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.

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 bewhose'' and forth'' should befourth''.

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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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