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Physics Letters B

www.elsevier.com/locate/physletb

Third family corrections to tri-bimaximal lepton mixing and a new sum rule

Stefan Antuscha'*, Stephen F. Kingb, Michal Malinskyb

a Max-Planck-Institut für Physik (Werner-Heisenberg-Institut) Föhringer Ring 6, D-80805 München, Germany b School of Physics and Astronomy, University of Southampton, SO16 1BJ Southampton, United Kingdom

ARTICLE INFO ABSTRACT

We investigate the theoretical stability of the predictions of tri-bimaximal neutrino mixing with respect to third family wave-function corrections. Such third family wave-function corrections can arise from either the canonical normalisation of the kinetic terms or renormalisation group running effects. At leading order both sorts of corrections can be subsumed into a single universal parameter. For hierarchical neutrinos, this leads to a new testable lepton mixing sum rule s = r cos S + 3a (where s, r, a describe the deviations of solar, reactor and atmospheric mixing angles from their tri-bimaximal values, and S is the observable Dirac CP phase) which is stable under all leading order third family wave-function corrections, as well as Cabibbo-like charged lepton mixing effects.

© 2008 Elsevier B.V. All rights reserved.

Article history:

Received 11 September 2008 Accepted 9 December 2008 Available online 12 December 2008 Editor: A. Ringwald

PACS: 14.60.Pq 12.15.Ff 12.60.Jv 11.30.Hv

1. Introduction

Since the discovery of neutrino masses and large lepton mixing angles, the flavour problem of Standard Model (SM) has received much attention. As the precision of the neutrino data has improved, it has become apparent that lepton mixing is consistent with the so-called Tri-bimaximal (TB) mixing pattern [1],

u tb =

/ J2 1 0 )

v 3 T3

V6 T3 T2

^ Ts 1 TrJ

• P m ,

where P m is the so far experimentally undetermined diagonal phase matrix encoding the two observable Majorana phase differences. Many models attempt to reproduce this as a theoretical prediction [2-9]. Since the forthcoming neutrino experiments will be sensitive to small deviations from TB mixing, it is important to quantify the "theoretical" uncertainty inherent in such TB mixing predictions.

In many classes of models TB mixing arises purely from the neutrino sector [10], subject to deviations due to charged lepton sector corrections [2,3]. If these charged lepton corrections are "Cabibbo-like" in nature (i.e. dominated by a 1-2 mixing), it leads to a predictive sum rule [2] which may be expressed in terms of the parameterisation in [11] as s = rcosS, where s and r describe the deviations of solar and reactor mixing angles from their tri-

* Corresponding author.

E-mail addresses: antusch@mppmu.mpg.de (S. Antusch), sfk@hep.phys.soton.ac.uk (S.F. King), malinsky@phys.soton.ac.uk (M. Malinsky).

bimaximal values, and S is the observable Dirac CP phase in the standard parameterisation [12].

Another source of theoretical uncertainty in TB mixing schemes is the renormalisation group (RG) running [13] of the relevant quantities from the high energy (usually the unification scale MG), where the theory is defined, to the electroweak scale M Z appropriate for experimental measurements. The dominant source of RG corrections to lepton mixing arises typically from the large tau lepton and third family neutrino Yukawa couplings, leading to relatively large wave-function corrections in the framework of supersymmetric models. Such RG corrections can be readily estimated analytically [14,15] for the TB mixing case with hierarchical light neutrinos considered here. Diagrammatically, such RG corrections correspond to loop diagrams involving third family matter and Higgs fields and their superpartners. Although suppressed by the loop factor of 1/16^2, they can be relevant since the loop factor is multiplied by a large logarithm of the ratio of energy scales.

Apart from RG effects there is another type of third family wave-function correction which emerges at tree-level in certain classes of models, and thus can potentially be rather large. These corrections modify the kinetic terms in the Lagrangian causing them to deviate from the standard (or canonical) form. Before the theory can be reliably interpreted, field transformations must be performed in order to return the kinetic terms back to canonical form which, however, leads to appropriate modifications of the Yukawa couplings. It is interesting that these effects are largest in many of the theories that predict TB mixing, especially those based on non-Abelian family symmetries spanning all three families of SM matter (see e.g. [16-18]). In such models the canonical normalisation (CN) corrections can in certain cases even exceed the effects due to RG running.

0370-2693/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2008.12.013

In this Letter we shall provide a unified treatment of all the above sources of theoretical corrections to the TB mixing, namely due to: (i) RG corrections, (ii) CN corrections and (iii) charged lep-ton corrections. We will present a novel testable neutrino mixing sum rule which, at leading order, is stable under all these effects.

2. General formalism

Suppose the original (before the RG and CN corrections are accounted for) charged lepton (l) and Majorana neutrino mass matrices Mi and mv are diagonalised by means of unitary transformations VlLMiVR = MD (we shall work in LR chirality basis) and V mv VvT = mD so that the uncorrected lepton mixing matrix obeys Upmns = VlLVV.

The effect of both CN and leading logarithmic RG corrections on the Mi and mv matrices can be described by a pair of transformation matrices PL R

Ml ^ PfM^R = Mi, mv^ Pfmv PL = mv

which induce a relevant change on V[ R ^ V[ R and VV ^ V[V so

that V'LMiViR = MD and mvT = mD and thus Upmns = vlvvr is the physical lepton mixing matrix (after global rephasing). One can always write PL, R = pL, R (1 + A PL, R) where, as we shall see, the constants pL,R have no effect on the mixing angles and A PL,R denote the corrections from the flavour non-universal part of the RG and CN effects to be identified later. Eq. (2) then implies

plpRv'L (1 + A PT )m¡ (1 + A pR )VR = M°,

p2LVV (1 + A PT )m v(1 + A Pl )VLV t = . (3)

If all the physical spectra are sufficiently hierarchical, the smallness

of A PL,R factors ensures only small differences between VVL and VLf (for f = l, v), in particular

V f = w[v[ = eiAWLf V [, (4)

where W are small unitary rotations in the unity neighborhood

with AW[ denoting their Hermitian generators. One can disentangle the left-handed and right-handed rotations in the charged lepton formula in (3) by considering MiMl with the result1

i )i=j,

□rW m i2+m 2 (V L A P¡V [)ij

+2m m (V rva pRv RR)^

where the eigenvalues ml2 of the original Mi matrix can, at leading order, be identified with the physical charged lepton masses. Similarly, the neutrino sector corrections obey (replacing Mi ^ mv, VlL ^ Vv, VlR ^ VI* and APR ^ APL in formula (5) above)

mv 2 - mv2

[(m v2 + m j 2)(V va pTV

+ 2m V m V(V [ A PTV jj.

From Eqs. (4), (5) and (6) one can write the corrected (i.e. physical) lepton mixing matrix Upmns = VlLVv^ in terms of the original

Upmns as Upmns = Upmns + aUpmns where

AU pmns ^ ^ AWlLU pmns - U PMNS AWv ^.

1 Notice that these formulae provide only the off-diagonal entries of AWf's. In-

deed, formulae (3) and (4) leave three phases of the diagonalization matrices W^v unconstrained so we shall conventionally put (AwL'v)ii = 0.

Due to the assumed hierarchy in the physical spectra, the first terms in Eqs. (5) and (6) dominate over the second (thus avoiding the ambiguity in the unknown structure of the right-handed (RH) rotations in the charged lepton sector) and so we shall neglect the latter and focus on the left-handed (LH) sector.

The RG effects in the supersymmetric case yield at leading order [13] PRG = rL 1 + APRG where

r[ = 1 -

16 n 2

3(TrYlYu - g2)ln — + Try\Yv ln —

V u u s ) Mz v v MN

(with g2 = gf + 1 g2) accounts for flavour-universal contribution while

A PRG =--Lr

L 16 n 2

Y*YT l^^ + Yí Yv ln^

Mz ' MN ' (9)

denotes the flavour non-trivial piece. In (8) and (9), MN denotes the mass of the lightest RH neutrino. Notice that since rL is close to 1 one can write at leading order

Prg ^ rL (1 + A Prg) (10)

rendering the rL factor irrelevant for the mixing angles.

Turning to CN effects, with the non-canonical LH lepton doublet (L) kinetic term written as iL^pKlL and KL = kL(1 + AKL), the CN transformation can be written in the form of (2) with (cf. [16-18])

PCN = (1 + APCN)/Vk[ where ApCn AK[.

At leading order, P™ fulfills (P™)-1^P™)-1 = KL.

Finally, one can combine both RG and CN effects under a single transformation satisfying at leading order

;Pcn pRg ;

(1 + A Pcn + A Prg)

yielding the assumed form of PL = pL (1 + A PL) with pL = rL/-Jkl and APL = APrg + APcn.

Using these results, the leading order corrections to lepton mixing from RG and CN effects can be calculated.

3. Third family corrections to tri-bimaximal lepton mixing

In this section we shall apply the formalism of Section 2 to TB lepton mixing, where we assume to begin with that this mixing originates entirely from the neutrino sector and subsequently extend the analysis to include corrections from charged lepton mixing [10]. Thus, let us assume first that the lepton mixing predicted by some underlying theory in the absence of RG and CN corrections happens to be exactly tri-bimaximal Upmns = VlLVv = Utb where VL = 1 while Vv^ = Utb. Including RG and CN corrections, VlL and VVvr then change according to (4) with the correction matrices given by Eqs. (5), (6).

Here we shall restrict ourselves to the dominant third family wavefunction corrections, so that from (9) one can write APrg = -2 diag(0, 0, nRG) with nRG = [y? ln(MG/Mz) + y?3 MMg/Mn)]/ 8n2. If the third family effects dominate also the form of KL in Eq. (11), AKl is a matrix controlled by the 33 entry and A P™ = -2 diag(0, 0, nCN) with nCN = (AKL)33. Then at the leading order A PL = - 2 diag(0, 0, n) is governed by a universal parameter n = nRG + nCN. This is the case, for instance, in all models in which a family symmetry spans all three SM matter families [16-18]. In such theories the third family corrections to the kinetic function KL have the same origin as the third family Yukawa couplings, and nCN can be as large as y? (in the SO(3)-type of models) or yT (for underlying SU(3) flavour symmetry). However, it is also possible that nCN ^ yT in certain classes of models, and the effect is strongly dependent on the details of the underlying theory [6,20].

■ ^ _ _ - -* - .....

_______ ......... 92S - 45

n = 35.26°

beyond loading order beyond leading order

approximation approximation

.............. 1 '■'«■'■ " ■ ■

0.0 Tf

Fig. 1. Renormalisation group and canonical normalization corrections to tri-bimaximal neutrino mixing. The shaded regions correspond to ^-values outside the linear approximation.

Given the diagonal form of A PL, and our assumption that VlL = 1, one gets from Eq. (5) AWlL = 0 and thus VlL = 1 so the corrected lepton mixing matrix is given by just VV = Utb + AUtb with AUTB = -iUTBAwf from Eq. (7). If the neutrino spectrum is hierarchical, the leading (first) term in (6) yields

A(U tb j

: -(UtbMuJbaP[Utb),, (no j sum.).

We can see that, at the leading order, the corrections to (UTB)ij for i < j are proportional to (UTB)ij itself, and thus for example both RG and CN corrections to 013 are zero due to (UTB)13 = 0.2 However the result A(UTB)13 = 0 arises only at leading order in

small quantities (n and ^Am^1 /Am^1 & m2/m3 & 1/5) and gets lifted at next-to-leading level. Restoring the second term in Eq. (6) one obtains

013 « -7=Am2i/Am2i « 4 x 10-2|n|.

All together, this yields at the leading order

(r- (1 - 12 n)

-76(1 -

v 76(1 +12 n)

7- (1 +1 n) 7-(1 - 1 n) -7-(1 +1 n)

0.04 x \n\e-is \

7(1 + 1 n) 7(1 - 1 n) )

which is unitary up to O(rj2). In terms of the deviations from the exact TB mixing parametrized [11] by sin012 = (1 + s)/V3, sin 023 = (1 + a)/J2 and sin 013 = r /72 one gets (so far without including charged lepton corrections):

r « 6 x 10-2|nl,

s = -n and a = — n. 6 ' 4 '

We see that in particular 013 is rather stable and the atmospheric 023 is changing faster with r than the solar 012, as shown in Fig. 1.

In any realistic model, the charged lepton mixing corrections entering Upmns must be taken into account. It is well known that if VlL is Cabibbo-like with 012 being the only non-negligible mixing angle, then (ignoring RG and CN effects) this gives rise to a particular pattern of corrections to 013 and 012 that obey (at the high

2 Note also that (13) implies that the Majorana phases have no effect on the corrections to the mixings angles.

scale) the relation s = r cos & with, e.g., r & 0C/3 for 012 & 0C/3 (0C is the Cabibbo angle) in many unified models [2].

We can include the leading order RG and CN corrections to s = r cos & by considering only the neutrino sector effects (encoded in AWV) for the individual terms. (This follows since for the Cabibbo-like VlL and A PL dominated by the 33 entry, it is still the case that AWlL = 0.) Therefore the previous example provides a good estimate of the relevant corrections at leading order in small quantities (including now also the charged lepton mixing 012). Neglecting the subleading correction in (14), from Eq. (16) one obtains s = r cos & + 6 n, which can be rewritten in terms of only measurable quantities in form of a new sum rule 2

s = r cos & + 3a. (17)

The new sum rule in Eq. (17) is (at leading order) stable under the considered theoretical corrections and additionally involves the deviation of atmospheric mixing from maximality [19]. In the considered scenario, the main sources of remaining uncertainties in formula (17) are the neglected (order 4%| r I) corrections to r cos & due to the subleading contribution (14), the higher order corrections to A Prg and A Pcn (all suppressed by the relevant Yukawa coupling ratios) and the higher order n-effects.

4. Conclusions

We have presented a unified formalism for dealing with both renormalisation group running effects and canonical normalisation corrections. Using this formalism we have investigated the third family wave-function corrections to the theoretical predictions of tri-bimaximal neutrino mixing. We found that at leading order both effects can be subsumed into a single universal parameter n. Including also the leading order Cabibbo-like charged lepton mixing corrections, which typically arise in unified flavour models, we have derived the theoretically stable sum rule s = r cos & + |a where s, r and a parametrize the deviations of the solar, reactor and atmospheric mixing angles from their tri-bimaximal values and & is the leptonic Dirac CP phase. Such a sum rule is testable in future high precision neutrino experiments [21].

Acknowledgements

We acknowledge partial support from the following grants: PPARC Rolling Grant PPA/G/S/ 2003/00096; EU Network MRTN-CT-2004-503369; EU ILIAS RII3-CT-2004-506222; NATO grant PST.CLG. 980066.

References

[1] P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 530 ( 2002) 167;

P.F. Harrison, W.G. Scott, Phys. Lett. B 535 (2002) 163;

P.F. Harrison, W.G. Scott, Phys. Lett. B 557 (2003) 76.

[2] S.F. King, JHEP 0508 (2005) 105;

I. Masina, Phys. Lett. B 633 (2006) 134;

S. Antusch, S.F. King, Phys. Lett. B 631 (2005) 42;

S. Antusch, P. Huber, S.F. King, T. Schwetz, JHEP 0704 (2007) 060.

[3] P.H. Frampton, S.T. Petcov, W. Rodejohann, Nucl. Phys. B 687 (2004) 31;

A. Dighe, S. Goswami, W. Rodejohann, Phys. Rev. D 75 (2007) 073023;

F. Plentinger, W. Rodejohann, Phys. Lett. B 625 (2005) 264;

R.N. Mohapatra, W. Rodejohann, Phys. Rev. D 72 (2005) 053001;

K.A. Hochmuth, S.T. Petcov, W. Rodejohann, arXiv: 0706.2975 [hep-ph].

[4] G. Altarelli, F. Feruglio, Y. Lin, Nucl. Phys. B 775 (2007) 31;

G. Altarelli, F. Feruglio, Nucl. Phys. B 741 (2006 ) 215;

G. Altarelli, F. Feruglio, Nucl. Phys. B 720 (2005) 64.

[5] E. Ma, arXiv: 0709.0507 [hep-ph];

E. Ma, Mod. Phys. Lett. A 22 (2007) 101;

E. Ma, Mod. Phys. Lett. A 21 (2006) 2931;

E. Ma, H. Sawanaka, M. Tanimoto, Phys. Lett. B 641 (2006) 301;

B. Adhikary, et al., Phys. Lett. B 638 (2006) 345;

S.L. Chen, M. Frigerio, E. Ma, Nucl. Phys. B 724 (2005) 423.

[6] I. de Medeiros Varzielas, G.G. Ross, Nucl. Phys. B 733 ( 2006 ) 31;

I. de Medeiros Varzielas, S.F. King, G.G. Ross, Phys. Lett. B 644 (2007) 153;

I. de Medeiros Varzielas, S.F. King, G.G. Ross, Phys. Lett. B 648 (2007) 201;

S.F. King, M. Malinsky, Phys. Lett. B 645 (2007) 351;

S.F. King, M. Malinsky, JHEP 0611 (2006) 071;

C. Luhn, S. Nasri, P. Ramond, Phys. Lett. B 652 (2007) 27.

[7] R.N. Mohapatra, S. Nasri, H.B. Yu, Phys. Lett. B 639 (2006) 318; R.N. Mohapatra, H.B. Yu, Phys. Lett. B 644 (2007) 346;

M.C. Chen, K.T. Mahanthappa, Phys. Lett. B 652 (2007) 34.

[8] A.H. Chan, H. Fritzsch, Z.z. Xing, arXiv: 0704.3153 [hep-ph]; Z.z. Xing, Phys. Lett. B 618 (2005) 141;

Shu Luo, Z.z. Xing, Phys. Lett. B 632 (2006) 341;

Z.z. Xing, Phys. Lett. B 533 (2002) 85;

S.K. Kang, Z.z. Xing, S. Zhou, Phys. Rev. D 73 ( 2006 ) 0130 01;

M. Hirsch, et al., Phys. Rev. D 75 (2007) 053006;

X.G. He, A. Zee, Phys. Lett. B 645 (2007) 427.

[9] S. Antusch, L.E. Ibanez, T. Macri, JHEP 0709 ( 2007 ) 087.

[10] As stated in the introduction, this situation is realised in many classes of TB mixing models. The precise meaning of the statement that the TB mixing comes entirely from the neutrino sector can also be expressed in a basis invariant way as discussed for example in: S.F. King, Nucl. Phys. B 786 (2007) 52.

[11] S.F. King, arXiv: 0710.0530 [hep-ph].

[12] Particle Data Group, W.M. Yao, et al., J. Phys. G 33 (2006) 1.

[13] P.H. Chankowski, Z. Pluciennik, Phys. Lett. B 316 (1993) 312; K.S. Babu, C.N. Leung, J. Pantaleone, Phys. Lett. B 319 (1993) 191; S.F. King, N.N. Singh, Nucl. Phys. B 591 (2000) 3;

S. Antusch, M. Drees, J. Kersten, M. Lindner, M. Ratz, Phys. Lett. B 519 (2001) 238;

S. Antusch, M. Drees, J. Kersten, M. Lindner, M. Ratz, Phys. Lett. B 525 (2002) 130;

S. Antusch, J. Kersten, M. Lindner, M. Ratz, Phys. Lett. B 538 ( 2002 ) 87; S. Antusch, M. Ratz, JHEP 0207 (2002) 059.

[14] S. Antusch, J. Kersten, M. Lindner, M. Ratz, M.A. Schmidt, JHEP 0503 (2005) 024;

S. Antusch, J. Kersten, M. Lindner, M. Ratz, Nucl. Phys. B 674 (2003) 401.

[15] A. Dighe, S. Goswami, W. Rodejohann, Phys. Rev. D 75 (2007) 073023; A. Dighe, S. Goswami, P. Roy, Phys. Rev. D 76 (2007) 096005.

[16] S.F. King, I.N.R. Peddie, Phys. Lett. B 586 (2004) 83.

[17] S.F. King, I.N.R. Peddie, G.G. Ross, L. Velasco-Sevilla, O. Vives, JHEP 0507 (2005) 049.

[18] S. Antusch, S.F. King, M. Malinsky, arXiv: 0708.1282 [hep-ph].

[19] S. Antusch, P. Huber, J. Kersten, T. Schwetz, W. Winter, Phys. Rev. D 70 (2004) 097302.

[20] S. Antusch, S.F. King, M. Malinsky, JHEP 0805 (2008) 066, arXiv: 0712.3759 [hep-ph].

[21] ISS Physics Working Group, arXiv: 0710.4947 [hep-ph].