1. 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,
Y. Uesaka, Y. Kuno, J. S, T. Sato, M.Yamanaka., Phys. Rev. D
+ arXiv: 1609?????

2. 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

3. 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

4. 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 ]

5. Basic Idea (PRL) m e e e － － － －

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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

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

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

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

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

17. 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

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

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

20. = 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

21. with the naïve estimate
Discovery reach
with the naïve estimate

22. 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

23. Discovery reach

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

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

26. 10 10 For run-time 1 year 3 10 s - muon at COMET experiment
Discovery reach
For run-time 1 year s 7 ～ × 10 18 10 19 -
muon at COMET experiment

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

28. 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 !!

29. Precise Estimate (PRD +) 1 Total Rate

30. 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)

31. 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

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

33. 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

34. High angular momentum is very important
Γ0 : previous calculation

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. Overlap of wave functions
enhancement
Line : current
Dashed: previous

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

44. 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

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

46. 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

47. 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

48. 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

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

50. 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

51. 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

52. Summary

53. 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

54. 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

55. Discussion (photonic dipole Interaction)

56. 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 ~

57. 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

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

59. 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 𝑞 𝜈 𝑒 −𝑖 𝑞 ⋅ 𝑥 − 𝑥′ 𝑞 0 2 − 𝑞 2 +𝑖𝜖
× 𝜓 𝑒 𝑝 1 , 𝑠 𝑥 𝜎 𝜇𝜈 𝜓 𝜇 1𝑆, 𝑠 𝜇 𝑥 𝜓 𝑒 𝑝 2 , 𝑠 𝑥′ 𝛾 𝜇 𝜓 𝑒 𝑛, 𝑠 𝑒 𝑥′ − 1↔2
Scattered electrons (Coulomb distorted)
Bound leptons
Infrared divergent , especially for high Z

60. Backup slides

61. 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 ~
-19

62. 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

63. 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

64. 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

65. 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

66. 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.

67. 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)

68. 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.

69. 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

70. 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

71. 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

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

73. 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

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

75. 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