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Group I: 49 diagrams obtained from the 1–loop muon vertex by inserting 1–, 2– and 3–loop lepton VP
subdiagrams, i.e., the internal photon line of Fig. 8 is replaced by the full propagator at three loops. The
group is subdivided into four g-i subclasses I(a), I(b), I(c) and I(d) as illustrated in Fig. 15. Results for this
µ
ℓ1
ℓ2
ℓ3
ℓ1
ℓ2
ℓ1 ℓ2
Ia Ib Ic Id
Fig. 15. Typical diagrams of subgroups Ia (7 diagrams), Ib (18 diagrams), Ic (9 diagrams) and Id (15 diagrams). The lepton
lines represent fermions propagating in an external magnetic field. ℓi denote VP insertions.
group have been obtained by numerical and analytic methods [126,118]. The numerical result [126]
A (8)
2 I
= 16.720 359 (20) ,
has been obtained by using simple integral representations.
Group II: 90 diagrams generated from the 2–loop muon vertex by inserting 1–loop and/or 2–loop lepton
VP subdiagrams as shown in Fig. 16. As for the previous case, results for this group have been obtained by
2
4
2
2 2
4
2 2
II
Fig. 16. Typical diagrams of group II (90 diagrams). The lepton lines as in Fig. 15. 2 and 4, respectively, indicate second (1–loop
subdiagrams) and fourth (2–loop subdiagrams) order lepton–loops.
numerical and analytic methods [126,118]. The result here is [126]
A (8)
2 II = −16.674 591 (68) .
Group III: 150 diagrams generated from the 3–loop muon vertex Fig. 10 by inserting one 1–loop electron
VP subdiagrams in each internal photon line in all possible ways. Examples are given in Fig. 17. This is a
2
2
III
2
2
4
2 2
Fig. 17. Typical diagrams of group III (150 diagrams). The lepton lines as in Fig. 15.
group which has been calculated numerically only. The result found in [126] reads
A (8)
2 III = 10.793 43 (414) .
Group IV: 180 diagrams with muon vertex containing LbL subgraphs decorated with additional radiative
corrections. This group is subdivided as shown in Fig. 18 into g-i subsets IV(a), IV(b), IV(c) and IV(d).
The calculation of the corresponding contribution is at the limit of present possibilities. The result has been
36
2
ℓ1