We show that the neutrino mixing pattern with the large mixing required for
the atmospheric neutrino problem and the small mixing angle MSW solution for
the solar neutrino problem can be naturally generated through radiative
magnification, even though all the mixing angles at the seesaw scale may be
small. This can account for the neutrino anomalies as well as the CHOOZ
constraints in the context of quark-lepton unified theories, where the quark
and lepton mixing angles are expected to be similar in magnitude at the high
scale. We also indicate the 4$\nu$ mixing scenarios for which this mechanism of
radiative magnification can provide a natural explanation.Comment: 14 pages RevTex, 2 eps figure

Amol S. Dighe

K.R.S. Balaji

M.K. Parida

R.N. Mohapatra

Physics Letters B

English

arXiv

We show that the neutrino mixing pattern with the large mixing required for the atmospheric neutrino problem and the small mixing angle MSW solution for the solar neutrino problem can be naturally generated through radiative magnification, even though all the mixing angles at the seesaw scale may be small. This can account for the neutrino anomalies as well as the CHOOZ constraints in the context of quark-lepton unified theories, where the quark and lepton mixing angles are expected to be similar in magnitude at the high scale. We also indicate the 4ν mixing scenarios for which this mechanism of radiative magnification can provide a natural explanation

Balaji, K. R .S.

Dighe, Amol S.

Mohapatra, R. N.

Parida, M. K.

Indian Academy of Sciences

Radiative magnification of neutrino mixings and a natural explanation of the neutrino anomalies

We show that the neutrino mixing pattern with the large mixing required forthe atmospheric neutrino problem and the small mixing angle MSW solution forthe solar neutrino problem can be naturally generated through radiativemagnification, even though all the mixing angles at the seesaw scale may besmall. This can account for the neutrino anomalies as well as the CHOOZconstraints in the context of quark-lepton unified theories, where the quarkand lepton mixing angles are expected to be similar in magnitude at the highscale. We also indicate the 4$\nu$ mixing scenarios for which this mechanism ofradiative magnification can provide a natural explanation.We show that the neutrino mixing pattern with the large mixing required for the atmospheric neutrino problem and the small mixing angle MSW solution for the solar neutrino problem can be naturally generated through radiative magnification, even though all the mixing angles at the seesaw scale may be small. This can account for the neutrino anomalies as well as the CHOOZ constraints in the context of quark-lepton unified theories, where the quark and lepton mixing angles are expected to be similar in magnitude at the high scale. We also indicate the 4$\nu$ mixing scenarios for which this mechanism of radiative magnification can provide a natural explanation.We show that the neutrino mixing pattern with the large mixing required for the atmospheric neutrino problem and the small mixing angle MSW solution for the solar neutrino problem can be naturally generated through radiative magnification, even though all the mixing angles at the seesaw scale may be small. This can account for the neutrino anomalies as well as the CHOOZ constraints in the context of quark-lepton unified theories, where the quark and lepton mixing angles are expected to be similar in magnitude at the high scale. We also indicate the 4 ν mixing scenarios for which this mechanism of radiative magnification can provide a natural explanation

Balaji, K.R.S.

Mohapatra, R.N.

Parida, M.K.

CERN Document Server

CERN-TH/2000-044IMSc/2000/02/04UMD-PP-00-054hep-ph/0002177Radiative magnication of neutrino mixings and a naturalexplanation of the neutrino anomaliesK. R. S. BalajiInstitute of Mathematical Sciences, Chennai 600 113, India.Amol S. DigheyTheory Division, CERN, CH-1211 Geneva 23, Switzerland.R. N. MohapatrazDepartment of Physics, University of Maryland, College park MD 20742, USA.M. K. ParidaxDepartment of Physics, North Eastern Hill University, Shillong 793022, India.AbstractWe show that the neutrino mixing pattern with the large mixing requiredfor the atmospheric neutrino problem and the small mixing angle MSW solu-balaji@imsc.ernet.inyAmol.Dighe@cern.chzrmohapat@physics.umd.eduxmparida@vsnl.com1tion for the solar neutrino problem can be naturally generated through radia-tive magnication, even though all the mixing angles at the seesaw scale maybe small. This can account for the neutrino anomalies as well as the CHOOZconstraints in the context of quark-lepton unied theories, where the quarkand lepton mixing angles are expected to be similar in magnitude at the highscale. We also indicate the 4 mixing scenarios for which this mechanism ofradiative magnication can provide a natural explanation.PACS numbers: 14.60.Pq, 12.15.Lk, 12.15.Ff, 12.60.-i2I. INTRODUCTIONA major theoretical challenge posed by the solutions to the atmospheric and solar neu-trino anomalies [1,2] is that the atmospheric neutrino data require a large  −  mixing,whereas the corresponding quark mixing between the second and the third generation is verysmall. This is not easy to understand in the context of quark-lepton unied theories. Whilethere are suggestions to understand a large mixing in the context of various kinds of uniedtheories [3] including the SO(10), where there is a natural quark-lepton unication [4], noconvincing natural model has yet emerged. It is therefore necessary to explore alternativepossibilities. One way to proceed is not to concentrate on a particular model, but to lookfor the features that a model should have in order to be able to predict the observed largemixing naturally.In a recent paper [5], we pointed out that for two Majorana neutrinos with the same CPparity that are nearly degenerate in mass, a small neutrino mixing at the high scale can bemagnied by the radiative corrections through the renormalization group running down tothe weak scale. In such theories, there would be no need to put special constraints on themixings in the theory at the high (e.g. seesaw) scale and indeed the quark and lepton mixingscan be very similar (as, say, would be predicted by the simple seesaw models). It is the goalof this paper to show that such a mixing pattern, which involves only small mixings at thehigh scale (), can explain the neutrino anomalies at the low scale as long as the conditionsoutlined in [5] are satised. It is then possible to explain the solar neutrino problem throughthe small angle MSW solution and the atmospheric neutrino problem through the largemixing angle, which gets generated through the radiative magnication.The paper is organized as follows. In Sec. II, we introduce the (Ω;;Ψ) parametrizationfor the mixing angles, and taking  = 0 at the high scale, show that the radiative correctionscan magnify Ψ while keeping Ω and  unaected. In Sec. III, we show that the condition  =0 is consistent with the current data from the solar, atmospheric and reactor experiments.In Sec. IV, we consider the possible 4 mixing schemes that can explain the LSND results in3addition, and identify the scenarios for which radiative magnication can provide a naturalexplanation through the quark-lepton unied theories. Sec. V concludes.II. RADIATIVE MAGNIFICATION FOR THREE NEUTRINO MIXINGIn the absence of CP violation in the lepton sector, the mixing matrix U at the scale can be parametrized asU = U12(Ω) U13() U23(Ψ) ; (1)where all the three rotation angles lie between 0 and =2. Note that the order of multi-plication of the rotation matrices is dierent from the conventional one [6], so the anglesΩ;;Ψ involved here should not be mistaken for the angles !; ;  used conventionally.Nevertheless, (1) is a perfectly valid way of parametrizing the mixing matrix, and is usefulfor addressing a certain class of problems (e.g. see [7]).At the low scale , the mixing matrix U can be written in general asU = U12(Ω) U13() U23( Ψ) : (2)The CHOOZ results [8] indicate a small Ue3, which corresponds to a small value forcos Ω cos Ψ sin  + sin Ω sin Ψ :This can be satised with the choice of  = 0 and a small sin Ω sin Ψ. That such a choicecan satisfy the solar and the atmospheric data is shown in Sec. III. With this motivation,we start with  = 0 at the high scale (this choice leads to  = 0, as we shall show in thissection), and show that the radiative corrections can magnify Ψ while keeping Ω and unaected.With only the Ω and Ψ mixings nonzero at the scale , the eective mass matrix Meffin the flavor basis isMeff = U Md Uy = U12(Ω) U23(Ψ) Md Uy23(Ψ) Uy12(Ω) ; (3)4where Md = Diag(m1; m2; m3). If the radiative corrections are included [9,10], we haveMeff ! Meff =0BBBBBB@pIe 0 00qI 00 0pI1CCCCCCAMeff0BBBBBB@pIe 0 00qI 00 0pI1CCCCCCA; (4)where I − 1  2 are the radiative corrections that appear due to the Yukawa couplingsof the charged leptons e;  and  respectively. Given the strong hierarchical pattern of thecharged lepton masses, we neglect the corrections due to e and , i.e. Ie = I = 1. Let usdene I  Diag(1; 1;pI ). Then from (3) and (4),Meff = I U12(Ω) U23(Ψ) Md U y23(Ψ) U y12(Ω) I : (5)Noting that [U12(Ω); I ] = 0, we getMeff = U12(Ω) [I U23(Ψ) Md U y23(Ψ) I ] U y12(Ω) : (6)The quantity in the square brackets in (6) is in a form where the rst row and column areeectively decoupled and the situation reduces to the two-generation mixing, which has beenconsidered in detail in [5]. This quantity can be written asI U23(Ψ) Md U y23(Ψ) I = U23( ~Ψ) Md U y23( ~Ψ) ; (7)where Md is a diagonal matrix. The new (2-3) mixing angle~Ψ is given bytan(2~Ψ) =tan(2Ψ)(1 + ) ;  (m3 − m2)C2Ψ + 2m(m3 − m2)C2Ψ ; (8)where m is the common mass of the quasi-degenerate neutrinos. Now, if  (m2 −m3)C2Ψ2m; (9)then   0, so that the mixing angle ~Ψ becomes large [5]. Since   1, for the condition(9) to be satised, 2 and 3 need to have the same CP parity. Thus the Ψ-mixing can bemagnied at the weak scale, which explains the atmospheric neutrino data (See Sec. III).5From (6) and (7),Meff = U12(Ω) U23(~Ψ) Md Uy23( ~Ψ) Uy12(Ω) : (10)This shows that the same (1-2) mixing angle Ω that was needed for diagonalizing Meff isalso needed for diagonalizing Meff [see (3) and (10)], and that a (1-3) mixing angle  is notrequired. Thus, Ψ = ~Ψ, Ω = Ω and  =  = 0 are the mixing angles at the low scale.As we shall see in Sec. III, we can explain the solar, atmospheric and the CHOOZ datawith Ψ  =4 and a small Ω (corresponding to the SMA solution for the solar neutrinos).In a typical quark-lepton unied theory, Ω would be small at the high scale. In the limitof neglecting the radiative corrections due to the second generation (i.e. I ! 1) that wehave considered here, the magnication of Ω due to radiative corrections is not possible.Also, if the CP parity of the neutrino 1 is opposite to that of 2 and 3 (which is requiredto ascertain the stability of a possible small nonzero ), a small Ω at the high scale willstay small even when the radiative corrections due to  are taken into account. Thus, thestability of a small Ω is guaranteed, and the small angle MSW scenario can be generatednaturally within the unication models.The radiative corrections from the second generation [i.e. I  Diag(1;qI; 1) 6=Diag(1; 1; 1)] modify (6) toMeff = I U12(Ω) [I U23(Ψ) Md U y23(Ψ) I ] U y12(Ω) I : (11)Since [U12(Ω); I] 6= 0, the value of Ω may now get modied and  may get generated. Thevalue of Ψ is also dierent from the value of ~Ψ as given in (8). But since[U12(Ω); I] =  sin Ω0BB@ 0 11 01CCA ; (12)and the values of Ω and  are both small, these dierences (Ω− Ω), (− ) and (Ψ− ~Ψ)are not expected to be large.6III. SATISFYING THE SOLAR, ATMOSPHERIC AND CHOOZ DATAIn the following, we show that our choice of parametrization (1) with  = 0 can explainthe solar and atmospheric anomalies and still be consistent with the stringent bounds comingfrom the CHOOZ experiment.We rst concentrate on the CHOOZ and the atmospheric data which share a commonmass scale m231  m232  10−3eV 2. In this case, the relevant probability expressions areP atm  4 jU3j2jU3j2 sin2 m231L4E!; (13)P atm  1 − 4jU3j2(1 − jU3j2) sin2 m231L4E!: (14)To satisfy the CHOOZ constraint [8]jUe3j2 < 0:03 for m231 > 2  10−3 eV2 (15)with  = 0, we needsin Ω sin Ψ < 0:17 : (16)This small value of jUe3j2 also guarantees Pee  1 [eq. (14)] and Pe  0 [eq. (13)] in theatmospheric neutrino data.A t to the L=E distribution of the atmospheric neutrinos [11] gives at 90% condencelevel, using (14),0:2 < jU3j2 < 0:8 : (17)This corresponds to0:45 < cos Ω sin Ψ < 0:9 : (18)By examing the mixing matrix as parametrized in (2), we can see that  = 0, Ψ  =4and a small Ω can easily satisfy the requirements (16) and (18). The smallness of Ω is forcedby the CHOOZ constraints and the large value of sin Ψ is required by the atmospheric P.7Let us now consider the solar neutrino solution. In the case of the solar neutrino anomaly,the SMA solution corresponds tojUe2j2  (0:5 2:5)  10−3 ; (19)whereas the other solutions { LMA, LOW and VO { correspond to jUe2j2 > 0:2. In theparametrization (2),jUe2j2 = sin2 Ω cos2 Ψ : (20)It is dicult to reconcile the smallness of Ω forced by the atmospheric and CHOOZ resultsto the jUe2j2 required for the LMA, LOW or VO solution. But in the case of the SMAsolution, (19) and (20) givesin Ω cos Ψ  0:02 0:05 ; (21)which can be satised simultaneously with (16) and (18). The region in the Ω− Ψ parameterspace that satises all the constraints (16), (18) and (21) is shown in Fig. 1. Our scheme thussupports the SMA solution: if we start with a small Ω at the high scale (which is naturalin the quark-lepton unied theories), it does not change much through radiative corrections(as we have shown in sec. II), and a small Ω is retained at the low scale.As pointed out in [5], the mechanism of radiative magnication does not need any ne-tuning, but is at work in a range of parameter space for any given model. As an exampleof the radiative magnication of Ψ, let us consider MSSM, where the parameter tan de-termines the magnitude of the radiative corrections. The value of tan  required to obtainany given magnied value of Ψ is shown in Fig. 2. This indicates the phenomenologicallyinteresting range of tan for radiative magnication.IV. FOUR NEUTRINO SCHEMESThe features of radiative magnication noted here can be used in order to identify the 4mixing scenarios in which the large atmospheric mixing can be naturally generated. Taking8into account that the recent atmospheric neutrino results disfavor (− s) oscillations [12],the 4 solution for all the anomalies (atmospheric [1], solar [2] and LSND [13]) is essentiallyof the form [14][e − s]−−− [ −  ] ; (22)where the [e−s] pair (m214  m2) and the [− ] pair (m223  m2atm) are separatedby m2es−  m2LSND. The solar neutrino puzzle is explained by the e $ s oscillationsand the atmospheric data are explained by the  $  oscillations. A small e−  mixingthen explains the LSND [13] observations.In (22), the neutrinos can be considered to be written in the increasing order of masses.With the current data, it is still possible to change the order of neutrinos within a bracket,or the order of the brackets themselves. The order within a bracket will not have anyinfluence on our conclusions, so we have only the two independent cases: (a) mes < mand (b) mes > m , where mes (m ) denotes the average mass of the [e − s] ([ −  ])pair. In the case (a),  and  are necessarily quasi-degenerate: taking m2LSND  1 eV2and m2atm  4  10−3 eV2, we get the degree of degeneracy ( mm ) for the  −  pair asmm< 2 10−3. Then the −  mixing angle  can be radiatively magnied, as we requirefor the atmospheric neutrino solution.In the case (b), the neutrinos  and  need not be quasi-degenerate, so the magnitudeof radiative corrections needed to magnify  is large. Accounting for the large  throughradiative magnication is then dicult. Thus, if radiative magnication is the reason forthe large  , then the case (a) is favored, i.e. mes < m on the grounds of naturalness.V. CONCLUSIONSWe have shown that, with the parametrization U = U12(Ω)  U13()  U23(Ψ) of thelepton mixing matrix,  = 0, Ψ  =4 and a small Ω at the low scale can satisfy all theconstraints from the solar, atmospheric and CHOOZ data. These mixing angles can be9generated at the high scale with  = 0 and small Ω and Ψ (which is natural in the quark-lepton unied theories), and magnifying Ψ through radiative corrections while keeping Ωand  unaected.Let us add a few words on the realization of this scenario of radiative magnicationin the unied theories. We have not given any specic model realization, rather we havepointed out a class of models that would be successful in generating a large lepton mixingnaturally, starting from a small mixing at the high scale. It is not hard to see that suchsmall mixing angle patterns can emerge at the high scale  in quark-lepton unied theoriesof type SU(2)L  SU(2)R  SU(4)c if the right-handed neutrino coupling is assumed tobe an identity matrix since the Dirac mass matrix for neutrinos that goes into the seesawmatrix is then identical to the up-quark mass matrix. Thus even though our discussion inthis paper is completely model independent, its realization in the context of unied theoriesis quite straightforward. Our work thus demonstrates a way to have a natural solution forthe neutrino anomalies in the quark-lepton unied theories.AcknowledgementsWe thank WHEPP-6, Chennai, India, where this work was initiated. The work of RNMis supported by the NSF Grant no. PHY-9802551. The work of MKP is supported bythe project No. 98/37/9/BRNS-cell/731 of the Govt. of India. A. D. would like to thankA. De Gouvea for helpful discussions. Balaji wishes to thank S. Uma Sankar for usefulclarications.10REFERENCES[1] Super-Kamiokande Collaboration: Phys. Rev. Lett. 82, 2644 (1999).[2] J. N. Bahcall, P. I. Krastev and A. Yu. Smirnov, Phys. Rev. D58, 096016 (1998); G.L. Fogli, E. Lisi, A. Marrone and G. Scoscia, Phys. Rev. D59, 033001 (1999); M.C.Gonzalez-Garcia, P.C. de Holanda, C. Pena-Garay, J.W.F. Valle, hep-ph/9906469 (tobe published in Nucl. Phys. B).[3] H. Fritzsch and Z. Xing, hep-ph/9903499 ; F. Vissani, hep-ph/9708483 ; G. K. Leontaris,S. Lola, C. Scheich and J. D. Vergados, Phys. Rev. D53, 6381 (1996) ; B. C. Allanach,hep-ph/9806294 ; V. Barger, S. Pakvasa, T. J. Weiler and K. Whisnant, Phys. Lett.B437, 107 (1998); A. J. Baltz, A. S. Goldhaber and M. Goldhaber, Phys. Rev. Lett.81, 5730 (1998); R. N. Mohapatra and S. Nussinov, Phys. Lett. B441, 299 (1998) ; Y.Nomura and Y. Yanagida, Phys. Rev. D59, 017303 (1999) ; S. K. Kang and C. S. Kim,Phys. Rev. D59, 091302 (1999) ; G. Altarelli and F. Feruglio, Phys. Lett. B439, 112(1998) and JHEP 9811, 021 (1998) ; R. Barbieri, L. J. Hall, G. L. Kane and G. G. Ross,hep-ph/9901228 ; A. S. Joshipura and S. Rindani, Phys. Lett. B464, 239 (1999); E.Ma, Phys. Lett. B456, 201 (1999); R. N. Mohapatra, A. Perez Lorenzana and C. Pires,hep-ph/9911395 (To appear in Phys. Lett. B).[4] C. Albright, K. S. Babu and S. Barr, hep-ph/9805266; B. Brahmachari and R. N.Mohapatra, Phys. Rev. D58, 15003 (1998); K. S. Babu, J. C. Pati and F. Wilczek,hep-ph/9812538.[5] K. R. S. Balaji, A. S. Dighe, R. N. Mohapatra and M. K. Parida, hep-ph/0001310.[6] T. K. Kuo and J. Pantaleone, Rev. Mod. Phys. 61, 937 (1989).[7] A. De Rujula, M. B. Gavela and P. Hernandez, hep-ph/0001124.[8] CHOOZ Collaboration: M. Apollonio et al, Phys. Lett. B420, 397 (1998); hep-ex/9907037.11[9] K. S. Babu, C. N. Leung and J. Pantaleone, Phys. Lett. B319, 191 (1993); P. H.Chankowski and Z. Pluciennik, Phys. Lett. B316, 312 (1993); M. Tanimoto, Phys.Lett. B360, 41 (1995).[10] N. Haba, N. Okamura and M. Sugiura, hep-ph/9810471; N. Haba, Y. Matsui, N. Oka-mura and M. Sugiura, hep-ph/9908429; J. Ellis and S. Lola, Phys. Lett. B458, 310(1999) ; J. A. Casas, J. R. Espinosa, A. Ibarra and I. Navarro, hep-ph/9905381.[11] SuperKamiokande Collaboration: Phys. Rev. Lett. 81, 1562 (1998).[12] K. Scholberg, for the SuperKamiokande Collaboration, hep-ex/9905016; T. Kajita, In-vited talk at PASCOS99, December ’99.[13] D. White, for the LSND Collaboration, Nucl. Phys. Proc. Suppl. 77, 207 (1999).[14] D. Caldwell and R. N. Mohapatra, Phys. Rev. D48, 3259 (1993); J. Peltoniemi andJ. W. F. Valle, Nucl. Phys. B406, 409 (1993); S. Bilenky, C. Giunti and W. Grimus,Euro. Phys. J. C1, 247 (1998); Phys. Rev. D57, 1920 (1998); D58, 033001 (1998); E.Ma and P. Roy, Phys. Rev. D52, 4780 (1995).12FIGURES0 20 40 60 8002468101214Ψ Ω(degrees)(degrees)____FIG. 1. The allowed parameter space for Ω and Ψ. The region below the solid line is allowed byCHOOZ, the region between the almost vertical dotted lines is allowed from the atmospheric dataand the region between the dashed lines corresponds to the SMA solution for the solar neutrinos.The shaded region is consistent with all the data.130 20 40 60 8051015202530Ψβ(degrees)__tan FIG. 2. The value of tan  required in MSSM to get a large Ψ from a small Ψ. Thesolid (dashed) curve stands for Ψ = 3o (10o). In each set, the lower (upper) curve denotesm2()=m2() = 2  10−3 (7  10−3) eV2 for the 2-3 pair. We have chosen =  1010.14

Crossref

Publications of the IAS Fellows

Radiative magnification of neutrino mixings and a natural explanation of
  the neutrino anomalies

Radiative magnification of neutrino mixings and a natural explanation of the neutrino anomalies

Abstract

Similar works

Full text

Available Versions

Indian Academy of Sciences

CERN Document Server

Crossref

Publications of the IAS Fellows