Study of the lineshape of the ${\chi }_{c1}(3872)$ state

A study of the lineshape of the ${\chi }_{c1}(3872)$ state is made using a data sample corresponding to an integrated luminosity of $3{\mathrm{fb}}^{-1}$ collected in $pp$ collisions at center-of-mass energies of 7 and 8 TeV with the LHCb detector. Candidate ${\chi }_{c1}(3872)$ and $\psi (2S)$ mesons from $b$-hadron decays are selected in the $J/\psi {\pi }^{+}{\pi }^{-}$ decay mode. Describing the lineshape with a Breit-Wigner function, the mass splitting between the ${\chi }_{c1}(3872)$ and $\psi (2S)$ states, $\mathrm{\Delta }m$, and the width of the ${\chi }_{c1}(3872)$ state, ${\mathrm{\Gamma }}_{\mathrm{BW}}$, are determined to be $\genfrac{}{}{0ex}{}{\mathrm{\Delta }m=185.598\pm 0.067\pm 0.068\mathrm{MeV},}{{\mathrm{\Gamma }}_{\mathrm{BW}}=1.39\pm 0.24\pm 0.10\mathrm{MeV},}$ where the first uncertainty is statistical and the second systematic. Using a Flatté-inspired model, the mode and full width at half maximum of the lineshape are determined to be $\genfrac{}{}{0ex}{}{\mathrm{mode}=3871.69_{-0.04-0.13}^{+0.00+0.05}\mathrm{MeV},}{\mathrm{FWHM}=0.22_{-0.06-0.13}^{+0.07+0.11}\mathrm{MeV}.}$ An investigation of the analytic structure of the Flatté amplitude reveals a pole structure, which is compatible with a quasibound $D^0{\overline{D}}^{*0}$ state but a quasivirtual state is still allowed at the level of 2 standard deviations.

Figure 1

Mass distributions for $J/\psi {\pi }^{+}{\pi }^{-}$ candidates in the $\psi (2S)$ region for (top) the low, (middle) mid and (bottom) high $p_{{\pi }^{+}{\pi }^{-}}$ bins. The left- (right-) hand plot is for 2011 (2012) data. The projection of the fit described in the text is superimposed.

Figure 2

Mass distributions for $J/\psi {\pi }^{+}{\pi }^{-}$ candidates in the ${\chi }_{c1}(3872)$ region for (top) the low, (middle) mid and (bottom) high $p_{{\pi }^{+}{\pi }^{-}}$ bins. The left- (right-) hand plot is for 2011 (2012) data. The projection of the fit described in the text is superimposed.

Figure 3

The coupling to the $D{\overline{D}}^{*}$ channels $g$ as a function of Flatté energy parameter $E_f$ (black points with error bars). The corresponding change in negative log likelihood, $\mathrm{\Delta }\mathrm{LL}$ is shown as well (red dots).

Figure 4

Comparison of the Flatté (solid, red) and Breit–Wigner (dotted, black) lineshapes. The left plot shows the raw lineshapes for the default fits. The location of the $D^0$ ${\overline{D}}^{*0}$ threshold is indicated by the blue vertical line. On the right the distributions are shown after applying smearing with the resolution function and adding background.

Figure 5

Distribution of the FWHM obtained for simulated experiments generated from the result of the Flatté model and fitted with the Breit-Wigner model (filled histogram). Both models account for the experimental resolution. The dashed red line shows the FWHM of the Flatté lineshape, while the solid blue line indicates the value of the Breit-Wigner width observed in the data.

Figure 6

The phase of the Flatté amplitude obtained from the fit to the data with $m_0=3864.5\mathrm{MeV}$ on sheets I (for $\mathrm{Im}E>0$) and II (for $\mathrm{Im}E<0$) of the complex energy plane. The pole singularity is visible at $E_{\mathrm{II}}=(0.06-0.13i)\mathrm{MeV}$. The branch cut is highlighted with the black line. The trajectory of the pole taken when the couplings to all but the $D{\overline{D}}^{*}$ channel are scaled down to zero is indicated in red.

Figure 7

The phase of the Flatté amplitude as obtained from the fit with a finite $D^{*0}$ width of ${\mathrm{\Gamma }}_{D^{*0}}=65.5\mathrm{keV}$ on sheets I (for $\mathrm{Im}E>-{\mathrm{\Gamma }}_{D^{*0}}/2$) and II (for $\mathrm{Im}E<-{\mathrm{\Gamma }}_{D^{*0}}/2$) of the complex energy plane. Since the ${\overline{D}}^{*0}$ meson is treated as an unstable particle, the $D^0$ ${\overline{D}}^{*0}$ branch cut indicated by the black solid line is located at $\mathrm{Im}E=-{\mathrm{\Gamma }}_{D^{*0}}/2$. The location of the pole is on the physical sheet with respect to the $D^0$ ${\overline{D}}^{*0}$ system.

Figure 8

Confidence regions for the pole position on sheets II and IV in the complex energy plane. The displayed uncertainties include statistical contributions and the modeling uncertainty. The poles are extracted at a Flatté mass point of $m_0=3864.5\mathrm{MeV}$. The shaded areas are the 1, 2 and $3\sigma$ confidence regions. The branch cut is shown as the blue line. The location of the branch cut singularity is indicated with a vertical bar at $E=0+0i$. The best estimates for the pole position is indicated by a cross. The black points indicate the samples from the pseudoexperiments procedure that lie outside the $3\sigma$ region.

Figure 9

Confidence regions for the pole position on sheet III in the complex energy plane. The displayed uncertainties include statistical contributions and the modeling uncertainty. The poles are extracted at a Flatté mass point of $m_0=3864.5\mathrm{MeV}$. The shaded areas are the 1, 2 and $3\sigma$ confidence regions. The branch cut is shown as the blue line. The location of the branch cut singularity is indicated with a vertical bar at $E=0+0i$. The best estimate for the pole positions is indicated by a cross. The confidence region for the pole on sheets II/IV is shown in outline for comparison. The black points indicate the samples from the pseudoexperiments procedure that lie outside the $3\sigma$ region.

Figure 10

Confidence regions for the pole on sheet II in the complex energy plane. The displayed uncertainties include statistical contributions and the uncertainty from the choice of the Flatté mass parameter $m_0$. Modeling uncertainties are not shown. The shaded areas are the 1, 2 and $3\sigma$ confidence regions. The branch cut is shown as the blue line. The location of the branch cut singularity is indicated with a vertical bar at $E=0+0i$. The black circles indicate the best estimates for the pole position for the different choices of $m_0$.

Figure 11

Confidence regions for the pole on sheet II in the complex energy plane, in the case that the mass scale is shifted up by 0.066 MeV, due to systematic uncertainty of the momentum scale. Only the statistical uncertainties are displayed. The shaded areas are the 1, 2 and $3\sigma$ confidence regions. The cross indicates the location of the pole found in the default fit, with the nominal momentum scale. The branch cut is shown as the blue line. The location of the branch cut singularity is indicated with a vertical bar at $E=0+0i$.