Three-photon exchange nuclear structure correction in hydrogenic systems

# Three-photon exchange nuclear structure correction in hydrogenic systems

Krzysztof Pachucki [ Faculty of Physics, Warsaw University, Pasteura 5, 02-093 Warsaw, Poland    Vojtěch Patkóš Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague 2, Czech Republic    Vladimir A. Yerokhin Center for Advanced Studies, Peter the Great St. Petersburg Polytechnic University, Polytekhnicheskaya 29, 195251 St. Petersburg, Russia
July 16, 2019
###### Abstract

The complete relativistic nuclear structure correction to the energy levels of ordinary (electronic) and muonic hydrogen-like atoms is investigated. The elastic part of the nuclear structure correction is derived analytically. The resulting formula is valid for an arbitrary hydrogenic system and is much simpler than analogous expressions previously reported in the literature. The analytical result is verified by high-precision numerical calculations. The inelastic nuclear structure correction is derived for the electronic and muonic deuterium atoms. The correction comes from a three-photon exchange between the nucleus and the bound lepton and has not been considered in the literature so far. We demonstrate that in the case of deuterium, the inelastic three-photon exchange contribution is of a similar size and of the opposite sign to the corresponding elastic part and, moreover, cancels exactly the model dependence of the elastic part. The obtained results affect the determination of nuclear charge radii from the Lamb shift in ordinary and muonic atoms.

###### pacs:
31.30.jr, 36.10.Ee, 14.20.Dh
preprint: Version 1.0

]www.fuw.edu.pl/ krp

## I Introduction

The determination of the nuclear charge radii from atomic spectra is a very interesting test of the Standard Model of fundamental interactions. The lepton universality, namely the identical interaction strength of all leptons, ensures that the nuclear charge radii derived from the ordinary (electronic) and the muonic atoms should be exactly the same. However, a series of experiments on H pohl:10 () and D pohl:16 () and (still unpublished) measurements on He and He pohl:priv () revealed significant discrepancies for the determined nuclear charge radii, as compared to those derived from the corresponding electronic atoms. In order to verify these discrepancies one should carefully examine all possible sources of uncertainties in the spectroscopic determinations of the nuclear charge radii.

The main theoretical uncertainty of the Lamb shift in light muonic atoms comes from our insufficient knowledge of the nuclear internal structure. The nuclear structure corrections are usually divided into the elastic and the inelastic part. The elastic part (also referred to as the finite nuclear size correction) is induced by a static distribution of the nuclear charge and can be obtained by solving the Dirac equation. The inelastic nuclear correction is much more complicated; it encompass the nuclear dipole polarizability and higher-order contributions. To deal with the nuclear corrections, one performs an expansion of the binding energy in powers of the fine structure constant and examines the expansion terms one after another.

The leading nuclear effect is of order and of pure elastic origin. The first-order nuclear-structure correction (often referred to as the two-photon exchange contribution) has both elastic and inelastic parts and was extensively studied both for the electronic and the muonic atoms friar:97:b (); pachucki:11:mud (); pachucki:15:mud (); hernandez:14 (); hernandez:18 (). One of the interesting results was a significant cancellation between the elastic and the inelastic nuclear contributions.

The next-order nuclear structure correction comes from the three-photon exchange between the bound lepton and the nucleus. Only the elastic part of this correction has been addressed in the literature so far friar:79:ap (). In the present work we demonstrate that the inelastic contribution is significant and partially cancels its elastic counterpart. We also derive formulas for the complete nuclear correction in deuterium. Our calculation is performed in the nonrecoil limit and neglects the magnetic dipole and electric quadrupole moments of the nucleus. The results obtained affect determinations of nuclear charge radii from the precision spectroscopy of ordinary and muonic atoms. However, they are not able to explain the previously reported discrepancy between the H-D and H-D isotope shift pohl:16 ().

We now introduce notations for the nuclear radii that will be extensively used throughout this paper. denotes the root mean square (rms) charge radius of an arbitrary nucleus, . We will use specific notations for several important nuclei: for the rms radius of the proton, for the rms radius of the deuteron, and for the deuteron structure radius . Since we neglect the finite nuclear mass effects, there is no term in . We define for an arbitrary nucleus, with the specific cases of for the proton, for the deuteron, and for the corresponding structure radius of the deuteron. is the third Zemach moment defined below by Eq. (15). We will also introduce two new effective nuclear radii of arbitrary nuclei, and , defined by Eqs. (62) and (66), respectively. The corresponding specific notations are and for the proton and and for the deuteron, respectively.

## Ii Leading finite nuclear size correction

In this section we rederive well-known results for the leading nuclear correction of order , which is of pure elastic (finite nuclear size) origin and induced by the one-photon exchange between the bound lepton and the nucleus. This derivation sets the ground for our further evaluation of higher-order corrections.

Let us assume that the nucleus is a scalar particle with the charge density in the momentum space. The electron-nucleus interaction potential in momentum space is then

 V(q2)=−ρ(q2)4πZαq2. (1)

The expansion coefficients of in ,

 ρ(q2)=1+ρ′(0)q2+12ρ′′(0)q4+…, (2)

can be interpreted in terms of momenta of the nuclear charge distribution and ,

 ρ′(0) = −⟨r2⟩6, (3) ρ′′(0) = ⟨r4⟩60. (4)

From the second term in the right-hand side of Eq. (2), one immediately obtains the leading finite nuclear size correction to the potential,

 δV=−ρ′(0)4πZαδ(3)(r), (5)

and to the energy level of a hydrogenic system,

 E(4)fns=⟨ϕ|δV|ϕ⟩=2π3Zα⟨r2⟩ϕ2(0), (6)

where, for states,

 ϕ2nS(0)=⟨δ(3)(r)⟩=(μZα)3πn3, (7)

and is the reduced mass of the atom.

In order to establish the importance of higher-order effects, we will need numerical values of the leading finite nuclear size effect in hydrogen and deuterium. The corresponding results, obtained assuming fm and  fm, are, for the electronic atoms,

 E(4)fns(2S−1S,H) = −1368396r2p=−967541Hz, (8) E(4)fns(2S−1S,D) = −1369513r2d=−6187818Hz, (9)

and for the muonic atoms,

 E(4)fns(2P−2S,μH) = −5.19745r2p=−3.67492meV, (10) E(4)fns(2P−2S,μD) = −6.07318r2d=−27.44022meV. (11)

We also observe that one of the relativistic corrections comes from the third term in Eq. (2),

 δ(2)V=12ρ′′(0)4πZα∇2δ(3)(r). (12)

Since its expectation value on states is singular, we will use dimensional regularization and combine this part with other corrections to obtain a finite result.

## Iii Two-photon exchange nuclear structure: muonic atoms

In this section we address the leading nuclear structure contribution in muonic atoms, which originates from the two-photon exchange between the bound lepton and the nucleus.

The elastic part can be obtained from the forward two-photon scattering amplitude at zero momentum

 (13)

with and . This leads to the so-called Friar correction friar:79:ap (),

 E(5)fns = −(4πZα)2ϕ2(0)2m∫d3q(2π)3ρ2(q2)−1−2q2ρ′(0)q6 (14) = −π3ϕ2(0)(Zα)2mr3Z,

where

 r3Z=∫d3r1∫d3r2ρ(r1)ρ(r2)|→r1−→r2|3. (15)

As pointed out in Refs. friar:97:b (); pachucki:11:mud (), it is important to consider the Friar correction together with the corresponding inelastic part, because of a cancellation between them, occurring both for the muonic and the ordinary atoms. For this reason, we do not separate out but absorb it in the total nuclear structure correction .

### iii.1 Muonic hydrogen

The inelastic two-photon exchange correction in H has been extensively studied in the literature (see Ref. birse:2012 () and references therein). It is also given by the forward scattering amplitude and can be parameterized in terms of two spin-independent structure functions of the proton. Using dispersion relations, these functions are usually expressed in terms of the cross section of the inelastic photon scattering off the proton, which is extracted from experiment. The main problem of this approach is that one of the dispersion relations involves subtractions that can only be obtained from theory, and this introduces the dominant uncertainty.

There is good agreement between different calculations of the two-photon exchange correction, with the final result of  meV assumed by the CREMA collaboration antognini:13:ap () in their determination of the proton charge radius. It is convenient to parameterize this result in terms of an effective radius , in analogy to Eq. (14),

 E(5)(μH)=−π3ϕ2(0)(Zα)2mr3pF, (16)

with

 r3pF=3.270(197)fm3. (17)

This parametrization will be used below in our calculation of the inelastic contribution in other muonic atoms, see Eq. (24).

### iii.2 Muonic atoms other than hydrogen

For all nuclei other than the proton, the inelastic contribution is dominated by the electric dipole polarizability. For muonic atoms, one may assume the nonrelativistic approximation, so the second-order correction due to the electric dipole nuclear excitation is

 E(5)pol0=α2⟨ϕϕN∣∣∣→d⋅→rr31EN+E0−HN−H0→d⋅→rr3∣∣∣ϕϕN⟩, (18)

where is the electric dipole operator divided by the elementary charge, and and are the nonrelativistic Coulomb Hamiltonian for the muon and the nucleus, respectively. To the leading order in , one may neglect the Coulomb interaction and replace to obtain a compact formula for the leading two-photon exchange contribution,

 E(5)pol0=−4πα23ϕ2(0)⟨ϕN∣∣∣→d√2mHN−EN→d∣∣∣ϕN⟩, (19)

which contributes meV to the transition energy in muonic deuterium pachucki:15:mud ().

There are many corrections to the leading contribution pachucki:11:mud (); pachucki:15:mud (); hernandez:14 (); hernandez:18 (), the most interesting of them being the one that partially cancels the Friar correction. To show this, following Ref. pachucki:11:mud (), we consider the muonic matrix element for the nonrelativistic two-photon exchange

 (20)

where is the nonrelativistic Hamiltonian for the muon (electron) in the nonrecoil limit, and is a position of the th proton with respect to the nuclear mass center. Using the on-mass-shell approximation, subtracting the leading Coulomb interaction, the finite nuclear size, and the electric dipole polarizability, and expanding in the small parameter , we obtain

 P = ≈ ∑i,jπ3mα2ϕ2(0)|→Ri−→R′j|3(1−15√2mE|→Ri−→R′j|+…).

The corresponding correction to the atomic energy is

 E(5)pol1=−∑∫dE∫d3Rd3R′ϕ∗N(→R)ϕE(→R)ϕ∗E(→R′)ϕN(→R′)P (22)

Let us consider only the first, -independent term. When , it corresponds to the elastic part, namely, the Friar correction given by Eq. (14). However, the inclusion of all excited states leads to

 E(5)pol1=−π3mα2ϕ2(0)Z∑i,j=1⟨ϕN||→Ri−→Rj|3|ϕN⟩, (23)

which is much different from Eq. (14), in particular it vanishes for deuterium.

There are further nuclear polarizability corrections which were extensively studied in the literature pachucki:11:mud (); pachucki:15:mud (); hernandez:14 (); hernandez:18 (). It is convenient to write the final result for the two-photon exchange nuclear structure correction separating out the contribution due to the two-photon exchange with individual nucleons,

 E(5)=E(5)pol−π3mα2ϕ2(0)[Zr3pF+(A−Z)r3nF], (24)

where . Such representation of the nuclear structure correction is particularly advantageous for calculating isotope shifts, since the individual nucleon contributions partially cancel each other in the difference, together with the corresponding uncertainties. Calculating the two-photon exchange with individual nucleons, we take the effective proton radius from Eq. (17), whereas for the neutron we assume the corresponding parameter to be four times smaller than that of the proton, , with uncertainty of 100%. This choice of is in agreement with results summarized in Ref. krauth:16:ap () but requires further investigations.

Despite the fact that the literature results for in D reported by different groups pachucki:11:mud (); pachucki:15:mud (); hernandez:14 (); hernandez:18 () are in good agreement with each other (see a summary in Ref. krauth:16:ap ()), one should bear in mind that a number of higher-order effects exist that have not yet been addressed in any of the previous studies. Specifically, it has not so far been possible to include nucleon relativistic corrections to the coupling of the nucleus to the electromagnetic field. We thus believe that all theoretical predictions of in D should bear an uncertainty whose relative value is approximately the ratio of the average nucleon binding energy to the nucleon mass, which is about 1%.

Summarizing our analysis of the existing literature results, we adopt the sum of entries labelled as “Our choice” in Table 3 in Ref. krauth:16:ap () as currently the best value of the two-photon nuclear polarizability correction to the transition energy in D, and ascribe the uncertainty of 1% to it,

 E(5)pol(2P1/2\rm--2S,μD)=1.6625(166)meV. (25)

The above uncertainty of is about 50% larger than the corresponding estimate of  meV given in Table 3 of Ref. krauth:16:ap (). Finally, we add the individual nucleon part in Eq. (24) and obtain the total two-photon nuclear structure correction to the transition energy in D,

 E(5)(2P1/2\rm--2S,μD)=1.7110(194) meV, (26)

which almost coincides with the corresponding result of  meV from Ref. krauth:16:ap (), as given by Eq. (17) of that work.

## Iv Two-photon exchange nuclear structure: electronic atoms

The elastic (finite nuclear size) part of the two-photon exchange nuclear structure correction for electronic atoms is given by the same formula as for the muonic atoms, Eq. (13).

### iv.1 Electronic hydrogen

We calculate the elastic part of the nuclear structure correction for hydrogen according to Eq. (13) and using the result for the third Zemach moment from Ref. borie:12 () obtained by averaging values measured in scattering experiments,

 rpZ=1.587(26)rp. (27)

The corresponding result for the transition is

 E(5)fns(2S\rm--1S,eH)=0.0307(15) \rm kHz. (28)

The inelastic part of the two-photon exchange nuclear structure correction was derived in the logarithmic approximation in Ref. khriplovitch:00 (),

 E(5)pol(2S\rm--1S,eH)=−mαϕ2(0)[5αp−βp]ln¯Epm, (29)

where is the average proton excitation energy and and are the static proton polarizabilities extracted from experiment. Using the same average proton excitation energy  MeV as in Ref. khriplovitch:00 () and the updated results for the proton polarizabilities demissie:16 (),

 αp = 10.65(35)(20)(30)\rm fm3, βp = 3.15(35)(20)(30)\rm fm3, (30)

we obtain the result for the transition of

 E(5)pol(2S\rm--1S,eH)=0.0567(85) \rm kHz, (31)

where, following Ref. khriplovitch:00 (), we assumed a 15% uncertainty due to the leading logarithmic approximation.

The total result for the two-photon nuclear structure correction in electronic hydrogen is

 E(5)(2S\rm--1S,eH)=E(5)fns+E(5)pol=0.0874(86) \rm kHz, (32)

which could be compared with the corresponding result of  kHz from Ref. mohr:16:codata ().

### iv.2 Electronic atoms other than hydrogen

Similarly to the muonic atoms, it is convenient to write the total two-photon exchange nuclear structure correction separating out the contribution due to the interaction with individual nucleons,

 E(5)=E(5)pol−mαϕ2(0)[π3αZr3pZ+Z[5αp−βp]ln¯Epm+(A−Z)[5αn−βn]ln¯Enm]. (33)

In the above formula, the first and the second terms in the brackets represent the elastic and the inelastic interactions with individual protons, respectively, whereas the third term comes from the inelastic interaction with individual neutrons. The parameters for the protons are the same as for hydrogen, whereas for the neutrons we use the experimental polarizabilities demissie:16 (),

 αn = 11.55(125)(20)(80)\rm fm3, βn = 3.65(125)(20)(80)\rm fm3, (34)

and the same value of  MeV as for the proton. We note that the elastic interaction of the bound electron with the nucleus as a whole is absorbed in , reflecting the fact that the third Zemach moment correction for a compound nucleus largely cancels out between the elastic and inelastic parts in the same way as in muonic atoms.

Similarly to the muonic atoms, the nuclear polarizability correction in Eq. (33) comes from the electric dipole polarizability, which, however, takes a very different form for the electronic atoms. Since in this case the nonrelativistic approximation is not valid, one should consider the complete two-photon exchange and keep the relativistic form of the matrix elements,

 E(5)pol = ie2ϕ2(0)∫dω2π∫d3k(2π)3ω2(δik−kikkω2)ω2−k2(δjl−kjklω2)ω2−k2 (35) ×Tr[(γj1⧸p−⧸k−mγi+γi1⧸p+⧸k−mγj)(γ0+I)4] ×⟨ϕN∣∣∣dk1EN−HN−ωdl∣∣∣ϕN⟩+… = E(5)pol1+E(5)pol2+E(5)pol3+…,

where . Assuming that the electron mass is much smaller than the nuclear excitation energy, the leading nuclear polarizability correction becomes

 E(5)pol1=−mα2ϕ2(0)23⟨ϕN∣∣∣→d1HN−EN[196+5ln2(HN−EN)m]→d∣∣∣ϕN⟩. (36)

The corresponding contribution to the transition in ordinary deuterium is  kHz friar:97:a ().

Various small corrections to the electric dipole polarizability for electronic atoms were considered by Friar in friar:97:b (). In particular, it was shown there that the Zemach contribution for deuterium vanishes in the same way as for the muonic deuterium. Furthermore, the higher-order terms in the expansion of Eq. (35) give rise to a correction

 (37)

which contributes 0.106 kHz to the transition in ordinary deuterium friar:97:b (). Another important correction is the one due to the magnetic suceptibility friar:97:b (),

 E(5)pol3=mα2ϕ2(0)23⟨ϕN∣∣∣→μ1(HN−EN)′[−16+ln2(HN−EN)m]→μ∣∣∣ϕN⟩, (38)

where is the magnetic moment operator divided by an elementary charge. It leads to a correction of  kHz to the transition in D friar:97:b ().

There were further corrections to the electric dipole polarizability considered in Ref. friar:97:b (). However, we are convinced that they were not treated correctly and, moreover, that there are many more relativistic corrections of the same order. For this reason we disregard the additional corrections from Ref. friar:97:b () and assume the total polarizability correction to be the sum of Eqs. (36), (37), and (38). Specifically, the result for the nuclear polarizability to the transition in electronic deuterium is

 E(5)pol(2S\rm--1S,D)=19.06(20)kHz. (39)

Adding the individual nucleon part contribution of  kHz, we obtain the total two-photon exchange nuclear structure contribution of

 E(5)(2S\rm--1S,D)=19.21(20)kHz, (40)

which could be compared with the sum of the nuclear polarizability correction and the third Zemach contribution from Ref. mohr:16:codata (),  kHz, perfect agreement of the numerical values being probably accidental.

## V Three-photon exchange elastic contribution

This contribution has been studied by different methods and a number of authors, of note analytically by Friar in Ref. friar:79:ap (), and numerically by solving the Dirac equation in the field of finite size nucleus indelicato:13 (). Here we present an alternative analytical approach, which leads to much simpler analytic formulas. A numerical verification of our formulas is given in Appendix B.

In the standard analytic approach, one applies the perturbation theory to the Dirac energies with the perturbing potential where and is the Coulomb potential from the finite size nucleus,

 δ(1)E = ⟨¯ψ|δV|ψ⟩ (41) δ(2)E = ⟨¯ψ|δV1(⧸p−γ0V0−m)′δV|ψ⟩ (42) δ(3)E = ⟨¯ψ|δV1(⧸p−γ0V0−m)′δV1(⧸p−γ0V0−m)′δV|ψ⟩ (43)

One can use the exact Dirac wave function and the reduced Dirac propagators to calculate the correction to the finite nuclear size friar:79:ap (), which we call the elastic three-photon exchange correction. However, we will not use the above formulas but employ a different approach, which we call the scattering amplitude approach. In this approach, the relativistic correction to the finite nuclear size is induced by the elastic three photon exchange. The corresponding correction can be divided into the low and the high energy momentum exchange parts, . These parts are calculated as follows.

### v.1 Three-photon exchange: low energy part

The low-energy part is again split into two parts

 EL = EL1+EL2, (44) EL1 = ⟨δV1(E−H)′δV⟩+⟨δ(2)V⟩, (45) EL2 = ⟨ϕ|18m2∇2(δV)+14m2→σ⋅→∇(δV)×→p|ϕ⟩ (46) +2⟨ϕ|δV1(E−H)′[−p48m3+πZα2m2δ3(r)]|ϕ⟩,

where is the nonrelativistic contribution proportional to , and is the relativistic part proportional to . All these matrix elements are calculated in dimensions. The following results are obtained for the states,

 EL1(nS) = (47) +(Zα)3[−1ϵ+4ln(Zα)]4[ρ′(0)]2⟨πδ(d)(r)⟩+4(Zα)6ρ′′(0)n5, EL2(nS) = ρ′(0)4(Zα)6n3[94n2−1n−52+γ−lnn2+Ψ(n)] (48) +(Zα)3[−1ϵ+4ln(Zα)]ρ′(0)⟨πδ(d)(r)⟩,

and for the states,

 EL(nP1/2) = (Zα)6(−94ρ′(0)+3ρ′′(0))R′n1(0)2, (49) EL(nP3/2) = (Zα)63ρ′′(0)R′n1(0)2, (50)

where

 R′n1(0)2=49n3(1−1n2). (51)

For all higher- states vanishes.

### v.2 Three-photon exchange: high energy part

We start by introducing the two potentials in -dimensions that will appear in the evaluation of the high energy-part ,

 Vd(r) = 4π∫ddq(2π)dei→q⋅→rρ(q2)q2, (52) V(2)d(r) = 4π∫ddq(2π)dei→q⋅→rρ(q2)q4. (53)

Their large asymptotics are

 Vd(r) = V(r)+\rm local terms, (54) V(2)d(r) = V(2)(r)+ρ′(0)V(r)+\rm local % terms, (55)

and in ,

 V(r) = 1r+\rm local terms, (56) V(2)(r) = −r2+ρ′(0)r+\rm local terms, (57)

where the local terms vanish outside the nucleus.

Now we proceed to the derivation of the high-energy part . It is given by the three-photon scattering amplitude with momenta ,

 EH = −(4πZα)3ϕ2(0)∫ddq1(2π)d∫ddq2(2π)dρ(q21)q41ρ(q22)q42ρ((→q1−→q2)2)(→q1−→q2)2 (58) ×Tr[(⧸p1+m)γ0(⧸p2+m)(γ0+I)4].

The above trace equates to , so we can split into the nonrelativistic and relativistic parts,

 EH=EH1+EH2. (59)

The nonrelativistic part is

 EH1 = −(4πZα)3ϕ2(0)4m2∫ddq1(2π)d∫ddq2(2π)dρ(q21)q41ρ(q22)q42ρ((→q1−→q2)2)(→q1−→q2)2 (60) = −ϕ2(0)(Zα)34m2∫ddrVd(r)[V(2)d(r)]2.

In order to calculate this integral, we split the integration region into and . The first integral is finite in but diverges at large , and in the second integral one can use the asymptotic form of potentials,

 EH1 = −ϕ2(0)(Zα)34m2{4π∫Λdrr2V(r)[V(2)(r)]2+∫ΛddrV(r)[V(2)(r)+ρ′(0)V(r)]2} (61) = ϕ2(0)(Zα)34m24π∫∞drln(r/rC)ddrr3{V(r)[V(2)(r)]2−1r(r2−ρ′(0)r)2} −ϕ2(0)(Zα)34m2ρ′(0)2{4πln(Λ/rC)+∫ΛddrV(r)3}.

The expression under the first integral is a local function of , so this integral is effectively over the nuclear size, which allows us to introduce an effective nuclear radius as

 lnrC1rC+2=36r4C∫∞drln(r/rC)ddrr3{V(r)[V(2)(r)]2−1r(r2−ρ′(0)r)2}. (62)

So, the first correction is represented in the following form

 EH1=4πϕ2(0)(Zα)34m2r4C36[14ϵ+52+γ+ln(rC1m)]. (63)

The relativistic part is

 EH2 = −(4πZα)3ϕ2(0)∫ddq1(2π)d∫ddq2(2π)dρ(q21)q41ρ(q22)q42ρ((→q1−→q2)2)(→q1−→q2)2→q1→q2 (64) = ϕ2(0)(Zα)3∫ddr{2πρ(r)V(2)d(r)−[Vd(r)]2}V(2)d(r),

and we proceed in a similar way as in the case of , namely

 EH2 = ϕ2(0)(Zα)3{4π∫Λdrr2[2πρ(r)V(2)(r)−[V(r)]2]V(2)(r) (65) −∫Λddr[V(r)]2(V(2)(r)+ρ′(0)V(r))} = −ϕ2(0)(Zα)34π∫∞drln(r/rC)ddrr3[2πρ(r)[V(2)(r)]2−[V(r)]2V(2)(r) −1r2(r2−ρ′(0)r)]−ϕ2(0)(Zα)3ρ′(0)[4πln(Λ/rC)+∫Λddr[V(r)]3].

The expression under the first integral is a local function, so we can introduce the second effective nuclear radius as

 (66)

So, the second correction is given by

 EH2=−4π(Zα)3ϕ2(0)r2C6[14ϵ−12+γ+ln(rC2m)]. (67)

### v.3 Three-photon elastic exchange: total result

The complete finite nuclear size correction for an arbitrary nucleus is given by the sum , with the result

 E(6)fns(nS) = −(Zα)6m3r2C23n3[94n2−3−1n+2γ−lnn2+Ψ(n)+ln(mrC2Zα)] (68) +(Zα)6m5r4C49n3[−1n+2+2γ−lnn2+Ψ(n)+ln(mrC1Zα)] +(Zα)6m5r4CC115n5, E(6)fns(nP1/2) = (Zα)6m(m2r2C6+m4r4CC45)1n3(1−1n2), (69) E(6)fns(nP3/2) = (70) E(6)fns(nLJ) = 0\rm forL>1, (71)

where and the effective nuclear charge radii and defined by Eqs. (62) and (66) encode the high-momentum contributions and are expected to be of the order of . Equations (68)-(71) are valid both for electronic and muonic atoms. However, in the case of the electronic atoms, the terms proportional to