Physics Letters B 553 (2003) 277–283

www.elsevier.com/locate/npe

Less suppressedµ→ eγ andτ → µγ loop amplitudes and extradimension theories

Bo Hea,1, T.P. Chenga, Ling-Fong Lib

a Department of Physics and Astronomy, University of Missouri, St. Louis, MO 63121, USAb Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Received 27 September 2002; received in revised form 15 November 2002; accepted 18 December 2002

Editor: H. Georgi

Abstract

Whenµeγ (or τµγ ) loop involves a vector boson, the amplitude is suppressed by more than two powers of heavy particlemasses. However we show that the scalar boson loop diagrams are much less damped. Particularly, the loop amplitude inwhich the intermediate fermion and scalar boson have comparable masses is as large as possible, as allowed by the decouplingtheorem. Such a situation is realized in the “universal extra dimension theory”, and can yield a large enough rate forµeγ to bedetectable in current experiments. Our investigation involves precise calculation of the scalar boson loop’s dependence on themasses of the intermediate states. 2002 Elsevier Science B.V. All rights reserved.

PACS: 13.35.Bv; 14.80.Cp; 12.15.Lk; 11.10.Kk

Keywords: Lepton-flavor; Muon; Tau; Extra-dimension; Scalar-boson

1. Introduction

1.1. Probing the physics beyond the SM through theµeγ loop effect

The Standard Model with massless neutrinos auto-matically conserves lepton flavors: the electron, muonand tau numbers. The ever stronger experimental ev-idence for neutrino oscillation [1] shows clearly thatlepton flavor is not conserved in nature. If we accom-

E-mail address: tpcheng@umsl.edu (T.P. Cheng).1 Present address: Department of Electric Engineering, Ohio

State University, Columbus, OH 43210, USA.

modate this feature simply by an introduction of neu-trino masses in the SM, other lepton flavor violatingprocesses such asµ → eγ would still have so smalla rate (a branching ratio� 10−40) that there is nohope for their detection in the foreseeable future. Thisis the case whether we have Dirac neutrino masseswith their small values inserted by hand, or the neutri-nos are Majorana particles with the smallness of theirmasses coming out of the seesaw mechanism. In thesmall Dirac mass scenario, theµeγ amplitude is sup-pressed by the neutrino mass differenceδm2

ν over thevector boson massM2

W , namely, suppressed by a lep-tonic GIM scheme [2]. In the seesaw scenario, super-heavy singlet neutrino states are present in the usual

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.doi:10.1016/S0370-2693(02)03258-6

278 B. He et al. / Physics Letters B 553 (2003) 277–283

left-handed flavor eigenstates. Their potentially signif-icant contribution is nevertheless muffled by the mass-suppressed mixing angles [3]. Thus a detection of theµeγ process would signal the physics beyond the SM,beyond any neutrino mixing mechanism.

We are particularly interested in the possibility ofthisµeγ amplitude being less damped in theories thatpredict new particles around TeV [3,4]. One is curi-ous whether the contribution by such heavy particlesto theµeγ amplitude could be as large as possiblewhile remaining compatible with the decoupling the-orem [5]. For example, the suppressions mentioned inthe last paragraph both involve at least two powers ofheavy particle mass in the denominator. Are there sit-uations in which the amplitude suppression is linear?In this Letter we report a precise calculation of theµeγ amplitude’s dependence on its intermediate parti-cle masses for the scalar boson loop diagrams. It showsthat, when the intermediate fermion and boson massesare comparable, such a potentially detectableµeγ rateis possible. While our result is applicable in any theorythat allows this lepton flavor nonconserving decay, inthis Letter we present it mainly in the context of “largeextra dimension theories” as their phenomenology haslately been under active discussion.

1.2. Large extra dimension theories

In the last few years there has been considerableinterest in theories having extra dimensions, whicheither have their compactification scales being muchlarger than the Planck’s length [6], or have strong cur-vature [7]. These theories can in principle generatethe observed gauge hierarchy, for example, by havinglarge extra-dimensional volume. Such theoretical sug-gestion would also provide an added impetus for theongoing experimental effort to test Newton’s gravitytheory at the millimeter scale.

The hallmark of extra dimensions is the existenceof Kaluza–Klein (KK) states: particles that can propa-gate in the compactified extra dimension have a towerof states with identical quantum numbers but ever in-creasing masses. For the simplest case of a scalar par-ticle in a five-dimensional spacetime, with the extra di-mension compactified into a circle (radiusR), we canexpand the fieldφ(xµ, x5) in harmonics

(1)φ(xµ, x5) =

∑n

φn(xµ

)eip5x

5.

Because the extra dimension is a circle, the positionsx5 and x5 + 2πR are identified, withφ(xµ, x5 +2πR) = φ(xµ, x5) leading to the quantization of theextra dimension momentum:p5 = n/R, wheren =0,1,2, . . . is the KK number. This implies a massspectrum ofM2(n)=M2(0)+n2/R2, where the zeromode bare massM(0) is expected to be much smallerthan the KK excitation energy of 1/R. Thus therewould be a tower of KK states associated with anyparticle that can propagate in an extra dimension.

1.3. Brane vs bulk particles: the universal extradimension theory

Different extra dimension theories often have dif-ferent particle assignments vis-a-vis whether they canpropagate in the full higher-dimensional space or not.Those do are “bulk particles”, and have associated KKstates, while the “brane particles” are those confined(on the brane) to the four-dimensional spacetime. Theoriginal large dimension theory has all the SM parti-cles stuck on the brane, and only graviton is a bulkfield [6]. Prior investigation [8] and later variations in-clude suggestions in which the neutrinos [9], the scalarbosons, or the vector gauge bosons [10], etc. propa-gate in the extra dimensions as well. Furthermore, withthe presence of brane, the translational invariance inthe extra-dimensional space is broken, and the corre-sponding extra dimension momentum (the KK num-ber) is not conserved.

Among the different modifications of the originallarge extra dimension model, the most appealing sug-gestion, to our thinking, has been that by Appelquist,Cheng and Dobrescu [11] who proposed thatall stan-dard model particles, as well as graviton, can prop-agate in the extra dimensions, thus all particles haveKK states. These are “universal extra dimensions”.Because brane’s presence is no longer required, trans-lational invariance, and KK number conservation, arerestored. There are no vertices involving only onenon-zero KK mode. This is the key feature that al-lows such a large extra-dimensional theory to pass allthe phenomenological tests. Appelquist et al. analyzedthe current electroweak data, computed some para-meters and concluded that the compactification sizeR could be 1/300 GeV for one extra dimension and1/400→ 1/800 GeV for two extra dimensions. Thesepredictions are in the range of current or near-future

B. He et al. / Physics Letters B 553 (2003) 277–283 279

experiments. Other discussions about the experimentsignatures of the universal extra dimensions can alsobe found in the literature [12]

1.4. KK particles in the µeγ loop

We are interested in estimating theµeγ rate asinduced by the loop diagram in which the intermediatevirtual particles (one fermion and one boson) may beKK states. We shall broadly distinguish two categoriesof models: in one category, theories do not have KKnumber conservation, and in such models it is possiblethat only one of the virtual particles is a heavy KKstate. Here the general situation corresponds to masslimits when either the fermion mass is much largerthan the boson mass, or the other way around. Inthe second category, we consider the KK numberconserving universal extra dimension theory. Hereboth the fermion and boson must have the same KKnumber and their masses are comparable. This is so,because their mass square difference is the same asthat between their zero modes, which is expected tobe much smaller than the KK excitation. In orderto consider the comparable mass case, we will haveto compute the one loop amplitude exactly in itsdependence of the intermediate particle mass ratio.

The gauge invariant decay amplitude forµ(p) →e(p− q)+ γ (q, ε) must have the form

T (µeγ )= ie

16πε∗λ(q)ue(p− q)σλρqρ

(2)× [A+(1+ γ5)+A−(1− γ5)

]uµ(p).

This corresponds to a dimension-five Lagrangian den-sity termψeσ

λρψµFλρ. The invariant amplitudesA±are induced by finite and calculable loop diagrams andproportional to an inverse mass power.

2. Vector loop amplitude

In the SM with one doublet of Higgs bosons andsmall Dirac neutrino masses, there is only one typeof loop diagrams (Fig. 1) for theµeγ decay. Theyhave a charged intermediate boson in the loop:µ− →(νiW

−γ ) → e− where the photon is emitted by the

chargedW boson in the loop (as denoted by thesubscriptγ ).

Fig. 1. µ → eγ as mediated by a vector loop. Contributions bydiagrams with photon emitted by the external leptons must also beincluded in the calculation.

The required exact mass calculation has been per-formed [3,4,13] giving, in theme = 0 approximation,the amplitudes ofAW− = 0 and

(3)AW+ = g2mµ

8πM2W

3∑i=1

U∗µiUeiF

(m2i

M2W

),

where the function2

F(z)= 1

6(1− z)4

(10− 43z+ 78z2 − 49z3

(4)+ 18z3 ln z+ 4z4)has limits

(5)F(z→ ∞)� 2

3+ 3

ln z

z,

(6)F(z→ 0)� 5

3− 1

2z,

(7)F(z→ 1)� 17

12+ 3

20(1− z).

The resultant amplitudes, after using the unitaritycondition of the mixing matrices

∑3i=1 U

∗eiUµi = 0,

are summarized in Table 1.The limitmi �MW is relevant to models in which

the neutrino is a bulk field while the vector bosonW is not. The amplitudeAW+ is suppressed by theheavy mass as(mµ/m

2i ) lnmi . Themi � MW case

includes the specific situation of massless neutrinosmi = 0, which leads to a vanishing amplitude, aslepton flavor must be conserved in the masslessneutrino limit. For neutrinos with small (zero mode)masses(mi)0 � (MW)0, as well as the case whenonly W has KK states, the amplitude is proportional

2 The sign in front of the log term was incorrectly written downin [3].

280 B. He et al. / Physics Letters B 553 (2003) 277–283

Table 1The vector loop amplitudesAW+

Limits AW+

mi �MW3g2mµ

8π∑3i=1

(1m2i

lnm2i

M2W

)U

∗µi

Uei

mi �MW − g2mµ

16πM4W

∑3i=1m

2i U

∗µiUei

mi �MW3g2mµ

160πM2W

∑3i=1

M2W

−m2i

M2W

U∗µiUei

to neutrino mass-squaredmµm2i /M

4W . This results in

a branching ratio so small that the decay cannot bedetected experimentally in the foreseeable future. Onemight think that the situation ofmi �MW could offera better chance of having a less suppressed amplitude.But this turns out not to be so. Since, for a given KKnumber, the mass-square-difference is given by that ofthe zero modes(M2

W − m2i )n =0 = (M2

W − m2i )0, the

amplitude is again suppressed bymµ(m2i )0/(M

4W)n

leading to an undetectably small branching ratio.

3. Scalar loop amplitude

Although the minimal SM needs only one doubletof Higgs particles, in most extensions more Higgs dou-blets are introduced. For example, in supersymmetrytwo Higgs doublets with opposite hypercharges are re-quired. Several versions of compactification of super-string theory leads to E6 grand unified theories whereeach generation of leptons and quarks has a pair of op-positely hypercharged Higgs scalar boson. Thus thereis strong motivation to consider theories with multi-ples of scalar bosons. Here we are interested how suchscalars, and their possible KK excitations, can con-tribute to theµeγ loop amplitude [14]. Just as in thevector loop case, there is the need to obtain the exactintermediate mass dependence. We have performedthis task and obtained the following result.

In Fig. 2(a) we have intermediate states of acharged scalar boson and a neutrino. Denote theYukawa couplings of the scalar bosonφ to leptonsliandlj by y±

li ,

Γ (φli lj )= li[y+ij (1+ γ5)+ y−

ij (1− γ5)]ljφ

(8)+ h.c.

(a)

(b)

Fig. 2.µ→ eγ as mediated by a loop having (a) a charged scalar,and (b) a neutral scalar, boson.

we find the amplitudes to be

Aφ+(a)= −

∑i

mµ

πM2φ

y+eiy

−µiG(r)

(9)−∑i

mi

πM2φ

y+eiy

+µiI (r),

Aφ−(a)= −

∑i

mµ

πM2φ

y−eiy

+µiG(r)

(10)−∑i

mi

πM2φ

y−eiy

−µiI (r),

wherer ≡ m2i

M2φ

− 1 and the two functions being:

(11)G(r)= 1

3r+ 3

2r2+ 1

r3− (1+ r)2

r4ln(1+ r),

I (r)= 1

r+ 2

r2− 2

r2ln(1+ r)

(12)− 2

r3ln(1+ r).

In Fig. 2(b) we have intermediate states of a neutralscalar boson and a charged lepton. The amplitudes are

Aφ+(b)=

∑i

mµ

πM2φ

y+eiy

−µiH(r)

(13)+∑i

mi

πM2φ

y+eiy

+µiK(r),

B. He et al. / Physics Letters B 553 (2003) 277–283 281

Table 2The scalar loop amplitudesA+

Limits A+ A(sum)+

mi �Mφ∑i

1πmi

y+ei y

+µi 36.8

∑i

1πmi(1)

y+ei y

+µi

mi �Mφ 4∑i ln

(Mφmi

)mi

πM2φ

y+ei y

+µi

2π3

∑i ln

[Mφ(1)

mi

]mi

M2φ(1)y+ei y

+µi

mi ≈Mφ13

∑i

1πMφ

y+eiy+µi

12.47∑i

1πMφ(1)

y+eiy+µi

Aφ−(b)=

∑i

mµ

πM2φ

y−eiy

+µiH(r)

(14)+∑i

mi

πM2φ

y−eiy

−µiK(r),

where the two functions are

(15)H(r)= 1

6r− 1

2r2 − 1

r3 + 1+ r

r4 ln(1+ r),

(16)K(r)= 1

r− 2

r2 + 2

r3 ln(1+ r).

We shall from now ignore theA− amplitudes asthey are similar to theA+ results. After making thesimplifying assumption that the masses of the chargedscalar bosonMφ and neutral leptonmi of Eq. (9)are the same as those for the neutral scalar bosonand charged lepton of Eq. (13), we add the twoamplitudes from (9) and (13):Aφ+(a)+A

φ+(b)= A+.

Various mass limits as shown in Eqs. (5)–(7) can betaken in a straightforward manner and we display theresults3 in Table 2. We have also listed, in the thirdcolumn, the results when we sum over the contributionof the whole tower of KK states [15] according tothe simple one-extra-dimension formula ofM(n) =n/R = nM(1) when M(0) � 0. Our purpose is todemonstrate that no qualitatively new feature appearsin such amplitude sums, which give overall numericalcoefficients and retain the same mass dependences.

The amplitudeA+ in themi �Mφ case, as well inthemi �Mφ limit, has only one power of heavy massin the denominator—they are as large as allowed bythe decoupling theorem, which requires the amplitudeto vanish when the heavy mass approaches infinity.

3 Subleading terms are dropped from the results in Table 2. Also,in themi �Mφ amplitude, the leading 1/mi terms from Fig. 2(a)and (b) actually cancel if the Yukawa couplings for the charged andneutral scalar bosons are identical. Since there is no reason to expectsuch an equality, we keep one of these dominant terms.

Themi �Mφ amplitude is somewhat more damped,by the heavy scalar massMφ asmiM

−2φ lnMφ .

4. Discussion

4.1. The chiral symmetry perspective

The structure of theµeγ amplitude in Eq. (2), sym-bolically written asψLσψRF or ψRσψLF, involvesflipping the fermion chirality. Thus the amplitude mustbe proportional to a fermion mass. Before discussingdetails, we observe that if we have bulk leptons prop-agating in the extra dimensions, then chiral symmetryis broken in the effective four-dimensional theory bytheir Kaluza–Klein states, which are necessarily mas-sive. Having a large chiral symmetry breaking, suchtheories offer from the outset the possibility for a lesssuppressedµeγ amplitude. This statement is validwhether the higher-dimensional theory has chiral sym-metry or not. Our basic assumption is that the zeromode fermion masses are negligibly small comparedto their KK excitation. The largest possible fermionmass that can bring about the chirality change in theµeγ amplitude is different in the vector and scalarloop diagrams.

For the vector loop contribution, we have assumedthat there are only left-handed charged current cou-plings.4 In such a situation, helicity change takes placeon the external lepton lines—hence the relevant massis that of the muon. Since the amplitude correspondsto a dimension-five operator, it must have an overalldimension of inverse-mass. Thus, in the vector loopamplitude, we expect a damping factor ofmµ/M

2W as

4 We have not considered neutral vector loop case as suchdiagrams would involve further suppressions at the flavor-changingvertices.

282 B. He et al. / Physics Letters B 553 (2003) 277–283

shown in Eq. (3). If this was the principal suppressionfactor, the resultant amplitude and decay rate wouldstill be large. The unitarity condition for the mixingmatrices

∑3i=1 U

∗eiUµi = 0 causes the actual ampli-

tude to be much more damped as the subleading termof the F -function in Eq. (4) generally has two ad-ditional powers of heavy masses in the denominator.Namely, a form of GIM mechanism [16] is operativehere.

The scalar loop amplitudes are less suppressedfor two reasons: (1) here the necessary chiralitychange can be effected by the large intermediatelepton mass, and (2) in the scalar case there is nocancellation mechanism

∑3i=1 U

∗eiUµi = 0, as in the

vector category, to further suppress the amplitude.For themi � Mφ case, the heavy lepton massmi

in the numerator flips the helicity, and its propagatorprovides two powers ofmi in the denominator, givingan over all 1/mi suppression. For themi ≈Mφ case,either the scalar or lepton propagator can provide themass power in the denominator. Because their massesare comparable, the resultant suppression is again1/mi.

4.2. A numerical estimate of the µeγ branching ratio

We find it particularly interesting that theµeγ am-plitude can be less suppressed when the intermedi-ate scalar and lepton masses are comparable, leadingto a possibly observable decay rate. From our expe-rience with the SM, we expect the Yukawa couplingto be small, on the order of gauge coupling times the(zero mode) mass ratio of lepton over gauge boson.In particular, there has been the suggestion of neutralscalar’s coupling to two charged fermions (i and j )being on the order ofg

√mimj/MW . This coupling

ansatz [17] has been studied extensively, and found tobe compatible with known phenomenology. With thisestimate of the Yukawa strength, the loop in Fig. 2(b)with the intermediate states being the KK states of atau-lepton and a scalar boson would yield a branchingratio of

B(µeγ )� α

πg4

(M4W

m2µM

2φ

)(y+eτ y

+τµ

)2

(17)� α

π

(me

mµ

)(mτ

Mφ

)2

.

For the first excited KK state withMφ = O (TeV)andme,µ,τ being zero mode lepton masses, Eq. (17)gives aB(µeγ ) =O(10−11), which is comparable tothe current experimental limit [18] ofB(µeγ )� 1.2×10−11. This means that it is entirely conceivable thatthe rate predicted by the scalar loop effect is withinreach of experimental detection in the near future [19].

4.3. Tau decays

The considerations in this Letter can be applieddirectly to radiative decays of the tau lepton:τ → µγ

andτ → eγ . To the extent that the final lepton massbeing negligibly small compared to the initial leptonmass, the tau decay results involve replacing the muonmassmµ in equations such as Eq. (3) by the tau leptonmassmτ . Thus an estimate of the branching ratio fortheτµγ decay yields

B(τµγ )� 0.17α

πg4

(M4W

m2τM

2φ

)(y+µτ y

+ττ

)2

(18)� 0.17α

π

(mµmτ

M2φ

)

(19)=O(10−10),

where 0.17 is the branching ratioB(τ → µνν) andwe have used the estimates ofy+

ττ = gmτ/MW andy+µτ = g

√mµmτ/MW. Since the existing limit [20]

B(τµγ ) = 1.1 × 10−6 is still much larger than thisestimate, it suggests that such a discovery would onlycome about after a significant increase in experimentaldetection efficiency has been achieved. Since ourmodel estimate suggestsB(τeγ ) � B(τµγ ), it iseven less likely that the decay [21]τ → eγ would beuncovered any time soon.

4.4. Conclusion

We have investigated the dependence by theµeγ

amplitude on heavy particle masses, finding markeddifference between vector and scalar loop contribu-tions. The vector loop amplitude, even in the compa-rable mass case, is strongly suppressed by powers ofneutrino mass divided by the heavy mass, leading tosuch a small rate that the decay is predicted to be un-observable in the foreseeable future. We have calcu-lated the precise mass-dependence of the scalar loops.

B. He et al. / Physics Letters B 553 (2003) 277–283 283

The loop amplitude with a single heavy fermion is lesssuppressed, not surprisingly, with one power of heavyfermion mass in the denominator. Interestingly, scalarloop amplitudes with approximately equal intermedi-ate scalar and fermion masses (as, for example, thecase in the universal extra dimension theory) are alsoless suppressed. Calculation using a plausible modelof Yukawa couplings shows that such linearly dampedamplitudes can lead to a decay rate accessible by thenext generation ofµeγ experiments.

Acknowledgements

B.H., T.P.C. and L.F.L. acknowledge the respec-tive support from the University of Missouri ResearchBoard and from US Department of Energy (GrantNo. DE-FG 02-91 ER 40682). T.P.C. also would like tothank Julia Thompson, Simon Eidelman, David Kraus,and Peter Truoel for information on the current experi-mental proposals for the relevant decay processed dis-cussed in this Letter.

References

[1] Super-Kamiokande Collaboration, Y. Fukuda, et al., Phys. Rev.Lett. 81 (1998) 1562;SNO Collaboration, Q.R. Ahmad, et al., Phys. Rev. Lett. 89(2002) 011301.

[2] S.T. Petcov, Sov. J. Nucl. Phys. 25 (1977) 340;T.-P. Cheng, L.-F. Li, Phys. Rev. Lett. 38 (1977) 381.

[3] T.-P. Cheng, L.-F. Li, Phys. Lett. B 502 (2001) 152;T.-P. Cheng, L.-F. Li, Phys. Rev. D 44 (1991) 1502.

[4] R. Kitano, Phys. Lett. B 481 (2000) 39.[5] T. Appelquist, J. Carazzone, Phys. Rev. D 11 (1975) 2856.[6] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429

(1998) 263;I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali,Phys. Lett. B 436 (1998) 257.

[7] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370.[8] I. Antoniadis, Phys. Lett. B 246 (1990) 377.[9] K.R. Dienes, E. Dudas, T. Gherghetta, Nucl. Phys. B 557

(1999) 25;N. Arkani-Hamed, S. Dimopoulos, G. Dvali, J. March-Russell,Phys. Rev. D 65 (2002) 024032;Y. Grossman, M. Neubert, Phys. Lett. B 474 (2000) 361.

[10] W.D. Goldberger, M.B. Wise, Phys. Rev. Lett. 83 (1999) 4922;A. Pomarol, Phys. Lett. B 486 (2000) 153;H. Davoudiasl, J.L. Hewett, T.G. Rizzo, Phys. Lett. B 473(2000) 43.

[11] T. Appelquist, H.-C. Cheng, B.A. Dobrescu, Phys. Rev. D 64(2001) 035002.

[12] T.G. Rizzo, Phys. Rev. D 64 (2001) 095010;B.A. Dobrescu, E. Pontón, Phys. Rev. Lett. 87 (2001) 031801;T. Appelquist, B.A. Dobrescu, E. Pontón, H. Yee, Phys. Rev.Lett. 87 (2001) 181802;T. Appelquist, B.A. Dobrescu, Phys. Lett. B 516 (2001) 85;K. Agashe, N.G. Deshpande, G.H. Wu, Phys. Lett. B 514(2001) 309;C. Macesanu, C.D. McMullen, S. Nandi, Phys. Rev. D 66(2002) 015009;J. Hewett, M. Spiropulu, hep-ph/0205106, Annu. Rev. Nucl.Part. Phys., in press.

[13] T.-P. Cheng, L.-F. Li, in: Gauge Theory of Elementary Parti-cle Physics—Problems and Solutions, Oxford Univ. Press, Ox-ford, 2000, p. 244.

[14] J.D. Bjorken, S. Weinberg, Phys. Rev. Lett. 38 (1977) 622;B. McWilliams, L.-F. Li, Nucl. Phys. B 179 (1981) 62.

[15] A. Ioannisian, A. Pilaftsis, Phys. Rev. D 62 (2000) 066001;A.E. Faraggi, M. Pospelov, Phys. Lett. B 458 (1999) 237;G.C. McLaughlin, J.N. Ng, Phys. Lett. B 493 (2000) 88.

[16] S.L. Glashow, J. Iliopoulos, L. Maiani, Phys. Rev. D 2 (1970)1285.

[17] T.-P. Cheng, M. Sher, Phys. Rev. D 35 (1987) 3484.[18] M.L. Brooks, et al., Phys. Rev. Lett. 83 (1999) 1521, hep-

ex/9905013.[19] See, for example, MEGA Collaboration, S. Ritt, et al., A re-

search proposal PSI-R-99-05, Search forµ+ → e+γ down to10−14 branching ratio,http://meg.psi.ch.

[20] CLEO Collaboration, S. Ahmed, et al., Phys. Rev. D 61 (2000)071101R.

[21] CLEO Collaboration, K.W. Edwards, et al., Phys. Rev. D 55(1997) R3919.