Two-photon exchange correction to the Lamb shift and hyperfine splitting of S levels

Abstract

We evaluate the two-photon exchange corrections to the Lamb shift and hyperfine splitting of S states in electronic hydrogen relying on modern experimental data and present the two-photon exchange on a neutron inside the electronic and muonic atoms. These results are relevant for the precise extraction of the isotope shift as well as in the analysis of the ground state hyperfine splitting in usual and muonic hydrogen.The discrepancy between the proton charge radius extractions from the Lamb shift in muonic hydrogen [1,2] and electron-proton scattering [3-5] triggered a lot of theoretical and experimental efforts both in scattering and spectroscopy, see Refs. [6,7] for recent reviews. Twophoton exchange (TPE) hadronic correction, see Fig. 1, is a limiting factor extracting radii from the muonic hydrogen spectroscopy [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Moreover, an accurate evaluation of two-photon corrections to hyperfine splitting (HFS) of ground state in electronic hydrogen in combination with an excellent experimental knowledge [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] (known with mHz accuracy) could help to analyse future precise measurements of 1S HFS in muonic hydrogen [41][42][43][44], which aim to decrease an uncertainty of 1S-level HFS from the level of 40 µeV [2] up to the level of 0.2 µeV. Though the two-photon correction is smaller than the modern accuracy of Lamb shift measurements in usual hydrogen, it can affect the precisely measurable 1S-2S transition [45,46] (with the experimental uncertainty 10-11 Hz), as well as the isotope shift [47,48] (with the experimental uncertainty 15 Hz), above the accuracy level of the difference between proton and deuteron charge radii [25,49]. In the latter references, the elastic Friar term [50] was accounted for and the inelastic correction was estimated in the leading logarithmic approximation [51][52][53]. Besides two-photon corrections, the more involved three-photon exchange contribution to the Lamb shift was recently evaluated in the nonrecoil limit neglecting magnetic dipole and electric quadrupole moments of the nucleus in Ref. [49]. FIG. 1: Two-photon exchange graph.In this paper, we provide a first complete dispersive calculation of α 5 two-photon exchange contribution to the Lamb shift in electronic hydrogen, summarize the current status of this correction to the hyperfine splitting of S states and provide an update of Ref.[40] for S-level HFS in µH. Additionally, we present contributions to the Lamb shift arising from the two-photon exchange on the neutron inside a nucleus.We evaluate the correction to the Lamb shift of S energy levels E LS following Refs. [8,13]. It can be expressed as a sum of three terms:the Born contribution E Born , the subtraction term E subt and the inelastic correction E inel . To evaluate the dimensionless forward unpolarized amplitude, we always normalize the TPE contributions to the energy E 0 :where M is the proton mass, m is the lepton mass, |ψ nS (0)| 2 = α 3 m 3 r /(πn...

The proton radius (puzzle?) and its relatives

“…Other estimates for just the polarizability term can be found in [146,147,143] obtaining similar numbers. 32 A more recent determination of the TPE in hydrogen using a dispersion relation analysis of the Mainz data without modeling of the Q 2 dependence of the unpolarized proton structure functions and using the proton charge radius from muonic hydrogen can be found in [144]. In Table 5, we display these results.…”

Fifth Force and Hyperfine Splitting in Bound Systems

“…The contributions of the X17 particle need to be compared to the proton structure effects, which have recently been analyzed in Refs. [36][37][38][39][40]. According to Ref.…”

Lattice QCD calculation of the two-photon exchange contribution to the muonic-hydrogen Lamb shift

“…From top to bottom, the results are refered to Refs. [23], [21], [19], [14], [18] and [13], respectively. with ⟨r 2 p ⟩ the square of the charge radius.…”

“…Figure 2.Theoretical results for the TPE contribution.From top to bottom, the results are refered to Refs [23],[21],[19],[14],[18]. and[13], respectively.…”