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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
Intro NRQED Numerical results FFK2017
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 ?
Intro NRQED Numerical results FFK2017
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
Intro NRQED Numerical results FFK2017
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)
Intro NRQED Numerical results FFK2017
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 ?
Intro NRQED Numerical results FFK2017
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
Intro NRQED Numerical results FFK2017
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)
Intro NRQED Numerical results FFK2017
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
Intro NRQED Numerical results FFK2017
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
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
Intro NRQED Numerical results FFK2017
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
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
)]
Intro NRQED Numerical results FFK2017
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
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
⟩
Intro NRQED Numerical results FFK2017
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
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
Intro NRQED Numerical results FFK2017
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
Intro NRQED Numerical results FFK2017
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
Intro NRQED Numerical results FFK2017
ENR - nonradiative contribution
0 2 4 6 8
0.05
0.10
0.50
1
Intro NRQED Numerical results FFK2017
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.
Intro NRQED Numerical results FFK2017
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