Amplitude analysis of $B^{-}\to D^{+}{\pi }^{-}{\pi }^{-}$ decays

The Dalitz plot analysis technique is used to study the resonant substructures of $B^{-}\to D^{+}{\pi }^{-}{\pi }^{-}$ decays in a data sample corresponding to $3.0{\mathrm{fb}}^{-1}$ of $pp$ collision data recorded by the LHCb experiment during 2011 and 2012. A model-independent analysis of the angular moments demonstrates the presence of resonances with spins 1, 2 and 3 at high $D^{+}{\pi }^{-}$ mass. The data are fitted with an amplitude model composed of a quasi-model-independent function to describe the $D^{+}{\pi }^{-}$ S wave together with virtual contributions from the $D^{*}(2007{)}^0$ and $B^{*0}$ states, and components corresponding to the $D_2^{*}(2460{)}^0$, $D_1^{*}(2680{)}^0$, $D_3^{*}(2760{)}^0$ and $D_2^{*}(3000{)}^0$ resonances. The masses and widths of these resonances are determined together with the branching fractions for their production in $B^{-}\to D^{+}{\pi }^{-}{\pi }^{-}$ decays. The $D^{+}{\pi }^{-}$ S wave has phase motion consistent with that expected due to the presence of the $D_0^{*}(2400{)}^0$ state. These results constitute the first observations of the $D_3^{*}(2760{)}^0$ and $D_2^{*}(3000{)}^0$ resonances, with significances of $10\sigma$ and $6.6\sigma$, respectively.

Figure 1

Results of the fit to the $B$ candidate invariant mass distribution shown with (left) linear and (right) logarithmic $y$-axis scales. Contributions are as described in the legend.

Figure 2

Distribution of $B^{-}\to D^{+}{\pi }^{-}{\pi }^{-}$ candidates in the signal region over (left) the DP and (right) the SDP.

Figure 3

The first seven unnormalized angular moments, from $⟨P_0⟩$ (a) to $⟨P_6⟩$ (g), for background-subtracted and efficiency-corrected data (black points) as a function of $m(D^{+}{\pi }^{-})$ in the range 2.0–4.0 GeV. The blue line shows the result of the DP fit described in Sec. 7.

Figure 4

Unnormalized angular moments $⟨P_7⟩$ and $⟨P_8⟩$ for background-subtracted and efficiency-corrected data (black points) as a function of $m(D^{+}{\pi }^{-})$ in the range 2.0–4.0 GeV. The blue line shows the result of the DP fit described in Sec. 7.

Figure 5

Zoomed views of the fourth and sixth unnormalized angular moments for background-subtracted and efficiency-corrected data (black points) as a function of $m(D^{+}{\pi }^{-})$. The blue line shows the result of the DP fit described in Sec. 7.

Figure 6

Signal efficiency across the SDP for $B^{-}\to D^{+}{\pi }^{-}{\pi }^{-}$ decays. The relative uncertainty at each point is typically 5%.

Figure 7

Square Dalitz plot distributions for (left) combinatorial background and (right) $B^{-}\to D^{(*)+}{K}^{-}{\pi }^{-}$ decays.

Figure 8

Real and imaginary parts of the S-wave amplitude shown in an Argand diagram. The knots are shown with statistical uncertainties only, connected by the cubic spline interpolation used in the fit. The leftmost point is that at the lowest value of $m(D^{+}{\pi }^{-})$, with mass increasing along the connected points. Each point labeled 1–13 corresponds to the position of a knot in the spline, at values of $m(D^{+}{\pi }^{-})=\{2.01,2.10,2.20,2.30,2.40,2.50,2.60,2.70,2.80,2.90,3.10,4.10,5.14\}\mathrm{GeV}$. The points at (0.5, 0.0) and (0.0, 0.0) are fixed. The anticlockwise rotation of the phase at low $m(D^{+}{\pi }^{-})$ is as expected due to the presence of the $D_0^{*}(2400{)}^0$ resonance.

Figure 9

Differences between the SDP distribution of the data and fit model, in terms of the normalized residual in each bin. No bin lies outside the $z$-axis limits.

Figure 10

Projections of the data and amplitude fit onto (top) $m(D^{+}{\pi }^{-}{)}_{\min }$, (middle) $m(D^{+}{\pi }^{-}{)}_{\max }$ and (bottom) $m({\pi }^{-}{\pi }^{-})$, with the same projections shown (right) with a logarithmic $y$-axis scale. Components are described in the legend.

Figure 11

Projections of the data and amplitude fit onto (left) $m(D^{+}{\pi }^{-})$ and (right) the cosine of the helicity angle for the $D^{+}{\pi }^{-}$ system in (top to bottom) the low mass threshold region, the $D_2^{*}(2460{)}^0$ region, the $D_1^{*}(2680{)}^0–D_3^{*}(2760{)}^0$ region and the $D_2^{*}(3000{)}^0$ region. Components are as shown in Fig. 10.