# Study of the lineshape of the $\chi_{c1}(3872)$ state

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## Abstract

A study of the lineshape of the $\chi_{c1}(3872)$ state is made using a data sample corresponding to an integrated luminosity of $3$fb$^{-1}$ collected in $pp$ collisions at centre-of-mass energies of $7$ and $8$TeV with the LHCb detector. Candidate $\chi_{c1}(3872)$ 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, $\Delta m$, and the width of the $\chi_{c1}(3872)$ state, $\Gamma_{\mathrm{BW}}$, are determined to be \begin{eqnarray*} \Delta m & = & 185.588 \pm 0.067 \pm 0.068 {\mathrm{MeV}} , \\ \Gamma_{\mathrm{BW}} & = & \phantom{00}1.39\phantom{0} \pm 0.24\phantom{0} \pm 0.10\phantom{0} {\mathrm{MeV}} , \end{eqnarray*} where the first uncertainty is statistical and the second systematic. Using a Flatt\'e-inspired lineshape, two poles for the $\chi_{c1}(3872)$state in the complex energy plane are found. The dominant pole is compatible with a quasi-bound $D^0\bar{D}^{*0}$state but a quasi-virtual state is still allowed at the level of $2$standard deviations.

## Figures and captions

 Mass distributions for ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \pi ^+ \pi ^-$ 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. Fig_1.pdf [195 KiB] HiDef png [443 KiB] Thumbnail [303 KiB] *.C file tex code Mass distributions for ${ J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \pi ^+ \pi ^-$ candidates in the $\chi _{ c 1} (3872)$ 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. Fig_2.pdf [192 KiB] HiDef png [474 KiB] Thumbnail [381 KiB] *.C file tex code The coupling to the $D \overline{ D }{} ^*$ channels $g$ as a function of Flatt\'e energy parameter $E_f$ (black points with error bars). The corresponding change in negative log likelihood, $\Delta\mathrm{LL}$ is shown as well (red dots). Fig3.pdf [15 KiB] HiDef png [90 KiB] Thumbnail [72 KiB] *.C file Comparison of the Flatt\'e (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. Fig_4.pdf [111 KiB] HiDef png [191 KiB] Thumbnail [178 KiB] *.C file tex code Distribution of the FWHM obtained for simulated experiments generated from the result of the Flatt\'e 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 Flatt\'e lineshape, while the solid blue line indicates the value of the Breit--Wigner width observed in data. Fig_5.pdf [43 KiB] HiDef png [111 KiB] Thumbnail [115 KiB] *.C file tex code The phase of the Flatt\'e amplitude obtained from the fit to the data with $m_0=3864.5\mathrm{ Me V}$ on sheets I (for $\operatorname{Im} E>0$) and II (for $\operatorname{Im} E<0$) of the complex energy plane. The pole singularity is visible at $E_{\mathrm{II}}=(0.06 - 0.13\:i)\mathrm{ Me V}$. 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. Fig6.pdf [1 MiB] HiDef png [3 MiB] Thumbnail [767 KiB] *.C file The phase of the Flatt\'e amplitude as obtained from the fit with a finite $D ^{*0}$ width of $\Gamma_ D ^{*0} =65.5\mathrm{ ke V}$ on sheets I (for $\operatorname{Im} E>-\Gamma_{ D ^{*0} }/2$) and II (for $\operatorname{Im} E<-\Gamma_{ D ^{*0} }/2$) 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 $\operatorname{Im} E=-\Gamma_{ D ^{*0} }/2$. The location of the pole is on the physical sheet with respect to the $D ^0$ $\overline{ D }{} {}^{*0}$ system. Fig7.pdf [1 MiB] HiDef png [3 MiB] Thumbnail [764 KiB] *.C file Confidence regions for the pole positions in the complex energy plane. The displayed uncertainties include statistical contributions and the modeling uncertainty. The poles are extracted at a Flatt\'e mass point of $m_0=3864.5\mathrm{ Me V}$. (Left) Sheets II and IV. (Right) Sheet III. 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 $E=0+0\:i$. The best estimates for the pole positions are indicated by crosses. In the right plot 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. Fig_8.pdf [163 KiB] HiDef png [449 KiB] Thumbnail [314 KiB] *.C file tex code 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 Flatt\'e mass parameter $m_0$. Modelling 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 $E=0+0\:i$. The black circles indicate the best estimates for the pole position for the different choices of $m_0$. Fig9_small.pdf [23 KiB] HiDef png [460 KiB] Thumbnail [251 KiB] *.C file Confidence regions for the pole on sheet II in the complex energy plane, in the case that the mass scale is shifted up by $0.066\mathrm{ Me V}$, 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 $E=0+0\:i$. Fig10.pdf [1022 KiB] HiDef png [181 KiB] Thumbnail [140 KiB] *.C file Animated gif made out of all figures. PAPER-2020-008.gif Thumbnail

## Tables and captions

 Results of the $\psi {(2S)}$ mass and scale factor $s_f$ obtained for the nominal fit model. The quoted uncertainties on the $\psi {(2S)}$ mass and $s_f$ are statistical. Table_1.pdf [74 KiB] HiDef png [98 KiB] Thumbnail [46 KiB] tex code Results for $\Delta m$ and $\Gamma_{\mathrm{BW}}$ and $\chi _{ c 1} (3872)$ signal yields. The quoted uncertainties are statistical. Table_2.pdf [82 KiB] HiDef png [81 KiB] Thumbnail [39 KiB] tex code Systematic uncertainties on the measurement of the mass difference $\Delta m$. Table_3.pdf [41 KiB] HiDef png [64 KiB] Thumbnail [28 KiB] tex code Results from the constrained Flatt\'e fit. The uncertainties are statistical. Table_4.pdf [58 KiB] HiDef png [30 KiB] Thumbnail [14 KiB] tex code Systematic uncertainty on the measurement of the Flatt\'e parameters. Table_5.pdf [69 KiB] HiDef png [50 KiB] Thumbnail [23 KiB] tex code Results of the fit with the Flatt\'e lineshape including statistical and systematic uncertainties. The Flatt\'e mass parameter $m_0=3864.5\mathrm{ Me V}$ is used. Table_6.pdf [59 KiB] HiDef png [39 KiB] Thumbnail [17 KiB] tex code

## Supplementary Material [file]

 Supplementary material full pdf supple[..].pdf [255 KiB] Figures: - Fig*.* : figures ins different formats - Figure*.* : final (ready-to-use) figures in different format FigS1.pdf [13 KiB] HiDef png [93 KiB] Thumbnail [68 KiB] *C file FigS2.pdf [14 KiB] HiDef png [145 KiB] Thumbnail [124 KiB] *C file FigureS1.pdf [121 KiB] HiDef png [181 KiB] Thumbnail [113 KiB] *C file FigureS2.pdf [41 KiB] HiDef png [182 KiB] Thumbnail [223 KiB] *C file lhcb-logo.pdf [4 KiB] HiDef png [269 KiB] Thumbnail [142 KiB] *C file

Created on 15 January 2021.