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

It is customary to apply the so-called narrow width approximation $\Gamma(B\to RP_3\to P_1P_2P_3)=\Gamma(B\to RP_3)Br(R\to P_1P_2)$ extract branching fraction of quasi-two-body decay $B\to RP_3$, with $R$ and $P_3$ being an intermediate resonant state a pseudoscalar meson, respectively. However, above factorization valid only in zero limit. We consider correction parameter $\eta_R$ from finite effects. Our main results are: (i) present general framework for computing show that it can be expressed terms normalized differential rate determined by its value at resonance. (ii) introduce form factor $F(s_{12},m_R)$ strong coupling involved $R(m_{12})\to P_1P_2$ when $m_{12}$ away $m_R$. find off-shell effects are small vector meson productions, but prominent $K_2^*(1430)$, $\sigma/f_0(500)$ $K_0^*(1430)$ resonances. (iii) evaluate theoretical QCD (QCDF) experimental parameterization (EXPP) three-body amplitudes. In general, $\eta_R^{\rm QCDF}$ EXPP}$ similar mesons, different tensor scalar A study rates enables us understand origin their differences. (iv) Finite-width corrections $Br(B^-\to RP)_{\rm NWA}$ obtained generally small, less than 10\%, they $B^-\to\sigma/f_0(500)\pi^-$ $B^-\to \overline K_0^{*0}(1430)\pi^-$ decays. The EXPP should contrasted predictions QCDF calculation as latter properly takes into account energy dependence weak