Dec 28, 2018 - 3.817 80(346) fm2 that is now consistent with the one obtained from the ordinary H-D ... radius obtained in different ways, such as fro...

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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 [1], measurements of parity violation in cesium [2], and the possible dependence of the fundamental constants on time [3]. 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 [6] 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 [7]. This led to extending the study of muonic systems to more complex nuclei, such as muonic deuterium (µD) [8] and helium (µHe) [9]. In the case of deuteron, the charge radius rd inferred from muonic measurements also deviates by 5.6σ from the CODATA14 world-averaged value [6] obtained in ordinary deuterium, and by 3.5σ from the radius extracted in the recent analysis [10] 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 [11] 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 [12], 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. [15], 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 ,

(1)

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 ,

(2)

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 [12] this part amounts to δTPE Epol = 1.715(23) meV

(3)

However, recent calculations [25] of the nucleon polarizability alter this value. Previously, the authors in Ref. [12], following Ref. [15], assumed that single-nucleon interactions amount to 0.0471(101) meV. On the other hand, based on dispersive calculations in Ref. [25] we obtained, through the proper scaling, δE1nucl =

m3r (µD) 0.3066(287) meV = 0.0448(42) meV, m3r (µH) 8 (4)

2 which is similar but more than twice as accurate. The reduced muon-nucleus mass is given by mr (µN) =

mµ mN , mµ + mN

(5)

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.

(6)

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 [26]. 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. [6]. 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. [21]). 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 [27], 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

(10)

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 [28], is δpot Epol = 0.0201 meV.

(13)

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. [29], α m2r e−x/2 8 + 12x − 26x2 4π 4 x + 2x3 + 8(x − 2)x (γ + ln x) ,

G2S (0, ~r) =

(16)

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)

(18)

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,

(19)

where, following Ref. [21], we assign 1 % uncertainty. Together with the inelastic three-photon-exchange correction δ3pe Epol = 0.008 75(92) meV, from Ref. [26], it gives the higher-order part δHO Epol = δvp Epol + δ3pe Epol of the nuclear polarizability, δHO Epol = 0.035 25(97) meV.

(20)

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

(21)

where most of the uncertainty comes from an insufficient understanding of electromagnetic interactions of nucleons inside the nucleus. Measurement in muonic deuterium [8] gives the experimental value of the Lamb shift, expt. ELS = 202.8785(31)stat(14)syst meV.

(22)

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 ,

(23a)

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.

(23b)

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 ,

(24)

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 [11], which, modified by the threephoton exchange in Ref. [26], is rd2 (eD) − rp2 (eH) = 3.820 70(31) fm2 .

(25)

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 [30] and the experimental measurement [8] 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 [28] 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. [15]) 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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