Precision tests of fundamental interactions with H2gron/PSAS2016/resources/Talks/Pachucki.pdf ·...

Intro NRQED Numerical results FFK2017

Precision tests of fundamentalinteractions with H2

K. Pachucki & M. Puchalski & D. Czachorowski & J. Komasa

University of Warsaw & Adam Mickiewicz University

Poland

PSAS 2016, Jerusalem, 23 May 2016


Motivation and challenges

Motivationsdetermining fundamental constants: α, me

determine nuclear properties: rch from comparison of measuredand calculated transition frequencies, like rp from H spectroscopy

search for existence of unknown yet interactions

Challengesaccurate calculations of QED and correlations

beyond static nucleus

systematic account for nuclear structure effects in muonic atoms

how come rp from (electronic) H differs by 4% from that of µH ?


The proton radius puzzle

[fm]ch

Proton charge radius R0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9

H spectroscopy

scatt. Mainz

scatt. JLab

p 2010µ

p 2013µ electron avg.

σ7.9

The proton rms charge radius measured with

electrons: 0.8770 ± 0.0045 fm

muons: 0.8409 ± 0.0004 fm

R. Pohl et al., Nature 466, 213 (2010).A. Antognini et al., Science 339, 417 (2013).

Randolf Pohl PSI2013, Sept. 9, 2013 – p. 3


Proton charge radius puzzle

δfsE = (2π α/3)φ2(0) 〈r2p 〉

The only solution which does not violate SM is the assumptionthat the hydrogen spectroscopy and e-p scatteringmeasurements, although in agreements, are both incorrect (atfew σ)

How it can be verified ?

muon-proton scatteringµHeH-spectroscopyHe+(1S − 2S)

D0(H2)


α charge radius from He spectroscopy

E(23S − 23P,4 He)centroid = 276 736 495 649.5(2.1) kHz,Florence, 2004

finite size effect: Efs = 3 387 kHz

since Efs is proportional to r2

∆rr

=12δEfs

Efs≈ 1

210

3 387= 1.5 · 10−3

electron scattering gives rHe = 1.681(4) fm, what corresponds toabout 2.5 · 10−3 relative accuracy

can theoretical predictions be accurate enough ∼ 10 kHz ?


rP from dissociation energy of H2

we aim for 10−6 cm−1 accuracy for H2 levels

the proton charge radius which contributes 1.2 10−4 cm−1

can be determined to 0.5% accuracy, discrepancy is at 4%

requires complete mα6 and leading mα7


H2 experimental data

D0(H2) = 36118.06962(37) cm−1 [1]∆v(1→ 0) = 4161.16632(18) cm−1 [2]∆v(1→ 0) = 4161.16636(15) cm−1 [3]

[1] J. Liu at al, 2009 (Frederic Merkt’s and Wim Ubachs’ labs)

[2] G.D. Dickenson et al, 2013 (VU Amsterdam)

[3] Mingli Niu et al, 2015 (VU Amsterdam)

current theoretical accuracy is of about 10−4 cm−1

the accuracy 10−6 cm−1 is feasible (relativistic nonad. andhigher order QED needs to be improved)


NRQED approach

Atomic energy levels are expanded in powers of the fine structureconstant α

E(α) = E (2) + E (4) + E (5) + E (6) + E (7) + · · · , E (n) ∼ m αn


NRQED approach


E(α) = E (2) + E (4) + E (5) + E (6) + E (7) + · · · , E (n) ∼ m αn

E (2) is a nonrelativistic energy corresponding to the Hamiltonian

H(2) =∑

a

~p 2a

2 m− Z α

ra+∑a>b

α

rab

All expansion terms are expressed in terms of expectation values ofsome effective Hamiltonian with the nonrelativistic wave function


NRQED approach


E(α) = E (2) + E (4) + E (5) + E (6) + E (7) + · · · , E (n) ∼ m αn

Leading relativistic correction

E (4) = 〈H(4)〉

H(4) =∑

a

−~p 4

a

8 m3 +π Z α

2 m2 δ3(ra) +Z α

4 m2 ~σa ·~ra

r 3a× ~pa

+∑a>b

∑b

−π α

m2 δ3(rab)− α

2 m2 pia

(δij

rab+

r iab r j

ab

r 3ab

)pj

b

−2π α3 m2 ~σa · ~σb δ

3(rab) +α

4 m2

σia σ

jb

r 3ab

(δij − 3

r iab r j

ab

r 2ab

)+

α

4 m2 r 3ab

×[2(~σa ·~rab × ~pb − ~σb ·~rab × ~pa

)+(~σb ·~rab × ~pb − ~σa ·~rab × ~pa

)]


NRQED approach


E(α) = E (2) + E (4) + E (5) + E (6) + E (7) + · · · , E (n) ∼ m αn

Leading QED correction

E (5) =

[16415

+143

lnα]α2

m2 〈δ3(r12)〉

+

[1930

+ ln(Z α)−2]

4α2 Z3 m2 〈δ

3(r1) + δ3(r2)〉

−143

mα5⟨

14π

P(

1(mα r12)3

)⟩− 2α

3πm2

⟨∑a

~pa (H0 − E0) ln[

2 (H0 − E0)

(Z α)2 m

]∑b

~pb

⟩


NRQED approach


E(α) = E (2) + E (4) + E (5) + E (6) + E (7) + · · · , E (n) ∼ m αn

Higher order effects mα6, mα7 · · ·

E (6) = 〈H(6)〉+ 〈H(4) 1(E0−H0)

′ H(4)〉

cancellation of singularities between the first and the second ordermatrix elements→ difficult in numerical calculations

E (7) known for hydrogenic systems→ challenging task for few electronatoms


H2: new results for nonadiabatic (nonrelativistic)energies

Ω =∑

i ni N ENREL

6 5580 -1.164 024 922 201 97 9486 -1.164 025 015 807 28 15531 -1.164 025 028 744 59 24211 -1.164 025 030 538 5

10 36642 -1.164 025 030 821 411 53599 -1.164 025 030 870 912 76601 -1.164 025 030 880 413 106764 -1.164 025 030 883 0∞ -1.164 025 030 883 8(2)

Bubin, Adamowicz -1.164 025 030 84(6)

accuracy of the extrapolated result δE = 4 · 10−8 cm−1


mα6 contribution

E (6) = ENR + E ′H + ER1 + ER2 − lnαπδ3(r)

E ′H , ER1, ER2 proportional to Dirac delta functions

calculation of ENR was challenging, requires high quality ofnonrelativistic wave function

low R (He) and large R (twice H) limits


ENR - nonradiative contribution

0 2 4 6 8

0.05

0.10

0.50

1


mα6 results

Results in cm−1v = 0, J = 0, ν(6) = −0.00207030

v = 0, J = 1, ν(6) = −0.00206134

v = 1, J = 0, ν(6) = −0.00187857

δνJ=1→0 = 0.00000896

δνv=1→0 = 0.000192

what presently limits theoretical predictions in H2 are combinedmα4 and nonadiabatic corrections.


International Conference on Precision Physics and Fundamental Physical Constants

FFK 2017

Faculty of Physics, University of Warsaw http://ffk2017.fuw.edu.pl

15–19.05 2017

Sonia Bacca (U Manitoba), Jan C. Bernauer (MIT), Dmitry Budker (UC Berkeley), Ali Eichenberg (METAS), Kield Eikema (VUA), Eric Hessels (York U), Bogumił Jeziorski (U Warsaw) , Jacek Komasa (AMU), Edmund Myers (FSU), Randolf Pohl (MPQ), Jonathan Sapirstein (U Notre Dame), Vladimir Shabaev (St. Petersburg St. U), Sven Sturm (MPIK), Peter G. Thirolf (LMU), Wim Vassen (VUA), Jochen Walz (Mainz U), Meng Wang (IOMP CAS)

Andrzej Czarnecki (University of Alberta, Edmonton, Canada) Simon I. Eidelman (Budker Institute of Nuclear Physics, Novosibirsk, Russia) Victor V. Flambaum (University of New South Wales, Sydney, Australia) Michal Hnatič (Safarik University, Kosice, Slovakia & JINR, Dubna, Russia) Masaki Hori (MPQ Munich, Germany & U. Tokyo, Japan) Dezső Horváth (Wigner Research Centre for Physics, Budapest, Hungary) Vladimir I. Korobov (JINR, Dubna, Russia) Estefania de Mirandes (BIPM, Sèvres, France) Vladimir Shabaev (St. Petersburg State University, Russia) Eberhard Widmann (SMI, Vienna, Austria) Savely G. Karshenboim (MPQ Munich, Germany; Pulkovo Observatory, Russia), co-chairman Krzysztof Pachucki (University of Warsaw, Poland), co-chairman

Fundamental physical constants Precision measurements in atomic and molecular physics Precision measurements in low energy particle physics and astrophysics Precision tests of fundamental interaction theories Exotic atoms

Conference Topics

Invited Speakers Include

Scientific Committee

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