Conversion of Bound Muons: Lepton Flavour and Number Violation

Conversion of Bound Muons: Lepton Flavour and Number Violation

Tanja Geib

+ Alexander Merle: Phys. Rev. D93 (2016) 055039→technical details onµ⁻–e⁻ + Stephen King, Alexander Merle, Jose Miguel No, Luca Panizzi: Phys. Rev. D93

(2016) 073007→complementarity ofµ⁻–e⁻with LHC

+ Alexander Merle, Kai Zuber: Phys. Lett. B764 (2017) 157→‘appetiser’µ⁻–e⁺ + Alexander Merle: arXiv:1612.00452→technical details onµ⁻–e⁺

Max Planck Institute for Physics

PSI Seminar, December 16, 2016

1 / 32

Today’s Agenda:

What happens in aµ– e conversion?

What are similarities and differences when considering µ⁻– e⁻ and µ⁻–e⁺ conversion?

How to tackleµ⁻–e⁻ conversion (using the example of a realisation via doubly charged scalars)?

Employing the complementarity between collider and low energy physics to increase the testability →Results based on the example case

How to tackleµ⁻–e⁺ conversion (using the example of a realisation via doubly charged scalars)?

Discovery potential forµ⁻– e⁺ conversion

Open issues→ where do we need to improve in order to get reliable predictions?

Summary and Outlook

2 / 32

Today’s Agenda:

What happens in aµ– e conversion?

What are similarities and differences when considering µ⁻– e⁻ and µ⁻–e⁺ conversion?

How to tackleµ⁻–e⁻ conversion (using the example of a realisation via doubly charged scalars)?

Employing the complementarity between collider and low energy physics to increase the testability →Results based on the example case

How to tackleµ⁻–e⁺ conversion (using the example of a realisation via doubly charged scalars)?

Discovery potential forµ⁻– e⁺ conversion

Open issues→ where do we need to improve in order to get reliable predictions?

Summary and Outlook

2 / 32

Today’s Agenda:

What happens in aµ– e conversion?

What are similarities and differences when considering µ⁻– e⁻ and µ⁻–e⁺ conversion?

How to tackleµ⁻–e⁻ conversion (using the example of a realisation via doubly charged scalars)?

Employing the complementarity between collider and low energy physics to increase the testability →Results based on the example case

How to tackleµ⁻–e⁺ conversion (using the example of a realisation via doubly charged scalars)?

Discovery potential forµ⁻– e⁺ conversion

Open issues→ where do we need to improve in order to get reliable predictions?

Summary and Outlook

2 / 32

Today’s Agenda:

What happens in aµ– e conversion?

What are similarities and differences when considering µ⁻– e⁻ and µ⁻–e⁺ conversion?

How to tackleµ⁻–e⁻ conversion (using the example of a realisation via doubly charged scalars)?

Employing the complementarity between collider and low energy physics to increase the testability →Results based on the example case

How to tackleµ⁻–e⁺ conversion (using the example of a realisation via doubly charged scalars)?

Discovery potential forµ⁻– e⁺ conversion

Open issues→ where do we need to improve in order to get reliable predictions?

Summary and Outlook

2 / 32

µ–e Conversion

What happens in a µ⁻– e^± conversion?? → experimentally a two-step process

BSM

µ⁻ e⁻

(Z,A) (Z,A)

BSM

µ⁻ e⁺

(Z,A) (Z−2,A)

First Step: µ⁻ is captured in an ‘outer’

atomic shell, and subsequently de-excites to the 1s ground state

Second Step: µ⁻ is captured by the nucleus and reemits ane^±

→ we only consider”coherent” conversion: initial and final state nucleus are in ground state

3 / 32

µ–e Conversion

What happens in a µ⁻– e^± conversion?? → experimentally a two-step process

BSM

µ⁻ e⁻

(Z,A) (Z,A)

BSM

µ⁻ e⁺

(Z,A) (Z−2,A)

First Step: µ⁻ is captured in an ‘outer’

atomic shell, and subsequently de-excites to the 1s ground state

Second Step: µ⁻ is captured by the nucleus and reemits ane^±

→ we only consider”coherent” conversion: initial and final state nucleus are in ground state

3 / 32

µ–e Conversion

What happens in a µ⁻– e^± conversion?? → experimentally a two-step process

BSM

µ⁻ e⁻

(Z,A) (Z,A)

BSM

µ⁻ e⁺

(Z,A) (Z−2,A)

First Step: µ⁻ is captured in an ‘outer’

atomic shell, and subsequently de-excites to the 1s ground state

Second Step: µ⁻ is captured by the nucleus and reemits ane^±

→ we only consider”coherent” conversion: initial and final state nucleus are in ground state

3 / 32

µ–e Conversion

What happens in a µ⁻– e^± conversion?? → experimentally a two-step process

BSM

µ⁻ e⁻

(Z,A) (Z,A)

BSM

µ⁻ e⁺

(Z,A) (Z−2,A)

First Step: µ⁻ is captured in an ‘outer’

atomic shell, and subsequently de-excites to the 1s ground state

Second Step: µ⁻ is captured by the nucleus and reemits ane^±

→ we only consider”coherent” conversion: initial and final state nucleus are in ground state

3 / 32

Energy Scales of the Process

muon boundin 1s state with binding energy _B ' ^m_m^µ

e ·13.6 eV·Z m_µ −−−−→^Z≤100 non-relativistic

consider ”coherent” process→ initial and final nucleus inground state

+ in good approximation: both nuclei at rest

⇒ E_e=m_µ−_B

| {z }

Eµ

∼ O(100MeV)

+E_i −E_f

| {z }

∼O(MeV)

∼ O(100MeV)

⇒ e^± is relativisticparticle under influence of Coulomb potential:

E_e'E_µ'm_µ andm_e'0

for 4-momentum transfer q⁰=p_e−p_µ In this set-up ⇒ q⁰² ' −m²_µ

BSM

µ⁻ e^±

(Z,A) (Z−2,A)

(Z,A)

−→

q^′

4 / 32

Energy Scales of the Process

muon boundin 1s state with binding energy _B ' ^m_m^µ

e ·13.6 eV·Z m_µ −−−−→^Z≤100 non-relativistic

consider ”coherent” process→ initial and final nucleus inground state

+ in good approximation: both nuclei at rest

⇒ E_e=m_µ−_B

| {z }

Eµ

∼ O(100MeV)

+E_i −E_f

| {z }

∼O(MeV)

∼ O(100MeV)

⇒ e^± is relativisticparticle under influence of Coulomb potential:

E_e'E_µ'm_µ andm_e'0

for 4-momentum transfer q⁰=p_e−p_µ In this set-up ⇒ q⁰² ' −m²_µ

BSM

µ⁻ e^±

(Z,A) (Z−2,A)

(Z,A)

−→

q^′

4 / 32

Energy Scales of the Process

muon boundin 1s state with binding energy _B ' ^m_m^µ

e ·13.6 eV·Z m_µ −−−−→^Z≤100 non-relativistic

consider ”coherent” process→ initial and final nucleus inground state

+ in good approximation: both nuclei at rest

⇒ E_e=m_µ−_B

| {z }

Eµ

∼ O(100MeV)

+E_i −E_f

| {z }

∼O(MeV)

∼ O(100MeV)

⇒ e^± is relativisticparticle under influence of Coulomb potential:

E_e'E_µ'm_µ andm_e'0

for 4-momentum transfer q⁰=p_e−p_µ In this set-up ⇒ q⁰² ' −m²_µ

BSM

µ⁻ e^±

(Z,A) (Z−2,A)

(Z,A)

−→

q^′

4 / 32

Energy Scales of the Process

muon boundin 1s state with binding energy _B ' ^m_m^µ

e ·13.6 eV·Z m_µ −−−−→^Z≤100 non-relativistic

consider ”coherent” process→ initial and final nucleus inground state

+ in good approximation: both nuclei at rest

⇒ E_e=m_µ−_B

| {z }

Eµ

∼ O(100MeV)

+E_i −E_f

| {z }

∼O(MeV)

∼ O(100MeV)

⇒ e^± is relativisticparticle under influence of Coulomb potential:

E_e'E_µ'm_µ andm_e'0

for 4-momentum transfer q⁰=p_e−p_µ In this set-up ⇒ q⁰² ' −m²_µ

BSM

µ⁻ e^±

(Z,A) (Z−2,A)

(Z,A)

−→

q^′

4 / 32

µ

– e

vs µ

– e

Conversion

μ

-e

conv.

LNV-Alterna2ves:

μ^--μ⁺ conversion K⁺ π⁺μ^-μ^-

LFV-Alterna2ves:

μ e+γ μ 3e

0νββ _conv. ^μ

^-e

from

TG, Merle, Zuber Phys.Lett. B764 (2017) 157

µ⁻– e⁻ occurs atsinglenucleon (∆Q= 0)

dominated by coherent process

µ⁻– e⁺

needs to occur attwo nucleons to achieve ∆Q= 2 → similar to 0νββ

around 40% of the process’

total areg.s.→g.s.

⇓

further investigations needed:

→ confirm/obtain the percentage that takes place ”coherently”

→ derive a more involved spectrum for the positrons

5 / 32

µ

– e

vs µ

– e

Conversion

μ

-e

conv.

LNV-Alterna2ves:

μ^--μ⁺ conversion K⁺ π⁺μ^-μ^-

LFV-Alterna2ves:

μ e+γ μ 3e

0νββ _conv. ^μ

^-e

from

TG, Merle, Zuber Phys.Lett. B764 (2017) 157

µ⁻– e⁻ occurs atsinglenucleon (∆Q= 0)

dominated by coherent process

µ⁻– e⁺

needs to occur attwo nucleons to achieve ∆Q= 2 → similar to 0νββ

around 40% of the process’

total areg.s.→g.s.

⇓

further investigations needed:

→ confirm/obtain the percentage that takes place ”coherently”

→ derive a more involved spectrum for the positrons

5 / 32

µ

– e

vs µ

– e

Conversion

μ

-e

conv.

LNV-Alterna2ves:

μ^--μ⁺ conversion K⁺ π⁺μ^-μ^-

LFV-Alterna2ves:

μ e+γ μ 3e

0νββ _conv. ^μ

^-e

from

TG, Merle, Zuber Phys.Lett. B764 (2017) 157

µ⁻– e⁻ occurs atsinglenucleon (∆Q= 0)

dominated by coherent process

µ⁻– e⁺

needs to occur attwo nucleons to achieve ∆Q= 2 → similar to 0νββ

around 40% of the process’

total areg.s.→g.s.

⇓

further investigations needed:

→ confirm/obtain the percentage that takes place ”coherently”

→ derive a more involved spectrum for the positrons

5 / 32

µ

– e

vs µ

– e

Conversion

μ

-e

conv.

LNV-Alterna2ves:

μ^--μ⁺ conversion K⁺ π⁺μ^-μ^-

LFV-Alterna2ves:

μ e+γ μ 3e

0νββ _conv. ^μ

^-e

from

TG, Merle, Zuber Phys.Lett. B764 (2017) 157

µ⁻– e⁻ occurs atsinglenucleon (∆Q= 0)

dominated by coherent process

µ⁻– e⁺

needs to occur attwo nucleons to achieve ∆Q= 2 → similar to 0νββ

around 40% of the process’

total areg.s.→g.s.

⇓

further investigations needed:

→ confirm/obtain the percentage that takes place ”coherently”

→ derive a more involved spectrum for the positrons

5 / 32

Improvements from Upcoming Experiments

Snapshot oncurrent limitsandsensitivities of upcoming experiments:

��^-�� ��^-�� ��^-�� ��^-�� ��^-��

��(μ^--�^-����������)

��-��

��-��

��-��

��-���

��-���

������ ����������� ���μ-�����������

���� ������������ �������

�������

�������

�������

������

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past: SINDRUM II for⁴⁸Ti (1993),

208Pb (1995), ¹⁹⁷Au (2006)

future: DeeMee for²⁸Si, COMET and Mu2e (taking data ∼2018) for²⁷Al, PRISM/PRIME for⁴⁸Ti

→ improvements can be transferred to µ⁻– e⁺ conversion

→ sensitivities on both processes will increase by several orders of magnitude in the foreseeable future

→ target both processes with the same experimental setup

⇒ it’s time to investigatethese bound muon conversions to describe them within a general frameworkindependent of the respective particle physics realisation

6 / 32

Improvements from Upcoming Experiments

Snapshot oncurrent limitsandsensitivities of upcoming experiments:

��^-�� ��^-�� ��^-�� ��^-�� ��^-��

��(μ^--�^-����������)

��-��

��-��

��-��

��-���

��-���

������ ����������� ���μ-�����������

���� ������������ �������

�������

�������

�������

������

������

������

past: SINDRUM II for⁴⁸Ti (1993),

208Pb (1995), ¹⁹⁷Au (2006)

future: DeeMee for²⁸Si, COMET and Mu2e (taking data ∼2018) for²⁷Al, PRISM/PRIME for⁴⁸Ti

→ improvements can be transferred to µ⁻– e⁺ conversion

→ sensitivities on both processes will increase by several orders of magnitude in the foreseeable future

→ target both processes with the same experimental setup

⇒ it’s time to investigatethese bound muon conversions to describe them within a general frameworkindependent of the respective particle physics realisation

6 / 32

Improvements from Upcoming Experiments

Snapshot oncurrent limitsandsensitivities of upcoming experiments:

��^-�� ��^-�� ��^-�� ��^-�� ��^-��

��(μ^--�^-����������)

��-��

��-��

��-��

��-���

��-���

������ ����������� ���μ-�����������

���� ������������ �������

�������

�������

�������

������

������

������

past: SINDRUM II for⁴⁸Ti (1993),

208Pb (1995), ¹⁹⁷Au (2006)

future: DeeMee for²⁸Si, COMET and Mu2e (taking data ∼2018) for²⁷Al, PRISM/PRIME for⁴⁸Ti

→ improvements can be transferred to µ⁻– e⁺ conversion

→ sensitivities on both processes will increase by several orders of magnitude in the foreseeable future

→ target both processes with the same experimental setup

⇒ it’s time to investigatethese bound muon conversions to describe them within a general frameworkindependent of the respective particle physics realisation

6 / 32

How to tackle µ

– e

conversion (using the example of a realisation via doubly charged scalars)?

7 / 32

Effective theory of a doubly charged scalar singlet

based on King, Merle, Panizzi JHEP 1411 (2014) 124

Minimal extension of SM:

onlyoneextra particle: S⁺⁺

→lightest of possible new particles (UV completion e.g. Cocktail model)

→reduction of input parameters

tree-level couplingto SM (to charged right-handed leptons)

→LNV and LFV!

effectiveDim-7 operator(necessary to generate neutrino mass) L=L_SM−V(H,S)

+ (DµS)^†(D^µS)+ fab(`Ra)<sup>c</sup>`RbS⁺⁺ +h.c.− ^g2_{4 Λ}^v43^ξS⁺⁺W_µ⁻W^−µ +h.c.

8 / 32

Effective theory of a doubly charged scalar singlet

based on King, Merle, Panizzi JHEP 1411 (2014) 124

Minimal extension of SM:

onlyoneextra particle: S⁺⁺

→lightest of possible new particles (UV completion e.g. Cocktail model)

→reduction of input parameters

tree-level couplingto SM (to charged right-handed leptons)

→LNV and LFV!

effectiveDim-7 operator(necessary to generate neutrino mass) L=L_SM−V(H,S)

+ (DµS)^†(D^µS) + fab(`Ra)<sup>c</sup>`RbS⁺⁺ +h.c.− ^g2_{4 Λ}^v43^ξS⁺⁺W_µ⁻W^−µ +h.c.

8 / 32

Effective theory of a doubly charged scalar singlet

based on King, Merle, Panizzi JHEP 1411 (2014) 124

Minimal extension of SM:

onlyoneextra particle: S⁺⁺

→lightest of possible new particles (UV completion e.g. Cocktail model)

→reduction of input parameters

tree-level couplingto SM (to charged right-handed leptons)

→LNV and LFV!

effectiveDim-7 operator(necessary to generate neutrino mass) L=L_SM−V(H,S)

+ (DµS)^†(D^µS) + fab(`Ra)<sup>c</sup>`RbS⁺⁺ +h.c.− ^g2_{4 Λ}^v43^ξS⁺⁺W_µ⁻W^−µ +h.c.

8 / 32

µ

– e

Conversion:

µ⁻–e⁻ conversion realised at one-looplevel

νa e⁻

−→k µ−

pµ

−→ −→^pe

−→

u

−→ −→

d W⁻ W⁺

↓ ↓

u

pµ−k

−→

e⁻ νa

µ⁻ pµ

−→ −→^k −→^pe W−

νa Zν, h⁰

−→q^′ k+q^′

−→

q

−→

q

−→

νa W⁻

−→k

pµ−k

−→

µ− pµ

−→

e⁻ pµ

−→ −→^pe Aν, Zν, h⁰

−→

q′ e⁻

−→

q q

−→

νa e⁻

−→k µ⁻

pµ

−→ −→^pe

−→

d

−→ −→

u W⁻ W⁺

↓ ↓

d

νa e⁻

µ⁻

W⁻ W⁻ pµ

−→ −→^k −→^pe

pµ−k

−→

p^µ−k+q^′

−→

Aν, Zν, h⁰

−→q^′ q

−→

q

−→

νa e⁻

µ⁻

W⁻ pµ

−→ −→^k −→^pe

pµ−k+q^′

−→

pµ+q^′

−→

µ⁻ Aν, Zν, h⁰

−→

q′

−→

q q

−→

l⁺ S⁻⁻

−→k

pµ−k

−→

µ⁻ pµ

−→

e⁻ pµ

−→ −→^pe Aν, Zν, h⁰ e⁻

−→

q q

−→ −→

q′

l⁺ e⁻

µ⁻

S⁻⁻

pµ

−→ −→^k −→^pe

pµ−k+q′

−→

pµ+q^′

−→

µ⁻ Aν, Zν, h⁰

−→q^′

−→

q q

−→

l+ S⁻⁻ S⁻⁻

−→k

p µ−k

−→

pµ−k+q^′

−→

A_ν, Z_ν, h0

−→

q′ µ⁻

pµ

−→

e− pe

−→

q

−→

q

−→

pµ−k

−→

e⁻ l⁺ µ−

pµ

−→ −→^k −→^pe S⁻⁻

l⁺ Aν, Zν, h⁰

−→q′ k+q′

−→

q

−→

q

−→

9 / 32

µ

– e

Conversion:

µ⁻–e⁻ conversion realised at one-looplevel

νa e⁻

−→k µ−

pµ

−→ −→^pe

−→

u

−→ −→

d W⁻ W⁺

↓ ↓

u

pµ−k

−→

e⁻ νa

µ⁻ pµ

−→ −→^k −→^pe W−

νa Zν, h⁰

−→q^′ k+q^′

−→

q

−→

q

−→

νa W⁻

−→k

pµ−k

−→

µ− pµ

−→

e⁻ pµ

−→ −→^pe Aν, Zν, h⁰

−→

q′ e⁻

−→

q q

−→

νa e⁻

−→k µ⁻

pµ

−→ −→^pe

−→

d

−→ −→

u W⁻ W⁺

↓ ↓

d

νa e⁻

µ⁻

W⁻ W⁻ pµ

−→ −→^k −→^pe

pµ−k

−→

p^µ−k+q^′

−→

Aν, Zν, h⁰

−→q^′ q

−→

q

−→

νa e⁻

µ⁻

W⁻ pµ

−→ −→^k −→^pe

pµ−k+q^′

−→

pµ+q^′

−→

µ⁻ Aν, Zν, h⁰

−→

q′

−→

q q

−→

l⁺ S⁻⁻

−→k

pµ−k

−→

µ− pµ

−→

e⁻ pµ

−→ −→^pe

Aν, Zν

−→q^′ e⁻

−→

q q

−→

l⁺ e⁻

µ⁻

S⁻⁻

pµ

−→ −→^k −→^pe

pµ−k+q^′

−→

pµ+q^′

−→

µ⁻ Aν, Zν

−→q′ q

−→

q

−→

l⁺ S−− S−−

−→k

pµ−k

−→

pµ−k+q′

−→

A_ν, Z_ν

−→

q′ µ⁻

pµ

−→

e⁻ pe

−→

q

−→

q

−→

pµ−k

−→

e− l⁺ µ⁻

pµ

−→ −→^k −→^pe S−−

l⁺ Aν, Zν −→q^′

k+q^′

−→

q

−→

q

−→

relevant diagrams

9 / 32

Different Contributions to µ

– e

Conversion

estimate nuclear radius: R =

∼O(10⁻¹⁵ m)

z}|{r₀ A^1/3 ∼ O(10⁻¹⁵ m) reduced Bohr radius: a₀

|{z}

O(10⁻¹⁰ m) me

mµ ∼ O(10⁻¹³ m)

estimate interaction range: r_γ→ ∞ andr_Z ≤10⁻¹⁸ m

⇒ for Z-exchange: µ⁻ has to bewithin nucleus! Probability?!

e⁻ µ⁻

q q

Z, γ

−→

q^′ ⇒

l⁺

S⁻⁻ S⁻⁻

γ

µ⁻ e⁻

q q photoniccontribution:

”long range”

l⁺

S⁻⁻ S⁻⁻

Z

µ⁻ e⁻

q q

non-photoniccontribution:

”short range”

⇒suppressed

⇒ contributions need to be treatedqualitatively differently!!

10 / 32

Different Contributions to µ

– e

Conversion

estimate nuclear radius: R =

∼O(10⁻¹⁵ m)

z}|{r₀ A^1/3 ∼ O(10⁻¹⁵ m) reduced Bohr radius: a₀

|{z}

O(10⁻¹⁰ m) me

mµ ∼ O(10⁻¹³ m)

estimate interaction range: r_γ→ ∞ andr_Z ≤10⁻¹⁸ m

⇒ for Z-exchange: µ⁻ has to bewithin nucleus! Probability?!

e⁻ µ⁻

q q

Z, γ

−→

q^′ ⇒

l⁺

S⁻⁻ S⁻⁻

γ

µ⁻ e⁻

q q photoniccontribution:

”long range”

l⁺

S⁻⁻ S⁻⁻

Z

µ⁻ e⁻

q q

non-photoniccontribution:

”short range”

⇒suppressed

⇒ contributions need to be treatedqualitatively differently!!

10 / 32

Different Contributions to µ

– e

Conversion

estimate nuclear radius: R =

∼O(10⁻¹⁵ m)

z}|{r₀ A^1/3 ∼ O(10⁻¹⁵ m) reduced Bohr radius: a₀

|{z}

O(10⁻¹⁰ m) me

mµ ∼ O(10⁻¹³ m)

estimate interaction range: r_γ→ ∞ andr_Z ≤10⁻¹⁸ m

⇒ for Z-exchange: µ⁻ has to bewithin nucleus! Probability?!

e⁻ µ⁻

q q

Z, γ

−→

q^′ ⇒

l⁺

S⁻⁻ S⁻⁻

γ

µ⁻ e⁻

q q photoniccontribution:

”long range”

l⁺

S⁻⁻ S⁻⁻

Z

µ⁻ e⁻

q q

non-photoniccontribution:

”short range”

⇒suppressed

⇒ contributions need to be treatedqualitatively differently!!

10 / 32

Different Contributions to µ

– e

Conversion

estimate nuclear radius: R =

∼O(10⁻¹⁵ m)

z}|{r₀ A^1/3 ∼ O(10⁻¹⁵ m) reduced Bohr radius: a₀

|{z}

O(10⁻¹⁰ m) me

mµ ∼ O(10⁻¹³ m)

estimate interaction range: r_γ→ ∞ andr_Z ≤10⁻¹⁸ m

⇒ for Z-exchange: µ⁻ has to bewithin nucleus! Probability?!

e⁻ µ⁻

q q

Z, γ

−→

q^′ ⇒

l⁺

S⁻⁻ S⁻⁻

γ

µ⁻ e⁻

q q photoniccontribution:

”long range”

l⁺

S⁻⁻ S⁻⁻

Z

µ⁻ e⁻

q q

non-photoniccontribution:

”short range”

⇒suppressed

⇒ contributions need to be treatedqualitatively differently!!

10 / 32

Different Contributions to µ

– e

Conversion

estimate nuclear radius: R =

∼O(10⁻¹⁵ m)

z}|{r₀ A^1/3 ∼ O(10⁻¹⁵ m) reduced Bohr radius: a₀

|{z}

O(10⁻¹⁰ m) me

mµ ∼ O(10⁻¹³ m)

estimate interaction range: r_γ→ ∞ andr_Z ≤10⁻¹⁸ m

⇒ for Z-exchange: µ⁻ has to bewithin nucleus! Probability?!

e⁻ µ⁻

q q

Z, γ

−→

q^′ ⇒

l⁺

S⁻⁻ S⁻⁻

γ

µ⁻ e⁻

q q photoniccontribution:

”long range”

l⁺

S⁻⁻ S⁻⁻

Z

µ⁻ e⁻

q q

non-photoniccontribution:

”short range”

⇒suppressed

⇒ contributions need to be treatedqualitatively differently!!

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Different Contributions to µ

– e

Conversion

estimate nuclear radius: R =

∼O(10⁻¹⁵ m)

z}|{r₀ A^1/3 ∼ O(10⁻¹⁵ m) reduced Bohr radius: a₀

|{z}

O(10⁻¹⁰ m) me

mµ ∼ O(10⁻¹³ m)

estimate interaction range: r_γ→ ∞ andr_Z ≤10⁻¹⁸ m

⇒ for Z-exchange: µ⁻ has to bewithin nucleus! Probability?!

e⁻ µ⁻

q q

Z, γ

−→

q^′ ⇒

l⁺

S⁻⁻ S⁻⁻

γ

µ⁻ e⁻

q q photoniccontribution:

”long range”

l⁺

S⁻⁻ S⁻⁻

Z

µ⁻ e⁻

q q

non-photoniccontribution:

”short range”

⇒suppressed

⇒ contributions need to be treatedqualitatively differently!!

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Photonic Contribution

e⁻ µ⁻

q q

Aν

−→

q^′

nucleus Γ^ν

M ∝^R d³rψ_jlm^e (p_e,r)Γ^νψ_j^µ

µlµmµ(p_µ,r)hN|qγ_νq|Ni

| {z }

Zeρ^(P)(r)δν0

→ wave functionsfor µ⁻ ande⁻ obtained by solving modified Dirac equation(+ Coulomb potential)

→ Mostgeneral (Lorentz-) invariantexpressionfor Γ^ν:

Γ^ν=

γ^ν−/q⁰q^0ν q⁰²

F1(q⁰²) +iσ^νρq⁰_ρ mµ

F2(q⁰²) +

γ^ν−q/⁰q^0ν q⁰²

γ5G1(q⁰²) +iσ^νρq⁰_ρ mµ

γ5G2(q⁰²)

with q⁰ =pe−pµ. In non-relativistic limit:

⇒ ψjlm andZeρ^(P)(r) factorise fromΓ⁰ on matrix element level

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