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New Process for Charged Lepton Flavor Violation Searches: m e e e in a muonic atom
Masato Yamanaka (KEK)
- - - -
Phys. Rev. Lett. 105, 121601 M. Koike, Y. Kuno, J. Sato, and M. Y.
Work in progress M. Koike, Y. Kuno, J. Sato, T. Sato, and M. Y.
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Introduction
Evidence of new physics
charged Lepton Flavor Violation (cLFV)
Should study many types of cLFV processes to understand new physics!!
Sensitive to high energy scale (> LHC energy)
Reaction ratio of each cLFV depends on model structure
One of the leading prove for fundamental physics and model discrimination
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Introduction
What is target ?
Flavor violation between and e m
What is advantage ?
Clean signal [ back-to-back dielectron ]
Sensitive to both photonic dipole interaction and 4-Fermi contact interaction
New idea for cLFV search
in a muonic atom m e e e - - - -
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Contents
1, Introduction
2, Reaction rate of m e e e - - - -
3, Numerical results
4, Improvement and enhancement
5, Summary
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Reaction rate of m e e e - - - -
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nucleus muon electron
electron 1s orbit
muon 1s orbit
Applications
Precision test of QED
m-e conversion reaction
Muon catalyzed fusion
Muonic atom
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electron 1s orbit
muon 1s orbit
cLFV vertex
nucleus muon electron
m e e e in muonic atom - - - -
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Reaction rate
cLFV vertex
m e e e in muonic atom - - - -
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cLFV vertex
Reaction rate
Overlap of wave functions
m e e e in muonic atom - - - -
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m >> m
∴
Approximation
Muon localization at nucleus position
Overlap = electron wave function at nucleus
Overlap of wave functions
m e
Overlap of wave functions
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Electron wave function
r : distance from nucleus
Z : atomic number of nucleus
Overlap of wave functions
Overlap of wave functions
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cLFV vertex
Reaction rate
Cross section for elemental interaction
m e e e in muonic atom - - - -
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cLFV effective coupling constant
Sensitive to the structure of new physics
[ Y. Kuno and Y. Okada Rev. Mod. Phys. 73 (2001) ]
Effective Lagrangian
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Dipole interaction mediated by photon
4-Fermi interaction mediated by heavy particle
Effective Lagrangian
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4-Fermi interaction mediated by heavy particle
(1) 4-Fermi interaction dominant case
(No contributions from dipole interaction)
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Branching ratio
Definition of branching ratio
Number of muonic atom
Event number of m e ー e e ー ー ー
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Branching ratio
Lifetime of free muon (2.197 10 s) -6 ×
Lifetime of bound muon 2.19 10 s for H -6 × 1
(7 - 8) 10 s for U -8 × 238
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Branching ratio
Coupling constants in effective Lagrangian
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Branching ratio
Enhancement factor from overlap of wave functions
∴ Positive charge attracts electron toward the nucleus position
Heavy nuclei provides great advantage
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Dipole interaction mediated by photon
(2) Dipole interaction dominant case
(No contributions from 4-Fermi interaction)
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Branching ratio
Coupling constants in effective Lagrangian
Enhancement factor from overlap of wave functions
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Branching ratio
Photon propagator in non-relativistic limit
Enhancement factor compared with 4-Fermi case
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~
(3) Both type interactions comparable case
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Ratio of 4-Fermi type BR and dipole type BR
One of the distinct features of m e ー e e ー ー ー
It is available for the discrimination of models
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Numerical results
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How many muons are required to break current cLFV limit and to discovery process ?
m e ー e e ー ー ー
Available inputs
BR( ) as a function of effective coupling constants
BR of other cLFV processes as a function of same effective coupling constants
Current limit of BR of other cLFV processes
How to estimate required muon number
m e ー e e ー ー ー
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Available inputs
m e ー e e ー ー ー BR( ) as a function of effective coupling constants
BR of other cLFV processes as a function of same effective coupling constants
Effective couplings are canceled in ratio of them
Required muon number is estimated from the limit
Current limit of BR of other cLFV processes
How to estimate required muon number
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(1) 4-Fermi interaction dominant case
(2) Dipole interaction dominant case
Br( ) < 1.0×10 -12 Limit from SINDRUM experiment
m e ー e e ー + ー
Br( ) < 1.2×10 -11 Limit from MEGA experiment
m + e + g
Ratio of branching ratios of each cLFV process
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Numerical result
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Current experimental bound in 4-Fermi interaction dominant case
Numerical result
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Current experimental bound in dipole interaction dominant case
Numerical result
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Numerical result To break world record and discovery cLFV, more than 10 muons are required 18
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Muon intensity in working and future experiments
For COMET experiment
More than 10 muons per year 3 10 (s) 18 7 × ~
However it could not be a first cLFV signal …???
m e ー e e ー ー ー must serve complemental information to shed light on the nature of cLFV
Conclusion of previous work
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Improvement and enhancement
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Improvement
Distorted wave function by nucleus potential
Shortcoming of previous calculation and improvement
Shortcoming(1):
Final electrons are described by plane wave
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Shortcoming of previous calculation and improvement
Shortcoming(2):
No information of muon position and nucleus potential
Approximation in previous work
Localized muon at nucleus position
Non-relativistic wave function of electron
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Improvement
Shortcoming of previous calculation and improvement
Muon position and distorted wave function
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Preliminary!!
Enhancement factor
New result
Enhancement factor = Reaction rate with distorted wave
Reaction rate with plane wave
New result (preliminary)
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To break world record and discovery cLFV, more than 10 - 10 muons are required 16 17
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Muon intensity in working and future experiments
For COMET experiment
More than 10 muons per year 3 10 (s) 18 7 × ~
First discovery of cLFV comes from in a muonic atom !? m e ー e e ー ー ー
Within the reach of next generation experiments
New result (preliminary)
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Summary
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Realistic wave functions lead large enhancement of reaction rate
in a muonic atom is one of the promising reactions to search for cLFV
m e ー e e ー ー ー
First signal of cLFV may be discovered from in next generation experiments !! m e ー e e ー ー ー
Clean signal [back-to-back energetic dielectron]
Reaction rate: proportional to (Z-1) 3
Cleaner experimental signature comparison with and m e ー e e ー + ー m + e + g
Sensitive to not only 4-Fermi int. but also photonic int.
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Back-up slides
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Electron wave function (initial state)
Initial state electron = electron in atomic orbital
Electron orbit radius >> Nucleus size
Wave function of Dirac particle in point Coulomb potential
Electron wave function in initial state =
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Muon wave function
Muon Bohr radius Nucleus size ~
Muon wave function penetrates into nucleus
Wave function has to be constructed taking into account nucleus size and Coulomb potential
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Muon wave function
Construction step of the muon wave function
(1) Deriving muon wave function with constant potential V
(2) Deriving muon wave function to be one in point Coulomb potential for radius
(3) Connecting these waves at an appropriate point
∞
0
V 0
Requirements
radius
Y(1)
dY(1)/dr
r = r0 Y(2)
dY(2)/dr =
r = r0
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Electron wave function (final state)
Requirements for constructing final state electrons
To be wave functions in point Coulomb potential at points being quite far from nucleus
To be consistent with total angular momentum of initial state
Constructing wave functions using partial wave expansion
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Electron wave function (final state)
Wave function concentrating on muon position compared with plane wave
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Reaction rate
Simplest case: scalar type 4-Fermi interaction
Amplitude
Distorted muon wave function
Distorted initial electron wave function
Distorted final electron wave functions
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Reaction rate
Cross section with distorted wave functions
Integrate over spatial coordinate of vertex and final state momentum by brute force!!
Simplest case: scalar type 4-Fermi interaction
Amplitude
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cLFV processes and experimental limit