A proposal of a New Charged Lepton Flavor Violation Experoment: m e e e in muonic atom - - - - MEee SATO, Joe (Saitama University) MPI Heidelberg 2016/Sep/11 M. Koike, Y. Kuno, J. S, M.Yamanaka, Phys. Rev. Lett. 105, 121601 Y. Uesaka, Y. Kuno, J. S, T. Sato, M.Yamanaka., Phys. Rev. D93 076006 + arXiv: 1609?????

Charged Lepton Flavor Violation (cLFV) via neutrino oscillation Introduction In Standard Model (SM) Charged Lepton Flavor Violation (cLFV) via neutrino oscillation But … Forever invisible Discovery of the cLFV signal One of the evidence for beyond the SM

Comparing LFV signals and their rates Introduction Supersymmetric model Extra dimension model [M.~Raidal et al., Eur. Phys. J. C 57 (2008)] [K.~Agashe, et al., Phys. Rev. D 74 (2006)] Comparing LFV signals and their rates Discrimination of new physics models Prove for structure of new physics Desire for many detectable cLFV processes

m e e e in muonic atom m Introduction New idea for cLFV search － － － － in muonic atom What is target ? Flavor violation between and e m What is advantage ? Sensitive to both photonic dipole cLFV operator and 4-Fermi contact cLFV operator Clean signal [ back-to-back dielectron ]

Basic Idea (PRL) m e e e － － － －

m e e e Muonic atom nucleus muon electron electron 1S orbit muon 1S orbit nucleus muon electron

m m e e e e e e in muonic atom Interaction rate LFV vertex － e － e － e － in muonic atom electron 1S orbit muon 1S orbit Interaction rate

m m m e e e e e e e in muonic atom Interaction rate LFV vertex m － e － e － e － in muonic atom electron 1S orbit muon 1S orbit Interaction rate m Overlap of wave function of and e

m ∴ m << m e e e Overlap of wave functions Muonic atom Approximation m << m Muon localization at nucleus position ∴ e m Overlap = electron wave function at nucleus

m e e e Muonic atom r : radial coordinate (distance from nucleus) Electron wave function r : radial coordinate (distance from nucleus) Z : atomic number of nucleus in muonic atom Overlap of wave functions

m m e e e e e e in muonic atom Interaction rate LFV vertex m － e － e － e － in muonic atom electron 1S orbit muon 1S orbit Interaction rate Cross section for elemental interaction

m e e e Effective Lagrangian cLFV effective coupling constant [ Y. Kuno and Y. Okada Rev. Mod. Phys. 73 (2001) ] cLFV effective coupling constant Sensitive to the structure of new physics

m e e e Effective Lagrangian 4-Fermi interaction type [ Y. Kuno and Y. Okada Rev. Mod. Phys. 73 (2001) ] 4-Fermi interaction type

m e e e 4-Fermi interaction dominant case Branching ratio Lifetime of free muon (2.197 10 s) × -6 1 2.19 10 s for H × -6 Lifetime of bound muon 238 (7 - 8) 10 s for U × -8

m e e e 4-Fermi interaction dominant case Branching ratio

m m e e e 4-Fermi interaction dominant case Branching ratio ( ) e e e Comparison two BRs probe for CP violating

m ∴ e e e 4-Fermi interaction dominant case Branching ratio Enhancement factor from overlap of wave functions Positive charge attracts muon and electron toward the nucleus position. ∴ Notable advantage for heavy nuclei

m e e e Effective Lagrangian Photonic interaction type [ Y. Kuno and Y. Okada Rev. Mod. Phys. 73 (2001) ] Photonic interaction type

m e e e Photonic interaction dominant case Branching ratio Photon propagator in non-relativistic limit Enhancement factor compared with 4-Fermi case

= m e e e Case : same order cLFV coupling ～ Ratio between photonic and 4-Fermi cross section One of the distinct features for the process

with the naïve estimate Discovery reach with the naïve estimate

m e How to get upper limit for BR( ) Discovery reach m e － How to get upper limit for BR( ) Calculate ratio of the BR to other limited cLFV BR 4-Fermi interaction dominant case Photonic interaction dominant case These ratios are independent on cLFV effective coupling

Discovery reach

Current experimental bound in 4-Fermi interaction dominant case Discovery reach Current experimental bound in 4-Fermi interaction dominant case

Current experimental bound in photonic interaction dominant case Discovery reach Current experimental bound in photonic interaction dominant case

10 10 For run-time 1 year 3 10 s - muon at COMET experiment Discovery reach For run-time 1 year 3 10 s 7 ～ × 10 18 10 19 - muon at COMET experiment

Discovery reach m e e e can be first signal of LFV !?

With these number of muons the process will be seen !! Discovery reach Project Intensity Reach COMET / PRISM 1018 – 1019 μ/year ν factories 1021 μ/year With these number of muons the process will be seen !!

Precise Estimate (PRD +) 1 Total Rate

Precise Estimate High Z nucleus is preferable All leptons are under strong Coulomb potentail Distorted wave function Out-going electron is very energetic >>me & High Z means high velocity for bound leptons Relativistic treatment Nuclei is not a point charge Numerical solution for Wave fns , integration for amplitude Solve Dirac Eq. , integrate Wave funs, numerically For trial, uniform charge density is assumed (not so important)

Calculating method Decay rate Γ Γ=2𝜋 𝑓 𝑖 𝛿( 𝐸 𝑓 − 𝐸 𝑖 ) 𝜓 𝑒 𝑠 1 𝒑 1 𝜓 𝑒 𝑠 2 ( 𝒑 2 ) 𝐻 𝜓 𝜇 𝑠 𝜇 1𝑠 𝜓 𝑒 𝑠 𝑒 (1𝑠) 2 use partial wave expansion to express the distortion 𝜓 𝑒 𝑠 𝒑 = 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 get radial functions by solving Dirac eq. numerically I show how to calculate the decay rate. We can write it down by this well-known formula. In order to express the distortion of final electrons, I use partial wave expansion. And, for calculating the partial waves and bound waves, I solve Dirac eq. numerically. 𝜓 𝒓 = 𝑔 𝜅 𝑟 𝜒 𝜅 𝜇 ( 𝑟 ) 𝑖𝑓 𝜅 𝑟 𝜒 −𝜅 𝜇 ( 𝑟 ) 𝑑 𝑔 𝜅 (𝑟) 𝑑𝑟 + 1+𝜅 𝑟 𝑔 𝜅 𝑟 − 𝐸+𝑚+𝑒𝜙 𝑟 𝑓 𝜅 𝑟 =0 𝑑 𝑓 𝜅 (𝑟) 𝑑𝑟 + 1−𝜅 𝑟 𝑓 𝜅 𝑟 + 𝐸−𝑚+𝑒𝜙 𝑟 𝑔 𝜅 𝑟 =0 𝜙 : nuclear Coulomb potential

m e e e Effective Lagrangian 4-Fermi interaction type [ Y. Kuno and Y. Okada Rev. Mod. Phys. 73 (2001) ] 4-Fermi interaction type

Reaction rate for contact interactions Reaction rate for contact ints. (Amplitude)2 up to J κ：(total angular + orbital) momentum for scattered electron cLFV interactions Physics !! Overlap of wave functions

High angular momentum is very important Γ0 : previous calculation

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, which means CLFV decay ratio to muonic atom decay, for contact process. It can be estimated by the experiment of mu->3e, SINDRUM result. Blue curve is previous result and red one is our result. This enhancement factor is about 7 in Pb case, so that we need 3.0*10^17 muonic atoms at least to update SINDRUM limit. atomic #, 𝒁 needed # of muonic atoms (𝑍=82) 2.1× 10 18 3.0× 10 17 7/14

Dirac Eq. : Final State Precise Estimate Previous calculation(1) Final electrons are described by plane wave Though Out-going electrons are highly relativistic and its wave functions are distorted by nuclear Coulomb potential, especially for high Z Improvement for scattered electrons Coulomb scattering wave function of Dirac Eq. with finite size nucleus Plane wave

Dirac Eq. : Final State Precise Estimate 𝜓 𝜅 𝑟 = 𝑔 𝜅 (𝑟) 𝜒 𝜅 ( 𝑟 ) 𝑖 𝑓 𝜅 (𝑟) 𝜒 −𝜅 ( 𝑟 ) solid line : large component, 𝑔 dotted line : small component, 𝑓 (black line : w.f. plane wave) Wave fun. near the center position becomes larger, which leads enhancement of the overlap and hence reaction rate Muon is located at r<O(10) fm Improvement for scattered electrons Coulomb scattering wave function of Dirac Eq. with finite size nucleus Plane wave

Dirac Eq. : Bound electron Precise Estimate Dirac Eq. : Bound electron Previous calculation (2) Non-relativistic electron wave function for bound state The bound electron is a relativistic Dirac particle Electron orbit radius Nucleus size > Improvement for bound electron Wave function of Dirac particle in point Coulomb potential NR wave function

Dirac Eq. : Bound electron Precise Estimate Dirac Eq. : Bound electron 𝜓 𝜅 𝑟 = 𝑔 𝜅 (𝑟) 𝜒 𝜅 ( 𝑟 ) 𝑖 𝑓 𝜅 (𝑟) 𝜒 −𝜅 ( 𝑟 ) solid line : large component, 𝑔 dotted line : small component, 𝑓 (black line : g: Non-Rel , f=0) Muon is located at r<O(10) fm More attracted, leading enhancement of the overlap Improvement for bound electron Wave function of Dirac particle in point Coulomb potential NR wave function

Dirac Eq. : Bound muon Precise Estimate Previous Calculation (3) Muon is localized at nucleus position Muon is not at center but spread around nucleus Improvement for bound muon Wave function of Dirac particle in Coulomb potential by finite size nucleus Localized muon wave function

Dirac Eq. : Bound muon Precise Estimate 𝜓 𝜅 𝑟 = 𝑔 𝜅 (𝑟) 𝜒 𝜅 ( 𝑟 ) 𝑖 𝑓 𝜅 (𝑟) 𝜒 −𝜅 ( 𝑟 ) solid line : large component, 𝑔 dotted line : small component, 𝑓 (black line : w.f. used in previous work) Density near the center position becomes smaller, leading decline of the overlap and hence the reaction rate Improvement for bound muon Wave function of Dirac particle in Coulomb potential by finite size nucleus Localized muon wave function

Overlap of wave functions enhancement Line : current Dashed: previous

Not only 1S but also 2S, …, contribute Rate : much larger

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, which means CLFV decay ratio to muonic atom decay, for contact process. It can be estimated by the experiment of mu->3e, SINDRUM result. Blue curve is previous result and red one is our result. This enhancement factor is about 7 in Pb case, so that we need 3.0*10^17 muonic atoms at least to update SINDRUM limit. atomic #, 𝒁 needed # of muonic atoms (𝑍=82) 2.1× 10 18 3.0× 10 17 7/14

m e e e Effective Lagrangian Photonic interaction type [ Y. Kuno and Y. Okada Rev. Mod. Phys. 73 (2001) ] Photonic interaction type

Upper limits of BR (photonic process) 𝐵𝑅( 𝜇 + → 𝑒 + 𝛾)<5.7× 10 −13 𝐵𝑅 𝜇 − 𝑒 − → 𝑒 − 𝑒 − < 𝐵 𝑚𝑎𝑥 (MEG, 2013) 𝑔 𝐿 𝑒 𝐿 𝜎 𝜇𝜈 𝜇 𝑅 𝐹 𝜇𝜈 Koike et al. (1s) 𝑩 𝒎𝒂𝒙 this work (1s) 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, so that the needed number of muonic atoms gets 4 times larger. What makes this difference? Preliminary 𝒁 needed # of muonic atoms (𝑍=82) 1.8× 10 18 6.6× 10 18 8/14

Suppresion factor for photonic interaction Phase shift effect of distortion (makes a momentum of 𝑒 − larger effectively) photonic process bound 𝜇 − emitted 𝑒 − No effect on photon by coulomb force No distortion for photon propergater I try to explain that easily. In contact process, we should estimate overlap integrals of 4 lepton wave functions. Since two scattering electrons overlap at one point, these extra momenta are canseled, so there are no momentum mismatches. On the other hand, in photonic process, verteces are separate and a photon propagator is not distorted. Therefore it generates momentum transfers to bound leptons. This makes overlap integrals smaller because high momentum component of a bound lepton is smaller than zero momentum component. Totally, the decay rate by the contact process is enhanced, but that by the photonic process is suppressed by distortion of final electrons. momentum transfers to bound leptons bound 𝑒 − emitted 𝑒 − make overlap integrals smaller Totally (combined with the effect to enhance the value near the origin), suppressed… 10/14

Phase shift effect of distortion (makes a momentum of 𝑒 − larger effectively) contact process photonic process bound 𝜇 − emitted 𝑒 − bound 𝜇 − emitted 𝑒 − I try to explain that easily. In contact process, we should estimate overlap integrals of 4 lepton wave functions. Since two scattering electrons overlap at one point, these extra momenta are canseled, so there are no momentum mismatches. On the other hand, in photonic process, verteces are separate and a photon propagator is not distorted. Therefore it generates momentum transfers to bound leptons. This makes overlap integrals smaller because high momentum component of a bound lepton is smaller than zero momentum component. Totally, the decay rate by the contact process is enhanced, but that by the photonic process is suppressed by distortion of final electrons. bound 𝑒 − emitted 𝑒 − bound 𝑒 − emitted 𝑒 − no momentum mismatches momentum transfers to bound leptons make overlap integrals smaller Totally (combined with the effect to enhance the value near the origin), enhanced !! suppressed… 10/14

Precise Estimate (PRD +) 1 Interaction type dependence How to discriminate ?

Discriminating method 1 ~ atomic # dependence of decay rates ~ 𝑍 dependence of Γ except 𝑍−1 3 𝚪 𝒁 𝒁−𝟏 𝟑 𝚪 𝐙=𝟐 Preliminary contact The first candidate of discriminating methods is atomic # dependence of the decay rates. As I showed, Coulomb effects makes the decay rates larger for contact process but smaller for photonic process. Thus I expect that it may be easy to discriminate contact and photonic processes using some muonic atoms with different atomic #. photonic 𝒁 The 𝑍 dependences are different between interactions. Compared to 𝑍−1 3 , that of contact process is larger, while that of photonic process is smaller. 12/14

Discriminating method 2 ~ energy and angular distributions ~ 𝐸 1 : energy of an emitted electron 𝐸 1 𝜃 𝜃 : angle between two emitted electrons 𝐸 2 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 𝐜𝐨𝐬𝛉 contact process photonic process Preliminary Second candidates are energy and angular distributions about emitted electron pair. That of photonic process widens to the energy direction rather than that of contact. 𝐄 𝟏 [𝐌𝐞𝐕] 𝐄 𝟏 [𝐌𝐞𝐕] 13/14

Summary

m e e e = New LFV process in muonic atom Summary m e New LFV process － － e － e － in muonic atom Clean signal (back to back electron with E m /2) = ～ e m Interaction rate Advantage : Large nucleus Discrimination among interactions : possible Making use of Z dependence, energy spectrum, position dependence

Detectable in on-going or future experiments Summary Detectable in on-going or future experiments Discussion with COMET peaple Unfortunately not a discovery mode …., though

Discussion (photonic dipole Interaction)

Numerical result Excluded Enhancement Upper bound with m 3e current limit Large enhancement of the reaction rate by the improvements Especially the improvements of electron wave functions -18 Example: upper bound for pb nucleus ~3.5 10

The reason of enhancement Wave function of initial electron approaches to nucleus Increase of overlap Wave function density of muon at nucleus becomes smaller Decrease of overlap Wave functions of final electron also approach to nucleus Increase of overlap As a total Enhancement

Cross section of cLFV elemental process cLFV effective Lagrangian Dipole photonic interaction

Photonic interaction: Long-range force Photon propagator : infrared divergent 𝑞 0 = 𝑝 1 − 𝐸 𝜇 = 𝐸 𝑒 − 𝑝 2 ℳ 𝑝 1 , 𝑠 1 ; 𝑝 2 , 𝑠 2 ;1𝑆, 𝑠 𝜇 ;𝑛, 𝑠 𝑒 ≡ 𝑒 𝑝 1 , 𝑠 1 ;𝑒 𝑝 2 , 𝑠 2 𝐻 𝜇 1𝑆, 𝑠 𝜇 ;𝑒 𝑛, 𝑠 𝑒 ∝ d 3 𝑥 d 3 𝑥′ d 3 𝑞 2𝜋 3 𝑞 𝜈 𝑒 −𝑖 𝑞 ⋅ 𝑥 − 𝑥′ 𝑞 0 2 − 𝑞 2 +𝑖𝜖 × 𝜓 𝑒 𝑝 1 , 𝑠 1 𝑥 𝜎 𝜇𝜈 𝜓 𝜇 1𝑆, 𝑠 𝜇 𝑥 𝜓 𝑒 𝑝 2 , 𝑠 2 𝑥′ 𝛾 𝜇 𝜓 𝑒 𝑛, 𝑠 𝑒 𝑥′ − 1↔2 Scattered electrons (Coulomb distorted) Bound leptons Infrared divergent , especially for high Z

Backup slides

Numerical result (previous work) The new process and m 3e are described by same operator Relation between upper bounds of the new process and m 3e Large enhancement by large nucleus charge Excluded Example: Upper bound for Pb nucleus ~ 5 10 -19

Distortion of emitted electrons 𝜅=−1 partial wave Pb case 208 𝑟 𝑔 𝐸 1/2 𝜅=−1 𝑟 [ MeV −3/2 ] 𝑍=82 𝐸 1/2 ≈48MeV distorted wave plane wave The answer is a distortion of emitted electrons. By nuclear Coulomb distortion, the wave function of a scattering electron is attracted to the center. Then you can rephrase distortion effect as combined with two effects. The first is enhancing value near the origin, which makes overlap of lepton wave functions larger. The second is giving phase shifts or extra momenta to emitted electrions. It is complicated to understand the effect. 𝑟 [fm] what the distortion makes overlap of w.f. 1. enhanced value near the origin 2. phase shift to boost momentum effectively complicated a little… 9/14

Phase shift effect of distortion (makes a momentum of 𝑒 − larger effectively) contact process photonic process bound 𝜇 − emitted 𝑒 − bound 𝜇 − emitted 𝑒 − I try to explain that easily. In contact process, we should estimate overlap integrals of 4 lepton wave functions. Since two scattering electrons overlap at one point, these extra momenta are canseled, so there are no momentum mismatches. On the other hand, in photonic process, verteces are separate and a photon propagator is not distorted. Therefore it generates momentum transfers to bound leptons. This makes overlap integrals smaller because high momentum component of a bound lepton is smaller than zero momentum component. Totally, the decay rate by the contact process is enhanced, but that by the photonic process is suppressed by distortion of final electrons. bound 𝑒 − emitted 𝑒 − bound 𝑒 − emitted 𝑒 − no momentum mismatches momentum transfers to bound leptons make overlap integrals smaller Totally (combined with the effect to enhance the value near the origin), enhanced !! suppressed… 10/14

Model-discriminating power After finding CLFV transition, “which CLFV interaction exists” would be important. Here, only 2 simple models will be considerd. model 1 : contact type ℒ 𝐼 = 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 Next I want to discuss the model-discriminating power of mue->ee. After finding this CLFV process, we would want to know what interaction is dominant and how to identify a correct model from experiments. Here after, I consider 2 simple CLFV interaction models. They generates these processes, respectively. model 2 : photonic type ℒ 𝐼 = 𝑔 𝑅 𝑒 𝐿 𝜎 𝜇𝜈 𝜇 𝑅 𝐹 𝜇𝜈 11/14

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 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, it is found that decay rate is enhanced for contact process and suppressed for photonic process. We can propose the model-discriminating method. Candidates are atomic # dependence of decay rate, and energy-angular distribution of emitted electrons. I want to emphasize that it is what can be seen for the first time by this improved calculations. Thank you for your attention. photonic process：decay rate suppressed (1/4 times in 𝑍=82) How to identify interaction types, found by this analyses atomic # dependence of the decay rate energy and angular distributions of emitted electrons 14/14

Radial functions (bound 𝒆 − ) Pb case 208 𝑍=81 (considering 𝜇 − screening) 𝑔 𝑒 1𝑠 𝑟 [ MeV 1/2 ] Type 𝐵 𝑒 (MeV) Rela 9.88× 10 −2 Non-rela 8.93× 10 −2 Next is bound electron. The blue dashed curve is a non-rela wave and the red solid curve is a relativistic wave. Relativistic effect enhances the value near the origin. 𝑟 [fm] Relativity enhances the value near the origin.

Upper limits of 𝐁𝐫 𝝁 − 𝒆 − → 𝒆 − 𝒆 − 𝜇𝑒𝑒𝑒 interaction 𝜇𝑒𝛾 interaction Br 𝜇 + → 𝑒 + 𝑒 − 𝑒 + <1.0× 10 −12 Br 𝜇 + → 𝑒 + 𝛾 < 5.7×10 −13 Br 𝜇 − 𝑒 − → 𝑒 − 𝑒 − <4.5× 10 −19 Br 𝜇 − 𝑒 − → 𝑒 − 𝑒 − <5.7× 10 −19 for Pb (𝑍=82) for Pb (𝑍=82)

Discriminating method 2 ~ energy and angular distributions ~ angular distributions (cos𝜃≈1) 𝐝𝚪 𝚪𝐝𝐜𝐨𝐬𝜽 contact (same chirality) contact (opposite chirality) photonic 𝐜𝐨𝐬𝜽 𝑔 5 has larger tail than 𝑔 1 due to Pauli principle.

Energy and angular disribution (g1) 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 𝑍=82 Muon is not point-like →distribution is spread 𝐸 1 : energy of scattered electrion 𝜃 : angle between scattered electrons 𝟏 𝚪 𝐝𝚪 𝐝 𝐄 𝟏 𝟏 𝚪 𝐝𝚪 𝐝𝐜𝐨𝐬𝛉 𝐸 1 [ 𝐌𝐞𝐕 −𝟏 ] 𝜃 𝐸 1 Coulomb DW Coulomb DW PLW PLW 𝐄 𝟏 [𝐌𝐞𝐕] 𝐜𝐨𝐬𝛉 𝑚 𝑒 ≤ 𝐸 1 ≤ 𝑚 𝜇 − 𝐵 𝜇 − 𝐵 𝑒 Almost back to back Peak at half value

Energy dependence of angular distribution (g1) 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 (Normalized by the maximum value) 𝑍=82 𝐸 𝑝 1 =0.57MeV 𝐸 𝑝 1 =23.0MeV At half value, almost back-to-back 𝐸 𝑝 1 =47.7MeV total

Angular dependence of energy distribution(g1) 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 𝑍=82 (Normalized by the muximum value) total cos𝜃=−1 cos𝜃=0 cos𝜃=1

Angle and energy distribution(g1) 𝑔 1 𝑒 𝐿 𝜇 𝑅 𝑒 𝐿 𝑒 𝑅 𝑍=82 “3D” plot

Angular dependence of energy distribution(g5) 𝑔 5 𝑒 𝑅 𝛾 𝜇 𝜇 𝑅 𝑒 𝐿 𝛾 𝜇 𝑒 𝐿 𝑍=82 (Normalized by the maximum value) total cos𝜃=−1 cos𝜃=0 cos𝜃=1

Angle and energy distribution(g5) 𝑔 5 𝑒 𝑅 𝛾 𝜇 𝜇 𝑅 𝑒 𝐿 𝛾 𝜇 𝑒 𝐿 𝑍=82 “3D” plot

Angular distribution(forward) 𝑍=82 𝒈 𝟓 𝒆 𝑹 𝜸 𝝁 𝝁 𝑹 𝒆 𝑳 𝜸 𝝁 𝒆 𝑳 difference 𝒈 𝟏 𝒆 𝑳 𝝁 𝑹 𝒆 𝑳 𝒆 𝑹 Electron emission with same chirality to same direction is suppressed by Pauli principle