nuclear structure corrections in light muonic atoms€¦ · q2 from 10-4 gev2 to 10-2 gev2 crema...

Sonia Bacca

Johannes Gutenberg University, Mainz

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Nuclear structure corrections in light muonic atoms

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What are muonic atoms?

Exotic atoms

2

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Exotic atoms

2

Electron replaced with a muon

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Exotic atoms

2


e-

Ordinary atoms

µ-

Muonic atoms

muon more sensitive to the nucleus

Hydrogen-like systems

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Exotic atoms

2


Can be used as a precision probe for the nucleus

e-

Ordinary atoms

µ-

Muonic atoms

muon more sensitive to the nucleus

Hydrogen-like systems

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Production and measurements

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e-The proton charge radius is measured from: electron-proton interactions: 0.8770 ± 0.0045 fm

eH spectroscopy e-p scattering

Pohl et al., Nature (2010)Antognini et al., Science (2013)

µ- muonic -proton interactions: 0.8409 ± 0.0004 fm 𝜇H Lamb-shift

Proton Radius Puzzle

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7 �

Is lepton universality violated?


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7 �


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Experimental efforts

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Higher precision electron scattering experiments

Q2 from 10-4 GeV2 to 10-2 GeV2

Repeat traditional measurement with windowless gas jet target (2019/20)

ISR measurement, not competitive, Phys.Lett. B 771 (2017) 194-198

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MUSE collaboration

Measure both e+/e- and 𝜇+/µ- to reduce uncertainties



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CREMA collaboration

CREMA collaboration currently measuring Lamb shift in light muonic atoms: Deuterium, Helions, etc.

MUSE collaboration

Measure both e+/e- and 𝜇+/µ- to reduce uncertainties



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µ-

Strong experimental program at PSI (Switzerland) from the CREMA collaboration to unravel the mystery by studying other muonic atoms:

𝜇D (results released) 𝜇4He+ (analyzing data)

𝜇3He+ (analyzing data)

𝜇3H (impossible?) 𝜇6Li2+, 𝜇7Li2+ (future)

Understanding the Proton Radius Puzzle

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µ-





�E2S�2P = �QED +AOPEhr2c i+ �TPE


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µ-






well known


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µ-






well known 'm4

r(Z↵4)

12

4


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µ-






well known 'm4

r(Z↵4)

12

4

unknown NS corrections


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µ-

CM

Theoretical derivation of TPE

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Perturbative potential: correction to the bulk Coulomb

µ-

CM


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Perturbative potential: correction to the bulk Coulomb

µ-

CM


Using perturbation theory at second order one obtains the expression for TPEup to order

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Take non-relativistic kinetic energy in muon propagator Neglect Coulomb force in the intermediate state Expand the muon matrix elements in powers of η

η=

Non relativistic term

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Take non-relativistic kinetic energy in muon propagator Neglect Coulomb force in the intermediate state Expand the muon matrix elements in powers of η

★ “virtual” distance traveled by the proton between the two-photon exchange

★ Uncertainty principle

★ ⌘ =

η=

Non relativistic term

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★ dominant term, related to the energy-weighted integral of the dipole response function

Theoretical derivation of TPE Non relativistic term

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★ Related to Zemach moment elastic contribution

�(1)Z3 =⇡

3mr(Z↵)2�2(0)

ZZd3Rd3R0|R�R0|3 ⇢p0(R)⇢p0(R

0)

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★ leads to energy-weighted integrals of three different response functions


★ Related to Zemach moment elastic contribution

�(1)Z3 =⇡

3mr(Z↵)2�2(0)

ZZd3Rd3R0|R�R0|3 ⇢p0(R)⇢p0(R

0)

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Consider the Coulomb force in the intermediate states Naively , actually logarithmically enhanced

Related to the dipole response function

Friar (1977), Pachucki (2011)

Coulomb term


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Coulomb term

Take the relativistic kinetic energy in muon propagator Related to the dipole response function

Relativistic terms


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Coulomb term

Take the relativistic kinetic energy in muon propagator Related to the dipole response function

Relativistic terms

Consider finite nucleon-size by including their charge distributions and obtain terms, e.g.,

Finite nucleon-size corrections

�(1)R1 = �8⇡mr(Z↵)2�2(0)

Z Zd3Rd3R0|R�R0|

2

�2⇢pp0 (R,R0)� �⇢np0 (R,R0)

�


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�TPE

= �AZem

+ �NZem

+ �Apol

+ �Npol

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�TPE

= �AZem

+ �NZem

+ �Apol

+ �Npol

�Apol

= �(0)D1

+ �(1)R3

+ �(1)Z3

+ �(2)R2 + �(2)Q + �(2)D1D3

+ �(0)C

+�(0)L + �(0)T + �(0)M + �(1)R1 + �(1)Z1 + �(2)NS

�AZem = ��(1)Z3 � �(1)Z1

Need to calculate and related uncertainties.�TPE

Friar an Payne (‘97)

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The uncertainty of the extracted radius depends on the precision of the TPE

Roughly: 95% 4% 1%

TPE needs to be know precisely, in order to exploit the experimental precision.

A matter of precision

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The uncertainty of the extracted radius depends on the precision of the TPE

Roughly: 95% 4% 1%

TPE needs to be know precisely, in order to exploit the experimental precision.

A matter of precision

Uncertainties comparison

Atom Exp uncertainty on ΔE2S-2PUncertainty on TPE prior to the discovery

of the proton radius puzzle

µ2H 0.003 meV 0.03 meV*

µ3He+ 0.08 meV 1 meV

µ4He+ 0.06 meV 0.6 meV

µ 6,7Li++ 0.7 meV 4 meV

*Leidemann, Rosenfelder ’95 using few-body methods

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Ab Initio Nuclear Theory

H|�i� = Ei|�i�

H = T + VNN (⇤) + V3N (⇤) + . . .

• Solve the Schrödinger equation for few-nucleons

HN

HN CM

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H|�i� = Ei|�i�

H = T + VNN (⇤) + V3N (⇤) + . . .


HN

HN CM

Hyper-spherical harmonics expansions for A=3,4,6,7 Barnea, Leidemann, Orlandini PRC 61 (2000) 054001

For A=2 we use an harmonic oscillator expansion

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H|�i� = Ei|�i�

H = T + VNN (⇤) + V3N (⇤) + . . .


HN

HN CM



• To compute the response functions, we use the Lorentz integral transform method directly or the Lanczos sum rule method.

Efros, et al., JPG.: Nucl.Part.Phys. 34 (2007) R459

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H|�i� = Ei|�i�

H = T + VNN (⇤) + V3N (⇤) + . . .


HN

HN CM



• To compute the response functions, we use the Lorentz integral transform method directly or the Lanczos sum rule method.

Efros, et al., JPG.: Nucl.Part.Phys. 34 (2007) R459

• We will use nuclear interactions derived from chiral effective filed theory (at various orders) and traditional potentials (AV18+UIX)

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4He

An example

S.B. and Saori Pastore, Journal of Physics G.: Nucl. Part. Phys. 41, 123002 (2014)

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An example

S.B. and Saori Pastore, Journal of Physics G.: Nucl. Part. Phys. 41, 123002 (2014)

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Muonic Deuterium

AV18 in agreement with Pachucki (2011)+ Pachucki, Wienczek (2015)J. Hernandez et al, Phys. Lett. B 736, 344 (2014)

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Deuteron radius puzzle

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μe

e

Pohl et al, Science 353, 669 (2016)


Hernandez et al., PLB 736, 334 (2014) Pachucki (2011)+ Pachucki, Wienczek (2015)

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μe

e

5.6�



Hernandez et al., PLB 736, 334 (2014) Pachucki (2011)+ Pachucki, Wienczek (2015)

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μe

e

5.6�

µH+iso: rp from µH and deuterium isotopic shift r2d -r2p: Parthey et al., PRL 104 233001 (2010)


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μe

e

5.6�

2.6�

µH+iso: rp from µH and deuterium isotopic shift r2d -r2p: Parthey et al., PRL 104 233001 (2010)


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J. Hernandez et al, Phys. Lett. B 778, 377 (2018)

µ2H

Only sightly mitigate the “small” proton radius puzzle (2.6 to 2 𝝈)

Order-by-order chiral expansion

Theory, PLB 2014

Statistical and systematic uncertainty analysis

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Higher order corrections in 𝛼

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Three-photon exchange

Pachucki et al., Phys. Rev. A 97 062511 (2018)

(Z𝛼)6 correction, negligible

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Higher order corrections in 𝛼

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Kalinowski, Phys. Rev. A 99 030501 (2019)One the many 𝛼6 corrections, supposedly the largest

Consistent within 1𝜎solves the small deuteron-radius puzzle

�TPE = �1.750+14�16 meV

�TPE = �1.7638(68) meV

Theory

Exp

Large deuteron-radius puzzle still unsolved! New data on electron scattering expected from MAMI and from the future MESA

Three-photon exchange

Pachucki et al., Phys. Rev. A 97 062511 (2018)

(Z𝛼)6 correction, negligible

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Uncertainties quantifications

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Uncertainties sources • Numerical • Nuclear model • Nucleon-size • Truncation of multiples • η-expansion• expansion in Z𝛼

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Uncertainties sources • Numerical • Nuclear model • Nucleon-size • Truncation of multiples • η-expansion• expansion in Z𝛼

10−4

10−3

10−2

10−1

100

0 5 10 15 20

Devia

tio

n

Kmax

leading dipole correction

4He

C.Ji et al., JPG: Part. Nucl. 45, 093002 (2018)

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Reduction of Uncertainties

Atom Exp uncertainty on ΔE2S-2P

Uncertainty on TPE prior to the discovery of the puzzle Uncertainty on TPE: ab initio

µ2H 0.003 meV 0.03 meV 0.02 meV

µ3He+ 0.08 meV 1 meV 0.3 meV

µ4He+ 0.06 meV 0.6 meV 0.4 meV

µ 6,7Li++ 0.7 meV 4 meV

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µ2H 0.003 meV 0.03 meV 0.02 meV



µ 6,7Li++ 0.7 meV 4 meV

1976 2017-6

-5.5

-5

-4.5

-4

δA

po

l [m

eV]

µ3He

+

Rinker Carlson et al.

2016

Nevo Dinur et al.

~ ~~~

time

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µ2H 0.003 meV 0.03 meV 0.02 meV



µ 6,7Li++ 0.7 meV 4 meV

1974 1976 1977-4

-3

-2

δA

po

l [m

eV]

µ4He

+

Bernabeau, Karlskog Rinker Friar Ji et al.2013

~ ~~~

time

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µ2H 0.003 meV 0.03 meV 0.02 meV



µ 6,7Li++ 0.7 meV 4 meV

What about µ6,7Li++?

Go to Poster Session, Contribution by Simone Li MuliThu 18:30

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Summary and Outlook

• Ab initio calculations have allowed to substantially reduce uncertainties in TPE

• Independently on the nature of the puzzle, these calculations are needed to support any spectroscopic measurement with muonic atoms

• In the future we will investigate the hyperfine splitting of muonic deuterium and the Lamb shift in muonic lithium atoms

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Summary and Outlook




Thanks to my collaboratorsN.Barnea, O.J. Hernandez, C.Ji, S.Li Muli, N.Nevo Dinur, A. Poggialini

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Summary and Outlook




Thanks to my collaboratorsN.Barnea, O.J. Hernandez, C.Ji, S.Li Muli, N.Nevo Dinur, A. Poggialini

Thank you for your attention!

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Backup Slides

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C.Ji et al., JPG: Part. Nucl. 45, 093002 (2018)

Uncertainties sources • Numerical • Nuclear model • Isospin symmetry breaking • Nucleon-size • Truncation of multiples • η-expansion• expansion in Z𝛼

nuclear structure corrections in light muonic atoms€¦ · q2 from 10-4 gev2 to 10-2 gev2 crema...

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