Weak Interaction Contributions in Light Muonic Atomsrnp/wiki/stuff/Workshop/Talks/503_Eides.pdf ·...

Weak Interaction Contributions in Light Muonic Atoms

Michael I. Eides

Department of Physics and AstronomyUniversity of Kentucky

USA

M.E., Phys. Rev. A 85, 034503 (2012)

ECT* Workshop on the Proton Radius PuzzleOctober 29 - November 2, 2012

Trento, Italy

Eides, ECT* 2012, Trento, Italy Weak Interaction November 2, 2012 1 / 18

Outline

1 Effective Weak Interaction Hamiltonian

2 Weak Interaction Contributions to Hyperfine SplittingHydrogenDeuteriumTritiumHelium Ions e3He+, µ3He+

Helium Ions e4He+, µ4He+

3 Weak Interaction Contribution to Lamb Shift

4 Conclusions


Outline





4 Conclusions


Outline





4 Conclusions


Outline





4 Conclusions


Effective Electroweak Theory

Fundamental Fermions

Effective low-energy field-theoretic weak interaction Hamiltonian dueto neutral currents for the fundamental fermions

HZ =4GF√

2

∫d3x

(∑i

ψiγµ(T3 − sin2 θWQ)ψi

)2

I θW is the Weinberg angleI Q is the charge operator in terms of proton chargeI T3 = T3(1− γ5)/2, T3 is the weak isospinI Summation goes over all species of fermions

∀ current contains a vector and an axial part

Axial parts of nucleon currents are renormalized by strong interactionsand should be multiplied by gA = 1.27


Effective Electroweak Theory

Nucleons

Lepton-nucleon Hamiltonian

HZ =GF

2√

2

∫d3x

[ψlγ

µγ5ψl − ψlγµ(1− 4 sin2 θW

)ψl

]×[gAψnγµγ

5ψn − ψnγµψn − gAψpγµγ5ψp

+ψpγµ(1− 4 sin2 θW

)ψp

],

I ψl is the lepton (electron or muon) fieldI ψp and ψn are the proton and neutron fields

Leading weak interaction contribution to HFS arises from interactionof axial currents

Only spatial components of axial neutral currents give nonzerocontribution


HFS Hamiltonian

Nucleons

HZ →gAGF

2√

2

∫d3x

(ψlγ

µγ5ψl

) (ψnγµγ

5ψn − ψpγµγ5ψp

)→ −gAGF

2√

2

∫d3x

(ψlγ

iγ5ψl

) (ψnγ

iγ5ψn − ψpγiγ5ψp

)Quantum mechanical Hamiltonian for a nucleus with Z protons andA− Z neutrons

HZ =gAGF

2√

2σl ·

(∑p

σp −∑n

σn

)δ(3)(r)

Leading weak interaction contribution to HFS is nonzero only in Sstates

All results below are valid both for light electronic and muonic atomsand ionsEides, ECT* 2012, Trento, Italy Weak Interaction November 2, 2012 5 / 18

Hyperfine Splitting

Hydrogen

Hamiltonian

HZ =gAGF

2√

2σl · σpδ

(3)(r)

Leading contribution in the nS-state (Eides, 1996)

∆EZ (nS) =gAGF

2√

2|ψn(0)|3(σe · σp)|F=1

F=0,

Coulomb-Schrodinger wave function at the origin:ψn(0) =

√(Zαmr )3/(πn3)

I mr = mlmp/(ml + mp)I Lepton spin operator: J = σl/2I Proton spin operator: I = σp/2I Total angular momentum: F = I + JI (σe · σp)|F=1

F=0 = 4


Hyperfine Splitting

Hydrogen

∆EZ (nS) =2gAGF√

2

(Zαmr )3

πn3.

∆EZ (2S) = 2.8× 10−4 meV

At least an order of magnitude smaller than the uncertainty in hyperfinesplitting due to proton structure contributions, Carlson, Nazaryan,Griffioen, 2011

EF =4

3gpα(Zα)3m3

r

mlmp≈ 182.44 meV

gp ≈ 5.58 . . . is the proton g -factor in nuclear magnetons

Dominant contribution to HFS scales as 1/n3

n3∆EZ (nS)

EF=

3

2√

2π

gAGFmµmp

gpα≈ 1.2 . . .× 10−5


Hyperfine Splitting

Hydrogen

∆EZ (nS) =2gAGF√

2

(Zαmr )3

πn3.

∆EZ (2S) = 2.8× 10−4 meV

At least an order of magnitude smaller than the uncertainty in hyperfinesplitting due to proton structure contributions, Carlson, Nazaryan,Griffioen, 2011

EF =4

3gpα(Zα)3m3

r

mlmp≈ 182.44 meV

gp ≈ 5.58 . . . is the proton g -factor in nuclear magnetonsDominant contribution to HFS scales as 1/n3

n3∆EZ (nS)

EF=

3

2√

2π

gAGFmµmp

gpα≈ 1.2 . . .× 10−5


Hyperfine Splitting

Deuterium

Deuteron: spin one loosely bound (p, n) system, (Z = 1, A = 1),predominantly S-state wave function

Hamiltonian

HZ =gAGF

2√

2σl · (σp − σn) δ(3)(r)

Spin wave function is symmetric, matrix element of thespin-antisymmetric deuteron nuclear factor is zero

〈σp − σn〉 = 0.

More accurately: ∃ admixture of D wave

D wave spin function is also symmetric with respect to spin variables

Weak interaction contribution to hyperfine splitting in electronic andmuonic deuterium in the leading nonrelativistic approximation is zero


Hyperfine Splitting

Tritium

Triton: spin one half (I = 1/2) system of one proton and twoneutrons (Z = 1, A = 3), T3 = 1/2− 1/2− 1/2 = −1/2

Predominantly a product of the S wave coordinate wave function anda completely antisymmetric spin-isospin wave function

In this approximation 〈σp − σn1 − σn2〉 = 2I

A more accurate analysis (Friar and Payne, 2005)

〈σp − σn1 − σn2〉 = 2I

(1− 4

3PS ′ − 2

3PD

)= 2cI, c ≈ 0.92


Hyperfine Splitting

Tritium

∆EZ (nS) =2cgAGF√

2

(Zαmr )3

πn3

∆EZ (n = 2) = 3.1× 10−4 meV.

EF =4

3gtα(Zα)3m3

r

mlmp= 239.919 . . . meV (1)

gt = 5.957924896(76) is the triton g -factor in nuclear magnetons


n3∆EZ (nS)

EF=

3

2√

2π

cgAGFmµmp

gtα≈ 1.0 . . .× 10−5


Hyperfine Splitting


Helion: spin one half (I = 1/2) system of two protons and a neutron(Z = 2, A = 3), T3 = 1/2 + 1/2− 1/2 = 1/2

Predominantly a product of the S wave coordinate wave function anda completely antisymmetric spin-isospin wave function

In this approximation 〈σp1 + σp2 − σn〉 = −2I

A more accurate analysis (Friar and Payne, 2005)

〈σp1 − σp2 − σn〉 = −2I

(1− 4

3PS ′ − 2

3PD

)= −2cI


Hyperfine Splitting


∆EZ (nS) = −2cgAGF√2

(Zαmr )3

πn3

∆EZ (n = 2) = −2.5 . . .× 10−3 meV

EF =4

3ghα(Zα)3m3

r

mµmp= −1370.8 . . . meV

gh = −4.255250613 is the helion g -factor in nuclear magnetons


n3∆EZ

EF= − 3

2√

2π

cgAGFmµmp

ghZα≈ 1.5 . . .× 10−5


Hyperfine Splitting


Spin of α-particle is zero

No hyperfine structure in e4He+, µ4He+

No weak interaction contribution to hyperfine structure


Lamb Shift

Leading weak interaction contribution to the Lamb shift arises frominteraction of vector currents

Only time components of vector currents give nonzero contributions

HZ →GF

2√

2

∫d3x

[ψlγ

µ(1− 4 sin2 θW

)ψl

] [ψnγµψn − ψpγµ(1− 4 sin2 θW )ψp

]→ GF

2√

2

∫d3x

[ψlγ

0(1− 4 sin2 θW

)ψl

] [ψnγ0ψn − ψpγ0

(1− 4 sin2 θW

)ψp

]Quantum mechanical Hamiltonian for a nucleus with Z protons and A− Zneutrons

HZ =GF

2√

2

(1− 4 sin2 θW

) [(A− Z )− Z

(1− 4 sin2 θW

)]δ(3)(r)


Lamb Shift

� Leading weak interaction contribution to the Lamb shift is nonzero onlyin S states

∆EZ (nS) =GF

2√

2

(1− 4 sin2 θW

) [(A− Z )− Z

(1− 4 sin2 θW

)] (mrZα)3

πn3

Muonic hydrogen: A = Z = 1 =⇒ additional suppression by a smallfactor 1− 4 sin2 θW ≈ 0.08

No additional suppression for all other light muonic systems

Dominant electron vacuum polarization contribution

∆Enl = −8α(Zα)2mr

3πn3Q

(1)nl (β)

β = me/(mrZα), Q(1)nl (β) is a known function


Lamb Shift

2P − 2S interval:

∆E (2P − 2S) = ∆E21 −∆E20 =α(Zα)2mr

3π(Q

(1)20 (β)− Q

(1)21 (β))

∆EZ (L, 2S)

∆E (2P − 2S)=

3GFm2r Z(1− 4 sin2 θW

) [(A− Z )− Z

(1− 4 sin2 θW

)]16√

2(Q(1)20 (β)− Q

(1)21 (β))

Muonic hydrogen: A = Z = 1, additional suppression an extra factor1− 4 sin2 θW ≈ 0.08

No suppression for A 6= Z

Muonic hydrogen

∆EZ (L, n = 2)

∆E (2P − 2S)≈ −1.7× 10−9.


Lamb Shift

Weak correction to the Lamb shift in muonic hydrogen is orders ofmagnitude smaller than the relative error of the Lamb shiftmeasurement (Pohl et al, 2010)

Much smaller than uncertainties of the proton structure corrections(Carlson, Vanderhaeghen, 2011)


Lamb Shift

Weak correction to the Lamb shift in muonic hydrogen is orders ofmagnitude smaller than the relative error of the Lamb shiftmeasurement (Pohl et al, 2010)

Much smaller than uncertainties of the proton structure corrections(Carlson, Vanderhaeghen, 2011)


Consclusions

Leading weak contributions to HFS and Lamb shift in light muonicatoms and ions are calculated

Leading correction to HFS in deuterium is zero

Corrections to Lamb shift in hydrogen are additionally suppressed bythe small factor (1− 4 sin2 θW )

In all cases weak corrections are much smaller than currentexperimental and theoretical errors


Consclusions






Consclusions






Consclusions






Date post:	21-Mar-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Weak Interaction Contributions in Light Muonic Atomsrnp/wiki/stuff/Workshop/Talks/503_Eides.pdf ·...

Documents