Neutrino masses & mixing from flavorantisymmetry

Newton NathPhysical Research Laboratory, Ahmedabad

(in collaboration with)

Anjan S. Joshipura

newton@prl.res.in

Nu HoRIzons VI @ HRI

17-19 March, 2016

Allahabad, India

Unknowns in Neutrino Physics :I 3-flavor neutrino oscillations can be described by 3-mixing ∠′s( θ12, θ23

and θ13), 2-(mass)2 differences (∆m221 and ∆m2

31) and one Dirac type CPphase δCP .

I Various ν-oscillations experiments have measured these parameters.

I The sign of ∆m231 i.e.

∆m231 > 0⇒ Normal Hierarchy (NH) or

∆m231 < 0⇒ Inverted Hierarchy (IH).

I The octant of θ23 i.e.θ23 > 45◦ ⇒ Higher Octant (HO) or

θ23 < 45◦ ⇒ Lower Octant (LO).

I The Dirac CP phase δCP , where δCP 6= 0o ,±180o ⇒ CP violation. There

already exists hints that CP phase may be nearly maximal.

Unknowns in Neutrino Physics :I 3-flavor neutrino oscillations can be described by 3-mixing ∠′s( θ12, θ23

and θ13), 2-(mass)2 differences (∆m221 and ∆m2

31) and one Dirac type CPphase δCP .

I Various ν-oscillations experiments have measured these parameters.

I The sign of ∆m231 i.e.

∆m231 > 0⇒ Normal Hierarchy (NH) or

∆m231 < 0⇒ Inverted Hierarchy (IH).

I The octant of θ23 i.e.θ23 > 45◦ ⇒ Higher Octant (HO) or

θ23 < 45◦ ⇒ Lower Octant (LO).

I The Dirac CP phase δCP , where δCP 6= 0o ,±180o ⇒ CP violation. There

already exists hints that CP phase may be nearly maximal.

Unknowns in Neutrino Physics :I 3-flavor neutrino oscillations can be described by 3-mixing ∠′s( θ12, θ23

and θ13), 2-(mass)2 differences (∆m221 and ∆m2

31) and one Dirac type CPphase δCP .

I Various ν-oscillations experiments have measured these parameters.

I The sign of ∆m231 i.e.

∆m231 > 0⇒ Normal Hierarchy (NH) or

∆m231 < 0⇒ Inverted Hierarchy (IH).

I The octant of θ23 i.e.θ23 > 45◦ ⇒ Higher Octant (HO) or

θ23 < 45◦ ⇒ Lower Octant (LO).

I The Dirac CP phase δCP , where δCP 6= 0o ,±180o ⇒ CP violation. There

already exists hints that CP phase may be nearly maximal.

Unknowns in Neutrino Physics :I 3-flavor neutrino oscillations can be described by 3-mixing ∠′s( θ12, θ23

and θ13), 2-(mass)2 differences (∆m221 and ∆m2

31) and one Dirac type CPphase δCP .

I Various ν-oscillations experiments have measured these parameters.

I The sign of ∆m231 i.e.

∆m231 > 0⇒ Normal Hierarchy (NH) or

∆m231 < 0⇒ Inverted Hierarchy (IH).

I The octant of θ23 i.e.θ23 > 45◦ ⇒ Higher Octant (HO) or

θ23 < 45◦ ⇒ Lower Octant (LO).

I The Dirac CP phase δCP , where δCP 6= 0o ,±180o ⇒ CP violation. There

already exists hints that CP phase may be nearly maximal.

Current status of the oscillation parameters1

Parameter best fit ± 1σ 3σ range

∆m221[10−5eV 2] 7.60+0.19

−0.18 7.11–8.18

|∆m231|[10−3eV 2] (NH) 2.48+0.05

−0.07 2.30–2.65|∆m2

31|[10−3eV 2] (IH) 2.38+0.05−0.06 2.20-2.54

sin2 θ12/10−1 3.23±0.16 2.78–3.75

θ23(deg)(NH) 48.9+1.8−7.2 38.8–53.3

θ23(deg)(IH) 49.2+1.5−2.3 39.4–53.1

sin2 θ13/10−2 (NH) 2.26±0.12 1.90–2.62sin2 θ13/10−2 (IH) 2.29±0.12 1.93–2.65

δ/π (NH) 1.41+0.55−0.40 0.0–2.0

δ/π (IH) 1.48±0.31 0.0–2.0

I It is natural to look for the group theoretical explanations forthese parameters.

I Flavor symmetry provide concrete framework to do this.

1PRD, 90, 093006 (2014)

Approaches

I Obtain residual symmetries Gν and Gl of the neutrinos andcharged leptons mass matrices based on the observed mixingpatter.

I Look for the flavor symmetry group Gf which contain Gν andGl as subgroups.

I Advantage of this approach is that mixing pattern iscompletely fixed by the choice of Gν and Gl without anydetail knowledge of the underlying theory.

I A general prescription can be formulated which leads to thedesired mixing from some underlying Lagrangian invariantunder Gf spontaneously broken to Gν and Gl .

Relating mixing angles to symmetry group G. 2

I Let Uν(Ul)diagonalize Mν (MlM†l ).

UTν MνUν = Diag.(mν1 , mν2 , mν3 ) , (1)

U†l MlM†l Ul = Diag.(m2

e , m2µ, m2

τ )

I Assume that Mν (MlM†l ) is invariant under some set of discrete

symmetries Si (Tl):

STi MνSi = Mν and T †l MlM

†l Tl = MlM

†l . (2)

I It is assumed that elements within Si and Tl commute among themselves.I Therefore, Si and Tl simultaneously diagonalized by unitary matrices Vν

and Vl respectively,

V †ν SiVν = si and V †l TlVl = tl , (3)

where si and tl correspond to diagonal matrices.

I These eqs. can be used to find UPMNS ,

UPMNS = U†l Uν = P∗l V†l VνPν (4)

Pl , Pν are diagonal phase matrices

2PLB, 727(2013)480-487

Three massive neutrinos:

I Choosing Det(Si ) = +1⇒ Si are discrete subgroups (DSG) of SU(3).

I Diagonal mass matrix is trivially invariant under,

s1 = Diag(1,−1,−1), s2 = Diag(−1, 1,−1) and s3 = s1s2 (5)

I Any two si defines Z2 × Z2 symmetry.

I Therefore, start with a group G, identify Gν = Z2 × Z2 and appropriate Gl

and use them to predict the observed mixing.

I In literature, G = A4, S4, A5, 4(96), 4(384), 4(150), 4(600) are

studied for the predictions of the mixing angles.3

3arXiv:1002.0211, 1205.5133, 1003.3552, 0804.2622, 1112.1340, 1301.1736

One massless neutrino

I Eq.(2) tells that, Det(Mν) = 0 , if Det(Si ) 6= ±1⇒ a massless ν

I Therefore, underlying group G containing Si necessarily belongs to U(3).

I Thus if ν-mass matrix is invariant under an element of a subgroup ofU(3) and not SU(3) then such invariance implies one massless neutrino.

I Here, the full residual symmetry for ν-mass matrix is Z2 × Z2 × ZN andZN in arbitrary basis is defined as,

S = Vνdiag(e2πi

k

N , 1, 1)V †ν (6)

here, k=1,2,...N-1 and e2πi

k

N 6= −1

I In PLB 727 (2013) 480-487, authors (AJ, KP) considered group seriesΣ(3N3), Σ(2N2) and S4(2) which contain Gν implying massless neutrino.

I Other possibilities like, a pair of degenerate neutrinos and a third massive

or massless neutrino have been discussed in PRD 88,093007 (2013) and

PRD 90, 036005(2014).

One massless neutrino

I Eq.(2) tells that, Det(Mν) = 0 , if Det(Si ) 6= ±1⇒ a massless ν

I Therefore, underlying group G containing Si necessarily belongs to U(3).

I Thus if ν-mass matrix is invariant under an element of a subgroup ofU(3) and not SU(3) then such invariance implies one massless neutrino.

I Here, the full residual symmetry for ν-mass matrix is Z2 × Z2 × ZN andZN in arbitrary basis is defined as,

S = Vνdiag(e2πi

k

N , 1, 1)V †ν (6)

here, k=1,2,...N-1 and e2πi

k

N 6= −1

I In PLB 727 (2013) 480-487, authors (AJ, KP) considered group seriesΣ(3N3), Σ(2N2) and S4(2) which contain Gν implying massless neutrino.

I Other possibilities like, a pair of degenerate neutrinos and a third massive

or massless neutrino have been discussed in PRD 88,093007 (2013) and

PRD 90, 036005(2014).

Flavor Antisymmetry4

I We assume that Majorana neutrino mass matrix Mν obeyflavor antisymmetry instead of flavor symmetry,

STν MνSν = −Mν (7)

I We also assume that Gf is some finite DSG of SU(3).

I Eq.(7) ⇒ Det(Mν) = 0 i.e. at least one of the νs remainsmassless.

I One can also determine all the allowed forms of Mν in a givenbasis for all possible Sν contained in SU(3).

I There exist only 4 possible Mν in a particular basis withdiagonal Sν .

I Two of these give one massless ν and two non degenerate ν.

I Other two give a massless and a pair of degenerate ν.

4AJ, JHEP 1511 (2015) 186

Allowed textures of Mν

I The unitary matrix Sν can be diagonalized by another unitarymatrix VSν as V †SνSνVSν = S̃ν = diag(λ1, λ2, λ3).

I In the diagonal basis of Sν , defining, M̃ν = V TSνMνVSν , eq.(7)

can be written as,

(M̃ν)ij(1 + λiλj) = 0, (i , j are not summed) (8)

I Only 2 form of S̃ν leads to ν mass matrices with 2 massive νs.

I These forms are,

S̃1ν = diag(λ,−λ∗,−1) and S̃2ν = diag(±i ,∓i , 1) (9)

I Structure of M̃ν is determined from eq.(8)

Cont...

I Assume that at least 1-off-diagonal elements of M̃ν 6= 0(say12 ele.)

I Eq.(8) ⇒ S̃1ν = diag(λ,−λ∗,−1) as a necessary condition.

I Various cases are, (I) λ = 1, (II) λ = ±i and λ 6= ±1, ± i

I Texture I: S̃1ν = diag(1,−1,−1) , M̃ν = m0

0 c se iβ

c 0 0se iβ 0 0

where here, c = cos θ and s = sin θ

I Texture II: S̃1ν = diag(±i ,±i ,−1) , M̃ν = m0

x1 y 0y x2 00 0 0

I Texture III: S̃1ν = diag(λ,−λ∗,−1) , M̃ν = m0

0 1 01 0 00 0 0

I Texture (I and III)⇒ one massless and a pair of degenerate νs

and texture (II) ⇒ one massless and 2 non degenerate νs.

Cont...

I M̃ν of texture(I) is diagonalize as V Tν M̃νVν = diag(m0,m0, 0)

where Vν =

1√2

−i√2

0

c√2

ic√2

−s

se−iβ

√2

ise−iβ

√2

ce−iβ

R12

I Assume that one of the diagonal elements of M̃ν 6= 0(say 11ele.)

I This ⇒ S̃ν = diag(±i , λ′,∓λ∗′) with |λ′| = 1

I λ′ = ∓i ⇒ Texture IV: S̃ν = diag(i ,−i , 1) ,

M̃ν = m0

x1 0 00 x2 00 0 0

I In this respect, two major group series ∆(3N2) and ∆(6N2)

have been studied to determine neutrino masses and mixingpatterns.5

5AJ, JHEP 1511 (2015) 186

A5 group

I The alternating group A5 is the even permutations of 5distinct objects.

I It has 60 different elements and five conjugacy classes:

1C1, 15C2, 20C3, 12C 15 and 12C 2

5 . (10)

I It has 5-irreducible representations: 1, 3, 3′, 4 and 5.I All representations except singlet are faithful.I The Abelian subgroups are Z2 (15 elements), Z3 (20

elements) and Z5 (24 elements).I 15 elements of Z2 contain 5 distinct Z2×Z2 subgroups.

I Generators in 3-dimensional representions are,

H = 12

[−1 µ− µ+µ− µ+ −1µ+ −1 µ−

]; E =

[0 1 00 0 11 0 0

]; f1 =

[1 0 00 −1 00 0 −1

]

here µ± = (−1±√

5)/2

Cont...

I 15 Z2 elements have eigenvalues (1, -1, -1).

I These 15 elements fall in texture I type which give onemassless and 2-degenerate ν.

I 20 Z3 and 24 Z5 elements have eigenvalues (1, η, η∗) withη 6= ±1 and |η|2 = 1

I In A5, we have 15 Sν and 59 Tl .

I Vl diagonalizing Tl(= Z3, Z5) can be chosen as,

Vl =

x1 z1 z∗1x2 z2 z∗2x3 z3 z∗3

I UPMNS = V †l (VSνVν)

I Product of Vl with any of the VSν give (µ− τ) symmetry fora given θ and β = 0.where µ− τ symmetry means |Uµi | = |Uτ i | for i=1,2,3

Case I: Sν = Z2 and Tl = Z5

θ 6= 0, β = 0 θ 6= 0, β 6= 0

|UPMNS | =

[0.2181 0.9616 0.16440.6901 0.1940 0.69720.6901 0.1940 0.6972

];

[0.2196 0.9603 0.17160.7118 0.1883 0.67660.6671 0.2055 0.7160

]

I L.H.S. |UPMNS | ⇒ |Uµ3| = |Uτ3| ⇒ maximal θ23 correct θ13

but not correct θ12.

I R.H.S |UPMNS | ⇒ non-maximal θ23 with correct θ13.

I Both of these are for a massless ν and a pair of degenerate νs.

I A small perturbation in the ν-mass matrix Mν leads to correctmass ratio and mixing ∠′s

I Similar results also obtain for Tl = Z3

Case II: Sν = Z2 × Z2 and Tl = Z3(or Z5)

I Mν has been considered anti-symmetric so far w.r.t. only 1 SνI This fails to determine M̃ν completely because of unknowns θ

and β

I We use here an additional residual symmetry, which determineM̃ν completely apart from an over all complex mass scale.

I We took S ′ν which commute with Sν .

I Mν should satisfy, STν MνSν = −Mν and S ′Tν MνS

′ν = Mν

I For a given Sν and S ′ν , Z2 × Z2 invariant Mν is given by,

Mν = m02

−µ+ 0 −10 −µ− 1−1 1 −1

I |UPMNS | obtain with this Mν gives (µ− τ) symmetry but not

correct θ13 or θ12.

I A small perturbation leads to correct mass ratio and mixingangles.

More cases

I At Oth order we have one massless and a pair of degenerateνs.

I With perturbation, anti-symmetry property gets broken andthat results 3 non-degenerate νs.

Oth order AfterCase Tl Sν

√× Perturbation

n=3 or 5

I Zn Z2 θ13, θ23 θ12√

II Zn Z2 × Z2 θ23 θ13, θ12√

III Z2 Z2 × Z2 θ13, θ23 θ12√

IV Z2 × Z2 Z2 θ13, θ23 θ12√

V Z2 × Z2 Z2 × Z2 θ13, θ23, θ12 ×

Summary

I We have mentioned here various unknowns in neutrino physics andcurrent status of neutrino oscillation parameters.

I Various group theoretical approaches are mentioned here to obtainneutrino masses and mixing.

I Majorana neutrino mass matrix obeying flavor symmetry with variousmass pattern are addressed.

I We mainly focussed on the consequences of assuming Majorana neutrinomass matrix display flavor anti-symmetry property.

I In this respect, we have studied A5 group and its various Abeliansubgroups.

I We have obtained correct θ13 and θ23 values in its 3σ range for most of

the (Sν ,Tl) pairs at Oth order and with small perturbation we are able to

obtain all parameters in its 3σ range.

Thank you

Thanks to:Srubabati Goswami (Supervisor)

Monojit Ghosh, Sushant K Raut, Shivani Gupta, Pomita Ghoshal