Neutrino Mixing & CP Violation

YonseiOctober 10, 2014

Sin Kyu Kang(Seoul-Tech)

Outline

• Current status of neutrino oscillations• Neutrino mixing matrix

- Structure of neutrino mixing matrix- Discrete flavor symmetry

• Leptonic CP violation- prediction of leptonic Dirac CP phase- How to measure LCPV

• Summary

I. Current status of neutrino oscillation

• Two big discoveries over past two decades :- Neutrinos are massive- Leptons mix

• Those discoveries come from the observation of neutrino oscillations

• Neutrino sources :- the sun, atmospheric, reactors and accelerators...

• Why leptons mix ?- Neutrino eigenstates involved in weak interactions

are not mass eigenstates:

𝜈𝜈𝑒𝑒𝑒𝑒− 𝐿𝐿

𝜈𝜈𝜇𝜇𝜇𝜇− 𝐿𝐿

𝜈𝜈𝜏𝜏𝜏𝜏− 𝐿𝐿 ∶

weak eigenstates𝜈𝜈𝑒𝑒𝜈𝜈𝜇𝜇𝜈𝜈𝜏𝜏

=𝑈𝑈𝑒𝑒1 𝑈𝑈𝑒𝑒2 𝑈𝑈𝑒𝑒3𝑈𝑈𝜇𝜇1 𝑈𝑈𝜇𝜇2 𝑈𝑈𝜇𝜇3𝑈𝑈𝜏𝜏1 𝑈𝑈𝜏𝜏2 𝑈𝑈𝜏𝜏3

𝜈𝜈1𝜈𝜈2𝜈𝜈3

mass eigenstates

• How do we probe neutrino mixing ? neutrino oscillation

Standard Parameterization of PMNS mixing matrix:

𝑈𝑈𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃=1 0 00 𝑐𝑐23 −𝑠𝑠230 𝑠𝑠23 𝑐𝑐23

𝑐𝑐13 0 −𝑠𝑠13𝑒𝑒−𝑖𝑖𝛿𝛿0 1 0

𝑠𝑠13𝑒𝑒𝑖𝑖𝛿𝛿 0 𝑐𝑐13

𝑐𝑐12 −𝑠𝑠12 0𝑠𝑠12 𝑐𝑐12 00 0 1

𝑒𝑒𝑖𝑖𝛼𝛼 00 𝑒𝑒𝑖𝑖𝛽𝛽0 0

P(να→να)=1 - sin22θ sin2(1.27 ∆m2 L/E)Oscillation Curve

Experimental steps to measure the mixing angles

θ23 → θ12 → θ13 → δ/ρ/σ~45° ~33° ~9° ~???

1998 2001 2012 20yy

SuperK SNODayaBayRENO

• Well measured by solar neutrino experiments (SK, SNO) and KamLAND

𝜃𝜃12

• SNO experiments using 8𝐵𝐵:

• in disagreement with vacuum oscillation

• MSW matter effect

ΦCC

ΦNC= 0.301 ± 0.033

Measurement of the solar νe deficit using an independent methodwith different systematic effects

P(νe→νe)=1 - sin22θ sin2(1.27 ∆m2 L/E)~ 1 − 1

2sin22θ ≥ 0.5

ξθθθ−∆

∆=

2cos2sin2tan 2

2

mm

m EAZEVW ρξ 710526.12 −×≈≡ [eV2] (ρ in g/cm3, E in MeV)

)(sin)(cos)(cos)(sin11 022022mmeee PP θθθθµ −−=−= ~0.3

• Confirmation of MSW effect

𝑃𝑃𝑒𝑒𝑒𝑒 ≈ �𝑐𝑐𝑐𝑐𝑠𝑠4𝜃𝜃13 1 − 𝑠𝑠𝑠𝑠𝑠𝑠22𝜃𝜃12/2 𝑓𝑓𝑐𝑐𝑓𝑓 𝑙𝑙𝑐𝑐𝑙𝑙 𝐸𝐸

𝑐𝑐𝑐𝑐𝑠𝑠4𝜃𝜃13𝑠𝑠𝑠𝑠𝑠𝑠2𝜃𝜃12 𝑓𝑓𝑐𝑐𝑓𝑓 ℎ𝑠𝑠𝑖𝑖ℎ 𝐸𝐸( Borexino 1308.0443)

Combined best fit : ∆m2 = (7.59 ± 0.21) x 10−5 eV2

tan2θ = 0.457 ⇒ θ = 34.06°χ2 / Ndof = 81.4 / 106

𝜃𝜃12

• Well measured by atmospheric neutrino experiments (SK) and accelerator experiments (MINOS, T2K)

𝜃𝜃23

𝜃𝜃23

• Measuring 𝜃𝜃13 : important role in determining CP violation & mass hierarchy

• Using reactor antineutrino oscillation

P νe → νe( )≈ 1− cos4 θ13 sin2 2θ12 sin2 1.27∆m12

2 LEν

− sin2 2θ13 sin2 1.27∆m13

2 LEν

Daya Bay

RENO

Double Chooz

𝜃𝜃13

𝜃𝜃13

G. L. Fogli et al, 1205.5354

Non-zero, relatively large 1-3 mixing

Substantial deviation of the 2-3 mixing from maximal

δCP ~ π

Weak dependenceon hierarch, mainly in 2-3 mixing quadrant

Global fits

Capozzi, Fogli et al, 1312.2878

δCP ~ 1.35 π

D. V. Forero, M.Tortola , J. W. F. Valle , 1405.7545

solid- NHdashed - IH

II. Neutrino Mixing Matrix

• Summary of Fit to neutrino data in 3-𝜈𝜈 framework

• Implications of global fit:

Ɵ12≃𝜋𝜋/4 ‒ƟC satisfied within 2σ.

Non-maximal & second octant of Ɵ23 is favored

Zero Ɵ13 is excluded at 10 σ.

The preferred global fit value of Ɵ13 :

Looks different from quark mixing matrix !!

• Implications of global fit:

The same 𝜃𝜃13 with completely different implications

∆m212

∆m322O(1)

sin2θ13 =~ ½ sin2θC

Quark Lepton ComplementarityGUT, family symmetry

Naturalness of mass matrix

~ ½cos2 2θ23 νµ − ντ - symmetry violation

θ13 + θ12 = θ23 Self-complementarity

𝑠𝑠13~12𝑠𝑠12𝑠𝑠23 Quark-lepton universality

(Vub = ½ VusVcb )

• How do we understand 𝜈𝜈 mixing matrix ?

• Before measuring 𝜃𝜃13 , tri-bimaximal mixing hypothesis :

- 𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 =

2√6

− 1√3

016

13

− 1√2

16

13

1√2

Harrison & Perkins & Scott (2002)

𝜈𝜈2 = −13𝜈𝜈𝑒𝑒 +

13𝜈𝜈𝜇𝜇 +

13𝜈𝜈𝜏𝜏

𝜈𝜈3 = −12𝜈𝜈𝜇𝜇 +

12𝜈𝜈𝜏𝜏

Sν

• Mixing appears as a result of different ways of the flavor symmetry breaking in the neutrino and charged lepton (Yukawa) sectors. (CS Lam, E. Ma,

Hernandez, Smirnov )

• This leads to different residual symmetries

𝐺𝐺𝐹𝐹

𝐺𝐺𝑙𝑙 𝐺𝐺𝜈𝜈Residual symmetriesof the mass matrices

𝑀𝑀𝜈𝜈𝑀𝑀𝑙𝑙 1ν?

T Sν

Symmetry transformationsin mass bases

• Mixing does not depend on values of masses

Z2 x Z2 Zm

𝑆𝑆𝜈𝜈 𝑀𝑀𝜈𝜈 𝑆𝑆𝜈𝜈𝑇𝑇 = 𝑀𝑀𝜈𝜈𝑇𝑇 𝑀𝑀𝑙𝑙 𝑇𝑇+ = 𝑀𝑀𝑙𝑙

• An example : 𝐴𝐴4 symmetry

- 𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 =

2√6

− 1√3

016

13

− 1√2

16

13

1√2

• Understanding from discrete flavor symmetries- In the basis where �𝑀𝑀𝑙𝑙 = 𝑀𝑀𝑙𝑙

+𝑀𝑀𝑙𝑙 is diagonal, 𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 renders𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 �𝑀𝑀𝜈𝜈 (𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 )𝑇𝑇 diagonal.

- setting 𝑈𝑈𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = (𝑢𝑢1,𝑢𝑢2,𝑢𝑢3), we construct group generators

𝑆𝑆1= 𝑢𝑢1𝑢𝑢1+-𝑢𝑢2𝑢𝑢2

+-𝑢𝑢3𝑢𝑢3+

𝑆𝑆2= −𝑢𝑢1𝑢𝑢1+ + 𝑢𝑢2𝑢𝑢2

+-𝑢𝑢3𝑢𝑢3+

𝑆𝑆3= −𝑢𝑢1𝑢𝑢1+-𝑢𝑢2𝑢𝑢2

+ + 𝑢𝑢3𝑢𝑢3+

• 𝑆𝑆1 = 13

1 −2 −2−2 −2 −12 −1 −2

𝑆𝑆2 = 13

−1 2 −22 −1 −2−2 −2 −1

𝑆𝑆3=−1 0 00 0 10 1 0

then, 𝑆𝑆𝑖𝑖2 = 𝐼𝐼, 𝑆𝑆𝑖𝑖𝑆𝑆𝑗𝑗 = 𝑆𝑆𝑘𝑘

𝑆𝑆 �𝑀𝑀𝜈𝜈𝑆𝑆𝑇𝑇 = �𝑀𝑀𝜈𝜈

• For charged lepton, 𝑇𝑇 �𝑀𝑀𝑒𝑒𝑇𝑇+ = �𝑀𝑀𝑒𝑒𝑇𝑇 = 𝐷𝐷𝑠𝑠𝐷𝐷𝑖𝑖(𝑒𝑒𝑖𝑖𝜙𝜙𝑒𝑒 , 𝑒𝑒𝑖𝑖𝜙𝜙𝜇𝜇 , 𝑒𝑒𝑖𝑖𝜙𝜙𝜏𝜏) & 𝑇𝑇𝑛𝑛 = 𝐼𝐼

• If 𝑆𝑆𝑖𝑖 and 𝑇𝑇 generate 𝐺𝐺𝐹𝐹, (𝑆𝑆𝑖𝑖 𝑇𝑇) belongs to 𝐺𝐺𝐹𝐹• If 𝐺𝐺𝐹𝐹 is finite , 𝑝𝑝 exist such that (𝑆𝑆𝑖𝑖𝑖𝑖𝑇𝑇)𝑝𝑝 = 𝐼𝐼

- 𝑆𝑆,𝑇𝑇 with (n=p=3) forms a discrete group A4

SiU = UTBM Si UTBM+

• In such a way, we can find various discrete family symmetries such as S3, S4, A5, D4…

• But, TB mixing ruled out due to the measurements of 𝜃𝜃13• Exact TB mixing may be leading order

• Alternative special form : bi-maximal mixing

𝑈𝑈𝐵𝐵𝑃𝑃 =

1√2

− 1√2

012

12

− 1√2

12

12

1√2

𝜃𝜃12 = 45°, 𝜃𝜃23 = 45°, 𝜃𝜃13 = 0°

• Since 𝜃𝜃13 ≈𝜋𝜋4− 𝜃𝜃12 , we can treat both 𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 and 𝑈𝑈𝐵𝐵𝑃𝑃 on

the same footing as leading order approximation

Neutrino Mixing Matrix

• Modifications of (Tri-)Bimaximal mixing

- Simple possible forms deviated from (Tri-)Bimaximal :

�𝑈𝑈 𝑇𝑇 𝐵𝐵𝑃𝑃 � 𝑈𝑈𝑖𝑖𝑗𝑗(𝜃𝜃)𝑈𝑈𝑖𝑖𝑗𝑗+(𝜃𝜃) � 𝑈𝑈 𝑇𝑇 𝐵𝐵𝑃𝑃

• 𝜃𝜃 possibly gives rise to non-zero 𝜃𝜃13 and deviation

from maximal for 𝜃𝜃12.

• We call those forms modified (T)BM parameterization

(He& Zee, Chao & Zheng, Kang & Kim)

• 𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 � 𝑈𝑈12 =

∎ ∎ 0∎ ∎ − 1

2

∎ ∎ 12

𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 � 𝑈𝑈13 =

∎ − 1√3

∎

∎ 1√3

∎

∎ 1√3

∎

• 𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 � 𝑈𝑈23 =

√2√3

∎ ∎1√6

∎ ∎1√6

∎ ∎

𝑈𝑈12� 𝑈𝑈𝐵𝐵𝑃𝑃 =∎ ∎ ∎∎ ∎ ∎12

12

1√2

• 𝑈𝑈23� 𝑈𝑈𝐵𝐵𝑃𝑃 =1√2

− 1√2

0∎ ∎ ∎∎ ∎ ∎

etc…

unchanged

Unchanged rows and columns may reflect the remnants of flavor symmetry residual symmetry

• Best fit achieved by ( Kang, Kim)

𝑼𝑼𝑻𝑻𝑻𝑻𝑻𝑻 � 𝑼𝑼𝟐𝟐𝟐𝟐~

√𝟐𝟐√𝟐𝟐

− 𝟏𝟏𝟐𝟐

− 𝟏𝟏𝟐𝟐𝝀𝝀

𝟏𝟏√𝟔𝟔

𝟏𝟏𝟐𝟐

+ 𝟏𝟏𝟐𝟐𝝀𝝀 𝟏𝟏

𝟐𝟐+ 𝟏𝟏

𝟐𝟐𝝀𝝀

𝟏𝟏√𝟔𝟔

𝟏𝟏𝟐𝟐− 𝟏𝟏

𝟐𝟐𝝀𝝀 𝟏𝟏

𝟐𝟐+ 𝟏𝟏

𝟐𝟐𝝀𝝀

(𝑐𝑐23~1, 𝑠𝑠23~𝜆𝜆)

• Other possible forms :

𝑈𝑈 𝑇𝑇 𝐵𝐵𝑃𝑃 � 𝑈𝑈𝑖𝑖𝑗𝑗(𝜃𝜃′′)𝑈𝑈𝑘𝑘𝑙𝑙 𝜃𝜃′, 𝜉𝜉 , 𝑈𝑈𝑖𝑖𝑗𝑗+ 𝜃𝜃′, 𝜉𝜉 � 𝑈𝑈𝑘𝑘𝑙𝑙+ 𝜃𝜃′′ � 𝑈𝑈 𝑇𝑇 𝐵𝐵𝑃𝑃,

𝑈𝑈𝑖𝑖𝑗𝑗+(𝜃𝜃′, 𝜉𝜉) � 𝑈𝑈 𝑇𝑇 𝐵𝐵𝑃𝑃 𝑈𝑈𝑘𝑘𝑙𝑙(𝜃𝜃′′) etc…

IV. Leptonic CP Violation

• Nonzero 𝜃𝜃13 opens up window to probe CP violation because CPV is measureable only when three mixing angles are non-zero.

• How can we measure it?• Is LCP phase predictable ?

Hints of CP violation

G. L. Fogli et. Al (2012)

D. V. Forero, M.Tortola , J. W. F. Valle , 1405.7545

• How can we predict possible size of LCPV?

-- It is conceivable that a LCP phase may be estimated in terms of some observables.

-- What observables can be responsible for prediction ofLCP phase, masses, mixing angles, or all of them ?

• We come up with a simple scheme to calculate the possible size of LCP phase in terms of two or three 𝜈𝜈mixing angles only, in the standard parameterization of neutrino mixing matrix.

• Among possible modifications of TBM, thefollowing 8 patterns lead us to predict Dirac CPphase in terms of neutrino mixing angles

Prediction of LCPV

𝑉𝑉 =

𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 � 𝑈𝑈23 𝜃𝜃, 𝜉𝜉 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐴𝐴)𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 � 𝑈𝑈13 𝜃𝜃, 𝜉𝜉 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐵𝐵)𝑈𝑈12+ 𝜃𝜃, 𝜉𝜉 � 𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐶𝐶)𝑈𝑈13+ 𝜃𝜃, 𝜉𝜉 � 𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐷𝐷)𝑈𝑈12+ 𝜃𝜃, 𝜉𝜉 � 𝑈𝑈𝐵𝐵𝑃𝑃 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐸𝐸)𝑈𝑈13+ (𝜃𝜃, 𝜉𝜉) � 𝑈𝑈𝐵𝐵𝑃𝑃 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐹𝐹)𝑈𝑈𝐵𝐵𝑃𝑃 � 𝑈𝑈23 𝜃𝜃, 𝜉𝜉 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐺𝐺)𝑈𝑈𝐵𝐵𝑃𝑃 � 𝑈𝑈13 𝜃𝜃, 𝜉𝜉 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐻𝐻)

(Kang & Kim, arXiv:1406.5014)

How can we get such patterns?• 𝐴𝐴4 symmetric model

- Introducing 6 heavy Higgs (3 singlets+ 1 triplet)

- Neutrino mass matrix in 𝐴𝐴4 basis,

• charged lepton mass matrix in 𝐴𝐴4 basis,

• Taking b=c, e=-f=x, the neutrino mass matrix in the flavor basis :

𝑀𝑀𝜈𝜈(𝑒𝑒,𝜇𝜇,𝜏𝜏) = 𝑈𝑈𝐶𝐶𝐶𝐶+ 𝑀𝑀𝜈𝜈

• Rotating by , we get

• Finally, we can see that is diagonalized by

• This is case A for neutrino mixing pattern.

• Any forms of neutrino mixing matrix should be equivalent to the standard parameterization

• 𝑈𝑈𝑃𝑃𝑇𝑇 = 𝑈𝑈23(𝜃𝜃23)𝑈𝑈13(𝜃𝜃13, 𝛿𝛿𝐷𝐷)𝑈𝑈12(𝜃𝜃12)𝑃𝑃𝜙𝜙

= 𝑐𝑐12𝑐𝑐13 −𝑠𝑠12𝑐𝑐13 𝑠𝑠13𝑒𝑒𝑖𝑖𝛿𝛿𝐷𝐷∗ ∗ −𝑠𝑠23𝑐𝑐13∗ ∗ 𝑐𝑐23𝑐𝑐13

�𝑒𝑒𝑖𝑖𝜙𝜙1

𝑒𝑒𝑖𝑖𝜙𝜙2𝑒𝑒𝑖𝑖𝜙𝜙3

= 𝑃𝑃𝛼𝛼 � 𝑉𝑉 � 𝑃𝑃𝛽𝛽 : 𝑉𝑉 neutrino mixing matrix proposed

Prediction of LCPV

𝑉𝑉𝑖𝑖𝑗𝑗𝑒𝑒𝑖𝑖(𝛼𝛼𝑖𝑖+𝛽𝛽𝑗𝑗) = 𝑈𝑈𝑖𝑖𝑗𝑗𝑃𝑃𝑇𝑇

Estimations of LCPV

𝑉𝑉 = �𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 � 𝑈𝑈23 𝜃𝜃, 𝜉𝜉 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐴𝐴)

𝑈𝑈𝑇𝑇𝐵𝐵𝑃𝑃 � 𝑈𝑈13 𝜃𝜃, 𝜉𝜉 (𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐵𝐵)• For

• Using 𝑉𝑉13 = 𝑈𝑈13𝑃𝑃𝑇𝑇 and �𝑉𝑉11𝑉𝑉12 = �𝑖𝑖11𝑆𝑆𝑆𝑆

𝑖𝑖12𝑆𝑆𝑆𝑆, we get

𝑠𝑠122 =�1 − 2

3 1−𝑠𝑠132(𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐴𝐴)

13 1−𝑠𝑠132

(𝑐𝑐𝐷𝐷𝑠𝑠𝑒𝑒 𝐵𝐵)

• From the explicit form of V for case A, we see that𝑉𝑉23+𝑉𝑉33𝑉𝑉22+𝑉𝑉32

= 𝑉𝑉13𝑉𝑉12

and 𝑉𝑉21 = 𝑉𝑉31

𝑉𝑉𝐴𝐴 =

26

13

−𝑆𝑆3𝑒𝑒𝑖𝑖𝜉𝜉

−16

13−

𝐶𝐶2𝑒𝑒−𝑖𝑖𝜉𝜉

−𝐶𝐶2−

𝑆𝑆3𝑒𝑒𝑖𝑖𝜉𝜉

−16

13

+𝐶𝐶2𝑒𝑒−𝑖𝑖𝜉𝜉

𝐶𝐶2−

𝑆𝑆3𝑒𝑒𝑖𝑖𝜉𝜉

𝑉𝑉𝐵𝐵 =

2𝐶𝐶6

13

−2𝑆𝑆6𝑒𝑒𝑖𝑖𝜉𝜉

−𝐶𝐶6−

𝑆𝑆2𝑒𝑒−𝑖𝑖𝜉𝜉

13

−𝐶𝐶2

+𝑆𝑆6𝑒𝑒𝑖𝑖𝜉𝜉

−𝐶𝐶6

+𝑆𝑆2𝑒𝑒−𝑖𝑖𝜉𝜉

13

𝐶𝐶2

+𝑆𝑆3𝑒𝑒𝑖𝑖𝜉𝜉

Estimations of LCPV𝑉𝑉𝑖𝑖𝑗𝑗𝑒𝑒𝑖𝑖(𝛼𝛼𝑖𝑖+𝛽𝛽𝑗𝑗) = 𝑈𝑈𝑖𝑖𝑗𝑗𝑃𝑃𝑇𝑇• Using , we can get

• Since 𝑉𝑉31 = 𝑉𝑉21,

𝑈𝑈13𝑃𝑃𝑇𝑇

𝑈𝑈12𝑃𝑃𝑇𝑇=𝑈𝑈23𝑃𝑃𝑇𝑇 + 𝑈𝑈33𝑃𝑃𝑇𝑇𝑒𝑒−𝑖𝑖(𝛼𝛼3 −𝛼𝛼2 )

𝑈𝑈22𝑃𝑃𝑇𝑇 + 𝑈𝑈32𝑃𝑃𝑇𝑇𝑒𝑒−𝑖𝑖(𝛼𝛼3 −𝛼𝛼2 )

𝑈𝑈3𝑖𝑖𝑃𝑃𝑇𝑇

𝑈𝑈2𝑖𝑖𝑃𝑃𝑇𝑇=𝑉𝑉3𝑖𝑖𝑉𝑉2𝑖𝑖

𝑒𝑒−𝑖𝑖(𝛼𝛼3 −𝛼𝛼2 )

𝑒𝑒−𝑖𝑖(𝛼𝛼3 −𝛼𝛼2 )=𝑖𝑖31𝑆𝑆𝑆𝑆

𝑖𝑖21𝑆𝑆𝑆𝑆

𝑈𝑈13𝑃𝑃𝑇𝑇

𝑈𝑈12𝑃𝑃𝑇𝑇=𝑈𝑈23𝑃𝑃𝑇𝑇𝑈𝑈31𝑃𝑃𝑇𝑇 + 𝑈𝑈33𝑃𝑃𝑇𝑇𝑈𝑈21𝑃𝑃𝑇𝑇

𝑈𝑈22𝑃𝑃𝑇𝑇𝑈𝑈31𝑃𝑃𝑇𝑇 + 𝑈𝑈32𝑃𝑃𝑇𝑇𝑈𝑈21𝑃𝑃𝑇𝑇

• Presenting 𝑈𝑈𝑖𝑖𝑗𝑗𝑃𝑃𝑇𝑇in terms of 𝜃𝜃𝑖𝑖𝑗𝑗 and 𝛿𝛿𝐷𝐷,

• Leptonic Jarlskog invariant :

Predicting LCPV

cos 𝛿𝛿𝐷𝐷 =1

tan 2𝜃𝜃23�

1 − 5𝑠𝑠132

𝑠𝑠13 2 − 6𝑠𝑠132

𝐽𝐽𝐶𝐶𝑃𝑃2 =(Im[𝑈𝑈11𝑃𝑃𝑇𝑇𝑈𝑈12𝑃𝑃𝑇𝑇∗𝑈𝑈21𝑃𝑃𝑇𝑇𝑈𝑈11𝑃𝑃𝑇𝑇∗])2

= 1122

8𝑠𝑠132 1 − 3𝑠𝑠132 − cos 2𝜃𝜃23𝑠𝑠132

Estimations of LCPV

• Taking the same procedure described for case A,

𝜉𝜉 = sin 2𝜃𝜃12 ,𝜛𝜛 = (𝑠𝑠132 (9𝑠𝑠122 -4)-3𝑠𝑠122 +1)2

• For cases G & H, we get

𝑠𝑠122 =�1 − 1

2 1−𝑠𝑠132(𝐶𝐶𝐷𝐷𝑠𝑠𝑒𝑒 𝐺𝐺)

12(1−𝑠𝑠132 )

(𝐶𝐶𝐷𝐷𝑠𝑠𝑒𝑒 𝐻𝐻)

• But they are not consistent with experimental data up to 3 𝜎𝜎

• Also, we get

𝑠𝑠232 =�1 − 1

2 1−𝑠𝑠132(𝐶𝐶𝐷𝐷𝑠𝑠𝑒𝑒𝑠𝑠 𝐶𝐶 & 𝐸𝐸)

12 1−𝑠𝑠132

(𝐶𝐶𝐷𝐷𝑠𝑠𝑒𝑒𝑠𝑠 𝐷𝐷 & 𝐹𝐹)

Numerical Results

Cases A and B Cases C and D

Cases E and F

Numerical Results• Contour plots for each values of 𝐽𝐽𝐶𝐶𝑃𝑃

Using 1 𝜎𝜎 datafor Case A

Numerical Results• Contour plots for each values of 𝐽𝐽𝐶𝐶𝑃𝑃

Using 3 𝜎𝜎 datafor Case A

Numerical Results• Contour plots for each values of 𝐽𝐽𝐶𝐶𝑃𝑃

Using 1 𝜎𝜎 datafor Case B

Numerical Results• Contour plots for each values of 𝐽𝐽𝐶𝐶𝑃𝑃

Using 3 𝜎𝜎 datafor Case B

Numerical Results• Contour plots for each values of 𝐽𝐽𝐶𝐶𝑃𝑃

Using 3 𝜎𝜎 datafor Cases C,D

Numerical Results• Contour plots for each values of 𝐽𝐽𝐶𝐶𝑃𝑃

Using 3 𝜎𝜎 datafor Cases E,F

Extension to M(T)BM• There exist more possible forms of neutrino mixing

matrix deviated from (T)BM mixing such as 𝑈𝑈 𝑇𝑇 𝐵𝐵𝑃𝑃 � 𝑈𝑈𝑖𝑖𝑗𝑗(𝜃𝜃′′)� 𝑈𝑈𝑘𝑘𝑙𝑙 𝜃𝜃′, 𝜉𝜉 , 𝑈𝑈𝑖𝑖𝑗𝑗+ 𝜃𝜃′, 𝜉𝜉 � 𝑈𝑈𝑘𝑘𝑙𝑙

+ 𝜃𝜃′′ � 𝑈𝑈 𝑇𝑇 𝐵𝐵𝑃𝑃

𝑈𝑈𝑖𝑖𝑗𝑗+(𝜃𝜃′, 𝜉𝜉) � 𝑈𝑈 𝑇𝑇 𝐵𝐵𝑃𝑃 𝑈𝑈𝑘𝑘𝑙𝑙(𝜃𝜃′′) etc…

• Among them, we consider

Extension to M(T)BM

• Taking the same method for M(T)BM,

Extension to M(T)BM

Extension to M(T)BMScatter plots of 𝐽𝐽𝐶𝐶𝑃𝑃 in terms of s2

23

Green (3σ ) & beige (1σ) : Case Ared (3σ) & coral (1σ) : Case B

Green (3σ) & beige (1σ ) : Case Cred (3σ) & coral (1σ) : Case D

How to measure CP violation ?

• To observe CPV, we need- neutrino appearance

- non-zero mixing θ13

=0 if 𝛼𝛼 = 𝛽𝛽

P(ν𝜇𝜇→νe)=

4c213[sin2 ∆23s2

12s213+c2

12(sin2∆13s213s2

23+sin2∆12s212(1-(1+s2

13)s223))]

-1/2c213sin2θ12s13sin2θ23cosδ[cos2∆13- cos2∆23-2cos2θ12sin2∆12]

+1/2c213sinδsin2θ12s13sin2θ23[sin2∆12-sin2∆13+sin2∆23]

Independent of δ

CP-even

CP-odd

∆CP(δ)≡ P(νµ→νe : δ)- P(�̅�𝜈µ→�̅�𝜈e : δ)

In vacuum

ν and anti-ν narrow beams tuned to 1st oscillation maximum

wide ν (anti-ν) beam to cover 1st and 2nd oscillation maxima

• CPV can be observed by

Asymmetry between Neutrino and Antineutrino

or Energy Dependence in the Neutrino appearance.

∆δ≡ P(νµ→νe : δ=π/2)- P(νµ→νe :δ=0)

It may not be easy to measure CP asymmetry !

P(νµ → νe) for sin22θ13=0.1

θ13

CP-even

CP-odd

Solar

Matter

ijijijij cs θθ cos sin == [ ] [ ]GeVEcmgEnGeVa eF νν ρ

×== −

352 106.722

)( eP νν µ → δδ −→−→ aa

νµ→νe in matter

change sign for NH → IH

Fake CP violation

Separating CPV From the Matter Effect

• Genuine CPV and the matter effect both lead to a difference between ν and ν oscillation.

• To disentangle them, one may make oscillation measurements at different L and/or E.

(Minakata & Nunokawa2001)

Summary

• In the past few years, neutrino oscillationparameters have been measured with goodprecision.

• The discovery of non-zero 𝜃𝜃13 opened up new eraof neutrino physics.

• (Tri-)bimaximal mixing pattern should be modifiedto accommodate measured neutrino mixing angles.

• LCP phase can be predicted in terms of neutrinomixing angles.

• Next generation oscillation experiments willaddress CPV and the mass hierarchy.

• Osc. probability modified depending on sign of ∆m2

• Matter effects amplified for long L, large Eν

∆−

=

213

1313

2cos

2sin2tan

mAm

m

θ

θθ

S. Parke

Search for CP violating effects• Difficult

measurement– Effect is small– Matter effects and

other osc. parameters need to be known

• Use wrong-sign rate ratio for µ+ vs µ− running– R = Nµ+ / Nµ−– Curves are δ = 0, δ

= ± π/2 Different Curves are δ = 0, δ = ± π/2Error bars reflect measurement with

1021 µ-decays @ 20 GeV

MatterEffect

Experimental Status

Solar Neutrinos Atmospheric ν Reactor ν, θ13 LSND effectoscillation observed

DavisGALLEXSuper-K

confirmedKAML ANDSNO

disappearance of νe

oscillation observedSuper-K

confirmedK2KMINOS

disappearance of νμ

no observation yet

intensive searchDoubleChoozDaya BayRENOT2K

disappearance of νe

not confirmed

contradicted byKARMENMiniBOONE

disappearance of νμ

D. V. Forero, M.Tortola , J. W. F. Valle , 1205.4018 v 4

solid- NHdashed - IH

M. Gonzalez-Garcia, M Maltoni, J. Salvado, T. Schwetz

Red – NHBlue - IH

sin2 θ23

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

MINOS, 1σ

SK (NH), 90%

Fogli et al, 1σ

SK (IH), 90%

QLC

Gonzalez-Garcia et al, 1σ

sin2 θ13

0.0 0.010 0.020 0.030 0.040

RENO, 1σ

Fogli et al, 1σ

T2K 90%

QLC

Double Chooz, 1σ

Daya Bay, 1σ

µ−τ breaking

mass ratio

Gonzalez-Garcia et al, 1σ

T2K

• Potential of existing equipment

T2K, NOvA,RENO Double Chooz, Daya Bay;

5 years each

Huber, Lindner, Schwetz, Winter, JHEP 0911 (2009) 044