Bounds on the width, mass difference and other properties of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3872</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>→</mml:mo><mml:msup><mml:mi>π</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi></mml:math>decays

We present results from a study of $X(3872)\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}J/\ensuremath{\psi}$ decays produced via exclusive $B\ensuremath{\rightarrow}KX(3872)$ decays. We determine the mass to be ${M}_{X(3872)}=(3871.85\ifmmode\pm\else\textpm\fi{}0.27(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.19(\mathrm{syst}))\text{ }\text{ }\mathrm{MeV}$, a 90% confidence level upper limit on the natural width of ${\ensuremath{\Gamma}}_{X(3872)}&lt;1.2\text{ }\text{ }\mathrm{MeV}$, the product branching fraction $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}{K}^{+}X(3872))\ifmmode\times\else\texttimes\fi{}\mathcal{B}(X(3872)\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}J/\ensuremath{\psi})=(8.63\ifmmode\pm\else\textpm\fi{}0.82(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.52(\mathrm{syst}))\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, and a ratio of branching fractions $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{K}^{0}X(3872))/\mathcal{B}({B}^{+}\ensuremath{\rightarrow}{K}^{+}X(3872))=0.50\ifmmode\pm\else\textpm\fi{}0.14(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.04(\mathrm{syst})$. The difference in mass between the $X(3872)\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}J/\ensuremath{\psi}$ signals in ${B}^{+}$ and ${B}^{0}$ decays is $\ensuremath{\Delta}{M}_{X(3872)}=(\ensuremath{-}0.71\ifmmode\pm\else\textpm\fi{}0.96(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.19(\mathrm{syst}))\text{ }\text{ }\mathrm{MeV}$. A search for a charged partner of the $X(3872)$ in the decays ${\overline{B}}^{0}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{X}^{+}$ or ${B}^{+}\ensuremath{\rightarrow}{K}^{0}{X}^{+}$, ${X}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}J/\ensuremath{\psi}$ resulted in upper limits on the product branching fractions for these processes that are well below expectations for the case that the $X(3872)$ is the neutral member of an isospin triplet. In addition, we examine possible ${J}^{PC}$ quantum number assignments for the $X(3872)$ based on comparisons of angular correlations between final state particles in $X(3872)\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}J/\ensuremath{\psi}$ decays with simulated data for ${J}^{PC}$ values of ${1}^{++}$ and ${2}^{\ensuremath{-}+}$. We examine the influence of $\ensuremath{\rho}$-$\ensuremath{\omega}$ interference in the $M({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})$ spectrum. The analysis is based on a $711\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ data sample that contains $772\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ meson pairs collected at the $\ensuremath{\Upsilon}(4S)$ resonance in the Belle detector at the KEKB ${e}^{+}{e}^{\ensuremath{-}}$ collider.