Abstract
We calculate a class of three-loop Feynman diagrams which contribute to the next-to-next-to-leading logarithmic approximation for the width difference ΔΓs in the {B}_s-{overline{B}}_s system. The considered diagrams contain a closed fermion loop in a gluon propagator and constitute the order αs2Nf, where Nf is the number of light quarks. Our results entail a considerable correction in that order, if ΔΓs is expressed in terms of the pole mass of the bottom quark. If the overline{mathrm{MS}} scheme is used instead, the correction is much smaller. As a result, we find a decrease of the scheme dependence. Our result also indicates that the usually quoted value of the NLO renormalization scale dependence underestimates the perturbative error.
Highlights
The i, j are colour indices and V ± A means γμ(1 ± γ5) while S ± P stands for (1 ± γ5)
Ref. [20] has calculated some of the matrix elements appearing at order ΛQCD/mb and these results went into the last terms of eqs. (1.6) and (1.7)
Comparing eq (5.2) with eq (5.3) we find that the MS result is quite stable, if we change the literal αs2Nf result to the non-abelianization approach (NNA) one, while the pole-scheme result is not
Summary
It has been suggested to trade Nf for β0, so that the αs2Nf term is replaced by a term of order αs2β0 (naive non-abelianization [25, 26]) In some applications this procedure gives a good approximation to the full αs term. In quantities involving effective four-quark operators, it is pure speculation whether the original αs2Nf term or its naively non-abelianized version ∝ αs2β0 approximates the full result in a better way, because neither term cancels the scheme dependence of the operator renormalization. [27] revealed that the αs2β0 term is not a good approximation to the full result In light of this finding we do not advocate the use of naive non-abelianization in our case. One can overcome the scheme-dependence issue by only keeping the αs2Nf terms of the NNLO correction to the RG-improved Wilson coefficients. A future NNLO calculation keeping higher powers of z terms will benefit from these results
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