  The prompt production of charmonium $\chi_{c}$ and $J/\psi$ states is studied
in proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV at
the Large Hadron Collider. The $\chi_{c}$ and $J/\psi$ mesons are identified
through their decays $\chi_{c}\rightarrow J/\psi \gamma$ and $J/\psi\rightarrow
\mu^+\mu^-$ using 36 pb$^{-1}$ of data collected by the LHCb detector in 2010.
The ratio of the prompt production cross-sections for $\chi_{c}$ and $J/\psi$,
$\sigma (\chi_{c}\rightarrow J/\psi \gamma)/ \sigma (J/\psi)$, is determined as
a function of the $J/\psi$ transverse momentum in the range $2 < p_{\mathrm
T}^{J/\psi} < 15$ GeV/$c$. The results are in excellent agreement with
next-to-leading order non-relativistic expectations and show a significant
discrepancy compared with the colour singlet model prediction at leading order,
especially in the low $p_{\mathrm T}^{J/\psi}$ region.
