Complex scaling and residual flavour symmetry in the neutrino mass matrix

Complex scaling and residual flavour symmetry in the neutrino mass matrix

PROBIR ROY

Centre for Astroparticle Physics and Space Science, Bose Institute, Kolkata 700 054, India (In collaboration with Rome Samanta and Ambar Ghosal of Saha Institute)

E-mail: probirrana@gmail.com Published online 9 October 2017

Abstract. Using the residual symmetry approach, we propose a complex extension of the scaling ansatz on the neutrino Majorana mass matrixM_ν which allows a nonzero mass for each of the three light neutrinos as well as a nonvanishingθ13. Leptonic Dirac CP violation must be maximal while atmospheric neutrino mixing need not be exactly maximal. Each of the two Majorana phases, to be probed by the search for 0νββdecay, has to be zero orπ and a normal neutrino mass hierarchy is allowed.

Keywords. Neutrinos; residual flavour symmetry; scaling ansatz.

PACS Nos 14.60.Pq; 12.60.Fr; 13.15.+g; 14.60.St

1. Introduction

In this contribution, we focus on the observed phe- nomenon of neutrino mixing and not so much on the dynamics of the generation of O(eV) light neutrino neutrino masses. Let us start with the neutrino flavour eigenstate fieldsνl (l = e, μ, τ). Then we have three left-chiral light neutrinos with the corresponding mass

eigenstate fields νi (i = 1,2,3), all of which are assumed to be Majorana in nature. The mass term in the neutrino Lagrangian relevant to laboratory energies is

−L^νmass= 1 2

ν¯_l^C(M_ν)lmνm+h.c. (1) with M_ν^∗ = M_ν = M_ν^T. Thus, there is a 3×3 unitary matrixU such that

U^TM_νU =M_ν^d ≡diag(m₁,m₂,m₃), (2) where each mi is real and positive. We shall assume throughout that all light neutrino masses are nonzero.

Experiments tell us that they are nondegenerate. For comparison, we may write the charged lepton mass matrix in our weak basis as

Mch=diag(me,m_μ,m_τ). (3) The unitary mixing matrixUof (2) can now be written in the PDG convention [1] as

U ≡UPMNS =

⎛

⎜⎝

c12c13 e^iα2s12c13 s13e^{−i(δ−β2)}

−s12c23−c12s23s13e^iδ e^iα2(c12c23−s12s13s23e^iδ) c13s23e^iβ2 s12s23−c12s13c23e^iδ e^iα2(−c12s23−s12s13c23e^iδ) c13c23e^iβ2

⎞

⎟⎠ (4)

with ci j ≡ cosθi j, si j ≡ sinθi j and θi j = [0, π/2] whileδ, α, β= [0,2π].

Let us quickly browse the neutrino fact file. The latest 3σranges [2] for the relevant neutrino mass parameters, obtained from oscillation data and cosmological obser- vations, are:

Solar:m²₂₁ ≡m²₂−m²₁:(7.02–8.09)×10⁻⁵eV², Atmospheric:|m²₃₁| ≡ |m²₃−m²₁|:(2.32–2.59)×10⁻³ eV²,

Planck:im_i <0.23 eV.

The 3σintervals of the mixing angles, introduced in (4), are [2]

θ12: 31.29^◦–35.91^◦ θ23: 38.3^◦–53.3^◦ θ13: 7.87^◦–9.11^◦.

In quoting the above numbers, we have not made the fine distinction between normal (m₃ > m₂ > m₁) and inverted (m2>m1>m3) types of mass ordering of the neutrinos, but we carefully do so in our numerical work reported later.

2. The problematic simple real scaling

Simple real scaling (SRS) was first proposed [3] as an ansatz and posited the relation (within our sign conven- tion)

M_ν^SRS _eμ −M_ν^SRS _eτ =

M_ν^SRS _μμ −M_ν^SRS _μτ =

M_ν^SRS _τμ −M_ν^SRS _ττ =k

(5) k being a real, positive, dimensionless scaling factor.

Given (5), the form ofM_ν^SRSfollows immediately to be

M_ν^SRS =

⎛

⎝ x −Y k Y

−Y k Z k² −Z k Y −Z k Z

⎞

⎠, (6)

wherex,Y,Z a prioriare unknown mass-dimensional quantities;x can be chosen to be real by absorbing an overall phase in the neutrino fields, but Y and Z are in general complex. The case k = 1 corresponds to a μτ interchange symmetric M_ν with the additional constraint M_μμ^ν = −M_μτ^ν . Since det M_ν^SRS = 0, one null eigenvalue of M_ν^SRSis implied. The corresponding eigenvector can be deduced to be the third column of UPMNS, i.e.

C₃^SRS=

⎛

⎜⎝

0 (1+k²)⁻¹²e^iβ2 k(1+k²)⁻¹²e^iβ2

⎞

⎟⎠ (7)

implying that m^SRS₃ = 0 = θ₁₃^SRS and s23 = (1+k²)^−1/2, c23 = k(1+k²)^−1/2 so that tanθ23 = k⁻¹ while θ₂₃^SRS is left undetermined (whereas other results change in an extension that we shall propose, the tanθ23 =k⁻¹relation will be seen to survive). The earlier mentioned unitary matrix is now given by

U^SRS=

⎛

⎜⎝

c12 s12e^iα2 0

−k(1+k²)⁻¹²s₁₂ k(1+k²)⁻¹²c₁₂e^iα2 (1+k²)⁻¹²e^iβ2 k(1+k²)⁻¹²s12 −k(1+k²)⁻¹²c12e^iα2 k(1+k²)⁻¹²e^iβ2

⎞

⎟⎠. (8)

This SRS proposal got knocked out by the experimental exclusion of a vanishingθ13at the 10σ level [4].

3. Residual flavour symmetry and generalized real scaling

Any flavour symmetry ofM_ν, implemented by the uni- tary transformationGand operating asνLα →G_αβνLβ, implies the following relations:

G^TM_νG= M_ν, U^†GU =d,

d_αβ = ±δαβ. (9)

Of the eight possible diagonal matricesd,d = ±I are trivial while the remaining six can be split into{da}and {−da}witha =1,2,3. Eachdacorresponds to aGi but there is a relationG_a = abcG_bG_c which leaves only two independentGs. We choose them to beG2,3with U^†G2U =d2 ≡diag(−1,1,−1),

U^†G3U =d3 ≡diag(−1,−1,1). (10) There is thus a residualZ2×Z₂ flavour symmetry [5]

inM_νcorresponding toG2,3: Z2:G^T₂M_νG2 =M_ν,

Z₂:G^T₃M_νG3 =M_ν. (11) We identifyZ₂as withZ^scaling₂ and equateG^scaling₃ with U^SRSd3U^SRS†, obtaining

G^scaling₃ =

⎛

⎜⎜

⎜⎜

⎝

−1 0 0 0 1−k²

1+k² 2k 1+k²

0 2k

1+k² −1−k² 1+k²

⎞

⎟⎟

⎟⎟

⎠. (12)

Moreover, equatingG^k₂withU^SRSd2U^SRS†, we obtain

G^k₂=

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

−cos 2θ12

ksin 2θ12

1+k² −sin 2θ12

1+k² ksin 2θ12

1+k² −(1−cos 2θ12)

k²(1+k²) −(1+cos 2θ12) k(1+k²)

−sin 2θ12

1+k² −(1+cos 2θ12)

k(1+k²) −(k²−cos 2θ12) 1+k²

⎞

⎟⎟

⎟⎟

⎟⎟

⎠ .

(13)

The most general form of M_ν, obeying the invariance (G^scaling₃ )^TM_νG^scaling₃ = M_ν is found to be not M_ν^SRS but its generalized real scaling (GRS) form

M_ν^GRS =

⎛

⎝ x −Y k Y

−Y k Z−W k⁻¹(k²−1) W

Y W Z

⎞

⎠ (14)

with an additional complex mass-dimensional element W. The latter needs to equal −Z k as a special case to yield M_ν^SRS. This M_ν^GRS can accommodate a nonzero m3; nevertheless, it is phenomenologically unaccept- able. The columnC3of (7) is still an eigenvector of the matrix (14) now with a nonzerom3. In consequence, the former is the third column of the corresponding unitary mixing matrix leading to a vanishingθ13and ruling out M_ν^GRS as a realistic neutrino mass matrix.

4. Complex extension of scaling: Our proposal We propose [6] a complex extension of the scaling ansatz through a nonstandard CP and flavour transformation νLα →i(G3)αβγ⁰ν^C_Lβleading to

G^scaling₃ M_νG^scaling₃ =M_ν^∗. (15) The most general complex extended scaling (CES) invariant form of M_ν, satisfying (15), is

M_ν^CES =

⎛

⎜⎜

⎜⎜

⎜⎝

x −y1k+iy2

k y1+i y2

−y1k+iy₂

k z1−wk²−1

k −i z2w−ik²−1 2k z2

y1+i y2 w−ik²−1

2k z2 z1+i z2

⎞

⎟⎟

⎟⎟

⎟⎠

. (16)

The mass matrix (16) is characterized by six real mass dimensional parameters x, y1,2,z1,2 andw apart from the scaling factork. It follows that [6]

G3U^∗=Ud˜, d˜ =diag(d˜1,d˜2,d˜3),

d˜a = ±1 fora =1,2,3. (17) By substituting G^scaling₃ from (12) and qualifying U byU^CES, we explicitly obtain from the first equation of (17) that

⎛

⎜⎜

⎜⎜

⎜⎝

−(U_e1^CES)^∗ −(U_e2^CES)^∗ −(U_e3^CES)^∗ 1−k²

1+k²(U_μ1^CES)^∗+ 2k

1+k²(U_τ^CES₁ )^∗ 1−k²

1+k²(U_μ2^CES)^∗+ 2k

1+k²(U_τ^CES₂ )^∗ 1−k²

1+k²(U_μ3^CES)^∗+ 2k

1+k²(U_τ^CES₃ )^∗ 2k

1+k²(U_μ^CES₁ )^∗−1−k²

1+k²(U_τ^CES₁ )^∗ 2k

1+k²(U_μ^CES₂ )^∗−1−k²

1+k²(U_τ^CES₂ )^∗ 2k

1+k²(U_μ^CES₃ )^∗− 1−k²

1+k²(U_τ^CES₃ )^∗

⎞

⎟⎟

⎟⎟

⎟⎠

=

⎛

⎜⎜

⎝

d˜1U_e1^CES d˜2U_e2^CES d˜3U_e3^CES d˜1U_μ^CES₁ d˜2U_μ^CES₂ d˜3U_μ^CES₃ d˜₁U_τ^CES₁ d˜₂U_τ^CES₂ d˜₃U_τ^CES₃

⎞

⎟⎟

⎠. (18)

From (18) one sees that, in the identification ofU^CES with U_PMNS, any combination with d˜₁ = 1 yields an imaginaryc12c13inUPMNSand is hence ruled out. Four other nontrivial possibilities remain, with

d˜1 = −1, d˜2 =η,

d˜3 =ξ (19)

andηa,b =1, ηc,d = −1, ξa,c=1 andξb,d = −1. Now a detailed comparison between the columns ofU^CESand U_PMNSof (4) leads to the results

e^−iα = −η,

e^i(2δ−β)= −ξ, (20) so that we obtain

α =π,0 forη= +1,−1,

2δ−β =π,0 forξ = +1,−1. (21)

On matching the remaining six elements ofUPMNS, one obtains six independent linear constraint conditions, as tabulated in table1.

The lowermost condition in table1, given the observed fact thatc12=0, yields the relations

Table 1. Constraint equations on elements of the mixing matrix.

Element ofU^CES Constraint condition

μ1 2kU_μ^CES₁ =(1−k²)U_τ^CES₁ −(1+k²)(U_τ^CES₁ )^∗ τ1 2kU_τ^CES₁ = −(1−k²)U_μ^CES₁ −(1+k²)(U_μ^CES₁ )^∗ μ2 2kU_μ^CES₂ =(1−k²)U_τ^CES₂ +η(1+k²)(U_τ^CES₂ )^∗ τ2 2kU_τ^CES₂ = −(1−k²)U_μ^CES₂ +η(1+k²)(U_μ^CES₂ )^∗ μ3 2kU_μ^CES₃ =(1−k²)U_τ^CES₃ +ξ(1+k²)(U_τ^CES₃ )^∗ τ3 2kU_τ^CES₃ = −(1−k²)U_μ^CES₃ +η(1+k²)(U_μ^CES₃ )^∗

Table 2. Output values obtained for normal mass ordering.

x y1 y2 z1 z2 w

(eV) (eV) (eV) (eV) (eV) (eV)

−0.20–+0.21 −0.12–+0.11 −0.05–+0.05 −0.17–+0.17 −0.18–+0.17 −0.16–+0.15

m1 m2 m3

(eV) (eV) (eV)

9.2×10⁻⁵–0.071 0.01–0.077 0.051–0.082

Table 3. Output values obtained for inverted mass ordering.

x y1 y2 z1 z2 w

(eV) (eV) (eV) (eV) (eV) (eV)

−0.44–+0.46 −0.16–+0.16 −0.14–+0.14 −0.01–+0.01 −0.01–+0.01 −0.05–+0.06

m1 m2 m3

(eV) (eV) (eV)

0.051–0.085 0.049–0.079 8.2×10⁻⁵–0.068

Figure 1. Variations of the light neutrino masses with the lightest massm1(normal ordering).

2kc23cosβ

2 = [k²(1+ξ)−1+ξ]s23cosβ 2, 2kc23sinβ

2 = [k²(1−ξ)−1−ξ]s23sinβ

2. (22)

Figure 2. Variations of the light neutrino masses with the lightest massm3(inverted ordering).

Sinceξ² =1, a multiplication of the two sides of the above equations implies that sinβ =0, i.e.

β =0 orπ. (23)

Figure 3. Plot of|M_ee^ν|vs. the lightest neutrino mass: the top two figures are ford˜aandd˜bwhile the last two figures are for d˜c(left) andd˜d(right).

From (21) and (23) we see that each of the two Majorana phases takes the value zero orπ, i.e., there is no Majo- rana CP violation in this model. Continuing further, we can identify four possibilities forβ,ξ and tanθ23: β =0, ξ =1⇒tanθ23 =k⁻¹

β =π, ξ = −1⇒tanθ23 =k⁻¹, β =0, ξ = −1⇒tanθ23 = −k,

β =π, ξ =1⇒tanθ23= −k. (24)

The last two possibilities can be excluded because when used on the fourth relation from the top listed in table1, these lead to the conclusion thatc12 = 0 – in contradiction with observation. Thus, we now unequiv- ocally have

tanθ23 =k⁻¹ (25)

and also from (20) and (21) cosδ=0, i.e. δ= π

2 or 3π

2 . (26)

In other words, Dirac CP violation is maximal.

5. Phenomenological discussion

We have tried to determine the six mass-dimensional unknown parameters x, y1,2,z1,2 andw of our model by inputting the observed 3σ ranges ofm²₂₁,|m²₃₁|, θ12,θ23andθ13as well as the cosmological upper bound onimi quoted in the neutrino fact file of §1. The out- put values depend on the type of mass ordering of the three neutrinos that is assumed. These are given in tables 2 and 3 for normal (m3 > m2 > m1) and inverted (m1 > m2 > m3) ordering respectively. The corre- sponding allowed mass bands for m3, m2 and m1 are shown in figures1and2against the central value of the lightest neutrino mass as the latter is varied continuously as a parameter.

We next turn our attention to neutrinoless double beta (0νββ) decay. The relevant constant here is theeeele- ment ofM_ν. The latter is given in our model by (M_ν)ee =c²₁₂c²₁₃m1+s₁₂² c²₁₃m2e^iα+s₁₃² m3e^i(β−2δ).

(27) There are four possibilities for the diagonal matrix d˜ that the complex-extended scaling ansatz embodies:

(1)d˜_a=diag(−1,+1,+1), (2)d˜_b=diag(−1,+1,−1), (3) d˜c = diag(−1,−1,+1), (4) d˜d = diag(−1,−1,

−1). The corresponding mass bands are for |M_ν|ee

as shown in figure 3. The upper and lower bands are for a normal (plotted against m1) and for an inverted (plotted against m3) mass ordering respectively. Only the extreme right corners of these plots will be accessi- ble to forthcoming experiments such GERDA.

6. Conclusion

• We have proposed a complex extended scaling ansatz onM_ν (see eq. (16)).

• We have obtained a 6-parameter form ofM_ν(see eq.

(16)).

• We have derived that Dirac (Majorana) CP violation should be maximal (absent).

• We have both normal and inverted types of mass ordering as allowed possibilities.

• Our neutrinos are mass hierarchical for most allowed values of the mass sumimi and approach quasi- degeneracy only when the latter is near the cosmo- logical upper bound 0.23 eV.

• We have interesting predictions on 0νββdecay.

Acknowledgements

This work has been supported by a Senior Scientist fel- lowship of the Indian National Science Academy.

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