Low-energy effective description of dark $Sp(4)$ theories

First of all, I wish to apologize to the authors for the delay in providing this report.

In this manuscript the authors study a dark $Sp(4)_c$ gauge theory with $N_f = 2$ fermions in the fundamental representation, which is a promising and fairly minimal candidate to accommodate Strongly Interacting Massive Particle (SIMP) dark matter. They discuss in detail the effective chiral Lagrangian for the pseudo-Goldstone bosons (where the pattern of spontaneous symmetry breaking is $SU(4)/Sp(4)$), including the anomalous Wess-Zumino/Witten terms as well as the lightest spin-1 resonances. They do so both for mass degenerate and mass non-degenerate ($m_u \neq m_d$) quarks. They also analyze the gauging of a dark $U(1)'$ symmetry, which often plays an important role in SIMP models where the associated gauge boson (dark photon) acts as mediator transferring entropy from the dark to the visible sector. Finally, in the last section they provide results of a lattice study of this theory in the case $m_u \neq m_d$ (the degenerate case had been studied before in Ref. [6]), which allows them to determine the low-energy constants and the range of validity of the chiral effective theory from first-principles calculations in the underlying $Sp(4)_c$ gauge/fermion theory. A long series of technical appendices completes the paper; Appendix D is especially useful, providing details on the lattice setup and the analysis of systematic effects.

SIMPs are compelling candidates for particle dark matter, and the fact that they find 'natural' embedding into dark QCD-like theories implies the need to confront the challenges inherent to strong dynamics in order to assess the viability of concrete models. In my opinion, the present manuscript does achieve the goal stated by the authors: by performing a thorough and self-consistent study of the $Sp(4)_c$ model, including genuinely novel aspects such as non-degenerate dark quark masses, it potentially opens the way for future studies in dark matter model building and phenomenology. In addition, the text is clearly written. As a result, I believe the paper deserves to be accepted for publication in SciPost Physics. However, there are a number of questions or comments that I ask the authors to consider in a revised version, with the goal of improving the quality of presentation:

1] One question with an eye to SIMP phenomenology. One of the main results of this work is the detailed study of the $m_u \neq m_d$ scenario, where the flavor symmetry is explicitly broken to $SU(2)_u \times SU(2)_d$, under which the diagonal Goldstone $\pi^C$ transforms as a singlet. The $\pi^C$ is also the lightest meson, as shown in section 6. From the discussion in Sec. 4 (page 17) I understand that $\pi^C$ is always unstable once a dark $U(1)'$ is gauged, regardless of the choice of the charge matrix $\mathcal{Q}$. If this is correct, then rendering such scenario compatible with cosmological/astrophysical constraints would seem to be an important challenge, deserving to be at least mentioned here. Do the authors agree with my reading?

2] A few comments regarding references. a) On the last line of page 1, [22] is cited for 'including dark vector resonances into the SIMP paradigm', however it seems to me that [33] deserves to be included here as well. b) On page 9, when introducing the $\pi^5$ interaction only the Witten paper [39] is cited, but I think it would be fair to also include the classic work by Wess and Zumino. c) While the authors have in mind the application to (SIMP) dark matter, in the broader context of the Introduction it may be worthwhile to reference also previous papers that studied the $SU(4)/Sp(4)$ coset in the context of electroweak symmetry breaking, such as 0902.1483 by Gripaios et al. and 1001.1361 by Galloway et al. This last point is merely a suggestion, which the authors are free to accept or decline.

3] I was somewhat confused by the treatment of the pseudoscalar singlet $\eta$ in section 3.2. In the QCD case one can add the $\eta'$ along similar lines, by extending ChPT to $U(3)_L \times U(3)_R$, however a 'hard breaking' mass term $\delta \mathcal{L} = - m_0^2 \eta^{\prime\,2} / 2$ is also included, to account for the effect of the anomaly of $U(1)_A$ with respect to QCD. Such treatment yields for example reasonable leading-order approximations for $\eta, \eta' \to \gamma\gamma$. Why is such an extra mass term not considered here?

4] In the discussion of the chiral Lagrangian with spin-1 resonances (section 3.3), I think there may be an issue with parameter normalizations. Equation (3.22) and those below it have identical form to those in [41], but there $F_\pi = \sqrt{2} f_\pi^{\rm here}\approx 132$ MeV. In addition, I believe there $g$ is also different from $g_{\rho \pi\pi}^{\rm here}$. I noticed this because plugging the KSRF relation into Eq. (3.23) does not seem to give $Z^2 = 1/2$ as it should. In the definition of $\rho_{\mu\nu}$ after equation (3.20), the sign convention for the gauge coupling seems to be opposite compared to the choice made in section 2.

5] Some suggestions about notation. a) I think it would be useful to give explicitly, in section 2, the general form $M = \mathrm{diag}\,(m_u, m_d, m_u, m_d)$ which is currently defined only implicitly. b) I found equation (3.5) slightly confusing, in the sense that it seems $\mathcal{L}_{\rm mass}$ is defined as $+(\mu^3/2) ( \mathrm{Tr}[ M \Sigma] + \mathrm{h.c.} )$, i.e. with the opposite sign compared to the preceding equation. Perhaps in Eq. (3.5) the LHS could just be removed? c) In equation (3.21), I suggest to replace $[ ... ]$ with $( ... )$. d) In Eq. (4.9) it is presumably assumed that $\mathcal{Q}$ is diagonal (in general, the last instance would be $\mathcal{Q}^T$). It would be helpful to state it explicitly.

6] In the last paragraph of page 25 it is stated that the choices $m_\rho/m_\pi \approx 1.15, 1.25, 1.4$ 'are suggested by existing phenomenological investigations of such theories as dark matter candidates [33]'. However, in Ref. [33] the chosen benchmarks ranged from $1.6$ to $1.9$. Later, on page 30, it is mentioned that ensembles with $m_\rho / m_\pi \gg 1.4$ are not available for computational limitations. This is of course very reasonable, but I think it would help the reader to point it out earlier, already on page 25.

7] On page 8, lines 4 to 7, one reads 'Third, we explore the various options for gauging ...', but isn't this done in section 4 actually, rather than in section 3?

8] The paper contains a rather large number of typos. I mention here those that I caught, and recommend the authors perform a further check. Pages 2, 11, 'Golstone' or 'Golstones'; page 3, 'are in the pseudoreal' $\to$ 'are in a pseudoreal'; page 7, 'all members of meson multiplet' $\to$ 'all members of a meson multiplet'; same page, 'see Tab. 2.3' presumably should be 'see Tab. 1'; page 9, 'the a five-dimensional'; page 12, 'iso-siglet'; page 13, 'heaver'; page 19, 'withh'; page 20, 'Comparison with (4.14)' $\to$ 'Comparison with (4.15)'?; page 24, 'axialcurrents'; page 25, 'remaing', 'degenaracy', 'predctions'; page 26, 'chosing'; page 27, 'mutliplicative'; same page, GMOR has not been defined yet; page 28, 'the the'; page 32, 'Note that for with'; page 33, 'unambgious'; page 36, 'off-diagional'; page 37, 'given in by'; page 42, 'projecting to zero momentum and study' $\to$ 'projecting to zero momentum and studying'; page 43, 'efffects', 'the the'; page 45, should it be '$m(\rho^M)/m_{\pi}^{\rm deg} > 2$' ? Also, 'wheres'; page 48, 'they are can be'.

In summary, I am of the opinion that once these aspects are addressed, the paper will be acceptable for publication in SciPost Physics.

1.In the paper, a variant of the SM extension with the use of the symplectic symmetry group, Sp(4), is formulated in sufficient detail and mathematically accurately.
2. Some estimates of the mass spectrum of the model obtained in lattice calculations are presented.
3. In the work, dark quarks are not representations of the EW group, but are connected to the SM through a new Abelian vector field that mixes with ordinary gauge fields (so-called “vector portal”). An explicit splitting of the quark masses is introduced, and the phenomenology of dark mesons is proposed to be considered within the framework of an analogue of the chiral perturbation theory of QCD, i.e. nonlinear implementation of the global symmetry of the model.

1.At the same time, the “vector portal” phenomenology is long and well studied in a bulk of papers.
2.The effective theory, which is discussed in the first five sections, is standard for this science. Apart from taking into account the effects of explicit symmetry breaking by the splitting of quark masses, nothing new has been done here. Exactly the same nonlinear chiral effective theory was built before. (Pseudo)vector mesons were taken into account in the same way in the framework of this scheme; pseudo-goldstones and eta meson - even more so. The classification of states is common for any implementation of symmetry. Accounting for the anomaly through the WZW interaction was done. The new vector field also does not represent a big innovation, since technically, for the effective Lagrangian there is no difference what kind of abelian field is there - new or the same as that already exists in the SM.
3. Although the authors refer to work [7], but also in [11] the WZW-term was already considered, as in other papers, so it should be noted the obvious negligence of the authors in providing links and discussing results in similar scenarios obtained by other authors.
4.Similarly, the lack of completeness of the reference list is also evident when reading peppy, but somewhat premature statements about future research on phenomenology in scenarios of this type, despite the fact that specific predictions of dark matter effects have long been studied in the Sp(4) extension of the SM. However, there are no references to these works (except for review [32]) in the article.

I hope that the authors will be able to conduct a more detailed and accurate analysis of the results obtained by other authors and expand the list of references. They will also be able to indicate, at least in advance, which physical effects and manifestations of the dark matter candidates they propose can be observed in direct experiments or indirect data in various energy regions.
It would also be interesting to see the effects of mixing new composite scalars with the Higgs taken into account within this scheme.
The article as a whole satisfies the criteria for publication in this journal and can be published after the specified additions and revision of the text.

As already noted, it is recommended to carry out a more complete analysis of the results obtained earlier
1. both in the study of the mathematical structure of symplectic extensions with a vector portal
2. and in the discussion of the phenomenological manifestations of dark matter and the possibility of their direct or indirect detection.
So, the list of references should be extended.
3. Some preliminary estimations of possible phenomenological manifestations of the DM candidates are also desirable, as is the analysis of scalar-Higgs boson mixing.