Coloring mixed QCD/QED evolution

\section*{Report 1}
We would like to acknowledge the work of the Referee in producing detailed comments. Unfortunately, it appears that some crucial aspects of the manuscript might have gone unnoticed when preparing the review. We have previously already commented in detail on the individual points that the referee prepared, so that we refrain from repetition.

In any case, we have seriously re-considered the manuscript in light of the report, since the questions by the referee indicated that our presentation was not clear enough. Consequently, we have made a significant effort to improve the presentation. In particular, we have
\begin{itemize}
\item refined the notation in Section II.B
\item added a number of clarifying comments, and a clarifying equation to Section II.B
\item extended the discussion of matrix-element corrections and their benefits in Section II.C, and clarified the notation w.r.t. Section II.B
\item clarified the differences between leading-color and fixed-color matrix element corrections.
\end{itemize}
These changes have, we believe, led to a significant clarification of the method and novel developments, and thus should address the (potential cause of the) questions raised by the referee.



\section{Report 2}

We would like to thank Referee 2 for the detailed and constructive feedback. In the following, we adress the requested changes one by one.

\begin{enumerate}
\item {\bf In Eq. (1), after the 1st term in the 2nd line, the dot (before +) should be removed.}

Done.

\item {\bf Plots in Figs. 1-5 are a bit too small - in the printed version of the paper they are barely legible.}

We increased the size of Figs. 1 \& 2, and the script size of Figs. 3-5 to improve legibility.

\item {\bf In Figs. 3-5, the authors do not specify the meaning of the 3rd axis (corresponding to colors). If it corresponds to the number of generated events, it would be better - in my opinion - to normalize the corresponding histograms to 1.}

We normalized the histograms to 1 and mention the meaning of the third axis in the caption of Fig. 3.

\item {\bf In Figs. 3-5, left plots: in the horizontal axis, $p_\perp$ should be divided by the physical unit (e.g., $p_\perp$/GeV), as the argument of the log-function should be dimensionless.}

We added the unit in the respective axis labels.

\item {\bf In Figs. 6-9, the physical unit (GeV, I suppose) for the invariant mass in the horizontal axis is missing.}

We added the units in Figs. 6-9.

\item {\bf In Fig. 8, there is a spike in the ratio distribution at the invariant mass value of 140 which probably results from statistical fluctuations - I would recommend generating a higher statistics sample in order to clarify this.}

Done. With increased statistics, the spurious spike is significantly reduced.

\item {\bf In Fig. 9, the ratio distribution in the RHS plot seems to be cut at the top - the scale in the vertical axis is too small.}

Done. We increased the upper scale plot limit.

\item {\bf On page 7, in the 3rd line of the 2nd paragraph the authors write: "the amplitude will be modified by a factor 1". Does this mean that it will be unmodified, or something is missing there?}

Yes, we updated the text to be clearer.

\item {\bf Two lines below the above, the phrase "with different but identical quark color indices" sounds like a contradiction.}

We clarified the text. We mean that color and anticolor index of the gluon are identical, and different from the $q \bar q$ color index.

\item {\bf On page 6, in footnote 2, the authors introduce the abbreviation FC without explanation of its meaning, similarly for the abbreviations LC and MEC on page 7.}

We do not use FC and LC in the text any longer, and introduce MEC before its further use.

\item {\bf Section II.A, in the 1st line of the 2nd paragraph: "evolution variables" $\rightarrow$ "evolution variable".}

Done.

\item {\bf Appendix A: in the bottom left box: "hisories" $\rightarrow$ "histories".}

Done.

\end{enumerate}

Referee report 2 suggested several improvements of the manuscript. We have performed the relevant changes and documented these in the author comments. Referee report 1 raised several questions which we have addressed by changing and improving notation. Again, these changes are more conveniently documented in the author comments. Below, we list additional minor changes mainly related to typos.

\section{Additional changes}
Besides the corrections mentioned above, we included the following minor corrections:
\begin{itemize}
\item Switch last two color flows in first line of Eq. 1.
\item Add brackets and a factor of 2 in Eq. 3.
\item Fix typesetting of sub-indices in Eq. 6.
\item Remove word "additional" below Eq. 20.
\item Remove one of two "both"s in caption of Fig. 9.
\item Add some missing arrow heads and fix spacing in Fig. 10.
\item Fix some umlauts in references.
\end{itemize}