Scholarly article on topic 'Underlying A4 symmetry for the neutrino mass matrix and the quark mixing matrix'

Underlying A4 symmetry for the neutrino mass matrix and the quark mixing matrix Academic research paper on "Physical sciences"

CC BY
0
0
Share paper
Academic journal
Physics Letters B
OECD Field of science
Keywords
{}

Abstract of research paper on Physical sciences, author of scientific article — K.S. Babu, Ernest Ma, J.W.F. Valle

Abstract The discrete non-Abelian symmetry A 4, valid at some high-energy scale, naturally leads to three degenerate neutrino masses, without spoiling the hierarchy of charged-lepton masses. Realistic neutrino mixing angles (one of which is necessarily maximal and the other large) are then automatically induced radiatively, and the correct mass splittings may be obtained in the context of softly broken supersymmetry. The quark mixing matrix is also calculable in a similar way. The mixing parameter U e3 is predicted to be imaginary, leading to maximal CP violation in neutrino oscillations. Neutrinoless double beta decay and τ→μγ should be in the experimentally accessible range.

Academic research paper on topic "Underlying A4 symmetry for the neutrino mass matrix and the quark mixing matrix"

Available online at www.sciencedirect.com

SCIENCE ^DIRECT8

Physics Letters B 552 (2003) 207-213

www. elsevier. com/locate/npe

Underlying A4 symmetry for the neutrino mass matrix and

the quark mixing matrix

K.S. Babua, Ernest Mab, J.W.F. Vallec

a Physics Department, Oklahoma State University, Stillwater, OK 74078, USA b Physics Department, University of California, Riverside, CA 92521, USA c Instituto de Física Corpuscular, CSIC, Universitat de Valencia, Edificio Institutos, Aptdo. 22085, E-46071 Valencia, Spain

Received 29 October 2002; received in revised form 25 November 2002; accepted 2 December 2002

Editor: G.F. Giudice

Abstract

The discrete non-Abelian symmetry A4, valid at some high-energy scale, naturally leads to three degenerate neutrino masses, without spoiling the hierarchy of charged-lepton masses. Realistic neutrino mixing angles (one of which is necessarily maximal and the other large) are then automatically induced radiatively, and the correct mass splittings may be obtained in the context of softly broken supersymmetry. The quark mixing matrix is also calculable in a similar way. The mixing parameter Ue3 is predicted to be imaginary, leading to maximal CP violation in neutrino oscillations. Neutrinoless double beta decay and t ^ iiy should be in the experimentally accessible range. © 2002 Elsevier Science B.V. All rights reserved.

It has often be said that the mixing pattern of neutrinos, which involves large angles, as evidenced by the atmospheric [1] and solar [2] neutrino data, is unexpected and difficult to understand, given that the quark charged-current mixing matrix VCKM involves only small angles. However, as shown below, both can be explained in a simple and unified way as small radiative corrections of a fixed pattern, valid at some high-energy scale as the result of an underlying symmetry, which we identify here as A4, the non-Abelian discrete symmetry group of the tetrahedron [3]. We show that at the high scale, neutrino masses are degenerate and VCKM is the identity matrix. We then calculate the radiative corrections down at the electroweak scale in the framework of softly broken supersymmetry [4,5] and obtain realistic versions of Mv and VCKM. The reason that neutrino mixing involves large angles is a simple consequence of degenerate perturbation theory, where a small off-diagonal term induces maximal mixing between two states of equal energy, whereas in the quark sector with hierarchical masses, the same small off-diagonal element induces only a small mixing.

Our starting point is the model of Ref. [3], but with the following two important improvements. (I) Instead of breaking A4 spontaneously at the electroweak scale, it is now broken at a very high scale. (II) Supersymmetry is added with explicit soft breaking terms which also break A4. The resulting theory at the electroweak scale is a specific version of the MSSM (Minimal Supersymmetric Standard Model), where the scalar lepton and quark

E-mail address: valle@ific.uv.es (J.W.F. Valle).

0370-2693/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0370-2693(02)03153-2

sectors are correlated with Mv and VCKM. In this way we also provide a theoretical framework for realizing the neutrino unification idea suggested in the first paper of Ref. [4], but with different specific predictions.

The non-Abelian discrete finite group A4 consists of 12 elements and has 4 irreducible representations. Three are one-dimensional, I, J_'. J_". and one is three-dimensional, 3, with the decomposition

1 x 3 = l+r+r + 3 + 3. (1)

The usual quark, lepton, and Higgs superfields transform under A4 as follows:

Qi = (ui,di), Li = (\>i, ef) ~ 3, <£1,2 ~ I, (2)

"C jC "C 1 ~C jC ~C i! "C jC "C AH

Mj, <2j, ej ~ J_, u2, a2, e2 ~ J_, w3, a3, e3 ~ J_ . (3)

We then add the following heavy quark, lepton, and Higgs superfields:

Uu Uf, DU Dl %, E?, Nf, xi ~3, (4)

which are all SU(2) singlets. The superpotential of this model is then given by

W = MuUiUC + fu QiUffo + huijkUiUCxk + MDDiDC + fdQiDCfa + hdijkDid]xk + meE^c

+ feLi Eci4> 1 + heijkEi ecj Xk + N? + fNU Nfh + l^lh + \Mx Xi Xi + hx Xl /2/3, (5)

where we have adopted the usual assignment of R parity to distinguish between the Higgs superfields, i.e., <j)1,2 and X, from the quark and lepton superfields. We have also forbidden the terms X.i NC NC, etc. by assigning

Xi ~ m, uC, dC, eC ~ m, (6)

and all others ~ 1 under a separate discrete symmetry Z3 (with m3 = 1 and 1 + m + m2 = 0). However, Z3 is allowed to be broken explicitly but only softly, which is uniquely accomplished in the above by MX = 0. The scalar potential involving Xi is given by

V = \MXX1 + hxX2X3|2 + |MxX2 + hxX3X1|2 + MxX3 + hxX1X2|2, (7)

which has the supersymmetric solution (V = 0)

<X1> = <X2> = <X3> = u = -Mx/hx, (8)

so that the breaking of A4 at the high scale MX does not break the supersymmetry. (Note that Eq. (8) is only possible because A4 allows the invariant symmetric product of 3 x 3 x 3, a highly nontrivial property not shared for example by the triplet representation of either SO(3) or SU(3).)

Consider now the 6 x 6 Dirac mass matrix linking (ei ,Ei) to (ec,Ec).

0 0 0 feV1 0 0

0 0 0 0 feV1 0

0 0 0 0 0 feV1

h\u h2u h3u me 0 0

h^u h2oM h3M2u 0 me 0

yh^u h2M2u h3oju 0 0 me )

where v1 = <&0>, and Eq. (17) of the first paper of Ref. [3] has been used, with similar forms for the quark mass matrices. The reduced 3 x 3 Dirac mass matrix for the charged leptons is then

Me = Ul

where hf = he[1 + (heu)2/M2]-1/2 and ' 11 1 \

1 a a2 1 . (11)

,1 a2 a '

This shows that charged-lepton masses are allowed to be all different, despite the imposition of the A4 symmetry, because there exist three inequivalent one-dimensional representations. The corresponding Yukawa couplings are not constrained by the A4 symmetry to be of the same order of magnitude, in the same way they are not constrained by the standard-model gauge symmetry. (Note that the permutation symmetry groups S3 and S4 have only two inequivalent one-dimensional representations and S3 has no three-dimensional representation [6].) Clearly, the up and down quark mass matrices are obtained in the same way, with the important conclusion that the charged-current mixing matrix VCKM is automatically equal to the identity matrix, because both are diagonalized by UL. Corrections to VCKM = 1 may then be ascribed to the structure of the soft supersymmetry breaking sector [5,7]. In the neutrino sector, the 6 x 6 Majorana mass matrix spanning (ve,vl,vt,N'CL,N2^,N^) is given by

MvN = (U » UMNV2), (12)

\UTfNV2 Mn J

where v2 = ($0}. Hence the 3 x 3 seesaw mass matrix for (ve,vl,vT) becomes

f2 v0 fo vo /100 \

Mv = q^UlUL = ^[0 0 1 . (13)

MN ^ Vo 1 0/

This shows that neutrino masses are degenerate at this stage.

Consider now the above as coming from an effective dimension-five operator [8]

j^kijViVjfltl (14)

where Xee = XlT = XTl = 1 and all other X's are zero, which is valid at some high scale. As we come down to the electroweak scale, Eq. (14) is corrected [9] by the wavefunction renormalizations of ve, vm, and vT, as well as the corresponding vertex renormalizations. Even if only the standard model is considered, this will lift the degeneracy of Eq. (13) because of the different charged-lepton masses. The resulting pattern, i.e., Xee = 1 + 2m°e, XlT = XTl = 1 + (m2 + m°)e, where e ~ 1/(16^2v2) ln(MN/MZ), is however not suitable for explaining the present data on neutrino oscillations. On the other hand, other radiative corrections exist in the context of softly broken supersymmetry with a general slepton mass matrix [4]. Given the structure of Xij at the high scale, its form at the low scale is necessarily fixed to first order as

(1 + 2See Sen + Set Sell, + Set \

Sei + Set 2SMr 1 + S11 + STT I , (15)

Sei + Set 1 + S11 + STT 2SMr '

where we have assumed all parameters to be real as a first approximation. (The above matrix is obtained by multiplying that of Eq. (13) on the left and on the right by all possible vi ^ Vj transitions.) Let us rewrite the above with So = S^ + Srr - 2SMr, S = 2SMT, S' = See - S^/2 - Sxx/2 - SMT, and S" = Sei + Set. Then

/1 + S0 + 2S + 2S' S" S" \

Xij = S" S 1 + So + S , (16)

V S" 1 + So + S S /

and the exact eigenvectors and eigenvalues are easily obtained:

(v \ / cos0 sin0/V2 sin0/V2\ / Ve\

V2 ) = -sin0 COS0/V2 COS 0/V2 , (17)

V3/ V 0 -I/V2 1/V2 / W/

Xl = 1 + So + 25 + 5' - \/S'2 + 28"2, X2 = 1 + 50 + 25 + 5' + V S'2 + 2S"2, X3 = —1 — So-

Note the remarkable fact that Eq. (17) is valid whatever values 80, 8, 8', and 8" may take. With 8"2/8'2 of order unity, this is also a very satisfactory description of present neutrino-oscillation data, i.e., sin2 20atm = 1, and

tan 0soi =

: = 0-44,

8"2 + 8'2 - 5'V8'2 + 28"2

if 8' < 0 and |8///8/| = 1.7. Note that for 8" = 8' Eq. (16) reproduces that proposed in the second paper of Ref. [3]. Whereas the latter was simply an ansatz, the form of Eq. (16) here is a necessary consequence of our assumption that radiative corrections are responsible for the splitting of the neutrino mass degeneracy enforced by the discrete A4 symmetry. Assuming that 8',8" ^ 8, we now have

Am?> ~ A«22 ~ 4Sm2,

Afflî7 ~ 4y/8'2 + 28"2ml,

where m0 is the common mass of all 3 neutrinos.

Note that Ue3 = 0 inEq. (17), which would imply the absence of CP violation in neutrino oscillations. However, if we do not assume Xij to be real, then it has one complex phase which cannot be rotated away. Without loss of generality, we now rewrite Eq. (16) as

1 + 2S + 2S'

S"* ^ 1+S S

where we have redefined 1 + 80 as 1, and 8, 8' are real. Although this mass matrix cannot be diagonalized exactly, if we assume that 8', Re 8" and (Im8")2/8 are all much smaller than 8 in magnitude, then Eqs. (17) to (22) are again valid (but only approximately) with the following changes:

i ImS" \[28

S' ^ S'+

(ImS ) 28

Note the important result that Ue3 is imaginary. Thus CP violation is predicted to be maximal in this model for neutrino oscillations. Using Eq. (22), we also have the relationship

Amh A mj2.

I Ue312

ReS'' 8

Note that | Ue3 \ is naturally of the order | Am12/Am2211/2 — 0.14. This result depends only on the form of Eq. (23), which is itself derived in the most general way.

It remains to be shown in the rest of this Letter that realistic values of 8, 8', and 8" are possible from the soft breaking of supersymmetry, without running into conflict with present limits on neutrinoless double beta (PP0v) decay and lepton flavor violating processes such as t ^ /¿y.

Let us calculate 8 in the context of supersymmetry. We show in Figs. 1 and 2 the wavefunction and vertex corrections respectively due to //L-TL mixing. Let the two scalar mass eigenstates have masses m 12 and their mixing angle be 0. For illustration, let us take the approximation that m 2 > /2 > M^2 = m 2, where / is the superpotential higgsino mixing term, while M1 ,2 denote the soft supersymmetry breaking gaugino mass parameters. We then obtain

sin0 cos 0

Fig. 1. Wavefunction contribution to S in supersymmetry.

Fig. 2. Vertex contribution to S in supersymmetry.

Using Eq. (22) and taking Am32 = 2.5 x 10 3 eV2 from atmospheric neutrino oscillations, we find S = 3.9 x

10-3(0.4 eV/mo)2. This implies that

sin 0 cos 0

0.4 eV

To the extent that this factor cannot be much greater than one, the common mass m0 as probed in neutrinoless double beta decay [10] cannot be much lower than the present upper bound of about 0.4 eV. This is in sharp contrast to the scenario proposed in the first paper of Ref. [4] where //0v decay is strongly suppressed.

Similarly, S" = Sei + STe = Sei + S*T is determined by eL-lL and eL-tl mixing. Using the experimental bound of |Ue31 < 0.16, we find ImS" < 8.8 x 10-4(0.4 eV/m0)2, and using Am12 ~ 5 x 10-5 eV2 from solar neutrino oscillations, we find ReS" < 5.5 x 10-5(0.4 eV/m0)2. These limits may be saturated mainly by eL-tl mixing, allowing eL-iL mixing to be much more suppressed. In other words, from the data on neutrino oscillations, we are able to make the direct connection in this model that flavor violation in the charged-lepton sector should be the greatest in the i-t sector and smallest in the e-i sector.

Using the same approximation which leads to Eq. (26), the t ^ iy amplitude is calculated to be

+¿2),.

A= ^icL' (smO cos0y-^exqv j!crXv(l + y5)r. 1536n2 m2

The resulting branching fraction is then

B(t ^ iy) = 4.8 x 10-6 sin2 0 cos2 0

100 GeV

Using the experimental upper limit of 1. 1 x 10-6 on this number, we require thus m2 > 102 GeV. Constraints from t ^ ey and / ^ ey are also similarly satisfied. Details will be given in a forthcoming comprehensive study of the correlation between the neutrino parameters and the pattern of soft supersymmetry breaking of this model.

Consider now the quark sector. Whereas the neutrino sector has only L-L scalar mixings, we now also have L-R and R-R scalar mixings. In a previous study [5], VCKM = 1 was obtained from proportional up and down quark mass matrices, and it was shown that a realistic VCKM could then be generated with L-R scalar quark mixings through gluino exchange. Here VCKM = 1 is obtained from our A4 symmetry for any set of arbitrary up and down quark masses, with the obvious implication that the above result also applies. (In the charged-lepton sector, the effect is smaller (because it comes from bino exchange) and does not significantly change the neutrino mixing angles except possibly for Ue3.) More details will be discussed in the forthcoming comprehensive study.

In conclusion, we have presented a concrete model based on the discrete symmetry A4 where quark and charged-lepton masses can be all different and yet neutrino masses are degenerate at some high scale where VCKM = 1 and the effective neutrino mass matrix in the ve, vM, vT basis is of the form

/ m0 0 0 \

Mv = I 0 0 m0 I . (30)

\ 0 m0 0 /

The parameter m0 naturally lies in the range where it can be probed in cosmology, neutrinoless double beta decay and tritium beta decay. Radiative corrections lift the neutrino degeneracy leading automatically to (A) sin220atm = 1, (B) Ue3 small and imaginary, and (C) large (but not maximal) solar mixing angle. These corrections can be ascribed to the structure of the soft supersymmetry breaking terms in the scalar sector, which also break the A4 symmetry explicitly and correlate the neutrino mass matrix with lepton flavor violating processes. Last but not least, a realistic quark mixing matrix VCKM may be obtained in a totally analogous way.

Acknowledgements

This work was supported in part by the US Department of Energy under Grants Nos. DE-FG03-94ER40837, DE-FG03-98ER41076, and DE-FG02-01ER45684, by a grant from the Research Corporation, by the European Commission Grant HPRN-CT-2000-00148, by the ESF Neutrino Astrophysics Network, and by the Spanish MCyT Grant PB98-0693. The authors wish to thank particularly the Institute for Nuclear Theory, University of Washington, for its hospitality during the mini-workshop on neutrinos (April 2002) where this work was initiated.

References

[1] Super-Kamiokande Collaboration, S. Fukuda, et al., Phys. Rev. Lett. 85 (2000) 3999, and references therein.

[2] Super-Kamiokande Collaboration, S. Fukuda, et al., Phys. Rev. Lett. 86 (2001) 5656, and references therein; See also, SNO Collaboration, Q.R. Ahmad, et al., Phys. Rev. Lett. 87 (2001) 071301;

Q.R. Ahmad, et al., Phys. Rev. Lett. 89 (2002) 011301; Q.R. Ahmad, et al., Phys. Rev. Lett. 89 (2002) 011302.

[3] E. Ma, G. Rajasekaran, Phys. Rev. D 64 (2001) 113012; E. Ma, Mod. Phys. Lett. A 17 (2002) 289;

E. Ma, Mod. Phys. Lett. A 17 (2002) 627.

[4] P.H. Chankowski, A. Ioannisian, S. Pokorski, J.W.F. Valle, Phys. Rev. Lett. 86 (2001) 3488; E. Chun, S. Pokorski, Phys. Rev. D 62 (2000) 053001.

[5] K.S. Babu, B. Dutta, R.N. Mohapatra, Phys. Rev. D 60 (1999) 095004.

[6] Y. Yamanaka, H. Sugawara, S. Pakvasa, Phys. Rev. D 25 (1982) 1895; P.H. Frampton, T.W. Kephart, Int. J. Mod. Phys. A 10 (1995) 4689.

[7] F. Gabbiani, E. Gabrielli, A. Masiero, L. Silvestrini, Nucl. Phys. B 477 (1996) 321.

[8] S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566; E. Ma, Phys. Rev. Lett. 81 (1998) 1171.

[9] K.S. Babu, C.N. Leung, J. Pantaleone, Phys. Lett. B 319 (1993) 191; P.H. Chankowski, Z. Pluciennik, Phys. Lett. B 316 (1993) 312.

[10] mo lies in the range where a hint for ßßov decay has been claimed by H.V. Klapdor-Kleingrothaus, et al., Mod. Phys. Lett. A 16 (2001) 2409;

See also comments by C.E. Aalseth, et al., Mod. Phys. Lett. A 17 (2002) 1475; And the reply by H.V. Klapdor-Kleingrothaus, hep-ph/0205228;

For a recent review see, for example, S.R. Elliott, P. Vogel, Annu. Rev. Nucl. Part. Sci. 52, in press.