Amplitude analysis of the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ decay and measurement of the ${\pi }^{-}{\pi }^{+}$ S-wave amplitude

An amplitude analysis of the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ decay is performed with a sample corresponding to $1.5\mathrm{f}{\mathrm{b}}^{-1}$ of integrated luminosity of $pp$ collisions at a centre-of-mass energy $\sqrt{s}=8$ TeV collected by the LHCb detector in 2012. The sample contains approximately six hundred thousand candidates with a signal purity of $95\mathrm{\%}$. The resonant structure is studied through a fit to the Dalitz plot where the ${\pi }^{-}{\pi }^{+}$ S-wave amplitude is extracted as a function of ${\pi }^{-}{\pi }^{+}$ mass, and spin-1 and spin-2 resonances are included coherently through an isobar model. The S-wave component is found to be dominant, followed by the $\rho (770{)}^0{\pi }^{+}$ and $f_2(1270){\pi }^{+}$ components. A small contribution from the $\omega (782)\to {\pi }^{-}{\pi }^{+}$ decay is seen for the first time in the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ decay.