Abstract
We consider the on-shell mass and wave function renormalization constants {Z}_m^{mathrm{OS}} and {Z}_2^{mathrm{OS}} up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters sqrt{1-{tau}^2} and sqrt{1-{tau}^2}/tau which extends the findings from ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order mathcal{O} (ϵ2) and mathcal{O} (ϵ) at two- and three-loop order which are crucial ingredients for a future four-loop calculation. Compact results for the expansions around the zero-mass, equal-mass and large-mass cases allow for a fast high-precision numerical evaluation.
Highlights
Bottom field, one has m1 = mb and m2 = mc
We consider the on-shell mass and wave function renormalization constants ZmOS and Z2OS up to three-loop order allowing for a second non-zero quark mass
Even in the case when finite electron mass effects are restored via massification [24] of virtual amplitudes computed for me = 0, the procedure employs the lepton’s wave-function renormalization constant with finite me effects to the relevant order in αem
Summary
At three-loop order, for the complicated master integrals only an expansion for x → 0 could be obtained. We use the results where ZmOS and Z2OS are expressed in terms of the 28 master integrals shown in figure 2. There it was noted that four master integrals (M20, M21, M22, M23) cannot be expressed in terms of HPLs at higher orders in = (4 − d)/2.
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