Production of the charmonium states<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>in proton nucleus interactions at<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msqrt><mml:mi>s</mml:mi></mml:msqrt><mml:mo>=</mml:mo><mml:mn>41.6</mml:mn><mml:mtext> </mml:mtext><mml:mtext> </mml:mtext><mml:mi>GeV</mml:mi></mml:math>

A measurement of the ratio ${R}_{{\ensuremath{\chi}}_{c}}=({\ensuremath{\chi}}_{c}\ensuremath{\rightarrow}J/\ensuremath{\psi}+\ensuremath{\gamma})/J/\ensuremath{\psi}$ in $p\mathrm{C}$, $p\mathrm{Ti}$, and $p\mathrm{W}$ interactions at $920\text{ }\text{ }\mathrm{GeV}/c$ ($\sqrt{s}=41.6\text{ }\text{ }\mathrm{GeV}$) in the Feynman-x range $\ensuremath{-}0.35&lt;{x}_{F}^{J/\ensuremath{\psi}}&lt;0.15$ is presented. Both ${\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and ${e}^{+}{e}^{\ensuremath{-}}$ $J/\ensuremath{\psi}$ decay channels are observed with an overall statistics of about 15 000 ${\ensuremath{\chi}}_{c}$ events, which is by far the largest available sample in pA collisions. The result is ${R}_{{\ensuremath{\chi}}_{c}}=0.188\ifmmode\pm\else\textpm\fi{}{0.013}_{\mathrm{st}}{\genfrac{}{}{0}{}{+0.024}{\ensuremath{-}0.022}}_{\mathrm{sys}}$ averaged over the different materials, when no $J/\ensuremath{\psi}$ and ${\ensuremath{\chi}}_{c}$ polarizations are considered. The ${\ensuremath{\chi}}_{c1}$ to ${\ensuremath{\chi}}_{c2}$ production ratio ${R}_{12}={R}_{{\ensuremath{\chi}}_{c}1}/{R}_{{\ensuremath{\chi}}_{c}2}$ is measured to be $1.02\ifmmode\pm\else\textpm\fi{}0.40$, leading to a cross section ratio $\frac{\ensuremath{\sigma}({\ensuremath{\chi}}_{c1})}{\ensuremath{\sigma}({\ensuremath{\chi}}_{c2})}=0.57\ifmmode\pm\else\textpm\fi{}0.23$. The dependence of ${R}_{{\ensuremath{\chi}}_{c}}$ on the Feynman-x of the $J/\ensuremath{\psi}$, ${x}_{F}^{J/\ensuremath{\psi}}$, and its transverse momentum, ${p}_{T}^{J/\ensuremath{\psi}}$, is studied, as well as its dependence on the atomic number, A, of the target. For the first time, an extensive study of possible biases on ${R}_{{\ensuremath{\chi}}_{c}}$ and ${R}_{12}$ due to the dependence of acceptance on the polarization states of $J/\ensuremath{\psi}$ and ${\ensuremath{\chi}}_{c}$ is performed. By varying the polarization parameter, ${\ensuremath{\lambda}}_{\mathrm{obs}}$, of all produced $J/\ensuremath{\psi}$'s by two sigma around the value measured by HERA-B, and considering the maximum variation due to the possible ${\ensuremath{\chi}}_{c1}$ and ${\ensuremath{\chi}}_{c2}$ polarizations, it is shown that ${R}_{{\ensuremath{\chi}}_{c}}$ could change by a factor between 1.02 and 1.21 and ${R}_{12}$ by a factor between 0.89 and 1.16.