Distribution of $\Delta M\,{=}\,M\left(\mu ^+ \,\mu ^- \,\gamma\right)\,{-}\,M\left(\mu ^+ \,\mu ^- \right)$ for selected candidates with $ 3\,{<}\, p_{\mathrm{T}}^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }\,{<}\, {15}\:{ {\mathrm{\,Ge V\!/}c} }$ for (a) converted and (b) non-converted photons. The lower solid curves correspond to the $\chi _{ c 0}$ , $\chi _{ c 1}$ and $\chi _{ c 2}$ peaks from left to right, respectively (the $\chi _{ c 0}$ peak is barely visible). The background distribution is shown as a dashed curve. The upper solid curve corresponds to the overall fit function.

Reconstruction and selection efficiency ratios in bins of $ p_{\mathrm{T}}^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }$ . The ratio of the $ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu}$ efficiency ( $\epsilon^{\chi _{ c 2} }_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }\,{/}\,\epsilon^{\chi _{ c 1} }_{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }$ ) is shown with red circles. The ratio of the photon reconstruction and selection efficiency times the $\chi _{c}$ selection efficiency ( $\epsilon^{\chi _{ c 2} }_{\gamma}\epsilon^{\chi _{ c 2} }_{\mathrm{sel}}\,{/}\,\epsilon^{\chi _{ c 1} }_{\gamma}\epsilon^{\chi _{ c 1} }_{\mathrm{sel}}$ ) is shown with blue triangles.

Ratio $\sigma(\chi _{ c 2} )\,/\,\sigma(\chi _{ c 1} )$ in bins of $ 2\,{<}\, p_{\mathrm{T}}^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }\,{<}\, {15}\:{ {\mathrm{\,Ge V\!/}c} }$ . The LHCb results, in the rapidity range $ 2.0\,{<}\, y^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }\,{<}\,4.5$ and assuming the production of unpolarised $\chi _{c}$ mesons, are shown with solid black circles and the internal error bars correspond to the statistical error; the external error bars include the contribution from the systematic uncertainties (apart from the polarisation). The lines surrounding the data points show the maximum effect of the unknown $\chi _{c}$ \ polarisations on the result. The upper and lower limits correspond to the spin states as described in the text. The CDF data points, at $\sqrt{s}\,{=}\, {1.96}\:{\mathrm{\,Te V} }$ in $ p\bar{p}$ collisions and in the $ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu}$ pseudo-rapidity range $ |\eta^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }|<1.0$ , are shown in (a) with open blue circles [6]. The two hatched bands in (b) correspond to the ChiGen Monte Carlo generator [15] and NLO NRQCD [3] predictions.

Animated gif made out of all figures.

Signal $\chi _{c}$ yields and fit quality from the fit to the converted and non-converted candidates in each $ p_{\mathrm{T}}^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }$ bin.

Polarisation weights in $ p_{\mathrm{T}}^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }$ bins for different combinations of $\chi _{ c 1}$ and $\chi _{ c 2}$ polarisation states $ |J,m_{\chi_{cJ}}\rangle$ with $ |m_{\chi_{cJ}}|=0,\cdots J$ . The polarisation axis is defined as the direction of the $\chi _{c}$ in the laboratory frame. Unpol. means the $\chi _{c}$ is unpolarised.

Summary of the systematic uncertainties (absolute values) on $\sigma(\chi _{ c 2} )\,/\,\sigma(\chi _{ c 1} )$ \ in each $ p_{\mathrm{T}}^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }$ bin.

Ratio $\sigma(\chi _{ c 2} )\,/\,\sigma(\chi _{ c 1} )$ in bins of $ p_{\mathrm{T}}^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }$ in the range $ 2\,{<}\, p_{\mathrm{T}}^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }\,{<}\, {15}\:{ {\mathrm{\,Ge V\!/}c} }$ and in the rapidity range $ 2.0\,{<}\, y^{ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} }\,{<}\,4.5$ . The first error is the statistical error, the second is the systematic uncertainty (apart from the branching fraction and polarisation) and the third is due to the $\chi _{c} \,\rightarrow\, { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \,\gamma$ branching fractions. Also given is the maximum effect of the unknown $\chi _{c}$ \ polarisations on the result as described in Sect. 3.2.