Study of the single top production in the $t$-channel with the CMS experiment at the LHC

We present a measurement of the inclusive single top $t$-channel production cross section in proton-proton collisions at the LHC, using data collected with the CMS experiment in 2011 and 2012 at a centre-of-mass energy of 7 TeV and 8 TeV, respectively. The 7 TeV analysis makes use of two different and complementary approaches. The first is a template fit analysis using background estimates determined from control samples in data. The second is based on multivariate techniques that probe the compatibility of the candidate events with the signal. The update of the template fit analysis with 8 TeV data is also presented together with the measurement of the $t$-channel single top quark and anti-quark production ratio. The cross section measurements are also used to determine the CKM matrix element $\lvert V_{\text{tb}}\rvert $