Unification Neutrino Inner Mixing Angle Space

Theta-13 Data and Geometry of Neutrino Mixing

E. M. Lipmanov 40 Wallingford Road # 272, Brighton MA 02135, USA

Abstract Neutrino masses and mixing are main unsolved Standard Model problems. A basic singular flavor- geometric hypothesis  neutrino mixing angles have a second physical meaning of direction angles of a vector in the macroscopic euclidean 3-dimensional space  is emphasized and further discussed in this short note, and applied for solutions of several fundamental flavor problems. It is surely a strange idea, but is interesting because its geometric explanation power is so strong that fundamental flavor problems of “why the numbers of particle flavors and dimensions of macroscopic space are equal, why two large solar and atmospheric neutrino mixing angles must be accompanied by one small reactor angle, why the new theta-13 data suggest a Pythagorean type relation between the three neutrino mixing angles and what is the phenomenological origin of the widely discussed in the literature empirical bimaximal mixing pattern” get instantaneous, simple and clear, almost tautological answers without free parameters. The idea of neutrino mixing vector makes right answers to the mentioned four flavor problems necessary and very natural. Qualitative solutions to topical flavor problems are geometrically visualized in the outer euclidean 3-space. In reverse, they support the initial singular generic flavor-geometric hypothesis.

1. Introduction At present flavor physics seems fundamental, but essentially not complete. There is no simple phenomenological explanation of charged lepton and neutrino mass ratios, mixing angles etc. Semi- empirical flavor phenomenology may be a needed precondition for new flavor theory similar to the early stage of quantum physics that explained new (and some old) seemingly strange experimental facts in terms of singular novel, but immediately interesting hypotheses  Plank, Einstein, De Broglie, Bohr. The mainstream approach in attempts to build true flavor theory is by a long path of different flavor symmetries, but still not resulted in completely renormalizable theory. An alternative, though less rigorous, approach may be by following the way of quantum mechanics formation 1 via preceded semi- empirical phenomenology2. Stimulated by the new theta-13 data [1-5], in this short note we emphasize a singular hypothesis in neutrino flavor mixing semi-empirical phenomenology  the three neutrino mixing double-angles have a second physical meaning of direction angles of a unit vector n (neutrino mixing vector) definitely oriented in the macroscopic euclidean 3-dimensional space relative to the experimental momentum vectors of flavor neutrino pℓ and its charged lepton partner pℓ. That hypothesis sheds light on many fundamental problems in neutrino mixing phenomenology without invoking new flavor symmetries beyond the implicit geometric symmetry of euclidean 3- dimensional macroscopic space. It is new and strange in relation to known particle mixing

1 Probably quantum mechanics formation way is a rule in new physics, while Einstein’s general relativity appearance was rather an historic exception. 2 As known, based on Plank constant semi-empirical phenomenology preceded the arrival of new paradigm of quantum theory by Heisenberg, Schrödinger, Born and Dirac.

2 framework, but is solidly related to known basic facts and opens a way for geometric visualization of particle (at least neutrino) mixing phenomena.

2. Geometric instructive power of a singular strange hypothesis Consider some outstanding topical questions in neutrino flavor phenomenology. Why the numbers of neutrino mixing angles, lepton flavors and the dimension (three) of macroscopic space are equal to each other? 2) Why the new Theta-13 data suggest a Pythagorean type relation [6] between the three neutrino mixing angles – solar angle θ12, atmospheric angle θ23 and reactor angle θ13 3) Why the two large neutrino mixing angles must be accompanied by one small mixing angle? 4) What is the phenomenological origin of the widely discussed in the literature bimaximal mixing? Andwhat physical geometric conditionsmay restore the broken mixing-angle symmetry of the low energy bimaximal mixing? 5) Why is the reactor mixing angle not zero? 6) Why there is an approximate complementarity relation between reactor and solar angles? 7) Why are the two large solar and atmospheric neutrino mixing angles different? All the listed above neutrino flavor problems can be answered in semi-empirical phenomenology guided by a singular hypothesis that the three neutrino mixing double-angles have second physical meaning of direction angles of a unit neutrino mixing vector n,

n = (cos 2θ12 cos 2θ23cos 2θ13), (1) in the outer macroscopic euclidean 3-dimensional space. It hints that neutrino mixing angles are primarily determined by some geometric feature of outer euclidean macroscopic space. The answer to the first question is that the number of neutrino mixing angles is ‘three’ by definition of a vector in 3-dimensional space. The number of lepton flavors must be 3 by definition of mixing angles and combinatory in the special case of three angles (and only three). As answer to the second question, neutrino mixing angles obey Pythagorean equation by geometric definition of direction angles of a vector in euclidean 3-space: 2 2 2 cos 2θ12 cos 2θ23cos 2θ13 (2) The answer to the third question (two large mixing angles are a necessary and sufficient condition for one small mixing angle) evidently follows from Eq. (2). It can be geometrically visualized by direction angles of a vector in macroscopic 3-space (imagine mixing vector n close to one coordinate axis). The answer to the fourth question is that bimaximal mixing appears as a violating geometric symmetry solution of Eq. (2) without nontrivial parameters3 (benchmark solution [7]). Complete restoration of that broken mixing-angle symmetry is described by another solution of Eq. (2) with three equal mixing angles (see (5)-(6)), which probably may be realized at some high energy scale. The answers to the above four questions spontaneously follow from the main hypothesis (1) without use of any mixing angle experimental data and free parameters. In reverse, the four neutrino flavor problems directly point to the one singular flavor-geometric hypothesis.

3 It should be noted that the last questions may be geometrically answered without solutions of Eq. (2), just by visualization of appropriate localizations of the mixing vector n relative to the neutrino momentum vector pℓ and the physical plain (pℓ pℓ). For example, bimaximal mixing pattern correspond to orientation of n parallel to pℓ. Complementarity relation between solar and reactor angles correspond to deviation of vector n from pℓ, but remaining in the plain (pℓ pℓ). Not zero theta-13 value is related to the vector n deviation from neutrino momentum that corresponds to well known not maximal solar angle beyond benchmark, etc. 3

The answers to the next three questions follow from the suggestion (1) plus one well established data indication that the deviation of solar angle θ12 from maximal value is not very small, in contrast to the atmospheric angle θ23 e.g. [9]. The answer to the fifth question is because of Eq. (2) and that the solar neutrino mixing angle is definitely not very close to maximal 45o. The sixth question follows from (2) and the atmospheric mixing angle is very close to maximal value. The answer to the seventh question follows from the specific orientation of the mixing vector relative to the two momentum vectors of flavor neutrino pℓ and its charged lepton partner pℓ in macroscopic euclidean space beyond leading benchmark approximation (Sec. 2). Thus all the above basic problems of neutrino mixing receive incredibly simple geometrically visualized answers by suggestion (1) and so should be related to geometry of the outer euclidean space. The known strange empirical fact that the charged current weak interactions prepare the initial free nonstationary neutrino quantum states as superpositions of not degenerate mass eigenstates with definite flavors, rather than stationary neutrino mass eigenstates, should be related to neutrino mixing vector and so to geometry of elementary particle space. Formally neutrino mixing vector n is considered a unit vector, but its length is not determined by data (indefinite scale) and physical meaning has only its direction in space. As reminder, localization of the mixing vector n in macroscopic space is in relation to experimentally defined two momentum vectors of flavor neutrino pℓ and its charged lepton partner pℓ, ℓ= e,  with the Z-axis of implied orthogonal coordinate system (X, Y, Z) parallel to the momentum vector pℓ and the (XZ)-plain coinciding with the (pℓ pℓ)-plain [6]. The neutrino mixing vector n is independent of the sort of flavor neutrino, namely it is independent of the labels e,  of the momentum vectors (pe, pe), (p, p), (p, p) and so it visualizes physical unity of the three flavor neutrinos in the outer macroscopic space. To conclude, the new concept of mixing vector relates the three neutrino mixing angles to geometry of macroscopic euclidean 3-space – exactly, or at least at leading benchmark approximation.

3. Geometric perspective of neutrino mixing A symmetric set of asymmetric mixing angle solutions of Eq. (2) without nontrivial parameters beyond ‘0’ and ‘1’ (benchmark solution system) is given by 2  2  2  (cos 2θ 12 cos 2θ 23cos 2θ 13) = (0, 0, 1) = c, 2  2  2  (cos 2θ 12 cos 2θ 23cos 2θ 13) = (0, 1, 0) = b, 2  2  2  (cos 2θ 12 cos 2θ 23cos 2θ 13) = (1, 0, 0) = a. 3) This system of three solutions can be geometrically visualized by a cube-model with three unit- vector edges a, b and c that are parallel to the implied orthogonal coordinate axes X, Y and Z in the macroscopic euclidean 3-space. The three solutions (3) correspond respectively to the three possible independent orientations of the neutrino mixing vector n parallel to the three coordinate axes (X, Y, Z). The set of solutions (3) is a symmetric one until the mixing angles are physically identified by experimental neutrino data. Physical identification of neutrino mixing angles by known experimental data singles out the upper line solution in the above set 2  2  2     cos (2θ 12) cos (2θ 23)cos (2θ 13) = 1; θ 12 = θ 23 = /4, θ 13 = 0.4)   Physical meaning of this solution is two θ 12 and θ 23 maximal solar and atmospheric and one  θ 13 zero reactor mixing angles of benchmark neutrino mixing pattern. The singling out of solution (4) by condition of approximate agreement with data is a spontaneous symmetry violation4 of Eq.

4 It is by the original meaning of the term “spontaneous symmetry violation”. 4

(2) and solution set (3). In geometric terms, it brakes the cube symmetry by singling out the edge c as neutrino mixing vector n = c = (0, 0, 1) at benchmark approximation. Neutrino mixing pattern (4) coincides with the widely discussed in the literature bimaximal neutrino mixing pattern, e. g. [8]. Hence Pythagorean Eq. (2) is factually the phenomenological origin of the bimaximal mixing. By summing up all three asymmetric solutions (3), universal geometrically symmetric solution is obtained 2  2  2  d = (a + b + c) = (3cos 2θ 12, 3cos 2θ 12, 3cos 2θ 12= (1, 1, 1),  (5) 2 k k o cos 2θ i j1/3, θ i j 27.4 , ij = 12, 23, 13. (6) It is completely symmetric under permutation of mixing angles. By geometric visualization, the double angels of solution (5)-(6) are equal to those of a diagonal in a cube5. Well known experimental data on particle mixing angles [9] show that solution (6) is in disagreement with neutrino mixing angles at low energies. Hence if solution (6) has physical meaning at all, it should describe neutrino mixing angles at some high energy scale where the violated geometric symmetry of the low energy bimaximal neutrino mixing get restored, and the cube-diagonal d replaces the cube-edge c as neutrino mixing vector6. If the Pythagorean Eq. (2) is an exact one, it suggests also a geometric approach to explanation of the empirical difference between the two large mixing angles, solar and atmospheric ones, which are equal at benchmark (4). In macroscopic space at benchmark approximation the mixing vector n (and coordinate Z-axis) have to be chosen parallel to the experimentally singled out neutrino momentum vector pℓ, and the (Z X)-plain coinciding with the physical (pℓpℓ)-plain. So, at benchmark approximation there is mirror symmetry of mixing vector location relative to the (pℓpℓ)-plain. If the solar and reactor angles receive small corrections, but the atmospheric o angle remains maximal <(nY) = 90 , the mixing vector n is still located in the (pℓ pℓ)-plain. Thus in that case the mirror symmetry remains exact. Mirror symmetry violating correction to the 90o benchmark atmospheric (nY)-angle means that the neutrino mixing vector n gets out from the

(pℓ pℓ)-plain, 2θ23< /2 or 2θ23> /2 that is a violation of mirror symmetry. It is difficult to see why the mirror symmetry should be violated and the atmospheric angle be deviated from maximal. If it is indeed not maximal by data [9], suppression of mirror symmetry violation is the physical reason of very small deviation from maximal value of atmospheric angle in contrast to the solar one. But if further accurate measurements will, nevertheless, establish maximal atmospheric angle [10], it would mean exact mirror symmetry of the mentioned type.

4. Theta-13 mixing angle from Pythagorean equation Eq. (2) defines the neutrino reactor mixing angle theta-13 through the experimental solar and atmospheric angles 2 2 2 sin 2θ13 cos 2θ12 cos 2θ23 (7)

Using recent results of the global 3 neutrino oscillation analysis [9] for solar angle θ12 at 2, 2 sin θ12 (8) o and maximum value for the atmospheric angle θ2345 , which is indicated by T2K data [10] and agrees with the global analysis [9] at 1, one gets o o θ13  . (9)

5 o (ad) = (bd) = (cd) cos54.7 . 6 Special interest of this possible symmetry restoration of bimaximal mixing is related to the important problem of high energy enhancement of lepton CP-violation because of the drastic increase of the theta-13 mixing angle from ~9o to ~27.4o. 5

As important results, the ranges (9) of reactor angle from Eq. (2) are in very good agreement with latest T2K e appearance data [1] and compatible with independent global analysis of reactor angle θ13 ranges [9] at 1(normal or inverted hierarchy) 2 o o sin θ13 0.0216 – 0.0260, θ13 8.6 – 9.3 . (10)

5. Conclusions The new experimental theta-13 data [1- 5] together with available data on solar and atmospheric angles suggest Pythagorean relation (2) for neutrino mixing angles. A singular hypothesis that this Pythagorean equation follows from geometry of neutrino mixing in known macroscopic 3-dimensional euclidean space is interesting because it leads to sound answers to many topical problems of neutrino semi-empirical mixing angle phenomenology  That hypothesis instantly answers four basic still mysterious flavor problems: a) Why the numbers of mixing angles, flavors and dimension’3’ of outer space are equal. As a rule, one thinks that an answer to this question belongs to future theory that may connect flavor and space geometry. Incredibly simple answer to this old problem on the level of phenomenology by a singular hypothesis without free parameters is a surprise. b) Why two large mixing angles, solar and atmospheric, must be accompanied by one small reactor angle, c) Why the widely discussed in the literature bimaximal mixing pattern is a leading approximation to neutrino mixing data and d) Why there is Pythagorean equation for neutrino mixing angles. Pythagorean relation for three neutrino mixing angles is compatible at 1-2 with the results of recent global 3 neutrino oscillation analysis [9]. Presented answers to the four fundamental flavor problems do not involve any experimental data parameters, based only on the one generic idea and geometry of macroscopic space. The answers to these four problems and considered in text three additional problems persistently support the starting generic flavor-geometric hypothesis. In semi-empirical phenomenology, this hypothesis solved the old problem of equal numbers of lepton flavors and dimensions of macroscopic space in analogy with solution of the problem of equal inertial and gravitating masses by Einstein’s equivalence principle. This phenomenological analogy may add to the stimuli for new fundamental flavor theory.

References [1] T2K Collaboration, K. Abe et al, arXiv:1311.4750; Phys. Rev. Lett. 107, 041801(2011). [2] DAYA-BAY Collaboration, F. P. An et al, Phys. Rev. Lett.108, 171803 (2012). [3] RENO Collaboration, J. K. Ahn et al, Phys. v. Lett.108, 191802 (2012). [4] MINOS Collaboration, P. Adamson et al, Phys. Rev. Lett.107, 181802 (2011). [5] DOUBLE CHOOZ Collaboration, Y. Abe et al, Phys. Rev. Lett., 108, 131801 (2012). [6] E. M. Lipmanov, arXiv:1212.1417. [7] E. M. Lipmanov, arXiv:1202.5043. [8] G. Altarelli, F. Feruglio, L. Merlo, arXiv:1205.5133; H. Georgi, S. L. Glashow, Phys. Rev. D61. 097301 (2000). [9] F. Capozzi, L. Fogli, E. Lisi, A. Marone, D. Montanino, A. Palazzo, arXiv:1312.2878. [10] K. Abe et al. (T2K Collaboration), Phys. Rev. Lett. 111,211803(2013).