Distinguishing muon LFV effective couplings using Nufact18 @ Virginia Tech Distinguishing muon LFV effective couplings using 𝜇 − 𝑒 − → 𝑒 − 𝑒 − in a muonic atom Joe Sato (Saitama University) I’m Yuichi Uesaka, a new postdoc at RCNP theory group. The title of today’s seminar is “charged lepton flavor violating process in a muonic atom”. Especially, I would like to focus on our recent work on mu- e- -> e- e- in a muonic atom. This was the theme of my doctoral thesis. This talk is based on two published paper and one work in progress. M. Koike, Y. Kuno, JS, & M. Yamanaka, Phys. Rev. Lett. 105, 121601 (2010). Y.Uesaka, Y. Kuno, JS, T. Sato & M. Yamanaka, Phys. Rev. D 93, 076006 (2016). Y.Uesaka, Y. Kuno, JS, T. Sato & M. Yamanaka, Phys. Rev. D 97, 015017 (2018). Y. Kuno, JS, T. Sato, Y.Uesaka & M. Yamanaka, in preparation

Contents 1. Introduction Charged Lepton Flavor Violation (CLFV) CLFV searches using muon 𝜇 − 𝑒 − → 𝑒 − 𝑒 − in a muonic atom 2. Transition probability of 𝜇 − 𝑒 − → 𝑒 − 𝑒 − Effective CLFV interactions Distortion of scattering electrons & Relativity of bound leptons Difference between contact & photonic processes ３. Distinguishment of CLFV interaction This is contents of my presentation. My talk is divided into 4 parts. First, I will introduce the current status of CLFV and the new CLFV process, me2ee. The aim of this research is a quantitative calculation for the rate of me2ee in a nuclear Coulomb potential. Second, I will talk about the way to analyze the transition probability. I will show the importance of taking into account distortion of scattering electron and relativity of bound electron. The result of the analysis gave us the possibility to distinguish among CLFV operators. So the next topic is ways to identify CLFV operator by experimental observables. I will tell you the details later. Finally, I will summarize my talk. Atomic # dependence of decay rates Energy-angular distribution of emitted electrons Asymmetry of emitted electrons by polarizing muon ４. Summary

Contents 1. Introduction Charged Lepton Flavor Violation (CLFV) CLFV searches using muon 𝜇 − 𝑒 − → 𝑒 − 𝑒 − in a muonic atom 2. Transition probability of 𝜇 − 𝑒 − → 𝑒 − 𝑒 − Effective CLFV interactions Distortion of scattering electrons & Relativity of bound leptons Difference between contact & photonic processes ３. Distinguishment of CLFV interaction Let me start. Atomic # dependence of decay rates Energy-angular distribution of emitted electrons Asymmetry of emitted electrons by polarizing muon ４. Summary

Charged Lepton Flavor Violation (CLFV) - A probe for new physics - lepton flavor violation for charged lepton = CLFV forbidden in SM contribution of neutrino mixing → very small cf. current experimental upper limit Br 𝜇→𝑒𝛾 ≲ 10 −54 Br< 4.2×10 −13 cannot be observed by current technology enhanced in many theories beyond SM CLFV is lepton flavor violation for the charged lepton. The process is forbidden in the standard model. As mentioned, the neutrino mass violates the strict conservation of lepton flavors, and the neutrino mixing may generate CLFV process via the higher order electroweak diagram. However the expected branching ratio due to this mechanism is very small, for example branching ratio of mu-e-gamma is smaller than 10-54. While many models beyond the standard model predict sizable CLFV process, much larger than neutrino-osci. effects. Therefore the study of the rare CLFV process is considered as a good tool to search for the physics beyond standard model. e.g. SUSY Searches for CLFV can access high energy physics with little SM backgrounds.

CLFV searches in muon rare decay Advantages of muon 1. high intensity 2. long lifetime current bounds L. Calibbi & G. Signorelli, arXiv:1709.00294 [hep-ph]. 𝜇 − - 𝑒 − conversion Among the current CLFV searches, strong limits are obtained from the muon decay. One of the advantage of muon process is that the high intensity muon is available and another is the long life time of muon. Here I show examples for muonic CLFV searches. Current upper limit of the branching ratio for Br(mu-e-gamma) is 10-13 by MEG exp. That for mu-eee is 10-12 by sindrum. Those two are leptonic process, the CLFV associated with quark is mu-e conversion 10-12 by sindrum II exp. mu-e conversion is a search using a muonic atom, which is an atom with one muon instead of one electron. We are looking for the coming experiments on the mu-e- conversion with higher sensitivity by COMET exp., DeeMe exp., and Mu2e exp. CLFV search using 𝜇-atom exploring 𝜇𝑒𝑞𝑞 interaction New experiments for “ 𝜇 − − 𝑒 − conversion” are planned with higher sensitivity than previous ones. (COMET, DeeMe @ J-PARC, Mu2e @ Fermilab)

𝝁 − 𝒆 − → 𝒆 − 𝒆 − in a muonic atom M. Koike, Y. Kuno, J. Sato, & M. Yamanaka, Phys. Rev. Lett. 105, 121601 (2010). New CLFV search using muonic atoms CLFV 𝜇 − 𝑒 − 𝐸 1 +𝑍𝑒 proposal in COMET R. Abramishvili et al., COMET Phase-I Technical Design Report (2016). 𝑒 − 𝑒 − 𝐸 2 Features clear signal : 𝐸 1 + 𝐸 2 ≃ 𝑚 𝜇 + 𝑚 𝑒 − 𝐵 𝜇 − 𝐵 𝑒 In this talk I will concentrate on the CLFV process of muonic atom proposed in 2010 by Koike and collaborators. The process is mu e decays into e e in muonic atom and has the following features. This processes can be identified by observing two electrons, where the sum of the energies of two electrons is given by initial lepton masses and binding energies, and this is supposed to be a clear signal. With this process, one can prove both the contact and the photonic CLFV mechanisms like m3e. The decay rate is proportional to cubic power of atomic number as I will explain in later slides. Actually this CLFV process is mentioned in the proposal of COMET. 2 CLFV mechanisms 𝜇 − 𝑒 − 𝜇 − 𝑒 − contact ( 𝜇𝑒𝑒𝑒 vertex ) 𝛾 ∗ photonic ( 𝜇𝑒𝛾 vertex ) 𝑒 − 𝑒 − 𝑒 − 𝑒 − (similar to 𝜇 + → 𝑒 + 𝑒 + 𝑒 − ) atomic # 𝑍 : large ⇒ decay rate Γ : large Γ∝ 𝑍−1 3

Comparison to other muonic CLFV Typical effective CLFV interactions 1. 𝜇𝑒𝛾 2. 𝜇𝑒𝑒𝑒 3. 𝜇𝑒𝑞𝑞 𝜇 𝜇 𝑒 𝑒 𝜇 𝑒 𝛾 𝑒 𝑒 𝑞 𝑞 1. 𝜇𝑒𝛾 2. 𝜇𝑒𝑒𝑒 3. 𝜇𝑒𝑞𝑞 𝜇 + → 𝑒 + 𝛾 ✓ - 𝜇 + → 𝑒 + 𝑒 − 𝑒 + 𝜇 − - 𝑒 − conv. 𝜇 − 𝑒 − → 𝑒 − 𝑒 − Let us compare the property of me2ee process with the others. Here I consider these three typical CLFV interactions. First one is meg interaction, second is leptonic 4-Fermi, and third is 4-Fermi interaction which includes quarks. The meg process is sensitive to only meg interaction. On the other hand, by the m3e process, we can prove leptonic contact interaction in addition to meg interaction. However, of course, meg process has stronger sensitivity for meg interaction, because the m3e diagram needs QED vertex besides exotic one. By the m-e conv., we can study CLFV interaction with light quarks. So, here goes our process. As mentioned, the property of me2ee is similar to m3e process.

Comparison to 𝝁 + → 𝒆 + 𝒆 + 𝒆 − 𝜇 − 𝑒 − → 𝑒 − 𝑒 − in a muonic atom 𝜇 + → 𝑒 + 𝑒 + 𝑒 − 𝑒 − 𝜇 𝑏 − 𝑒 − 𝜇 + 𝑒 + 𝑒 𝑏 − 𝑒 − 𝑒 + difference 1 : signal 2 𝑒 − s 1 𝑒 − & 2 𝑒 + s I told you that the me2ee is similar to m3e. Here let me note the difference between them. I show two differences. The one is obvious: the signal is different. Searching me2ee looks for a pair of electrons with fixed total energy. On the other hand, for m3e, you have to detect 3 electrons and reconstruct muon mass. The other is about the interference property. By the difference, it is possible to test the relative phase of CLFV couplings. Therefore the me2ee process is certainly complemental compared to the previous CLFV searches. difference 2 : interference among CLFV couplings 𝐶 𝑋𝑌 𝑍 𝑒 Γ 𝑍 𝑃 𝑋 𝜇 𝑒 Γ 𝑍 𝑃 𝑌 𝑒 Γ 𝜇e→𝑒𝑒 ∝ 𝐶 𝑅𝑅 𝑆 +4 𝐶 𝐿𝐿 𝑉 2 + 𝐶 𝐿𝐿 𝑆 +4 𝐶 𝑅𝑅 𝑉 2 +4 𝐶 𝑅𝐿 𝑉 − 𝐶 𝐿𝑅 𝑉 2 Γ 𝜇→𝑒𝑒𝑒 ∝ 𝐶 𝑅𝑅 𝑆 2 + 𝐶 𝐿𝐿 𝑆 2 + 𝐶 𝑅𝑅 𝑉 2 + 𝐶 𝐿𝐿 𝑉 2 +8 𝐶 𝑅𝐿 𝑉 2 +8 𝐶 𝐿𝑅 𝑉 2

(Rough) Estimation of decay rate “flux” Γ= 𝜎 𝑣 rel ∫d𝑉 𝜌 𝜇 𝜌 𝑒 Suppose nuclear Coulomb potential is weak, Γ 𝜇 − 𝑒 − → 𝑒 − 𝑒 − = 2𝜎 𝑣 rel 𝜓 1𝑆 𝑒 0 2 Phys. Rev. Lett. 105,121601 (2010). (sum of two 1𝑆 𝑒 − s) 𝜎 : cross section of 𝜇 − 𝑒 − → 𝑒 − 𝑒 − 𝑣 rel : relative velocity of 𝜇 − & 𝑒 − (free particles’) 𝜓 1𝑆 𝑒 𝑟 = 𝑚 𝑒 𝑍−1 𝛼 3 𝜋 exp − 𝑚 𝑒 𝑍−1 𝛼 𝑟 So let us review the simple estimation of the rate of me2ee, which is given in this paper. Suppose the nuclear Coulomb potential is weak, we can divide the cross section for free particle from overlap of lepton wave functions, and the rate can be written by this formula. The formula contains several factors. The factor of two means that there are two electrons in 1s state. The sigma indicates a cross section of free muon and electron into two electrons. The overlap of bound muon and electron is represented as the value at the origin of electron wave function because bound muon is much more compact than bound electron due to the heavy mass. The wave function of bound electron can be written like this by solving the Schrödinger equation. Here, for electron, the charge of potential is Z-1 rather than Z, considering the screening of muon. As result, the decay rate is proportional to Z to 3rd power. I want to emphasize that the Z dependence does not depend on the CLFV operator, due to the factorization. ： wave function of 1𝑆 bound electron (non-relativistic) Γ∝ 𝑍−1 3 (the same 𝑍 dependence in the both contact & photonic cases)

Branching ratio of CLFV decay How many muonic atoms decay with CLFV, compared to created # ? Γ∝ 𝑍−1 3 BR 𝜇 − 𝑒 − → 𝑒 − 𝑒 − ≡ 𝜏 𝜇 Γ 𝜇 − 𝑒 − → 𝑒 − 𝑒 − due to existence prob. of bound 𝑒 − at the origin cf. 2.2μs for a muonic H (𝑍=1) 𝜏 𝜇 : lifetime of a muonic atom 80ns for a muonic Pb (𝑍=82) BR with CLFV coupling fixed on allowed maximum e.g. BR<5.0× 10 −19 for Pb 𝑍=82 if contact process is dominant BR increases with atomic # 𝑍. Let me explain the branching ratio of CLFV process estimated by Koike et al. The branching ratio is given by the ratio between transition rate of CLFV process and the total decay rate of a muonic atom which includes DIO and muon capture. The atomic number dependence of branching ratio is shown here. The strength of the CLFV interaction is estimated by using the upper limit of m->3e or m->eg obtained by experiments. For example, the branching ratio is about 5*10-19 for Pb if we assume that contact process is dominant. As we have mentioned, CLFV decay rate is proportional to the cubic of atomic #. A muonic atom with larger atomic number is favored to search for me2ee. Using muonic atoms with large 𝑍 is favored to search for 𝜇 − 𝑒 − → 𝑒 − 𝑒 − . Phys. Rev. Lett. 105,121601 (2010).

To improve calculation for decay rate previous formula of CLFV decay rate by Koike et al. Γ 𝜇 − 𝑒 − → 𝑒 − 𝑒 − = 2𝜎 𝑣 rel 𝜓 1𝑆 𝑒 0 2 ∝ 𝑍−1 3 Note “𝑍 dependence” comes from only 𝜓 1𝑆 𝑒 0 2 always Γ∝ 𝑍−1 3 emitted 𝑒 − s are expected to be back-to-back with equal energies used approximations (𝑍𝛼≪1) In atoms with large 𝑍, spatial extension of bound lepton ← small orbital radius ≫ wave length of emitted 𝑒 − The results of previous slide is obtained by the following way. The decay rate is expressed by mue to ee cross section times incident relative velocity, which is essentially mue to ee transition probability, times probability of bound electron at the origin. The cubic atomic # dependence of transition rate is due to the electron wave function in this formula, which does not depend on the mechanism of CLFV, because the CLFV interaction is factorized from overlap among wave functions. This formula is derived with following approximation. Spatial extension of bound lepton is larger than the wave length of emitted electron and therefore one can factorize the bound electron wave function at the origin. Bound electron wave can be treated in non-relativistic approximation. And the Coulomb interaction of emitted electron with a nucleus can be neglected and plane wave approximation is used for electron wave functions. It is questionable to use these approximations for larger atomic number. Therefore the approximate analysis should be improved for quantitative estimation of transition rate. bound lepton : non-relativistic ← relativistic (especially, 𝑒 − ) emitted 𝑒 − : plane wave ← Coulomb distortion More quantitative estimation is needed ! (important for large 𝑍)

Contents 1. Introduction Charged Lepton Flavor Violation (CLFV) CLFV searches using muon 𝜇 − 𝑒 − → 𝑒 − 𝑒 − in a muonic atom 2. Transition probability of 𝜇 − 𝑒 − → 𝑒 − 𝑒 − Effective CLFV interactions Distortion of scattering electrons & Relativity of bound leptons Difference between contact & photonic processes ３. Distinguishment of CLFV interaction Let me move on to the improved estimation for the rate of me2ee. Atomic # dependence of decay rates Energy-angular distribution of emitted electrons Asymmetry of emitted electrons by polarizing muon ４. Summary

Effective Lagrangian for 𝝁 − 𝒆 − → 𝒆 − 𝒆 − ℒ 𝐼 = ℒ 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 + ℒ 𝜇→𝑒𝛾 ℒ contact =− 4 𝐺 𝐹 2 [ 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 + 𝑔 2 𝑒 𝑅 𝜇 𝐿 𝑒 𝑅 𝑒 𝐿 + 𝑔 3 𝑒 𝑅 𝛾 𝜇 𝜇 𝑅 𝑒 𝑅 𝛾 𝜇 𝑒 𝑅 + 𝑔 4 𝑒 𝐿 𝛾 𝜇 𝜇 𝐿 𝑒 𝐿 𝛾 𝜇 𝑒 𝐿 + 𝑔 5 𝑒 𝑅 𝛾 𝜇 𝜇 𝑅 𝑒 𝐿 𝛾 𝜇 𝑒 𝐿 + 𝑔 6 𝑒 𝐿 𝛾 𝜇 𝜇 𝐿 𝑒 𝑅 𝛾 𝜇 𝑒 𝑅 ]+[𝐻.𝑐.] ℒ 𝜇→𝑒𝛾 =− 4 𝐺 𝐹 2 𝑚 𝜇 𝐴 𝑅 𝑒 𝐿 𝜎 𝜇𝜈 𝜇 𝑅 𝐹 𝜇𝜈 + 𝐴 𝐿 𝑒 𝑅 𝜎 𝜇𝜈 𝜇 𝐿 𝐹 𝜇𝜈 +[𝐻.𝑐.] 𝜇 𝑒 𝛾 ∗ 𝑒 𝜇 We start from an effective CLFV interaction Lagrangian which generates mue->ee process. Here I wrote down general terms which respect Lorentz invariance and electromagnetic U(1) gauge symmetries. Operators with a dimension higher than 6 are ignored. There are two parts, contact and photonic. The contact interactions indicate vertices among a muon and 3 electrons, which generate a short range process. In four terms g1 to g4 of contact interactions, final electrons have the same chirality, and in g5 and g6 terms, final electrons have the opposite chirality. While the photonic interactions represent vertices among a muon, an electron, and a photon, which generate a long range process together with a standard electromagnetic vertex. In our study, coupling constants g1 to g6, AL and AR are estimated from maximum branching ratio from other CLFV experiments. contact interaction photonic interaction constrained by 𝜇→𝑒𝑒𝑒 constrained by 𝜇→𝑒𝛾

Effective Lagrangian for 𝝁 − 𝒆 − → 𝒆 − 𝒆 − ℒ 𝐼 = ℒ 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 + ℒ 𝜇→𝑒𝛾 ℒ contact =− 4 𝐺 𝐹 2 [ 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 + 𝑔 2 𝑒 𝑅 𝜇 𝐿 𝑒 𝑅 𝑒 𝐿 + 𝑔 3 𝑒 𝑅 𝛾 𝜇 𝜇 𝑅 𝑒 𝑅 𝛾 𝜇 𝑒 𝑅 + 𝑔 4 𝑒 𝐿 𝛾 𝜇 𝜇 𝐿 𝑒 𝐿 𝛾 𝜇 𝑒 𝐿 + 𝑔 5 𝑒 𝑅 𝛾 𝜇 𝜇 𝑅 𝑒 𝐿 𝛾 𝜇 𝑒 𝐿 + 𝑔 6 𝑒 𝐿 𝛾 𝜇 𝜇 𝐿 𝑒 𝑅 𝛾 𝜇 𝑒 𝑅 ]+[𝐻.𝑐.] ℒ 𝜇→𝑒𝛾 =− 4 𝐺 𝐹 2 𝑚 𝜇 𝐴 𝑅 𝑒 𝐿 𝜎 𝜇𝜈 𝜇 𝑅 𝐹 𝜇𝜈 + 𝐴 𝐿 𝑒 𝑅 𝜎 𝜇𝜈 𝜇 𝐿 𝐹 𝜇𝜈 +[𝐻.𝑐.] 𝜇 𝑒 𝛾 ∗ 𝑒 𝜇 We start from an effective CLFV interaction Lagrangian which generates mue->ee process. Here I wrote down general terms which respect Lorentz invariance and electromagnetic U(1) gauge symmetries. Operators with a dimension higher than 6 are ignored. There are two parts, contact and photonic. The contact interactions indicate vertices among a muon and 3 electrons, which generate a short range process. In four terms g1 to g4 of contact interactions, final electrons have the same chirality, and in g5 and g6 terms, final electrons have the opposite chirality. While the photonic interactions represent vertices among a muon, an electron, and a photon, which generate a long range process together with a standard electromagnetic vertex. In our study, coupling constants g1 to g6, AL and AR are estimated from maximum branching ratio from other CLFV experiments. contact interaction photonic interaction constrained by 𝜇→𝑒𝑒𝑒 constrained by 𝜇→𝑒𝛾

Our formulation for decay rate Γ= 𝑓 𝑖 2𝜋 𝛿( 𝐸 𝑓 − 𝐸 𝑖 ) 𝜓 𝑒 𝒑 1 , 𝑠 1 𝜓 𝑒 𝒑 2 , 𝑠 2 𝐻 𝜓 𝜇 1𝑠, 𝑠 𝜇 𝜓 𝑒 1𝑠, 𝑠 𝑒 2 use partial wave expansion to express the distortion 𝜓 𝑒 𝒑,𝑠 = 𝜅,𝜇,𝑚 4𝜋 𝑖 𝑙 𝜅 ( 𝑙 𝜅 ,𝑚,1/2,𝑠| 𝑗 𝜅 ,𝜇) 𝑌 𝑙 𝜅 ,𝑚 ∗ ( 𝑝 ) 𝑒 −𝑖 𝛿 𝜅 𝜓 𝑝 𝜅,𝜇 𝜅 : index of angular momentum get radial functions by solving “Dirac eq. with 𝜙” numerically The decay rate is calculated by this standard formula. This delta function means just the energy conservation. In this calculation, since we treat that the nucleus is a very heavy potential source, the 3-momentum is not needed to be conserved. In calculating the transition density, we need lepton wave functions in both initial and final states. In order to express the distortion of final electrons, we use the partial wave expansion. Assuming that the potential is spherical, we can analytically integrate the angular part by the Racah algebra. For calculating the radial part of the partial waves and bound waves, we solve Dirac eq. with Coulomb potential numerically. 𝑑 𝑔 𝜅 (𝑟) 𝑑𝑟 + 1+𝜅 𝑟 𝑔 𝜅 𝑟 − 𝐸+𝑚+𝑒𝜙 𝑟 𝑓 𝜅 𝑟 =0 𝜙 : nuclear Coulomb potential 𝑑 𝑓 𝜅 (𝑟) 𝑑𝑟 + 1−𝜅 𝑟 𝑓 𝜅 𝑟 + 𝐸−𝑚+𝑒𝜙 𝑟 𝑔 𝜅 𝑟 =0 𝜓 𝒓 = 𝑔 𝜅 𝑟 𝜒 𝜅 𝜇 ( 𝑟 ) 𝑖𝑓 𝜅 𝑟 𝜒 −𝜅 𝜇 ( 𝑟 )

Contact process bound 𝜇 − bound 𝑒 − scattering 𝑒 − scattering 𝑒 − overlap of bound 𝜇 − , bound 𝑒 − , and two scattering 𝑒 − s 𝑟 2 𝑔 𝜇 1𝑠 𝑟 𝑔 𝑒 1𝑠 𝑟 𝑔 𝐸 1/2 𝜅=−1 𝑟 𝑔 𝐸 1/2 𝜅=−1 𝑟 wave functions shift scat. 𝑒 − : plane → distorted to the center bound 𝑒 − : non-relativistic → relativistic First, let us consider contact process. Here bound muon and electron contact and electron pair emit at one point. I show typical transition density, overlap of wave functions. You can see that distortion and relativistic effect enhance the transition rate. It is because these effects make wave functions shift to the center and the overlap gets larger. transition rate increases! 𝑟 [fm]

Upper limits of BR (contact process) 𝐵𝑅( 𝜇 + → 𝑒 + 𝑒 − 𝑒 + )<1.0× 10 −12 𝐵𝑅 𝜇 − 𝑒 − → 𝑒 − 𝑒 − < 𝐵 𝑚𝑎𝑥 (SINDRUM, 1988) 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 this work (1s+2s+…) this work (1s) 𝑩 𝒎𝒂𝒙 Koike et al. (1s) This figure shows the current upper limit of branching ratio for contact process. It is estimated by the experiment of mu->3e, SINDRUM result. Blue curve is previous result and red one is our result. By including the Coulomb distortion and relativistic effect, the branching ratio is enhanced compared to the previous analysis. This enhancement factor is about 7 in Pb case. atomic #, 𝒁 inverse of 𝐵 𝑚𝑎𝑥 (𝑍=82) 2.1× 10 18 3.0× 10 17

Photonic process bound 𝜇 − bound 𝑒 − scattering 𝑒 − scattering 𝑒 − 𝛾 ∗ 𝑒 − bound 𝑒 − scattering 𝑒 − 𝑒 − 𝑒 − scattering 𝑒 − 𝑟 2 𝑔 𝜇 1𝑠 𝑟 𝑔 𝐸 1/2 𝜅=−1 𝑟 𝑗 0 𝑞 0 𝑟 𝑟 2 𝑔 𝑒 1𝑠 𝑟 𝑔 𝐸 1/2 𝜅=−1 𝑟 𝑗 0 𝑞 0 𝑟 scat. 𝑒 − : plane → distorted scat. 𝑒 − : plane → distorted bound 𝑒 − : non-relativistic → relativistic On the other hand, the situation is quite different in photonic process. The diagram has two vertices: the one is of CLFV, and the other is of QED. Transition densities are shown again for each vertex. It may be a little hard to see, but the distortion of scattering electron makes overlap integral decrease. 𝑟 [fm] 𝑟 [fm] distortion of scattering 𝑒 − overlap integral decreases

Upper limits of BR (photonic process) 𝐵𝑅( 𝜇 + → 𝑒 + 𝛾)<4.2× 10 −13 𝐵𝑅 𝜇 − 𝑒 − → 𝑒 − 𝑒 − < 𝐵 𝑚𝑎𝑥 (MEG, 2016) 𝐴 𝑅 𝑒 𝐿 𝜎 𝜇𝜈 𝜇 𝑅 𝐹 𝜇𝜈 Koike et al. (1s) 𝑩 𝒎𝒂𝒙 this work (1s) this work (1s+2s+…) This is a figure showing upper limit of branching ratio for photonic process. It is limited by MEG experiment. The result is quite different from that of contact process. In Pb case, the decay rate is a quarter as large as Koike's result. 𝒁 inverse of 𝐵 𝑚𝑎𝑥 (𝑍=82) 1.8× 10 18 6.6× 10 18

Effect of distortion enhanced !! scat. 𝑒 − : distorted wave (Assuming momentum conservation at each vertex) bound 𝜇 − emitted 𝑒 − bound 𝜇 − emitted 𝑒 − bound 𝑒 − emitted 𝑒 − bound 𝑒 − emitted 𝑒 − contact process photonic process momentum transfers to bound leptons When Coulomb interaction of two electrons is switched on, electron wave function is enhanced near the origin due to the attractive Coulomb potential and local momentum in this region is increased compared with the asymptotic momentum. For contact interaction, zero-momentum transfer to the bound leptons is achieved even if the local momentum increases. The overlap integral increases because of increase of the wave function at origin. For the one-photon exchange interaction, the propagated photon remains the same as PLW, while the local momentum of electrons increases in DW. Then the overlap integral of both vertices decrease because of non-zero momentum transfer to the bound leptons. This suppression effect is larger than enhancement due to the increase of electron wave function at origin. make overlap integrals smaller Totally (combined with the effect to enhance the value near the origin), enhanced !! suppressed…

Contents 1. Introduction Charged Lepton Flavor Violation (CLFV) CLFV searches using muon 𝜇 − 𝑒 − → 𝑒 − 𝑒 − in a muonic atom 2. Transition probability of 𝜇 − 𝑒 − → 𝑒 − 𝑒 − Effective CLFV interactions Distortion of scattering electrons & Relativity of bound leptons Difference between contact & photonic processes ３. Distinguishment of CLFV interaction We found the different modification of distortion between photonic and contact operators. Utilizing the result, I would like to discuss the distinguishment of CLFV interaction by experimental observables after we get CLFV signals in future. Atomic # dependence of decay rates Energy-angular distribution of emitted electrons Asymmetry of emitted electrons by polarizing muon ４. Summary

Distinguishing method 1 ~ atomic # dependence of decay rates ~ 𝑍 dependence of Γ 𝚪 𝒁 𝚪 𝐙=𝟐𝟎 contact 𝐴 𝐿 : 𝑔 1 =1:100 photonic The first possibility of discriminating methods is atomic # dependence of the decay rates. Here Gamma is decay rate including the Coulomb effects. As I have shown, the Coulomb effects make the decay rates larger for contact process but smaller for photonic process. Thus it may be possible to discriminate contact and photonic processes observing CLFV of muonic atoms with different atomic #. 𝒁 The 𝑍 dependences are different among interactions. That of contact process is strongly increasing, while that of photonic process is moderately increasing.

Distinguishing method 2 ~ energy and angular distributions ~ 𝐸 1 𝐸 2 𝜃 12 𝐸 1 : energy of an emitted electron 𝜃 : angle between two emitted electrons 𝑍=82 𝟏 𝚪 𝐝 𝟐 𝚪 𝐝 𝐄 𝟏 𝐝𝐜𝐨𝐬𝜽 𝟏 𝚪 𝐝 𝟐 𝚪 𝐝 𝐄 𝟏 𝐝𝐜𝐨𝐬𝜽 [ 𝐌𝐞𝐕 −𝟏 ] [ 𝐌𝐞𝐕 −𝟏 ] 𝐜𝐨𝐬𝛉 contact 𝐜𝐨𝐬𝛉 photonic The second possibility is energy and angular distributions of emitted electron pair. As mentioned before, since the total momentum of emitted electrons is not need to be conserved, we have information in the distribution. In both contact and photonic interactions, the most favored kinematics is that two electrons have the same energies and emit back-to-back. However the distributions are a little different between contact and photonic interactions. 𝐄 𝟏 [𝐌𝐞𝐕] 𝐄 𝟏 [𝐌𝐞𝐕] The distributions are (a little) different among interactions.

Model distinguishing power We can distinguish “contact” or “photonic”. method 1. 𝑍-dep. of decay rates method 2. energy-angular distribution Γ 𝑍 Γ 𝑍=20 𝒁 Here we have two methods to distinguish between contact and photonic interactions. Next you may need a way to identify between left and right of CLFV operator. For example, g1 and g2 type interactions cannot be identified in the usual way. We need some technique. Can we distinguish “left” or “right” ? e.g. 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 & 𝑔 2 𝑒 𝑅 𝜇 𝐿 𝑒 𝑅 𝑒 𝐿

Distinguishing method 3 ~ electron asymmetry from polarized muon ~ 𝑒 𝐿 − 𝑒 𝐿 − 𝑝 1 ℒ 𝐶𝐿𝐹𝑉 = 𝑒 𝐿 𝜇 𝑅 𝑒 𝑒 In preparation 𝑃 cos 𝜃 1 = 𝑃 ⋅ 𝑝 1 𝜇 − dΓ dcos 𝜃 1 = Γ 2 1+𝛼𝑃cos 𝜃 1 𝛼<0 is expected 𝑝 1 ( ℒ 𝐶𝐿𝐹𝑉 = 𝑒 𝑅 𝜇 𝐿 𝑒 𝑒 ) 𝛼>0 is expected 𝑒 𝐿 − Measurement of angular distribution asymmetry Determination of dominant interaction !? cf : 𝜇 + → 𝑒 + 𝛾, 𝜇 + → 𝑒 + 𝑒 + 𝑒 − with polarized muon Y. Kuno & Y. Okada, Phys. Rev. Lett. 77, 434 (1996). Y. Okada, K. Okumura & Y. Shimizu, Phys. Rev. D 61, 094001 (2000).

d 5 Γ d 𝐸 1 d Ω 1 d Ω 2 ∝ 1+ 𝐹 1 𝑃 ⋅ 𝑝 1 + 𝐹 2 𝑃 ⋅ 𝑝 2 + 𝐹 𝐷 𝑃 ⋅ 𝑝 1 × 𝑝 2 Final state is determined by 4 parameters, say, ( 𝐸 1 , 𝜃 1 , 𝜃 2 , 𝜃 12 ) 𝜃 1 : angle between 𝑃, 𝑝 1 2 are fixed for examples 𝜃 2 : 𝑃, 𝑝 2 𝜃 12 : 𝑝 1 , 𝑝 2 Example 1 : 𝜃 1 = 𝜃 2 = 𝜃 12 /2 𝜃 12 𝐴𝑠𝑦𝑚. = 𝐹 𝐸 1 , 𝐸 2 , c 12 +𝐹 𝐸 2 , 𝐸 1 , c 12 cos 𝜃 12 /2 = 𝐹 𝑆 𝐸 1 , 𝐸 2 , c 12

𝐹 𝑆 𝑔 1 𝑔 3 c 12 𝜖 1 𝐴 𝑅 𝑔 5

Example 2 : 𝜃 1 =𝜋− 𝜃 2 = 𝜋 2 − 𝜃 12 2 𝜃 12 𝐴𝑠𝑦𝑚. = 𝐹 𝐸 1 , 𝐸 2 , c 12 −𝐹 𝐸 2 , 𝐸 1 , c 12 sin 𝜃 12 /2 = 𝐹 𝐴 𝐸 1 , 𝐸 2 , c 12

𝐹 𝐴 𝑔 1 𝑔 3 c 12 𝜖 1 𝐴 𝑅 𝑔 5

・ In all cases there is asymmetry Useful to determine the parity violation of effective couplings Especially for photonic interaction ・ Relativistic treatment is important ・ 𝑔 1 type In non-relativistic limit , exactly 0 Even if relativistic , if nuclear is point like the asymmetry is 0 ∵ asymmetry ∝∫𝑑𝑟 𝑔 𝜇 𝑟 𝑓 𝑒 𝑟 − 𝑓 𝜇 𝑟 𝑔 𝑒 𝑟 𝑗 1 𝑝𝑟 ・Distortion is very important ・ 𝑔 5 type In any case , non-zero ・ Shape of Assymetry can determine the interaction !?

Contents 1. Introduction Charged Lepton Flavor Violation (CLFV) CLFV searches using muon 𝜇 − 𝑒 − → 𝑒 − 𝑒 − in a muonic atom 2. Transition probability of 𝜇 − 𝑒 − → 𝑒 − 𝑒 − Effective CLFV interactions Distortion of scattering electrons & Relativity of bound leptons Difference between contact & photonic processes ３. Distinguishment of CLFV interaction Let me summarize my talk. Atomic # dependence of decay rates Energy-angular distribution of emitted electrons Asymmetry of emitted electrons by polarizing muon ４. Summary

Summary 𝜇 − 𝑒 − → 𝑒 − 𝑒 − process in a muonic atom interesting candidate for CLFV search Our finding Distortion of emitted electrons Relativistic treatment of a bound electron are important in calculating decay rates. Distortion makes difference between 2 processes. contact process ：decay rate Enhanced (7 times Γ 0 in 𝑍=82) photonic process：decay rate suppressed (1/4 times Γ 0 in 𝑍=82) me->ee process in a muonic atom is an interesting candidate for CLFV search. In calculation of decay rate, important effects are distortion of emitted electrons and relativistic treatment of a bound electron. As the result, we found that decay rate is enhanced for contact process and suppressed for photonic process compared to the previous analysis. We have proposed model-discriminating methods. Candidates are atomic # dependence of decay rate, and energy-angular distribution of emitted electrons. Using muon polarization is an interesting subject because it has possibility to identify the parity-violating component of CLFV interaction. How to discriminate interactions, found by this analyses atomic # dependence of the decay rate energy and angular distributions of emitted electrons asymmetry of electron emission by polarized muon

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Coulomb prevents the contact process? Use the simple Hamiltonian (a muon & an electron in nuclear potential) 𝐻=− 𝑖=𝜇,𝑒 𝛁 𝑖 2 2 𝑚 𝑒 − 𝑖=𝜇,𝑒 𝑍𝛼 𝒓 𝑖 + 𝛼 𝒓 𝜇 − 𝒓 𝑒 Assume that the form of the wave function is 𝜓 𝑎,𝑏 𝑍 𝜇 , 𝑍 𝑒 𝒓 𝜇 , 𝒓 𝑒 = 𝑁 𝑎,𝑏 𝑍 𝜇 , 𝑍 𝑒 exp − 𝑚 𝜇 𝑍 𝜇 𝛼 𝒓 𝜇 exp − 𝑚 𝑒 𝑍 𝑒 𝛼 𝒓 𝑒 × 1−𝑏exp −𝑎 𝒓 𝜇 − 𝒓 𝑒 find the parameter set to minimize the energy Our calculation is based on the independent particle picture of the initial state. I’m afraid you may doubt that the contact interaction really happens, because the repulsive Coulomb force seems to suppress the overlap between muon and electron. In order to answer it, I tried to analyze the bound wave function by the variational method. Here I use the simple Hamiltonian which describes the system of a muon and an electron in a Coulomb potential by a point charge. Let us assume that the form of the wave function is like this. I want to know how large the suppression factor to overlap between muon and electron is. There are four variational parameters: Zm, Ze, a, b. In my analysis shown in the previous slides, the extra factor is not taken into account. As a result of the variational method, it is found that Zm~Z, Ze~Z-1, and b~0. We can safely neglect the extra factor, actually. 𝑍 𝜇 ≃𝑍 𝑏≃0 𝑍 𝑒 ≃𝑍−1 We can safely neglect the additional factor.

Radial wave function (bound 𝒆 − ) Pb case 208 𝑍=81 (considering 𝜇 − screening) 𝑔 𝑒 1𝑠 𝑟 [ MeV 1/2 ] Type 𝐵 𝑒 (MeV) Relativistic 9.88× 10 −2 Non-relativistic 8.93× 10 −2 Before the main result, let us see the wave functions. This is that of the bound electron. The blue dashed curve is a non-relativistic wave and the red solid curve is a relativistic wave. The important notice is that relativistic effect enhances the value near the origin. This relativistic effect is expected to enhance the overlap with muon wave function. 𝑟 [fm] Relativity enhances the value near the origin.

Radial wave function (scattering 𝒆 − ) e.g. 𝜅=−1 partial wave Pb case 208 𝑟 𝑔 𝐸 1/2 𝜅=−1 𝑟 𝑍=82 [ MeV −3/2 ] 𝐸 1/2 ≈48MeV shifted by nuclear Coulomb potential distorted wave plane wave Next is scattering electrons. The wave function of a scattering electron is shifted to the origin because of attractive nuclear Coulomb potential. There are two effects of the distortion. The first is enhancing value near the origin, which makes overlap of lepton wave functions larger. The second is giving phase shifts or extra local momenta to emitted electrons. 𝑟 [fm] ① enhanced value near the origin ② local momentum increased effectively

Radial wave function (bound 𝝁 − ) Backup Radial wave function (bound 𝝁 − ) Pb case 208 𝑍=82 𝑟 𝑔 𝜇 1𝑠 𝑟 ,𝑟 𝑓 𝜇 1𝑠 𝑟 [ MeV −1/2 ] 𝑟 𝑔 𝜇 1𝑠 𝑟 : solid 𝐵 𝜇 :10.5 MeV 𝑟 𝑓 𝜇 1𝑠 𝑟 : dotted Before main results, I’ll show radial wave functions of leptons. The horizontal axis indicates distance from the center of nucleus. This is radial wave function of 1s bound muon in Pb case. As you can see, a bound muon go round close to a charge radius of nucleus. Therefore it is important to consider finite nuclear radius in constructing the Coulomb potential. 𝑟 [fm] (radius of 208 Pb ) It is important to consider finite nuclear charge radius.

Effect of finite size of muon wave Γ/ Γ 0 ( Γ 0 : the rate in the previous approx. ) scattering 𝑒 − : plane wave bound lepton : nonrelativistic contact ( 𝑔 1 - 𝑔 4 ) contact ( 𝑔 5 - 𝑔 6 ) photonic ( 𝐴 𝑅 - 𝐴 𝐿 ) 𝑍 Momentum fluctuation of bound muon overlap integral small

Decay rate Γ 𝜇 − 1𝑆 𝑒 − 𝛼 → 𝑒 − 𝑒 − Backup Decay rate 𝐸 1 𝜃 𝐸 2 Γ 𝜇 − 1𝑆 𝑒 − 𝛼 → 𝑒 − 𝑒 − = 1 2 𝑚 𝑒 𝑚 𝜇 − 𝐵 𝜇 1𝑆 − 𝐵 𝑒 𝛼 d 𝐸 1 −1 1 d cos𝜃 d 2 Γ d 𝐸 1 dcos𝜃 𝐸 1 : energy of an emitted electron 𝜃 : angle between two emitted electrons differential decay rate : 𝑃 𝑙 : Legendre polynomial d 2 Γ d 𝐸 1 dcos𝜃 = 𝜅 1 , 𝜅 2 , 𝜅 1 ′ , 𝜅 2 ′ ,𝐽,𝑙 𝑀 𝐸 1 , 𝜅 1 , 𝜅 2 ,𝐽 𝑀 ∗ 𝐸 1 , 𝜅 1 ′ , 𝜅 2 ′ ,𝐽 ×𝑤 𝜅 1 , 𝜅 2 , 𝜅 1 ′ , 𝜅 2 ′ ,𝐽,𝑙 𝑃 𝑙 cos𝜃 Eventually, the decay rate can be written by the differential decay rate like this form. E1 is energy of one emitted electron and theta is angle between two electrons. We can calculate analytic form of M and w. M is transition amplitude which can be determined by the structure of operators. w is weighting factor which can be determined by geometry. In general, interaction terms interfere, but in this talk, I will show only special cases with just one CLFV coupling. 𝑀 𝐸 1 , 𝜅 1 , 𝜅 2 ,𝐽 = 𝑖=1,⋯,6 𝑔 𝑖 𝑀 contact 𝑖 𝐸 1 , 𝜅 1 , 𝜅 2 ,𝐽 + 𝑗=𝐿,𝑅 𝑔 𝑗 𝑀 photo 𝑗 𝐸 1 , 𝜅 1 , 𝜅 2 ,𝐽 contact photonic

Discriminating method 2 Backup Discriminating method 2 𝐸 1 𝜃 𝜃 : angle between two emitted electrons 𝐸 2 angular distribution (cos𝜃≈1) 𝑍=82 𝐝𝚪 𝚪𝐝𝐜𝐨𝐬𝜽 contact (same chirality) contact (opposite chirality) 𝐜𝐨𝐬𝜽 𝑒 − pair has same chirality 𝑒 − pair cannot emit same momentum (due to Pauli principle)

Contribution from all bound 𝒆 − s Backup Contribution from all bound 𝒆 − s normalize the contribution of 1𝑆 𝑒 − to 1 contact 𝑔 1 1S 2S 2P 3S 3P 3D 4S Total 1 0.17 6.2× 10 −3 5.1× 10 −2 3.1× 10 −3 2.3× 10 −9 2.1× 10 −2 1.25 photonic 𝑔 𝐿 1S 2S 2P 3S 3P 3D 4S Total 1 0.15 7.3× 10 −3 4.3× 10 −2 2.6× 10 −3 2.4× 10 −5 1.8× 10 −2 1.21 it is sufficient to consider about 𝑆 electrons for both cases

終状態のkinematicsを決めるパラメータは4つ ( 𝐸 1 , 𝜃 1 , 𝜃 2 , 𝜃 12 ) 非対称度の測定 d 5 Γ d 𝐸 1 d Ω 1 d Ω 2 = 1 8 𝜋 2 d 2 Γ d 𝐸 1 d c 12 1+𝐹 𝐸 1 , 𝐸 2 , c 12 𝑃 ⋅ 𝑝 1 +𝐹 𝐸 2 , 𝐸 1 , c 12 𝑃 ⋅ 𝑝 2 終状態のkinematicsを決めるパラメータは4つ ( 𝐸 1 , 𝜃 1 , 𝜃 2 , 𝜃 12 ) 𝜃 1 : 𝑃− 𝑝 1 の角度 𝜃 2 : 𝑃− 𝑝 2 2つを固定して図を作成 𝜃 12 : 𝑝 1 − 𝑝 2 例 : 𝜃 1 = 𝜃 2 = 𝜃 12 /2 𝜃 12 𝐴𝑠𝑦𝑚. = 𝐹 𝐸 1 , 𝐸 2 , c 12 +𝐹 𝐸 2 , 𝐸 1 , c 12 cos 𝜃 12 /2 = 𝐹 𝑆 𝐸 1 , 𝐸 2 , 𝑐 12

𝐹 𝑆 𝐸 1 , 𝐸 2 , c 12 𝑔 1 𝑔 3 𝑔 5 𝐴 𝑅 cos 𝜃 12 cos 𝜃 12 𝜖 1 𝜖 1

𝐹 𝐴 𝐸 1 , 𝐸 2 , c 12 𝑔 1 𝑔 3 𝑔 5 𝐴 𝑅 cos 𝜃 12 cos 𝜃 12 𝜖 1 𝜖 1