  With the goal of increasing the precision of NLO QCD predictions for the
$pp\to t\bar{t} \gamma$ process in the di-lepton top quark decay channel we
present theoretical predictions for the ${\cal R}=
\sigma_{t\bar{t}\gamma}/\sigma_{t\bar{t}}$ 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 $t\bar{t}$ and $t\bar{t}\gamma$ production are employed. They
are based on matrix elements for $e^+\nu_e \mu^- \bar{\nu}_\mu b\bar{b}$ and
$e^+\nu_e \mu^- \bar{\nu}_\mu b\bar{b}\gamma$ 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.
