Theory of the Lamb shift in hydrogen and light hydrogen-like ions
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Theory of the Lamb shift in hydrogen and light hydrogen-like ions
Vladimir A. Yerokhin
Center for Advanced Studies, Peter the Great St. Petersburg Polytechnic University, Polytekhnicheskaya 29, St. Petersburg 195251, Russia
Krzysztof Pachucki
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
Vojtˇech Patk´oˇs
Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague 2, Czech Republic
Theoretical calculations of the Lamb shift provide the basis required for the determination of the Rydbergconstant from spectroscopic measurements in hydrogen. The recent high-precision determination of the protoncharge radius drastically reduced the uncertainty in the hydrogen Lamb shift originating from the proton size.As a result, the dominant theoretical uncertainty now comes from the two- and three-loop QED effects, whichcalls for further advances in their calculations. We review the present status of theoretical calculations of theLamb shift in hydrogen and light hydrogen-like ions with the nuclear charge number up to Z = 5 . Theoreticalerrors due to various effects are critically examined and estimated. I. INTRODUCTION
Hydrogen atom plays a special role in modern physics. Asthe simplest atomic system, hydrogen is often considered tobe an ideal testing ground for exploring limits of the theorybased on predictions of the bound-state quantum electrody-namics (QED). One of the important tests of theory is thecomparison of the proton charge radius values obtained fromthe Lamb shift in electronic and muonic hydrogen. The . σ discrepancy between these values, known as the proton radiuspuzzle [1, 2], attracted large attention of the scientific com-munity. This discrepancy could indicate violation of the lep-ton universality and existence of interactions not accountedfor in the Standard Model. Such a possibility is still open, al-though recent experiments on electronic hydrogen [3–6] hintat existence of unknown systematic effects in hydrogen mea-surements rather than at new physics.Another important role of hydrogen is that comparison oftheory and experiment for its transition energies is used [7] fordetermining the Rydberg constant, which is one of the mostaccurately known fundamental constants today. If one adoptsthe proton charge radius determined from the muonic hydro-gen [2], the uncertainty of the Rydberg constant is defined bythe currently available theory of the hydrogen Lamb shift.Precise spectroscopy of light hydrogen-like ions may alsoprovide determinations of the Rydberg constant in the fore-seeable future. Such determinations will be independent onthe proton radius and systematic effects in the hydrogen spec-troscopy. Helium isotopes look most promising in this re-spect, because of high-precision results for nuclear radii ex-pected soon from experiments on muonic helium [8]. Wemention here the ongoing projects of measuring the S – S transition energy in He + pursued in Garching [9] and in Am-sterdam [10], which require improved theoretical predictionsfor the helium Lamb shift.Motivated by the needs outlined above, in the present workwe summarize the presently available theory for the Lambshift of hydrogen and light hydrogen-like ions with the nuclearcharge up to Z = 5 . This summary is intended as an update of the CODATA review of the hydrogen theory [7]. In particular,we perform a reanalysis of results available for the higher-order two-loop QED corrections, which presently define thetheoretical uncertainty of the Lamb shift. Results for the nu-clear recoil effect are significantly improved by taking intoaccount recent nonperturbative calculations [11, 12]. The nu-clear finite size and nuclear polarizability effects are reformu-lated, according to recent theoretical developments [13, 14].Relativistic units m = ~ = c = 1 are used throughout thispaper (where m is the electron mass). In these units the elec-tron rest mass energy mc = 1 , so that all energy correctionsappear to be dimensionless. In order to convert any energycorrection in relativistic units to arbitrary units, it is sufficientto multiply it by R /α , where R = hcR ∞ is the Rydbergenergy and R ∞ is the Rydberg constant. While m = 1 in ourunits, we will write m explicitly when it enters dimensionlessratios, such as m/M and m r /m . II. BINDING ENERGY
We consider the binding energy E njl of an electronic statewith quantum numbers n , j , and l in a light hydrogen-likeatom. If the atomic nucleus has a nonzero spin I , the en-ergy level | njl (cid:11) is splitted by the interaction with the nuclearmagnetic moment according to values of the total angular mo-mentum F , | njlF (cid:11) . In this case, we define the binding energy E njl as a centroid averaged over all hyperfine-structure (hfs)components, E njl = P F (2 F + 1) E njlF P F (2 F + 1) . (1)The interaction with the dipole nuclear magnetic moment, re-sponsible for the hyperfine structure, does not contribute to E njl in the first order. There is, however, a second-order hfseffect that shifts (slightly) the centroid energy E njl . It mani-fests itself as a nuclear-spin dependent recoil correction and isaddressed in Sec. IV.The goal of the present paper is to summarize the presentlyavailable theory for the binding energy E njl of the S , S ,and P / states of light hydrogen-like atoms. The hyperfinesplitting of energy levels will not be discussed. For the nS states it was investigated in detail in Ref. [15]; a review of thehfs of the higher- l states is available in Ref. [16].The binding energy of a light hydrogen-like atom is usuallyrepresented as a sum of three contributions, E njl = E D + E M + E L , (2)where E D is the Dirac point-nucleus biding energy in the non-recoil limit, E M is the correction containing the dominant partof the nuclear recoil effect, and E L is the Lamb shift. We notethat the total recoil effect is thus distributed between E M and E L ( E M being the dominant part and smaller corrections be-ing ascribed to the Lamb shift E L ). This distribution is notunique and done differently in the literature.The Dirac point-nucleus nonrecoil binding energy E D isgiven by E D = r − ( Zα ) N − , (3)where N = p ( n r + γ ) + ( Zα ) , (4) γ = p κ − ( Zα ) , n r = n − | κ | is the radial quantum num-ber, n is the principal quantum number, and κ = ( l − j )(2 j +1) is the angular momentum-parity quantum number.The leading recoil correction E M is E M = mM ( Zα ) N − (cid:16) mM (cid:17) ( Zα ) n m r m , (5)where M is the nuclear mass and m r = mM/ ( m + M ) isthe reduced mass. All further recoil corrections are ascribedto the Lamb shift E L . The first part of E M comprises thecomplete m/M recoil effect to orders ( Zα ) and ( Zα ) and,in addition, corrections of order ( Zα ) and higher that can beobtained from the Breit Hamiltonian. The second part of E M is the nonrelativistic recoil correction of second and higherorders in m/M . In the nonrelativistic limit, the sum E D + E M reduces to the Schr¨odinger energy eigenvalue, E D + E M = m r m ( Zα ) n + . . . , (6)where . . . represents contributions of order ( Zα ) and higher.Our choice of E M (and, therefore, our definition of theLamb shift E L ) follows Ref. [17] and differs slightly fromthe popular definition [18] based on the Barker-Glover for-mula [19] and, as a consequence, from the definition of theCODATA review [7] ( cf. Eqs. (25) and (26) therein). The rea-son for this difference was the need for a simple and concisedefinition valid for an arbitrary nucleus, whereas the Barker-Glover formula is valid only for the spin- / nucleus. Bothdefinitions are equivalent through orders ( m/M )( Zα ) n and ( m/M ) n ( Zα ) , with n ≥ . The difference is that our present definition of Eq. (5) does not contain any contributionof order ( m/M ) ( Zα ) (which depends on the nuclear spin)or any spurious higher-order terms. The correction of order ( m/M ) ( Zα ) is included into the Lamb shift; it is given bythe first line of Eq. (41).Another difference in definitions in the literature is associ-ated with the off-diagonal hfs correction, which is small butrelevant on the level of the experimental interest for the l > states [20]. In the old Lamb-shift measurements (in particular,Ref. [21]), this correction was subtracted from the experimen-tal result. Reviews [7, 17] do not discuss it, thus excludingit from the definition of the Lamb shift. The review [16] in-cludes this correction [see Eq. (30) therein] but ascribes it tothe hyperfine splitting. A part of the off-diagonal hfs correc-tion shifts the centroid energy E njl and thus needs to be in-cluded into the definition of the Lamb shift. The correspond-ing contribution is given by Eq. (42).We now turn to examining various effects that contribute tothe Lamb shift E L . III. QED EFFECTSA. One-loop QED effects
The one-loop QED effects for the point nuclear charge arerepresented as E QED1 = απ ( Zα ) n (cid:16) m r m (cid:17) × h F SE ( Zα ) + F VP ( Zα ) i , (7)where the functions F SE ( Zα ) and F VP ( Zα ) correspond tothe one-loop self-energy and vacuum-polarization, respec-tively.The Zα expansion of the electron self-energy is given by F SE ( Zα ) = L A + A + ( Zα ) A + ( Zα ) (cid:20) L A + L A + G SE , pnt ( Zα ) (cid:21) , (8)where L = ln (cid:2) ( m/m r )( Zα ) − (cid:3) and G SE ( Zα ) = A + . . . is the remainder that contains all higher-order expansion termsin Zα . The coefficients of the Zα expansion in Eq. (8) arewell known. They are discussed, e.g., in review [22] and sum-marized in Table I. Numerical results for the remainder func-tion are obtained by Jentschura and Mohr [23, 24] and listedin Table II. Results for Z = 0 correspond to the coefficient A ; they were taken from Ref. [25].The Zα expansion of the vacuum-polarization correction isgiven by F VP ( Zα ) = − δ ℓ + 548 π ( Zα ) δ ℓ + ( Zα ) × (cid:20) − L δ ℓ + G Ueh ( Zα ) + G WK ( Zα ) (cid:21) , (9)where G Ueh ( Zα ) and G WK ( Zα ) are the higher-order remain-der functions induced by the Uehling and Wichmann-Krollparts of the vacuum polarization, respectively. Numerical re-sults for the remainder functions are listed in Table II. TheWichmann-Kroll part of the vacuum polarization was calcu-lated with help of the approximate potential based on the ana-lytical expansions of Whittaker functions from Ref. [26]. Theuncertainty due to approximations in the potential is negligi-ble at the level of current interest. In the limit Z → , resultsfor the higher-order remainders are (see review [22] for de-tails) G Ueh ( Z = 0 , S ) = 415 (cid:16) ln 2 − (cid:17) , (10) G Ueh ( Z = 0 , S ) = − , (11) G Ueh ( Z = 0 , P / ) = − , (12) G WK ( Z = 0) = (cid:16) − π (cid:17) δ ℓ . (13)The vacuum-polarization induced by the µ + µ − virtualpairs is given by [27, 28] E µ VP = (cid:18) mm µ (cid:19) απ ( Zα ) n (cid:16) m r m (cid:17) (cid:18) − (cid:19) δ ℓ , (14)where m µ is the muon mass.The hadronic vacuum-polarization correction is of the sameorder as the muonic vacuum polarization and is given by [29] E hadVP = 0 .
671 (15) E µ VP . (15) B. Two-loop QED effects
The two-loop QED correction is expressed as E QED2 = (cid:16) απ (cid:17) ( Zα ) n (cid:16) m r m (cid:17) F QED2 ( Zα ) , (16)where the function F QED2 is given by F QED2 ( Zα ) = B + ( Zα ) B + ( Zα ) (cid:2) B L + B L + B L + G QED2 ( Zα ) (cid:3) , (17)and G QED2 ( Zα ) = B + . . . is the remainder that containsall higher-order expansion terms in Zα .The two-loop QED correction is conveniently divided intothree parts: the two-loop self-energy (SESE), the two-loopvacuum-polarization (VPVP), and the mixed self-energy andvacuum-polarization (SEVP), F QED2 = F SESE + F SEVP + F VPVP . (18)Coefficients of the Zα expansion of the individual two-loopcorrections for the states under consideration are summarizedin Table III, for details see recent studies [25, 30–34] and references to earlier works therein. We note that the analyt-ical result for the B coefficient derived in Ref. [30] wasincomplete; one missing piece was added later in Ref. [25]and another, in Ref. [34]. The listed value of B forthe S states differs from that given in Refs. [7, 17] by − /
36 + 133 π /
864 = − . . . . , which is the light-by-light contribution from Ref. [34]. Numerical values for thedelta-function correction to the Bethe logarithm N ( nS ) and N ( nP ) that enter B can be found in Refs. [25, 35].The two-loop higher-order remainder G QED2 is only partlyknown up to now. Its Zα expansion has the form G QED2 ( Zα ) = B + ( Zα ) (cid:2) B L + B L + . . . (cid:3) . (19)The dominant part of the coefficient B comes from thetwo-loop self-energy. It was calculated for the S and S states by Pachucki and Jentschura [31], with the result B (1 S, SESE) = − . . , (20) B (2 S, SESE) = − . . , (21)where the uncertainty comes from omitted contributions.The complete n dependence of B ( nS ) was calculated inRefs. [25, 32]. For the nP states, the coefficient B wascalculated in Ref. [36]. The results for the SESE and SEVPcorrections and the P / state are B (2 P / , SESE) = − . , (22) B (2 P / , SEVP) = − .
016 571 . . . . (23)We use opportunity to correct a mistake in Ref. [36] for theVPVP correction (given by Eqs. (A3) and (A6) of that work).The corrected results are B ( nP / , VPVP) = − (cid:18) − n (cid:19) , (24) B ( nP / , VPVP) = − (cid:18) − n (cid:19) . (25)The logarithmic coefficients B and B in Eq. (19) wererecently investigated in Ref. [37]. The leading logarithmiccoefficient B was derived as B (SESE) = (cid:16) − (cid:17) π δ ℓ , (26) B (SEVP) = − π δ ℓ , (27) B (VPVP) = 0 . (28)The next coefficient B was obtained for the nP states, withthe result B (SESE , nP ) = (cid:16) −
49 ln 2 (cid:17) π n − n , (29) B (SEVP , nP ) = 5216 π n − n , (30) B (VPVP , nP ) = 0 . (31)Ref. [37] also reported the n dependence of B ( nS ) . TABLE I: Coefficients of the Zα expansion of the one-loop electron self-energy in Eq. (8).Term S S P / A
41 43 δ ℓ A − ln k ( n, l ) + δ ℓ − m/m r κ (2 l +1) (1 − δ ℓ ) − .
867 726 964 − .
637 915 413 − .
126 644 388 ( m/m r ) A (cid:0) − (cid:1) π δ ℓ .
291 120 908 9 .
291 120 908 0 A − δ ℓ − − A (cid:16) ln 2 + ln n + ψ ( n + 1) − ψ (1) − − n (cid:17) δ ℓ .
572 222 222+ (cid:20) n − n (cid:0) + δ j, / (cid:17) + 8 − l ( l +1) /n l +3) l ( l +1)(4 l − (cid:21) (1 − δ ℓ ) TABLE II: Results for the higher-order remainder functions G SE , G Ueh , and G WK in Eqs. (8) and (9). Z S S P / Self-energy:0 − .
924 149 46 (1) − .
840 465 09 (1) − .
998 904 40 − .
290 24 (2) − .
185 15 (9) − .
973 45 (19) − .
770 967 (5) − .
644 66 (5) − .
949 40 (5) − .
299 170 (2) − .
151 93 (2) − .
926 37 (2) − .
859 222 (1) − .
691 27 (1) − .
904 12 (1) − .
443 372 (1) − .
255 033 (8) − .
882 478 (8)
Vacuum-polarization, Uehling:0 − .
633 573 − .
825 556 − .
064 286 − .
618 724 − .
808 872 − .
064 006 − .
607 668 − .
796 118 − .
063 768 − .
598 207 − .
785 075 − .
063 567 − .
589 838 − .
775 230 − .
063 399 − .
582 309 − .
766 322 − .
063 262
Vacuum-polarization, Wichmann-Kroll:0 .
056 681 0 .
056 681 0 .
055 721 0 .
055 721 0 .
000 002 .
054 823 0 .
054 824 0 .
000 006 .
053 978 0 .
053 983 0 .
000 012 .
053 178 0 .
053 188 0 .
000 020 .
052 418 0 .
052 437 0 .
000 030
Calculations of the SESE part of the higher-order remain-der, G SESE , were carried out to all orders in Zα for hydrogen-like ions with Z ≥ [38, 39]. The latest results were ob-tained in Ref. [40] for Z < and in Ref. [41] for Z ≥ .The extrapolation of the all-order S results down to Z = 1 reported in Ref. [40] showed only a marginal agreement withthe analytical value (20). A possible reason for this could bea large contribution from the unknown logarithmic coefficient B .In the present work, we merge together the numerical andanalytical results, in order to obtain the presumably best val-ues for the higher-order remainder. Specifically, for the S state, we fit the numerical all-order data for Z ≥ fromRefs. [40, 41] to the form G SESE (1 S ) = B + B ( Zα ) ln ( Zα ) − + b ( Zα ) ln( Zα ) − + ( Zα ) pol( Zα ) , (32)where B and B are given by Eqs. (20) and (26), and pol( Zα ) denotes a polynomial in Zα . b and the coeffi- cients of the polynomial are fitting parameters. The uncer-tainty was obtained by varying ( i ) B within its error bars(20), ( ii ) numerical data within their error bars, and ( iii ) thelength of the polynomial and the number of data points in-cluded. The higher-order remainder for the S state was ob-tained by adding to G SESE (1 S ) the difference G SESE (2 S ) − G SESE (1 S ) , as fitted in Ref. [41]. For the P / state, wemerged together the analytical results (22) and (29) and nu-merical data from Ref. [41]. The uncertainty was obtained byquadratically adding the error of the B coefficient and onehalf of the leading logarithmic B contribution. The obtainedresults for the higher-order SESE remainder are summarizedin Table IV.Calculations of the SEVP and VPVP corrections were per-formed in Ref. [42] to all orders in Zα . Results for the higher-order remainder G SEVP listed in Table IV were obtained fromTables I and IV of Ref. [42], after subtracting contributions ofthe leading Zα -expansion coefficients and keeping in mindthat the light-by-light (LBL) contribution was not included innumerical calculations and thus should not be subtracted. Theuncertainty of the SEVP contribution comes from the missingLBL contribution. It was estimated for the S states as one halfof the LBL B contribution, calculated in Ref. [34]. For the P states, we assume the uncertainty to be negligible.The results for the higher-order remainder G VPVP listed inTable IV were obtained from Tables II and III of Ref. [42],after subtracting contributions of the leading Zα -expansioncoefficients summarized in Table III. The uncertainty due toomitted higher-order K¨all´en-Sabry contributions is assumedto be negligible at the level of present interest.For the S state of hydrogen, our result for the two-loophigher-order remainder is G QED2 = − , which isslightly lower than the value adopted by CODATA 2016 of − [7]. C. Higher-order QED effects
The Zα expansion of the three-loop QED correction isgiven by E QED3 = (cid:16) απ (cid:17) ( Zα ) n (cid:16) m r m (cid:17) h C + ( Zα ) C + ( Zα ) (cid:16) C L + C L + . . . (cid:17)i , (33)The leading-order contribution C was obtained in Refs. [43,44] and is given by C = (cid:20) −
568 a ζ (5)24 − π ζ (3)72 −
84 071 ζ (3)2304 −
71 ln − π ln π ln 2108 + 1591 π −
252 251 π (cid:21) δ ℓ + (cid:20) −
100 a ζ (5)24 − π ζ (3)72 − ζ (3)18 −
25 ln π ln
218 + 298 π ln 29 + 239 π −
17 101 π −
28 2595184 (cid:21) m/m r κ (2 ℓ + 1) (1 − δ ℓ ) , (34)where a = P ∞ n =1 / (2 n n ) = 0 .
517 479 061 . . . . For thenext-order contribution C , there are only partial results upto now [45, 46]. Following Ref. [7], we do not include partialresults and estimate the uncertainty due to absence of this termas C = ± δ ℓ . The leading logarithmic contribution C was derived in Ref. [37] as C = − B , (35) where B is the leading-order two-loop coefficient summa-rized in Table III. Ref. [37] presented results also for thesingle-logarithmic contribution C for the nP states and thedifference C ( nS ) − C (1 S ) . IV. NUCLEAR RECOIL
The dominant part of the nuclear recoil effect is accountedfor by E M in Eq. (5) and by the reduced-mass prefactors inprevious formulas. Beyond that, there are a number of furtherrecoil corrections. The first one is the nuclear recoil correctionof order ( Zα ) ≥ and of first order in m/M , E REC = mM ( Zα ) π n (cid:20)(cid:16) m r m (cid:17) ln( Zα ) − D + (cid:16) m r m (cid:17) D + ( Zα ) D + ( Zα ) G REC ( Zα ) (cid:21) , (36)where G REC ( Zα ) is the higher-order remainder containingall higher orders in Zα . Coefficients of the Zα expansion inEq. (36) are reviewed in Ref. [22] and summarized in Table V.The higher-order remainder G REC has an expansion of theform G REC ( Zα ) = D ln ( Zα ) − + D ln ( Zα ) + D + . . . , (37)where D = − / δ ℓ [47, 48] and the next two coeffi-cients were obtained by fitting numerical results in Refs. [11,12] D (1 S ) = 2 .
919 (10) , D (1 S ) = − .
32 (10) , (38) D (2 S ) = 3 .
335 (10) , D (2 S ) = − .
26 (6) , (39) D (2 P / ) = 0 .
149 (5) , D (2 P / ) = − .
035 (15) . (40)Numerical, all-order in Zα results for the higher-order re-mainder G REC are obtained in Refs. [11, 12] and summa-rized in Table VI. In the present review we do not includeresults for the finite nuclear size correction to E REC obtainedin Refs. [11, 12], since this effect is partly included in calcula-tions of nuclear polarizability summarized in the next section.The relativistic recoil corrections of second order in themass ratio is [18, 49, 50], E REC , = (cid:16) mM (cid:17) ( Zα ) n (cid:20) n − l + 1 + 12 δ ℓ δ I, / − ( Zα ) 2 π (cid:16) mM ln mM (cid:17) δ ℓ (cid:21) . (41)The first part of this correction ∝ ( Zα ) depends on the nu-clear spin I , which is the consequence of the choice of thedefinition of the point-like particle with a spin I . For I > such a definition is not commonly established, so we ascribean uncertainty of ± δ ℓ relative to the square brackets in theabove formula. This part agrees with the ( Zα ) ( m/M ) term TABLE III: Coefficients of the Zα expansion of the two-loop QED effects in Eq. (17). ζ ( n ) denotes the Riemann zeta function, ψ ( n ) is thedigamma function, γ E is Euler’s constant, N ( nL ) is a delta-function correction to the Bethe logarithm, defined by Eq. (4.21a) of Ref. [25].Term S S P / SESE B h − − π + π ln 2 − ζ (3) i δ ℓ .
409 244 1 .
409 244 0 .
114 722 ( m/m r ) − h − + π − π ln 2 + ζ (3) i m/m r κ (2 l +1) (cid:0) − δ l (cid:1) B unknown − .
265 06 (13) − .
265 06 (13) B − δ ℓ − / − / B
62 169 (cid:16) − ln 2 + n − n − ln n + ψ ( n ) + γ E (cid:17) δ ℓ − .
639 669 0 .
461 403 1 / n − n δ ℓ B
61 43 N ( nL ) + (cid:20) + π − ln 2 − π ln 2 + ln ζ (3) 48 .
388 913 40 .
932 915 0 .
202 220+ (cid:16) − ln 2 (cid:17)(cid:16) + n − n − ln n + ψ ( n ) + γ E (cid:17)(cid:21) δ ℓ + n − n (cid:16) + δ j, / − ln 2 (cid:17) δ ℓ SEVP B (cid:16) − + π (cid:17) δ ℓ .
142 043 0 .
142 043 − .
005 229 ( m/m r )+ (cid:16) − π (cid:17) j ( j +1) − l ( l +1) − / l ( l +1)(2 l +1) mm r (1 − δ ℓ ) B unknown .
305 370 1 .
305 370 B B
62 845 δ ℓ /
45 8 / B h − + π + ln 2 − (cid:16) + n − n − ln n + ψ ( n ) + γ E (cid:17)i δ ℓ .
436 241 0 .
995 812 − .
044 444 − δ ℓ VPVP B − δ ℓ − / − /
81 0 B (cid:16) − π + ln 2 (cid:17) π δ ℓ . . B B B − δ ℓ − .
541 728 − .
541 728 contained in Eq. (25) of the CODATA review [7]. The sec-ond part of this correction ∝ ( Zα ) is the Erickson formula(see the last line of Eq. (27) in Ref. [7]) expanded in m/M .This formula is derived for the spin- / nucleus; its depen-dence on nuclear spin is not known. However, we assume thecorresponding uncertainty to be negligible.An additional recoil contribution arises for the P (andhigher- l ) states because of mixing of the fine-structure sub-levels by the hyperfine-structure (hfs) interaction. This contri-bution is also known as the off-diagonal hfs shift. It dependson the nuclear spin I and the nuclear magnetic moment µ andis given, for the nP states [16, 20], by E REC , hfs ( nP ) = (cid:18) mm p (cid:19) α ( Zα ) n × (cid:16) µµ N (cid:17) I ( I + 1)81 ( − j +1 / δ ℓ , (42)where µ N = | e | / (2 m p ) is the nuclear magneton and m p isthe proton mass. This correction shifts the P / centroid energy by − . kHz for hydrogen, by − . kHz for deu-terium, and by − . kHz for He. We note that this correc-tion was not included in the definition of the energy levels inthe CODATA review [7] and needed to be accounted for to-gether with the hyperfine structure. Corrections to Eq. (42)are assumed to be suppressed by α/π , which is included intouncertainty.Furthermore, there is the radiative recoil correction [47, 51–53] E RREC = mM (cid:16) m r m (cid:17) α ( Zα ) π n δ ℓ (cid:20) ζ (3) − π ln 2+ 35 π − π ( Zα ) ln ( Zα ) − (cid:21) . (43)Following Ref. [54], we ascribe to this correction an uncer-tainty of Zα ) ln( Zα ) − relative to the square brackets inthe above equation. TABLE IV: Results for the two-loop higher-order remainder G QED2 in Eq. (17). Z S S P / SESE0 − . . − . . − . − . . − . . − .
37 (31) − . . − . . − .
28 (31) − . . − . . − .
20 (33) − . . − . . − .
13 (34) − . . − . . − .
06 (35)
SEVP0 − . − . . − . . − .
016 (6) − . . − . . − .
015 (5) − . . − . . − .
011 (2) − . . − . . − .
007 (2) − . . − . . − .
004 (1)
VPVP0 − . − .
76 (2) − . − . − . − . − . − . − . − . − . − . − . − . − . − . V. NUCLEAR SIZE AND POLARIZABILITY
It is customary in the literature to consider separately thefinite nuclear size (fns) effect (also known as the elastic partof the nuclear structure) and the nuclear polarizability (alsoknown as the inelastic nuclear structure). To a large extent,the separate treatment is due to the fact that the fns correctioncan be obtained numerically from the Dirac equation, whereascalculations of the nuclear polarizability are much more com-plicated. However, it was shown [13, 55, 56] that for lightatoms, there is significant cancelation between the fns effectsand the polarizability corrections. Moreover, it turned out thatsome of the nuclear model dependence of the individual cor-rections cancels out in the sum. Because of this, it is desirableto keep these contributions together and address them on thesame footing. We thus consider the sum of the fns correction E fns and the polarizability correction E pol , E nucl = E fns + E pol = X i ≥ E ( i )nucl , (44)where the upper index i indicates the order in Zα . A. ( Zα ) nuclear contribution The leading-order nuclear contribution comes solely fromthe finite nuclear size. It is given for an arbitrary hydrogen-like system by a simple formula, E (4)nucl = E (4)fns = 23 ( Zα ) n (cid:16) m r m (cid:17) R C δ ℓ , (45) where R C is the root-mean-square (rms) charge radius of thenucleus R C = Z d r r ρ ( r ) , (46)and ρ ( r ) is the nuclear charge distribution.The higher-order nuclear contributions are specific for eachnucleus. We start our consideration with hydrogen, which isa special case since proton is the only non-composite (one-nucleon) nucleus. B. ( Zα ) nuclear contribution for hydrogen If we assume that the nucleus has a fixed charge densitydistribution, then the ( Zα ) nuclear correction is given by thetwo-Coulomb exchange amplitude. The resulting fns correc-tion is [57] E (5)fns = −
13 ( Zα ) n (cid:16) m r m (cid:17) R Z δ ℓ , (47)where R Z is the third Zemach moment R Z = Z d r Z d r ρ ( r ) ρ ( r ) | ~r − ~r | . (48)The numerical value for the proton is R pZ ≡ R Z (H) =1 . fm, which is the average of two results derived fromthe electron-positron scattering [58, 59].A more detailed consideration shows, however, that a nu-cleus cannot generally be treated as a rigid body, because itis polarized by the surrounding electron. This gives rise tothe so-called nuclear polarizability contribution. The protonpolarizability correction is usually calculated as the forwardtwo-photon exchange amplitude, expressed via dispersion re-lations in terms of the inelastic scattering amplitude, which inturn is accessible in experiments.The recent evaluation of the proton ( Zα ) nuclear contri-bution [14] yields the result of − . kHz for the hy-drogen S state, which agrees with the previous (elastic + po-larizability) value adopted by CODATA [7] of − . kHz.The result [14] can be conveniently parameterized in terms ofthe effective proton radius R pF , which is introduced in anal-ogy with Eq. (47), E (5)nucl (H) = −
13 ( Zα ) n (cid:16) m r m (cid:17) R pF δ ℓ , (49)with R pF = 1 .
947 (75) fm . (50)We note that for the proton there is no cancelation betweenthe elastic and polarizability contributions, in contrast to thecomposite nuclei. TABLE V: Coefficients of the Zα expansion of the nuclear recoil correction in Eq. (36).Term S S P / D
51 13 δ ℓ D − ln k + (cid:2) − − n + ln n + ψ ( n + 1) − ψ (1) (cid:3) δ ℓ − − [ l ( l + 1)(2 l + 1)] − (1 − δ ℓ ) D (cid:0) − (cid:1) π δ ℓ + 2 π h − l ( l +1) n i [(4 l − l + 3)] − (1 − δ ℓ ) − − Z S S P / .
720 (3) 14 .
899 (3) 1 .
509 7 (2) .
390 (1) 15 .
010 (1) 1 .
307 39 (5) . . .
192 04 (2) . . .
112 68 (2) . . .
053 21 (2) C. ( Zα ) nuclear contribution for composite nuclei For compound nuclei consisting of several nucleons, theZemach fns correction (47) cancels out in a sum with the cor-responding nuclear structure contribution [13]. However, itsurvives in the contribution induced by the interaction withindividual nucleons. In the result, we write the total nuclearstructure correction E (5)nucl (known also as the two-photon ex-change correction) for a composite nuclei as E (5)nucl = E (5)pol − α ( Zα ) n (cid:2) Z R pF + ( A − Z ) R nF (cid:3) δ ℓ , (51)where the first term E (5)pol is the intrinsic nuclear polarizabilityand the second term is the contribution of individual nucle-ons. In the above equation, R pF is the effective proton radiusgiven in Eq. (50), R nF is an analogous effective radius for theneutron, and A is the mass number. We extract R nF from thecalculation of Tomalak (Table II of Ref. [14]), with the result R nF = 1 .
43 (16) fm . (52)The nuclear polarizability correction E (5)pol is dominated bythe electric dipole excitations and is given by [13, 56, 60] E (5)pol = − α φ (0) 23 (cid:28) φ N (cid:12)(cid:12)(cid:12)(cid:12) ~d H N − E N (cid:20) H N − E N ) m (cid:21) ~d (cid:12)(cid:12)(cid:12)(cid:12) φ N (cid:29) − π α φ (0) Z X i,j =1 h φ N || ~R i − ~R j | | φ N i + many small corrections , (53) where ~d is the electric dipole operator divided by the elemen-tary charge, H N and E N are the nuclear Hamiltonian and itseigenvalue, φ N and φ are the nuclear and electronic wavefunctions, and ~R i is the position vector of i th proton in thenucleus. The second term in Eq. (53) is the remainder of theZemach fns correction (47) for a composite nuclei.For atoms with Z ≤ , the nuclear polarizability correctionhas been investigated only for deuterium, helium, and someneutron-rich isotopes of Li and Be. For deuteron, the two-photon nuclear polarizability was calculated in Ref. [55] andrecently reanalysed in Ref. [13], E (5)pol (D) = − . δ ℓ n h kHz ± . (54)For helium, the nuclear polarizability correction was calcu-lated in Ref. [61], with the result E (5)pol ( He) = − . δ ℓ n h kHz ± , (55) E (5)pol ( He) = − . δ ℓ n h kHz ± . (56)For stable isotopes with Z = 3 , 4, and 5, we use the fol-lowing estimate E (5)pol ≈ − E fns ± , (57)which was obtained in Ref. [17] basing on an analysis of avail-able results throughout the whole Z sequence. D. ( Zα ) nuclear contribution The ( Zα ) nuclear contribution arises from the three-photon exchange between electron and the nucleus. The cor-responding fns correction is known in the nonrecoil limit andis given for the nS and nP / ( κ = 1 ) states by [13, 57] E (6)fns = ( Zα ) n R C ( − (cid:20) n − − n + 2 γ E − ln n n ) + ln (cid:0) mR C Z α (cid:1)(cid:21) δ ℓ + 16 (cid:18) − n (cid:19) δ κ ) , (58)where R C is the effective nuclear charge radii that encodesthe high-momentum contribution (for exact definition seeRef. [13]). The effective nuclear radii R C has the numericalvalue close to R C and depends on the model of the nuclearcharge distribution. We use the result obtained in Ref. [13]for the exponential model, R C /R C = 1 .
068 497 , (59)which does not depend on nuclear charge. It was shown inRef. [13] that the dependence on R C in Eq. (58) cancels outin the sum with the corresponding nuclear polarizability cor-rection, so the model dependence of R C does not contributeto the uncertainty.The ( Zα ) nuclear polarizability is practically unknown forthe electronic atoms. The only available results are estimatesfrom Ref. [13] for hydrogen E (6)pol (H) = 0 . δ ℓ n h kHz ± , (60)and deuterium E (6)pol (D) = − . δ ℓ n h kHz ± . (61)It is remarkable that for hydrogen, the three-photon nu-clear polarizability dominates over the two-photon polariz-ability. The reason for this is that E (6)pol ∝ ( Zα ) R C whereas E (5)nucl (H) ∝ ( Zα ) R C , so that the two-photon exchange iseffectively suppressed by a parameter mR C / ( Zα ) ≪ . Forall atoms other than hydrogen, the two-photon exchange isdominated by the electric dipole polarizability ∝ ( Zα ) R C and, therefore, the three-photon polarizability is smaller thanthe two-photon one, as usually expected. We estimate the un-certainty due to the unknown three-photon nuclear polariz-ability for nuclei with Z = 2 − to be 10% of the corre-sponding two-photon polarizability. E. Radiative fns correction
The leading radiative fns correction is of order α ( Zα ) andnonzero only for S states (see review [22] for details), E (5)fns , rad = 23 α ( Zα ) n (cid:16) m r m (cid:17) R C (cid:0) − (cid:1) δ ℓ . (62)The next-order radiative fns correction for the S states isknown only partially [35, 62, 63], E (6)fns , rad ( nS ) = 23 α ( Zα ) π n R C h −
23 ln ( Zα ) − + ln ( mR C ) i . (63)In the above formula we keep only the squared logarithms anddo not include some higher-order terms derived in Ref. [62],because the term ∝ ln( Zα ) − is not known and expected to be of similar magnitude as the omitted terms. The result forthe P states [35, 62, 63] is E (6)fns , rad ( nP / ) = 16 α ( Zα ) π n R C (cid:18) − n (cid:19) × "
89 ln( Zα ) − −
89 ln 2 + 166135 + 4 n n − N ( nP ) . (64)The uncertainty of Eqs. (63) and (64) was evaluated by com-paring with results of the more complete treatment [63]. F. Nuclear self-energy
The nuclear self-energy correction was derived in Ref. [64],with the result E NSE = (cid:16) mM (cid:17) Z ( Zα ) πn × (cid:20) ln (cid:18) Mm ( Zα ) (cid:19) δ ℓ − ln k ( n, l ) (cid:21) . (65)It should be noted that there is some ambiguity associated withthis correction since the nuclear self-energy contributes notonly to the Lamb shift but to the nuclear charge radius andthe nuclear magnetic moment. Specifically, addition of an ar-bitrary constant in the brackets of Eq. (65) is equivalent tochanging the definition of the nuclear charge radius. This im-plies that the presently used definition of the nuclear chargeradius (through the slope of the Sachs form-factor) is ambigu-ous on the level of a constant in the brackets of Eq. (65). Thisissue was pointed out in Ref. [64] (together with the sugges-tion for a rigorous definition of the nuclear charge radius) butdid not attracted attention of the community up to now. Inorder to quantify this ambiguity, we ascribe to E NSE an un-certainty of 0.5 in the square brackets, as in Ref. [54]. Thenumerical value of this uncertainty is . kHz for the hydro-gen S state, which can be disregarded at present but mightbecome relevant in the future. VI. NUMERICAL RESULTS
In order to obtain numerical results for the Lamb shift andthe transition energies, we need to specify values of fun-damental constants and nuclear parameters. In the presentreview we use the charge radii of the proton and thedeuteron as derived from the muonic atoms [2, 65] ( R p =0 . fm and R d = 2 . fm) and the corre-sponding value of the Rydberg constant from Ref. [16], c R ∞ = 3 289 841 960 248 . . kHz . (66)It should be mentioned that the exact values of R p , R d , and R ∞ are under debates at present. In particular, the Rydbergconstant of Eq. (66) differs from the value recommended by0CODATA 2014 [7] by . σ . On the level of the present exper-imental accuracy, this controversy is relevant only for hydro-gen and deuterium and can be disregarded for heavier atoms.The nuclear charge radii for elements with Z > are takenas follows. For He and He, we use values by Sick [66, 67];for Li and Li isotopes, values from Ref. [68]; for otheratoms, values from Ref. [69]. The nuclear masses are takenfor hydrogen from Ref. [70], for deuterium and helium iso-topes from Ref. [7], and for all other nuclei from Ref. [71].Nuclear magnetic moments are taken from Ref. [72]. Thefine-structure constant is [7] α = 1 / .
035 999 139 (31) . (67)The individual contributions to the Lamb shift for two ex-perimentally most interesting cases, H and He + , are listed inTable VII. The results for the QED and the leading fns correc-tion are presented in the nonrecoil limit (i.e., with m r → ).The contribution due to the reduced mass in all formulas issummed up and tabulated separately as the relativistic reducedmass (RRM) correction. The uncertainty of the fns correc-tion is due to the uncertainty of the nuclear charge radius R C ,whose values are specified in the table. The total results for theLamb shift E L are given with two uncertainties. The first oneis the theoretical uncertainty, whereas the second one comesfrom the uncertainty of the nuclear charge radius.We observe that for the hydrogen Lamb shift, the theoreti-cal uncertainty is twice larger than the uncertainty due to theproton charge radius (as extracted from muonic hydrogen).The two largest theoretical uncertainties come from (i) thetwo-loop self-energy and (ii) the three-loop QED correction.As compared to the previous CODATA review [7], the mainchange is due to our reanalysis of the two-loop QED effects;it shifted the theoretical value by one half of the previous un-certainty and improved the accuracy by a factor of 1.5. For helium, the uncertainty of the Lamb shift is presentlydominated by the uncertainty from the nuclear radius. Butthis is likely to change once the results of the muonic heliumexperiment are evaluated [5, 8].Table VIII presents theoretical results for the S – S and S – P / transition energies in hydrogen and light hydrogen-like ions. Theoretical predictions are given with two uncer-tainties. The first one is the theoretical uncertainty, whereasthe second one is induced by uncertainties of nuclear radiiand masses. The uncertainty due to the Rydberg constant R ∞ is not included. Theoretical predictions are compared withavailable experimental results for the S – P / Lamb shift inhydrogen, helium and lithium. We do not present a compar-ison with the hydrogen S – S experimental results [73, 74]since the value of the Rydberg constant (66) is derived fromthe comparison of theory and these experiments. For the samereason we do not include the uncertainty due to Rydberg con-stant in the theoretical predictions.In summary, theoretical calculations of the Lamb shift inhydrogen and light hydrogen-like ions are required for thedetermination of the Rydberg constant. In the present workwe summarized the present status and recent developments oftheoretical calculations of QED and nuclear effects, criticallyevaluating uncertainties of all contributions. Acknowledgments
The authors are grateful to A. Kramida for pointing out amistake in an early version of the manuscript. V.A.Y. ac-knowledges support by the Ministry of Education and Sci-ence of the Russian Federation Grant No. 3.5397.2017/6.7.K.P. acknowledges support by the National Science Center(Poland) Grant No. 2017/27/B/ST2/02459. V.P. acknowl-edges support from the Czech Science Foundation - GA ˇCR(Grant No. P209/18-00918S). [1] R. Pohl, A. Antognini, F. Nez, F. D. Amaro, F. Biraben, J. a.M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L. M. P.Fernandes, A. Giesen, T. Graf, T. W. H¨ansch, P. Indelicato,L. Julien, C.-Y. Kao, P. Knowles, E.-O. Le Bigot, Y.-W. Liu,J. A. M. Lopes, L. Ludhova, C. M. B. Monteiro, F. Mulhauser,T. Nebel, P. Rabinowitz, J. M. F. dos Santos, L. A. Schaller,K. Schuhmann, C. Schwob, D. Taqqu, J. a. F. C. A. Veloso,F. Kottmann,
Nature (London) , , 213.[2] A. Antognini, F. Nez, K. Schuhmann, F. D. Amaro, F. Biraben,J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, M. Diepold,L. M. P. Fernandes, A. Giesen, A. L. Gouvea, T. Graf,T. W. H¨ansch, P. Indelicato, L. Julien, C.-Y. Kao, P. Knowles,F. Kottmann, E.-O. L. Bigot, Y.-W. Liu, J. A. M. Lopes, L. Lud-hova, C. M. B. Monteiro, F. Mulhauser, T. Nebel, P. Rabinowitz,J. M. F. dos Santos, L. A. Schaller, C. Schwob, D. Taqqu, J. F.C. A. Veloso, J. Vogelsang, R. Pohl, Science , , 417.[3] A. Beyer, L. Maisenbacher, A. Matveev, R. Pohl, K. Khabarova,A. Grinin, T. Lamour, D. C. Yost, T. W. H¨ansch, N. Ko-lachevsky, T. Udem, Science , , 79.[4] H. Fleurbaey, S. Galtier, S. Thomas, M. Bonnaud, L. Julien,F. Biraben, F. Nez, M. Abgrall, J. Gu´ena, Phys. Rev. Lett. , , 183001.[5] R. Pohl, talk at , May 13-18, Bad Honnef, Ger-many, 2018.[6] E. Hessels, talk at Precision Measurements and FundamentalPhysics: The Proton Radius Puzzle and Beyond , July 23-27,2018 Mainz, Germany, 2018.[7] P. J. Mohr, D. B. Newell, B. N. Taylor,
Rev. Mod. Phys. , , 035009.[8] S. Schmidt, M. Willig, J. Haack, R. Horn, A. Adamczak, M. A.Ahmed, F. D. Amaro, P. Amaro, F. Biraben, P. Carvalho, T.-L.Chen, L. M. P. Fernandes, T. Graf, M. Guerra, T. W. H¨ansch,M. Hildebrandt, Y.-C. Huang, P. Indelicato, L. Julien, K. Kirch,A. Knecht, F. Kottmann, J. J. Krauth, Y.-W. Liu, J. Machado,M. Marszalek, C. M. B. Monteiro, F. Nez, J. Nuber, D. N. Pa-tel, E. Rapisarda, J. M. F. dos Santos, J. P. Santos, P. A. O. C.Silva, L. Sinkunaite, J.-T. Shy, K. Schuhmann, I. Schulthess,D. Taqqu, J. F. C. A. Veloso, L.-B. Wang, M. Zeyen, A. An-tognini, R. Pohl, arXiv:1808.07240 .[9] T. Udem, private communication, 2018.[10] K. Eikema, private communication, 2017. TABLE VII: Individual contributions to the Lamb shift E L , in MHz. Abbreviations are as follows: “SE” is the one-loop self-energy, “Ue” isthe Uehling one-loop vacuum polarization, “WK” is the Wichmann-Kroll one-loop vacuum-polarization, “Ue( µ had)” is the Uehling muon andhadronic vacuum polarization, “SESE” is the two-loop self-energy, “SEVP” is the electron self-energy with vacuum-polarization insertions,“VPVP” is the two-loop vacuum-polarization, “QED(ho)” is the three-loop QED correction, “RRM” is the relativistic reduced mass correction(see text), “REC” is the recoil correction E REC , “REC(ho)” is the sum of higher-order recoil corrections E REC , , E REC , hfs , and E RREC ,“FNS” is the leading-order fns correction E (4)nucl , “NUCL5” is the ( Zα ) nuclear correction E (5)nucl , “NUCL6” is the ( Zα ) nuclear correction E (6)nucl , “FNS(rad)” is the radiative fns correction E fns , rad , “NSE” is the nuclear self-energy correction E NSE . S S P / Z = 1 , H , R C = 0 .
840 87 (39) fm , M/m = 1 836 .
152 673 346 (81) SE .
453 556 (1) 1 072 .
958 455 − .
858 661 (1) Ue − .
170 186 − .
897 303 − .
000 347 WK .
002 415 0 .
000 302 0
Ue( µ had) − .
008 48 (8) − .
001 06 (1) 0
SESE .
335 0 (13) 0 .
292 48 (16) 0 .
027 253 (4)
SEVP .
288 39 (16) 0 .
036 015 (20) − .
001 241
VPVP − .
895 224 − .
236 911 − .
000 003
QED(ho) .
001 83 (96) 0 .
000 23 (12) − .
000 216
RRM − .
765 917 − .
633 931 0 .
011 741
REC .
402 830 0 .
340 469 − .
016 656
REC(ho) .
013 16 (74) − .
003 227 (92) − .
001 335 (4)
FNS .
107 6 (10) 0 .
138 45 (13) 0
NUCL5 − .
000 109 (1) − .
000 014 0
NUCL6 .
001 07 (39) 0 .
000 140 (49) 0 .
000 001
FNS(rad) − .
000 135 (1) − .
000 017 0
NSE .
004 63 (16) 0 .
000 585 (20) 0 .
000 001 (20)
Total .
770 4 (18)(10) 1 044 .
994 66 (23)(13) − .
839 463 (21)(0) Z = 2 , He + , R C = 1 . fm , M/m = 7 294 .
299 541 36 (24) SE
111 054 .
170 69 (1) 14 257 .
035 60 (2) − .
794 17 (2) Ue − .
099 45 − .
952 77 − .
022 109 WK .
152 06 0 .
019 01 0 .
000 002
Ue( µ had) − .
135 7 (12) − .
016 97 (15) 0
SESE .
569 (64) 4 .
095 9 (80) 0 .
440 50 (25)
SEVP .
956 8 (88) 0 .
617 9 (11) − .
020 086
VPVP − .
047 28 − .
756 393 (1) − .
000 210
QED(ho) .
029 (31) 0 .
003 6 (38) − .
003 468
RRM − .
915 19 − .
393 65 0 .
046 950
REC .
676 28 2 .
533 38 − .
130 835
REC(ho) − .
121 (10) − .
020 0 (13) 0 .
000 549
FNS .
82 (34) 8 .
853 (42) 0
NUCL5 − .
034 6 (32) − .
004 33 (40) 0
NUCL6 .
152 0 (35) 0 .
020 66 (43) 0 .
000 354
FNS(rad) − .
017 43 (39) − .
002 179 (50) 0 .
000 007
NSE .
018 75 (65) 0 .
002 372 (82) 0 .
000 005 (82)
Total
107 693 .
18 (7)(34) 13 837 .
035 (9)(42) − .
482 51 (26)(0) [11] V. A. Yerokhin, V. M. Shabaev,
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Z R C [fm] M/m S – S S – P / H 0.84087 (39) 1 836.152 673 346 (81) .
413 1869 (18)(10) 1 .
057 834 12 (23)(13)1 .
057 847 (9) a D 2.12562 (78) 3 670.482 967 85 (13) .
407 5345 (17)(52) 1 .
059 219 91 (21)(65) He + .
006 31 (7)(34) 14 .
041 517 (9)(42)14 .
041 13 (17) b He + . .
043 96 (1)(20) Li
22 206 430 .
550 (1)(26) 62 .
734 2 (1)(32)62 .
765 (21) c Li
22 206 719 .
625 (1)(26) 62 .
723 1 (1)(33) Be
39 482 224 .
239 (4)(24) 179 .
771 9 (5)(30) B
61 697 635 .
70 (1)(14) 404 .
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