Extrapolation of the Zalpha-Expansion and Two--Loop Bound--State Energy Shifts

Quantum electrodynamic (QED) effects that shift the binding energies of hydrogenic energy levels have been expressed in terms of a semi-analytic expansion in powers of Zalpha and ln[(Zalpha)^{-2}], where Z is the nuclear charge number and alpha is the fine-structure constant. For many QED effects, numerical data are available in the domain of high Z where the Zalpha expansion fails. In this Letter, we demonstrate that it is possible, within certain limits of accuracy, to extrapolate the Zalpha-expansion from the low-Z to the high-Z domain. We also review two-loop self-energy effects and provide an estimate for the problematic nonlogarithmic coefficient B_60.


Introduction
The subject of the current Letter is the investigation of QED radiative corrections in bound hydrogenlike systems which provide one of the most stringent and accurate available tests of quantum field theory and are amenable to high-precision spectroscopy on which the determination of fundamental constants is based [1]. The purpose of this investigation is twofold: first, to demonstrate that the Zα-expansion which is inherently valid only at small Z, can be extrapolated by "deferred" Padé approximants to the domain of high Z, albeit with a certain loss of accuracy in the theoretical predictions. The second purpose is to provide a brief review of logarithmic two-loop higher-order binding corrections to the Lamb shift of hydrogenic states. The calculation of these effects has recently been completed [2,3], but results have been provided only for the total effect which is the sum of the two-loop self-energy, two-loop vacuum polarization and combined effects. For a comparison to numerical calculations which are currently being pursued [4], it is helpful to analyze the coefficients that relate to specific sets of gauge-invariant diagrams. This second purpose is actually a prerequisite for carrying out the extrapolation of the two-loop self-energy from low to high Z, wherefore we commence this Letter with the endeavour of providing some "mini-review" of the two-loop hydrogenic energy shifts.

Brief Review of Two-Loop Hydrogenic Energy Shifts
We use natural Gaussian units with = c = ǫ 0 = 1 and e 2 = 4πα, as it is customary for QED bound-state calculations. The two-loop radiative shift of a hydrogenic S state, within the Zα-expansion, reads where and the ellipsis denotes higher-order terms.
This section is a brief review of the known two-loop coefficients B 40 , B 50 , B 63 , B 62 and B 61 . The Feynman diagrams which contribute to the two-loop bound-state energy shifts are shown in Fig. 1.
For S states, the contributions to B 40 can be evaluated by considering the form-factor approach described e.g. in Sec. VIII.B.1-VIII.B.3 of [5] or in Sec. 1 of [6]. Detailed information about the F ′ 1 (0) form factor slope and the magnetic form factor F 2 (0) attributable to the different sets of diagrams (2LSE) and (SVPE) shown in Fig. 1 are given in Eqs. (16), (17) and (18) of Ref. [6].
We recall the following n-independent results for the B 40 -coefficients of S states, Note that the distribution of B 40 -contributions among the different sets of diagrams in Fig. 1 [notably (2LSE) and (SVPE)] is different from the separation into a "self-energy" correction [Eq. (A24) of [1]] and a "magnetic moment contribution" [Eq. (A25) of [1]]. We also recall that the first treatment of the leading two-loop self-energy coefficient B 40 was completed in [7].
The evaluation of the relativistic correction B 50 due to the two-loop self-energy has represented a considerable challenge [8,9]. Diagrams involving a closed fermion loop were studied in [10,11]. It might be useful to point out that the contribution of the diagram (SVPE) is the sum of the contribution labeled E II and E V in Ref. [10]. The coefficients read: These results are in agreement with the data presented in Eqs. (A28)-(A29) of [1].
The coefficients of sixth order in Zα have recently been analyzed in [2] (see also the references therein). The triple logarithm B 63 originates exclusively from the set (2LSE). The diagrams of the set (SEVP) have been calculated in Sec. VII of [2], and the results for the double and single logarithms originating from these diagrams can be obtained by adding the contributions labeled E 1 V P and E 2 V P in Ref. [2], and the additional logarithm implicitly contained in the Eq. (40) ibid. which is proportional to B  [2]]. In total, the results read as follows: The total result for B 62 as well as its n-dependence were obtained in Refs. [2,12,13,14]. Here, we give the formulas for the particular sets of diagrams, for the case n = 1 as well as the difference to a state of general n: where Ψ denotes the logarithmic derivative of the gamma function, and C = 0.577216 . . . is Euler's constant.
The slightly shifted results for N (1S) also explains a discrepancy in an intermediate step of the calculation of radiative corrections to the muonium hyperfine splitting [15,16].

Extrapolation of the Zα-Expansion
We start our consideration with the one-loop self-energy which is the dominating radiative correction in hydrogenlike bound systems. We write the (real part of the) one-loop self-energy shift ∆E where F (Zα) is a dimensionless quantity which depends on the principal quantum number n, the total electron spin+angular momentum j and the electron orbital angular momentum l, and of course on the parameter Zα.
The semi-analytic expansion of F (Zα) about Zα = 0 for P states and states with higher angular momenta gives rise to the expression [21], where the ellipsis again denotes omitted higher-order terms. The A 60 coefficient has proven to be by far the most difficult to evaluate [22,23,24,25,26], and for 2P states, results have become available recently [17,18]. Refs. [17,18]. The extrapolated semi-analytic Zα-expansions are closer to the numerical data for high Z than the "raw" Zα-expansion. Numerical data at high Z are taken from Refs. [19,20].
The semi-analytic expansion (10) is generally assumed to converge to the function F (Zα) for low Z, at least in an asymptotic sense. This is confirmed by recent numerical evaluations [27,28] for S and P states and the successful consistency check with available analytic results [17,26]. In many cases, an asymptotic expansion valid a priori for small expansion parameter Zα can be extrapolated to large coupling, if it is combined with a suitable convergence acceleration or resummation method (the latter in the case of a divergent input series [29]). The logarithms in Eq. (10) make a power series expansion about Zα = 0 impossible. However, an extrapolation is still possible if we expand about Zα = α = 0 and use the fact that the nonperturbative function F (Zα), in the range of small Z, is very well represented by the first terms in its asymptotic expansion, as suggested by Fig. 1 of Ref. [27]. The logarithms in (10), when expanded about Zα = α, give rise to an infinite power series in the variable g = (Z − 1) α. We proceed as follows: . Numerical data are found in Ref. [4]. Much better agreement between numerical and analytic data is achieved for negative B 60 .
for all radiative corrections studied in the sequel, we start from the semi-analytic expansion and take into account all known coefficients. We then expand in g and evaluate the diagonal [2/2]-Padé approximant to the resulting power series (for the definition and a comprehensive discussion of Padé approximants we refer to [30]). This could be characterized a "deferred" approximant which is evaluated only after one has "advanced" to the point Z = 1 from the "starting point" Z = 0 (or equivalently Zα = 0). Formally, the semi-analytic Zα-expansion is performed about the point Z = 0. In re-expanding the perturbation series about a different point in the complex plane, we follow ideas outlined in [31] which were originally applied to the problem of calculating the autoionization width of atomic resonances in an external electric field.
The choice of the [2/2]-Padé approximant is motivated by the paradigm of finding a compromise between the necessity to harvest the information contained in the logarithms, which give rise to power series terms of arbitrarily high order, and at the same time to avoid spurious singularities which may be incurred when the degree of the Padé approximant is increased to an excessively high order. The [2/2]-deferred Padé approximant about Z = 1 to the function F [see also Eq. (3) of [32]] has five parameters p 0 , . . . , p 2 , q 1 , . . . , q 2 , which are determined by the requirement that the power series expansion of [2/2] F (g) about g = 0 reproduce the power series expansion of F (Zα) with Zα = α + g, also about g = 0, up to the order O(g 4 ). As discussed in [30], this condition alone defines the Padé approximant uniquely. The "one" in the denominator of (11) is the so-called Baker convention.
Because we do not observe a factorially divergent alternating-sign pattern in the g-expansion, we do not employ the delta transformation [29,Ch. 8] which has proven to be superior to Padé approximants in a number of applications where factorial divergence is observed (e.g., [32]). We rely on the robust Padé approximants, while stressing that it may be possible to find better extrapolation algorithms that harvest the analytic structure of (10) and give rise to logarithmic terms naturally. As yet, we have been unable to find such algorithms. Fig. 2-3 show that the extrapolated semi-analytic expansions have a somewhat better agreement with medium and high-Z numerical data than the known terms of the Zα-expansion alone. We observe that the high-Z results for the energy correction given by the irreducible set of twoloop self-energy insertions into the bound electron propagator (see Ref. [4]) can only be made consistent with our extrapolated Zα-expansion if we assume that the coefficient B 60 (1S 1/2 ) is negative and rather large in magnitude.

Conclusions
We have presented a "mini-review" of recent two-loop self-energy calculations [2,3,12] in Sec. 2, clarifying the distribution of sixth-order (in Zα) two-loop binding corrections to the Lamb shift over the set of diagrams shown in Fig. 1. Results for the two-loop coefficients, including excited S states, are provided in Eqs. (3)- (8). The distribution of the logarithmic corrections over distinct sets of diagrams needs to be clarified in order to allow for an accurate comparison to numerical calculations which are currently being pursued [4]. In Sec. 3, we present a crude extrapolation scheme for the extrapolation of the Zα-expansion from low Z to high Z. The scheme follows ideas outlined in [31] and is based upon an expansion in the variable g where g is defined as Zα = α + g. The "deferred" Padé approximant is then evaluated in terms of the variable g, i.e. after the coupling parameter Zα has acquired the value α = 0, "starting" from Zα = 0. This deferment circumvents the problems introduced by the logarithms in Eqs. (2) and (10); however, we stress here that it would be highly desirable to find better extrapolation algorithms that harvest the analytic structure of (2) and (10) and give rise to logarithmic terms naturally.
Although the extrapolation scheme has problems (in some cases, we observe spurious poles in the Padé approximant at medium-Z values), we have observed rather consistent improvement over the "raw" Zα-expansion with this scheme for a number of states and QED effects which we studied using the "deferred" Padé-extrapolation scheme. Details will be presented elsewhere. Based on our extrapolations of the two-loop effect in Fig. 3, we would like to advance the tentative estimate B 60 (1S 1/2 ) ≈ −100(50).