Bi-pair Neutrino Mixing

0.13 MB·

by  Teruyuki Kitabayashi

Bi-pair Neutrino Mixing
Teruyuki Kitabayashi∗ and Masaki Yasu`e†
Department of Physics, Tokai University,
4-1-1 Kitakaname, Hiratsuka,
Kanagawa 259-1292, Japan
(Dated: January,2011)
Anewtypeofneutrinomixingnamedbi-pairneutrinomixingisproposedtodescribethecurrent
neutrino mixing pattern with a vanishing reactor mixing angle and is determined by a mixing
matrixwithtwopairsofidenticalmagnitudesofmatrixelements. Asaresult,wepredictsin2θ12 =
1 1/√2( 0.293) forthesolarneutrinomixingandeithersin2θ23 =tan2θ12 orcos2θ23 =tan2θ12
− ≈
1 fortheatmosphericneutrinomixing. WedetermineflavorstructureofamassmatrixM,leadingto
1 diagonal masses of m1,2,3, and find that (cid:12)(cid:12)Mµµ Mee/t212(cid:12)(cid:12): Mµτ :(cid:12)(cid:12)Mττ Mee/t212(cid:12)(cid:12) =t223 : t23 :1
0 forthenormalmasshierarchyifm1=0,where−tij=tanθij (|i,j=1|,2,3)an−dMij (i,j=e,µ,τ|)st|and
2 for flavorneutrinomasses. For theinverted mass hierarchy,thebi-pairmixingschemeturnsout to
n satisfy the strong scaling ansatz requiring that Mµµ : Mµτ : Mττ =1: t23 :t223 if m3=0.
| | | | | | | |
a
J PACSnumbers: 12.60.-i,13.15.+g,14.60.Pq,14.60.St

1
1
More than ten years have passed since the first con- These predictions are consistent with the 2σ data al-

] firmation of the neutrino oscillations by the Super- though sin2θ12 slightly exceeds the allowed range of the
h Kamiokande collaboration, who observed the oscillation 1σ data.
p
of atmospheric neutrinos [1]. Subsequent experimental In this short note, we would like to find new mixing
-
p observationshavealsoconfirmedthesolarneutrinooscil- schemes [10], which may well describe the solar neutrino
e lation [2, 3]. These oscillations really occur in terrestrial mixing. To do so, we demand that at least two of the
h neutrinos[4,5]. Theotheroscillationarisingfromreactor magnitudes of matrix elements be equalto eachother as
[
neutrinomixing hasnotyetbeen observed[6]. Theoreti- in the tri-bimaximal case, which shows

1 cally,mixingsofthreeflavorneutrinosνe,µ,τ canaccount U = U = U ,
v for the neutrino oscillations if neutrinos are massive and | 12| | 22| | 32|
3 are characterizedby three mixing angles θ12,23,13 associ- |U21|=|U31|,
7
ated with the mixings of ν -ν , ν -ν and ν -ν , respec- U = U . (3)
0 e µ µ τ e τ | 23| | 33|
tively. The Pontecorvo-Maki-Nakagawa-Sakata(PMNS)
2 We have parameterized the PMNS mixing matrix U to
unitary matrix [7] parameterized by these mixing angles
1. is used to convert mass eigenstates of neutrinos into the be U =K0U0:
0
1 flavor neutrinos. c12 s12 0
1 The observed results of the mixing angles are summa- U0 =  −c23s12 c23c12 s23 ,
: rized as [8]: s23s12 s23c12 c23
v  − 
i K0 = diag(1,eiφ2/2,eiφ3/2), (4)
rX sin2θ12 = 0.304+−00..002126 (0.27−0.35), where cij =cosθij, sij =sinθij (i,j=1,2,3) and φ2,3 are
a CP-violating Majorana phases.
sin2θ23 = 0.50+−00..0076 (0.39−0.63), As long as θ13 = 0 is maintained, it is not difficult
to searchfor alternative relations to Eq.(3) for the given
sin2θ13 = 0.01+−00..001161 (≤0.04), (1) values of Eq.(1). There are only two possibilities, which
shows
for the 1σ range, where the values in the parentheses U = U ,
12 32
denote the 2σ range. There is a theoreticalprediction of | | | |
U = U , (5)
these mixing angles based on the tri-bimaximal mixing | 22| | 23|
scheme [9], which yields as the case (1), and
U = U ,
12 22
1 1 | | | |
sin2θ = , sin2θ = , sin2θ =0. (2) U = U , (6)
12 3 23 2 13 | 32| | 33|
as the case (2). These equations in turn provide useful
relationship among the mixing angles:
∗Electronicaddress: [email protected] sinθ12 = cosθ12 , (7)
†Electronicaddress: [email protected] cosθ12 √1+cos2θ12
2
as well as diag(m ,m ,m ) is the diagonal neutrino mass matrix.
1 2 3
OneimportantpointisthatU shouldcontainredundant
sin2θ23 =tan2θ12, (8) phases,whicharedenotedbyγ andalsobyρ(tobeused
inEq.(16)),totakecareofgeneralphasestructureofM
ν
for the case (1), and [14, 15]. For UPDG as the standard parameterization of
U adopted by the Particle Data Group (PDG) [16], M
cos2θ =tan2θ , (9) ν
23 12 is shifted to a modified mass matrix M after ρ and γ
present in U are transferred to M.
for the case (2). Numerically, these relations predict
Owing to the rephasing ambiguity in the charged lep-
1 ton sector, one can choose three flavor masses to be real
sin2θ = 1 0.293,
12 − √2 ≈ numbers, where d can be taken to be positive without
loss of generality. As a result, we obtain
√2 1 0.414 the case (1)
sin2θ23 = (cid:26) 2 −√2≈0.586···the case (2) ,(10) κ a eiα b t eiβ b
− ≈ ··· a| | | | − 23 | |
Mν = d κe e ,
which are consistent with the 2σ data. | | | |
f
Our prediction on sin2θ is slightly inconsistent with  
the 1σ data as in the simil2a3r situation to that of sin2θ12 f =e4iγ d + 1−t223κ e2iγ e , (15)
e
in the tri-bimaximal mixing. However, it is well known | | t23 | |
thatthecorresponding2-3mixinginchargedleptons(la-
where κ take care of the sign of a and e. The mixing
beled by θℓ ) can produce additional contribution to θ a,e
23 23 angles θ is given by
without affecting θ and θ . Namely, we obtain that 12
12 13
2eiξ b
θ =θν θℓ , (11) tan2θ = | | ,(16)
23 23− 23 12 c23 e2iγ d t23κe e κae2iρ a
| |− | |− | |
whereθν isgivenbyθ inEq.(10). Therefore,ifcharged α+β
23 23 ξ =ρ+γ+α =ρ+ , (17)
leptons have a mass matrix Mℓ described by (cid:18) 2 (cid:19)
m 0 0 whichdeterminesthephaseρexpressedintermsofflavor
e
Mℓ = 0 , (12) neutrino masses for a given value of θ12.
∗ ∗
0 After redundant phases are removed from U, U be-
 ∗ ∗ comes UPDG and, accordingly, M is shifted to
ν
appropriatecorrectionautomaticallycomesinθ sothat
23
θ23 can be shifted to the 1σ region. Other corrections e2iρκa|a| eiξ|b| −t23eiξ|b|
may arise from the renormalization effect [11] if the bi- M = e2iγ d κe e . (18)
| | | |
pair mixing is generated at a higher scale such as the e−2iγf
 
seesaw scale, where the seesaw mechanism [12] gets ac-
We finally reach M given by
tive.
Having understood that the bi-pair mixing is another
t 0 0
23
candidatepredictingthereasonablevaluesofθ andθ , −
we discuss its implication of flavor structure o1f2the ne2u3- M = e−i(α−β)|d|I +κe|e| 00 01 1−1t223 
trinomasses. Ithasbeendiscussedthatanymodelswith  t23 
sinθ13 = 0 should be described by the following flavor A 1 t23
neutrino mass matrix Mν with general phase structure +ei(ρ+α+2β) b  1 0 −0 , (19)
[13] | | t 0 0
23
 − 
a eiα b t23eiβ b A = t212−1, (20)
Mν = d| | − e | | , c23t12
f
  where t12 = tanθ12 and I is the unit matrix and a in
1 t2 Eq.(18) is eliminated by Eq.(16) to yield Eq.(19).|T|his
f =e4iγd+e2iγ − 23e, (13)
t massmatrixcertainlygivesM forthetri-bimaximalneu-
23
trino mixing if t2 = 1 and t2 = 1/2 giving A = 1.
where a, b, d and e are complex numbers, α and β are The bi-pair neut2r3ino mixing 1g2ives A = 1/ t w−ith
23
phases, γ is given by t2 =1/√2andsinθ =σtanθ forthec−ase(|1),|where
23 23 12
σ(= 1) takes care of the sign of sinθ , and A = t2
β α ± 23 − 23
γ = − , (14) with t2 = √2 and cosθ = tanθ for the case (2).
2 23 23 12
More transparent flavor structure for the bi-pair neu-
and t = tanθ . This result is a consequence of the trino mixing can be found if neutrino mass hierarchies
23 23
direct calculation of UTM U = M , where M = are taken into account.
ν diag diag
3
Neutrino masses are calculated to be It is noted that Eq.(23) may satisfy the strong scaling
ansatz [18] since sinθ = 0 and m = 0. It is evident
eiξ b 13 3
m e−iϕ1 = κ e2iρ a tanθ | |, that the resulting mass matrix does satisfy the strong
1 a 12
| |− c23 scalingansatzrequiringtherelationofEq.(29)(aswellas
m e−iϕ2 = κ e2iρ a + 1 eiξ|b|, (21) Eq.(24)). Therefore, when m3 6= 0, the bi-pair neutrino
2 a mixing provides an example of the approximate strong
| | tanθ c
12 23
scaling ansatz, where Eq.(29) is approximately satisfied.
1
m3e−iϕ3 = e−i(α−β)|d|+ t κe|e|, If m1 = 0 for the normal mass hierarchy, the relation
23 Eq.(27) including M can be predicted. 1
ee
where the CP-violating Majorana phases φ are given Insummary,wehavefoundthatthebi-pairmixingwell
2,3
by φ =ϕ ϕ and φ =ϕ ϕ . Let us consider that reproducesthecurrentneutrinomixingsandisdescribed
2 2 1 3 3 1
neutrinosex−hibiteitherm1=0−leadingtothenormalmass by UBP:
hierarchyorm =0leadingtotheinvertedmasshierarchy
3
c s 0
as in the minimal seesaw model [17], where det(M)=0 12 12
is satisfied. We, then, find that, for the normal mass UBP =  −t212 t12 t12 , (30)
s t s t /c
hierarchy,  12 12 − 12 12 12 
for the case (1), and
Bei(α+β) b ei(α+β) b t ei(α+β) b
23
| | | | − | |
 Bei(α+β)|b| 
c s 0
t2 κ e 12 12
M =  +t1223κe|e| Bei(tα2e+β|)||b| , UBP =  −st12122t12 −st1122 t12t/12c12 , (31)
 +t12123κe|e|  for the case (2), where s212 is predicted to be: s212 =
B = tanθ12 = |t23|···(sinθ23 =σtanθ12) , (22) 1−1/√2. The bi-pair mixing scheme turns out to be
c23 (cid:26)1···(cosθ23 =tanθ12) complementary to the tri-bimaximal mixing scheme in a
sense that
where ρ = (α+β)/2 (mod π) from m1 = 0 and, for the the bi-pair mixing predicts sin2θ12 = 0.293, which
•
inverted mass hierarchy, well describes the solar neutrino mixing while it
predicts sin2θ = 0.414/0.586, which gives slight
23
κae2iρ a eiξ b t23eiξ b deviationoftheatmosphericneutrinomixingangle
| | | | − | |
M = −t213κe|e| κe|e| , (23) from the 1σ region, and
t κ e
23 e
 − | |  the tri-bimaximal mixing predicts sin2θ = 0.5,
23
•
wheree−i(α−β) d = κ /t e fromm =0,thus,lead- whichwelldescribestheatmosphericneutrinomix-
ing to α=β (mo|d|π)−andeρ2i3s|d|etermine3d so as to satisfy ing while it predicts sin2θ12 = 0.333, which gives
Eq.(16), which is used to express this mass matrix in slight deviation of the solar neutrino mixing angle
terms of b and e. from the 1σ region.
| | | |
We observe that flavor structure of M for the bi-pair
We have clarified the flavor structure of the neutrino
neutrino mixing shows
mass matrix giving sinθ = 0, which is described by
13
Eq.(19) as long as the parameterization of UPDG is
M : M =1: t , (24)
| eµ| | eτ| | 23| adopted. For the bi-pair mixing, in the simplest cases
arg(Meµ)=arg(Meτ) (mod π), (25) with m1 = 0 for the normal mass hierarchy and m3 = 0
for the inverted mass hierarchy, the phase structure is
and
subject to Eqs.(25) and (28). We have also predicted
Eq.(27) for the normal mass hierarchy, which should be
for the normal mass hierarchy,
• comparedwithEq.(29)forthestrongscalingansatzvalid
M (sinθ =σtanθ ) in the inverted mass hierarchy.
|Mee|=(cid:26)|Meeτµ|···(cosθ2233 =tanθ121)2 , (26) Finally, we point out that the results of Eqs.(24) and
| |··· (25) for both normal and inverted mass hierarchies and
M M
M ee : M : M ee ofEq.(27)forthenormalmasshierarchyandEq.(29)for
(cid:12)(cid:12) µµ− t212 (cid:12)(cid:12) | µτ| (cid:12)(cid:12) ττ − t212 (cid:12)(cid:12) the inverted mass hierarchy are not only valid in the bi-
(cid:12)(cid:12) =t223 :|t(cid:12)(cid:12)23|:1, (cid:12)(cid:12) (cid:12)(cid:12) (27) pair mixing scheme and but also valid for any models
arg(M )=arg(M ) (mod π), (28)
ee eµ
for the inverted mass hierarchy,
• 1 If m23 ≫ m22 is further imposed, the condition of b≈ 0 leading
|Mµµ|:|Mµτ|:|Mττ|=1:|t23|:t223. (29) |tMo µMτe|e:|≈Mτ0τ|s≈hotu22l3d:b|te23s|a:ti1s.fied and Eq.(27) becomes |Mµµ| :
4
with sinθ =0,where θ aresimply regardedas free mass matrix as well as phenomenological implication of
13 12,23
parameters. The bi-pair mixing scheme provides a good the bi-pair mixing scheme based on Majorana CP viola-
exampleofthesepropertiesoftheflavorneutrinomasses. tion[19]fromEq.(21)andonleptogenesis[20]inafuture
Wewilldiscussthedetailedfeatureofourflavorneutrino publication [21].
[1] Y. Fukuda et al., [Super-Kamiokande Collaboration], Seroˆdio, “Leptonic mixing, family symmetries and neu-
Phys. Rev. Lett. 81 (1998) 1562; Phys. Rev. Lett. 82 trino phenomenology”, arXiv:1101.0602 [hep-ph].
(1999) 2430; T. Kajita, Nucl. Phys. Proc. Suppl. 77 [11] Seeforexample,S.Antusch,J.Kersten,M.Lindnerand
(1999) 123. See also, T. Kajita and Y. Totsuka, Rev. M. Ratz, Nucl. Phys. B 674 (2003) 401; S. Antusch, J.
Mod. Phys. 73 (2001) 85. Kersten,M.Lindner,M.RatzandM.A.Schmidt,JHEP
[2] J.N.Bahcall,W.A.Fowler,I.IbenandR.L.Sears,Astro- 0503 (2005) 024; R.N. Mohapatra, M.K. Parida and G.
phys.J. 137(1963) 344; J. Bahcall, Phys.Rev.Lett.12 Rajasekaran, Phys. Rev. D 71 (2005) 057301; J.W. Mei
(1964)300;R.Davis,Jr.,Phys.Rev.Lett.12(1964)303; andZ.Z.Xing,Phys.Rev.D69(2004)073003;J.W.Mei,
R.Davis,Jr.,D.S.HarmerandK.C.Hoffman,Phys.Rev. Phys. Rev. D 71 (2005) 073012; S. Luo, J.W. Mei and
Lett.20(1968) 1205; J.N. Bahcall, N.A.Bahcall and G. Z.Z. Xing, Phys. Rev. D 72 (2005) 053014; S. Luo and
Shaviv, Phys. Rev. Lett. 20 (1968) 1209; J.N. Bahcall Z.Z. Xing,Phys. Lett. B 632 (2006) 341.
and R.Davis, Jr., Science 191 (1976) 264. [12] P.Minkowski,Phys.Lett.B67(1977)421; T.Yanagida,
[3] Y. Fukuda et al., [Super-Kamiokande Collaboration], in Proceedings of the Workshop on Unified Theories and
Phys.Rev.Lett.81(1998)1158;[Erratum-ibid81(1998) BaryonNumberintheUniverseeditedbyA.Sawadaand
4279]; B.T. Clevel et al., [Super-Kamiokande Collabora- A. Sugamoto (KEK Report No.79-18, Tsukuba, 1979),
tion], Astrophys. J. 496 (1998) 505; W. Hampel et al., p.95; Prog. Theor.Phys.64(1980) 1103; M.Gell-Mann,
[GNOCollaboration],Phys.Lett.B447(1999)127;Q.A. P. Ramond and R. Slansky, in Supergravity edited by P.
Ahmed et al., [SNO Collaboration], Phys. Rev.Lett. 87 vanNieuwenhuizenandD.Z.Freedmann(North-Holland,
(2001) 071301; Phys.Rev. Lett.89 (2002) 011301. Amsterdam 1979), p.315; R.N. Mohapatra and G. Sen-
[4] S.H.Ahnetal.,[K2KCollaboration], Phys.Lett.B511 janovi´c, Phys. Rev. Lett. 44 (1980) 912. See also, P.
(2001) 178; Phys.Rev.Lett. 90 (2003) 041801. Minkowski,inProceedings ofXI International Workshop
[5] K. Eguchi et al., [KamLAND collaboration], Phys. Rev. on Neutrino Telescopes in Venice edited by M. Baldo
Lett.90(2003) 021802. K.Inoue,NewJ.Phys.6(2004) Ceolin (Papergraf S.p.A., Italy, 2005)), p.7, “Neutrino
147;S.Abeetal.,[KamLANDcollaboration],Phys.Rev. oscillations, a historical overview and its projection”,
Lett.100 (2008) 221803. arXiv:hep-ph/0505049.
[6] M.Apollonioetal.,[CHOOZCollaboration],Euro.Phys. [13] K. Yudaand M. Yasu`e, Phys.Lett. B 693 (2010) 571.
J. C 27 (2003) 331; X. Guo et al., [Daya-Bay Col- [14] T. Baba and M. Yasu`e, Phys.Rev.D 75 (2007) 055001;
laboration], “A precision measurement of the neutrino Phys.Rev.D77(2008) 075008; Prog. Theor.Phys.123
mixing angle θ13 using reactor antineutrinos at Daya (2010) 659.
Bay”, arXiv:hep-ex/0701029; F. Ardellier et al., [Dou- [15] Z.Z.XingandY.-L.Zhou,Phys.Lett.B693(2010)584.
ble Chooz Collaboration], “Double Chooz: A search for [16] S. Eidelman et al. (Particle Data Group), Phys. Lett. B
the neutrino mixing angle θ13”, arXiv:hep-ex/0606025; 592 (2004) 149. See also, L.-L. Chau and W.-Y. Keung,
C. Palomares, “Double-Chooz Neutrino Experiment”, Phys. Rev.Lett. 53 (1984) 1802.
arXiv:0911.3227 [hep-ex]; J. K. Ahn et al., [RENO Col- [17] L. Lavoura and W. Grimus, JHEP 09 (007) 2007;
laboration], “RENO:AnExperimentforNeutrinoOscil- T. Endoh, S.Kaneko, S.K. Kang, T. Morozumi, and
lationParameterθ13 UsingReactorNeutrinosatYongg- T.Tanimoto, Phys. Rev. Lett. 89 (2002) 231601; P.H.
wang”, arXiv:1003.1391 [hep-ex]. Frampton,S.L.Glashow,andT.Yanagida,Phys.Lett.B
[7] B.Pontecorvo,Sov.Phys.JETP7(1958)172[Zh.Eksp. 548(2002)119;M.RaidalandA.Strumia,Phys.Lett.B
Teor. Fiz. 34 (1958) 247]; Z. Maki, M. Nakagawa and S. 553(2003)72;V.Barger,D.A.Dicus,H-J.He,andT.Li
Sakata, Prog. Theor. Phys. 28 (1962) 870. Phys.Lett.B583(2004)173;R.G.Felipe,F.R.Joaquim,
[8] T. Schwetz, M. Tortola and J.W.F. Valle, New J. Phys. and B.M. Nobre, Phys. Rev. D 70 (2004) 085009, For a
10 (2008) 113011. review, see, for example, W.L. Guo, Z.Z. Xing, and S.
[9] P.F.Harrison,D.H.Perkins,andW.G.Scott,Phys.Lett. Zhou, Int.J. Mod. Phys. E16 (2007) 1.
B 530 (2002) 167; Z.Z. Xing, Phys. Lett. B 533 (2002) [18] R.N.MohapatraandW.Rodejohann,Phys.Lett.B644
85; P.F. Harrison and W.G. Scott, Phys. Lett. B 535 (2007) 59; A. Blum, R. N. Mohapatra, and W. Rodejo-
(2002) 163. hann, Phys.Rev.D 76 (2007) 053003.
[10] For earlier references, H. Fritzsch and Z.-Z. Xing, Phys. [19] S.M. Bilenky, J. Hosek and S.T. Petcov, Phys. Lett. B
Lett. B 372 (1996) 265; V.D. Barger, S. Pakvasa, T.J. 94(1980)495;J.SchechterandJ.W.F.Valle,Phys.Rev.
Weiler and K. Whisnant, Phys. Lett. B 437 (1998) 107; D 22 (1980) 2227; M. Doi, T. Kotani, H. Nishiura, K.
Y. Kajiyama, M. Raidal, A. Strumia, Phys. Rev. D 76 Okudaand E. Takasugi, Phys. Lett. B 102 (1981) 323.
,Phys. Rev. D 76 (2007) 117301; W. Rodejohann, Phys. [20] M. FukugidaandT. Yanagida,Phys.Lett.B174(1986)
Lett. B 671 (2009) 267; C. H. Albright, A. Dueck and 45.
W. Rodejohann, Euro. Phys. J. C 70 (2010) 1099. See [21] T. Kitabayashi and M. Yasu`e, work in progress.
also,I.deMedeirosVarzielas, R.Gonz´alezFelipeandH.