Corrections for Tribimaximal, Bimaximal ... - home.kias.re.krhome.kias.re.kr/psec/NCTS2014/PPR_SKICONF_Kumar Sumit.pdf · [email protected] February 11, 2014 S.K.Garg (Yonsei Univ.-Seoul)

Corrections for Tribimaximal, Bimaximal and Democraticneutrino mixing matrices

Sumit K. Garg(Yonsei University Seoul)

Seoul, South Korea 120-749

[email protected]

February 11, 2014

S.K.Garg (Yonsei Univ.-Seoul) JHEP 1310 (2013) 128 February 11, 2014 1 / 30


Neutrinos in SM

Standard Model(SM) is a theoretical framework which tells how thesefundamental particles interact with each other. It is based on internalsymmetries of nature. Gauge Group: SU(3)× SU(2)L × U(1)Y

1 Quarks: Q(3, 2, 16 ), uR(3, 1, 2

3 ),dR(3, 1,− 1

3 )

2 Leptons: L(1, 2,− 12 ),

eR(1, 1,−1)

Gauge particles belongs to adjoint representations of symmetry group.


Neutrinos in SM

A fermion spinor can be decomposed into its left handed and righthanded part i.e. Ψ = ΨL + ΨR

In SM ΨL transform like a SU(2) doublet while ΨR transform assinglet. Thus mass term ΨLΨR is not allowed.

Thus scalar Φ introduce Yukawa term ΨΨ̄Φ in the theory and thusgenerating masses of fermions.

−LΨ = hΨLΨRΦ + h.c. =⇒ mf = hv√2

(1)

SM contains only left handed neutrinos. Thus neutrinos remain massless in Standard Model.


Neutrino Oscillations

Solar, atmospheric and reactor neutrino experiments- Neutrinos areMassive and flavor mix !!

However the mass scale of neutrino is extremely small(∼ meV ) ascompared to other SM fields.

Magnitude of flavor mixing - θ23, θ12 >> θ13.


Seesaw Mechanism

SM - effective field theory i.e. low energy limit of more fundamental theory.

Seesaw Mechanism relates the smallness of neutrino mass to a newscale(M) in theory.

Neutrinos are Majorana particles

1 Seesaw : YLνcΦ + Mνcνc + h.c .

Mν ∼ Yv2

M⇒ M ∼ 1014±1GeV ≈ MGUT !!

The nature of neutrinos can be tested in Neutrinoless Double betadecay experiment.

Nil Results so far =⇒ |mββ | =∑

i U2eimνi ≤ (0.14− 0.38)eV .


Mixing schemes

Different mixing schemes were studies to explain neutrino mixing data.

UTBM =

√

23

√13

0

−√

16

√13

√12

−√

16

√13

−√

12

, UBM =

√

12

√12

0

− 12

12

√12

12

− 12

√12

UDC =

√

12

√12

0√16

−√

16

−√

23

−√

13

√13

−√

13

. (2)

All of them predict reactor mixing angle to be zero i.e. θ13 = 0◦

Solar mixing angle is maximal for BM and DC while its θ12 = 35.3◦

for TBM case.

Atmospheric mixing angle is maximal for BM and TBM while itsθ23 = 54.7◦ for DC case.


Non zero θ13

However recent experimental observations favors non zero value of θ13.

The Daya Bay experiment in China- θ13 6= 0◦ with 5.2σ significance.

sin2 2θ13 = 0.092± 0.016(stat)± 0.05(syst)

The T2K experiment: νµ → νe , 13 mixing angle at 90% CL.

5◦(5.8◦) < θ13 < 16◦(17.8◦) - Normal(Inverted) mass hierarchy.

These results goes in well with Double Chooz, MINOS and RENO.

Recent Global fit[1]: θ13 = 9◦, θ12 = 33.3◦ and θ23 = 40◦(50◦).

1 . M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz,Global fit to three neutrino mixing: critical look at present precision,JHEP 1212, 123 (2012) [arXiv:1209.3023 [hep-ph]].


Perturbations

However these mixing schemes are in conflict with recent experimental fit.

Thus these special structures need perturbations in order to beconsistent with neutrino mixing data.

The PMNS matrix is obtained by mismatch between the matriceswhich diagonalize charged lepton and neutrino matrix and is given by

UPMNS = U†l Uν

Thus the corrections which modify the original predictions mayoriginate from Charged lepton, neutrino or from both sectors.

In this work we looked into perturbations which can come from bothsectors.


Perturbation Matrix

A 3× 3 Unitary matrix can be parametrized by 3 mixing angles and 1physical phase.

Here we deal with CP conserving case only i.e. phase, δ = 0.

Thus correction matrix, U = R23 · R13 · R12

R12 =

cosα sinα 0− sinα cosα 0

0 0 1

, R23 =

1 0 00 cos β sin β0 − sin β cos β

, R13 =

cos γ 0 sin γ0 1 0

− sin γ 0 cos γ

The correlations between mixing angles are weakened or hiddened in afull 3 mixing scenario.

The corresponding perturbed PMNS matrix: UPMNSijkl = R l

ij · U · R rkl .


χ2 Analysis

We define a function χ2 which is a measure of deviation from thecentral value of mixing angles.

χ2 =

3∑i=1

{θi (P)− θiδθi

}2 (3)

The corresponding χ2 for different mixing schemes underconsideration are given as

χ2 for θ23 ∈ [38.52◦, 42.13◦] χ2 for θ23 ∈ [49.1◦, 51.7◦]BM DC TBM BM DC TBM

354.5 413.6 109.9 364.0 357.6 119.4


Numerical Section

Here we look into R lij · U · R r

kl rotation cases for which ij = kl andij 6= kl .

The correction parameters α, β, γ ∈ [−0.5, 0.5] - small rotation limit.

We enforced the condition χ2 < χ2i (i= TBM, DC and BM) for

selecting data points.

The results are presented in terms of χ2 vs perturbation parametersand θ13 over θ12 − θ23 plane.

The regions with low χ2 are marked with different colors in mixingangle plots for studying correlations between both figures.


R12 · U · R13

The neutrino mixing angles for small perturbation parameters(O(θ2))are given by

sin θ13 ≈ |αU23 + γU11 + αγU21|, (4)

sin θ23 ≈ |U23 + γU21 − (α2 + γ2)U23 − αγU11

cos θ13|, (5)

sin θ12 ≈ |U12 + αU22 − α2U12

cos θ13|. (6)

The perturbations parameters enters at leading order in all mixingangles and thus exhibit strong correlations.


R12 · U · R13

χ2 < 3 in all mixing schemes. BM∼ 1σ while TBM, DC ∼ 2σ.


R12 · U · R23


sin θ13 ≈ |αU23 + βU12 + αβU22|, (7)

sin θ23 ≈ |U23 + βU22 − (α2 + β2)U23 − αβU12

cos θ13|, (8)

sin θ12 ≈ |U12 + αU22 − (α2 + β2)U12 − αβU23

cos θ13|. (9)

Like previous case, perturbations parameters enters at leading order inall mixing angles and thus exhibit strong correlations.


R12 · U · R23

χ2 < [3, 10], tiny region for TBM. DC is excluded while TBM, BM ∼ 3σ.


R13 · U · R12

The neutrino mixing angles in the small rotation limit are given by

sin θ13 ≈ |γU33|, (10)

sin θ23 ≈ | U23

cos θ13|, (11)

sin θ12 ≈ |U12 + αU11 + γU32 − (α2 + γ2)U12 + αγU31

cos θ13|. (12)

θ23 doesn’t receive corrections and thus sticks to its original value.

|γ| ∼ 0.175− 0.245, θ13 → 3σ range for TBM and BM.


R13 · U · R12

χ2 > 10 for all cases. TBM and BM ∼ 3σ while DC is excluded.


R13 · U · R23


sin θ13 ≈ |βU12 + γU33 + βγU32|, (13)

sin θ23 ≈ |U23 + βU22 − β2U23

cos θ13|, (14)

sin θ12 ≈ |U12 + γU32 − (γ2 + β2)U12 − βγU33

cos θ13|. (15)

All mixing angles ∼ O(θ) and thus show nice correlations amongthemselves.


R13 · U · R23

χ2 < 3 for BM and consistent at 1σ. TBM and DC ∼ 3σ.


R23 · U · R12

In this case there are no corrections from perturbation parameters to13 mixing angle i.e. θ13 = 0. So we will not discuss it any further.


R23 · U · R13

In small rotation limit, neutrino mixing angles are given by

sin θ13 ≈ |γU11|, (16)

sin θ23 ≈ |U23 + βU33 + γU21 − (β2 + γ2)U23 + βγU31

cos θ13|, (17)

sin θ12 ≈ | U12

cos θ13|. (18)

θ12 doesn’t get any corrections and thus TBM is preferred.

TBM-|γ| ∼ 0.154− 0.213 for θ13 in its 3σ range.


R23 · U · R13

χ2 ∈ [3, 10] in a very tiny region for TBM and is consistent at 3σ level.


R12 · U · R12


sin θ13 ≈ |α1U23|, (19)

sin θ23 ≈ |(α21 − 1)U23

cos θ13|, (20)

sin θ12 ≈ |U12 + α1U22 + α2U11 − (α21 + α2

2)U12 + α1α2U21

cos θ13|.(21)

θ23 ∼ O(θ2) and thus remains close to its unperturbed value.


R12 · U · R12

χ2 < 10 for TBM and BM. All cases are consistent at 3σ only while DCremains far away from its 2σ boundary.


R13 · U · R13

The neutrino mixing angles under small rotation limit are given by

sin θ13 ≈ |γ1U33 + γ2U11 + γ1γ2U31|, (22)

sin θ23 ≈ |U23 + γ2U21 − γ22U23

cos θ13|, (23)

sin θ12 ≈ |U12 + γ1U32 − γ21U12

cos θ13|. (24)

Corrections parameters enters at leading order and thus mixing anglesshow good correlations among themselves.


R13 · U · R13

χ2 < 3 for TBM ∼ 1σ while BM and DC are only consistent at 3σ level.


R23 · U · R23

In small rotation limit, neutrino mixing angles are given by

sin θ13 ≈ |β2U12|, (25)

sin θ23 ≈ |U23 + β1U33 + β2U22 − (β21 + β2

2)U23 + β1β2U32

cos θ13|,(26)

sin θ12 ≈ |(β22 − 1)U12

cos θ13|. (27)

θ12 receives very small corrections and thus TBM is preferred.


R23 · U · R23

χ2 < 3 for TBM and is allowed at 2σ. BM, DC are excluded.


Conclusions

BM mixing: R12 · U · R13 and R13 · U · R23 can fit Neutrino mixingangles at 1σ level.

TBM: R13 · U · R13 is consistent at 1σ level.

DC: This mixing scheme is not viable at 1σ level.

These results may help in restricting vast number of possible modelsand thus provide a guideline for the neutrino model building physics.


Corrections for Tribimaximal, Bimaximal ... - home.kias.re.krhome.kias.re.kr/psec/NCTS2014/PPR_SKICONF_Kumar Sumit.pdf · [email protected] February 11, 2014 S.K.Garg (Yonsei Univ.-Seoul)

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