Re-evaluation of neutrino mixing pattern according to latest T2K result

0.13 MB·English

by  Ya-juan Zheng

EPJ manuscript No.
(will be inserted by the editor)
Re-evaluation of neutrino mixing pattern according to latest
T2K result
2
Ya-juan Zheng1,2 and Bo-Qiang Ma1,3a
1
0 1 School of Physics and StateKey Laboratory of Nuclear Physics and Technology, Peking University,Beijing 100871, China
2 2 Department of Physics, Shandong University,Jinan, Shandong250100, China
n 3 Center for High Energy Physics, PekingUniversity,Beijing 100871, China
a
J
Received: date/ Revised version: date
5

] Abstract. Were-evaluateneutrinomixing patternsaccording to thelatest T2K result for alarger mixing
h angle θ13, and find that the PMNS mixing matrix has larger deviations from bimaximal (BM) and tribi-
p maximal (TB) mixing patterns than previously expected. We also find that several schemes connecting
- PMNSandCKMmixingmatricescanaccommodatethelatestT2Kresultnicely.Asnecessaryupdatesto
p
former works, wemake newtriminimal expansions of PMNS mixingmatrix based on BM and TB mixing
e
h patterns. We also propose a new mixing pattern with a self-complementary relation between the mixing
[ angles θ1ν2+θ1ν3 ≃ 45◦, and find such a new mixing pattern in leading order can provide a rather good
description of thedata.
3
v
PACS. 14.60.Pq – 12.15.Ff – 14.60.Lm
0
4
0
1 Introduction
4
.
6
Neutrino oscillations observed in the solar [1], atmospheric [2], reactor [3] and accelerator [4] neutrino experiments
0
1 have provided plentiful information on neutrino masses and neutrino mixings. Among the knowns and unknowns of
1 neutrino physics, the nonzero but nontrivial smallest mixing angle θ13 has received a lot of attentions, therefore the
: measurement of θ is important to enrich our understanding of neutrino properties. It may lead to the measurement
v 13
of CP violation in the lepton sector and the study of its origin. In addition, the precision measurement of θ would
i 13
X greatly enhance our knowledge to the solar, atmospheric, the secondary effects of long baseline, and the supernova
r neutrino oscillations as well.
a
In the last decade, international collaborations have been established to focus on the measurement of θ , e.g.,
13
DoubleChoozinFrance,ReactorExperimentforNeutrinoOscillations(RENO)inSouthKorea,DayaBayExperiment
in China, NOνA in United States, and Tokai to Kamioka (T2K) experiment in Japan. With the appearance of the
latestresultfromT2K[5],whichindicatesarelativelylargeθ ,weneedtore-evaluateourunderstandingonneutrino
13
masses and mixings, especially on the neutrino mixing pattern [6,7].
Ever since the establishment of neutrino oscillation, theorists have been working on the explanation of neutrino
flavor mixings. From bimaximal (BM) mixing, trimaximal (TB) mixing to the tribimaximal mixing, various neutrino
mixingpatternshavebeenbroughtupandlaterevaluatedbytheexperimentaldata.Ontheotherhand,theunderlying
unification and symmetry concerning leptons and quarks are speculated from some clues of low energy experimental
data, such as the quark-lepton complementarity (QLC) [8–19].As the first result from T2K offers new bounds on the
smallestneutrinomixingangles,weaimtomakeananalysisontheimplicationsandconsequencesfromtheobservation
of a nonzero and also not so small value of θ .
13
The remaining part of this paper is organized as follows. In Sec. 2, we check experimental status of quark and
leptonmixings basedonpreviousfits togetherwiththe latestT2Kresultofalargermixingangleθ .Itisremarkable
13
that the deviations of the lepton mixing matrix from bimaximal and tribimaximal mixing patterns are expressed in
termsoftheWolfensteinparameterλofquarkmixingmatrix.ThenweexamineinSec.2.1severalschemesconnecting
the mixing matrices of quarks and leptons, and find that some of them are possible to accommodate the latest T2K
data. In Sec. 3, as a necessary update, we make new expansions of PMNS mixing matrix based on both bimaximal
andtribimaximalmixingpatterns,withamorereasonablehierarchystructureinpowersoftheWolfensteinparameter
a e-mail: [email protected]
2 Ya-juan Zheng, Bo-Qiang Ma: Re-evaluation of neutrinomixing pattern according to latest T2K result
λinthe quarkmixingmatrix.InSec.4fromthe phenomenologypointofview,weproposeanewmixing patternwith
a self-complementary relation between the mixing angles θν +θν 45 . The new mixing matrix is more closer to
12 13 ≃ ◦
the data thanpreviouslyknownmatricessuchasBMandTB mixing patterns.Thenwe provideasummary inSec.5.
2 Experimental status of quark and lepton mixings
Quarkandlepton mixings are phenomenologicallydescribed by the Cabibbo-Kobayashi-Maskawa(CKM) [20,21]ma-
trix and Pontecorvo-Maki-Nakagawa-Sakata(PMNS) [22,23]matrix respectively. The standard parameterization[24]
ofquarkandleptonmixingmatricesisexpressedbythreemixinganglesθ ,θ ,θ andoneCP-violatingphaseangle
12 13 23
δ. As shown below, the elements of the mixing matrix in the first row and third column adopt very simple forms
c c s c s e iδ
 12 13 12 13 13 − 
V = c s s eiδ s c s s s eiδ+c c s c , (1)
− 12 23 13 − 12 23 − 12 23 13 12 23 23 13
−c12c23s13eiδ+s12s23 −s12c23s13eiδ−c12s23 c23c13 
where s = sinθ ,c = cosθ (i,j = 1,2,3). For describing the quark mixing matrix V , another simple form of
ij ij ij ij CKM
parametrization,i.e., the Wolfenstein parametrization was proposed with the defination that s =λ, s =Aλ2 and
12 23
s eiδ =Aλ3(ρ+iη) [25] . Its explicit form at the accuracy of (λ4) is
13
O
1 1λ2 λ Aλ3(ρ iη)
V = − λ2 1 1λ2 Aλ−2 + (λ4), (2)
CKM − − 2 O
Aλ3(1 ρ iη) Aλ2 1 
− − −
and we adopt the following inputs given by the Particle Data Group [26]:
λ=0.2257+0.0009, A=0.814+0.021, ρ¯=0.135+0.031, η¯=0.349+0.015, (3)
0.0010 0.022 0.016 0.017
− − − −
where ρ¯=ρ 1ρλ2+O(λ4) and η¯=η 1ηλ2+O(λ4). The correspondingquark mixing angles couldbe obtained by
calculatingth−e2moduliofthemixingmat−rix2elementsθ1q2 =13.04◦+00..005593◦◦,θ2q3 =2.37◦+00..008851◦◦ andθ1q3 =0.20◦+00..002203◦◦ [27].
One advantage of the Wolfenstein parametrization is that one m−ay estimate the o−rder of magnitude of a−ny element
fromthehierarchicalfeatureofλ.ItisnaturaltorecallthatthedeviationofV canbe consideredasexpansionsin
CKM
ordersofλfromtheunitmatrixI.ToassumefirstlythattheV quarkmixingmatrixtakestheformofunitmatrix,
CKM
it is easy to get three 0 quark mixing angles in the unit mixing pattern. Hence, we can obtain the corresponding
◦
deviation of each mixing angle:
∆I = θq θI 2.37 , sin∆I (λ2),
∆2I3 =|θ2q3−θ2I3|≃13.0◦4 , sin∆I23 ∼ O(λ), (4)
∆1I2 =|θ1q2−θ1I2|≃0.20 ,◦ sin∆1I2 ∼ O(λ3),
13 | 13− 13|≃ ◦ 13 ∼ O

where λ 0.2 denotes the Wolfenstein parameter.
≃
For the neutrino sector, with the latest global fit of experimental data given in [28,29], the three mixing angles of
the PMNS mixing matrix U read as
PMNS
sin2θ =0.312(1+0.128) (2σ),
12 0.109
−
sin2θ =0.466(1+0.292) (2σ),
23 0.215
−
sin2θ =0.016 0.010 (1σ), (5)
13
±
which indicate that
θ1ν2 ≃33.957◦−+22..144334◦◦, θ2ν3 ≃43.050◦−+57..883349◦◦, and θ1ν3 ≃7.27◦−+22..802142◦◦, (6)
but the CP violating phase δ remains unconstrained. As the recent T2K collaboration results [5] give more robust
indication of a relatively large θ for δ =0:
13 CP
0.03<sin22θ <0.28, Normal Hierarchy (NH),
13
0.04<sin22θ <0.34, Inverted Hierarchy (IH), (7)
13
the smallest mixing angle θ could be naturally produced
13
4.987 <θν <15.974 (NH), 5.769 <θν <17.834 (IH). (8)
◦ 13 ◦ ◦ 13 ◦
Ya-juan Zheng, Bo-Qiang Ma: Re-evaluation of neutrinomixing pattern according to latest T2K result 3
The best-fit values are sin22θ =0.11 (NH) or sin22θ =0.14 (IH), i.e.,
13 13
θ1ν3 =9.685◦+64..268998◦◦ (NH) or θ1ν3 =10.986◦+65..824188◦◦ (IH). (9)
− −
In recent years, lots of efforts have been devoted to explore a new theoretical framework to accommodate tiny
neutrino masses and large flavor mixing angles. Among those studies which have tried to parametrize the PMNS
matrix with only constant numbers, the proposal of bimaximal (BM) [30–35] and tribimaximal (TB) [36–40] mixing
patterns are considered to be the most successful parametrizations and are mostly discussed. The specific forms are
written as
√2/2√2/2 0
 
U = 1/2 1/2 √2/2 P , (10)
BM ν
−
 1/2 1/2 √2/2
−
2/√6 1/√3 0
 
U = 1/√6 1/√3 1/√2 P , (11)
TB ν
−
 1/√6 1/√31/√2
−
where P = Diag e iα/2,e iβ/2,1 includes two CP-violating phases if the three neutrinos are Majorana fermions.
ν − −
{ }
These two scenarios give the prediction of neutrino mixing angles
θBM =45 , θBM =45 , θBM =0 ,
23 ◦ 12 ◦ 13 ◦
θTB =45 , θTB =35.26 , θTB =0 , (12)
23 ◦ 12 ◦ 13 ◦
respectively in the standard parametrization of the 3 3 neutrino mixing matrix. In these two cases the Dirac CP-
×
violating phase δ is absent, which leads to no CP violation in neutrino oscillations.
With the above experimental results as well as phenomenological deduction, combined with the quark-lepton
complementarity (QLC) relations [8–19] given by
θq +θν =45 , θq +θν =45 , (13)
12 12 ◦ 23 23 ◦
we can directly compare the deviation of PMNS lepton mixing matrix from BM and TB mixing patterns in terms of
three mixing angles:
∆BM = θν θBM 2.950 , sin∆BM (λ2),
 23 | 23− 23 |≃ ◦ 23 ∼ O
∆BM = θν θBM 11.043 , sin∆BM (λ), (14)
 12 | 12− 12 |≃ ◦ 12 ∼ O
∆BM = θν θBM 9.685 (10.986 ), sin∆BM (λ),
13 | 13− 13 |≃ ◦ ◦ 13 ∼ O

and
∆TB = θν θTB 2.950 , sin∆TB (λ2),
 23 | 23− 23 |≃ ◦ 23 ∼ O
∆TB = θν θTB 1.303 , sin∆TB (λ3), (15)
 12 | 12− 12 |≃ ◦ 12 ∼ O
∆TB = θν θTB 9.685 (10.986 ), sin∆TB (λ).
13 | 13− 13 |≃ ◦ ◦ 13 ∼ O

From above we find that the mixing angle ∆ν (λ), which serves as a larger deviation from BM and TB mixing
13 ∼ O
patterns according to the latest T2K result of larger θ for neutrino oscillation. This situation is different from our
13
expectation ∆ν (λ2) or (λ3) with small θ , which make it necessary to update the corresponding study [52] on
13 ∼ O 13
expansions based on the deviation.
2.1 Relating quarks and leptons
In this section, we aim to explore the seemingly independent mixing patterns between quark and lepton sectors. The
PMNS matrix depends generallyon the chargedlepton sector whose diagonalizationleads to a chargedlepton mixing
matrix Ul† which should be combined with the neutrino mixing matrix Uν [43,44]:
UPMNS =Ul†Uν. (16)
4 Ya-juan Zheng, Bo-Qiang Ma: Re-evaluation of neutrinomixing pattern according to latest T2K result
SincetheCKMmatrixisquitenearunitmatrix,andthePMNSmatrixhasbeenconsideredtobeclosetobimaximal
matrix and tribimaximal mixing matrix, it is interesting to assume that the deviation of the PMNS matrix from the
BM and TB mixings can be described by the CKM matrix as generally discussed in reference [43,44], that is,
U V =U , or V U =U , (17)
PMNS CKM BM/TB CKM PMNS BM/TB
which is equivalent to
UPMNS =UBM/TBVC†KM, or UPMNS =VC†KMUBM/TB. (18)
Adopting similar interpretation, Ref. [45] provides explicit discussions of the following two cases for the relation
between CKM and PMNS mixing matrices with U the bimaximal mixing matrix:
ν
UPMNS =UBMVC†KM, and UPMNS =VC†KMUBM. (19)
In the first case, substituting the corresponding matrices of U and V one can calculate the moduli of each
BM CKM
mixing element of U and give
PMNS
√2
sPMNS = Aλ2 (λ λρ 1)2+(λη)2 =0.48λ2. (20)
| 13 | 2 p − −
Thus, we get the explicit value
1.39 <θ <1.40 , (21)
◦ 13 ◦
assuming no CP violation. It is obvious that this region is excluded by the T2K results. Thus this scheme is not
preferable judging by present experimental data.
In the second case, U could be produced similarly
13
√2
sPMNS = λ [Aλ2(1 ρ) 1]2+(Aλ2η)2 =0.68λ, (22)
| 13 | 2 p − −
and consequently
8.79 <θ <8.84 . (23)
◦ 13 ◦
Although the second case provides θ values more closer to the experimental data, it is proved in the original paper
13
that this case does not accommodate the QLC relations, i.e., Eq. (13), very well [45].
We shouldpoint outthat the multiplying relationshipbetweenPMNS andCKMmatrices,suchasEq.(17),is not
independent from different parametrizations of the charged lepton CKM matrix. The reason is that different choices
of phase factors in V can produce differences in the predicted U . If we choose the leptonic mixing matrix U
CKM PMNS ν
to be tribimaximal pattern, for another choice of charged lepton mixing matrix Ul† in the case of UPMNS = Ul†UTB,
an Ansatz have been discussed [46,47] using a new parametrization[48]:
1 λ2/2 λeiδ hλ3
Ul† = −−λe−iδ 1−λ2/2 (f +he−iδ)λ2+O(λ4) , (24)
 fλ3e−iδ (f +heiδ)λ2 1 
−
where the parameters A, ρ and η in the Wolfenstein parametrization [25] are replaced by f, h and δ in the new
Wolfenstein-like one [48]. And the corresponding parameters read as:
f =0.749+−00..003347, h=0.309−+00..001172, λ=0.22545±0.00065, δ =(89.6+−20..9846)◦. (25)
As we will see below, this matrix could be a candidate for the small deviation of the PMNS matrix from the TB
mixing pattern.
By substituting the corresponding matrix in UPMNS = Ul†Uν, the leptonic mixing matrix corrected by the contri-
butions from Ul† can be written, up to order of λ3, as [46,47]
λeiδ λ2(1+hλ) λeiδ λ2(1 2hλ) λ(hλ2 eiδ)
− −
 − √6 − √6 √3 − 2√3 √2 
UPMNS =UTB+−λq23e−iδ− λ2(1−22f√−62he−iδ) −λe√−3iδ − λ2(1−2f2−√23hλe−iδ) λ2(1+22f√+22he−iδ)Pν
 λ2(f+heiδ+2fλe−iδ) λ2(f+heiδ fλe−iδ) λ2(f+heiδ) 
 − 
√6 − √3 √2
+ (λ4) . (26)
O
Ya-juan Zheng, Bo-Qiang Ma: Re-evaluation of neutrinomixing pattern according to latest T2K result 5
In this case, Ref. [46] adopts an alternative explicit form for tribimaximal mixing matrix with negative 21, 23 and
31 matrix elements, which is physically equivalent to Eq. (11). Thus, the smallest mixing angle θ can be obtained
13
by [46]
λ
sinθ = (1 hλ2cosδ) ,
13
√2 −
withanon-vanishingθ =9.2 .Suchascheme,whichhasbeenproposedbyAhn-Cheng-Oh[46,47],providesaviable
13 ◦
relation to connect quark and lepton mixing matrices in agreement with the new T2K results.
3 Triminimal expansion
AftertheaboveinitiativestudyofT2Kresults,wefoundthattherelativelylargeθ makesitnecessarytoupdateand
13
expand the former works on triminimal expansion [51,52]. If the flavor mixing matrices are parametrized according
to the hierarchicalstructure of mixing, it may reveal more physical information about the underlying theory. A good
choice of this idea is the triminimal parametrization [49–52]with an approximation as the basis matrix to the lowest
order. That is to express a mixing angle in the mixing matrix as the sum of a zeroth order angle θ0 and a small
perturbation angle ǫ with
θ =θ0 +ǫ , θ =θ0 +ǫ , θ =θ0 +ǫ . (27)
12 12 12 23 23 23 13 13 13
Withthedeviationsǫ ,onecanexpandthemixingmatrixinpowersofǫ whiledifferentchoicesofθ0 leadtodifferent
ij ij ij
basis.
According to new results of neutrino mixing angles, we can see that the deviation of PMNS mixing matrix from
TB mixing pattern is more close to that of CKM mixing matrix from unit matrix if measured in terms of mixing
angles. To be explicit, the comparison of deviation hierarchy is listed as follows for corresponding mixing angles:
sin∆TB sin∆I (λ) and sin∆TB sin∆I (λ2). From this aspect, the deviation hierarchy from BM and
13 ∼ 12 ∼ O 23 ∼ 23 ∼ O
TB mixing patterns of different mixing angles can serve as the triminimal expansion basis in constructing the PMNS
matrix.
3.1 Expansion on bimaximal pattern
To realize the triminimal expansion on BM mixing pattern, we assume that ǫL = λ and ǫL = Aλ2 considering
the suggestion of QLC [52], and ǫLeiδL = λZ from the constraint of new T2K12resu−lt as estim23ated−in Sec. 2. To the
13
third order in ǫL, the expansion is [52]
ij
UPMNS = UBM+ǫL12−−21√1222 −√221212 000+ǫL2301212 −−02211 −√02√222+ǫL13−−22110eeiiδδLL −−22110eeiiδδLL e−00iδL
√2 √2 0 0 0 0 √2 √2 0
+ (ǫL)2−14 − 41 0+(ǫL)2 1 1 √2+(ǫL )2−04 −04 √2
12  −441 −414 0 23 −441 −144 −−√442 13  0 0 −−√442
0 0 0 0 0 0 0 0 0
+ ǫLǫL 1 1 0+ǫLǫLeiδL1 1 0+ǫLǫLeiδL 1 1 0
12 23 2 2 12 13 2 −2 23 13 −2 −2
1 1 0 1 1 0  1 1 0
2 2 2 −2 2 2
+ (ǫL)3 √1122 −1√122 00+(ǫL )3 01 01 0√2+(ǫL )3 1 e0iδL 1 e0iδL −16e0−iδL
12 12 12 23 −12 12 −12 13 12 12
−112 −112 0 −112 112 √122  112eiδL 112eiδL 0 
0 0 0 0 0 0 √2 √2 0
+ ǫL(ǫL)2 1 1 0+(ǫL)2ǫL  1 1 0+ǫL(ǫL)2 04 −04 0
12 23 4 4 12 23 −4 4 12 13
 1 1 0  1 1 0  0 0 0
−4 −4 −4 4
0 0 0 00 0 0 0 0
+ (ǫL)2ǫLeiδL1 1 0+ǫL(ǫL)200 √2+(ǫL )2ǫLeiδL1 1 0
12 13 4 4 23 13 − 4 23 13 4 4
1 1 0 00 √2  1 1 0
4 4 4 4 4
6 Ya-juan Zheng, Bo-Qiang Ma: Re-evaluation of neutrinomixing pattern according to latest T2K result
0 0 0
+ ǫLǫLǫLeiδL 1 1 0+ (ǫL)4 . (28)
12 23 13  21 −12 0 O(cid:0) ij (cid:1)
−2 2
Since the hierarchy is ǫL (ǫL )2, ǫ eiδL ǫL, then to the third order in ǫL, we have
23 ∼ 12 13 ∼ 12 12
UPMNS = UBM+ǫL12−−21√2212 −√212221 000+ǫL2301122 −−02211 −√02√222+ǫL13−−22110eeiiδδLL −−22110eeiiδδLL e−00iδL
√2 √2 0 √2 √2 0 0 0 0 0 0 0
+ (ǫL)2−14 − 41 0+(ǫL)2−04 −04 √2+ǫLǫL 1 1 0+ǫLǫLeiδL1 1 0
12  −441 −144 0 13  0 0 −−√442 12 23122 221 0 12 13 221 −−221 0
+ ǫLǫLeiδL 01 01 00+(ǫL )3 √1122 −1√122 00+(ǫL)3 1 e0iδL 1 e0iδL −61e0−iδL
23 13 −2 −2 12 12 12 13 12 12
 12 21 0 −112 −112 0 112eiδL 112eiδL 0 
√2 √2 0 0 0 0
+ ǫL(ǫL)2 04 −04 0+(ǫL )2ǫLeiδL1 1 0+ (ǫL)4
12 13  0 0 0 12 13 14 41 0 O(cid:0) ij (cid:1)
4 4
√2 √2 Ze i2δL
 − 2 2 − − 
= UBM−λ −21(1−Z)−21(1−Z) 0
 1(1+Z) 1(1+Z) 0 
2 2
√2(1+Z2e i2δL) √2(1+Z2e i2δL) 0
+ λ2− 41(A+Z −1) − 41(A+Z −1) √2(A+ 1Z2e i2δL)
 −−122(A+Z−+ 221) 122(A+Z−+ 221) −√222(A− 212Z2e−−i2δL) 
√2(1 +Z2e i2δL) √2(1 +Z2e i2δL) 1e i4δL
+ λ3 1 (6AZ−+64A3+3Z+−Z3e i2δL 1) 1 (6AZ+46A3+3Z+−Z3e i2δL 1) −6 0− 
11122(−6AZ+6A+3Z+Z3e−−i2δL−+1) 11122(−6AZ+6A+3Z+Z3e−−i2δL−+1) 0 
+ λ4 . (29)
O(cid:0) (cid:1)
3.2 Expansion on tribimaximal pattern
If we start expansion with an alternative TB mixing form, the triminimal parametrization of the PMNS matrix is
obtained as [49,50] to the second order in ǫL:
ij
1 2 0 0 0 0 0 0 e iδL
U =U +ǫL −√13 √61 0+ǫL  1 1 1 +ǫL  1 eiδL 1 eiδL −0 
PMNS TB 12 −√3 −√6 23 √6 −√3 √2 13 −√3 −√6
 √13 √16 0 √16 −√13 −√12 −√13eiδL −√16eiδL 0 
1 1 0 0 0 0 1 1 0
+(ǫL )2 −1√6 −2√13 0+(ǫL)2 1 1 1 +(ǫL )2−0√6 −20√3 1 
12 2√6 −2√3 23 2√6 −2√3 −2√2 13 −2√2
−2√16 2√13 0 −2√16 2√13 −2√12  0 0 −2√12
0 0 0 0 0 0 0 0 0
+ǫL ǫL  1 1 0+ǫL ǫLeiδL 1 1 0+ǫL ǫLeiδL 1 1 0+ ((ǫL)3), (30)
12 23√13 √16 0 12 13 √16 −√13 0 23 13 −1√3 −1√6 0 O ij
√3 √6 √6 −√3 √3 √6
whereǫL,ǫL,andδL arethesameparametersasparametrizationofthePMNSmatrixjustlikeintheBMtriminimal
23 13
expansion case, whereas ǫL is not. This set of expansion parameters is certainly better than the one in the previous
12
sectionifconvergencyisthecriteriaoftheexpansion.WiththereplacementsǫL =Bλ3,ǫL = Aλ2andǫLeiδL =λZ,
the hierarchy is ǫL (ǫL)3 and ǫL (ǫL)2. The parameters A and λ are th1e2 same Wo2lf3enst−ein parame1t3ers as those
12 ∼ 13 23 ∼ 13
Ya-juan Zheng, Bo-Qiang Ma: Re-evaluation of neutrinomixing pattern according to latest T2K result 7
in the BMcase,arguedfromthe aspect ofQLC relations,andB is a new parameter,which is of order (1), adjusted
to fit the data according to the estimate in Sec. II. Thus to the third order of ǫL: O
13
1 2 0 0 0 0 0 0 e iδL
U =U +ǫL −√13 √61 0+ǫL  1 1 1 +ǫL  1 eiδL 1 eiδL −0 
PMNS TB 12 −√3 −√6 23 √6 −√3 √2 13 −√3 −√6
 √13 √16 0 √16 −√13 −√12 −√13eiδL −√16eiδL 0 
1 1 0 0 0 0
+(ǫL )2−0√6 −20√3 1 +ǫLǫLeiδL 1 1 0+ ((ǫL )4)
13  0 0 −−22√√122 23 13 −√1√33 −√1√66 0 O 13
 0 0 e−i2δL √16Z2e−i2δL 2√13Z2e−i2δL 0 
=U +λ 1 Z 1 Z 0 λ2 1 A 1 A 1 (2A+Z2e i2δL)
TB −√3 −√6 − √6 −√3 2√2 −
−√13Z −√16Z 0   √16A −√13A −2√12(2A−Z2e−i2δL)
1 B 2 B 0
 −√3 √6 
+λ3 1 (B AZ) 1 (B AZ)0 + (λ4). (31)
−1√3(B −AZ) −1√6(B −AZ) 0 O
 √3 − √6 − 
It is novel that the deviations of the lepton sector are explicitly illustrated in orders of Wolfenstein parameter λ
from the quark sector, along with the idea to understand both the quark and lepton mixing patterns in a unified
manner [51,52].
4 A phenomenological proposal of new mixing pattern
From previous global fits of neutrino mixing angles,
θ1ν2 ≃33.957◦−+22..144334◦◦, θ2ν3 ≃43.050◦−+57..883349◦◦, (32)
together with the latest T2K implication:
θ1ν3 =9.685◦+64..268998◦◦ (NH) or θ1ν3 =10.986◦+65..824188◦◦ (IH), (33)
− −
we can find a self-complementary relation between the mixing angles
θν +θν θν 45 . (34)
12 13 ≃ 23 ≃ ◦
ThisleadstoaproposalforanewmixingpatternwhichisclosertotheexperimentaldatathanBMandTMpatterns.
To construct such a new mixing pattern, we begin with the assumption that
sin2θ = 1,
23 2
sin2θ = 1,
12 3
sin2θ =sin2(θ θ )= 1 √2. (35)
13 23− 12 2 − 3
Thus the new mixing matrix could be given as
 √√23q12 + √32 √13q12 + √32 √13 − √16 
UNM =−13 + √62 − √16 −√62 + 16 + √13 √22q12 + √32
−31 + √62 + √16 −√62 + 16 − √13 √22q12 + √32
0.8047 0.5690 0.1691
 
0.5059 0.5083 0.6969 . (36)
≃ −
 0.3106 0.64640.6969
−
One can easily see that for this new mixing pattern, the moduli of the mixing matrix elements are compatible
with the results from global fits [27–29]. Generally, the mixing matrix is expanded as below. With the replacements
8 Ya-juan Zheng, Bo-Qiang Ma: Re-evaluation of neutrinomixing pattern according to latest T2K result
ǫL = Bλ3, ǫL = Aλ2, and ǫLeiδL = λ3Z or λ2Z defined according to the hierarchical structure, the expansion
c1o2uld be obta2i3ned−to the order o13f (λ3): ′ ′
O
s0 c0 c0 c0 0
 − 12 13 12 13 
U =U +ǫ s0 s0 s0 eiδ c0 c0 c0 s0 s0 eiδ s0 c0 0
PMNS NM 12 12 13 23 − 12 23 − 12 13 23 − 12 23
s0 s0 c0 eiδ+c0 s0 c0 s0 c0 eiδ+s0 s0 0
12 13 23 12 23 − 12 13 23 12 23
0 0 0
+ǫ  s0 c0 c0 eiδ+s0 s0 s0 s0 c0 eiδ s0 c0 c0 c0 
23 − 13 12 23 23 12 − 12 13 23 − 23 12 13 23
 c012s023s013eiδ+c023s012 s012s023s013eiδ−c012c023 c012s023
c0 s0 s0 s0 c0 e iδ
+ǫ  c0−c102s103eiδ c0−s102s103eiδ 1s30−s0 + (λ4), (37)
13 − 12 13 23 − 13 23 12 − 13 23 O
 c0 c0 c0 eiδ s0 c0 c0 eiδ s0 c0 
− 12 13 23 − 12 13 23 − 13 23
where s0 =sinθ , c0 =cosθ with θ denotes the complementary mixing angles proposed in Eq. (35). We see that
ij ij ij ij ij
this expansion based on the new mixing pattern is much simpler compared with the triminimal expansions based on
either BM or TM mixing patterns in the above section. The leading order basis matrix U is much closer to the
NM
experimental fits.
5 Summary
In summary, we re-analyzed the neutrino mixing patterns according to the latest T2K result for a larger mixing
angle θ , and found that the deviation of the neutrino mixing pattern from bimaximal (BM) and tribimaximal (TB)
13
patterns become larger than previously expected. We also examined relations connecting quark and lepton mixing
matrices and it turned out that several schemes can still accommodate the latest T2K result nicely. As a necessary
update of previous works, we made new triminimal expansions of the PMNS mixing matrix based on BM and TB
mixingpatternsintermsofamorereasonableλhierarchy.Fromthephenomenologicalpointofview,wealsoproposed
a new mixing pattern with a self-complementary relation between the mixing angles θν +θν 45 , and concluded
12 13 ≃ ◦
that such a mixing pattern works well in providing rather good descriptions to the data at least with a precision of
(λ2). Thus the new T2K result can enrich our understanding of neutrino properties.
O
ThisworkispartiallysupportedbyNationalNaturalScienceFoundationofChina(GrantsNo.11021092,No.10975003,
No. 11035003,and No. 11120101004)and by Peking University Visiting Scholar Programfor Graduate Students.
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