Abstract:We calculate the long-distance effect generated by the four-quark operators with c-quarks in the B → K ( * ) ℓ + ℓ − decays. At the lepton-pair invariant masses far below thecc-threshold, q 2 ≪ 4m 2 c , we use OPE near the light-cone. The nonfactorizable softgluon emission from c-quarks is cast in the form of a nonlocal effective operator. The B → K ( * ) matrix elements of this operator are calculated from the QCD light-cone sum rules with the B-meson distribution amplitudes. As a byproduct, we also predict the charm-loop contribution to B → K * γ beyond the local-operator approximation. To describe the charmloop effect at large q 2 , we employ the hadronic dispersion relation with ψ = J/ψ, ψ(2S), ... contributions, where the measured B → K ( * ) ψ amplitudes are used as inputs. Matching this relation to the result of QCD calculation reveals a destructive interference between the J/ψ and ψ(2S) contributions. The resulting charm-loop effect is represented as a q 2 -dependent correction ∆C 9 (q 2 ) to the Wilson coefficient C 9 . Within uncertainties of our calculation, at q 2 below the charmonium region the predicted ratio ∆C 9 (q 2 )/C 9 is ≤ 5% for B → Kℓ + ℓ − , but can reach as much as 20% for B → K * ℓ + ℓ − , the difference being mainly caused by the soft-gluon contribution.

A technique to sum up the regular corrections appearing under the analytic continuation from the spacelike momentum region to the timelike one is proposed. A perturbative part of the inclusive semileptonic decay width of the τ -lepton in analyzed in detail.

We describe some ways how higher order corrections can reveal themselves if integrated over the infrared region. We show that in different renormalization group (RG) schemes and for some observables one has no factorial divergences. We argue that for treating things in the infrared region it is preferable to start with a RG scheme without the infrared Landau pole in the running coupling constant. The uncertainties for the τ lepton width resulting from accounting for higher order corrections are discussed.With new high order corrections of perturbation theory hardly available anymore in cases like e + e − annihilation or τ lepton width [1] it is tempting to speculate on the general structure of series within perturbation theory (PT) [2,3,4]. Some attention has been recently paid to possible factorial divergences in PT series for observables that include integration over an infrared region in momentum space [5,6,7,8]. At the level of diagrams within PT it happens to any observable and the factorial divergence due to the simple bubble chain diagrams is predicted. In fact this prediction is not well justified in QCD because of the choice of a particular subset of diagrams and even of the special contributions (nonabelianization [9]). Next terms of PT expansion can cancel these divergences because such terms become large at small momenta and can not be treated as corrections. In QED there is a formal parameter for ordering diagrams within the 1/N f expansion and the statement about the factorial growth of coefficients can be confirmed by direct computation [10] but it is known also that QED within the 1/N f expansion has practically no sense at all and using it as a guide for general structure of QCD is not well grounded.In this paper we consider some observables that are represented by integrals over the infrared region and give several ways to define them using the freedom of choice of the RG scheme. The main conclusion we draw is that the results of integration can easily be made well defined without any nonperturbative (in a strictly defined sense) contributions. These results are ambiguous to the same degree as any ordinary PT series, numerically it can be important because in the infrared region the coupling constant becomes large in most of "natural" RG schemes. However it can be made small as well by some particular choice of extrapolation to low momenta.The paper is organized as follows. First we define our main object as an integral of some PT expansion over infrared region and show that the corresponding expression 1

In this work radiative corrections in the total hadronic decay rate of the τ lepton and some moments of its differential distributions are studied employing perturbative QCD and the operator product expansion. We calculate quadratic quark mass corrections in the strange mass to the decay rate ratio R τ to the order O(α 3 s m 2 ) and find that they contribute appreciably to the Cabibbo suppressed decay modes of the τ -lepton. Using the results of a recent experimental analysis, we obtain m s (1 GeV) = 200 ± 40 exp ± 30 th MeV.