PHYSICAL REVIEW D, VOLUME 64, 037701

Anomalous magnetic moment of the muon and radiative lepton decays Xavier Calmet, Harald Fritzsch, and Dirk Holtmannspo¨tter Ludwig-Maximilians-University Munich, Sektion Physik, Theresienstraße 37, D-80333 Munich, Germany 共Received 4 April 2001; published 10 July 2001兲 The leptons are viewed as composite objects, exhibiting anomalous magnetic moments and anomalous flavor-changing transition moments. The decay ␮ →e ␥ is expected to occur with a branching ratio of the same order as the present experimental limit. DOI: 10.1103/PhysRevD.64.037701

PACS number共s兲: 12.60.Rc, 13.10.⫹q, 13.35.Bv

Recently an indication was found that the anomalous magnetic moment of the muon ␮ ⫹ is slightly larger than expected within the standard model 关1兴. The deviation is of the order of 10⫺9 , ⫺9

⌬a ␮ ⫽a ␮ 共 expt兲 ⫺a ␮ 共 SM 兲 ⫽ 共 4.3⫾1.6兲 ⫻10

.

共1兲

For a review of the contribution of the standard model to the anomalous magnetic moment of the muon, see Ref. 关2兴. The observed effect 共2.6 ␴ excess兲 does not necessarily imply a conflict with the standard model, in view of the systematic uncertainties in the theoretical calculations due to the hadronic corrections 关3兴. If this result is confirmed by further experimental data, it might be interpreted as the first signal towards an internal structure of the leptons 共see e.g. Ref. 关4兴兲, although other interpretations 共vertex corrections due to new particles兲 are also possible 关5兴. A new contribution to the magnetic moment of the muon can be described by adding an effective term L e f f to the Lagrangian of the standard model as follows: L⫽

e eff ¯ 共 A⫹B ␥ 5 兲 ␴ ␮ ␯ ␮ F ␮ ␯ , ␮ 2⌳

共2兲

where ␮ is the muon field, F ␮ ␯ the electromagnetic field strength, ⌳ the compositeness scale and A and B are constants of order one. We have included a ␥ 5 -term in view of a possible parity violation of the confining interaction. The constants in L e f f depend on the dynamical details of the underlying composite structure. If the latter is analogous to QCD, where such a term is induced by the hadronic dynamics, the constant is of the order one, and the BNL result would give ⌳⬇2.5⫻107 GeV using ⌬a ␮ ⫽

冉 冊

m␮ , ⌳

共3兲

assuming 兩 A 兩 ⫽1. The ␥ 5 -term does not contribute to the anomalous magnetic moment. The magnetic moment term 共2兲 has the same chiral structure as the lepton mass term. Thus one expects that the same mechanism which leads to the small lepton masses (m ␮ Ⰶ⌳), e.g. a chiral symmetry, leads to a corresponding suppression of the magnetic moment. In this case the effective Lagrangian should be written as follows: L⫽

e eff m ␮ ¯ 共 A⫹B ␥ 5 兲 ␴ ␮ ␯ ␮ F ␮ ␯ . ␮ 2⌳ ⌳

0556-2821/2001/64共3兲/037701共3兲/$20.00

共4兲

The contribution of the compositeness to the magnetic moment is in this case given by ⌬a ␮ ⫽

冉 冊 m␮ ⌳

2

.

共5兲

Using the central value of ⌬a ␮ , one obtains ⌳⬇1.61 TeV, i.e., ⌳ is much smaller due to the chiral symmetry argument 关6兴. The 95% confidence level range for ⌳ is 关4兴 1.22 TeV⬍⌳⬍3.19 TeV.

共6兲

If the leptons have a composite structure, the question arises whether effects which are absent in the standard model, in particular flavor-changing transitions, e.g. the decays ␮ →e ␥ or ␶ → ␮␥ , arise. In this paper we shall study flavor changing magneticmoment type transitions which indeed lead to radiative decays of the charged leptons on a level accessible to experiments in the near future. We start by considering the limit m e ⫽m ␮ ⫽0, i.e., only the third lepton ␶ remains massive. Neutrino masses are not considered. In this limit the mass matrix for the charged leptons has the structure m l ⫺ ⫽m ␶ diag(0,0,1) and exhibits a ‘‘democratic symmetry’’ 关7,8兴. Furthermore there exists a chiral symmetry SU(2) L 丢 SU(2) R acting on the first two lepton flavors. The magnetic moment term induced by compositeness, being of a similar chiral nature as the mass term itself, must respect this symmetry. We obtain L⫽

e eff m ␶ ¯␺ M ˜ 共 A⫹B ␥ 5 兲 ␴ ␮ ␯ ␺ F ␮ ␯ . 2⌳ ⌳

共7兲

˜ is given by M ˜ Here ␺ denotes the vector (e, ␮ , ␶ ) and M ⫽diag(0,0,1). Once the chiral symmetry is broken, the mass matrix receives non-zero entries, and after diagonalization by suitable transformations in the space of the lepton flavors it takes the form M ⫽diag(m e ,m ␮ ,m ␶ ). If after symmetry breaking the ˜ were mass matrix M and the magnetic moment matrix M identical, the same diagonalization procedure which leads to a diagonalized mass matrix would lead to a diagonalized magnetic moment matrix. However there is no reason why ˜ and M should be proportional to each other after symmeM try breaking. The matrix elements of the magnetic moment operator depend on details of the internal structure in a different way than the matrix elements of the mass density operator. Thus in general the magnetic moment operator will

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©2001 The American Physical Society

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PHYSICAL REVIEW D 64 037701

the mass density and the magnetic moment operators due to the internal substructure. If the substructure were turned off (⌳→⬁), the effects should not be present. The simplest ansatz for the transition terms between the leptons flavors i and j is const冑m i m j /⌳. It obeys the constraints mentioned above: it vanishes once the mass of one of the leptons is turned off, it is symmetric between i and j and it vanishes for ⌳→⬁. In this case the magnetic moment operator has the general form

not be diagonal once the mass matrix is diagonalized and vice versa. Thus there exists flavor-non-diagonal terms 共for a discussion of analogous effects for the quarks see Ref. 关7兴兲, e.g., terms proportional to ¯e ␴ ␮ ␯ (A⫹B ␥ 5 ) ␮ . These flavornon-diagonal terms must obey the constraints imposed by the chiral symmetry, i.e., they must disappear once the masses of the light leptons involved are turned off. For example, the e⫺ ␮ transition term must vanish for m e →0. Furthermore the flavor changing terms arise due to a mismatch between

Leff⫽

e m␶ ¯␺ 2⌳ ⌳

冉

me m␶ C e␮ C e␶

C e␮

冑m e m ␮

⌳

⌳ m␮ m␶

⌳

冑m e m ␶

冑m e m ␮

C ␮␶

C ␮␶

冑m ␮ m ␶ ⌳

Here C i j are constants of the order one. In general one may introduce two different matrices 共with different constants C i j ) both for the 1-term and for the ␥ 5 -term, but we shall limit ourselves to the simpler structure given above. Based on the flavor-changing transition terms given in Eq. 共8兲, we can calculate the branching ratios for the decays ␮ →e ␥ , ␶ → ␮␥ and ␶ →e ␥ . We find ⌫ 共 ␮ →e ␥ 兲 ⫽e 2

m␮ 4␲

⌫ 共 ␶ → ␮␥ 兲 ⫽e 2

m␶ 4␲

m␶ ⌫ 共 ␶ →e ␥ 兲 ⫽e 4␲ 2

冉冑 冊 冉 冊 冉 冊 冉冑 冊 冉 冊 冉 冊 冉冑 冊 冉 冊 冉 冊 m ␮m e ⌳

2

m ␶m ␮ ⌳

2

m ␶m e ⌳

2

m␮ ⌳

2

m␶ ⌳

2

m␶ ⌳

2

m␶ ⌳

2

m␶ ⌳

2

m␶ ⌳

2

,

共9兲

,

共10兲

.

共11兲

The corresponding branching ratios are, taking the constants equal to one, Br共 ␮ →e ␥ 兲 ⬇2.8⫻10⫺10,

共12兲

Br共 ␶ → ␮␥ 兲 ⬇6.1⫻10⫺10,

共13兲

Br共 ␶ →e ␥ 兲 ⬇3.0⫻10⫺12,

共14兲

using the central value of ⌬a ␮ to evaluate ⌳. One obtains the following ranges for the branching ratios:

关1兴 Muon 共g⫺2兲 Collaboration, H. N. Brown et al., Phys. Rev. Lett. 86, 2227 共2001兲. 关2兴 A. Czarnecki and W. J. Marciano, Phys. Rev. D 64, 013014 共2001兲; J. Calmet, S. Narison, M. Perrottet, and E. de Rafael,

C e␶

冑m e m ␶ ⌳

冑m ␮ m ␶ ⌳ 1

冊

˜ 共 A⫹B ␥ 5 兲 ␴ ␮ ␯ ␺ F ␮ ␯ . M

共8兲

1.5⫻10⫺9 ⬎Br共 ␮ →e ␥ 兲 ⬎4.6⫻10⫺12,

共15兲

3.3⫻10⫺9 ⬎Br共 ␶ → ␮␥ 兲 ⬎1.0⫻10⫺11,

共16兲

1.6⫻10⫺11⬎Br共 ␶ →e ␥ 兲 ⬎5.0⫻10⫺14,

共17兲

using the 95% confidence level range for ⌳ 共6兲. These ranges are based on the assumption that the constants of order one are fixed to one. The upper part of the range for the ␮ →e ␥ decay given in Eq. 共15兲 is excluded by the present experimental limit Br( ␮ →e ␥ )⬍1.2⫻10⫺11 关9兴. Our estimates of the branching ratio should be viewed as order of magnitude estimates. In general we can say that the branching ratio for the ␮ →e ␥ decay should lie between 10⫺13 and the present limit. The decay ␶ → ␮␥ processes at a level which cannot be observed, at least not in the foreseeable future. The decay ␶ →e ␥ is, as expected, much suppressed compared to ␶ → ␮␥ decay and cannot be seen experimentally. The experiment now under way at the PSI should be able to detect this decay. If it is found, it would be an important milestone towards a deeper understanding of the internal structure of the leptons and quarks. We thank Z. Xing for useful discussions.

Rev. Mod. Phys. 49, 21 共1977兲. 关3兴 F. J. Yndurain, hep-ph/0102312. 关4兴 K. Lane, hep-ph/0102131. 关5兴 L. Everett, G. L. Kane, S. Rigolin, and L. T. Wang, Phys. Rev.

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Lett. 86, 3484 共2001兲; J. L. Feng and K. T. Matchev, ibid. 86, 3480 共2001兲; E. A. Baltz and P. Gondolo, ibid. 86, 5004 共2001兲; D. Chakraverty, D. Choudhury, and A. Datta, Phys. Lett. B 506, 103 共2001兲; T. Huang, Z.-H. Lin, L.-Y. Shan, and X. Zhang, hep-ph/0102193; D. Choudhury, B. Mukhopadhyaya, and S. Rakshit, Phys. Lett. B 507, 219 共2001兲; S. Komine, T. Moroi, and M. Yamaguchi, ibid. 506, 93 共2001兲; S. N. Gninenko and N. V. Krasnikov, hep-ph/0102222; K. Cheung, Phys. Rev. D 64, 033001 共2001兲; P. Das, S. K. Rai, and S. Raychaudhuri, hep-ph/0102242; T. W. Kephart and H. Pa¨s, hep-ph/0102243; E. Ma and M. Raidal, hep-ph/0102255; Z. Xiong and J. M. Yang, hep-ph/0102259; A. Dedes and H. E. Haber, J. High Energy Phys. 05, 006 共2001兲; Z. Xing, Phys.

关6兴 关7兴 关8兴 关9兴

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