Weak phaseαfromB0→a1±(1260)π∓

Abstract

Quasi two-body decays ${B}^{0}(t)\ensuremath{\rightarrow}{a}_{1}^{\ifmmode\pm\else\textpm\fi{}}(1260){\ensuremath{\pi}}^{\ensuremath{\mp}}$ identified by four charged pions determine a phase ${\ensuremath{\alpha}}_{\mathrm{eff}}$, which is equal to the weak phase $\ensuremath{\alpha}$ in the limit of vanishing penguin amplitudes. Applying flavor SU(3) to these decays and to $B\ensuremath{\rightarrow}{a}_{1}K$ and $B\ensuremath{\rightarrow}{K}_{1}\ensuremath{\pi}$, with ${K}_{1}$ an admixture of ${K}_{1}(1270)$ and ${K}_{1}(1400)$, we derive expressions providing bounds on $\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{eff}}$. Higher precision in $\ensuremath{\alpha}$ may be achieved by an overall fit to a complete set of SU(3) related measurements. A method is sketched applying isospin symmetry to time-dependent invariant mass distributions in $B\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$.