M uon as a Probe of New Physics

1. Muon as a Probe of New Physics O.M.Boyarkin and G.G. Boyarkina BGPU, Minsk The neutrino plays a key role in particle physics and cosmology. For example, neutrino oscillations, whose existence was conclusively confirmed at the SNO and KamLAND neutrino telescopes, are a source of a number of exotic phenomena in the lepton sector, such as resonance transitions, partial and complete flavor violation, CP and CPT violation. Gnd Unified Theory Grand Unified Theory M. Raidal, Phys.Rev.Lett. 93, 161801 (2004).

2. The discovery that the neutrino has a mass makes it a natural candidate for the role of a hot-dark-matter particle and enables us to estimate the mean matter density of the Universe, the age of the Universe, and its further fate. Not only does the introduction of heavy neutrinos in the theory help us to explain the smallness of the mass of the left-handed neutrinos via the "see-saw" mechanism, but this also makes it possible to relate baryogenesis with leptogenesis. Also, the neutrino is widely in applied investigations (O. M . Boyarkin, Physics of Massive Neutrinos, 2nd ed. (URSS,Moscow,2006) [in Russian]). Neutrino astronomy Neutrino geotomography However, neutrino astronomy and neutrino geotomography are both still in their infancy and will reach maturity only after the properties of the neutrinos are conclusively established and after high resolution neutrino telescopes are created.

3. The SM supplemented with an right handed neutrino singlet Let’s focus our attention upon the neutrino dipole magnetic moment for the electron neutrino Laboratory experiments give M.C.Gonzalez-Garcia and M.Maltoni, Phys.Rep. 460, 1 (2008). A global fit to data on solar neutrinos that is supplemented with the model of the Sun's magnetic field yields By introducing extra particles in the SM, we may increase the neutrino MM to values on the same order of magnitude as the upper experimental limits. For example, the SM supplemented with a singly charged Higgs boson possessing zero vacuum expectation value gives

4. gauge group Physical Higgs bosons R.N . Mohapatra and J.C. Pati, Phys. Rev. D 11, 566 (1975); R.N. Mohapatra and G. Senjanovic, Phys. Rev.D 23, 165 (1981); O.M.Boyarkin, Phys.Rev. D 50, 2247 (1994). The left-right model involves a set of extra particles not present in the SM. Physical Higgs bosons Available experimental data admit that some of these bosons have masses at the electroweak scale and that their couplings are on the same order of magnitude as the electroweak couplings.

5. 0 and It was shown that one of these heavy neutrinos can have a mass at the electroweak scale ( O.M.Boyarkin,G.G. Boyarkina, Phys.Rev . D70, 113010 (2004)) It is obvious that, in the left-right model, the contributions of the aforementioned particles lead to changes in the neutrino dipole magnetic moment R.E. Shrock, Nucl.Phys. B 206, 359 (1982); M.A. B.Beg,W.J.Marciano, and M.Ruderman, Phys.Rev. D 17, 1395 (1978)

6. Setting we then obtain the constraints In the left-right model, the neutrino can also possess a Dirac nature. Upon spontaneous symmetry breaking, the theory has two singly charged and six neutral physical Higgs bosons. We note that there are presently no experimental limits on the neutrino dipole magnetic moment of heavy neutrinos.

7. The objective of the present study is to examine the effect of the neutrino dipole magnetic moment on the properties of the muon In microscopic physics, interest in muons is associated primarily with the construction of muon colliders. First Muon Collider, will make it possible to reach a c.m. energy of at a luminosity of Next Muon Collider ---- , O.M. Boyarkin, Introduction to Physics of Elementary Particles, New York, 2007.

8. 2. Radiative decay of the muon Without allowance for the electromagnetic properties of the neutrino, these decay widths were calculated within the Standard Model T.P.Cheng and L.F.Li, Phys.Rev.D 16, 1425 (1977); S.T.Petkov,Yad. Fiz. 25, 641 (1977) in various supersymmetric Grand Unified Theories F.Borzumati and A.Masiero, Phys. Rev. Lett. 57, 961 (1986); K.Huitu, J.Maalampi, M.Raidal, and A.Santamaria, Phys.Lett.B 430,355 (1998), and in the left-right model involving a Majorana neutrino A.G.Akeroyd, M.Aoki, and Y.Okada, Phys.Rev.D 76, 013004(2007).

9. to MEGA Collaboration was organized at the Los Alamos Meson Physics Facility (LAMPF, USA) Since 2006, MEG Collaboration at the Paul Scherrer Institut (Switzerland) has been performing the investigations of this decay. The goal of the MEG experiment is to improve the sensitivity in measuring to If the radiative muon decay is detected in that experiment, it will be possible to measure the angular distribution of decay products and to obtain thereby additional criteria for establishing a true model of electroweak interactions.

10. We assume the Dirac nature of a neutrino where We assume the Dirac nature of a neutrino Fig.1

11. For the sake of simplicity, we assume that only the electron and muon neutrinos are mixed. Let us consider the vertex function which describes the diagram involving the exchange of boson and a light neutrino. It is obvious that, since theories involving spontaneous symmetry breaking are renorma- lizable, the vertex function (20) must be finite. Indeed, the total bare Lagrangian of the left-right model does not involve any multipole moments of the neutrino. These moments arise only owing to radiative corrections. We employ a simpler method --- namely, we perform calculations in the unitary gauge, where one takes into account the contributions of physical particles and uses Dyson's procedure, in which the expansion of the integrand in a power series in external momenta is followed by the subtraction of divergent quanti- ties.

12. (3) The results of the calculations show that a dominant contribution comes from the diagrams in which the virtual lines are associated with heavy neutrinos. Moreover, only the diagrams involving -boson exchange contribute substantially among all diagrams of the type in Fig.1 (3)

13. In the left-right model, radiative muon decay also receives contributions from the diagrams in Fig.2. Among these, dominant contributions come from the diagrams involving heavy-neutrino exchanges. Fig.2 Fig.2

14. The corresponding decay width is given by (4) (4) Because the sign of is opposite to the sign of , the interference between the diagrams associated with charged gauge bosons and the diagrams generated by singly charged Higgs bosons is destructive.

15. In order to find the expression for the radiative decay width of the muon in the SM supplemented with an right-handed neutrino singlet, we must set to zero in (4) and perform the following substitutions there: Setting we obtain a hopelessly small value of branching ratio The inclusion of the neutrino MM does not change situation. Indeed, one can easily see that, upon making in (3) the substitutions andsetting the result becomes Using the equations obtained, one can find the expression for the radiative decay width in the model relying on the same gauge group as the SM and involving two doublets of Higgs fields (two Higgs doublet model) An analysis shows that the branching ratio for radiative muon decay is so small that this process cannot be detected experimentally. It follows that, from the point of view of the SM involving a Dirac neutrino, the radiative muon decay is an unobservable effect.

16. It turns out that, at identical values of the model parameters, the contributions of gauge bosons are on average four orders of magnitude greater than those of the neutrino dipole magnetic moment. In order to obtain values in the range it is necessary in this case to have quasidegeneracy in the heavy neutrino sector. If, for example, the upper experimental limit on the branching ratio for radiative muon decay is obtained at and

17. However, there is one important case where the contributions of the neutrino dipole magnetic moment play a leading role. Since and have opposite signs, they can cancel each other. One can readily show that, under the condition the width vanishes. In this case, an analysis reveals that, if only the term is taken into account in (3), a value of or less can be obtained for branching ratio for radiative muon decay in the experimentally allowed region of the parameters of the left-right model. But if is on the same order of magnitude as , may reach values of order O.M.Boyarkin, G.G.Boyarkina, Phys. Atom. Nuclei, 72, 607 (2009).

18. 3.Anomalous magnetic moment of the muon Investigations of the dipole magnetic moment of particles has always given impetus to the development of microscopic physics. For example, precise measurements of the anomalous magnetic moment of the electron, together with measurements of the hyperfine structure of hydrogen and the Lamb shift, played an important role in the development of QED and the theory of renor-malizations. The detection of nucleon anomalous magnetic moments proved to be a compelling argument in favor of the Yukawa π-meson theory of nuclear forces. It was expected that measurements of the muon AMM would also be a crucial point in particle physics.

19. A compilation of the major experimental efforts in measuringmuon AMM over the last five decades is given in Table 1.

20. In order to take full advantage of the experimental precision achieved in the E-0821 experiment, the contributions to the muon AMM from all sectors of the SM (or its extensions), must be calculated approximately to the same degree of precision. (5) J. P. Miller, E. de Rafael, and B. L. Roberts, Rep. Prog. Phys. 70, 795 (2007) Introducing the quantity we arrive at Thus, the SM predictions prove to be 3.4σbelow the experimental value. Similar calculations based on the SM and performed in F. Jegerlehner, hep-ph/0703125; M. Davier, Nucl. Phys. B (Proc.Suppl.) 169, 288 (2007); K. Hagiwara et al., Phys. Lett. B 649, 173 (2007) yield somewhat different results, but all of them lead to deviations from the BNL result in excess of 3σ.

21. (J. P. Leveille, Nucl. Phys. B 137, 63 (1978) ) The results of previous investigations of the (g-2)µ anomaly within the the LRM An analysis of the effect of the gauge and bosons revealed that the - boson contributions are negative; at the same time, we know that, for the result of the E-0821 experiment to be explained, the -boson mass must be about 100 GeV, but this contradicts experimental data. At the same time, it was established in that, at specific values of the parameters of the LRM, the contributions of physical Higgs bosons account for the BNL results. Thus, the effect of extra particles on the (g-2)µ anomaly has not yet studied only for three heavy neutrinos. Let us proceed to determine the contributions of the neutrino dipole magnetic moment to the muon AMM. Majorana neutrino ( G. G. Boyarkina and O. M. Boyarkin, Phys. Rev. D 67, 073023 (2003) ) where -- neutrino mixing matrix.

22. The Feynman diagrams determining the neutrino contribution to the muon AMM can be obtained from the vertex diagrams in Fig.1 upon replacing the electron line by a muon line and the photon line by an external-lectromagnetic-field line. Since there are four singly charged Higgs bosons in the LRM involving a Majorana neutrino, the number of the vertex diagrams corresponding to Fig.1 b is then doubled. Let us first estimate the diagrams associated with the charged gauge bosons. An analysis shows that a dominant contribution to the muon AMM comes from the diagram proportional to Employing we arrive at the conclusion that, for the observed value of the muon AMM to be explained , must reach values around However, such values are much greater than the theoretical predictions.

23. Thus, in the case of the Majorana neutrino, values that the off-diagonal matrix elements of the neutrino dipole magnetic moment must have for the (g-2)ano-maly to be explained are greatly in excess of theoretical predictions. Estimation of the diagrams associated with singly charged Higgs bosons leads to the analogous result. the Dirac neutrino The theory predicts that the diagonal elements of the neutrino dipole magnetic mo-ment can be much greater than its off-diagonal elements. It could then be expected that large values of would provide an explanation for the (g-2)anomaly. Obviously, only the diagrams involving charged gauge bosons must be taken into ac-count. From the calculations, it follows that the diagrams involving a virtual boson are again dominant. The corresponding correction to the muon AMM is given by (6) where

24. However, the integrals on the right-hand side of (6) prove to be so small that, despite large values of , the observed value of the muon AMM can – not be explained by the presence of the neutrino dipole magnetic moment. • O.M.Boyarkin, G.G. Boyarkina and V.V. Makhnach, Phys. Rev. D77, 033004 (2008).

25. CONCLUSION 1.It has been shown that, both in the SM supplemented with an right-handed neutrino singlet and in the SM supplemented with two Higgsdoublets, the branching ratio for the radiative muon decay is negligible. In other words, this decay is unobservable process from the point of view of the SM. . • 2.In the LRM we can always obtain the upper limits on muon radiative decay at • definite parameter values which are in agreement with current experimental • data. By this there are two kind of diagrams: diagrams involving the neutrino • electromagnetic vertex and diagrams disregarding the electromagnetic • neutrino properties ( , ). At identical model parameter values the • contributions of the latter are four orders of magnitude greater than that of the • former. Note the contributions of and have opposite signs! • and compensate each other. Then the contributions of the diagrams • taking into account the electromagnetic neutrino properties become dominant and we • could obtain the limit on the neutrino MM. • and don’t compensate each other. The limits on heavy neutrino masses • may be found. O. M. Boyarkin, G. Boyarkina, Phys.Atom. Nucl. 72, 607 (2009)

26. 3. The calculations within the LRM have shown that, for the BNL result to be explained, the dipole magnetic moment of the heavy neutrino must be about This value proves to be a few orders of magnitude larger than the value predicted by the theoretical predictions. 4. Within the LRM only the Higgs bosons among all extra particles (with respect to the SM) may provide an explanation for the (g-2)-anomaly.