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By Williams, Loyalka

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The double logarithmic B62 -coefficient is spin-independent, so that we have ∆fs B62 = 0. Here, we evaluate the fine-structure differences ∆fs B61 = B61 (nP3/2 ) − B61 (nP1/2 ) , ∆fs B60 = B60 (nP3/2 ) − B60 (nP1/2 ) . 2) We follow the convention that ∆fs X ≡ X(nP3/2 ) − X(nP1/2 ) denotes the “fine-structure part” of a given quantity X. For ∆fs B61 and ∆fs B60 , we provide complete results. It is perhaps worth noting that the two-loop self energy for bound states has represented a considerable challenge for theoretical evaluations.

6) is not applicable in this energy domain; we therefore have to keep the √ numerator of the integrand ω 2 + β 2 in unexpanded form. However, we can expand the denominator 1 − ω 2 of the integrand in powers of ω; because 0 < ω < ǫ (with ǫ small), this expansion in ω is in fact an expansion in β – although the situation is somewhat problematic in the sense that every term in the ω-expansion gives rise to terms of arbitrarily high order in the β-expansion [see also Eq. 10) below]. 9) corresponds to the expansion into the (Zα)-expansion in the low-energy part.

Of crucial importance was the development of convergence acceleration methods which were used extensively for the evaluation of remaining one-dimensional integrals which could not be done analytically. These integrals are analogous to expressions encountered in previous work [1, 2]. The numerically evaluated contributions involve slowly convergent hypergeometric series, and – in more extreme cases – infinite series over partial derivatives of hypergeometric functions, and generalizations of Lerch’s Φ transcendent [91, 92].

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