Finite-width effects in three-body <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi></mml:math> decays

Abstract

It is customary to apply the so-called narrow width approximation $\mathrm{\ensuremath{\Gamma}}(B\ensuremath{\rightarrow}R{P}_{3}\ensuremath{\rightarrow}{P}_{1}{P}_{2}{P}_{3})=\phantom{\rule{0ex}{0ex}}\mathrm{\ensuremath{\Gamma}}(B\ensuremath{\rightarrow}R{P}_{3})\mathcal{B}(R\ensuremath{\rightarrow}{P}_{1}{P}_{2})$ to extract the branching fraction of the quasi-two-body decay $B\ensuremath{\rightarrow}R{P}_{3}$, with $R$ and ${P}_{3}$ being an intermediate resonant state and a pseudoscalar meson, respectively. However, the above factorization is valid only in the zero width limit. We consider a correction parameter ${\ensuremath{\eta}}_{R}$ from finite-width effects. Our main results are as follows: (i) We present a general framework for computing ${\ensuremath{\eta}}_{R}$ and show that it can be expressed in terms of the normalized differential rate and determined by its value at the resonance. (ii) We introduce a form factor $F({s}_{12},{m}_{R})$ for the strong coupling involved in the $R({m}_{12})\ensuremath{\rightarrow}{P}_{1}{P}_{2}$ decay when ${m}_{12}$ is away from ${m}_{R}$. We find that off-shell effects are small in vector meson productions, but prominent in the ${K}_{2}^{*}(1430)$, $\ensuremath{\sigma}/{f}_{0}(500)$, and ${K}_{0}^{*}(1430)$ resonances. (iii) We evaluate ${\ensuremath{\eta}}_{R}$ in the theoretical framework of QCD factorization (QCDF) and in the experimental parametrization (EXPP) for three-body decay amplitudes. In general, ${\ensuremath{\eta}}_{R}^{\mathrm{QCDF}}$ and ${\ensuremath{\eta}}_{R}^{\mathrm{EXPP}}$ are similar for vector mesons, but different for tensor and scalar resonances. A study of the differential rates enables us to understand the origin of their differences. (iv) Finite-width corrections to $\mathcal{B}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}RP{)}_{\mathrm{NWA}}$ obtained in the narrow width approximation are generally small, less than 10%, but they are prominent in ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\sigma}/{f}_{0}(500){\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{\overline{K}}_{0}^{*0}(1430){\ensuremath{\pi}}^{\ensuremath{-}}$ decays. The EXPP of the normalized differential rates should be contrasted with the theoretical predictions from QCDF calculation as the latter properly takes into account the energy dependence in weak decay amplitudes. (v) It is common to use the Gounaris-Sakurai model to describe the line shape of the broad $\ensuremath{\rho}(770)$ resonance. After including finite-width effects, the PDG value of $\mathcal{B}({B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\rho}{\ensuremath{\pi}}^{\ensuremath{-}})=(8.3\ifmmode\pm\else\textpm\fi{}1.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ should be corrected to $(7.9\ifmmode\pm\else\textpm\fi{}1.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ in EXPP and $(7.7\ifmmode\pm\else\textpm\fi{}1.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ in QCDF. (vi) For the very broad $\ensuremath{\sigma}/{f}_{0}(500)$ scalar resonance, we use a simple pole model to describe its line shape and find a very large width effect: ${\ensuremath{\eta}}_{\ensuremath{\sigma}}^{\mathrm{QCDF}}\ensuremath{\sim}2.15$ and ${\ensuremath{\eta}}_{\ensuremath{\sigma}}^{\mathrm{EXPP}}\ensuremath{\sim}1.64$. Consequently, ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\sigma}{\ensuremath{\pi}}^{\ensuremath{-}}$ has a large branching fraction of order ${10}^{\ensuremath{-}5}$. (vii) We employ the Breit-Wigner line shape to describe the production of ${K}_{0}^{*}(1430)$ in three-body $B$ decays and find large off-shell effects. The smallness of ${\ensuremath{\eta}}_{{K}_{0}^{*}}^{\mathrm{QCDF}}$ relative to ${\ensuremath{\eta}}_{{K}_{0}^{*}}^{\mathrm{EXPP}}$ is ascribed to the differences in the normalized differential rates off the resonance. (viii) In the approach of QCDF, the calculated $CP$ asymmetries of ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{f}_{2}(1270){\ensuremath{\pi}}^{\ensuremath{-}},\ensuremath{\sigma}/{f}_{0}(500){\ensuremath{\pi}}^{\ensuremath{-}},{K}^{\ensuremath{-}}{\ensuremath{\rho}}^{0}$ decays agree with the experimental observations. The nonobservation of $CP$ asymmetry in ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\rho}(770){\ensuremath{\pi}}^{\ensuremath{-}}$ can also be accommodated in QCDF.

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