Extracting the CP-violating phases of trilinear R-parity violating couplings from $\mu \to eee$

It has recently been shown that by measuring the transverse polarizations of the final particles in $\mu^- \to e^- e^+ e^-$, it is possible to extract information on the phases of the effective couplings leading to this decay. We examine this possibility within the context of R-parity violating Minimal Supersymmetric Standard Model (MSSM) in which the $\mu^- \to e^- e^+e^-$ process can take place at a tree level. We demonstrate how a combined analysis of the angular distribution of the emitted positrons and their transverse polarization can determine the CP-violating phases of the trilinear R-parity violating Yukawa couplings.

they can be responsible for the creation of the baryon asymmetry in the universe [6].
The paper is organized as follows. In section 2, we review the general effective Lagrangian that can lead to µ → eee and µ → eγ. We outline the information that can be derived on the effective couplings without the measurement of the spin of the final particles. In section 3, we review the contributions that the effective couplings receive from the R-parity and lepton flavor violating trilinear Yukawa couplings. We then briefly review bounds on these couplings. In section 4, we introduce the P-odd asymmetry, A, and the transverse polarization. In section 5, we briefly discuss the feasibility of measuring the transverse polarization of the final particles. In section 6, we discuss the results that can be derived by combined analysis of A and the transverse polarization. Results are summarized in section 7.

Effective Lagrangian in a general framework
The effective Lagrangian leading to µ → eee can in general be written as [7] L eff = B 1 (μ L e R )(ē R e L ) + B 2 (μ R e L )(ē L e R )+ C 1 (μ L e R )(ē L e R ) + C 2 (μ R e L )(ē R e L )+ G 1 (μ R γ ν e R )(ē R γ ν e R ) + G 2 (μ L γ ν e L )(ē L γ ν e L ) −A R (μ L [γ µ , γ ν ] q ν q 2 e R ) (ēγ µ e) − A L (μ R [γ µ , γ ν ] q ν q 2 e L ) (ēγ µ e) + H.c. (1) Notice that by using the identities (σ µ ) αβ (σ µ ) γδ ≡ 2ǫ αγ ǫ βδ and (σ µ ) αβ = ǫ βδ (σ µ ) δγ ǫ γα (where ǫ 11 = ǫ 00 = 0 and ǫ 10 = −ǫ 01 = 1) and employing the fact that the fermions anti-commute, we can rewrite the terms on the first line of Eq. (1) as This effective Lagrangian leads to [7] Br(µ → eee) = 1 32G 2 F |B 1 | 2 + |B 2 | 2 + 8(|G 1 | 2 + |G 2 | 2 ) By studying the energy distribution of the final particles, more information can be derived on the effective couplings. For example, let us consider the contributions from A L and A R which come from a virtual photon exchange. When the invariant mass of an electron positron pair goes to zero, the virtual exchanged photon goes on shell. As a result, the corresponding Dalitz plot should have a peak at (P e − +P e + ) 2 = 0 whose height is given by |A L | 2 +|A R | 2 . The combinations that can be derived by studying the energy distributions of the final particles (see for example [7] and references therein). By studying the angular distributions of the final particles relative to the polarization of the initial muon, one can . A combined analysis of angular and energy distribution therefore yields the absolute values of all the effective couplings, B 1 , B 2 , A L and A R as well as the CP-violating phases Notice however that there is still some information in Eq. (1) that cannot be derived by the methods described above. In particular, the CP-violating phase arg[B 1 B * 2 ] cannot be derived by these methods. Further information can be obtained by studying the transverse polarization of e − in µ + → e + e − e + [3].
Notice that the terms on the last line of Eq. (1) come from the effective coupling of the photon which gives rise to The present bound on this LFV rare decay is very strong Br(µ → eγ) < 1.2 × 10 −11 [1] which implies |A L | 2 + |A R | 2 < 3.3 × 10 −27 GeV −2 . Thus, from Eq. (3), we observe that the contribution from A L and A R to µ → eee cannot be larger than 7 × 10 −14 so if Br(µ → eee) turns out to be close to its present bound, we can safely neglect the contributions from A L and A R .

Effects of trilinear R-parity violating couplings
Within the R-parity conserving MSSM, the slepton mass matrix as well as the trilinear Aterm involving the sleptons include LFV sources which can induce the effective Lagrangian in Eq. (1). By relaxing the R-parity conservation, new sources of LFV emerge. In particular, the R-parity violating Yukawa couplings in the superpotential add nine new sources of LFV. That is each nonzero element of λ ijk is a source of LFV. Some of these couplings can contribute to µ → eee at tree level. Throughout this letter, we will focus on the effects of λ and set other LFV sources equal to zero for simplicity. The nonzero elements of λ ijk contain nine phases out of which three can be absorbed by rephasingL i . In this basis, each of the bilinear R-parity terms, µ ′ iL iĤu can be considered as a new source for CP-violation. In this letter, we will investigate the possibility of deriving the CP-violating phases of λ ijk from the transverse polarization of the final particles in µ → eee.
The contributions to the effective couplings from λ ijk have been calculated in [2] and the results are as follows: Notice that while the couplings B 1 and B 2 receive a contribution at a tree level, A R and G 1 receive contributions only at the one-loop level. If λ ijk are the only sources of LFV, up to one-loop level, all other couplings vanish. As discussed in the previous section, the values of gives bounds on λ's. In our analysis, we pick up values of λ that respect these bounds. The bounds that we use are summarized in Table 1.
The third column shows the observable from which the bound is extracted. Notice that the bound on λ 12k comes from the numerical value of the CKM matrix. At first sight, this might seem counterintuitive. Remember however that the value of V ud is extracted by comparing Γ(n → peν e ) with Γ(µ → eν e ν µ ). The unitarity condition of the CKM matrix combined by the extracted values of the CKM matrix elements yields a bound on the contribution from λ 21k . For a full review of the bounds see [10].

Transverse polarization and asymmetry
Let us define P-odd asymmetry, A, as A nonzero A violates parity. In fact, A is sensitive to the P-odd combinations such as The polarizations of the final electron in µ + → e + 1 e − e + 2 can be defined as Finally, θ is the angle between the momentum of e − and the polarization of the muon: As shown in [3], while sT− Of course, similar formulas hold for the transverse polarization of the final positron in µ − → e − e + e − .
Let us suppose Br(µ → eee) is measured and found to be close to 10 −12 . Let us moreover suppose that the Dalitz plots reveal that B 1 and B 2 give the dominant contribution to this decay as it is expected in the case that LFV effects originate from λ ijk . If the MEG experiment at PSI reports a null result (i.e., Br(µ → eγ) < 10 −13 ), within the context of MSSM (to be tested at the LHC), such a set of conditions attests our assumption that λ ijk is the prime source for LFV. If MEG also finds a signal for µ → eγ, other LFV terms, such as slepton masses or the trilinear soft supersymmetry breaking terms, might contribute to Br(µ → eγ) but as we discussed earlier, even in this case, we can neglect the contribution from A L and A R to Br(µ → eee). LFV terms in slepton masses or the trilinear soft supersymmetry breaking terms can contribute to other effective couplings in (1) but the effect will be loop suppressed and negligible. Notice that although within the model under our study, G 1 and A R are given by the same combinations of λ ijk , G 1 is enhanced by a factor of log(m 2 ν /m 2 l ) so we do not neglect its contribution [see Eq. (5)]. Neglecting the effects of A R , the formulas for A, sT− 1 and sT− 2 will have forms: and in which P µ is the polarization of the initial muons.
Moreover the longitudinal polarization is given by Notice when the electron is emitted in the direction perpendicular to the spin of the muon, the transverse polarization is maximal. In our analysis, we will set θ = π/2 which experimentally means we study the data from the polarimeter collecting electrons with momentum perpendicular to s µ . If more than a single polarimeter is installed, more data can of course be collected. The average polarization can be defined as It is noteworthy that if the values of Br(µ → eee), A and sT− 3 | θ= π 2 are measured, the numerical values of |B 1 |, |B 2 | and |G 1 | can be derived.

Feasibility of measuring the transverse polarization
The typical experimental setups devoted to the study of muon decay (such as the MEG experiment or the experiment at TRIUMF described in [13]) can be summarized as follows.
A proton beam collides on a target producing pions. Charged pions are stopped in the target and decay at rest into a neutrino and a muon. Muons at production are 100 % polarized up to negligible correction due to the neutrino mass [13]. Muons exit the first target and are transmitted to a second target where they stop and then decay at rest. Based on the setup of the experiment, muons of either positive or negative sign can be selected to be transmitted to the second target. Negative muons would form bound states with atoms in the second target so we focus on the decay of positive muons which decay as free states. When muons decay, they are still highly polarized. The degree of depolarization from the production to decay depends on the setup of experiment. Especially if only the muons produced at the surface of the first target are collected (as it is done both in MEG [12] and in the experiment described in [13]), the depolarization will be quite negligible. For example, in the TRIUMF experiment described in [13], the polarization remains above 99% until the muon decay in the second target. For the purpose of the present analysis we can safely replace P µ = 100%.
Notice that unlike the case of [13], even a moderate accuracy in knowledge of P µ will be sufficient to perform the present analysis so from this aspect, it is easier to carry out the present measurement [13]. The angular distribution of the final particles are described by (θ, φ): where the energy-momentum conservation implies k ′ = m e k m e + k(1 − cos θ) .
A cumbersome but straightforward calculation shows that 2π 0 dσ d cos θ dφ cos φdφ = (12) Thus, by measuring the partial cross section, dσ/(d cos θ dφ), and taking the above integrals, a and b (up to an overall phase) can be determined and the spin of e 1 can be therefore reconstructed.
There are two problems that complicate the measurement: (1) In the lab frame where e − Remember that, while B 1 and B 2 receive nonzero contributions at a tree level, G 1 receives a contribution only at a loop level. As long as G 1 has a negligible value, for given A and Br(µ → eee), |B 1 | and |B 2 | are fixed. It is straightforward to check that, for |G 1 | → 0, As seen in the figures, the majority of points lie inside an oval-shaped region whose boundaries are given by Eq. (14). These are the points for which the contribution from loop suppressed |G 1 | 2 can be neglected. As seen from Fig. 1, there are only few points (about 2 percent of all points) lying outside the oval-shaped region. At points with A < −1/3, is expected from Eqs. (9) and (10). In Fig. 2, we have removed the points for which (5c)) so unlike the case of (ϕ 212 = 0, ϕ 321 = 0), in this case, the nonzero phase is not given From Fig. 2, we observe that for |A| < 0.2, by simultaneous measurements of A and sT− 2 with reasonable accuracy, even without independent knowledge of |λ ijk |, solutions with (ϕ 321 = π/2,ϕ 212 = 0) and (ϕ 321 = π/4,ϕ 212 = 0) can be distinguished (see points denoted by violet ⊲ and pink ). Notice that all points denoted by red △ and blue ⊳ corresponding respectively to (ϕ 321 = 0,ϕ 212 = π/2) and (ϕ 321 = 0,ϕ 212 = π/4) lie above the horizontal axis. Solutions with positive and negative sin ϕ 212 are distinguishable but deriving the value of ϕ 212 without knowledge of |λ ijk | does not seem to be practical. We have found that the contributions from the phases that enter only at loop level (i.e., ϕ 322 , ϕ 313 and ϕ 323 ) to sT− 2 are smaller than 0.1. Thus, in deriving the values of ϕ 212 and ϕ 321 from sT− 2 , the potential contributions from the rest of the phases can be ignored.
As discussed above, an upper bound on |G 1 | can considerably simplify the analysis and solve the degeneracies. Although |G 1 | 2 (more precisely, |G 1 | 2 + |C 2 | 2 /16) can in principle be extracted from the energy distribution of final particles, within the present model, its value will most probably be too small to be measured so in practice only an upper bound on |G 1 | can be extracted as we have assumed in deriving Fig. 2. Notice that That is while if G 1 and G 2 (or C 1 and C 2 ) gave the main contribution to µ → eee, we would expect that sT− 3 | θ= π 2 /A = 1. The ratio of longitudinal polarization to A can therefore be regarded as a cross-check for the smallness of |G 1 |.
To draw Figs 1 and 2, we have used the spectrum at the α benchmark [15] as the input: i.e., We have set mν µ = mν τ = 285 GeV. For two reasons, we expect the results to be robust against varying the input masses: (i) Varying m 2 νµ and m 2 ντ is respectively equivalent to rescaling λ 211 and λ 311 (see Eqs. (5b) and (5c)). (ii) Both A and sT− 2 are defined as ratios so the dependence on the supersymmetry scale disappears. We have re-drawn the diagram for different benchmarks. As expected, the results are not sensitive to the input for the mass spectrum.
It is noteworthy that if the only sources of LFV are the λ ijk 's giving rise to Br(µ → eee), Br(µ → eγ) will be smaller than 10 −13 so if the MEG experiment reports a µ → eγ signal, sources for µ → eγ other than λ ijk 's must exist.
As seen in Figs. 1 and 2 be employed, so with a few hundred µ + → e + e − e + decays, the statistical error in the measurement of A will be reasonably small and below 0.1. However, we expect only a fraction of the emitted electrons, r, to enter the polarimeters so for establishing nonzero sT− 2 , the total number of µ + → e + e − e + decays has to be larger than 100/r. That is if r ∼ 10%, the total number of µ + → e + e − e + has to be larger than a few thousand.
In the above analysis, we have employed information on the R-parity violating couplings from only the LFV rare decays. The R-parity violating couplings can in principle be directly measured by accelerators. |λ i11 | leads to a resonant production ofν i in a e − e + collider (e − e + →ν i ) so |λ i11 | can be derived provided that |λ i11 | > 10 −5 and the center of mass energy is equal to the mass ofν i [16,10]. Moreover, the λ couplings can lead toν [17,10] whereχ 0 1 andχ + 1 are the lightest neutralino and chargino. Thus, by measuring the decay length and the flavor of the final charged leptons, |λ ijk | can in principle be extracted. If |λ ijk | < O(10 −5 ), the decay length will be too small to be resolved [10]. This method is therefore sensitive only to the values of |λ ijk | smaller than O(10 −5 ). 2 If |λ|'s are all smaller than 10 −5 , each |λ ijk | might be extracted will be too small (Br(µ → eee) < 10 −16 ). Let us now consider the range, m susy ∼ 100 GeV, λ 211 , λ 311 > 10 −3 and λ ijk ∼ 10 −5 with ijk = 211, 311. In this range, e − e + →ν i yields |λ i11 | and Br(µ → eee) is close to the present bound. Moreover, for ijk = 211, 311,ν i → l + j l − k andχ 0 1 →ν i l + j l − k will have a resolvable decay length but the point is that the decay modes involving λ 211 and λ 311 will dominate and lead to a decay length too short to be observable: e.g., Γ(ν µ → e + e − )/Γ(ν µ → τ + τ − ) ∼ 10 4 . Thus, even in case that the R-parity conserving decay modes are kinematically forbidden, extracting the decay lengths will be quite challenging. Let us however suppose that these experimental difficulties are partly solved and certain |λ ijk | (but not necessarily all) are measured. Such achievement might not be out of reach if λ i11 ∼ 10 −4 and the rest of λ's are of order of 10 −5 . Complementary information can then be derived from Br(µ → eee), A and s T − 2 : Neglecting the |G 1 | 2 effects, Br(µ → eee) and A give |B 1 | and |B 2 | which to leading order correspond to |λ 311 | · |λ 321 | and |λ * 211 λ 212 + (m 2 νµ /m 2 νµ )λ * 311 λ 312 |, respectively. Thus, if λ 311 is extracted from e + e − →ν τ at ILC, the measurement of Br(µ → eee) and A gives |λ 321 |. If |λ 211 |, |λ 212 |, |λ 311 | and |λ 312 | are all derived by accelerators, this method will give the phase of λ 212 . The measurement of sT− 2 will then yield the phase of λ 321 .

Conclusions and discussion
The trilinear R-parity violating Yukawa couplings, λ ijkLiLjÊk /2 can lead to µ → eee. In particular, λ 321 , λ 311 , λ 211 , λ 312 and λ 212 contribute to µ → eee at tree level. By rephasing the lepton fields, we can go to a basis in which λ 311 , λ 312 and λ 211 are real. This exhausts the freedom to rephase the other fields so λ 321 and λ 212 can in general be complex and can be considered as sources for CP-violation. Their phases can induce transverse polarization We have also studied the P -odd asymmetry, A defined in Eq. (6) and discussed the information that from a combined analysis of A and sT− 2 can be obtained. For the majority of the λ configurations, the tree level effects dominate so the effective coupling |G 1 | is much smaller than |B 1 | and |B 2 | and its effects can therefore be neglected. In this case, |G 1 | will be too small to be measured but an upper bound can be put on |G 1 | by studying the energy distribution of the final particles in µ → eee or as discussed in the present paper by combining information on A, Br(µ → eee) and sT− 3 . We have noticed that if There are established techniques to measure the transverse polarization of the positron based on the azimuthal distribution of the photon pair produced by the annihilation of the positron on the polarized electrons in a thin magnetized target [14]. We have shown that a similar setup can be employed to measure the transverse polarization of the electrons, too. In fact, the azimuthal distribution of the final electrons in Möller scattering, e − 1 e − 2 → e − 3 e − 4 with polarized e − 2 is sensitive to the polarization of e − 1 . The challenges before this measurement are similar to the ones in [14] and can be overcome by similar methods.
One can repeat similar discussion for three body LFV decays of τ lepton such as τ → µµµ or τ → eee. The measurements of the angular distribution and polarization of the final particles in these decays can provide complementary information on the λ ijk couplings. Such a study will be presented elsewhere.  Figure 1: Transverse polarization of the electron in µ + → e + e − e + versus the P-odd asymmetry A defined in Eq. (6). The input values for masses correspond to the ones at the α benchmark in [15] with mν µ ≃ mν τ ≃ 285 GeV. Random values between 10 −5 up to bounds in Table 1 are assigned to the λ couplings and points at which Br(µ → eee) ∈ 5 · 10 −13 (1 ± 10%) are selected. To calculate sT  Fig. 1, except that we have removed the points at which 8|G 1 | 2 /(|B 1 | 2 + |B 2 | 2 ) > 0.05.