Search for the decay<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msubsup><mml:mi>a</mml:mi><mml:mn>1</mml:mn><mml:mo>±</mml:mo></mml:msubsup><mml:msup><mml:mi>ρ</mml:mi><mml:mo>∓</mml:mo></mml:msup></mml:math>
Search for the decay<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>→</mml:mo><mml:msubsup><mml:mi>a</mml:mi><mml:mn>1</mml:mn><mml:mo>±</mml:mo></mml:msubsup><mml:msup><mml:mi>ρ</mml:mi><mml:mo>∓</mml:mo></mml:msup></mml:math>
We present a search for the rare $B$-meson decay ${B}^{0}\ensuremath{\rightarrow}{a}_{1}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\rho}}^{\ensuremath{\mp}}$ with ${a}_{1}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$. We use $(110\ifmmode\pm\else\textpm\fi{}1.2)\ifmmode\times\else\texttimes\fi{}{10}^{6}\text{ }\text{ }\mathit{\ensuremath{\Upsilon}}(4S)\ensuremath{\rightarrow}B\overline{B}$ decays collected with the BABAR detector at the PEP-II asymmetric-energy $B$ Factory at SLAC. We obtain an upper limit of $30\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\text{ }\text{ }(90%\mathrm{C}.\mathrm{L}.)$ for the branching fraction product $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{a}_{1}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\rho}}^{\ensuremath{\mp}})\mathcal{B}({a}_{1}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}})$, where we assume that …