Measuring CP-violating phases through studying the polarization of the final particles in μ→eee
aa r X i v : . [ h e p - ph ] J u l Measuring CP-violating phases through studying the polarization of the final particlesin µ → eee Yasaman Farzan ∗ School of physics, Institute for Research in Fundamental Sciences (IPM),P.O. Box 19395-5531, Tehran, Iran (Dated: June 10, 2018)It is shown that the polarizations of the final particles in µ + → e + e − e + provide us with informa-tion on the CP-violating phases of the effective Lagrangian leading to this Lepton Flavor Violating(LFV) decay. PACS numbers: 11.30.Hv, 13.35.BvKeywords: Lepton Flavor, Muon decay, CP-violation, angular distribution
I. INTRODUCTION
In the context of the “old” Standard Model (SM) with zero neutrino masses, the lepton flavor is conserved. As aresult, the LFV decays such as µ → eee and µ → eγ are strictly forbidden within the “old” SM. Within the “new” SMwith sources of LFV in the neutrino mass matrix, such decays are in principle allowed but their rates are suppressed bypowers of the neutrino mass and are therefore beyond the reach of any searches in the foreseeable future [1]. However,a variety of beyond SM scenarios can lead to rates for these processes exceeding the present experimental bounds [2]:Br( µ → eγ ) < . × − and Br( µ − → e − e + e − ) < . × − . (1)Notice that the bound on µ → eee is even stronger than the famous bound on µ → eγ . In the context of modelslike R-parity conserving MSSM in which the new particles can appear only in even numbers in each vertex, theseprocesses can only take place at the loop level. Since µ → eee is a three body decay, in such a model, Br( µ → eee ) issuppressed relative to Br( µ → eγ ) by a factor typically of order of e / (16 π ) log( m µ /m e ) [3]. However, in the modelsthat new particles can appear in odd numbers at each vertex ( e.g., in R-parity violating MSSM) the process µ → eee can take place at tree level and as a result, its rate can even exceed that of µ → eγ [4].It is rather well-known that by measuring the angular distribution of the final particles relative to the spin of theinitial muon in µ → eγ and µ → eee , one can derive information on the chiral nature of the effective Lagrangianleading to this process [4, 5, 6]. In the case of µ → eee , as shown in the literature [4, 7, 8], the angular distributionof the final particles relative to the spin of the initial muon also yields information on certain combinations of theCP-violating phases.Recently, it has been shown in [9, 10, 11] that if we measure the polarization of the emitted particles in µ → eγ and µN → eN , we can derive information on the CP-violating parameters of the theory. It was pointed out in [9, 10] thatby measuring the spin of the more energetic final positron in µ + → e + e − e + , some information on the CP-violatingphases can be derived. The analysis was performed in the framework of the models such as R -parity conserving MSSM,in which the dominant contribution to µ → eee comes from a penguin diagram (i.e., µ + → γ ∗ e + → e + e − e + ). In thisletter, we focus on the case that µ + → e + e − e + happens at the tree level through the exchange of heavy particles. Weshow that the transverse polarizations of the emitted particles in µ + → e + e − e + provide us with information on theCP-violating properties of the effective Lagrangian. The information derived from the spins of the emitted electronsand positrons are complementary to each other. In this letter, we focus on µ + → e + e − e + . Similar arguments holdfor µ − → e − e + e − .We show that the method proposed here is sensitive to a combination of the phases in the effective Lagrangianwhich is different from those that can be derived by methods discussed in the literature [4, 7, 8]. The effectivenessof each method depends on the relative magnitude of the different terms in the effective Lagrangian which in turndepends on the details of the underlying model. ∗ Electronic address: [email protected]
II. THE RATE OF µ + → e + e − e + Consider a general beyond SM scenario leading to µ + → e + e − e + . After integrating out the heavy states, the effectcan be described by an effective Lagrangian of form L = L + L (2)where L = B (¯ µ γ e )(¯ e − γ e ) + B (¯ µ − γ e )(¯ e γ e )+ C (¯ µ γ e )(¯ e γ e ) + C (¯ µ − γ e )(¯ e − γ e )+ G (¯ µγ ν γ e )(¯ eγ ν γ e ) + G (¯ µγ ν − γ e )(¯ eγ ν − γ e ) + H . c . (3)and L = A L ¯ µ [ γ µ , γ ν ] 1 + γ eF µν + A R ¯ µ [ γ µ , γ ν ] 1 − γ eF µν + H . c . (4)Notice that by using the identities ( σ µ ) αβ ( σ µ ) γδ ≡ ǫ αγ ǫ βδ and (¯ σ µ ) αβ = ǫ βδ ( σ µ ) δγ ǫ γα (where ǫ = ǫ = 0 and ǫ = − ǫ = 1) and employing the fact that the fermions anti-commute, we can rewrite the terms on the first line ofEq. (3) as − B µ γ γ ν e )(¯ e − γ γ ν e ) − B µ − γ γ ν e )(¯ e γ γ ν e ) . In the literature, it has been shown that by studying the angular distribution of the final particles relative to thespin of the initial muon, one can derive information on Re[ A R B ∗ − A L B ∗ ], Re[ A R G ∗ − A L G ∗ ], Im[ A R B ∗ + A L B ∗ ]and Im[ A R G ∗ + A L G ∗ ] (see e.g., [4, 7, 8]). However, by this method the phases of C and C cannot be derived.Moreover, if A L and A R are zero or suppressed, this method loses its effectiveness.By studying the energy spectrum of the emitted particles, it will be possible to differentiate between the differentterms in L and L . For example, the A L and A R couplings lead to a sharp peak in the distribution of the square ofthe invariant mass of a e − e + pair [ i.e., in the distribution of ( P e − − P e + ) ] at ( m µ / . Such a peak does not appearif the dominant contribution comes from Eq. (3). The A L and A R couplings are generally loop suppressed but, aswe see below, terms in L can appear in a wide range of models at tree level. In the present letter, we only considerterms in L .The B i and C i couplings in L can originate from the exchange of a heavy neutral scalar field (or fields). Letus demonstrate this through a simple toy model. Consider two heavy complex fields, φ and φ with the followingcouplings g µL φ ¯ µ γ e + g µR φ ¯ µ − γ e + g eL φ ¯ e γ e + g eR φ ¯ e − γ e + H . c . It is straightforward to show that the effective Lagrangian resulting from integrating out the heavy states φ and φ is of form (3) with B = g µL g ∗ eL m φ , B = g µR g ∗ eR m φ , C = C = 0 . If we swap φ and φ in the third and fourth terms ( i.e., taking L = [ g µL φ ¯ µ (1 + γ ) e + g µR φ ¯ µ (1 − γ ) e + g eL φ ¯ e (1 + γ ) e + g eR φ ¯ e (1 − γ ) e + H . c . ] / C = g µL g ∗ eR /m φ , C = g µR g ∗ eL /m φ and B = B = 0 . Taking φ and φ real , we find that B , B , C and C are all nonzero. It can similarly be shown that G and G canoriginate from the exchange of a doubly charged scalar field. The R-parity violating MSSM at the tree level leads tocouplings of form B and B [4]. That is while in this model, the A L and A R couplings are loop suppressed.Notice that while the B i couplings are chirality conserving, the C i and G i couplings are chirality-flipping. Noticethat under the parity transformation, B i and C i transform as B ⇔ B , G ⇔ G and C ⇔ C . Thus, | B − B | , | G − G | and | C − C | can be considered as measures of the parity violation. Under the CP transformation B ⇒ ηB ∗ , B ⇒ ηB ∗ , G ⇒ ηG ∗ , G ⇒ ηG ∗ , C ⇒ ηC ∗ and C ⇒ ηC ∗ , ~s µ + θ − φ + ~P e − ~P e +1 ˆ T − ˆ T − ~s µ + θ + φ − ~P e +1 ~P e − ˆ T +3 ˆ T +1 (a) (b)FIG. 1: These figures schematically depict the direction of the momenta of the final particles in the LFV decay µ + → e +1 e − e +2 relative to the spin of the anti-muon in its rest frame. Both figures correspond to a single decay a) illustrating ˆ T − , θ − and φ + and b) illustrating ˆ T + , θ + and φ − . where η is a pure phase that comes from the freedom in the definition of the CP-conjugate of the electron and themuon. By rephasing the electron and/or the muon field, either of these couplings can be made real. Thus, theLagrangian in Eq. (3) contains five physical CP-violating phases.From the formulas derived in the appendix, we find that the differential decay rate of µ + → e + e − e + is Z π X ~s e − ,~s e +1 ~s e +2 d Γ( µ + → e +1 e − e +2 ) dE e dE e +1 d Ω dφ + = 1128 π h ( | B | + | B | )( − m µ + 4 m µ E e +1 + 3 m µ E e − E e m µ − m µ E e +1 − m µ E e E e +1 )+ ( | B | − | B | ) P µ cos θ − ( m µ − m µ E e +1 − m µ E e + 8 m µ E e E e +1 + 4 m µ E e +1 + 4 m µ E e − m µ E e − m µ E e E e +1 ) /E e + (cid:2) ( | C | + 16 | G | )(1 + P µ cos θ − ) + ( | C | + 16 | G | )(1 − P µ cos θ − ) (cid:3) E e m µ ( m µ − E e ) (cid:3) (5)where d Ω is the differential solid angle determining the orientation of the emitted electron. We have summed overthe spins of the final particles and integrated over φ + which determines the azimuthal angle of the emitted positrons(see Fig. 1).Integrating over the energies of the final particles, we find X ~s e − ,~s e +1 ,~s e +2 d Γ d cos Ω = 0 . π m µ (cid:20) ( | B | + | B | + | C | + | C | | G | + 8 | G | )+ P µ cos θ − ( | B | − | B | | C | − | C | | G | − | G | )) (cid:21) . (6)The total rate of R ( d Γ /d Ω) d Ω is given by ( | B | + | B | + ( | C | + | C | ) /
2) + 8( | G | + | G | ). From the bound onBr( µ → eee ) ( see Eq. 1), we find | B | + | B | + | C | + | C | | G | + | G | ) < . From Eq. (5), we observe that the dependence of the coefficients of | B | + | B | and | C | + | C | + 16( | G | + | G | )on E e +1 and E e are different. As a result, by studying the energy spectrum of the final particles, it will be possibleto extract | B | + | B | and | C | + | C | + 16( | G | + | G | ). If in addition to the energy spectrum, the angulardistribution of the electron relative to ~s µ is measured, it will be possible to extract the parity violating combinations | B | − | B | and | C | − | C | + 16( | G | − | G | ). Of course the larger the polarization of the initial muon, thehigher the sensitivity of the angular distribution to these combinations. Thus, in principle by measuring the spectrumof the emitted particles and angular distribution of the final electron, it will be possible to derive the absolutevalues of the couplings. The angular distribution of the final positrons also give information on | B | − | B | and | C | − | C | + 16( | G | − | G | ) (see the appendix). Thus, the following combinations can be measured from thestudy of the angular distribution plus energy spectrum measurements: | B | , | B | , | C | + 16 | G | and | C | + 16 | G | . (7)However, without sensitivity to the spins of the final particles, it is not possible to derive information on the phasesof the couplings. In sect. II A, we explore what can be learned from the polarization of the emitted electron. Insect. II B, we discuss the polarization of the positron. A. Polarization of the emitted electron
Consider the decay µ + → e + e − e + in the rest frame of the muon, where the final electron makes an angle of θ − with the spin of the initial muon. Let us take the ˆ T − direction parallel to the momentum of the emitted electron andˆ T − ≡ ˆ T − × ~s µ / | ˆ T − × ~s µ | (see Fig. 1). The spin of the electron is determined by ( c e − d e − ) with ( | c e − | + | d e − | ) / = 1.That is the components of the spin of the electron areˆ T − · ~s e − = | c e − | − | d e − | , ˆ T − · ~s e − = 2Re[ c ∗ e − d e − ] and ˆ T − · ~s e − = 2Im[ c ∗ e − d e − ] . (8)From the formulas in the appendix, we find that the differential decay rate in the rest frame of the muon is X ~s e +1 ,~s e +2 d Γ( µ + → e +1 e − e +2 ) d cos Ω = Z π Z m µ / Z m µ / m µ / − E e X ~s e +1 , ~s e +2 | M | E e +1 E e ( m µ − E e +1 − E e ) dE e +1 dE e dφ + = m µ π (cid:2) (0 . | B | | c e − | + | B | | d e − | + ( | C | / | G | ) | d e − | + ( | C | / | G | ) | c e − | ]+ (0 . P µ cos θ − [ | B | | d e − | / − | B | | c e − | / | C | / | G | ) | d e − | − ( | C | / | G | ) | c e − | ] − (0 . P µ sin θ − (Re[ G C ∗ d e − c ∗ e − ] + Re[ G C ∗ d ∗ e − c e − ])+ 0 . B B ∗ d e − c ∗ e − ] P µ sin θ − ] , (9)where P µ is the polarization of the initial muon and d Ω represents the differential solid angle of the momentum ofthe electron relative to the spin of the muon. To obtain this equation, we have summed over the spins of the emittedpositrons and integrated over the azimuthal angle that the plane containing the momenta of these positrons makeswith the plane of the spin of the anti-muon and the momentum of the electron. See the appendix for the details.Notice that there is no interference term between the chirality-flipping and chirality-conserving couplings.From Eq. (24) in the appendix, we find that the longitudinal polarization of the electron is h s T − i ≡ d Γ /d Ω | ce − =1 d e − =0 − d Γ /d Ω | ce − =0 d e − =1 d Γ /d Ω | ce − =1 d e − =0 + d Γ /d Ω | ce − =0 d e − =1 = P − − P µ cos θ − P − P − + P µ cos θ − P − , (10)where P − = | B | − | B | − | C | | C | | G | − | G | P − = | B | | B | | C | | C | | G | + 8 | G | P − = | B | + | B | + | C | | C | | G | + 8 | G | P − = − | B | | B | | C | − | C | − | G | + 8 | G | . (11) h s T − i is sensitive only to the absolute values of the couplings. Notice that by measuring h s T − i and its angulardistribution, one can derive the same combinations that are listed in Eq. (7) i.e., the combinations that can beextracted from the angular distribution and energy spectrum (without measuring the spin of the final particles).Derivation of these combinations from the measurement of h s T − i can be considered as a cross-check for the derivationfrom the energy spectrum. Notice that even in the P µ = 0 limit, h s T − i is still non-vanishing and provides us withinformation on the parity violating combination P . That is while in this limit, the angular distribution of the electronis uniform and has no sensitivity to parity violation in the effective Lagrangian (3) (see Eq. (5)).The transverse polarizations are h s T − i ≡ d Γ /d Ω | ce − =1 / √ d e − =1 / √ − d Γ /d Ω | ce − =1 / √ d e − = − / √ d Γ /d Ω | ce − =1 / √ d e − =1 / √ + d Γ /d Ω | ce − =1 / √ d e − = − / √ = Re [(2 . B ∗ B − G ∗ C − G C ∗ ] sin θ − P µ P − + P − P µ cos θ − (12)and h s T − i ≡ d Γ /d Ω | ce − =1 / √ d e − = i/ √ − d Γ /d Ω | ce − =1 / √ d e − = − i/ √ d Γ /d Ω | ce − =1 / √ d e − = i/ √ + d Γ /d Ω | ce − =1 / √ d e − = − i/ √ = Im [(2 . B ∗ B − G ∗ C − G C ∗ ]] sin θ − P µ P − + P − P µ cos θ − . (13)The transverse polarizations are maximal for electrons emitted in the direction perpendicular to the spin of the muon: i.e., θ − = π/ . Notice that h s T − i and h s T − i provide independent information on the real and imaginary parts of thesame combination; i.e., (2 . B ∗ B − G ∗ C − G C ∗ ). Notice that both the moduli and the phase of this combinationcontain extra information beyond the combinations listed in Eq. (7). In the context of various models G i and C i can vanish. For example, as explained in the previous section, if the effective Lagrangian comes from integratingout neutral scalars, the G i couplings will vanish. Within such models, h s T − i ∝ Re[ B B ∗ ] and h s T − i ∝ Im[ B B ∗ ].Considering that | B i | can be independently measured, the derivation of arg[ B B ∗ ] from both h s T − i and h s T − i can beconsidered as a cross-check.In the limit of unpolarized muon ( i.e., P µ → h s T − i and h s T − i vanish. This is understandable because in the P µ = 0 limit, there is no preferred directions so we cannot define ˆ T − and ˆ T − (see Fig. 1). Thus, to derive theCP-violating phase, arg[ B B ∗ ], it is necessary to have a source of polarized muon which is quite feasible. For exampleif the muons are produced by the decay of pions at rest, they will be almost 100 % polarized. In fact, this is the casefor the on-going MEG experiment.By measuring the polarization of the emitted electron, it is not possible to derive the values of all the CP-violatingphases of the Lagrangian (3). Measurement of the angular distribution of the positrons does not provide any furtherinformation. As we shall see next, the transverse polarizations of the emitted positrons provide complementaryinformation on the phases. B. Polarization of the positron
Consider the decay µ + → e +1 e − e +2 in the rest frame of the muon where one of the final positrons makes anangle of θ + with the spin of the initial muon. Let us take the ˆ T +3 direction parallel to the momentum of e +1 andˆ T +2 ≡ ˆ T +3 × ~s µ / | ˆ T +3 × ~s µ | (see Fig. 1). The spin of e +1 is determined by ( c e +1 d e +1 ) with ( | c e +1 | + | d e +1 | ) / = 1. Thecomponents of the spin areˆ T +3 · ~s e +1 = | c e +1 | − | d e +1 | , ˆ T +1 · ~s e +1 = 2Re[ c ∗ e +1 d e +1 ] and ˆ T +2 · ~s e +1 = 2Im[ c ∗ e +1 d e +1 ] . (14)From the formula in the appendix, we find that the differential decay rate in the rest frame of the muon is X ~s e − ,~s e +2 d Γ( µ + → e +1 e − e +2 ) d cos Ω = Z π Z m µ / Z m µ / m µ / − E e X ~s e − ,~s e +2 | M | E e +1 E e ( m µ − E e +1 − E e ) dE e dE e +1 dφ − = (0 . m µ π (cid:20) ( | C | | c e +1 | + | C | | d e +1 | ) − ( | C | | c e +1 | − | C | | d e +1 | ) P µ cos θ + | B | + | B | + [ | B | ( | c e +1 | − | d e +1 | / − | B | ( | d e +1 | − | c e +1 | / P µ cos θ + + 16( | G | | c e +1 | + | G | | d e +1 | ) −
163 ( | G | | c e +1 | − | G | | d e +1 | ) P µ cos θ + + (Re[ B C ∗ c e +1 d ∗ e +1 ] + Re[ B C ∗ c ∗ e +1 d e +1 ]) P µ sin θ + + 24(Re[ G B ∗ c ∗ e +1 d e +1 ] − Re[ G B ∗ c e +1 d ∗ e +1 ]) P µ sin θ + i (15)where d Ω represents the differential solid angle of the momentum of e +1 relative to the spin of the muon.From the above equation, we find that the longitudinal polarization of e +1 is h s T +3 i = d Γ /d Ω | ce +1 =1 d e +1 =0 − d Γ /d Ω | ce +1 =0 d e +1 =1 d Γ /d Ω | ce +1 =1 d e +1 =0 + d Γ /d Ω | ce +1 =0 d e +1 =1 = P +1 + P +2 P µ cos θ + / P +3 + P +4 P µ cos θ + / , (16)where P +1 = | C | − | C | + 16 | G | − | G | P +2 = −| C | − | C | − | G | − | G | + 4 | B | + 4 | B | P +3 = | C | + | C | + 16( | G | + | G | ) + 2( | B | + | B | ) P +4 = | C | − | C | − | G | + 16 | G | + 2 | B | − | B | . (17)The transverse polarizations are h s T +1 i = d Γ /d Ω | ce +1 =1 / √ d e +1 =1 / √ − d Γ /d Ω | ce +1 =1 / √ d e − = − / √ d Γ /d Ω | ce +1 =1 / √ d e +1 =1 / √ + d Γ /d Ω | ce +1 =1 / √ d e +1 = − / √ = Re [ B C ∗ + B ∗ C + 24 G ∗ B − G B ∗ ]) sin θ + P µ P +3 + P +4 P µ cos θ + / h s T +2 i = d Γ /d Ω | ce +1 =1 / √ d e +1 = i/ √ − d Γ /d Ω | ce +1 =1 / √ d e +1 = − i/ √ d Γ /d Ω | ce +1 =1 / √ d e +1 = i/ √ + d Γ /d Ω | ce +1 =1 / √ d e +1 = − i/ √ = Im [ B C ∗ + B ∗ C + 24 G ∗ B − G B ∗ ] sin θ + P µ P +3 + P +4 P µ cos θ + / . (19)Like the case of the electron discussed in sect. II A, the longitudinal polarization, h s T +3 i , does not give informationon the CP-violating phases and can be used only as a cross-check for the derivation of the combinations listed inEq. (7) by the methods discussed earlier. The ratio of h s T +1 i and h s T +2 i gives arg [ B C ∗ + B ∗ C + 24 G ∗ B − G B ∗ ].Considering that the absolute value of this combination is also unknown, |h s T +2 i| + |h s T +1 i| provides an independentpiece of information. We have integrated over φ − which means the measurement of the direction of the emittedelectron is not necessary for this analysis. III. CONCLUSIONS AND PROSPECTS
A large variety of the beyond standard models predict a sizeable rate for µ + → e + e − e + exceeding the presentexperimental bound. In principle, by studying the energy spectrum of the final particles and their angular distribution,it is possible to derive the form of the terms in the effective Lagrangian leading to this process and extract informationon the absolute values of the couplings (see Eq. (7)). The effective Lagrangian responsible for µ + → e + e − e + caninclude new CP-violating phases. In order to derive information on the CP-violating phases, we have suggested tomeasure the polarization of the emitted particles. In this letter, we have focused on the effective Lagrangian in Eq. (3)that can result from integrating out heavy scalar fields with LFV couplings at the tree level. The rest of the terms ( i.e., A L and A R ) are expected to be loop suppressed and are neglected in this analysis. We have shown that the transversepolarization of the emitted electron in µ + → e + e − e + is sensitive to arg[2 . B B ∗ − G ∗ C − G C ∗ ] [see Eqs. (12,13)].That is while the transverse polarizations of the emitted positron is given by arg[ B C ∗ + B ∗ C + 24 G ∗ B − G B ∗ ].From Eqs. (12,13,18,19), we observe that if the initial muon is unpolarized ( i.e., P µ = 0) the transverse polarizationsof the emitted particles vanish. Thus, in order to derive the CP-violating phases, a source of polarized muons isrequired.In sum, the effective Lagrangian in Eq. (3) includes six new couplings and five physical phases. By measuring theenergy spectrum of the final particles and the angular distributions relative to the spin of the initial muon, one canderive the CP-conserving combinations listed in Eq. (7): i.e., four out of the six CP-conserving quantities. Neglectingthe loop suppressed A L and A R couplings, the angular distribution cannot provide information on the phases. Thetransverse polarizations of the emitted particles provide four independent pieces of information on the phases andcouplings. This information is not enough to reconstruct all the couplings but considerably reduces the degeneracyin the parameter space.We have also discussed the longitudinal polarization of the emitted particles. The longitudinal polarizations donot depend on the phases of the effective couplings. It is noteworthy that even in the P µ = 0 limit, the longitudinalpolarizations of the positron, h s T +3 i , and the electron, h s T − i , are nonzero and respectively yield information on theparity violating combinations | C | − | C | + 16 | G | − | G | and | C | − | C | + 16 | G | − | G | + 2( | B | − | B | ).Remember that in the P µ = 0 limit, there is no preferred direction so the angular distribution of the final particles isuniform and does not yield information on the parity-violating combinations.There are running and/or under construction experiments that aim to probing signals for µ → eγ [12] and µ − e conversion on nuclei [13] several orders of magnitudes below the present bounds on their rates. However, as shownin [4], it is possible that while µ + → e + e − e + is round the corner, the rates of µ → eγ and µ − e conversion aretoo low to be probed. In fact as shown in [4], the three experiments provide us with complementary informationon the parameters of the effective LFV Lagrangian. If the muons are produced from the decay of pions at rest (likethe case of the running MEG experiment [12]), the initial muons in µ → eee will be polarized. On the other hand,there are well-established techniques to measure the polarization of the emitted particles in this energy range. Infact, such polarimetry has been used to measure the Michel parameters since 80’s [14]. As a result, if the rate of µ + → e + e − e + is close to the present bound and a hypothetical experiment finds statistically large number of such aprocess, performing the analysis proposed in this paper sounds possible.In this letter, we have focused on the LFV three-body decay of the anti-muon, µ + → e + e − e + . The same discussionapplies to the decay of the muon, µ − → e − e + e − . In this mode, the transverse polarizations of the electrons wouldgive arg[ B C ∗ + B ∗ C + 24 G ∗ B − G B ∗ ] and the transverse polarizations of the emitted positrons would givearg[2 . B B ∗ − G ∗ C − G C ∗ ]. The method of measuring the polarization described in [14] is based on studyingthe distribution of the photon pair from the annihilation of the emitted positron on an electron in a target. If thismethod is to be employed, only the polarization of the positron can be measured. Thus, to derive both combinations,the experiment has to run in both muon and anti-muon modes.The three-body LFV decay modes of the tau lepton such as τ → e ¯ ee or τ → µ ¯ µµ can also shed light on theunderlying theory. Recently it has been shown that by studying the angular distribution of the final particles in τ → µ ¯ µµ at the LHC, one can discriminate between various models [15]. Discussions in the present letter also applyto the decay modes τ → e ¯ ee and τ → µ ¯ µµ . Appendix
In this appendix, we derive the decay rate of µ + → e +1 e − e +2 . We first derive the decay rate into an electron ofdefinite spin, summing over the spins of e +1 and e +2 . We then concentrate on the spin of e +1 and derive the decay rateinto a positron of definite spin, summing over the spins of the electron and the other positron.With effective Lagrangian (3), we find M ( µ + → e +1 e − e +2 ) = B ¯ µ γ e ¯ e − γ e − B ¯ µ γ e ¯ e − γ e + B ¯ µ − γ e ¯ e γ e − B ¯ µ − γ e ¯ e γ e + C ¯ µ γ e ¯ e γ e − C ¯ µ γ e ¯ e γ e + C ¯ µ − γ e ¯ e − γ e − C ¯ µ − γ e ¯ e − γ e − G ¯ µc − γ γ e ∗ e T c γ e − G ¯ µc γ γ e ∗ e T c − γ e . (20) Decay rate into e − with a given spin Consider the decay µ + → e +1 e − e +2 in the rest frame of the muon. Since we are interested in the spin of theelectron, it is convenient to use the coordinate system defined as ˆ T − ≡ ~P e − / | ~P e − | , ˆ T − ≡ ( ˆ T − × ~s µ ) / | ˆ T − × ~s µ | andˆ T − ≡ ˆ T − × ˆ T − . In this coordinate system, P µ + = ( m µ , , , P e +1 = E e +1 (1 , sin α cos φ + , sin α sin φ + , cos α ) P e − = ( E e , , , E e ) P e +2 = ( m µ − E e +1 − E e , − E e +1 sin α cos φ + , − E e +1 sin α sin φ + , − E e − E e +1 cos α ) (21)where the electron mass is neglected (see Fig. 1-a). Writing the kinematics and neglecting effects of O ( m e /m µ ) ≪ α = m µ − m µ E e +1 − m µ E e + 2 E e +1 E e E e +1 E e . (22)Summing over the spins of the emitted positrons, we find that(2 π ) Z π (2 E e +1 )(2 E e +2 ) X ~s e +1 ,~s e +2 | M | dφ + = ( | B | | c e − | + | B | | d e − | ) E e +1 h ( m µ − E e +1 (1 − cos α )) + (1 − cos α )( m µ − E e − E e +1 ) i + ( | B | | c e − | − | B | | d e − | ) E e +1 h cos α [ m µ − E e +1 (1 − cos α )] − (1 − cos α )( E e +1 cos α + E e ) i P µ cos θ − + (cid:2) ( | C | + 16 | G | ) | d e − | (1 + P µ cos θ − ) + ( | C | + 16 | G | ) | c e − | (1 − P µ cos θ − ) (cid:3) E e +1 [ m µ − E e (1 − cos α )] − P µ Re[ B B ∗ d e − c ∗ e − ] E e +1 (1 − cos α )( E e +1 (1 − cos α ) − m µ ) sin θ − + 8 P µ (Re([ G C ∗ d e − c ∗ e − ] + Re[ G C ∗ d ∗ e − c e − ]) E e +1 ( E e (1 − cos α ) − m µ ) sin θ − , (23)where ( c e − d e − ) determines the spin of the emitted electron [see Eq. (8)].The differential rate of the decay into an electron in a direction that makes an angle of θ − with the spin of theinitial muon is X ~s e +1 ,~s e +2 d Γ( µ + → e +1 e − e +2 ) d Ω = Z π Z m µ / Z m µ / m µ / − E e X ~s e +1 ,~s e +2 | M | E e E e +1 ( m µ − E e − E e +1 ) dE e +1 dE e dφ + , where d Ω is the differential solid angle determining the orientation of the emitted electron. The factor E e E e +1 ( m µ − E e − E e +1 ) comes from the momentum-space volume for a three body decay [ i.e., from δ ( P µ − P e − P e +1 − P e +2 ) d P e d P e +1 d P e +2 ]. Inserting | M | from Eq. (23), we obtain X ~s e +1 ,~s e +2 d Γ( µ + → e +1 e − e +2 ) d Ω= m µ π ) " ( | c e − | | B | + | d e − | | B | ) Z / Z / / − y ( − y + 4 x − y − x − xy ) dx dy + P µ cos θ − ( | B | | c e − | − | B | | d e − | ) Z / Z / / − y [1 − x − y + 2 xy − x + 2 y − y (1 − x − y + 2 xy + y )] dx dy + (( | C | + 16 | G | ) | d e − | (1 + P µ cos θ − ) + ( | C | + 16 | G | ) | c e − | (1 − P µ cos θ − )) Z / Z / / − y y (1 − y ) dx dy − P µ (Re[ G C ∗ d e − c ∗ e − ] + Re[ G C ∗ d ∗ e − c e − ]) sin θ − Z / Z / / − y y (1 − y ) dx dy − P µ Re[ B B ∗ d e − c ∗ e − ] sin θ − Z / Z / / − y (2 x + 2 y −
1) 2 x − y dx dy . (24)The integrals are all finite and lead to the numbers in Eq. (9) Decay rate into e +1 with a given spin Let us now concentrate on one of the positrons, e +1 , whose momentum makes an angle of θ + with ~s µ (see Fig.1-b). To perform this analysis, it is convenient to work in the following coordinate system: ˆ T +3 ≡ ~P e +1 / | ~P e +1 | ,ˆ T +2 ≡ ˆ T +3 × ~s µ / | ˆ T +3 × ~s µ | and ˆ T +1 ≡ ˆ T +2 × ˆ T +3 . In the rest frame of the muon and in this coordinate system, P µ + = ( m µ , , , P e − = E e (1 , sin α cos φ − , sin α sin φ − , cos α ) P e +1 = ( E e +1 , , , E e +1 ) P e +2 = ( m µ − E e +1 − E e , − E e sin α cos φ − , − E e sin α sin φ − , − E e +1 − E e cos α ) (25)where α is given by Eq. (22).Summing over the spins of e +2 and e − , we find(2 π ) (2 E e )(2 E e +2 ) Z X ~s e − ,~s e +2 | M | dφ − = ( | B | | c e +1 | + | B | | d e +1 | ) E e [ m µ − E e +1 (1 − cos α )] + ( | B | | d e +1 | + | B | | c e +1 | )(1 − cos α ) E e ( m µ − E e − E e +1 )+ 16( | G | | c e +1 | + | G | | d e +1 | ) E e ( m µ − E e (1 − cos α ))+ ( | C | | c e +1 | + | C | | d e +1 | ) E e ( m µ − E e (1 − cos α ))+ P µ h ( | B | | c e +1 | − | B | | d e +1 | )[ m µ − E e +1 (1 − cos α )] cos θ + + P µ ( | B | | c e +1 | − | B | | d e +1 | )(1 − cos α )( E e cos α + E e +1 ) i E e cos θ + + P µ ( | B | | c e +1 | − | B | | d e +1 | ) cos φ − E e sin α (1 − cos α ) sin θ + + P µ ( | C | | c e +1 | − | C | | d e +1 | ) E e cos α ( m µ − E e (1 − cos α )) cos θ + + 16 P µ ( | G | | c e +1 | − | G | | d e +1 | ) E e ( m µ − E e (1 − cos α )) cos α cos θ + + P µ Re[ B C ∗ c e +1 d ∗ e +1 ] m µ E e sin θ + (1 + cos α )+ P µ Re[ B C ∗ c ∗ e +1 d e +1 ] m µ E e sin θ + (1 + cos α )+ 4 P µ h Re[ G B ∗ c ∗ e +1 d e +1 ] − Re[ G B ∗ d ∗ e +1 c e +1 ] i E e ( m µ − E e (1 − cos α ))(1 − cos α ) sin θ + (26)where ( c e +1 d e +1 ) determines the spin of the emitted positron [see Eq. (14)]. The differential rate of the decay into apositron in a direction that makes an angle of θ + with the spin of the initial muon is X ~s e − ,~s e +2 d Γ( µ + → e +1 e − e +2 ) d Ω = Z π Z m µ / Z m µ / m µ / − E e +1 X ~s e − ,~s e +2 | M | E e E e +1 ( m µ − E e − E e +1 ) dE e dE e +1 dφ − where d Ω is the differential solid angle determining the orientation of e +1 . Inserting | M | from Eq. (23), we obtain X ~s e − ,~s e +2 d Γ( µ + → e +1 e − e +2 ) d Ω= m µ π ) " ( | d e +1 | | B | + | c e +1 | | B | ) Z / Z / / − y (1 − x − y )(2 x + 2 y − dx dy + ( | c e +1 | | B | + | d e +1 | | B | ) Z / Z / / − y y (1 − y ) dx dy + ( | C | | c e +1 | + | C | | d e +1 | ) Z / Z / / − y x (1 − x ) dx dy + 16( | G | | c e +1 | + | G | | d e +1 | ) Z / Z / / − y x (1 − x ) dx dy + P µ cos θ + ( | B | | d e +1 | − | B | | c e +1 | ) Z / Z / / − y (2 x + 2 y − − y − − x − y + 2 xy y )] dx dy + P µ cos θ + ( | C | | c e +1 | − | C | | d e +1 | ) Z / Z / / − y (1 − x ) 1 − x − y + 2 xy y dx dy P µ cos θ + ( | B | | c e +1 | − | B | | d e +1 | ) Z / Z / / − y y (1 − y ) dx dy + 16 P µ cos θ + ( | G | | c e +1 | − | G | | d e +1 | ) Z / Z / / − y (1 − x ) 1 − x − y + 2 xy y dx dy + 2 P µ (Re[ B C ∗ c e +1 d ∗ e +1 ] + Re[ B C ∗ c ∗ e +1 d e +1 ]) sin θ + Z / Z / / − y (2 xy + 0 . − x − y ) dx dy + 4 P µ (Re[ G B ∗ c ∗ e +1 d e +1 ] − Re[ G B ∗ c e +1 d ∗ e +1 ]) sin θ + Z / Z / / − y (1 − x )(2 x + 2 y − x dx dy . (27) Acknowledgments
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