Evidence for M-theory based on fractal nearly tri-bimaximal neutrino mixing
EEvidence for M-theory based on fractal nearly tri-bimaximal neutrino mixing
Hui-Bin Qiu ∗ Department of Physics, Nanchang University, JiangXi, Nanchang 330031, China (Dated: May 14, 2018)Developing a theory that can describe everything in the universe is of great interest, and is closelyrelevant to M-theory, neutrino oscillation and charge-parity (CP) violation. Although M-theoryis claimed as a grand unified theory, it has not been tested by any direct experiment. Here weshow that existing neutrino oscillation experimental data supports one kind of high dimensionalunified theory, such as M-theory. We propose a generalization of the tri-bimaximal neutrino mixingansatz, and we find that the latest neutrino oscillation experimental data constraints dimension ina range between 10.46 and 12.93 containing 11, which is an important prediction of M-theory. Thisansatz naturally incorporates the fractal feature of the universe and leptonic CP violation into theresultant scenario of fractal nearly tri-bimaximal flavor mixing. We also analyze the consequences ofthis new ansatze on the latest experimental data of neutrino oscillations, and this theory matches theexperimental data. Furthermore, an approach to construct lepton mass matrices in fractal universeunder permutation symmetry is discussed. The proposed theory opens an unexpected window onthe physics beyond the Standard Model.
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Introduction. – M-theory, one of the most promisingtheories beyond the Standard Model, is suffering frompseudoscience questions [1] because of the lack of di-rect experimental evidence, and causes wide discussions[2]. Recently acquired Neutrino oscillation experimentaldata might provide promising chances to either supportor decline the M-theory. However, the relationship be-tween neutrino oscillation and M-theory has not beenfully established yet. This is because the dimensionsof these two theories are not identical. Neutrino the-ory is a low-dimensional theory while the M-theory is11-dimensional [3, 4]. Usually, the high-energy M-theoryhas to be shrunk to 4 dimensions, forming a low-energytheory to match the experimental data such as those fromLarge Hadron Collider (LHC). However, none of thesepredictions have been supported by the LHC data yetbecause the low-dimension M-theory has not been com-pletely developed.Here, we expand the neutrino oscillation theory to11-dimension using nonextensive statistics [5, 6]. Thismethod has succeeded in many fields such as general-izations of relativistic and quantum equations [7], trans-verse momenta distributions at LHC experiments [8],dissipative optical lattices [9], plasmas [10], etc (seehttp://tsallis.cat.cbpf.br/biblio.htm, for a regularly up-dated bibliography). Nonextensive statistics is based onthe fractal principle [11]. We bring it to modify thetri-bimaximal neutrino mixing pattern, which allows toincorporate CP violation and the fractal feature of theuniverse into the resultant scenario of fractal nearly tri-bimaximal flavor mixing. Results show that the dimen-sion of a neutrino system using the nonextensive statisticsis located between 10.46 and 12.93, which well covers the11 predicted by the M-theory.
Constraints on dimension and mixing factors. – In or-der to obtain the dimension range of neutrino system, we analyze the latest neutrino oscillation experimentaldata with fractal nearly tri-bimaximal neutrino mixingtheory (see Appendixes). In detail, adopting theoreticalformula (see Eq. (15) in Appendixes) sin θ chz = 1 − c ,in which c ≡ cos q θ , combining with experimental data[12] sin θ chz = (8 . ± . × − , we obtain the allowedrange of space-time dimension (there is an intimate re-lation q = d f between q and fractal dimension d f whenthe Euclidean dimension is one [13]): 10.46 ≤ q ≤ . φ which is the source of leptonic CP vi-olation in neutrino oscillations, so there is a set S chz,q = { . ≤ q ≤ . , −∞ < φ < + ∞} , which can be ex-pressed in Fig. 1 with the red strip area. For sake ofseeing the limit of theoretical formula (see Eq. (15) inAppendixes) sin θ atm = 1 − s , in which s ≡ sin q θ ,and experimental data [12] sin (2 θ ) = 0 . +0 . − . fornormal mass hierarchy and sin (2 θ ) = 1 . +0 . − . for inverted mass hierarchy on the range of q and φ ,we do the corresponding calculation and find that q can take any real number, namely, −∞ < q < + ∞ ,and this moment, theoretical formula and experimen-tal data also have no limit on φ . So, there is a set S atm,q = {−∞ < q < + ∞ , −∞ < φ < + ∞} which canbe expressed in Fig. 1 with the blue strip area. Thereis relationship S atm,q ⊃ S chz,q , seeing Fig. 1. Take theintersection of these two sets we conclude that the rangeof space-time dimension that our theory combined withthe latest neutrino oscillation experimental data allowedis between 10.46 and 12.93 containing 11, which is animportant prediction of M-theory. We can also see thatthe allowed range of space-time dimension will be fur-ther restricted with the improvement of the experimen-tal accuracy. The neutrino oscillation experimental databecomes the first evidence of M-theory, which will effec-tively eliminate the people’s question to M-theory [1]. a r X i v : . [ h e p - ph ] D ec q- ∞ -15 -10 -5 0 5 10 15 + ∞ φ - ∞ -0.8-0.6-0.4-0.200.20.40.60.8+ ∞ TC1331707182 × S atm,q = { - ∞ < q < + ∞ , - ∞ < φ < + ∞ }S chz,q = { 10.46 ≤ q ≤ ∞ < φ < + ∞ }S sun,q ( φ ) ⊂ ( S chz,q ∩ S atm,q )S atm,q -S chz,q S chz,q -S sun,q ( φ )S sun,q ( φ ) FIG. 1. The space of dimension q and phase φ . Thefigure expresses three sets and relationship among them: S chz,q = { . ≤ q ≤ . , −∞ < φ < + ∞} , S atm,q = {−∞ < q < + ∞ , −∞ < φ < + ∞} ⊃ S chz,q , S sun,q ( φ ) ⊂ S chz,q , where, the range of q in set S chz,q is obtained basedon experimental data sin θ chz = (8 . ± . × − , and φ is not limited now, so it can take any real number. S atm,q is decided by experimental data sin (2 θ ) = 0 . +0 . − . fornormal mass hierarchy and sin (2 θ ) = 1 . +0 . − . for in-verted mass hierarchy, and after checked there is S atm,q ⊃ S chz,q . In fact, q in S atm,q can take all real number dueto the fact that the experimental upper and lower limitsof sin (2 θ ) are automatically satisfied. For upper limit,sin θ atm = 1 − s ≤ , ∀ q ∈ R , and for lower limit one hassin θ atm ≥ . φ is also not limited, so it can take any real num-ber too On the basis of meet the above conditions S sun,q ( φ )is decided by experimental data sin (2 θ ) = 0 . ± . S sun,q ( φ ) ⊂ ( S chz,q ∩ S atm,q ). Therefore, the value of q in S sun,q ( φ ) is { . ≤ q ≤ . } , and the value of φ is decidedby experimental data sin (2 θ ) = 0 . ± . θ sun ≤ . S sun,q ( φ ) is expressed by the yellowarea in the figure. With the purpose of obtaining the rangeof φ , we adopt theoretical formula (seeEq. (15) in Appendixes) sin θ sun = (cid:0) − s − sc cos q φ + s c cos q φ − s c cos q φ (cid:1) to analyze the experimental data [12] sin (2 θ ) =0 . ± . φ is depending on parameter q . The topand bottom limit of φ under the typical q values arein Table I. The φ set under the q that allowed by alltheoretical formula and experimental data, S sun,q ( φ ),can be expresses with the yellow area in Fig. 1, andthere is relationship S sun,q ( φ ) ⊂ S chz,q .In conclusion, the set of q and φ allowed by theoreticalformula and experimental data is the intersection of sets S chz,q , S atm,q and S sun,q ( φ ), namely, S sun,q ( φ ), i.e. theyellow area in Fig. 1. Change on mixing factors. – Next, we inves-tigate the change of range of φ after the di-mension increased with the theoretical formula(see Eq. (15) in Appendixes) sin θ sun = (cid:0) − s − sc cos q φ + s c cos q φ − s c cos q φ (cid:1) under the cases of q = 1 and 10 . ≤ q ≤ .
93, respec-tively. From Table 1 and Fig. 2 we find that when q = 1,0.49 ≤ φ q =1 ≤ .
27 the order of magnitude is 1; but when10 . ≤ q ≤ .
93, the range of φ increases with the in-crease of q namely, from − . ≤ φ q =10 . ≤ . − ≤ φ q =12 . ≤ . So, the order of magnitude of φ rangeincreases 3 order after the dimension increased, whicheliminates the question of small range of φ values.Specifically, when q = 11, − ≤ φ q =11 ≤ Prediction on CP violation. – To examine the theoryproposed in this paper, we give a prediction of thestrength of CP or T violation in neutrino oscillations.No matter whether neutrinos are Dirac or Majorana par-ticles, the strength of CP or T violation in neutrino os-cillations is measured by a universal parameter J whichis defined as [14]: Im (cid:16) V αi V βj V ∗ αj V ∗ βi (cid:17) = J (cid:80) γ,k ( ε αβγ ε ijk ),in which the Greek subscripts run over ( e, µ, τ ), and theLatin subscripts run over (1 , , J = sc sin q φ (cid:0) c + s ρ q ( φ ) (cid:1) . Theprediction of the strength of CP or T violation in neu-trino oscillations under typical q values are in Table 1,and especially, when q = 11, − ≤ J q =11 ≤ ν µ → ν e and ν e → ν µ transitions or by the CP-violating asymmetry between ν µ → ν e and ¯ ν µ → ¯ ν e transitions in a long-baseline neu-trino oscillation experiment, when the terrestrial mattereffects are under control or insignificant. TABLE I. The range of φ and strength of CP or T viola-tion. q = 1 is the ideal one-dimensional case; q = 10 . .
93 are dimension lower and upper limits allowed byexisting neutrino oscillation experimental data, respectively; q = 11 is the prediction of M-theory. After the dimension in-creased, the range of phase spanned from 0 . ≤ φ q =1 ≤ . − . ≤ φ q =11 ≤ .
81, increasing 3 orders; The pre-dicted strength of CP violation is − . ≤ J q =11 ≤ . q φ min φ max J min J max .
00 0 .
49 1 .
27 0 . . . − .
79 1537 . − . . . − .
81 2162 . − . . . − .
47 6372 . − . . φ -6000 -4000 -2000 0 2000 4000 6000 s i n θ s un φ range spanned TC1331707213 × -1 q=01.00q=10.46q=11.00q=12.93 φ -1 0 1 s i n θ s un × -1 FIG. 2. The mixing factors sin θ sun against parameter φ under different values of q in fractal nearly tri-bimaximalneutrino mixing patterns and the top and bottom limits ofexperimental data. The two horizontal magenta lines arethe top and bottom limits of experimental data sin (2 θ ) =0 . ± . q →
1, andthat time the experimental data sin (2 θ ) = 0 . ± . . ≤ φ q =1 ≤ .
27. Because the experimental datasin θ chz = (8 . ± . × − limits 10 . ≤ q ≤ .
93, theline of q = 1 is not true. The blue solid line is the experi-mental lower limit case ( q = 10 . (2 θ ) = 0 . ± .
021 limits − . ≤ φ q =10 . ≤ .
79. The purple solid line is theexperimental upper limit case ( q = 12 . (2 θ ) = 0 . ± . − . ≤ φ q =12 . ≤ .
47. The red solid line isthe M-theory predicted case ( q = 11), that time the upperlimit of experimental data sin (2 θ ) = 0 . ± .
021 limits − . ≤ φ q =11 ≤ .
81. To facilitate observing details,subgraph is the full figure’s part of − . ≤ φ ≤ .
5. Thefigure shows the order of magnitude of φ range increases 3orders after the dimension increased. Further discussions and remarks. – Our findings reveala strong association between neutrino oscillation and M-theory at the point of 11 dimensions of space-time. Thiswould mean that the neutrino oscillation experiment isthe initial robust evidence of M-theory, broking the spellthat the M-theory has no experimental evidence, elimi-nating pseudoscience questions [1], and opening an un-expected window on the physics beyond the StandardModel. However, we should realize that in spite of theM-theory have part truth, but not completely developedyet, and there may be other way. Fractal theory andpractice [15] have illuminated that the world is of fractal.The definition of fractal dimension is more universal thanthe one of Euclidean dimension. Euclid dimension is justa special case of fractal dimension, and there is intimaterelation q = d f between q and fractal dimension d f when φ -6000 -4000 -2000 0 2000 4000 6000 J × -3 -2024681012 dimension increase φ range spanned TC1331707212 q=01.00q=10.46q=11.00q=12.93 φ -1 0 1 J × -3 FIG. 3. The strength of CP or T violation J against pa-rameter φ under different values of q in fractal nearly tri-bimaximal neutrino mixing patterns. Black dotted line is thelimit case of q →
1, and that time the experimental datasin (2 θ ) = 0 . ± .
021 limits 0 . ≤ φ q =1 ≤ .
27. Basedon the figure as well as the numerical calculations, one ob-tains 0 . ≤ J q =1 ≤ . q →
1, in theory,sin θ chz = 1 − cos θ = 0 . θ chz = (8 . ± . × − , so the lineof q = 1 is not true. The blue solid line is the experimen-tal lower limit case ( q = 10 . (2 θ ) = 0 . ± .
021 limits − . ≤ φ q =10 . ≤ .
79. Based on the figure as well asthe numerical calculations, one obtains − . ≤ J q =10 . ≤ . q = 12 . (2 θ ) = 0 . ± .
021 limits − . ≤ φ q =12 . ≤ .
47. Based on the figure as well as the numer-ical calculations, one obtains − . ≤ J q =12 . ≤ . q = 11),that time the upper limit of experimental data sin (2 θ ) =0 . ± .
021 limits − . ≤ φ q =11 ≤ .
81. Based onthe figure as well as the numerical calculations, one obtainsprediction − . ≤ J q =11 ≤ . − . ≤ φ ≤ . − . ≤ J q =11 ≤ . the Euclidean dimension is one [13]. As the q → θ chz is not consistent with the ex-perimental data. Although existing neutrino oscillationexperiment data limits the range of space-time dimensionbetween 10.46 and 12.93 (see Fig. 1), the range of space-time dimension will be narrowed down with the increas-ing of experimental accuracy, and we expect an exclusionof 12 dimension. In addition, we find the order of mag-nitude of φ range increases 3 orders after the dimensionincreased (see Fig. 2). Moreover, this theory yields aprediction (see Fig. 3) which can be determined by theT-violating asymmetry between ν µ → ν e and ν e → ν µ transitions or by the CP-violating asymmetry between ν µ → ν e and ¯ ν µ → ¯ ν e transitions in a long-baseline neu-trino oscillation experiment, when the terrestrial mattereffects are under control or insignificant. Note that ourscenario predicts that − ≤ J q =11 ≤ φ = 0, J q =11 = 0, namely, there is no CP viola-tion. Therefore, our theory can be applied whether CPis violated or not.Finally, let us remark that the fractal nearly tri-bimaximal mixing pattern and its possible extensions re-quire some peculiar flavor symmetries to be imposed onthe charged lepton and neutrino mass matrices. It islikely that the fractal nearly tri-bimaximal neutrino mix-ing pattern under discussion serves as the more compli-cated flavor mixing matrix that scientists are looking for[16], and one of the nearly tri-bimaximal neutrino mix-ing patterns is its leading-order approximation. We ex-pect that more delicate neutrino oscillation experimentsin the near future will be able to verify the fractal nearlytri-bimaximal mixing pattern, from which one may getsome insight into the underlying flavor symmetry and itsbreaking mechanism responsible for the origin of bothlepton masses and leptonic CP violation. APPENDIXES
The mixing factors of solar, atmospheric and CHOOZneutrino oscillations read:sin θ sun = 4 | V e | | V e | , sin θ atm = 4 | V µ | (cid:16) − | V µ | (cid:17) , sin θ chz = 4 | V e | (cid:16) − | V e | (cid:17) . (1) A. Fractal nearly tri-bimaximal neutrino mixing
The tri-bimaximal neutrino mixing pattern U v = V can be constructed from the product of two modified Eu-ler rotation matrices: R ( θ x ) = c x s x − s x c x
00 0 1 ,R ( θ y ) = c y s y − s y c y , (2)where s x ≡ sin q θ x , c y ≡ cos q θ y , and so on. Functionssin q u and cos q u can be defined with exp q ( u ) which isthe one-dimensional q-exponential function that natu-rally emerges in nonextensive statistics [5] spawned byfractal thought [11]. For a pure imaginary iu , one de-fines exp q ( iu ) as the principal value ofexp q ( iu ) = [1 + (1 − q ) iu ] − q ) , exp ( iu ) ≡ exp ( iu ) . (3)The above function satisfies [6]:exp q ( ± iu ) = cos q ( u ) ± i sin q ( u ) , (4)cos q ( u ) = ρ q ( u ) cos (cid:26) q − q − u ] (cid:27) , (5)sin q ( u ) = ρ q ( u ) sin (cid:26) q − q − u ] (cid:27) , (6) ρ q ( u ) = (cid:104) − q ) u (cid:105) − q )] , (7)exp q ( iu ) exp q ( − iu ) = cos q ( u ) + sin q ( u ) = ρ q ( u ) . (8)Note that exp q [ i ( u + u )] (cid:54) = exp q ( iu ) exp q ( iu ) for q (cid:54) = 1[5]. Then we obtain: V = R ( θ y ) ⊗ R ( θ x )= c x s x − s x c y c x c y s y s x s y − s y c x c y . (9)The general form of the corresponding neutrino mass ma-trix M ν is M ν = V m m
00 0 m V T = c x m + s x m − c x c y s x ( m − m ) c x s x s y ( m − m ) − c x c y s x ( m − m ) c y s x m + c x c y m + s y m − c y s y (cid:0) s x m + c x m − m (cid:1) c x s x s y ( m − m ) − c y s y (cid:0) s x m + c x m − m (cid:1) s x s y m + c x s y m + c y m . (10)Taking q = 1, θ x = arctan (cid:0) (cid:14) √ (cid:1) ≈ . ◦ and θ y = 45 ◦ ,the results in usual space-time are reproduced[16].To make CP violation and the fractal feature of theuniverse be naturally incorporated into V , we adopt thefollowing complex rotation matrices: R ( θ, φ ) = c se iφq − se − iφq c
00 0 1 , (11)where c ≡ cos q θ , s ≡ sin q θ , and e iφq = exp q ( iφ ). In thiscase, we obtain the lepton flavor mixing of the followingpattern: V = R † ( θ, φ ) ⊗ V = √ (cid:0) c + se iφq (cid:1) √ (cid:0) c − se iφq (cid:1) − √ se iφq − √ (cid:0) c − se − iφq (cid:1) √ (cid:0) c + se − iφq (cid:1) √ c √ − √ √ . (12) V represents a fractal nearly tri-bimaximal flavor mixingscenario, if the rotation angle θ is small. The parameter φ in V are the source of leptonic CP violation in neutrinooscillations. B. Constraints on dimension, mixing factors and CPviolation
A proper texture of M l which may lead to the flavormixing pattern V is M l = C l C ∗ l B l
00 0 A l , (13)where A l = m τ , B l = m µ − m e , and C l = √ m e m µ e iφq .Then the mixing angle θ in V readstan q ( θ ) = sin q θ cos q θ = (cid:114) m e m µ . (14)It is easy to prove that when q →
1, the results in usualspace-time are recovered, namely [16], C l = √ m e m µ e iφ ,tan (2 θ ) = 2 √ m e m µ m µ − m e .In the next step we calculate the mixing factors of so-lar, atmospheric and reactor neutrino oscillations. Ac-cording to this theory, one obtainssin θ sun = (cid:0) − s − sc cos q φ + s c cos q φ − s c cos q φ (cid:1) , sin θ atm = 1 − s , sin θ chz = 1 − c . (15)Note when q →
1, the results in usual space-time are recovered [16]:sin θ sun = (cid:0) − sin θ − sin θ cos θ cos φ + sin θ cos θ cos φ − θ cos θ cos φ (cid:1) , sin θ atm = 1 − sin θ, sin θ chz = 1 − cos θ. (16)In this scenario, adopting experimental data [12]sin θ chz = (8.5 ± × − , one obtains 10.46 ≤ q ≤ . ≤ sin θ atm ≤ . (2 θ )= 0 . +0 . − . for normal mass hierarchy andsin (2 θ ) = 1 . +0 . − . for inverted mass hierar-chy; in addition, to make sin θ sun ≤ .
867 to accordwith the experimental data sin (2 θ ) = 0 . ± . − . ≤ φ q =10 . ≤ .
79 or − . ≤ φ q =12 . ≤ .
47, which are much betterthan the usual space-time case (0 . ≤ φ q =1 ≤ . q is close to 11 and the inti-mate relation q = d f between q and fractal dimension d f when the Euclidean dimension is one [13], we assume q = 11, then this scenario gives the predicted values ofsin θ chz = 0 . θ atm = 0 . (2 θ ) =(8 . ± . × − and sin (2 θ ) = 0 . +0 . − . for nor-mal mass hierarchy (sin (2 θ ) = 1 . +0 . − . for invertedmass hierarchy), respectively; the range of parameter − . ≤ φ q =11 ≤ .
81 limited by current datasin (2 θ ) = 0 . ± .
021 is also much better than thatin usual space-time (0 . ≤ φ q =1 ≤ . θ sun as the function of q and φ is shown in Fig. 2, where the two horizontal lines arethe top and bottom limits of experimental data. As canbe seen from the figure, in φ = 0 case, sin θ sun verysensitively dependent on φ .The strength of CP or T violation in neutrino oscil-lations is measured by a universal parameter J which isdefined as [14]: Im (cid:0) V αi V βj V ∗ αj V ∗ βi (cid:1) = J (cid:88) γ,k ( ε αβγ ε ijk ) . (17)Considering the lepton mixing scenario proposed above,one has J = 16 sc sin q φ (cid:0) c + s ρ q ( φ ) (cid:1) . (18)Obviously, when q →
1, the result in usual space-time isrecovered [16]: J = 16 sc sin φ. (19)Based on Figs. 2 and 3 as well as the numerical calcula-tions, one obtains the table I.The strength of CP or T violation J in fractal nearlytri-bimaximal neutrino mixing patterns is predicted as: − . ≤ J q =11 ≤ . φ , but unfortunately at present, there is no ex-perimental information on the Dirac and Majorana CP violation phases in the neutrino mixing matrix is avail-able [12]. ∗ Corresponding author: [email protected][1] G. Ellis and J. Silk, Nature , 321 (2014).[2] D. Castelvecchi, Nature , 446 (2015).[3] C. Hull and P. Townsend, Nuclear Physics B , 109(1995), ISSN 0550-3213.[4] E. Witten, Nuclear Physics B , 85 (1995), ISSN 0550-3213.[5] C. Tsallis,
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