This article presents measurements of the $t$-channel single top-quark ($t$) and top-antiquark ($\bar{t}$) total production cross sections $\sigma(tq)$ and $\sigma(\bar{t}q)$, their ratio $R_{t}=\sigma(tq)/\sigma(\bar{t}q)$, and a measurement of the inclusive production cross section $\sigma(tq + \bar{t}q)$ in proton--proton collisions at $\sqrt{s} = 7$ TeV at the LHC. Differential cross sections for the $tq$ and $\bar{t}q$ processes are measured as a function of the transverse momentum and the absolute value of the rapidity of $t$ and $\bar{t}$, respectively. The analyzed data set was recorded with the ATLAS detector and corresponds to an integrated luminosity of 4.59 fb$^{-1}$. Selected events contain one charged lepton, large missing transverse momentum, and two or three jets. The cross sections are measured by performing a binned maximum-likelihood fit to the output distributions of neural networks. The resulting measurements are $\sigma(tq)= 46\pm 6\; \mathrm{pb}$, $\sigma(\bar{t}q)= 23 \pm 4\; \mathrm{pb}$, $R_{t}=2.04\pm 0.18$, and $\sigma(tq + \bar{t}q)= 68 \pm 8\; \mathrm{pb}$, consistent with the Standard Model expectation. The uncertainty on the measured cross sections is dominated by systematic uncertainties, while the uncertainty on $R_{t}$ is mainly statistical. Using the ratio of $\sigma(tq + \bar{t}q)$ to its theoretical prediction, and assuming that the top-quark-related CKM matrix elements obey the relation $|V_{tb}|\gg |V_{ts}|, |V_{td}|$, we determine $|V_{tb}|=1.02 \pm 0.07$.
Predicted and observed events yields for the 2-jet and 3-jet channels considered in this measurement. The multijet background is estimated using data-driven techniques (see Sec. VB); an uncertainty of $50\%$ is applied. All the other expectations are derived using theoretical cross sections and their uncertainties (see Secs. VA and VC in the paper).
Differential t-channel top-quark production cross sections and normalized differential t-channel top-quark production cross sections as functions of PT(TOP).
Detailed list of the contribution of each source of uncertainty to the total uncertainty on the measured values of $\sigma(tq)$, $\sigma(\bar{t}q)$, $R_t$, and $\sigma(tq+\bar{t}q)$. The evaluation of the systematic uncertainties has a statistical uncertainty of $0.3\,\%$. Uncertainties contributing less than $1.0\,\%$ are marked with "$<1$" in the paper. To provide numerical values for this table in HEPdata, these uncertainties are approximated with $\pm 0.5\,\%$. This approximation is applied to all measurements for the following uncertainties$:$ JES statistical, JES physics modeling, JES mixed detector and modeling, JES close-by-jets, JES pileup, $b$-JES, jet vertex fraction, mistag efficiency and $W+\;$jets shape variation. For the measurement of $\sigma(tq)$ the approximation is applied in addition to the following uncertainties$:$ JES flavor response, $c$-tagging efficiency, $t\bar{t}$ generator + parton shower and $t\bar{t}$ ISR/FSR. For the measurement of $\sigma(\bar{t}q)$ the approximation is applied in addition to these uncertainties$:$ JES flavor response, $b/\bar{b}$ acceptance, and $t\bar{t}$ ISR/FSR. For the measurement of $R_t$ the approximation is applied in addition to these uncertainties$:$ JES detector, $b$-tagging efficiency, $c$-tagging efficiency, $b/\bar{b}$ acceptance and $tq$ scale variations. For the measurement of $\sigma(tq+\bar{t}q)$ the approximation is applied in addition to these uncertainties$:$ JES flavor response, $c$-tagging efficiency, $b/\bar{b}$ acceptance, $t\bar{t}$ generator + parton shower and $t\bar{t}$ ISR/FSR.