probability Eq. 7, which is a function of the variables and the densities.The Earth is approximated by 5 concentric spheres, each of uniform density and composition as has been doneby Giunti et al. [22]. To find the average distance that neutrinos travel through each layer as a function of theirmomentum and zenith angle, a small Monte-Carlo simulation has been developed. The simulation assumes a flatzenith angle distribution in each bin, smears it with the average angular correlation between the charged lepton andthe neutrino (55 and 20 degrees for sub-GeV and multi-GeV neutrinos, respectively [2]), and then tracks the neutrinosback through the Earth, computing the distance traveled through each layer. The values found are given in Table I.The algebraic expressions for the eigenvectors and eigenvalues of the propagation matrix as a function of its coeffi-cients are fairly complex. We have used Mathematica [23] to generate the expression in computer-readable form andsystematically checked that the orthogonality conditions are satisfied. This is done to verify that solutions are notproduced artificially due to machine accuracy.B. ResultsThe values of the parameters resulting in the lowest χ2 are given in Table II. Table III gives the expected value foreach of the experimental results for the best fit parameters. The minimum χ2 value found is 15.6, corresponding toa probability of 7.6%.The variables are somewhat correlated (as expected) such that changes in their values generally affect the width ofthe “good” region for the others. Figure 1 shows the region in parameter space allowed at 95 % and 99 % C.L. foreach combination of off-diagonal versus diagonal parameter.Since the value found for ǫ23 has to be significantly larger than what is allowed by the present non-observation ofthe decay τ → ρ0µ, the new interactions must either only apply to neutrinos, or suffer significant SU(2)L violation.It has recently been argued [24], however, that such violation could be in excess of present experimental bounds.3V. CONCLUSIONSWe have searched for, and found coupling values of a lepton family number violating interaction which would explainboth the solar and atmospheric neutrino anomalies through an MSW-like mechanism in the Sun and in the Earth.The fit to the data is not as good as for a purely neutrino oscillation solution, and the interaction must either onlyapply to neutrinos or undergo significant SU(2)L violation. It is however clear that such an interaction could play amajor role in explaining the anomalies.It should be noted that in order to have significant flavor change, neutrinos have to traverse large amounts ofmatter. This results in a large predicted difference between the atmospheric ratio of ratios for the first two and lastthree bins of the Super-Kamiokande multi-GeV azimuthal dependence histogram. Increased statistics may make thiseffect visible.VI. ACKNOWLEDGEMENTSThis work was finished while working at Fermi National Accelerator Laboratory. The author would like to thankJ. Govaerts and G. Gre´goire for useful discussions during this work.[1] N. Hata and P. Langacker, Phys. Rev. D 56, 6107 (1997).[2] Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81, 1562 (1998). to Phys. Rev. Lett. (1998).[3] M. Drees, S.Pakvasa, X.Tata, T. ter Veldhuis, Phys. Rev. D 57, 5335 (1998).[4] O.C.W. Kong, hep-ph/9808304 (1998).[5] B. Mukhopadhyaya, S. Roy and F. Vissani, hep-ph/9808265 (1998).[6] E.J. Chun, S.K. Kang, C.W.Kim and U. W. Lee, hep-ph/9807327 (1998).[7] L. Wolfenstein, Phys. Rev. D 17, 2369 (1978).[8] S.P. Mikheev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985).[9] M.M. Guzzo, A. Masiero and S.T. Petcov, Phys. Lett. B 260, 154 (1991).[10] E. Roulet, Phys. Rev. D 44, 935 (1991).[11] V. Barger, R.J.N. Phillips and K. Whisnant, Phys. Rev. D 44, 1629 (1991).[12] P.I. Krastev and J.N.Bahcall, talk given at the Symposium on Flavor Changing Neutral Currents: Present and FutureStudies (FCNC 97), Santa Monica, CA, 19-21 Feb 1997.[13] J.N. Bahcall and M. Pinsonneault, Rev. Mod. Phys. 67, 781 (1995).[14] B. T. Cleveland et al., Proc. of the 7th International Workshop on Neutrino Telescopes, Venice, Feb 27 - Mar 1 1996.[15] Sage Collaboration, J.N. Abdurashitov et al., Phys. Lett. B 328, 234 (1994).[16] Gallex Collaboration, W. Hampel et al., Phys. Lett. B 388, 384 (1996).[17] Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81, 1158 (1998).[18] G.L. Fogli, E. Lisi, A. Marrone and G. Scioscia, hep-ph/9808205 (1998).[19] P. Roy, talk given at the Pacific Particle Physics Phenomenology Workshop, Seoul, 31 Oct.-2 Nov., 1997.[20] J.E. Kim, P. Ko and D. Lee, Phys. Rev. D 56, 100 (1997).[21] MINUIT - Function Minimization and Error Analysis, (CERN Program Library Long Writeup D506), Application SoftwareGroup, Computing and Networks Division, CERN, Geneva, Switzerland.[22] C. Giunti, C.W. Kim and M. Monteno, hep-ph/9709439 (1997).[23] Mathematica 3.0, Copyright 1988-1996 Wolfram Research, Inc.[24] S. Bergmann and Y. Grossman, hep-ph/9809524 (1998).4a bc d-3-2.5-2-1.5-1-0.50-3 -2 -1 0-3-2.5-2-1.5-1-0.50-3 -2 -1 0-3-2.5-2-1.5-1-0.50-3 -2 -1 0-3-2.5-2-1.5-1-0.50-3 -2 -1 0FIG. 1. Allowed regions in parameter space for a) ǫ13 versus ǫ22, b) ǫ23 versus ǫ22, c) ǫ13 versus ǫ33, and d) ǫ23 versus ǫ33.Both 95 and 99 % C.L. regions are indicated.TABLE I. Average distance (in km) traveled by neutrinos through each Earth layer (1=core) for each zenith and momentumbin.Sample Bin No. cos θ Layer 1 Layer 2 Layer 3 Layer 4 Layer 5Sub-GeV 1 [−1.0, −0.6] 245.0 1780.8 3663.3 630.4 1131.52 [−0.6, −0.2] 149.4 1130.1 2623.4 505.4 983.73 [−0.2, 0.2] 81.8 649.9 1731.3 373.3 791.14 [0.2, 0.6] 37.6 318.5 1000.0 243.8 565.75 [0.6, 1.0] 10.9 100.9 396.4 112.1 291.6Multi-GeV 1 [−1.0, −0.6] 247.5 2148.3 5157.8 811.3 1237.82 [−0.6, −0.2] 8.2 236.4 2398.9 775.0 1671.93 [−0.2, 0.2] 0.1 12.0 466.8 286.5 1029.44 [0.2, 0.6] 0.0 0.2 34.1 36.1 237.75 [0.6, 1.0] 0.0 0.0 0.4 0.8 11.75TABLE II. Fit variables, maximum allowed absolute value derived from the present limits on R-parity violating couplings(assuming SU(2)L symmetry), and lowest χ2 results. The corresponding χ2 value is also given.Variable Maximum Absolute Value Best Fit Valueǫ13 0.421.10−1 0.45.10−2ǫ23 0.708.10−2 0.19ǫ22 0.164 0.05ǫ33 0.460 0.414χ2 15.6TABLE III. Fit results compared with the data. The first column indicates each of the 13 constraints and the second onegives the observed results: for the solar data the fraction of observed rate over the rate predicted by the Bahcall- PinsonneaultStandard Solar Model 95, and for the atmospheric data the “ratio of ratios” (see text) for each zenith angle bin. The thirdcolumn shows the predicted results for the model described in this paper.Solar NeutrinosExperiment Type Observed Flux Fraction Best Fit Flux FractionHomestake 0.273 ± 0.021(exp) +0.05−0.03(theo) 0.368Gallium 0.507 ± 0.049(exp)± 0.028(theo) 0.424Super-Kamiokande 0.366 +0.017−0.014(exp)+0.074−0.046(theo) 0.372Atmospheric NeutrinosSub-GeV (cosθ range) Observed Ratio of Ratios Best Fit Ratio of Ratios[−1.0,−0.6] 0.41 ± 0.07(stat) ± 0.08(syst) 0.50[−0.6,−0.2] 0.63 ± 0.07(stat) ± 0.08(syst) 0.50[−0.2, 0.2] 0.63 ± 0.07(stat) ± 0.08(syst) 0.58[0.2, 0.6] 0.82 ± 0.07(stat) ± 0.08(syst) 0.79[0.6, 1.0] 0.71 ± 0.07(stat) ± 0.08(syst) 0.96Multi-GeV (cosθ range) Observed Ratio of Ratios Best Fit Ratio of Ratios[−1.0,−0.6] 0.42 ± 0.14(stat) ± 0.12(syst) 0.51[−0.6,−0.2] 0.52 ± 0.13(stat) ± 0.12(syst) 0.54[−0.2, 0.2] 0.70 ± 0.12(stat) ± 0.12(syst) 0.90[0.2, 0.6] 0.72 ± 0.12(stat) ± 0.12(syst) 1.00[0.6, 1.0] 0.87 ± 0.16(stat) ± 0.12(syst) 1.006

A Supersymmetric Solution to the Solar and Atmospheric Neutrino Anomalies

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