Nov 14, 2014 - â0.00127. 10 O(mÂµÎ±6 Ã ln Î±). V. (2). VP; c. (Âµ). D. â0.00454. 11 O(mrÎ±4 Ã m2 r r2 p). V(2); c. (p). D ; r2 p. â5.1975 r2 p fm2. 12 O(mrÎ±5 Ã m2 r r2.
The energy levels of muonic atoms are very sensitive to effects of quantum electrodynamics. (QED), nuclear ... should not be treated using perturbation theory; instead the Uehling potential should be added to the nuclear .... The Wichmann-Kroll 
Dec 1, 2015 - 4.1.4 Static potential (vacuum polarization): Î´EL â¼ O(mrÎ±5) . . . . . . . 19. 4.1.5 Static potential (light-by-light): Î´E. V. (0,4). LbL. L. â¼ O(mrÎ±5) . . . . . . . . . 20. 4.2 Corrections from the 1/mÂµ potentials without vacuu
Jul 9, 2014 - 2He)+ is based on the use of quasipotential method in quantum electrodynamics [15â17], where the .... three-loop vacuum polarization in the third order perturbation theory in Fig. 2(c) shows ... 3. In the energy spectrum it gives the
Oct 15, 2011 - Fine structure interval (2P3/2 â 2P1/2) for muonic deuterium can be written in the form. [66â68]:. âEfs = E(2P3/2) â E(2P1/2) = (82). = Âµ3(ZÎ±)4.
Aug 21, 2014 - Introduction. The energy levels of muonic atoms are very sensitive to effects of quantum electrodynamics ... should not be treated using perturbation theory; instead the Uehling potential should be added to .... published results are g
Jul 4, 2005 - The energy levels of muonic atoms are very sensitive to effects of quantum electrodynamics. (QED), nuclear ... should not be treated using perturbation theory; instead the Uehling potential should be added to the ... The Wichmann-Kroll
Sep 6, 2012 - We calculate the amplitude T1 for forward doubly-virtual Compton scattering in heavy-baryon ... DATA value  and from the results of recent electron .... the review: Ref. .) In the case of forward V2CS, the standard amplitudes in
Jan 27, 2014 - is due to lack of quasielastic data at low energies and forward angles. .... butions and quotes an uncertainty limit well within the ... data is sparse. A consequence of this, phrased in terms of fitting procedures, is that small chang
Feb 28, 2016 - data on the deuteron structure functions Fd. 2 and Rd. High energy charged lepton scattering is a well-established tool to investigate the structure of the deuteron. The contribution of hadronic deuteron polarizability of order (ZÎ±)5
Nov 24, 2010 - 213 (2010)], we reexamine the theory on which the quantum electrodynamic (QED) predictions are based. In particular, we update the theory of the. 2Pâ2S Lamb shift, by calculating the self-energy of the bound muon in the full Coulomb+
Sep 23, 2016 - correction which we denote further as the â2-loop vpâ correction. The potential of loop- after-loop VP effect has the form [10, 31]:. V C vpâvp. (r) =.
Jun 4, 2014 - V (2); c(Âµ). D. â0.00127. O(mÂµÎ±6 Ã ln Î±). V (2). VP ; c(Âµ). D. â0.00454. O(mrÎ±4 Ã m2 rr2 p). V (2); c(p). D ; r2 p. â5.1975 r2 p fm2. O(mrÎ±5 Ã m2.
Regina, Sask. S4S 0A2, Canada, and. G. Lambiaseâ¡, G. Scarpetta. Dipartimento di Scienze Fisiche âE.R. Caianielloâ. Universit`a di Salerno, 84081 Baronissi ...
The Lamb shift measurement and theory are now both a dynamically developing field and we give a .... radius from hydrogen atom spectroscopy. It is necessary ...
May 3, 2017 - Now, reducing the system to 2 coupled differential equations, we would solve them .....  Glen W. Erickson, Physical Review Letters, vol.
We apply a Casimir energy approach to evaluate the self-energy or one-photon radiative correction for an electron in a hydrogen orbital. This linking of the Lamb shift to the. Casimir effect is obtained by treating the hydrogen orbital as a one-elect
Oct 16, 2012 - of Q2 (the subtraction function) be introduced. One accounts for the nucleon Born terms, and the remainder of the unknown subtraction function ...
Dec 4, 2013 - of the pion cloud is paramagnetic in one case and diamagnetic in the other ... where N(x) and MN is the nucleon field and mass respectively, .... ing on the O(p3) corrections (i.e., VVCS amplitude corresponding to ..... and by the UK Sc
Nov 10, 2011 - related set-ups based on cavities, the impact of a thermal phase reservoir is considered. A thorough ... In contrast to single-frequency environments, typical reservoirs for ... tion employed as a threshold current detector, similar.
Nov 23, 2016 - (Dated: revised version, Oct. 25, 2016). We give a pedagogical description of the method to extract the charge radii and Rydberg constant from laser spectroscopy in regular hydrogen (H) and deuterium (D) atoms, that is part of the CODA
Apr 18, 2006 - arXiv:physics/0604133v1 [physics.atom-ph] 18 Apr 2006. EPJ manuscript No. ... particular may be a muon (mµ ≃ 207 me; κ ≃ 1.5 Z), a pion (mπ ...
Nov 20, 2012 - verifications of quantum electrodynamics (QED) . Comparison of the measured transition frequencies in hydrogen with theory is limited by the uncertainty of the proton structure [9, 10]. Here, the main uncertainty ..... Wichmann-Krol
May 18, 2016 - School of Physics and Optical Information Technology,. Jiaying University, 514015 MeiZhou, China. (Dated: May 27, 2016). Abstract ... Controversies continuously associate with quantum mechanics since its foundation in the early twentie
Deuteron charge radius from the Lamb-shift measurement in muonic deuterium Marcin Kalinowski∗
arXiv:1812.10993v3 [physics.atom-ph] 7 Mar 2019
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland (Dated: March 8, 2019) The deuteron charge radius is calculated from the measurement of the Lamb shift in muonic deuterium, taking into account the electron vacuum polarization correction to the nuclear-structure effects. This correction is unexpectedly large and gives a mean-square charge-radii difference rd2 − rp2 = 3.817 47(346) fm2 , which is now consistent with that obtained from the ordinary H-D isotope shift in the 1S-2S transition. This suggests that the long-standing discrepancy in the proton charge radius obtained from electronic and muonic systems is due to an underestimated uncertainty in ordinary hydrogen spectroscopy.
Atomic measurements are at the frontier of low-energy tests of fundamental interactions, which include the search for electric dipole moment in molecules such as thorium monoxide , measurements of parity violation in cesium , and the possible dependence of the fundamental constants on time . So far, none of these methods has indicated any new physics. Recently, an approach based on the comparison of the nuclear charge radius obtained in different ways, such as from muonic and electronic systems, has shown promising results. Due to the very precise theoretical description of the hydrogenic spectra, the charge radius can be extracted from the corresponding spectroscopic experiments. The value of the proton radius rp obtained from measurements in muonic hydrogen (µH) [4, 5], which is a bound system of the muon and the proton, is in 5.6σ discrepancy with the world-averaged value  obtained from ordinary hydrogen. Because every relevant contribution in the current theory was taken into account, such disagreement may suggest unknown effects or unknown interactions that could not be explained by a straightforward modification of the standard model . This led to extending the study of muonic systems to more complex nuclei, such as muonic deuterium (µD)  and helium (µHe) . In the case of deuteron, the charge radius rd inferred from muonic measurements also deviates by 5.6σ from the CODATA14 world-averaged value  obtained in ordinary deuterium, and by 3.5σ from the radius extracted in the recent analysis  of spectroscopy measurements in ordinary deuterium only. Because the determination of the deuteron charge radius depends on the proton charge radius, the discrepancy in the rp affects results for rd . Therefore, we think that a better way to compare electronic and muonic systems is to combine the results for µD and µH into a mean-square charge-radii difference rd2 − rp2 that can be matched against the similar value inferred from very precise measurements  of the ordinary H-D isotope shift in the 1S-2S transition. In this approach, the proton contribution cancels out and the difference depends mostly on the deuteron structure radius. According to the latest estimate , the mean-square charge-radii difference rd2 − rp2 deviates by 2σ between muonic and electronic
systems. Several recent experiments in ordinary hydrogen [13, 14] favor the smaller proton size and agree with muonic measurements [4, 5], which seems to resolve the discrepancy. In this work we show that, in the case of deuteron, incorporating a missing theoretical contribution resolves the 2σ discrepancy mentioned above. Natural units (~ = c = ε0 = 1) are used throughout. Theoretical prediction of the 2P1/2 -2S1/2 splitting, known as the Lamb shift, in muonic deuterium can be expressed, following Ref. , as the sum of the precisely calculated QED contribution [16, 17] in the point-nucleus limit, the part proportional to the mean square charge radius rd2 of the deuteron [16, 18, 19] and the nuclear polarizability contribution ∆Epol [20–24], with the total splitting expressed as ELS = 228.7766(10) meV + ∆Epol − 6.11025(28)rd2 meV fm−2 ,
where ∆Epol is the main limiting factor in the precise theoretical description. Nuclear polarizability can be split into two terms th ∆Epol = δTPE Epol + δHO Epol ,
where δTPE Epol contains terms from the two-photon exchange, which are of fifth order in the fine-structure constant α, and additionally the Coulomb distortion correction. According to the latest analysis  this part amounts to δTPE Epol = 1.715(23) meV
However, recent calculations  of the nucleon polarizability alter this value. Previously, the authors in Ref. , following Ref. , assumed that single-nucleon interactions amount to 0.0471(101) meV. On the other hand, based on dispersive calculations in Ref.  we obtained, through the proper scaling, δE1nucl =
2 which is similar but more than twice as accurate. The reduced muon-nucleus mass is given by mr (µN) =
mµ mN , mµ + mN
where mµ is the muon mass and mN denotes the mass of the appropriate nucleus. Henceforth mr ≡ mr (µD). Replacing the single-nucleon interaction contribution to Eq. (3) with the result of Eq. (4) gives the new value of the α5 two-photon exchange correction, δTPE Epol = 1.713(21) meV.
All contributions, excluding the Coulomb distortion correction, that are of higher order than α5 constitute δHO Epol . They were not included in the calculation of th ∆Epol in Refs. [12, 15], and the only higher-order contribution that has been calculated is the three-photon exchange . Unfortunately, its value is too small to resolve the 2σ discrepancy. We report the calculation of the missing contribution, which comes from the unexpectedly large electron vacuum polarization (eVP) correction to the dominant nuclear-structure term. The leading nuclear polarizability correction is described by the two-photon exchange between the muon and the nucleus. The dominating term comes from the nonrelativistic limit, where, because the distance from the proton to the nuclear center of mass is very small compared to that of the muon, the leading contribution comes from the electric dipole excitations ~ ·∇ ~ α δE = ψ φN R r 1 ~ ·∇ ~ α ψ φN , (7) R × EN + E0 − HN − H0 r
where H0 = p2 /(2mr ) − α/r is the nonrelativistic Coulomb Hamiltonian for the muon with reduced mass ~ is the position mr , HN is the deuteron Hamiltonian, R of the proton with respect to the nuclear center of mass, ψ is the muon wave function, and φN is the nuclear wave function. All values of the fundamental physical constants are from Ref. . The average nuclear excitation energy E is much larger than the atomic one, so we perform expansion in the large parameter E/(mr α2 ) in Eq. (7). The leading term is the dipole polarizability r Z 4πα2 2 2mr 2 ~ δEpol0 = dE ψ (0) |hφN |R|Ei| . (8) 3 E ET It contributes to the Lamb shift by δEpol0 = 1.910 meV, which is at least an order of magnitude larger than any other nuclear-structure effect (see Table I in Ref. ). Therefore, we considered the eVP correction δvp Epol only to this dominating term.
The leading electron vacuum polarization correction δvp Epol to Eq. (7) is of the order α6 and has two components, δpot Epol and δwf Epol . The first corresponds to the modification of the photon propagator, which effectively replaces one of the Coulomb potentials V = −α/r with the term δV from the Uehling potential , Z α 2α ∞ −2r me ξ Vvp = V + δV = − 1+ , dξ ρ(ξ) e r 3π 1 (9) where ρ(ξ) is a dimensionless function ρ(ξ) =
p 2ξ 2 + 1 ξ2 − 1 . 2ξ 4
Neglecting the Coulomb distortion and deuteron quadrupole moment, and approximating ψ(r) with ψ(0), the leading correction in α is expressed as Z ∞ Z 4mr α3 2 2 ~ dE |hφN |R|Ei| ψ (0) dξ ρ(ξ) δpot Epol =2 9π ET 1 Z d3 p 4π 4π × , (11) (2π)3 p2 + 4 m2e ξ 2 p2 + 2 mr E where E denotes the nuclear excitation energy and the combinatorial factor 2 at the beginning is due to the modification of one of the two Coulomb potentials. q The result r of Eq. (11) depends on the large parameter Em 2m2e ∼ 20. From the first two terms of the expansion, we obtain r Z 2mr 8α3 2 2 ~ dE |hφN |R|Ei| ψ (0) (12) δpot Epol = 9 E ET # " r 2mr 5 3π me 2mr E + 2 ln − + × ln 2mr me 3 4 mr E The numerical value, calculated with the AV18 potential , is δpot Epol = 0.0201 meV.
The second correction δwf Epol is the result of perturbing the muon wave function ψ in Eq. (7) with the potential δV defined in Eq. (9), Z ˜ ψ(0) = − d3 r G2S (0, ~r)δV (r)ψ(r), (14)
where G2S (0, ~r) is a special case of the reduced Coulomb Green’s function, defined as 1 ~r2 , Gn (~r1 , ~r2 ) = ~r1 (15) ′ (H0 − En ) where the prime in the denominator denotes the exclusion of the state n with the corresponding energy En . The explicit form of formula (15) for the 2S atomic state was derived in Ref. , α m2r e−x/2 8 + 12x − 26x2 4π 4 x + 2x3 + 8(x − 2)x (γ + ln x) ,
G2S (0, ~r) =
3 where x = mr αr. After integration, Eq. (14) gives the value of the perturbed wave function of the 2S state at the origin α ˜ ψ(0). (17) ψ(0) = 0.72615 π
The contribution to the Lamb shift is obtained through ˜ the substitution ψ 2 (0) → ψ ∗ (0)ψ(0) in Eq. (8), δwf Epol = 2
˜ ψ(0) δEpol0 = 0.0064 meV, ψ(0)
where the factor 2 is from the perturbation of the left and right wave functions. The total electron vacuum polarization correction to the nuclear structure is the sum of terms in Eqs. (13) and (18), δvp Epol = 0.0265(3) meV,
where, following Ref. , we assign 1 % uncertainty. Together with the inelastic three-photon-exchange correction δ3pe Epol = 0.008 75(92) meV, from Ref. , it gives the higher-order part δHO Epol = δvp Epol + δ3pe Epol of the nuclear polarizability, δHO Epol = 0.035 25(97) meV.
The total correction, as expressed in Eq. (2), with the α5 contribution from Eq. (6) and the higher-order terms from Eq. (20), gives th = 1.748(21) meV, ∆Epol
where most of the uncertainty comes from an insufficient understanding of electromagnetic interactions of nucleons inside the nucleus. Measurement in muonic deuterium  gives the experimental value of the Lamb shift, expt. ELS = 202.8785(31)stat(14)syst meV.
The mean-square charge radius of deuteron is obtained through Eq. (1), with the updated theoretical polarizability from Eq. (21), rd2 = 4.52453(53)prot(346)rest fm2 ,
where (53)prot is the uncertainty from the proton polarizability only and (346)rest is the remainder. The new deuteron charge radius is rd = 2.127 10(82) fm.
In order to reliably compare this result with electronic measurements, we use the proton charge radius rp = 0.840 87(39) fm inferred from muonic hydrogen experiments [4, 5] to calculate the mean-square charge-radii difference, rd2 (µD) − rp2 (µH) = 3.817 47(346) fm2 ,
where the dependence on the proton polarizability cancels out, as does the uncertainty from the proton in Eq. (23a). This result is consistent with the very precise value obtained from the ordinary H-D isotope shift in the 1S-2S transition , which, modified by the threephoton exchange in Ref. , is rd2 (eD) − rp2 (eH) = 3.820 70(31) fm2 .
Agreement in the deuteron charge radius suggests that we have sufficient knowledge of the nuclear-structure effects in the Lamb shift to perform calculations of the nuclear polarizability contributions in heavier elements, such as 3 He, 4 He, 6 Li, and 7 Li. We note, however, that the spin-dependent part of the nuclear polarizability is not well understood, which is reflected in the recently observed 5σ discrepancy between the theoretical prediction  and the experimental measurement  of the 2S hyperfine splitting in muonic deuterium. In general, to reduce the uncertainty further and increase the accuracy of the test, we should better understand electromagnetic interactions of nucleons inside the nucleus. We note that the electron vacuum polarization is not the only radiative correction to the nuclear-structure effects. Muonic and nuclear self-energy (SE) corrections are present as well, but we argue that they are significantly smaller than the eVP correction in Eq. (19). The nuclear SE is not only small, but also it is partially included in the heuristic proton-neutron potential  and effective electromagnetic moments of the nucleus. The µSE is of the order α π relative to the leading term in Eq. (8) but does not have the mr /me enhancement, in contrast to the correction discussed in this work. Moreover, in the point nucleus limit, the value of µSE is very small (see Table I in Ref. ) in comparison with the electron vacuum polarization, which is the leading term in the Lamb shift of muonic systems. It indicates that radiative corrections to the nuclear-structure effects, other than the eVP, can be neglected with the current level of precision. In summary, we calculated the electron vacuum polarization correction to the leading nuclear polarizability effect in muonic deuterium, which, combined with other recent results [12, 25, 26], gives a new muonic meansquare charge-radii difference. Its value is in agreement with the very precise result from the ordinary H-D isotope shift in the 1S-2S transition. This consistency is strong evidence for the correctness of measurements in muonic hydrogen and deuterium. Therefore, it suggests that the current disagreement in the determination of the proton charge radius is caused exclusively by underestimated uncertainty in ordinary hydrogen spectroscopy. The author would like to thank Krzysztof Pachucki for useful discussions and Oscar Javier Hernandez and Sonia Bacca for correcting our numerical calculations. This work was supported by the National Science Center (Poland) Grant No. 2017/27/B/ST2/02459.
       
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