Low energy effective theories of composite dark matter with real representations

1. The paper gives a very detailed account of how to compute
quantities of physical interest for SIMP models where the dark gauge
group has real representations, which is less studied than the case of
SU(N).

2. They overcome a technical obstacle in the determination of the WZW
interactions, necessary for 3-to-2 scattering which is essential to
the SIMP paradigm, in which the standard approach for finding these
interactions does not work.

3. The authors give a rather complete characterization of the spectrum
and symmetry-breaking patterns of such models.

1. The phenomenology section is disappointing, in that the authors
only investigate the new affects on freeze-out from the $\eta'$ state,
ignoring the vector mesons, which as far as we know could have more
important effects.

2. The new effects of the $\eta'$ are relatively small and require
tuning its mass to be close to that of the lighter mesons. Overall
the parameter space allowed by the relic density and self-interaction
constraints is small, making one wonder whether further investigation
of these models is very motivated.

3. The authors discuss CMB/BBN constraints on the decays of the
$\eta'$ quite superficially, and they have chosen a benchmark model
which appears to violate those constraints.

The theoretical part of the paper is a worthwhile addition to the SIMP
literature, paving the way for more detailed studies of models of the
kind the authors have studied. Overall I feel it merits publication.

1. In Fig.3, it seems strange to name the isosinglets as $\rho$
and the isotriplets as $\omega$, contrary to the standard model.
Is there some rationale for this choice?

2. Can the authors comment on whether there could be $N$-quark
antisymmetrized bound state DM candidates, analogous to baryons in the
standard model?

3. The prime is missing on $\eta$ in a number of places in section 4,
which leads to confusion.

4. The benchmark model values chosen below Eq.\ (4.12) are in the
region of Fig.4, Ref.[91], that is excluded by lack of thermalization
between the SIMP sector and the SM. The authors should clarify this.

5. The prime is missing on $\eta$ in a number of places in section 4,
which leads to confusion.

6. Grammatical/spelling errors: ``who's'' should be ``whose'' and
``forth'' should be ``fourth''.

Ask for minor revision

The paper deals with a detailed construction of low -energy effective theories for composite dark matter for SO(Nc) color groups in the dark sector. General effective interations for dark pions, dark photon and vector mesons in hidden local symmetry framework are provided and some phenomenological study for dark pion dark matter was discussed.

There is a limited discussion on the phenomenological application of the construction to dark pions and dark \eta' only. A new point in this work is to show the effects of light dark \eta' on the relic density for dark pions. But, an unnecessary assumption for decoupling dark vector mesons was made.

There are confusing statements on light \eta' whose mass was assumed to be comparable to dark pions.

BBN constraints on the decay of dark \eta' were discussed but the parameter choice of this paper is not compatible with BBN.

The paper provides low -energy effective theories for composite dark matter for SO(Nc) color groups in the dark sector including dark vector mesons in hidden local symmetry scheme. However, there is a limited discussion on the phenomenological application for dark pions and it is unnatural to keep dark \eta's meson as light as dark pions in this work, even if there are anomalous couplings of \eta's to dark gluons as in QCD. I recommend a publication of the paper in a certain form, but questions in the following must be addressed properly.

At least, a brief discussion on inclusion of vector mesons for dark matter should be made. For instance, in the construction of dark fermions in real representations of SO(Nc), the important differences for vector mesons from the previous work should be discussed.

The authors argued that for a large Nc, dark \eta' has anomalous couplings to dark gluons suppressed but it becomes light enough to affect the annihilation of dark pions. It would be unnatural to have a light dark \eta' due to the absence of symmetry, unlike dark pions whose mass is protected by the approximate non-anomalous global symmetry. So, a more justfication for light dark \eta's scenarios is needed. In particular, dark \eta' mass was taken to be very close to dark pion masses, although there is no obvious reason for that.

In the paragraph below eq.(4.12), it was argued that \eta'-> 4f decay mode dominates the lifetime due to larger m_{Z'} suppression. But, from eqs.~(4.11) and (4.12), \eta'->2f looks dominant. In the same paragraph, the lifetime of dark \eta' is much longer than 1sec. There is a brief comment on the BBN constraint in this case. So, a more concrete discussion on BBN constraint should be made, by choosing a realistic lifetime of \eta' and discussing whether or not the decay rate of \eta's is relevant in the Boltzmann equation.

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