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TRANSCRIPT
Yuichi Uesaka
Collaborators
T. Sato 1,
February 1, 2018 @ KEKçè«ã»ã³ã¿ãŒJ-PARCå宀掻åç·æ¬ç 究äŒ
( ðâðâ â ðâðâ in muonic atoms )
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).
(Osaka U.)
YU, Y. Kuno, J. Sato, T. Sato & M. Yamanaka, Phys. Rev. D 97, 015017 (2018).
ãã¥ãŒãªã³ååäžã§ã®CLFVéçš ðâðâ â ðâðâ
Contents
1. Introduction
2. Transition probability of ðâðâ â ðâðâ
Charged Lepton Flavor Violation (CLFV)
CLFV searches using muon
Distortion of scattering electrons & Relativity of bound leptons
ïŒ. Summary
ðâðâ â ðâðâ in a muonic atom
ïŒ. Distinguishment of CLFV interaction
Asymmetry of emitted electrons by polarizing muon
Atomic # dependence of decay rates
Energy-angular distribution of emitted electrons
Effective CLFV interactions
Difference between contact & photonic processes
Contents
1. Introduction
2. Transition probability of ðâðâ â ðâðâ
Charged Lepton Flavor Violation (CLFV)
CLFV searches using muon
Distortion of scattering electrons & Relativity of bound leptons
ïŒ. Summary
ðâðâ â ðâðâ in a muonic atom
ïŒ. Distinguishment of CLFV interaction
Asymmetry of emitted electrons by polarizing muon
Atomic # dependence of decay rates
Energy-angular distribution of emitted electrons
Effective CLFV interactions
Difference between contact & photonic processes
Charged Lepton Flavor Violation (CLFV)- æ°ç©çæ¢çŽ¢ã®æååè£ -
⢠In the standard model (SM), lepton flavors are conserved.
ðâ â ðâðððð
process where lepton flavors are not conserved ïŒ LFV process
LFV in charged lepton sector ïŒ CLFV
ðâ â ðâðð
allowed forbidden (CLFV)
ðâ â ðâðŸ
ðâ â ðâð+ðâ
e.g. SUSY
⢠The contribution of ð osci. is tiny.
Br ð â ððŸ < 10â54expected branching ratio
Various CLFV modes have been searched, but not found yet.
⢠Many âmodels beyond the SMâ predict CLFV.
L. Calibbi & G. Signorelli, arXiv:1709.00294 [hep-ph].
CLFV searches in muon rare decay
1. high intensity 2. long lifetime
Advantages of muon
current bounds
ðâ-ðâ conversion
CLFV search using muonic atom
exploring ðððð interaction
ðâðâ â ðâðâ in a muonic atom
+ðð
ðâ
ðâ
C
L
F
V
ðâ
ðâ
Features
⢠2 CLFV mechanisms
⢠atomic # ð : large â decay rate Î : large
M. Koike, Y. Kuno, J. Sato & M. Yamanaka,
Phys. Rev. Lett. 105, 121601 (2010).
ðž1
ðž2
ðâ
ðâ
ðâ
ðâ
ðâ
ðâ ðâ
ðâ
ðŸâ
Î â ð â 1 3
New CLFV search
using muonic atoms
contact ( ðððð vertex )
photonic ( ðððŸ vertex )
⢠clear signal : ðž1 + ðž2 â ðð +ðð â ðµð â ðµð
R. Abramishvili et al.,
COMET Phase-I Technical Design Report
(2016).
proposal in COMET
(similar to ð+ â ð+ð+ðâ)
Branching ratio of CLFV decay
Phys. Rev. Lett. 105,121601 (2010).
BR ðâðâ â ðâðâ â¡ ÇððÎ ðâðâ â ðâðâ
Çðð : lifetime of a muonic atom
How many muonic atoms decay with CLFV, compared to created # ?
BR with CLFV coupling fixed on allowed maximum
2.2ÎŒs for a muonic H (ð = 1)
80ns for a muonic Pb (ð = 82)
cf.
Î â ð â 1 3
due to existence prob.
of bound ðâ at the origin
BR increases with atomic # ð.
Using muonic atoms with large ðis favored to search for ðâðâ â ðâðâ.
e.g. BR < 5.0 Ã 10â19 for Pb ð = 82
if contact process is dominant
To improve calculation for decay rate
Îðâðââðâðâ = 2ðð£rel ð1ðð 0 2 â ð â 1 3
previous formula of CLFV decay rate by Koike et al.
emitted ðâs are expected to be back-to-back with equal energies
More quantitative estimation is needed ! (important for large ð)
Note
⢠emitted ðâ : plane wave
⢠spatial extension of bound lepton
⢠bound lepton : non-relativistic
â small orbital radius
â relativistic (especially, ðâ)
â Coulomb distortion
used approximations (ððŒ ⪠1)
â« wave length of emitted ðâ
In atoms with large ð,
âð dependenceâ comes from only ð1ðð 0 2 always Î â ð â 1 3
energy & angular distribution of emitted ðâ pair
lepton wave function ïŒ relativistic Coulomb
Improvement and expectation
this work
⢠Coulomb distortion of emitted ðâ
⢠finite orbit-size of bound leptons
⢠relativistic effects for bound leptons
the improvement containsâŠ
other than quantitative modifications
How are CLFV decay rates modified ?
+
+
model-dependence
Contents
1. Introduction
2. Transition probability of ðâðâ â ðâðâ
Charged Lepton Flavor Violation (CLFV)
CLFV searches using muon
Distortion of scattering electrons & Relativity of bound leptons
ïŒ. Summary
ðâðâ â ðâðâ in a muonic atom
ïŒ. Distinguishment of CLFV interaction
Asymmetry of emitted electrons by polarizing muon
Atomic # dependence of decay rates
Energy-angular distribution of emitted electrons
Effective CLFV interactions
Difference between contact & photonic processes
âðŒ = âðððð¡ððð¡ + âðâððŸ
ð
ð ð
ð
ðŸâ
ð
ð
ð
ð
contact interaction photonic interaction
Effective Lagrangian for ðâðâ â ðâðâ
constrained by ð â ððð constrained by ð â ððŸ
âðâððŸ = â4ðºð¹
2ðð ðŽð ðð¿ð
ðððð ð¹ðð + ðŽð¿ðð ððððð¿ð¹ðð + [ð». ð. ]
âcontact = â4ðºð¹
2[ð1 ðð¿ðð ðð¿ðð + ð2 ðð ðð¿ ðð ðð¿
+ð5 ðð ðŸððð ðð¿ðŸððð¿ + ð6 ðð¿ðŸððð¿ ðð ðŸ
ððð ] + [ð». ð. ]
+ð3 ðð ðŸððð ðð ðŸððð + ð4 ðð¿ðŸððð¿ ðð¿ðŸ
ððð¿
Our formulation for decay rate
Î = 2ð
ð
Ò§ð
ð¿(ðžð â ðžð) ððð1,ð 1ðð
ð2,ð 2 ð» ðð1ð ,ð ððð
1ð ,ð ð2
get radial functions by solving âDirac eq. with ðâ numerically
ððð (ð)
ðð+1 + ð
ððð ð â ðž +ð + ðð ð ðð ð = 0
ððð (ð)
ðð+1 â ð
ððð ð + ðž âð + ðð ð ðð ð = 0 ð ð =
ðð ð ðð ð( Æžð)
ððð ð ðâð ð( Æžð)
use partial wave expansion to express the distortion
ððð,ð
=
ð ,ð,ð
4ð ððð (ðð , ð, 1/2, ð |ðð , ð)ððð ,ðâ ( Æžð)ðâðð¿ð ðð
ð ,ð
ð : nuclear
Coulomb potential
ð : index of angular momentum
Upper limits of BR (contact process)
2.1 Ã 1018 3.0 Ã 1017
this work (1s)
ðµð (ð+ â ð+ðâð+) < 1.0 à 10â12
(SINDRUM, 1988)
atomic #, ð
ðµð ðâðâ â ðâðâ < ðµððð¥
ð©ððð
ð1 àŽ¥ðð¿ðð àŽ¥ðð¿ðð
this work
(1s+2s+âŠ)
Koike et al. (1s)
inverse of ðµððð¥ (ð = 82)
ðµð (ð+ â ð+ðŸ) < 4.2 à 10â13
(MEG, 2016)
ð
ðµð ðâðâ â ðâðâ < ðµððð¥
ð©ððð
ðð¿ àŽ¥ðð¿ððððð ð¹ðð
Upper limits of BR (photonic process)
this work (1s)
Koike et al. (1s)
1.8 Ã 1018 6.6 Ã 1018
inverse of ðµððð¥ (ð = 82)
this work
(1s+2s+âŠ)
Contents
1. Introduction
2. Transition probability of ðâðâ â ðâðâ
Charged Lepton Flavor Violation (CLFV)
CLFV searches using muon
Distortion of scattering electrons & Relativity of bound leptons
ïŒ. Summary
ðâðâ â ðâðâ in a muonic atom
ïŒ. Distinguishment of CLFV interaction
Asymmetry of emitted electrons by polarizing muon
Atomic # dependence of decay rates
Energy-angular distribution of emitted electrons
Effective CLFV interactions
Difference between contact & photonic processes
ð
ðª ð
ðª ð = ðð
ð dependence of Î
ðð¿: ð1 = 1: 100
The ð dependences are different among interactions.
That of contact process is strongly increasing,
while that of photonic process is moderately increasing.
Distinguishing method 1~ atomic # dependence of decay rates ~
contact
photonic
~ energy and angular distributions ~
Distinguishing method 2
ð = 82ð
ðª
ðððª
ððððððšð¬ðœ[ðððâð]
ðž1 : energy of an emitted electron
ð : angle between two emitted electrons
ððšð¬ð
ðž1
ðž2
ð
ðð[ððð]
ððšð¬ð
ðð[ððð]
[ðððâð]
photonic
ð
ðª
ðððª
ððððððšð¬ðœ
contact
The distributions are (a little) different among interactions.
Model distinguishing power
method 1. ð-dep. of decay rates method 2. energy-angular distribution
We can distinguish âcontactâ or âphotonicâ.
Can we distinguish âleftâ or ârightâ ?
Î ð
Î ð = 20
ð
e.g. ð1 ðð¿ðð ðð¿ðð & ð2 ðð ðð¿ ðð ðð¿
d2Î
dðž1dcosð1=1
2
dÎ
dðž11 + ðŒ ðž1 ðcosð1
ðð ð ⡠නð3ð2
ð 1,ð 2,ð ð
ððð1,ð 1ðð
ð2,ð 2 âðŒ ðð1ð ,ð ððð
1ð ,ð ð2
ð¿ ðžð â ðžð
ðŒ ðž1 ðcosð1 =ðâ â ðâ
ðâ + ðâ
asymmetry factor ðŒ
( ð ð =â, â )
angular distribution of one emitted electroncosð1 = ð â Æžð1
ð : polarization vector
( integrate the angle of the other electron )
~ asymmetry of electron emission by polarized muon ~
Distinguishing method 3
ð
ðð¿â
ðâ
ð1
ðâð1
ð¬ð dependence of electron asymmetry
âðððð¡ððð¡ââ
âðððð¡ððð¡ââ
âðâðð¡ð
ðŒ ðž1
ðž1 MeV
ð = 82
dÎ
ÎdE1
ðž1 MeV
cf. ðž1 distributionMeVâ1
ðððð¡ððð¡ðâðð¡ð
ðŽð ðŽð¿
Contents
1. Introduction
2. Transition probability of ðâðâ â ðâðâ
Charged Lepton Flavor Violation (CLFV)
CLFV searches using muon
Distortion of scattering electrons & Relativity of bound leptons
ïŒ. Summary
ðâðâ â ðâðâ in a muonic atom
ïŒ. Distinguishment of CLFV interaction
Asymmetry of emitted electrons by polarizing muon
Atomic # dependence of decay rates
Energy-angular distribution of emitted electrons
Effective CLFV interactions
Difference between contact & photonic processes
contact process ïŒdecay rate Enhanced (7 times Î0 in ð = 82)
ðâðâ â ðâðâ 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 discriminate interactions, found by this analyses
atomic # dependence of the decay rate
photonic processïŒdecay rate suppressed (1/4 times Î0 in ð = 82)
Summary
Distortion makes difference between 2 processes.
asymmetry of electron emission by polarized muon
BACKUP
There is possibility to test the relative phase of couplings.
Comparison to ð+ â ð+ð+ðâ
ððâ
ððâ
ðâ
ðâ
ð+
ð+
ðâ
ð+
ðâðâ â ðâðâ in a muonic atom ð+ â ð+ð+ðâ
difference 1 : signal
difference 2 : interference among interactions
2 ðâs 1 ðâ & 2 ð+s
(approximately) two-body decay three-body decay
Interference appears differently.
Backup
Radial wave function (bound ðâ)
ð [fm]
[MeVâ1/2]
ð = 82Pb case208
ððð1ð ð , ððð
1ð ð
(radius of 208Pb )
ðµð: 10.5 MeVððð
1ð ð : solid
ððð1ð ð : dotted
It is important to consider finite nuclear charge radius.
Backup
Radial wave function (bound ðâ)
ð [fm]
[MeV1/2]
Type ðµð(MeV)
Relativistic 9.88 Ã 10â2
Non-
relativistic8.93 Ã 10â2
ð = 81Pb case208
(considering ðâ screening)
Relativity enhances the value near the origin.
ðð1ð ð
Backup
Radial wave function (scattering ðâ)
ð [fm]
[MeVâ3/2]ðððž1/2
ð =â1 ð
shifted by nuclear Coulomb potential
ð = 82
ðž1/2 â 48MeV
e.g. ð = â1 partial wave
distorted wave
plane wave
Pb case208
â enhanced value near the origin
â¡ local momentum increased effectively
Backup
Effect of distortion
bound ðâ
(Assuming momentum conservation at each vertex)scat. ðâ : plane wave
photonic process
bound ðâ
bound ðâ emitted ðâ
emitted ðâbound ðâ emitted ðâ
emitted ðâ
contact process
Backup
Effect of distortion
photonic process
bound ðâ
bound ðâ emitted ðâ
emitted ðâ
momentum transfers to bound leptons
bound ðâ emitted ðâ
bound ðâ emitted ðâ
contact process
make overlap integrals smaller
no momentum mismatches
Totally (combined with the effect to enhance the value near the origin),
enhanced !! suppressedâŠ
scat. ðâ : distorted wave (Assuming momentum conservation at each vertex)
Backup
Contact process
ðâ
ðâ
ðâ
ðâ
overlap of bound ðâ, bound ðâ, and two scattering ðâs
bound ðâ
scattering ðâ scattering ðâ
bound ðâ
ð [fm]
transition rate increases!
bound ðâ : non-relativistic
â relativistic
ð2ðð1ð ð ðð
1ð ð ððž1/2ð =â1 ð ððž1/2
ð =â1 ð
scat. ðâ : plane â distortedwave functions shift
to the center
208Pb
Backup
Photonic process
bound ðâ bound ðâ
scattering ðâ scattering ðâ
ðŸâ
ð [fm] ð [fm]
scat. ðâ : plane â distorted
ð2ðð1ð ð ððž1/2
ð =â1 ð ð0 ð0ð ð2ðð1ð ð ððž1/2
ð =â1 ð ð0 ð0ð
bound ðâ : non-relativistic
â relativistic
ðâ
ðâðâ
ðâ
overlap integral decreasesdistortion of scattering ðâ
scat. ðâ : plane â distorted
208Pb
Backup
(Rough) Estimation of decay rate
Îðâðââðâðâ = 2ðð£rel ð1ðð 0 2
ð : cross section of ðâðâ â ðâðâ
ïŒ wave function of 1ð bound electron (non-relativistic)
ð1ðð ÔŠð¥ =
ðð ð â 1 ðŒ 3
ðexp âðð ð â 1 ðŒ ÔŠð¥
Phys. Rev. Lett. 105,121601 (2010).
Î â ð â 1 3
ð£rel : relative velocity of ðâ & ðâ
Î = ðð£relâ« dððððð
(the same ð dependence in the both contact & photonic cases)
(sum of two 1ð ðâs)
Backup
Suppose nuclear Coulomb potential is weak,
âfluxâ
(free particlesâ)
Î ðâ 1ð ðâ ðŒ â ðâðâ
=1
2නðð
ððâðµð1ðâðµð
ðŒ
dðž1නâ1
1
d cosðd2Î
dðž1dcosð
ð ðž1, ð 1, ð 2, ðœ =
ð=1,â¯,6
ðððcontactð ðž1, ð 1, ð 2, ðœ +
ð=ð¿,ð
ðððphotoð
ðž1, ð 1, ð 2, ðœ
d2Î
dðž1dcosð=
ð 1,ð 2,ð 1â² ,ð 2
â² ,ðœ,ð
ð ðž1, ð 1, ð 2, ðœ ðâ ðž1, ð 1
â² , ð 2â² , ðœ
à ð€ ð 1, ð 2, ð 1â² , ð 2
â² , ðœ, ð ðð cosð
differential decay rate :
ðž1
ðž2
ð
ðž1 : energy of an emitted electron
ð : angle between two emitted electrons
ðð : Legendre polynomial
Decay rate
Backup
angular distribution (cosð â 1)
ððª
ðªðððšð¬ðœ
ððšð¬ðœ
ðâ pair has same chirality
ð = 82
ð : angle between two emitted electronsðž1
ðž2
ð
contact (same chirality)
contact (opposite chirality)
ðâ pair cannot emit same momentum
Discriminating method 2
(due to Pauli principle)
Backup
Contribution from all bound ðâs
1S 2S 2P 3S 3P 3D 4S Total
1 0.157.3Ã 10â3
4.3Ã 10â2
2.6Ã 10â3
2.4Ã 10â5
1.8Ã 10â2
1.21
1S 2S 2P 3S 3P 3D 4S Total
1 0.176.2Ã 10â3
5.1Ã 10â2
3.1Ã 10â3
2.3Ã 10â9
2.1Ã 10â2
1.25
contact ð1
photonic ðð¿
normalize the contribution of 1ð ðâ to 1
it is sufficient to consider about ð electrons for both cases
Backup