With the goal of increasing the precision of NLO QCD predictions for the $ppto tbar{t} gamma$ process in the di-lepton top quark decay channel we present theoretical predictions for the ${cal R}= sigma_{tbar{t}gamma}/sigma_{tbar{t}}$ cross section ratio. Results for the latter together with various differential cross section ratios are given for the LHC with the Run II energy of $sqrt{s} = 13$ TeV. Fully realistic NLO computations for $tbar{t}$ and $tbar{t}gamma$ production are employed. They are based on matrix elements for $e^+ u_e mu^- bar{ u}_mu bbar{b}$ and $e^+ u_e mu^- bar{ u}_mu bbar{b}gamma$ processes and include all resonant and non-resonant diagrams, interferences, and off-shell effects of the top quarks and the $W$ gauge bosons. Various renormalisation and factorisation scale choices and parton density functions are examined to assess their impact on the cross section ratio. Depending on the transverse momentum cut on the hard photon a judicious choice of a dynamical scale allows us to obtain $1%-3%$ percent precision on ${cal R}$. Moreover, for differential cross section ratios theoretical uncertainties in the range of $1%-6%$ have been estimated. Until now such high precision predictions have only been reserved for the top quark pair production at NNLO QCD. Thus, ${cal R}$ at NLO in QCD represents a very precise observable to be measured at the LHC for example to study the top quark charge asymmetry or to probe the strength and the structure of the $t$-$bar{t}$-$gamma$ vertex. The latter can shed some light on possible new physics that can reveal itself only once sufficiently precise theoretical predictions are available.
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