Dispersion relations and rescattering effects inBnonleptonic decays

Abstract

Recently, the final state strong interactions in nonleptonic B decays were investigated in a formalism based on hadronic unitarity and dispersion relations in terms of the off-shell mass squared of the B meson. We consider an heuristic derivation of the dispersion relations in the mass variables using the reduction LSZ formalism and find a discrepancy between the spectral function and the dispersive variable used in the recent works. The part of the unitarity sum which describes final state interactions is shown to appear as spectral function in a dispersion relation based on the analytic continuation in the mass squared of one final particles. As an application, by combining this formalism with Regge theory and SU (3) flavour symmetry we obtain constraints on the tree and the penguin amplitudes of the decay B 0 → π + π − .

Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings

“…As shown for the first time in [9], the optimal mapping, which ensures the convergence of the corresponding power series in the entire doubly cut Borel plane, is given by the functioñ…”

“…II, the nature of the singularities of the Borel transform in the u plane depend only on the first two coefficients, β 1 and β 2 , of the β function, which are scheme independent. This means that the first singularities of the function B(u, C) defined in (23) are expected to have the same location at u = −1 and u = 2, and their nature to be described by the same relations (9). Another argument in favor of this property, put forward in [18], is that the behavior of the Borel transform near the first singularities is dictated by the limit of vanishing coupling, where the MS and the C scheme coincide.…”

“…In the present paper we exploit the analytic structure of the Adler func-tion in the Borel plane, which encodes the large-order behavior of the perturbative expansion. We then apply a procedure of series acceleration by conformal mappings, proposed in [9] and investigated further in Refs. [10]- [16].…”

Dispersion relations and Omnès representations for $K\to \pi \pi$ decay amplitudes

“…system, where p 2 n = s is the total energy squared, we see that the matrix elements in (8) depend only on s and the physical masses. In the two-particle approximation of the unitarity sum, the integral in (8) can be performed exactly [17]. According to the discussion above we can write…”

“…We follow to some extent the dispersive treatment of B → ππ decay considered in [17,18]. However, the different masses of the decaying particles in the two processes require specific treatments.…”