Associated Higgs boson production with top quarks at the CERN Large Hadron Collider: NLO QCD corrections

We present in detail the calculation of the $O\left({\alpha }_s^3\right)$ inclusive total cross section for the process $p\overrightarrow{p}t\overline{t}h,$ in the standard model, at the CERN Large Hadron Collider with a center-of-mass energy $\sqrt{s_H}=14\mathrm{}\mathrm{TeV}.$ The calculation is based on the complete set of virtual and real $O\left({\alpha }_s\right)$ corrections to the parton level processes $q\overline{q}\to t\overline{t}h$ and $g\overrightarrow{g}t\overline{t}h,$ as well as the tree level processes $(q,\overline{q})\overrightarrow{g}t\overline{t}h+(q,\overline{q}).\mathrm{}$ The virtual corrections involve the computation of pentagon diagrams with several internal and external massive particles, first encountered in this process. The real corrections are computed using both the single and the two cutoff phase space slicing method. The next-to-leading order QCD corrections significantly reduce the renormalization and factorization scale dependence of the Born cross section and moderately increase the Born cross section for values of the renormalization and factorization scales above $m_t.$