Exploring the Flavor Symmetry Landscape

First of all, we would like to thank the referees for their time and effort in reviewing the manuscript and apologize for the delay in our response. In the following, we address the points raised.


"I would like the authors to clarify the advantage to the correlations between EFT operators kept in their approach, particularly if one can exploit these correlations to devise experimental search strategies to improve discovery potential at current/and future machines."

The word ``correlate'' appears in several places in our article and was perhaps used ambiguously. The correlation we refer to should not be understood in the statistical sense but rather in the possibility of connecting the results of direct and indirect searches. Such a connection is only possible when one has a concrete picture of a Beyond the Standard Model scenario. Indeed, only through explicit BSM models we can interpret current experimental results and future projections all within the same parameter space we use for both indirect and direct searches, in our case the $m_*/g_*$ plane. Understanding how indirect searches affect each other and how they affect direct searches is not possible within a model agnostic EFT framework. In order to address this potential confusion we made a few adjustments in the new version of our manuscript. In particular, we added some clarification at the end of the Introduction and replaced some of the ``correlate'''s in the text, with more suitable terms.


"Second, given the focus on FCC-hh in the paper, I think comments on the indirect potential of FCC-ee are in order, given that the machine is part of the same program and is proposed to arrive much earlier."

We agree with the referee that the indirect FCC-ee potential is definitely an important aspect to be investigated, as we explicitly mention in the paper at the end of section 7.1. Yet, we decided not to include any discussion on FCC-ee in our study mainly for two reasons. The first one is brevity: while interesting, assessing the indirect potential of such a machine would require a separate and in-depth study that was outside the scope of our work. The second reason is that the logic we followed was to compare how the \emph{indirect} constraints available now and in the near future -- say within the next 10-15 years -- would influence direct searches. Since FCC-ee is neither a ``near future'' experiment nor a machine capable of such direct searches, we decided not to include it in our discussion.


"Finally, since the optimal flavor symmetries identified in this work appear very close to $U(2)^n$ symmetries appearing in the literature that have long been recognized to provide a good compromise for simultaneously passing flavor, collider, and EW bounds in a wide variety of contexts (i.e. in EFT analyses and in concrete models), I think a more exhaustive inclusion of references would be appropriate."

In the paper, we often refer to previous literature and specifically to previous studies on "U(2)^n" symmetric models.

In particular, in Sec.m4.3.1 we write

\begin{quote}
Models with pRU are somewhat reminiscent of the $U(2)_{\rm RC}^3$ scenarios studied in \cite{Barbieri:2012bh, Barbieri:2012tu, Barbieri:2012uh}. But in principle, the two approaches differ significantly, both in the underlying symmetry hypothesis as well as in the way that the symmetry is broken. Indeed, in \cite{Barbieri:2012bh, Barbieri:2012tu, Barbieri:2012uh} a $U(2)^3$ symmetry is approximately shared by both the composite sector and the fundamental fermions $q_L,u_R,d_R$, and a priori may be broken by any of the bilinear couplings $\overline\psi{\cal O}_\psi$ allowed by gauge invariance. The authors decide to focus on the minimal set of symmetry-breaking spurions that can reproduce the SM masses and mixing angles, but other choices may be made.
\end{quote}

We mention again these models in the Summary in Sec. 7, namely

\begin{quote}
A number of the models we have studied, the optimal ones in particular, are not new and have previously appeared in the literature.$^{26}$
\end{quote}

with footnote $26$

\begin{quote}
More precisely the puRU model was discussed in \cite{Redi:2012uj} while the $U(2)^2$ models of \cite{Barbieri:2012bh,Barbieri:2012tu,Barbieri:2012uh} practically coincide with our pRU and pLU models with a further restriction of their parameters.
\end{quote}

If the referee feels that any important references are missing, either here or elsewhere in the paper, we would be grateful for their suggestions.

- On pg 19, We added the explicit definition of the CKM matrix in terms of the matrices $U_u$ and $U_d$

- On pg 39, We changed the sentence above eq. (4.78) for clarifications on the constraint from $C_7^\prime$.

- In section 4.3.3, we clarified some of the logic. In particular, we modified a sentence at the end of pg 39 to better clarify the relation between pRU and puRU constraints, and at the end of the section to clarify the discussion on the ``optimal" range for the $\varepsilon$ parameters.

- In section 6, a section of the discussion on Partial lepton universality was reworked to include the interplay between $\mu\to e \gamma$ and $\mu \text{Au} \to e \text{Au}$ measurements in constraining flavor changing dipole operators. Similarly in the conclusions, the last bullet point of section 7.2 was modified to include this discussion.

- Section 7 was slightly restructured. In particular, section 7.1 ``Main results" was divided into two subsections and a small introduction to section 7 was added.

- In figures 5 to 9, we added a line with the bounds coming from direct searches of heavy spin-1 resonances. A discussion of these constraints was added at the end of App. C.6. These bounds are referenced in the summaries of the various models when relevant.

- In App. A.1, we made some small changes and corrections, in both the text and footnote 30.

- In App. C.2, the discussion was significantly changed. In particular, we added the study of the contribution to the $T$ parameter coming from RG effects. The appendix was further split into subsections, separately discussing the phenomenology for different representations of the composite fermions.