The Lepton mixing matrix and the Cabibbo texture · The Lepton mixing matrix and the Cabibbo texture (XXI DAE-BRNS High Energy Physics Symposium) December 8-12, 2014 Subhankar Roy

The Lepton mixing matrix and the Cabibbotexture

(XXI DAE-BRNS High Energy Physics Symposium)December 8-12, 2014

Subhankar Roy

Department of Physics, Gauhati University, Guwahati, Assam.

December 9, 2014

Subhankar Roy (Department of Physics, Gauhati University, Guwahati, Assam.)The Lepton mixing matrix and the Cabibbo texture December 9, 2014 1 / 19

Motivation

A model-independent ansatz → lepton and quark mixing →“Cabibbo angle”.

(λ, A) + ( ψ, δ)

θ23 and δCP .


CKM mixing

VCKM =

1− 12 λ2 λ Aλ3(ρ− iη)−λ 1− 1

2 λ2 Aλ2

Aλ3(1− ρ− iη) −Aλ2 1

(1)

(λ, A, ρ, η) → Standard Wolfenstein parameters.

L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).

θc = θCKM12 ≈ 130, (2)

θCKM23 ≈ 2.40, (3)

θCKM13 ≈ 0.20. (4)

λ = sin θc .


Lepton mixing → UPMNS

θPMNS13 ≈ 90 ∼ O(θc), (5)

θPMNS12 ≈ 34.630, (6)

θPMNS23 ≈ 490 (430). (7)

“Neutrino oscillations refitted”, D.V. Forero, M. Trtola, and J. W. F. Valle, arXiv:1405.7540 [hep-ph]

Bi-Large (BL) mixing:

sin θν13 = λ, (8)

sin θν12 = sin θν

23 = ψλ; (9)

Where, λ = sin θc .

S. Boucenna, S. Morisi, M. Tortola, and J. Valle, Phys.Rev. D86, 051301 (2012), arXiv:1206.2555 [hep-ph].


BiLarge neutrino mixing

UBL ≈

c(1− λ2

2 ) ψλ(1− λ2

2 ) λ

−cψλ(1 + λ) c2 − λ3ψ2 ψλ(1− λ2

2 )

λ2ψ2 − λc2 −cψλ(1 + λ) c(1− λ2

2 )

(10)

c = cos sin−1 ψλ,

ψ→ O(1) coefficient =?.

3 V. Bouchard, J. J. Heckman, J. Seo, and C. Vafa, Journal of High Energy Physics 1, 061 (2010),

3 G.-J. Ding, S. Morisi, and J. Valle, Phys.Rev.D87,053013 (2013), arXiv:1211.6506 [hep-ph].


Charged lepton Diagonalizing matrix (SO(10) based):

Ml ∼ Md .

Ul1 ≈

1− 12 λ2 λ e−iδ 0

−λ e iδ 1− 12 λ2 0

0 0 1

(Type-1) (11)

Ul2 ≈

1− 12 λ2 λ e−iδ 0

−λ e iδ 1− 12 λ2 Aλ2

Aλ3e iδ −Aλ2 1

, (Type-2) (12)


Charged lepton Diagonalizing matrix (SU(5) motivated):

Ml ∼ MTd .

Ul3 ≈

1− 12 λ2 −λ e−iδ 0

λ e iδ 1− 12 λ2 0

0 0 1

(Type-3) (13)

Ul4 ≈

1− 12 λ2 −λ e−iδ Aλ3e−iδ

λ e iδ 1− 12 λ2 −Aλ2

0 Aλ2 1

(Type-4) (14)

4 H. Fritzsch and Z.-z. Xing, Phys.Rev. D57, 594 (1998), arXiv:hep-ph/9708366 [hep-ph]


θl12 = θν13 = θc

?? (15)


UPMNS = Ul†.Uν. (16)


Example: Type-4 parametrization

U4 = Ul4†.UBL.Iφ , (17)

Example: Type-4 parametrization.

s213 ≈ λ2

(s2 + 2s cos δ + 1

), (18)

s212 ≈ s2 + λ2

(c4 + s4 − s2

)+ 2c2λs cos δ, (19)

s223 ≈ s2 + λ2

(2Acs + s4 + 2s3 cos δ− s2 − 2s cos δ

), (20)

Jcp ≈ −c2s3λ sin δ. (21)

s = ψλ, c = cos sin−1 ψλ

ψ, δ =?


Figure: The parametrization of ψ and δ, and prediction on s223 and Jcp are illustrated for

Type-4 BL. For all the cases, ψ and δ are first parametrized with respect to best -fit, 1σ, 2σ and3σ ranges of s2

12 and s213 which are then are used to predict s2

23 and Jcp . In the above illustrationwe fix λ and A at their central values: λ = 0.22551 and A = 0.813



Table

Type ψ δ/π sin2 θ23 δCP/π Jcp

1 2.9521+0.2087−0.2043 1.764+0.0476

−0.0428 0.4585+0.08543−0.08646 1.2308+0.0692

−0.0717, 0.0250+0.0137−0.0105

2 2.9521+0.2087−0.2043 1.764+0.0476

−0.0428 0.4174+0.0921−0.0937 1.2159+0.0754

−0.0733, 0.0250+0.0137−0.0105

3 2.9522+0.2087−0.2201 0.7644+0.0476

−0.0427 0.4585+0.0855−0.08641 1.2303+0.0717

−0.0713 0.0250+0.0137−0.0105

4 2.9522+0.2087−0.2201 0.7644+0.0476

−0.0427 0.50+0.0927−0.0935 1.2303+0.0717

−0.0713 0.0250+0.0137−0.0105

Table: Summary of the results corresponding to four BL models. ψ and δ corresponds to the central ± 3σ range of s212, s2

13,

λ and A. We have taken, s212 = [0.278, 0.375], s2

13 = [0.0177, 0.0297], λ = [0.22551− 0.001, 0.22551 + 0.001] and

A = [0.813− 0.029, 0.813− 0.040]. The other observables s223, δcp (the Dirac type CP phase) and Jcp (Jarkslog invariant

parameter) are the theoretical predictions for each model. This is to be noted that the best result of the Type-4 BL model isconsistent with the maximal mixing prediction.


Type 1

Type 2

Type 3

Type 4

3Σ

1Σ

0.3 0.4 0.5 0.6 0.7 0.8

sin2Θ23

BL

Model

s


Type 1

Type 2

Type 3

Type 4

1Σ 1Σ

0.0 0.5 1.0 1.5 2.0

∆CP� Π

BL

Mo

del

s


The PMNS matrix: (Type-4 parametrization)

U4(λ,A, ψ, δ) ≈ −u∗(1 + λ){u(λ− 1) + ψλ2}c (ψ− u∗c2)λ + ψλ3 λ− ( λ2 + u∗ψ)λ2

cλ2 {(λ

2 − 2)(u + ψ)− 2λ(ψ + cAλ)} c2(1− λ2

2 )− ψλ2{u + (ψ + cA)λ} ψλ(1− λ2) + (cA− u)λ2

{ψ2 + cA(u + ψ)λ}λ2 − c2 −c{ψ + (ψ + cA)λ}λ −ψAλ3 + c(1− λ2

2 )

,

(22)

where, u = e iδ, c = cos sin−1(ψλ).


Summary

UPMNS = Ul†.UBL, Where, Ul 1,2,3,4 → CKM-Like, UBL → BL mixing.

Four models → best-fit θ12 and θ13.→ definite predictions for θ23 and δCP .

Type-4 BL model → maximal atmospheric mixing.

“Theory-inspired” bottom-up approach to the flavor problem.

“ θc as the universal seed of quark and leptonmixings” .


The present work is done in collaboration with:S. Morisi,DESY, Platanenallee 6, D-15735 Zeuthen, Germany;

N. N. Singh,Deptt. of Physcs, Manipur University, Imphal, Manipur,India.

J. W. F. Valle,AHEP Group, Institut de Fısica Corpuscular – C.S.I.C./Universitat de Valencia, ParcCientific de Paterna.

C/ Catedratico Jose Beltran, 2 E-46980 Paterna (Valencia) - SPAIN

arXiv:1410.3658 [hep-ph]



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The Lepton mixing matrix and the Cabibbo texture · The Lepton mixing matrix and the Cabibbo texture (XXI DAE-BRNS High Energy Physics Symposium) December 8-12, 2014 Subhankar Roy