Bloch-Nordsieck cancellations beyond logarithms in heavy particle decays.

We investigate the one-loop radiative corrections to the semileptonic decay of a charged particle at finite gauge boson mass. Extending the Bloch-Nordsieck cancellation of infrared logarithms, the subsequent non-analytic terms are also found to vanish after eliminating the pole mass in favor of a mass defined at short distances. This observation justifies the operator product expansion for inclusive decays of heavy mesons and implies that infrared effects associated with the summation of the radiative corrections are suppressed by at least three powers of the mass of the heavy decaying particle. PACS numbers: 12.38.Cy, 12.39.Hg, 13.20.He ∗ On leave of absence from St.Petersburg Nuclear Physics Institute, 188350 Gatchina, Russia † On leave of absence from Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 Munich The problem of the infrared (IR) behavior of amplitudes is inherent to gauge theories since by construction they contain massless bosons. Historically, already the first studies of the problem revealed both complexities and simplicities. On the one hand, if a finite gauge boson mass λ is employed as IR regulator, the perturbative amplitudes contain logarithms in the mass, lnλ, which apparently do not allow for the limit λ → 0. On the other hand, as shown first by Bloch and Nordsieck [1], these singularities cancel if one considers inclusive processes. This summation over final states is an integral part of any calculation in gauge theories. With the advent of QCD the problem of the IR behavior of amplitudes became even more acute since the effective coupling blows up in the infrared. Thus, if an amplitude is perturbatively sensitive to infrared momenta, it is contributed in fact by nonperturbative effects as well and cannot be evaluated reliably. From this point of view the famous prediction of QCD that the total cross section of ee into hadrons measures quark charges rests on the Bloch-Nordsieck (BN) cancellation. Indeed, all the IR logarithms disappear from the total cross section and it is determined by short distance physics. Technically, ee annihilation is treated within the operator product expansion (OPE). Then, at least naively, IR contributions are suppressed by four powers of the large scale. The BN type cancellations ensure that this counting is not violated by divergencies of perturbative expansions, explicit or through its divergence in large orders. In this sense there is a correspondence between IR cancellations and the OPE. The example of ee annihilation into hadrons is still special since there are no colored particles in the initial state. In the general situation, it is known from the work of Kinoshita and Lee and Nauenberg (KLN) [2] that to remove all IR singularities, summation over degenerate initial states might also be needed. This summation does not necessarily reflect the actual experimental situation and in this sense some IR logarithms can survive. The OPE has been applied more recently in the context of heavy quark physics and the inclusive decay widths of heavy hadrons in particular. In this case there is a charged particle in the initial state. Though it is generally accepted that with only one colored particle in the initial state, the BN theorem is sufficient to ensure the absence of explicit soft divergencies, it is not a priori clear whether one can rely on extended BN type cancellations in the sense of IR power counting, implying the validity of OPE, or whether one has to invoke the more general KLN cancellation, which would require modifications of the standard OPE. In particular, the standard OPE states that there are no corrections to the decay width linear in the inverse mass of the heavy quark. This conclusion is not trivial in view of the recent observation [3, 4] that the pole mass of a charged particle itself does receive such corrections. It has been argued, therefore, that the widths are to be expressed in terms of a (unphysical) quark mass normalized far off-mass-shell, which is infrared stable in the linear approximation [3]. In this Letter we test the counting of IR effects implied by the OPE by an explicit calculation of the IR sensitive contributions to the decay of a charged particle. The infrared contributions are singled out by non-analytic terms in the gauge boson mass, some of which are associated with higher dimensional operators. The ones which are not should disappear either after summation over final states or final and initial states. In the latter case the IR counting implied by the OPE is violated. The leading lnλ