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The prompt production of the charmonium $\chi_{c1}$ and $\chi_{c2}$ mesonshas been studied in proton-proton collisions at the Large Hadron Collider at acentre-of-mass energy of $\sqrt{s}=7$ TeV. The $\chi_c$ mesons are identifiedthrough their decays $\chi_c\to J/\psi\,\gamma$ with $J/\psi \to \mu^+ \mu^-$using 36 $\mathrm{pb^{-1}}$ of data collected by the LHCb detector in 2010. Theratio of the prompt production cross-sections for the two $\chi_c$ spin states,$\sigma(\chi_{c2})/\sigma(\chi_{c1})$, has been determined as a function of the$J/\psi$ transverse momentum, $p_{\mathrm{T}}^{J/\psi}$, in the range from 2 to15 GeV/$c$. The results are in agreement with the next-to-leading ordernon-relativistic QCD model at high $p_{\mathrm{T}}^{J/\psi}$ and lieconsistently above the pure leading-order colour singlet prediction.

Measurement of the cross-section ratio $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ for prompt $\chi_c$ production at $\sqrt{s}=7$ TeV