Two-photon exchange correction to the Lamb shift and hyperfine splitting of S levels
TTwo-photon exchange correction to the Lamb shift and hyperfine splitting of S levels
Oleksandr Tomalak
1, 2, 3 Institut f¨ur Kernphysik and PRISMA Cluster of Excellence,Johannes Gutenberg Universit¨at, Mainz, Germany Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Fermilab, Batavia, IL 60510, USA (Dated: May 13, 2019)We evaluate the two-photon exchange corrections to the Lamb shift and hyperfine splitting ofS states in electronic hydrogen relying on modern experimental data and present the two-photonexchange on a neutron inside the electronic and muonic atoms. These results are relevant for theprecise extraction of the isotope shift as well as in the analysis of the ground state hyperfine splittingin usual and muonic hydrogen.
The discrepancy between the proton charge radius ex-tractions from the Lamb shift in muonic hydrogen [1, 2]and electron-proton scattering [3–5] triggered a lot of the-oretical and experimental efforts both in scattering andspectroscopy, see Refs. [6, 7] for recent reviews. Two-photon exchange (TPE) hadronic correction, see Fig. 1,is a limiting factor extracting radii from the muonic hy-drogen spectroscopy [8–25]. Moreover, an accurate eval-uation of two-photon corrections to hyperfine splitting(HFS) of ground state in electronic hydrogen in com-bination with an excellent experimental knowledge [26–40] (known with mHz accuracy) could help to analysefuture precise measurements of 1S HFS in muonic hy-drogen [41–44], which aim to decrease an uncertaintyof 1S-level HFS from the level of 40 µ eV [2] up to thelevel of 0 . µ eV. Though the two-photon correction issmaller than the modern accuracy of Lamb shift mea-surements in usual hydrogen, it can affect the preciselymeasurable 1S-2S transition [45, 46] (with the experi-mental uncertainty 10-11 Hz), as well as the isotope shift[47, 48] (with the experimental uncertainty 15 Hz), abovethe accuracy level of the difference between proton anddeuteron charge radii [25, 49]. In the latter references, theelastic Friar term [50] was accounted for and the inelasticcorrection was estimated in the leading logarithmic ap-proximation [51–53]. Besides two-photon corrections, themore involved three-photon exchange contribution to theLamb shift was recently evaluated in the nonrecoil limitneglecting magnetic dipole and electric quadrupole mo-ments of the nucleus in Ref. [49]. FIG. 1: Two-photon exchange graph.
In this paper, we provide a first complete dispersivecalculation of α two-photon exchange contribution tothe Lamb shift in electronic hydrogen, summarize the current status of this correction to the hyperfine splittingof S states and provide an update of Ref. [40] for S-levelHFS in µ H. Additionally, we present contributions to theLamb shift arising from the two-photon exchange on theneutron inside a nucleus.We evaluate the correction to the Lamb shift of S en-ergy levels E LS following Refs. [8, 13]. It can be expressedas a sum of three terms:E LS = E Born + E subt + E inel , (1)the Born contribution E Born , the subtraction term E subt and the inelastic correction E inel . To evaluate the di-mensionless forward unpolarized amplitude, we alwaysnormalize the TPE contributions to the energy E :E = | ψ nS (0) | M m , (2)where M is the proton mass, m is the lepton mass, | ψ nS (0) | = α m r / ( π n ) is the non-relativistic squaredwave function of the hydrogen atom at origin with thereduced mass of the lepton and proton bound state m r = M m/ ( M + m ), α is the fine-structure constant, andn is the principal quantum number. The inelastic con-tribution E inel can be expressed as an integral over theunpolarized proton structure functions F and F [13]:E inel E = − α ˆ ∞ dQ Q ˆ ∞ ν inelthr dν γ × (cid:18) (cid:101) γ (˜ τ , τ l ) F ( ν γ , Q ) ν γ + (cid:101) γ (˜ τ , τ l ) F ( ν γ , Q )4 M τ P (cid:19) , (3)with the photon energy ν γ = ( p · q ) /M , the virtuality Q = − q and kinematical notations: τ l = Q m , τ P = Q M , ˜ τ = ν γ Q . (4)The photon-energy integration starts from the pion-nucleon inelastic threshold ν inelthr : ν inelthr = m π + m π + Q M , (5) a r X i v : . [ h e p - ph ] M a y where m π denotes the pion mass. The weighting func-tions ˜ γ and ˜ γ are given by [13]˜ γ ( τ , τ ) = √ τ γ ( τ ) − √ τ γ ( τ ) τ − τ , ˜ γ ( τ , τ ) = 1 τ − τ (cid:18) γ ( τ ) √ τ − γ ( τ ) √ τ (cid:19) . (6)The contribution E subt from the forward Comptonscattering subtraction function T subt1 (cid:0) , Q (cid:1) is definedaccording to Refs. [8, 13, 15, 20] asE subt E = 4 αM ˆ ∞ dQ Q γ ( τ l ) √ τ l T subt1 (cid:0) , Q (cid:1) , (7)with γ ( τ ) = (1 − τ ) (cid:16) (1 + τ ) / − τ / (cid:17) + τ / , (8)and determined mainly by the value of the magnetic po-larizability β M entering the low-energy expansion of T as T subt1 (cid:0) , Q (cid:1) = β M Q + O (cid:0) Q (cid:1) . (9)The left part of the TPE effect from proton form fac-tors is called the Born correction E Born [8, 13]:E
Born E =4 α ˆ ∞ dQ Q (cid:18) − γ ( τ l ) √ τ l (cid:0) F − (cid:1) + 16 M m G (cid:48) E (0)( M + m ) Q + m M − m (cid:18)(cid:18) γ ( τ P ) √ τ P − γ ( τ l ) √ τ l (cid:19) (cid:0) G M − (cid:1) − (cid:18) γ ( τ P ) √ τ P − γ ( τ l ) √ τ l (cid:19) G E − τ P (cid:0) G M − (cid:1) τ P (1 + τ P ) (cid:33)(cid:33) , (10)with the Dirac (F D ), Sachs electric (G E ) and magnetic(G M ) form factors. The kinematical factor γ is given by γ ( τ ) = (1 + τ ) / − τ / − τ / . (11)In this term, we expand the electric form factor in termsof charge radius at low momentum transfer followingevaluation of the Zemach correction in Ref. [39], andthe third Zemach moment contribution in Refs. [54, 55],and connect regions of large and small momentum trans-fer. We take an average of Refs. [2, 4] for a central valueof the charge radius and estimate its uncertainty as halfthe difference between results in Refs. [2, 4] and providethe evaluation for the charge radius from the muonic hy-drogen spectroscopy r µ HE [2]. We evaluate the Born con-tribution exploiting form factors of Refs. [3, 4] and takethe unpolarized proton structure functions from the fitof Refs. [56–58]. E e HLS (1S) HzBorn, E e HBorn -44.1(9.6)Born, E e H r µ HE -39.9(6.8)Subtraction, E e Hsubt e Hinel − e HLS = E e HBorn + E e Hsubt + E e Hinel − µ HLS (1S) µ eVBorn, E µ HBorn -166.1(19.5)Born, E µ H r µ HE -148.9(12.8)Subtraction, E µ Hsubt [20] 18.5(10.0)Inelastic, E µ Hinel [13] − µ HLS = E µ HBorn + E µ Hsubt + E µ Hinel − α TPE contributions to the Lamb shiftof S energy levels in electronic and muonic hydrogen.
We present results for TPE corrections to the Lambshift (LS) of the ground state in electronic hydrogen inTab. I. The Born TPE is around 1.3 times larger thanthe leading third Zemach momen correction [8]. The con-tribution from the subtraction function in electronic hy-drogen is roughly two times smaller than the Born cor-rection and larger than the estimate of Ref. [59], wherethe smaller value of the proton magnetic polarizability β M = (1 . ± . × − fm , compared to the currentp.d.g. quotation β M = (2 . ± . × − fm [67], wasused and the Q -dependence of the subtraction functionwas assumed but not remove well-justified by data or the-ory. The inelastic correction to the Lamb shift is almosttwice larger than the Born contribution and 1.3 timeslarger than the result in the logarithmic approximation[8]. Our estimate is 1.1 times smaller than the calculationof Ref. [9] and agrees with an update of Ref. [59] withinuncertainties. In Ref. [9], the inelastic contribution wasdescribed by the Regge model. The model of structurefunctions as a sum of resonances with nonresonant back-ground was used in Ref. [59], while the result in Tab. Iis based mainly on the fit of precise JLAB experimentaldata in the resonance region of Refs. [56, 57]. Note thatthe inelastic two-photon effect in electronic hydrogen isin agreement within errors with the dispersive calculationof Ref. [60] which is based mainly on the photoabsorp-tion cross section data modified by empirical elastic formfactors. The sum of inelastic and subtraction correctionsis closer to the logarithmic approximation of Ref. [11]than to the full heavy-baryon effective field theory cal-culation of Ref. [12]. Moreover, we present results forthe muonic hydrogen in Tab. I. The Born correction inmuonic hydrogen is accidentally in a reasonable agree-ment with Ref. [8], where the dipole parametrization ofproton form factors was used, and slightly smaller thanthe previous estimate of Ref. [13], where we have com-bined proton state contributions in Ref. [13] for compar-ison, due to our implementation of the expansion at lowmomentum transfer with the smaller charge radius value.Indeed, E µ HBorn differs by 34 . µ eV substituting the chargeradius of Ref. [2] versus Ref. [4].Studying the isotope shift in light atoms, it is instruc-tive to know also the two-photon effect due to the scatter-ing on a single neutron [49, 61]. We repeat the Lamb shiftcalculation without the subtraction of pure Coulomb partand leading charge radius ( ∼ G (cid:48) E (0)) contribution in Eq.(10) in case of the neutron. Note that a special care hasto be taken applying these results to nuclei, since we nor-malize to the energy E of Eq. (2) which changes going tothe nucleus. We exploit form factors from Refs. [62–66],use the fit of Christy and Bosted [56] for the unpolarizedstructure functions, and estimate the subtraction func-tion following Ref. [20] with the neutron magnetic polar-izability β M = (3 . ± . × − fm from p.d.g. [67] andthe Reggeon residue according to Refs. [58, 68, 69]. Wepresent results in Tab. II. The Born correction in en and E e nLS (1S) HzBorn, E e nBorn e nsubt e ninel − e nLS = E e nBorn + E e nsubt + E e ninel − µ nLS (1S) µ eVBorn, E µ nBorn µ nsubt µ ninel − µ nLS = E µ nBorn + E µ nsubt + E µ ninel − α TPE contributions to the Lamb shiftfrom electromagnetic interaction with neutron E e nLS , E µ nLS . µn systems has a different sign compared to ep and µp .For a neutron with zero charge, the elastic Friar term isrelatively small compared to the positively charged pro-ton, and the main contribution comes from the neutronmagnetic form factor resulting in a positive sign and rel-atively small uncertainty. We obtain the central valueaveraging over the form factor parametrizations and es-timate the uncertainty as a difference between the largestand smallest results. As in Ref. [49], the inelastic correc-tions for proton and neutron coincide within errors. Wedouble the uncertainty for the inelastic contribution incase of the neutron compared to the proton. Note thatthe resulting two-photon exchange effect in µ H is roughlyfour times larger than in µn system: E µ HLS ≈ µ nLS , as ithas been estimated in Refs. [49, 70]. The main uncer-tainty in the two-photon correction is due to the pureknowledge of the forward Compton scattering subtrac-tion function. However, it can be improved exploitingthe chiral perturbation theory predictions [15, 17, 19],constraints at high energy [22, 71] as well as the phe-nomenological studies of the difference between the sub-traction function for protons and neutrons [69, 72], andby improved extraction of the neutron magnetic polariz- ability [73–76].For the hyperfine splitting correction E HFS , we use def-initions of Refs. [25, 38–40]. The result is given by a sumof the Zemach E Z , the recoil E R and the polarizabillityE pol terms: E HFS = E Z + E R + E pol , (12)where the contributions relative to the leading Fermisplitting E F : E F = 8 πα µ P | ψ nS (0) | M m , (13)with the proton magnetic moment µ P are given byE Z E F = 8 αm r π ∞ ˆ d QQ (cid:32) G E (cid:0) Q (cid:1) G M (cid:0) Q (cid:1) µ P − (cid:33) , (14)E R E F = απ ∞ ˆ d Q Q (2 + ρ ( τ l ) ρ ( τ P )) F D (cid:0) Q (cid:1) √ τ P √ τ l + √ τ l √ τ P G M (cid:0) Q (cid:1) µ P + 3 απ ∞ ˆ d Q Q ρ ( τ l ) ρ ( τ P ) F P (cid:0) Q (cid:1) √ τ P √ τ l + √ τ l √ τ P G M (cid:0) Q (cid:1) µ P − απ ∞ ˆ d QQ (cid:32) mM ρ ( τ l ) ( ρ ( τ l ) −
4) F (cid:0) Q (cid:1) µ P − m r Q (cid:33) − ∆ Z , (15)E pol E F = 2 απµ P ∞ ˆ d Q Q ∞ ˆ ν inelthr d ν γ ν γ (2 + ρ ( τ l ) ρ (˜ τ )) g (cid:0) ν γ , Q (cid:1) √ ˜ τ √ τ l + √ τ l √ τ − απµ P ∞ ˆ d Q Q ∞ ˆ ν inelthr d ν γ ν γ τ ρ ( τ l ) ρ (˜ τ ) g (cid:0) ν γ , Q (cid:1) √ ˜ τ √ τ l + √ τ l √ τ + απ ∞ ˆ d QQ mM ρ ( τ l ) ( ρ ( τ l ) −
4) F P (cid:0) Q (cid:1) µ P , (16)with ρ ( τ ) = τ − (cid:112) τ (1 + τ ), F P is the Pauli form fac-tor, g (cid:0) ν γ , Q (cid:1) and g (cid:0) ν γ , Q (cid:1) are the spin-dependentinelastic proton structure functions. For the electronichydrogen, the Zemach correction E Z is obtained by scal-ing with the reduced mass in hydrogen to the muonichydrogen from the averaged over electric and magneticradii result of Ref. [39]. The recoil E R and polarizabil-ity E pol contributions are evaluated following the samesteps as in Ref. [39] for µ H. The proton spin structurefunctions parametrization is based on Refs. [77–81].In Tab. III, we provide the hyperfine-splitting TPEcontributions as well as extractions from the exper-imental data exploiting radiative corrections of Refs.[25, 37, 82–89]. The experimental value of the hyperfinesplitting in muonic hydrogen is taken from Ref. [2] and inelectronic hydrogen from Refs. [26–36]. All correctionsto the hyperfine splitting in electronic hydrogen are three E e HHFS (1S) kHzZemach, E e HZ − e HR e Hpol e HHFS = E e HZ + E e HR + E e Hpol − e HHFS from 1S HFS in eH − µ HHFS (1S) µ eVZemach, E µ HZ [39] − µ HR [39] 154(1)Polarizability, E µ Hpol [39] 66(16)E µ HHFS = E µ HZ + E µ HR + E µ Hpol [39] − µ HHFS from 2S HFS in µ H − µ HHFS from 1S HFS in e H − α TPE contributions to the hyperfinesplitting of S energy levels in hydrogen and muonic hydrogen.In experimental extractions, the first uncertainty is the errorof radiative corrections and measurement, and the second onecontains a possible α E HFS error from higher orders. orders of magnitude above the Lamb shift contributions.As well as in muonic hydrogen [39], they slightly differ tothe previous estimates of Ref. [23] due to the inclusion ofthe recent form factor measurements [3, 4]. Theoreticalestimates of the hyperfine-splitting correction are withinerrors of the phenomenological extraction from measure-ments.Additionally, we provide an update of Ref. [40] forthe absolute value of the hyperfine-splitting energy E µ HHFS in muonic hydrogen removing axial-vector mesons [90]from the analysis and accounting for the vacuum polar-ization graphs with elastic and inelastic proton structurein higher-order radiative corrections:E µ HHFS (1S) = 182 . ± .
012 meV , (17)E µ HHFS (2S) = 22 . ± . . (18)An improved calculation of two-photon diagrams withQED corrections on fermion lines, graphs with three ex-changed photons [91] as well as evaluation of the two-photon contributions in non-forward kinematics can re-duce the uncertainty further.We presented the current knowledge of the TPE cor-rection to S energy levels. The Lamb shift results canbe useful in future extractions of the isotope shift, whilethe contributions to the hyperfine splitting can help totune and analyze forthcoming 1S HFS measurements in µ H [41–44].We acknowledge Krzysztof Pachucki for the advicegiven during the manuscript preparation and MarcinKalinowski for useful discussion. This work was sup-ported in part by a NIST precision measurement grantand by the U. S. Department of Energy, Office of Science,Office of High Energy Physics, under Award No. DE-SC0019095. 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