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Carbon-bearing molecules, particularly CO, have been widely used as tracersof molecular gas in the interstellar medium (ISM). In this work, we aim tostudy the properties of molecules in diffuse, cold environments, where CO tendsto be under-abundant and/or sub-thermally excited. We performed one of the mostsensitive (down to $\mathrm{\tau_{rms}^{CO} \sim 0.002}$ and$\mathrm{\tau_{rms}^{HCO^+} \sim 0.0008}$) sub-millimeter molecular absorptionline observations towards 13 continuum sources with the ALMA. CO absorption wasdetected in diffuse ISM down to $\mathrm{A_v< 0.32\,mag}$ and \hcop was down to$\mathrm{A_v < 0.2\,mag}$, where atomic gas and dark molecular gas (DMG) startsto dominate. Multiple transitions measured in absorption toward 3C454.3 allowfor a direct determination of excitation temperatures $\mathrm{T_{ex}}$ of4.1\,K and 2.7\,K, for CO and for \hcop, respectively, which are close to thecosmic microwave background (CMB) and provide explanation for their beingundercounted in emission surveys. A stronger linear correlation was foundbetween $\mathrm{N_{HCO^+}}$ and $\mathrm{N_{H_2}}$ (Pearson correlationcoefficient P $\sim$ 0.93) than that of $\mathrm{N_{CO}}$ and$\mathrm{N_{H_2}}$ (P $\sim$ 0.33), suggesting \hcop\ being a better tracer ofH$_2$ than CO in diffuse gas. The derived CO-to-\h2 conversion factor (the COX-factor) of (14 $\pm$ 3) $\times$ 10$^{20}$ cm$^{-2}$ (K \kms)$^{-1}$ isapproximately 6 times larger than the average value found in the Milky Way.

Revealing The CO X-factor In Dark Molecular Gas through Sensitive ALMA Absorption Observations