Abstract
We report calculations of the one-loop self-energy correction to the bound-electron g factor of the 1s and 2s states of light hydrogen-like ions with the nuclear charge number Z ≤ 20. The calculation is carried out to all orders in the binding nuclear strength. We find good agreement with previous calculations and improve their accuracy by about two orders of magnitude.
Abstract
The bound-electron g factor in light hydrogen-like and lithium-like ions has been measured with a high accuracy, which reached to 3 × 10 −11 in the case of C 5+ [1] . Such measurements have yielded one of the best tests of the boundstate QED theory [2] and significantly improved the precision of the electron mass [1, 3] . Further advance of the experimental accuracy toward the 10 −12 level is anticipated in the near future [4] .
Abstract
One of the dominant effects in the bound-electron g factor is the one-loop electron self-energy. Its contribution to the total g factor value is so large that the effect needs to be calculated to all orders in the nuclear binding strength parameter Zα even for ions as light as carbon (Z is the nuclear charge number, α is the fine-structure constant). The numerical error in the evaluation of the electron self-energy is currently the second-largest source of uncertainty for the hydrogen-like ions (the largest error stemming from the two-loop electron self-energy [5, 6] ). The error needs to be descreased in order to match the anticipated experimental precision.
Abstract
The numerical accuracy of the one-loop self-energy is also relevant for the determination of the electron mass [1, 3] . The self-energy values actually used in the electron-mass determinations were obtained by an extrapolation of the high-and medium-Z numerical results down to Z = 6 (carbon) and 8 (oxygen). Clearly, this situation is not fully satisfactory and a direct numerical calculation would be preferable.
Abstract
All-order (in Zα) calculations of the electron self-energy to the bound-electron g factor have a long history. First calculations of this correction were accomplished two decades ago [7] [8] [9] . The numerical accuracy of these evaluations was advanced in the later works [10, 11] , which was crucial at the time as it brought an improvement of the electron mass determination. This correction was revisited again in Refs. [12, 13] . In the present work, we aim to advance the numerical accuracy of the one-loop electron self-energy and bring it to the level required for future experiments.
Abstract
We consider the one-loop self-energy correction to the g factor of an electron bound by the Coulomb field of the point-like and spinless nucleus. This correction can be represented [10, 11] as a sum of the irreducible (ir) and the vertex+reducible (vr) parts,
Abstract
The irreducible part is
Abstract
where Σ(ε) = Σ(ε) − δm is the (renormalized) one-loop selfenergy operator (see, e.g., [13] ) and |δ g a is the perturbed wave function
Abstract
with δV g = 2m[r × α] z being the effective g-factor operator
Abstract
[13] that assumes that the spin projection of the reference state is m a = 1/2. The vertex+reducible part is
Abstract
where I(ω) is the operator of the electron-electron interaction (see, e.g., [11] ), ω is the energy of the virtual photon, ∆ ab = ε a − ε b , and a proper covariant identification and cancellation of ultraviolet and infrared divergences is assumed. The integration contour C in Eq. (4) is the standard Feynman integration contour; it will be deformed for a numerical evaluation as discussed below. The vertex+reducible contribution is further divided into three parts: the zero-potential, one-potential, and manypotential contributions, ∆g vr = ∆g 
Abstract
This separation is induced by the following identity, which splits the integrand according to the number of interactions with the binding Coulomb field in the electron propagators,

The bound-electron g factor in light hydrogen-like and lithium-like ions has been measured with a high accuracy, which reached to 3 × 10 −11 in the case of C 5+ [1] . Such measurements have yielded one of the best tests of the boundstate QED theory [2] and significantly improved the precision of the electron mass [1, 3] . Further advance of the experimental accuracy toward the 10 −12 level is anticipated in the near future [4] .

One of the dominant effects in the bound-electron g factor is the one-loop electron self-energy. Its contribution to the total g factor value is so large that the effect needs to be calculated to all orders in the nuclear binding strength parameter Zα even for ions as light as carbon (Z is the nuclear charge number, α is the fine-structure constant). The numerical error in the evaluation of the electron self-energy is currently the second-largest source of uncertainty for the hydrogen-like ions (the largest error stemming from the two-loop electron self-energy [5, 6] ). The error needs to be descreased in order to match the anticipated experimental precision.

The numerical accuracy of the one-loop self-energy is also relevant for the determination of the electron mass [1, 3] . The self-energy values actually used in the electron-mass determinations were obtained by an extrapolation of the high-and medium-Z numerical results down to Z = 6 (carbon) and 8 (oxygen). Clearly, this situation is not fully satisfactory and a direct numerical calculation would be preferable.

All-order (in Zα) calculations of the electron self-energy to the bound-electron g factor have a long history. First calculations of this correction were accomplished two decades ago [7] [8] [9] . The numerical accuracy of these evaluations was advanced in the later works [10, 11] , which was crucial at the time as it brought an improvement of the electron mass determination. This correction was revisited again in Refs. [12, 13] . In the present work, we aim to advance the numerical accuracy of the one-loop electron self-energy and bring it to the level required for future experiments.

We consider the one-loop self-energy correction to the g factor of an electron bound by the Coulomb field of the point-like and spinless nucleus. This correction can be represented [10, 11] as a sum of the irreducible (ir) and the vertex+reducible (vr) parts,

The irreducible part is

where Σ(ε) = Σ(ε) − δm is the (renormalized) one-loop selfenergy operator (see, e.g., [13] ) and |δ g a is the perturbed wave function

with δV g = 2m[r × α] z being the effective g-factor operator [13] that assumes that the spin projection of the reference state is m a = 1/2. The vertex+reducible part is

where I(ω) is the operator of the electron-electron interaction (see, e.g., [11] ), ω is the energy of the virtual photon, ∆ ab = ε a − ε b , and a proper covariant identification and cancellation of ultraviolet and infrared divergences is assumed.

The integration contour C in Eq. (4) is the standard Feynman integration contour; it will be deformed for a numerical evaluation as discussed below. The vertex+reducible contribution is further divided into three parts: the zero-potential, one-potential, and manypotential contributions,

This separation is induced by the following identity, which splits the integrand according to the number of interactions with the binding Coulomb field in the electron propagators,

where

is the free-electron propagator, and

is the one-potential electron propagator.

In the present work, we will be concerned mainly with the numerical evaluation of ∆g

vr , since all other contributions were computed to the required accuracy in our previous investigations [11, 13] .

After performing integrations over the angular variables analytically as described in Ref. [11] , we obtain the result that can be schematically represented as

where x, y, and z are the radial integration variables, |κ| is the absolute value of the angular momentum-parity quantum number of one of the electron propagators, and f |κ| is the integrand. Summations over other angular quantum numbers are finite and absorbed into the definition of f |κ| .

The approach of the present work is to split ∆g

where κ a is an auxiliary parameter and C LH,a and C LH,b are two integration contours used for the evaluation of the two parts of Eq. (8) . In the present work we used κ a = 120, which corresponds to the maximal value of |κ| used in Ref. [13] , and C LH,a being the same contour as used in that work. So, the numerical evaluation of ∆g (2+) vr,a was mostly analogous to the one reported in Ref. [13] , but we had to improve the accuracy of numerical integrations by several orders of magnitude. In the updated numerical integrations, the extended Gauss-log quadratures [14] were employed, alongside with the standard Gauss-Legendre quadratures.

We found it impossible to extend the partial-wave expansion significantly beyond the limit of κ a = 120 within the same numerical scheme as used in Ref. [13] . The reason is that the integration contour C LH,a used there, as well as in our previous works [10, 11] , involved computations of the Whittaker functions of the first kind M α,β (z) and their derivatives for large complex values of the argument z. The algorithms we use [15] for computing M α,β (z) become unstable for large α (needed for large κ's) and large and complex z, even when using the quadruple-precision arithmetics. For this reason, in order to compute ∆g (2+) vr,b , we had to switch to the contour C LH,b , which was originally introduced by P. J. Mohr in his calculations of the one-loop self-energy [16, 17] . The crucial feature of this contour is that it involves the computation of the Whittaker functions M α,β (z) and W α,β (z) of the real arguments z only. For real arguments, the computational algorithms were shown [17] to be stable even for very large κ's (and, hence, α's). Specifically, the contours C LH,a and C LH,b consist of two parts, the low-energy and the high-energy ones. The lowenergy part extends along (∆, 0) on the lower bank of the cut of the photon propagator of the complex ω plane and along (0, ∆) on the upper bank of the cut. The high-energy part consists of the interval (∆, ∆ + i∞) in the upper half-plane and the interval (∆, ∆ − i∞) in the lower half-plane. The difference between C LH,a and C LH,b is only in the choice of the parameter ∆. For C LH,a , we use ∆ = Zα ε a (the same choice as in our previous works [10, 11, 13] ), whereas for C LH,b , we use ∆ = ε a (the Mohr's choice). Detailed discussion of the integration contour and the analytical properties of the integrand can be found in the original work [16] .

We found that the price to pay for using the contour C LH,b was the oscillatory behavior of the integrand as a function of the radial variables for ω ∼ ε a . Because of this, we had to employ very dense radial grids for numerical integrations, which made computations rather time-consuming.

The largest error of the numerical evaluation of Eq. (8) comes from the termination of the infinite summation over |κ| and the estimation of the tail of the expansion. In the present work, we performed the summation over |κ| before all integrations and stored the complete sequence of partial sums, to be used for the extrapolation performed on the last step of the calculation. The convergence of the expansion was monitored; in the cases when the series converged to the prescribed accuracy (i.e., the relative contribution of several consecutive expansion terms was smaller than, typically, 10 −11 for ∆g (2+) vr,a and 10 −6 for ∆g

vr,b ), the summation was terminated. This approach reduced the computation time considerably as compared to our previous scheme [11] , where the summation over |κ| was performed after all integrations. If the convergence of the partial-wave expansion had not been reached, the summation was extended up to the upper cutoff κ max = 450.

The remaining tail of the series was estimated by analyzing the |κ|-dependence of the partial-wave expansion terms after all integrations. We fitted the last m expansion terms (typically, m = 20) to the polynomial in 1/|κ| with 1-3 fitting parameters,

The uncertainty of the extrapolation was estimated by varying the cutoff parameter κ max by 20% and multiplying the resulting difference by a conservative factor of 1.5. This procedure usually led to the expansion tail estimated with an accuracy of about 10%. We observed an interesting feature, namely, that the tail of the expansion, with a high accuracy, is the same for the 1s and for the 2s states. E.g., for Z = 4, we find the expansion tail of δg(1s) = −1.88 (19) × 10 −12 and δg(2s) = −1.88 (19) × 10 −12 ; for Z = 16, we obtain δg(1s) = −3.00 (27) × 10

and δg(2s) = −3.01 (27) × 10 −11 . We do not know the reason for this but such an agreement shows a high degree of consistency of our numerical calculations for the 1s and 2s states.

Our numerical results for the self-energy correction to the bound-electron g factor of the 1s and 2s states of hydrogenlike ions are presented in Table I . The values for the irreducible part ∆g ir are taken from our previous investigations (from Ref. [13] for Z ≤ 12 and from Ref. [11] otherwise). Using results of Ref. [11] , we introduced small corrections that accounted for a different value of the fine-structure constant used in that work. In Table I we also present values of the higher-order remainder function H(Zα), obtained after separating out all known terms of the Zα expansion [5, 6] from our numerical results, In the present work, we performed direct numerical calculations for ions with Z ≥ 4. For smaller Z, numerical cancelations in determining the higher-order remainder become too large to make numerical calculations meaningful. Instead of direct calculations, we extrapolated the numerical values presented in Table I for H(Zα) down towards Z → 0. Doing this, we assumed the following ansatz for H(Zα), which was inspired by the expansion of the one-loop self-energy for the Lamb shift,

For the 2s -1s difference, we use the form (10) with c 12 = 0, assuming the leading logarithm to be state-independent. The extrapolated results are presented in Table III . The uncertainties quoted for our fitting results are obtained under the assumption that the logarithmic terms in the next-to-leading order of the Zα expansion of H(Zα) comply with Eq. (10). If we introduce, e.g., a cubed logarithmic term into Eq. (10), our estimates of uncertainties would increase by about a factor of 2. In summary, we reported calculations of the one-loop selfenergy correction to the bound-electron g factor of the 1s and 2s state of light hydrogen-like ions, performed to all orders in the binding nuclear strength parameter Zα. The relative accuracy of the results obtained varies from 1 × 10 −10 for Z = 4 to 3 × 10 −9 for Z = 20. Our results agree well with the previously published values but their accuracy is by about two orders of magnitude higher. [9] a extrapolation of the numerical data from [10] . 

{'paper_id': '118917758', 'title': 'One-loop electron self-energy for the bound-electron $g$ factor', 's2_url': 'https://api.semanticscholar.org/CorpusID:118917758', 'source': 'S2ORC'}