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Probing physics beyond the standard model from lepton sector

a r X i v :h e p -p h /0204100v 1 9 A p r 2002

J.Hisano a

a Institute of Cosmic Ray Research,University of Tokyo,Kashiwa 277-8582,Japan b

Theory Group,KEK,Tsukuba 305-0801,Japan

In this review we discuss physics of the lepton sector,the anomalous dipole moment of muon,the charged lepton-?avor violation,and the electric dipole moments of charged leptons,from viewpoints of the minimal supersymmetric standard model and the extensions.

1.Introduction

The Standard Model (SM)is the most success-ful model to explain physics below the weak scale.The recent results for sin 2βor sin 2φ1by Belle [1]and Babar [2]are converging,and they are con-sistent with the Kobayashi-Maskawa mechanism [3].Also,the precision measurements of the elec-troweak parameters suggest the light SM Higgs boson such as m h <～196GeV (95%CL)[4].Now we are waiting for signature of physics beyond the SM.

We have the clue for the physics beyond the SM in the neutrino oscillation experiment re-sults.The atmospheric neutrino result by the superKamiokande experiment is established [5].The combined result of the superKamiokande [6]and SNO [7]experiments shows a strong evidence for the appearance of νμor ντin the solar neu-trino.At present,the large-angle MSW solution is the most favored,which will be checked by the Kamland experiment soon [8].These neutrino os-cillation results can be explained by introduction of small neutrino masses.

Introduction of small neutrino masses gives us a new scenery for physics beyond the SM.While the promising candidate for origin of the small neu-trino masses is the see-saw mechanism [9]by in-troduction of the right-handed neutrinos,the at-mospheric neutrino result implies that the right-handed neutrino mass should be smaller than ～10?15GeV,which is much smaller than the Planck scale.This is a good motivation for introduc-

tion of Supersymmetry (SUSY).A quadratically-divergent contribution to the Higgs boson mass proportional to the right-handed neutrino masses should be canceled even if the correction propor-tional to the Planck mass is vanishing by some mysterious physics.

Nowadays,the minimal supersymmetric stan-dard model (MSSM)is one the most promising extension of standard model (SM),and many ex-periments are searching for the possible evidence of the low-energy supersymmetry.In this review we discuss the possibilities in physics of charged lepton,paying attention to the dipole-moment operators;

L =e m l

try is not exact in nature,and the problem is how large is the charged LFV.The small neutrino masses themselves,expected from the neutrino oscillation results,cannot give any prediction for the charged LFV processes accessible in near fu-ture.In the MSSM,the charged LFV is supplied by the SUSY breaking masses of sleptons,and the magnitude depends on the origin of the SUSY breaking and interaction beyond the MSSM,such as in see-saw mechanism or the supersymmetric grand uni?ed models(SUSY GUTs).

When diagonal parts of L ij and/or R ij have imaginary part,CP is violating and the elec-

tric dipole moments(EDM)are predicted;d l

i =

em l

i Im(R ii?L ii).The EDMs are also supplied by

the SUSY breaking slepton masses in the MSSM. We organize this review as follows.In the next section we summarize the current status of the muon(g?2).In section2we discuss depen-dence of the charged LFV processes on the SUSY breaking models,and show the branching ratios of the charged LFV processes in the supersym-metric see-saw model,using the neutrino oscilla-tion data.Section3is for the EDMs of charged leptons.Section4is summary.

2.Muon anomalous magnetic moment The latest result for the anomalous magnetic moment of muon(BNL’98&’99+CERN’77[10])is a exp

=(116592023±151)×10?11,while the SM

prediction is a SM

μ=(116592768±65)×10?11.

The contents of the SM contribution are listed in Table1.The deviation of the measurement

from the SM prediction is a NP

μ(≡a exp

μ?a SMμ)=

255±164×10?11,and it is1.6σaway.At present the signi?cance of the deviation is small.The experimental error is expected to be improved by a factor2in BNL’00data,and the ultimate goal may be～40×10?11.

Before going to the SUSY contribution to the muon g?2,we review the error in the SM prediction.The largest ambiguities in the SM prediction come from the leading hadronic vac-

uum polarization contribution a Had

μ(VP1)and the

hadronic light-by-light(LbyL)scattering contri-

bution a Had

μ(LbyL).The a Had

(VP1)in Table1is

derived by M.Davier and A.Hocker[12]from the Table1

The SM contribution to the muon g?2.

116584705.7(2.9)

6924(62)

-100(6)

86(19)

195

?43(4)

a SM

e+e?hadronic cross section and the hadronicτdecay data,including perturbative QCD calcula-tion in the high q2part.1This estimation will be further improved by high quality data for the e+e?hadronic cross section by CMD2in Novosi-birsk[19],KLOE in Fascati[20],and BES in Bei-jing[21].The Babar may contribute to it by mea-surement via the initial state radiation of hard photon[22].The CLEO and LEP data for the tau decay are also important.

On the other hand,the estimate of the LbyL scattering contribution relies on the model calcu-lation.The LbyL contribution comes from three type diagrams in Fig.1,and the value in Table1is the average value for the latest results of Refs.[14] and[15];

a Had

(LbyL)=(89.6±15.4)×10?11[14],(2)

a Had

(LbyL)=(83±32)×10?11[15].(3)

The dominant contribution in a Had

(LbyL)comes from the pion-pole diagram.This diagram was reevaluated by several groups[23][14][15],and the sign problem has been?xed now.However, still they rely on the model calculation since the diagram is divergent.They are based on the chiral perturbation or the ENJL model.The vector-meson dominance is assumed and the phe-nomenological parametrization of the pion form factorπγ?γ?is introduced in order to regularize the divergence.

In Ref.[24]the pion-pole contribution is eval-uated in a model-dependent way,based on

the

Figure1.The light-by-light scattering contribu-

tions to the muon g?2.

chiral perturbation theory.The result is follow-

ing;

a Had

(LbyL)|π0pole=3

π 3

mμ

+(1

mμ

+?C ,

whereΛis the ultraviolet cuto?(Λ～4πFπ).The

largest term proportional to log2Λ/mμis?xed by

the gauge invariance and chiral anomaly.χis a

counter term to regularize the two-loop diagram.

While it can be determined by the leptonic de-

cay of the psuedescalar mesons,the sensitivity is

low at present.Furthermore,?C,which is a piece

not enhanced by log,cannot be evaluated with-

out explicit models.The uncertainty due to?C is

δaμ=31×10?11?C.

While the model-dependent calculations(2)

and(3)seem to be converged,we do not have a

strategy to derive the pion-pole contribution pre-

cisely enough in a model-dependent way.Also,we

have a subtle problem in the light-by-light contri-

bution,whether the inclusion of the quark loop is

double-counting or not.Thus,the calculation of

the light-by-light contribution on base of QCD is

strongly desired.

If the hadronic contribution is well-controlled,

the muon g?2is so sensitive to physics beyond

the SM[25],as mentioned in Introduction.Before

closing this section,we discuss it from a viewpoint

of the MSSM.The nature of two Higgs doublet

model in the MSSM can enhance the contribu-

tion,and the contribution proportional to tanβ

[26],which comes from Fig.(2),is given as

a SUSY

μ?

5α2+αY

m2S

tanβ

?1.3×10?9

100GeV

SUSY breaking in the MSSM comes from physics at high energy scale,such as the minimal super-

gravity model,A t～M3where M3is the SU(3)C gaugino mass.If sign of the two gaugino masses

is required to have the same

sign,the anomaly

mediated SUSY breaking model[32]will be dis-

favored[27].The consistency of the muon g?2 and the Yukawa uni?cation(Y t=Y b=Yτ)at the

GUT scale is also interesting since the Yukawa

uni?cation favorsμM3<0[33].

Third is the tanβenhanced processes.If the

muon g?2deviation is observed,it will give nor-malizations of the processes induced by dipole op-

erators.Especially,the LFV processes,such as

μ→eγ,have a direct relation to it[34].The processes generated by the Yukawa coupling may

be also enhanced.For example,the counting rate

of the neutralino dark matter would be enhanced [35].

3.Lepton-?avor violation in the charged-

lepton sector

While the lepton-?avor violation is observed in the neutrino oscillation experiments,this does

not mean sizable LFV processes in the charged-lepton sector exit.The charged LFV processes

induced by the small neutrino masses,expected

from the neutrino oscillation results,are sup-pressed by the GIM mechanism,as Br(μ→

eγ)<～10?48(mν/1eV)4,even if the neutrino mix-ing is maximal.On the other hand,if the SM is supersymmetrized,the situation is changed.The

SUSY breaking slepton masses are not necessary aligned to the lepton masses,and it may lead to sizable lepton-?avor violating.

Let us asuume that(m2?

L )12in the left-handed

slepton mass matrix is non-vanishing.In this case,μ→eγis generated by diagrams in Fig.(3), and the approximate formula is given as Br(μ→

eγ)?3×10?5(a SUSY

μ/10?9)2((m2?

)12/m2S)2[34].

The diagrams in Fig(3)are so similar to the dia-grams in Fig.(2)contributing to the muon g?2, and the muon g?2gives the normalization of the branching ratio ofμ→eγ.

In Table2we summarize the current exper-imental bounds to the charged LFV processes, the sensitivities in the present activities,and Figure3.The contributions in the MSSM to the μ→eγ,which are enhanced by tanβ.Here,it is assumed that the left-handed sleptons have the LFV masses.

prospects in the future experiments such as the PRISM project[45]and the front ends of neu-trino factories under consideration at CERN[40]. The charged LFV processes are radiative-induced in the MSSM as far as the R party is not bro-ken.Thus,the branching ratio ofμ→3e and the μ-e conversion rate in nuclei are approximately given as Br(μ→3e)/Br(μ→eγ)?7×10?3and R(μ?Ti(Al)→e?Ti(Al))/Br(μ→eγ)?5(3)×10?3.(See the detailed calculation of theμ-e con-version rate in nuclei is given in Refs.[46].)From these simple formulas,the naive current bound on (m2?

)12/m2S is<～6×10?4(δa SUSY

/10?9)?1,and PSI and MECO/BNL(PRISM and NuFACT) may reach to～10?5(10?6).These experiments are stringent tests of the lepton-?avor symmetry in the MSSM.

The charged LFV in the MSSM depends on the origin of the SUSY breaking term in the MSSM and the interaction of physics beyond the MSSM.The SUSY breaking model is classi?ed to two types by degeneracy or non-degeneracy of the sfermion masses.The later may predict the large LFV rates and sometimes the broad pa-rameter region has been excluded already.Here, we will concentrate on the SUSY breaking mod-els where the degeneracy of the sfermion masses is predicted by assuming the hidden sector,such as the gravity-[47],gauge-[48],anomaly-[32], gaugino-mediation[49]models.

The magnitude of the charged LFV processes in these models depends on the scale of the SUSY breaking mediation(M M)and the scale of the physics with LFV interaction(M LF V).The well-

5 Table2

Current experimental bounds to the charged LFV processes,the sensitivities in the present activities, and prospects in the future experiments.

Current bound Future

1.0×10?6[36]10?(8?9)[37]

1.2×10?11[38]10?15[40]

1.0×10?12[41]10?(15?16)[40]

6.1×10?13[42]10?18[40][45]

large branching ratio of τ→μγ.In Fig.

(5)we present Br (τ→μγ)in this model.Here,we use m 2

ντ=2×10?3eV 2and U 23=1/

√ 2.

Here we take m νe =0and assume the canon-ical generational structure for the right-handed neutrino masses.For the SUSY breaking pa-rameters,the universal gaugino mass M 1/2=200GeV,m 0=200GeV,and A 0=200GeV.The horizontal line is for the right-handed tau neu-trino mass M N τ.A broad region has been ex-cluded already,and the future experiments may

cover almost the region above M N τ>～1011

GeV.

If M N τ<～O(1011

)GeV,the Dirac mass for the tau neutrino is smaller than O(1)GeV,which is much smaller than the top quark mass.

Figure 6.Br (μ→eγ)in the supersymmet-ric see-saw model,using the atmospheric neutrino result and the large-angle MSW solution.We as-sume the gravity-mediation model.

In this section we discussed the charged LFV.After the SUSY particles are discovered at LHC or lepton colliders,the LFV slepton decay is im-portant [54].The e +e ?linear collider and muon collider have sensitivity for the ?τ-?μmixing beyond the current proposed τ→μγsearch.4.EDMs

In this section we discuss the EDMs of charged leptons.The current experimental bounds and the sensitivities of the future experiments are listed in Table 3.While the EDMs are suppressed

7 in the SM as d e(dμ)<10?40(10?38)e cm,they

are sensitive to the MSSM.The relative phases of the F-term SUSY breaking parameters,the A and B terms and the gaugino masses,con-tribute to the EDMs.In this section,we assume for simplicity that the sfermion masses are?avor-independent and the CP-violating phases of the SUSY breaking parameters are zero at the SUSY-breaking mediation scale,and consider the EDMs radiatively-induced in physics beyond the MSSM. In the minimal SUSY SU(5)GUT,the pre-dicted EDMs are very small.The quark and lepton masses are given by the up-type and down-type quark Yukawa couplings at the GUT scale.As the result,the EDMs of electron and muon are proportional to a Jarskog invariant,～f2b f2c f4t Im[V11V?12V22V?21],where V is the CKM matrix at the GUT scale.This situation is simi-lar to the SM.Thus,the EDMs are suppressed so much.

We know that the minimal SUSY SU(5)GUT cannot explain the quark and lepton masses for the?rst and second generations,and the exten-sion is needed.Also,it does not have the right-handed neutrinos.These extension

may change the prediction for the EDM drastically[51].Let us consider that the SUSY SU(5)GUT with the right-handed neutrinos.In this case,the EDM of electron(muon)may be proportional to a Jarskog

invariant,～f2ν

τf2t Im[V31(2)V?33U1(2)3U?33].Here,

we assume for simplicity that the right-handed neutrino masses are degenerate and U is the MNS matrix at the GUT scale.The relative phases between U and V contribute to the EDMs.In Fig.(7)we show the Br(μ→eγ)and the EDMs of electron and muon.We asuume the maxi-mal CP violating phases.See Ref.[60]for the input parameters in this?gure.Since the left-handed and right-handed sleptons get the LFV

masses as(m2?

L )ij∝U i3U?j3and(m2?E)ij∝V3i V?3j,

Br(μ→eγ)and the EDMs have a strong cor-relation.From this?gure it is found that the prediction may be accessible in the future experi-ments,and U13is an important parameter for the electron EDM.

In the supersymmetric see-saw model,if the right-handed neutrino masses are exactly degen-erate,the EDMs of charged leptons are also

10-10

10-12

10-14

10-16

10-18

10-20

100200300400500

(GeV)

(

)

M =50GeV

Experimental bound

tan =30,10,3

μ > 0

m = 0.07eV

ντ

M =100GeV

d eμ

-26

-27

-31

-29

-30

-28

-26

-27

-25

-28

-24

U13

0.1

Figure7.Br(μ→eγ)and the EDMs of elec-tron and muon in the SUSY SU(5)GUT with the right-handed neutrinos.

suppressed,similar to the minimal SUSY SU(5) GUT.The non-degeneracy of the right-handed neutrino masses may enhance the EDMs dras-tically[59],and the muon(electron)EDM can reach to10?26(10?31)e cm.

Other CP violating observable in the lepton sector is the T-odd asymmetry in the polarized muon decay to3e.While it comes from interfer-ence between the photon-penguin diagram and the Z penguin and box diagrams,the photon-penguin diagram tends to be dominant in the μ→3e process and the T-odd asymmetry is suppressed.In the minimal SUSY SU(5)GUT and the supersymmetric see-saw model the T-odd asymmetry may reach to10%if the photon-penguin contribution is suppressed[61][62].

5.Summary

In this review we discuss physics of the lep-ton sector from viewpoints of the minimal super-symmetric standard and the extensions.While the muon g?2is sensitive to the MSSM,the understanding of the systematic error in the the SM prediction,especially the light-by-light con-tribution,is very serious when the experimental

Table3

The current experimental bounds to the electric dipole moments of charged leptons and the prospects in the future experiments.

Current bound

Future

1.6×10?27e[55]10?33[56] (3.7±3.4)×10?19[57]10?26[40][45]

JH thanks Prof.S.Ohta for comment on the lat-tice calculation of the light-by-light contribution to the muon g?2.

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