Geometrical model for non-zero �13

Jun-Mou Chen, Bin Wang, and Xue-Qian Li

School of Physics, Nankai University, Tianjin, 300071, China(Received 29 June 2011; published 3 October 2011)

Based on Friedberg and Lee’s geometric picture by which the tribimaximal Pontecorvo-Maki-

Nakawaga-Sakata leptonic mixing matrix is constructed, namely, corresponding mixing angles corre-

spond to the geometric angles among the sides of a cube. We suggest that the three realistic mixing angles,

which slightly deviate from the values determined for the cube, are due to a viable deformation from the

perfectly cubic shape. Taking the best-fitted results of �12 and �23 as inputs, we determine the central value

of sin22�13 should be 0.0238, with a relatively large error tolerance; this value lies in the range of

measurement precision of the Daya Bay experiment and is consistent with recent results from the T2K

Collaboration.

DOI: 10.1103/PhysRevD.84.073002 PACS numbers: 14.60.Pq

I. INTRODUCTION

Neutrino oscillation observations have revealed evi-dence that neutrinos are massive. Neutrinos are producedvia weak interaction as flavor eigenstates �f ¼ ð�e; ��; ��Þand can be written in the mass basis �m ¼ ð�1; �2; �3Þ,which are really the physical states. These two bases arerelated by a unitary matrix U�, i.e., �f ¼ U��m. The

mixing in the lepton sector is named as the Pontecorvo[1]-Maki-Nakawaga-Sakata [2] (PMNS) matrix, which canaccount for the currently available data on the observationof solar, atmospheric neutrino oscillations and the reactor

and accelerator neutrino experiments [3]. In the standardmodel, the weak charged currents are

J � ¼ �li��ð1� �5ÞðUy

l U�Þij�j; (1)

where i, j ¼ 1, 2, 3 and correspond to physical particles.The mixing matrix

UPMNS ¼ Uyl U� (2)

is a 3� 3 unitary matrix and can be parameterized by threemixing angles �12, �23, and �13, and one CP phase � [3],

UPMNS ¼c12c13 s12c13 s13e

�i�

�s12c23 � c12s23s13ei� c12c23 � s12s23s13e

i� s23c13

s12s23 � c12c23s13ei� �c12s23 � s12c23s13e

i� c23c13

0BB@

1CCA; (3)

where cij � cos�ij, sij � sin�ij. If neutrinos are Majoranaparticles, there would be an additional diagonal matrixdiagðei�1=2; ei�2=2; 1Þ multiplied to the above UPMNS ma-trix, which is not relevant for neutrino oscillations. Theparametrization Eq. (3) can be rewritten as a product ofthree rotations Rij in the ij plane through angles �ijand a diagonal CP phase matrix U�¼diagðei�=2;1;e�i�=2Þ,

UPMNS ¼ R23ð�23ÞUy�R13ð�13ÞU�R12ð�12Þ; (4)

with

R23 ¼1 0 0

0 c23 s23

0 �s23 c23

0BB@

1CCA; R13 ¼

c13 0 s13

0 1 0

�s13 0 c13

0BB@

1CCA;

R12 ¼c12 s12 0

�s12 c12 0

0 0 1

0BB@

1CCA: (5)

There have been numerous phenomenological Ansatzefor the entries of UPMNS, for example, the democratic [4],the bimaximal [5], and the tribimaximal Ansatze [6].Among them, the tribimaximal mixing is closer to theexperimentally observed mixing patterns, and the matrixis given by

Utribi ¼2=

ffiffiffi6

p1=

ffiffiffi3

p0

�1=ffiffiffi6

p1=

ffiffiffi3

p1=

ffiffiffi2

p

1=ffiffiffi6

p �1=ffiffiffi3

p1=

ffiffiffi2

p

0BB@

1CCA; (6)

which suggests �12 ¼ sin�1ð1= ffiffiffi3

p Þ, �23 ¼ �=4, �13 ¼ 0.As noted, the CP phase ei� is always associated with s13[Eq. (3)]; thus, null �13 would imply that one cannotobserve CP violation at lepton sector in the frameworkof the standard model even though � is not zero. Obviously,there is no priori that the CP violation should appear at thelepton sector, but only nonzero �13 can intrigue an enthu-siasm to explore CP violation at the lepton sector. Once the�13 is determined to be nonzero as the T2K experiment and

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our theoretical prediction made in this work suggest, thenext step would be searching for CP violation at the leptonsector.

Indeed, the tribimaximal symmetry is well manifestedby the data. A rigorous symmetry would demand �13 to bezero; however, it is not the whole story because this elegantsymmetry is to be broken, and a nonzero �13 is expected.The question is if it is not zero, what is its size, which is themain concern of the recent studies.

The unbroken tribimaximal matrix Eq. (6) can be furtherwritten as a sequential product of two independent rota-tions on 12 and 23 planes:

R23ð�=4Þ ¼1 0 0

0 1=ffiffiffi2

p1=

ffiffiffi2

p

0 �1=ffiffiffi2

p1=

ffiffiffi2

p

0BB@

1CCA;

R12ðsin�1ð1= ffiffiffi3

p ÞÞ ¼2=

ffiffiffi6

p1=

ffiffiffi3

p0

�1=ffiffiffi3

p2=

ffiffiffi6

p0

0 0 1

0BB@

1CCA;

(7)

and R13 becomes a 3� 3 unit matrix, i.e., Utribi ¼R23ð�=4ÞR12ðsin�1ð1= ffiffiffi

3p ÞÞ. Friedberg and Lee [7] propose

a geometrical interpretation for the tribimaximal symmetryas shown in Fig. 1. For readers’ convenience, let us brieflyintroduce Friedberg and Lee’s geometrical model andtheir conventions [7]. The charged leptons in the basisLc ¼ ðSe; S�; S�ÞT correspond to the three mutually per-

pendicular sides of a cube, and the neutrino basis Ln ¼ðS�1

; S�2; S�3

ÞT corresponds to another coordinate system

(see Fig. 1). These two coordinate systems are relatedto each other by rotations. One can perform two indepen-

dent rotations to associate them. Ry23ð�=4Þ and

R12ðsin�1ð1= ffiffiffi3

p ÞÞ transform the two independent basesinto a common one. These practical operations are de-

scribed below. Ry23ð�=4Þ mixes the second and third com-

ponents of the basis Lc and keeps the first one invariant to

get a new basis ðS1; S2; S3ÞT , while R12ðsin�1ð1= ffiffiffi3

p ÞÞmixes the first and second components of Ln, retainingthe third one invariant to reach the same basis ðS1; S2; S3ÞT .The mathematical expressions for relating ðS1; S2; S3ÞTwith the charged lepton basis ðSe; S�; S�ÞT and neutrinobasis ðS�1

; S�2; S�3

ÞT are shown as follows:

S1

S2

S3

0BB@

1CCA ¼ Ry

23ð�=4ÞSe

S�

S�

0BB@

1CCA;

S1

S2

S3

0BB@

1CCA ¼ R12ðsin�1ð1= ffiffiffi

3p ÞÞ

S�1

S�2

S�3

0BB@

1CCA:

(8)

Following the convention given in Ref. [7], when wediscuss the geometry structure, we abbreviate the sides Sl(S�i

) as l (�i) without causing any confusion. Here Se;�;�

and S�1;�2;�3just refer to the corresponding geometrical

quantities marked in Fig. 1 and are by no means thephysical states.

Comparing Utribi ¼ R23ð�=4ÞR12ðsin�1ð1= ffiffiffi3

p ÞÞ with

UPMNS ¼ Uyl U�, it appears that the two rotations

Ry23ð�=4Þ and R12ðsin�1ð1= ffiffiffi

3p ÞÞ correspond to the mixing

matrices for the charged leptons and neutrinos, respec-tively. It is noted that in Eq. (8), we only concern themixing parts; thus, inserting �0��ð1� �5Þ betweenððS1; S2; S3ÞTÞy and ðS1; S2; S3ÞT , which is irrelevant toour geometrical settings, we just derive the Lagrangian ofweak interaction. We also would like to point out that inEq. (8), the high symmetry is assumed, and all quantitiesare indeed corresponding to the zeroth order ones [7], andthen later when we introduce a deformation of the cube tobreak the tribimaximal symmetry, the concerned quantitieswould turn into the physical ones.Concretely, in Fig. 1, sidesOX,OY, andOZ represent e,

�, and �; and �1, �2, and �3 correspond to OX0, OB, andOZ0, respectively. The lineOZ0 resides on the planeOZAYand spans an angle of �=4 with respect to the OZ axis,whereas line OX0, line OZ0, and line OB compose three-dimensional mutually perpendicular coordinate axes, andaccording to the right-hand rule, we have an OB�OZ0 �OX0 system. The angle spanned between OX0 and OXis �12. Therefore, the two rectangular coordinate systemstransform from each other by two rotations about the axesOX and OZ0, respectively. The right-handed rotation

Ry23ð�=4Þ about the OX axis brings � to OA and � to �3,

and a second right-handed rotation R12ðsin�1ð1= ffiffiffi3

p ÞÞ, with�12 ¼ sin�1ð1= ffiffiffi

3p Þ, turns �1 into e and �2 into OA. Then,

after performing the two successive operations, the basisðS1; S2; S3ÞT shown above can be directly read out asðe;OA; �3ÞT .

FIG. 1. Geometric representation of the tribimaximal mixingin Eqs. (6) and (8).

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Although the tribimaximal mixing Ansatz is close to theexperimental data and exhibits a striking symmetry, it isnot the exact form of the PMNS matrix. Moreover, thissymmetry demands �13 to be zero. If the tribimaximalsymmetry is not exact, with the angles �23 and �12 obvi-ously deviate from the values determined by the symmetry,one has sufficient reason to believe that �13 should not bezero. In fact, the previous measurements set a lower boundfor �13 as sin

22�13 < 0:15 [3], and will be more preciselymeasured at the upcoming reactor experiments Daya Bay[8] and Double Chooz [9].

It would be interesting to investigate how to break thetribimaximal symmetry from a theoretical aspect.Friedberg and Lee suggest to break the symmetry fromthe charged lepton side [10], whereas He and his collabo-rators break the symmetry from the neutrino sector [11].Since the whole mixing matrix is a product of the twounitary matrices that, respectively, diagonalize the charged

lepton and neutrino mass matrices as UPMNS ¼ Uyl U�,

breaking from either side is just like climbing up MountEverest from the south or north side as Lee comments [12].Their schemes to break the symmetry are algebraic.

Instead, in this work, we propose to break the symmetrybased on Friedberg and Lee’s geometrical picture. Namely,we let the cube be slightly deformed and the nonzero �13value would emerge. Concretely, by deforming the geo-metric representation of the tribimaximal mixing, the an-gles would deviate from their ideal values; by fitting themto the data, we determine the deformation scale of the cube,and then by the new geometric shape �13 is no longer zero.

The work is organized as follows. In Sec. II, we presentour geometrical model of deforming the cube to get the �13as a function of the other two mixing angles. Then, inSec. III, we present our numerical results. The last sectionis devoted to our conclusion and some discussions.

II. THE DEFORMED CUBE MODEL

It is noticeable that the angle between lines OA and OBand that between OY and OA in the cube are precisely thetwo mixing angles of the tribimaximal matrix �12 and�23, respectively. For a perfect symmetry, which corre-

sponds to a complete cube, we have �12 ¼ sin�1ð1= ffiffiffi3

p Þand �23 ¼ �=4, which are determined by the geometry. Itis then viable that a deformation would lead to the morerealistic form of the PMNS matrix, and, thus, the nonzero�13 would emerge. After this deformation, �12 and �23 arenot the values given above anymore, but dependent on theform of the deformation. A cube is a kind of polyhedronwith high symmetry described by a certain group, so adeformation of a cube should be regarded as a symmetrybreaking.

Now, let us demonstrate how to deform the abovecube. For choosing the deformation scheme, we set threeprinciples:

(i) There are three rotation axes for a cube as presentedin Fig. 2, i.e., EE0, FF0, and GG0. Apparently, theaxis GG0 is related with the mixing angle �12(ffAOB), and the axis FF0, which is parallel to theside OA, is related to �23 (ffAOY). Then, the restsymmetry axis EE0 may be related to the zero �13 inthe tribimaximal mixing. Thus, after the supposeddeformation, the three symmetries would be broken,and the value of the deformation angle is related to�13. For simplicity, we just choose the deformationangle to be �13.

(ii) For the tribimaximal mixing, �23 ¼ �=4 and�13 ¼ 0, there exists the �� � symmetry [13].The global fit [14] gives �23 ¼ 42:8�, which appar-ently breaks the �� � symmetry. Thus, in thedeformation, �23 (ffAOY) should be changed from�=4 to some values in order to break the �� �symmetry.

(iii) In Ref. [14], the global fits of �12 and �23 are 34.4�

and 42.8�, respectively. Thus, for the deformation,�12 (ffAOB) and �23 (ffAOY) should be changed

toward smaller values than sin�1ð1= ffiffiffi3

p Þ and �=4,respectively.

Considering the above principles, the simplest and mostdirect scheme to deform the cube is to slide the bottom faceparallel to the top face, and a small angle would emerge,and this angle is identified as �13. The length of each side isunchanged during the slide. This operation is explicitlyillustrated in Fig. 3. With the parallel slide, the bottom facebecomes EFGH. To be consistent with Friedberg andLee’s picture and the principles we proposed above, weidentify �12 ¼ ffFAG, �23 ¼ ffBAF, and �13 ¼ ffEAE0.E0 is the point of intersection between side AE0 and planeEFGH. AE1 and E0E1 are perpendicular to the diagonalline EG.

FIG. 2 (color online). The three classes of symmetry axes for acube, i.e., EE0, FF0, and GG0.

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Setting ffE0EG � � and in the rectangular triangleRt�AE1G, one has

AG2¼AE21þE1G

2

¼ cos2�13þsin2�13sin2�þð ffiffiffi

2p þsin�13 cos�Þ2: (9)

In �AEF,

AG2¼1þ4cos2�23�4cos�23 cosð���12

�sin�1ð2sin�12 cos�23ÞÞ; (10)

and in �EE0F,

E0F2 ¼ E0E

2 þ EF2 � 2E0E � EF cosffE0EF (11)

4cos2�23 � cos2�13 ¼ sin2�13 þ 1� 2 sin�13 cos

��þ �

4

�:

(12)

The geometrical relationship of the sides and angles inthe deformed cube would determine Eq. (9), (10), and (12).From these equations, we can get an analytical expressionof �13, with respect to the other two mixing angles �12 and�23 as

sin2�13¼4cos4�23�4cos2�23

þ4cos2�23cos2ð�12þsin�1ð2sin�12cos�23ÞÞþ1:

(13)

As we have mentioned above, a cube is a highly sym-metric polyhedron that could be represented by a global S4group [15]. This group has 24 elements classified in fiveconjugate classes. As shown in Fig. 2, a cube has threekinds of rotation axes, h ¼ 2, h ¼ 3, and h ¼ 4, corre-sponding to FF0, GG0, and EE0, respectively. All therotation axes in the same h are equivalent.

It is notable that the angle between the axes of h ¼ 2 andh ¼ 4 is �=4, and the angle between the axes of h ¼ 2 and

h ¼ 3 is sin�1ð ffiffiffiffiffiffiffiffi1=3

p Þ. In other words, the two angles areexactly that in the tribimaximal form of the PMNS matrix.Another angle corresponding to �13 must be an anglebetween h ¼ 4 and h ¼ 4 itself, so �13 ¼ 0.With the deformation, the symmetry of the cube is

broken, and �13 acquires a nonzero value. We can thenview �13 as the parameter representing the deviation fromthe cubic symmetry. It is then viable to define �13 as theangle between the ‘‘new’’ h ¼ 4 axis and the ‘‘old’’ one.

III. NUMERICAL RESULTS

Ref. [14] presents the updated global fit to the three-generation neutrino mixing:

�12 ¼ 34:4� 1:0�; �23 ¼ 42:8þ4:7��2:9 : (14)

Using the data as inputs, we obtain the numerical valueof �13:

sin 22�13 ¼ 0:0238; �13 ¼ 4:44�: (15)

The errors of the fit would cause a theoretical uncertaintyto �13:

sin22�13 ¼ 0:0238þ0:0762�0:0238: (16)

The errors are rather large, and, in fact, to makesin22�13 � 0, the lower bound shown in the above expres-sion is set. This expression indicates that our prediction on�13 is somehow sensitive to the input data and that in orderto get more precise values of �13, more precise values ofthe input are needed.Two reactor neutrino experiments, Daya Bay [8] and

Double Chooz [9], aiming to directly measure �13 areexpected to reach a very high precision. We illustrate arelation of the expected sensitivity of the Daya Bay and

FIG. 3 (color online). The sketch of the shift of the bottom facerelative to the top face.

FIG. 4 (color online). Comparison of the toy model (dashedline) with the Daya Bay expected sensitivity limit to sin22�13 asa function of running time.

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Double Chooz as a function of the running time in Figs. 4and 5, respectively, where we mark the central value ofsin22�13 calculated in this work.

Apparently, because of the high precision, the sin22�13value from our model would be probed at the first run of theDaya Bay experiment.

We show sin22�13 as a function of �12 for �23 ¼ 42:8� inFig. 6, and the dependence on �23 for �12 ¼ 34:4� in Fig. 7.Particle Data Group (PDG) presents an upper bound of �13as sin22�13 < 0:15 at CL ¼ 90% [3], which is also markedin Figs. 6 and 7. From Figs. 6 and 7, we can see thatthe theoretically predicted value of sin22�13 is sensitive

to both �12 and �23. By the updated data, �12 and �23are constrained within the range (31:9�–36:5�) and(40:2�–48:3�), respectively.

IV. DISCUSSIONS AND CONCLUSIONS

�23 ¼ �=4 and �13 ¼ 0 imply the so-called �� �symmetry [13] embedding in the neutrino mass matrix,i.e., the mass matrix in the flavor basis has an obvious�� � �� permutation symmetry. This leads to the mass

matrix with the form

M ¼A B BB C DB D C

0@

1A: (17)

In Ref. [11], the authors discussed the soft breaking of the�� � symmetry that arises from the Majorana mass termof the heavy right-handed neutrinos in the minimal seesawmodel. From their �� � symmetry breaking model, theyderived a relation among the mixing angles and Dirac CPphase

�23 � �4 ¼ ��13 cot�12 cos�: (18)

For the case that the Dirac CP phase � ¼ 0 and substitut-ing the experimental fits �12 ¼ 34:4� and �23 ¼ 42:8� intoEq. (18), we obtain the value of �13 as

sin 22�13 ¼ 0:00276; �13 ¼ 1:51�: (19)

Instead, in a parallel work, Friedberg and Lee [10]suggested that one can break the �� � symmetry at thecharged lepton side in terms of a perturbation method,and they also showed that the breaking may lead to anonzero �13.

FIG. 5 (color online). Comparison of the toy model (dashedline) with the Double Chooz expected sensitivity limit tosin22�13 as a function of running time.

FIG. 6 (color online). sin22�13 as a function of �12 for �23 ¼42:8� from the toy model (solid line). The limit sin22�13 < 0:15,CL ¼ 90% from PDG is plotted as the dashed line.

FIG. 7 (color online). sin22�13 as a function of �23 for �12 ¼34:4� from the toy model (solid line). The limit sin22�13 < 0:15,CL ¼ 90% from PDG is plotted as the dashed line.

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In this work, by deforming the cube that corresponds to afull tribimaximal form of the mixing matrix according tothe proposed principles, we derive the analytic relationamong the three lepton mixing angles, and, taking theexperimental data as inputs, we deduce the value of un-known mixing angle �13. The result gives sin22�13 ¼0:0238, i.e., �13 ¼ 4:44�. As noticed, our theoretical pre-diction favors smaller �13. As �13 is to be measured at theDouble Chooz and Daya Bay experiments, our result in-dicates that in the future, there would be a great opportu-nity to fix the mysterious �13. The recent measurement ofthe T2K collaboration [16] indicates that sin22�13 falls in arather wide range of 0:03ð0:04Þ< sin22�13 < 0:28ð0:34Þ,

and the central value of our theoretical prediction is con-sistent with the lower bound set by the collaboration, andthe error range is comparable. This value also does notcontradict the newmeasurement byMain Injector NeutrinoOscillation Search [17].

ACKNOWLEDGMENTS

We thank Dr. Ye Xu for helpful discussion on theexperimental issues. This work is supported by theNational Natural Science Foundation of China, underContract No. 11075079.

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