Yuichi Uesaka

Collaborators

T. Sato 1,

December 20, 2016 @ Particle Physics Theory Seminar, Osaka U.

Charged lepton flavor violating process,

𝝁−𝒆− → 𝒆−𝒆− in a muonic atom

Y. Kuno 1, J. Sato 2, M. Yamanaka 3

1Osaka U., 2Saitama U., 3Kyoto Sangyo U.

YU, Y. Kuno, J. Sato, T. Sato & M. Yamanaka, Phys. Rev. D 93, 076006 (2016).

(Nuclear Theory, Osaka U.)

YU, Y. Kuno, J. Sato, T. Sato & M. Yamanaka, in preparation.

Contents

1. Introduction

2. Formulation

Charged Lepton Flavor Violation (CLFV)

Partial wave expansion

４. Summary

𝜇−𝑒− → 𝑒−𝑒− in a muonic atom

３. Results

Decay rates (Branching ratios)

Model-discriminating power

Relativistic treatment for bound leptons

CLFV search using muons

1. Introduction

• predicted in many theories beyond SM

Charged Lepton Flavor Violation (CLFV)

• contribution of neutrino oscilation → very small

Br 𝜇 → 𝑒𝛾 < 10−54 cannot be observed by current technology

- A probe for new physics -

(does not contaminate the search for new physics)

e.g. SUSY• forbidden in SM

If seen, that is an evidence of new physics !! (not 𝜈 osci.)

𝒆− 𝝁− 𝝉− 𝝂𝒆 𝝂𝝁 𝝂𝝉 𝒆+ 𝝁+ 𝝉+ 𝝂𝒆 𝝂𝝁 𝝂𝝉 others

+1 0 0 +1 0 0 -1 0 0 -1 0 0 0

0 +1 0 0 +1 0 0 -1 0 0 -1 0 0

0 0 +1 0 0 +1 0 0 -1 0 0 -1 0

𝐿𝑒

𝐿𝜇

𝐿𝜏

• lepton flavor #, 𝐿𝑒, 𝐿𝜇 , 𝐿𝜏 ( Please distinguish them from lepton #, 𝐿 = 𝐿𝑒 + 𝐿𝜇 + 𝐿𝜏. )

1. Introduction

lepton flavor violation in charged lepton sector = CLFV

CLFV search using muons

BR < 1.0 × 10−12 by SINDRUM

BR < 7 × 10−13 (𝜇−Au → 𝑒−Au) by SINDRUM II

BR < 5.7 × 10−13 by MEG

Examples of CLFV processes using muons

a) 𝜇+ → 𝑒+𝛾

b) 𝜇+ → 𝑒+𝑒−𝑒+

c) 𝜇−𝑁 → 𝑒−𝑁

1. long lifetime and simple kinematics

BR ： Branching Ratio

Phys. Rev. Lett. 110, 201801 (2013).

Nucl. Phys. B 299, 1 (1988).

Eur. Phys. J. C 47, 337 (2006).

New experiments for “𝜇− − 𝑒− conversion” are planned with higher sensitivity than previous ones.

Advantages of using muon for rare process

(COMET, DeeMe @ J-PARC, Mu2e @ Fermilab)

1. Introduction

(≡ signal rate / total decay rate)

(∼ 108−9 muons per a second)2. high intensity muon beam

𝝁−𝒆− → 𝒆−𝒆− in a muonic atom

+𝑍𝑒

𝜇−

𝑒−

C

L

F

V

𝑒−

𝑒−

Features

• 2 type CLFV interactions

• atomic # 𝑍 : large⇒ decay rate Γ : large

M. Koike, Y. Kuno, J. Sato & M. Yamanaka,

Phys. Rev. Lett. 105,121601(2010)

𝐸1

𝐸2

𝜇−

𝑒−

𝑒−

𝑒−

𝜇−

𝑒− 𝑒−

𝑒−

𝛾∗

Γ ∝ 𝑍 − 1 3

New CLFV search

using muonic atoms

𝜇𝑒𝑒𝑒 vertex

𝜇𝑒𝛾 vertex

• clear signal : two 𝑒−s (𝐸1 + 𝐸2 ≃ 𝑚𝜇 +𝑚𝑒 − 𝐵𝜇 − 𝐵𝑒)

R. Abramishili et al.,

COMET Phase-I Technical Design Report,

KEK Report 2015-1 (2015)

proposal in COMET

1. Introduction

(similar to 𝜇+ → 𝑒+𝑒+𝑒−)

(Rough) Estimation of decay rate

Γ𝜇−𝑒−→𝑒−𝑒− = 2𝜎𝑣rel 𝜓1𝑆𝑒 0 2

𝜎 : cross section of 𝜇−𝑒− → 𝑒−𝑒−

： wave function of 1𝑆 bound electron (non-rela)

𝜓1𝑆𝑒 Ԧ𝑥 =

𝑚𝑒 𝑍 − 1 𝛼 3

𝜋exp −𝑚𝑒 𝑍 − 1 𝛼 Ԧ𝑥

Phys. Rev. Lett. 105,121601 (2010).

Γ ∝ 𝑍 − 1 3

𝑣rel : relative velocity of 𝜇− & 𝑒−

Γ = 𝜎𝑣rel∫ d𝑉𝜌𝜇𝜌𝑒

(the same 𝑍 dependence in the both contact & photonic cases)

(sum of two 1𝑆 𝑒−s)

1. Introduction

Suppose nuclear Coulomb potential is weak,

“flux”

(free particles’)

Branching ratio

Phys. Rev. Lett. 105,121601 (2010).

Br 𝜇−𝑒− → 𝑒−𝑒− ≡ ǁ𝜏𝜇Γ 𝜇−𝑒− → 𝑒−𝑒−

ǁ𝜏𝜇 : lifetime of a muonic atom

How many muons decay, compared to created # ?

Constraint from previous CLFV experiments

2.2μs for a muonic H (𝑍 = 1)

80ns for a muonic Pb (𝑍 = 82)

cf.

1. Introduction

increasing as atomic # 𝑍 is larger

Using muonic atom with large 𝑍is favored.

can product 𝒪 1018−19 muonic atoms.

cf. Future experiment (COMET Phase-II)

To improve the previous estimation

Γ𝜇−𝑒−→𝑒−𝑒− = 2𝜎𝑣rel 𝜓1𝑆𝑒 0 2 ∝ 𝑍 − 1 3

Koike et al.

• emitted 𝑒− : plane wave

• spacial extension of bound lepton

• bound lepton : non-rela

( valid when nuclear Coulomb potential is sufficiently small)

In atoms with large 𝑍,

← small orbital radius

← relativistic (especially, 𝑒−)

← Coulomb distortion

not able to estimate differential decay rate

“𝑍 dependence” doesn’t depend on CLFV int. always Γ ∝ 𝑍 − 1 3

More quantitive estimation is needed ! (important for large 𝑍)

approximations (𝑍𝛼 ≪ 1)

1. Introduction

Note

≫ wave length of emitted 𝑒−

lepton wave function ： relativistic Coulomb

Improvement and expectation

this work

we can estimate energy & angular distribution of emitted 𝑒− pair

• Coulomb distortion of emitted 𝑒−

• finite orbit-size of bound leptons

• relativistic effects for bound leptons

we may get different results, depending on CLFV interaction

influence for Γ𝜇−𝑒−→𝑒−𝑒−

？

suppressed

enhanced

the improvement contains…

other expectation

Totally, how CLFV decay rates are changed ?

1. Introduction

2. Formulation

ℒ𝐼 = ℒ𝑐𝑜𝑛𝑡𝑎𝑐𝑡 + ℒ𝑝ℎ𝑜𝑡𝑜

+[ℎ. 𝑐. ]

𝜇

𝑒 𝑒

𝑒

+[ℎ. 𝑐. ]

𝛾∗

𝑒

𝜇

𝑒

𝑒

contact interaction

(short-range process)

photonic interaction

(long-range process)

Effective Lagrangian

2. Formulation

𝑒−

𝑒−

+𝑍𝑒𝒑1

𝒑2CLFV

𝜇−𝑒− → 𝑒−𝑒−

initial state final state

bound 𝝁− (𝟏𝑺) & bound 𝒆− 2 scattering 𝒆−s (distorted wave)

・ignore interaction among 𝑒−s

𝐸1 + 𝐸2 = 𝑚𝜇 +𝑚𝑒 − 𝐵𝜇 − 𝐵𝑒

・ignore nuclear recoil energy

𝝁−𝒆− → 𝒆−𝒆− process

𝜇−

2. Formulation

Decay rate

Γ = 2𝜋

𝑓

ҧ𝑖

𝛿(𝐸𝑓 − 𝐸𝑖) 𝜓𝑒𝑠1 𝒑1 𝜓𝑒

𝑠2(𝒑2) 𝐻 𝜓𝜇𝑠𝜇 1𝑠 𝜓𝑒

𝑠𝑒(1𝑠)2

get radial functions by solving Dirac eq. numerically

𝑑𝑔𝜅(𝑟)

𝑑𝑟+1 + 𝜅

𝑟𝑔𝜅 𝑟 − 𝐸 +𝑚 + 𝑒𝜙 𝑟 𝑓𝜅 𝑟 = 0

𝑑𝑓𝜅(𝑟)

𝑑𝑟+1 − 𝜅

𝑟𝑓𝜅 𝑟 + 𝐸 −𝑚 + 𝑒𝜙 𝑟 𝑔𝜅 𝑟 = 0

𝜓 𝒓 =𝑔𝜅 𝑟 𝜒𝜅

𝜇( Ƹ𝑟)

𝑖𝑓𝜅 𝑟 𝜒−𝜅𝜇( Ƹ𝑟)

use partial wave expansion to express the distortion

𝜓𝑒𝑠 𝒑 =

𝜅,𝜇,𝑚

4𝜋 𝑖𝑙𝜅(𝑙𝜅, 𝑚, 1/2, 𝑠|𝑗𝜅 , 𝜇)𝑌𝑙𝜅,𝑚∗ ( Ƹ𝑝)𝑒−𝑖𝛿𝜅𝜓𝑝

𝜅,𝜇

𝜙 : nuclear Coulomb potential

2. Formulation

Γ 𝜇− 1𝑆 𝑒− 𝛼 → 𝑒−𝑒−

=1

2න𝑚𝑒

𝑚𝜇−𝐵𝜇1𝑆−𝐵𝑒

𝛼

d𝐸1න−1

1

d cos𝜃d2Γ

d𝐸1dcos𝜃

𝑀 𝐸1, 𝜅1, 𝜅2, 𝐽 =

𝑖=1,⋯,6

𝑔𝑖𝑀contact𝑖 𝐸1, 𝜅1, 𝜅2, 𝐽 +

𝑗=𝐿,𝑅

𝑔𝑗𝑀photo𝑗

𝐸1, 𝜅1, 𝜅2, 𝐽

d2Γ

d𝐸1dcos𝜃=

𝜅1,𝜅2,𝜅1′ ,𝜅2

′ ,𝐽,𝑙

𝑀 𝐸1, 𝜅1, 𝜅2, 𝐽 𝑀∗ 𝐸1, 𝜅1

′ , 𝜅2′ , 𝐽

× 𝑤 𝜅1, 𝜅2, 𝜅1′ , 𝜅2

′ , 𝐽, 𝑙 𝑃𝑙 cos𝜃

differential decay rate :

𝐸1

𝐸2

𝜃

Energy & angular distribution

2. Formulation

𝐸1 : energy of an emitted electron

𝜃 : angle between two emitted electrons

𝑃𝑙 : Legendre polynomial

3. Results

Radial wave function (bound 𝝁−)

𝑟 [fm]

[MeV−1/2]

𝑍 = 82Pb case208

𝑟𝑔𝜇1𝑠 𝑟 , 𝑟𝑓𝜇

1𝑠 𝑟

(radius of 208Pb )

𝐵𝜇: 10.5 MeV

3. Results

𝑟𝑔𝜇1𝑠 𝑟 : solid

𝑟𝑓𝜇1𝑠 𝑟 : dotted

Radial wave function (bound 𝒆−)

𝑟 [fm]

[MeV1/2]

Type 𝐵𝑒(MeV)

Rela 9.88 × 10−2

Non-rela 8.93 × 10−2

𝑍 = 81Pb case208

(considering 𝜇− screening)

Relativity enhances the value near the origin.

𝑔𝑒1𝑠 𝑟

3. Results

Radial wave function (scattering 𝒆−)

𝑟 [fm]

[MeV−3/2]𝑟𝑔𝐸1/2

𝜅=−1 𝑟

attracted by nuclear Coulomb potential

𝑍 = 82

𝐸1/2 ≈ 48MeV

e.g. 𝜅 = −1 partial wave

distorted wave

plane wave

Pb case208

① enhanced value near the origin

② local momentum increased effectively

3. Results

Contact process

𝜇−

𝑒−

𝑒−

𝑒−

overlap of bound 𝜇−, bound 𝑒−, and two scattering 𝑒−s

bound 𝜇−

scattering 𝑒− scattering 𝑒−

bound 𝑒−

𝑟 [fm]

transition rate UP!

bound 𝑒− : non-rela → rela

𝑟2𝑔𝜇1𝑠 𝑟 𝑔𝑒

1𝑠 𝑟 𝑔𝐸1/2𝜅=−1 𝑟 𝑔𝐸1/2

𝜅=−1 𝑟

scat. 𝑒− : plane → distorted

wave functions move

to the center

3. Results

Upper limits of BR (contact process)

2.1 × 1018 3.0 × 1017

this work (1s)

needed # of muonic atoms (𝑍 = 82)

𝐵𝑅(𝜇+ → 𝑒+𝑒−𝑒+) < 1.0 × 10−12

(SINDRUM, 1988)

atomic #, 𝒁

𝐵𝑅 𝜇−𝑒− → 𝑒−𝑒− < 𝐵𝑚𝑎𝑥

𝑩𝒎𝒂𝒙

𝑔1 ഥ𝑒𝐿𝜇𝑅 ഥ𝑒𝐿𝑒𝑅

this work

(1s+2s+…)

Koike et al. (1s)

3. Results

YU, Y. Kuno, J. Sato, T. Sato & M. Yamanaka, Phys. Rev. D 93, 076006 (2016).

Photonic process

bound 𝜇− bound 𝑒−

scattering 𝑒− scattering 𝑒−

𝛾∗

𝑟 [fm] 𝑟 [fm]

scat. 𝑒− : plane → distorted scat. 𝑒− : plane → distorted

𝑟2𝑔𝜇1𝑠 𝑟 𝑔𝐸1/2

𝜅=−1 𝑟 𝑗0 𝑞0𝑟 𝑟2𝑔𝑒1𝑠 𝑟 𝑔𝐸1/2

𝜅=−1 𝑟 𝑗0 𝑞0𝑟

bound 𝑒− : non-rela → rela

𝑒−

𝑒−𝜇−

𝑒−

overlap integral downdistortion of scattering 𝑒−

3. Results

𝐵𝑅(𝜇+ → 𝑒+𝛾) < 5.7 × 10−13

(MEG, 2013)

𝒁

𝐵𝑅 𝜇−𝑒− → 𝑒−𝑒− < 𝐵𝑚𝑎𝑥

𝑩𝒎𝒂𝒙

𝑔𝐿 ഥ𝑒𝐿𝜎𝜇𝜈𝜇𝑅𝐹𝜇𝜈

Upper limits of BR (photonic process)

this work (1s)

Koike et al. (1s)

1.8 × 1018 6.6 × 1018

needed # of muonic atoms (𝑍 = 82)

3. Results

Phase shift effect of distortion

photonic process

bound 𝜇−

bound 𝑒− emitted 𝑒−

emitted 𝑒−

momentum transfers to bound leptons

bound 𝜇− emitted 𝑒−

bound 𝑒− emitted 𝑒−

contact process

make overlap integrals smaller

(makes a momentum of scattering 𝑒− larger effectively)

no momentum mismatches

Totally (combined with the effect to enhance the value near the origin),

enhanced !! suppressed…

3. Results

𝜇−

𝑒−

𝑒−

𝑒−

𝜇−

𝑒− 𝑒−

𝑒−

𝛾∗

𝜇−

𝑒−

𝑒−

𝑒−

After finding CLFV, can the CLFV source be decided ?

ℒ𝐼 = 𝑔1 𝑒𝐿𝜇𝑅 𝑒𝐿𝑒𝑅 + ℎ. 𝑐.

ℒ𝐼 = 𝑔5 𝑒𝑅𝛾𝜇𝜇𝑅 𝑒𝐿𝛾𝜇𝑒𝐿 + ℎ. 𝑐.

ℒ𝐼 = 𝑔𝑅𝑒𝑅𝜎𝜇𝜈𝜇𝑅𝐹𝜇𝜈 + ℎ. 𝑐.

Model 2 : contact (opposite chirality)

Model 3 : photonic

𝑔𝑅 ≠ 0, 𝑔𝑒𝑙𝑠𝑒 = 0

Model 1 : contact (same chirality)

𝑔5 ≠ 0, 𝑔𝑒𝑙𝑠𝑒 = 0

𝑔1 ≠ 0, 𝑔𝑒𝑙𝑠𝑒 = 0

Here, only 3 very simple model will be considered.

Model-discriminating power

3. Results

Discriminating method 1

𝑍 dependence of Γ/Γ0

The 𝑍 dependences are different among interactions.

Compared to 𝑍 − 1 3, that of contact process is larger,

while that of photonic process is smaller.

~ atomic # dependence of decay rates ~

3. Results

𝒁

𝚪 𝒁

𝚪𝟎 𝒁 contact (same chirality)

contact (opposite chirality)

photonic

~ energy and angular distributions ~

Discriminating method 2

𝑍 = 82𝟏

𝚪

𝐝𝟐𝚪

𝐝𝐄𝟏𝐝𝐜𝐨𝐬𝜽[𝐌𝐞𝐕−𝟏]

𝐸1 : energy of an emitted electron

𝜃 : angle between two emitted electrons

𝐜𝐨𝐬𝛉

𝐸1

𝐸2

𝜃

𝐄𝟏[𝐌𝐞𝐕]

𝐜𝐨𝐬𝛉

𝐄𝟏[𝐌𝐞𝐕]

[𝐌𝐞𝐕−𝟏]

photonic process

𝟏

𝚪

𝐝𝟐𝚪

𝐝𝐄𝟏𝐝𝐜𝐨𝐬𝜽

3. Results

contact processcontact process

4. Summary

contact process ：decay rate Enhanced (7 times in 𝑍 = 82)

𝜇−𝑒− → 𝑒−𝑒− process in a muonic atom

Our finding

interesting candidate for CLFV search

• Distortion of emitted electrons

• Relativistic treatment of a bound electronare important in calculating decay rates.

energy and angular distributions of emitted electrons

How to identify interaction types, found by this analyses

atomic # dependence of the decay rate

photonic process：decay rate suppressed (1/4 times in 𝑍 = 82)

Conclusion

Distortion makes difference between 2 processes.

4. Summary

further analysis for experimental setup

Future work

estimation of background

calculation using various models beyond SM

moderate setting of experiment

• main noise : Decay In Orbit (DIO)

• What atom is favored ?

• Is a better muon beam pulse or DC ?

interference among interactions

4. Summary