Hadronic vacuum polarization correction to the bound-electron g factor

The hadronic vacuum polarization correction to the g factor of a bound electron is investigated theoretically. An eﬀective hadronic Uehling potential obtained from measured cross sections of e − e + annihilation into hadrons is employed to calculate g factor corrections for low-lying hydrogenic levels. Analytical Dirac-Coulomb wave functions, as well as bound wave functions accounting for the ﬁnite nuclear radius are used. Closed formulas for the g factor shift in case of a point-like nucleus are derived. In heavy ions, such eﬀects are found to be much larger than for the free-electron g factor.


I. INTRODUCTION
Precision Penning-trap experiments on the g factor of hydrogenlike and few-electron highly charged ions allow a thorough testing of quantum electrodynamics (QED), a cornerstone of the standard model describing electromagnetic interactions.The g factor of hydrogen-like silicon (Z = 14) has been measured with a 5 × 10 −10 relative uncertainty [1,2], allowing to scrutinize bound-state QED theory (see e.g.[3][4][5][6][7][8][9][10][11][12][13]).Two-loop radiative effects and shifts due to nuclear structure and recoil are observable in such measurements.The high accuracy which can be achieved on the experimental as well as theoretical side also enables the determination of fundamental physical constants such as the electron mass m e [14][15][16][17][18]. Recently, it was shown that g factor studies can also help in the search for new physics, i.e. the coupling strength of a hypothetical new interaction can be constrained through the comparison of theoretical and experimental results [12,19,20].
Further improved tests and possible determinations of fundamental constants [21][22][23] call for an increasing accuracy on the theoretical side.The evaluation of twoloop terms up to order (Zα) 5 (with Z being the atomic number and α the fine-structure constant) has been finalized recently [24,25], increasing the theoretical accuracy especially in the low-Z regime.First milestones have been also reached in the calculation of two-loop corrections in stronger Coulomb fields, i.e. for larger values of Zα [26,27].As the experiments are advancing towards heavy ions [28,29], featuring smaller and smaller characteristic distance scales for the interaction between the bound electron and the nucleons, the effects of other forces may need to be considered as well.
Motivated by these prospects, in this article we investigate vacuum polarization (VP) corrections due to the virtual creation and annihilation of hadrons.The dominant VP contribution arises from virtual e − e + pair creation, which has been widely investigated in the lit-erature [5,6,30,31] and is well understood.The other leptonic VP effect is due to virtual muons, the contribution of which is suppressed by the square of the electronto-muon mass ratio [32].The hadronic VP effect, which arises due to a superposition of different virtual hadronic states, is comparable in magnitude to muonic VP, however, it requires a completely different description since the virtual hadrons interact via the strong force.An effective approach to take into account such effects for the free-electron g factor is described in e.g.Ref. [33], in which hadronic VP is characterized by the cross section of hadron production via e − e + annihilation.Following this treatment, we apply the known empirical parametric hadronic polarization function for the photon propagator from Ref. [34] to account for the complete hadronic contribution in case of the bound-electron g factor.
While in case of the free electron, the hadronic correction only appears on the two-loop level, as a correction to the electrons electromagnetic self-interaction (see Fig. 1a), in case of a bound electron it appears already as a one-loop effect (see Fig. 1b).Furthermore, the hadronic VP is boosted by approximately ∼ Z 4 , i.e. by the fourth power of the nuclear charge number, and thus, as we will see later, for heavier ions above Z = 14 its contribution is larger than in case of a free electron [35].FIG.1: Feynman diagrams representing the leading hadronic VP corrections to the free-electron g factor (1a) and the bound-electron g factor (1b).Double lines represent electrons in the electric field of the nucleus and wavy lines with a triangle depict the interaction with the external magnetic field.For the free electron, it is a two-loop process where the self-interaction of the electron is perturbed by the effective hadronic polarization function (shaded bubble).For the bound electron, it is a one-loop correction where the Coulomb interaction with the nucleus (cross) is perturbed by the effective hadronic polarization function.

arXiv:2303.07973v1 [hep-ph] 14 Mar 2023
An effective potential constructed from the parametrized VP function, the hadronic Uehling potential, has been derived in Ref. [36].We calculate the perturbative correction to the g factor due to this radial potential employing analytical Dirac-Coulomb wave functions, as well as numerically calculated wave functions accounting for a finite-size nucleus.Analytical formulas are presented, and numerical results are given for hydrogenic systems from H to U 91+ .We note that such an approach assumes an infinitely heavy nucleus, i.e. nuclear recoil effects are excluded in our treatment.
We use natural units with = c = 1 for the reduced Planck constant and the speed of light c, and α = e 2 , where α is the fine-structure constant and e is the elementary charge.Three-vectors are denoted by bold letters.

II. g FACTOR CORRECTIONS
Generally speaking, the g factor describes the coupling of the electron's magnetic moment µ to its total angular momentum J .The corresponding first-order Zeeman splitting ∆E due to the electron's interaction with an external homogeneous magnetic field B is where µ B = e/(2m e ) is the Bohr magneton of the electron and g is its g factor, which depends on the electron configuration.
On the other hand, the relativistic interaction of an electron with the external magnetic field can be derived from the minimal coupling principle in the Dirac equation.In first-order perturbation theory, this leads to the energy shift where α are the usual Dirac matrices given in terms of the gamma matrices by α i = γ 0 γ i [37] and A is the vector potential for the magnetic field, such that B = ∇ × A. Choosing the magnetic field to be directed along the z axis, one can see that a possible choice for the vector potential is A = [B × r]/2, where r is the position vector.Together with Eq. ( 1) and ( 2), one can derive the following general expression for the g factor [30]: where n is the principal quantum number of the bound state, j = |κ| − 1/2 is the total angular momentum quantum number and κ is the relativistic angular momentum quantum number.The functions G nκ (r), F nκ (r) are the radial components in the electronic Dirac wave function, where m is the magnetic quantum number and r = |r|.
The spherical spinors Ω ±κm (θ, ϕ) make up the angular components and are the same for any central potential V (r) [38].
A straightforward approach for calculating the g factor shift ∆g VP due to vacuum polarization (VP), is to solve the radial Dirac equation numerically with the inclusion of the VP effect, and then substituting the perturbed functions G VP nκ (r), F VP nκ (r) into Eq.( 3).The difference between the pertubed and the unperturbed g factor gives the corresponding shift ∆g VP = g VP − g . (5) However, we will apply a different method to investigate the hadronic g factor shift.As shown in Ref. [31], owing to the properties of Dirac wave functions, the g factor in Eq. ( 3) can be expressed through the energy eigenvalues E nκ , if the potential V (r) does not depend on the electron mass m e .This formula was used successfully, e.g., to investigate the finite nuclear size effect in Ref. [31].We apply this new approach to investigate the vacuum polarization effect, described by an effective potential.Having a small perturbation δV (r) to the nucleus potential (like the hadronic Uehling potential [36]), the g factor shift can be shown to be [31] ∆g VP = − κ 2 j(j + 1)m e r ∂δV (r) ∂r . ( For the relativistic ground state and a point-like nucleus, this expectation value can be evaluated further to obtain where γ = 1 − (Zα) 2 and ∆E 1s = δV 1s is the corresponding energy shift in first-order perturbation theory.
Since the second term on the right-hand side of Eq. ( 8) is Zα times smaller than the first term, the g factor shift can be approximated for light ions (Zα 1) with the formula: A similar expression also appeared in Ref. [23,31] in a different context, studying the finite size effect.However, we will investigate the applicability of this formula as an approximation for calculating the g factor shift due to VP effects for light ions.A. Leptonic vacuum polarization correction to the g factor The leptonic VP correction to the bound-electron g factor is well known.The corresponding diagrams are shown in Fig. 2 and can be divided into two groups: the electric loop (EL) and the magnetic loop (ML) contribution.The vacuum polarization effect in the EL contribution (Fig. 2a and Fig. 2b) is equivalent to a perturbation in the interaction between the bound electron and the nucleus, and thus can be described by an effective perturbing potential δV EL (r).This allows the usage of perturbation theory and the simple inclusion of hadronic VP effects to the bound-electron g factor shift, using Eq. ( 7).As can be seen in Ref. [6,39], the ML contribution (Fig. 2c) is Zα times smaller than EL in the leading order, and is not the subject of the current work.
The vacuum loop in the EL contribution can be expanded in powers of the nuclear coupling strength Zα, which corresponds to a free loop interacting with the nucleus.Due to Furry's theorem, only odd powers of Zα contribute [39,40].The leading term in this expansion is described by the Uehling potential δV Ue (r) and the contributions of higher order in Zα are summarized to the Wichmann-Kroll potential δV WK (r), such that the effective perturbing potential is given by δV EL (r) = δV Ue (r) + δV WK (r) [39].The diagrams in Fig. 2a and Fig. 2b contribute equally to the EL correction.In this paper, we will investigate the leading contribution to the vacuum polarization due to the Uehling potential: δV EL (r) ≈ δV Ue (r).
In case of leptonic vacuum loops, the well-known leptonic Uehling potential is given by [41] δV Ue where ρ(x) denotes the nuclear charge distribution normalized to unity, m l is the mass of the virtual particle in the fermionic loop and K 1 (x) is given by The g factor shift of a bound electron in the ground state can be calculated analytically for a point-like nucleus and was already derived in [30].We will show that one arrives to the same result using the approach in Eq. (7).Using the leptonic Uehling potential for a point-like nucleus (ρ(x) = δ (3) (x)) [30], and the radial components of the electronic wave function in the ground state [32], one obtains from Eq. ( 7) I abc is a modification of the base integral given in Ref. [30], see Appendix A, and s = m e /m l is the ratio of the electron and the loop particle masses.
The leading order Zα expansion is given by For s = 1, this is exactly the same result as in Ref. [30], however, obtained with a different method.In the case of muonic VP, thus s = m e /m µ , the results for a finite size nucleus were obtained numerically in Ref. [39].
In the next Subsection, we will use this approach to derive an analytic expression for the hadronic VP correction to the bound-electron g factor.

B. Hadronic vacuum polarization correction to the g factor
As discussed in [33,34,36], the hadronic vacuum polarization function can be constructed semi-empirically from experimental data of e − e + annihilation cross sections.The whole hadronic polarization function is parametrized for seven regions of momentum transfer and is given e.g. in Ref. [34].In Ref. [36], it was found that only the first region of parametrization is significant for the hadronic energy shift calculations.This is also clear from the physical point of view, since atomic physics is dominated by low energies around eV -keV.Thus, we will use the analytic hadronic Uehling potential introduced in Ref. [36] for our calculations.For a point-like nucleus it is given by with the coefficients B 1 = 0.0023092 and C 1 = 3.9925370 GeV −2 [34,36] and the exponential integral E 1 (x) which can be generalized for n = 0, 1, 2, ... by [42] E The values for B 1 and C 1 are taken from the most recent parametrization in Ref. [34] and will be used for the calculations.The error of numerical results is estimated by comparison with an older parametrization in Ref. [43] like has been done in Ref. [36].
The corresponding hadronic Uehling potential for an extended nucleus with spherical charge distribution ρ(x) is obtained by the convolution [36] δV had.fns (r) = d 3 x ρ(x) δV had.point (r − x) where x = |x| and As in our previous work [36], we will consider the homogeneously charged sphere as the model for the extended nucleus with root-mean-square (RMS) radii taken from Ref. [44].The charge distribution ρ(r) is given by where θ(x) is the Heaviside step function and the effective radius R is related to the RMS nuclear charge radius R rms via R = 5/3 R rms .The correspondig hadronic Uehling potential is given analytically in [36], see Appendix B.
Let us turn to the evaluation of the leading hadronic VP contribution to the bound-electron g factor, depicted in Fig. 1b.In the low-energy limit, the hadronic Uehling potential is given by [45] Using Eq. ( 7) and the non-relativistic expectation value of the delta function, the leading order in Zα of the hadronic g factor shift for general ns states is found to be [46] ∆g had.non−rel.(ns For the 1s state, a fully relativistic expression for the point-like nucleus can be given.Using the hadronic Uehling potential in Eq. ( 15) and the relativistic wave function of the ground state, one obtains with Eq. ( 7): where λ = Zαm e and ∆E had.point (1s) is the analytical energy shift for a point-like nucleus given in Ref. [36], with 2 F 1 (a, b; c; z) being the hypergeometric function [42].The expansion of this expression up to 6 th order in Zα is given by and it coincides with the non-relativistic approximation in Eq. ( 21) to order (Zα) 4 .
A similar relativistic calculation for the 2s state yields The leading orders of Eq. ( 24) and Eq. ( 25) satisfy the non-relativistic relationship in Eq. ( 21 C. Hadronic vacuum polarization correction to the reduced g factor Additionally, we investigate hadronic effects on the weighted difference of the g factor and the bound-electron energy E of H-like ions, called reduced g factor, put forward in Ref. [23] for a possible novel determination of the fine-structure constant, and for testing physics beyond the standard model [20].It was shown there that the detrimental nuclear structure contributions featuring uncertainties can be effectively suppressed in the above combination of the g factor and level energy of the hydrogenic ground state.The question arises whether the same can be said about the hadronic VP corrections investigated in the present article. The hadronic VP correction to the reduced g factor for a point-like nucleus can be found analytically using Eq. ( 22) and Eq. ( 23).The leading order Zα expansion is given by ghad.
point (1s) = ∆g had.point (1s) − Thus, the leading term of order (Zα) 4 in ∆g had.point (1s) cancels such that the hadronic VP contribution to the reduced g factor is indeed small for practical purposes.This also supports the approximation in Eq. ( 9).Therefore, we may conclude that hadronic effects do not hinder the extraction of α or detailed tests of QED and standard model extensions via the measurement of g.

D. Hadronic vacuum polarization correction to the weighted g factor difference of H-and Li-like ions
Another quantity of interest is the weighted difference of the g factors of the Li-like and H-like charge states of the same element, where g(2s) is the g factor of the Li-like ion and g(1s) is the g factor of the H-like ion.For light elements, the parameter Ξ can be calculated to great accuracy by [21,47] This weighted (or specific) difference was introduced to suppress uncertainties arising from the nuclear charge radius and further nuclear structural effects [48].Therefore, bound-state QED theory can be investigated more accurately in g factor experiments combining H-and Lilike ions than with the individual ions alone.
As we have seen, the hadronic VP correction to δ Ξ g for a point-like nucleus can be found analytically.We approximate ∆g had.point (2s) of the Li-like ion with the expression in Eq. ( 25) for the H-like ion.Since there are no electron-electron interactions in this approximation, we have to neglect the terms of relative orders 1/Z and 1/Z 2 in Eq. (30).We note that the residual weight exactly cancels the first two leading orders (Zα) 4 and (Zα) 5 : Therefore, we can conclude that hadronic VP effects are also largely cancelled in the above specific difference.A similar conclusion can be drawn for the case of the specific difference introduced for a combination of H-and B-like ions [22].This result is well understood, since nuclear and hadronic VP contributions are both short-range effects with a similar behavior.

III. NUMERICAL RESULTS
As mentioned in Ref. [36], the hadronic VP contribution to the energy shift is about 1/0.665 ≈ 1.5 times smaller than the muonic VP contribution in the case of the Uehling term.This can be also confirmed for the g factor shift.Comparing the non-relativistic approximation for the hadronic g factor shift ∆g had.VP non−rel.,point in Eq. ( 20) with the first term of the expression for the muonic g factor shift ∆g muonic VP non−rel.,point in Eq. ( 14), yields for hydrogen in the ground state ∆g had.VP non−rel.,point(1s) = −1.092(14) × 10 −16 = 0.664(9) ∆g muonic VP non−rel.,point(1s) .
The values for the hadronic g factor shift with an extended nucleus were calculated numerically using two different methods, both yielding the same results within the given uncertainties.The first method consists of calculating the expectation value in Eq. (7) with the FNS hadronic Uehling potential and the semi-analytic wave functions of a homogeneously charged spherical nucleus given in Ref. [49].As a consistency check, these results were reproduced by using the approach of solving the radial Dirac equation numerically with the inclusion of the FNS potential, and substituting the resulting large and small radial wave function components into Eq.( 3) and Eq. ( 5).The results for the hydrogenlike systems H, Si, Ca, Xe, Kr, W, Pb, Cm and U are given in Table I.A diagrammatic representation is shown in Fig. 3.We note that for Z = 14 and above, the magnitude of the hadronic vacuum polarization terms considered in this work exceed in magnitude the hadronic contribution to the free-electron g factor [50].However, it is important to mention that the uncertainty of the leading finite nuclear size correction to the g factor is approximately an order of magnitude larger than the hadronic VP effect for all elements considered (see e.g.[23]), hindering the identification of the effect.
The errors given in Table I and II are based on the uncertainty of the nuclear root-mean-square radii R rms given in Ref. [44] and an uncertainty for the parameters 1 and C 1 as described in Section II B. The total error is dominated by the assumed uncertainty of B 1 and C 1 .Owing to the closed analytical expression for the hadronic Uehling potential, numerical uncertainties are negligible.For the results ∆g had.approx,fns using the approximate formula in Eq. ( 9), the hadronic energy shifts ∆E approx rel.,fns from Ref. [36] and their respective uncertainties are utilized.For Z = 92, the hadronic energy shift, which is not given in Ref. [36], was calculated using the same method.
One can see that the non-relativistic approximation in Eq. ( 21) represents a lower bound for the hadronic g factor shift and is not sufficient for large atomic numbers Z.On the other hand, the analytic expression for the relativistic g factor shift in case of a point-like nucleus in Eq. ( 22) represents an upper bound and differs also significantly from the numerical results for extended nuclei.We conclude that the effects due to a finite size nucleus need to be included in a precision calculation of the hadronic VP effect.At the present time, the uncertainty stemming from the assumed nuclear charge distribution model limits the accuracy to about 1% [36].At the same time, the absence of more precise parametrizations of the hadronic polarization function in the low-energy regime limits the accuracy also to about 1%, see Table I.Thus, the given errors include, to a great part, all possible limitations of the uncertainty of the hadronic g factor shift.
The simple approximate formula in Eq. ( 9) is found to be a good approximation for atomic numbers below Z = 14.The error is less than 1% for atomic numbers up to Z = 36.
As shown in Section II C and II D, the hadronic VP contribution to the reduced and the weighted g factor in case of a point-like nucleus is at least Zα times smaller than the regular hadronic g factor shift, see Eq. ( 24).In fact, numerical results for extended nuclei confirm that the hadronic contribution to both quantities does not differ significantly from zero for small atomic numbers below Z = 36 at the current level of accuracy.To see this, note that the numerical results for the finite-size reduced and weighted g factor can be obtained from Table I and II via ghad.
Even for larger atomic numbers, hadronic effects do not constrain high-precision tests of QED via the measurement of the reduced and weighted g factor.Recently, a high-precision measurement of the g factor difference of two Ne isotopes was performed [12].It was shown that QED effects mostly cancel, whereas nuclear effects like the nuclear recoil are well observable.In the following, we investigate hadronic VP contributions to the bound-electron g factor of the isotopes 20 Ne 9+ and 22 Ne 9+ in the ground state.
First, we calculate the hadronic VP correction to the g factor difference stemming from the different nuclear size of the isotopes.Nuclear recoil effects are excluded for now, and nuclear charge radii are taken from Ref. [44].Using R rms = 3.0055 (21) fm for 20 Ne 9+ and R rms = 2.9525 (40) This is approximately a third of the hadronic contribution of the free electron given in Extended Table 1 in Ref. [12].Thus, we conclude that at the given level of accuracy, hadronic effects of the bound electron also do not hinder the precise calculation of the isotopic shift of 20 Ne 9+ and 22 Ne 9+ .
To estimate also the hadronic VP correction stemming from the different nuclear mass of the isotopes including nuclear recoil effects, we use the non-relativistic formula [35] Thus, also the nuclear recoil effect to the hadronic VP contribution cannot be resolved at the given level of accuracy.

IV. SUMMARY
Hadronic vacuum polarization corrections to the bound-electron g factor have been calculated, employing a hadronic polarization function constructed from empirical data on electron-positron annihilation into hadrons.We have found that for a broad range of H-like ions, this one-loop effect is considerably larger than hadronic VP for the free electron (see Fig. 1a).Hadronic effects will be observable in future bound-electron g factor experiments once nuclear charge radii and charge distributions TABLE I: Results for the hadronic VP contribution to the g shift of bound electron in the ground state arising from the Uehling potential in the EL diagram (Fig. 1b) using different approaches: the non-relativistic approximation ∆g had.non−rel.,point in Eq. ( 21), the relativistic formula for a point-like nucleus ∆g had.
rel.,point in Eq. ( 22), the approximate formula ∆g had.approx,fns using the hadronic energy shift with an extended nucleus from [36] in Eq. ( 9), and the full relativistic result for an extended nucleus ∆g had.rel.,fns using the analytical finite-size Uehling potential with numerical finite-size wave functions in Eq. ( 7).Root-mean-square nuclear charge radii Rrms are taken from [44].

Z
Rrms FIG.3: Comparison of analytical and numerical results for the hadronic g factor shift of the bound electron in the ground state of H-like ions with atomic numbers Z obtained in this work, see Table I.The green solid line represents the analytical expression for a point-like nucleus ∆g had.rel.,point in Eq. ( 21), while the red dashed line represents the non-relativistic expression ∆g had.non−rel.,point in Eq. (20).The full numerical results for extended nuclei ∆g had.rel.,fns (crosses) are compared to the approximation ∆g had.approx,fns in Eq. ( 9) (circles) with hadronic energy shifts for extended nuclei taken from Ref. [36].will be substantially better known.We have also found that the hadronic effect does not pose a limitation on testing QED or physics beyond the standard model, and determining fundamental constants through specific differences of g factors for different ions, or through the reduced g factor.Finally, the analytic hadronic Uehling potential proves to be very useful and can be applied to further atomic systems, e.g.positronium, or the hyperfine structure.

FIG. 2 :
FIG.2: Feynman diagrams representing the VP correction to the bound-electron g factor.Double lines represent electrons in the electric field of the nucleus and wavy lines with a triangle depict the interaction with the external magnetic field.

TABLE II :
Results for the hadronic VP contribution to the g factor shift of the bound electron in the 2s state.