A study of collider signatures for two Higgs doublet models with a Pseudoscalar mediator to Dark Matter

In this submission the authors study the LHC phenomenology of a model containing two Higgs doublets, a pseudoscalar singlet and a (fermionic) dark matter particle. This model provides a gauge invariant and renormalizable version of pseudoscalar mediation, which is a relevant case among DM simplified models. The authors apply the CONTUR method to scan over two-dimensional planes in the model parameter space and test where the new physics would have significantly contributed to the measurements implemented in the code, which primarily include `SM-like' cross sections, with a few involving $E_T^{\rm miss}$. The study begins from a reference point in parameter space and investigates the effects of varying in turn several model parameters, such as the masses of the BSM Higgs bosons, the DM mass, the mixing between $a$ and $A$, and the mixing between $h$ and $H$.

While CONTUR is an interesting method to test new physics at the LHC, the present manuscript contains a routine (if technically sound) application that I do not expect will have a major impact on future studies of the 2HD + a + DM model. The main reason is that the present analysis does not consider several measurements which are expected or already known to have important impact, examples being mono-Higgs and additional $E_T^{\rm miss}$ measurements concerning Section 4, and measurements of the 125 GeV Higgs properties concerning Section 7. The authors do clearly state these omissions, but the resulting physical picture is lacking key information. In addition, the paper provides limited insight beyond the CONTUR figures, as it presents a series of departures from the initial benchmark but refrains from analyzing any of them in depth. For these reasons, the manuscript clearly does not meet the demanding criteria for acceptance in SciPost Physics.

Nevertheless, in my opinion the paper could be considered for publication in the second-tier journal SciPost Physics Core. It can be viewed as a `CONTUR status report' - an application of this developing framework to an important problem, technically sound and containing some new findings despite the issues mentioned above. If the authors are interested in this possibility, I ask them to address the comments or questions concerning the text listed below.

1) In section 4, line 11, the authors mention that for $M_A = M_{H^\pm}$ there are extra constraints on $\sin\theta$ from electroweak precision, favoring smaller values. Is it obvious that the choice $\sin\theta = 0.35$ made in Section 4.3 is consistent with these constraints?

2) In Section 4.2, fifth-to-last line: when writing `... in this region the $H$ decays dominantly to $ah$', do the authors mean $A\to ah$ perhaps?

3) In Section 4.3, sixth line: `$M_{H^\pm} = M_H$' should read `$M_{H^\pm} = M_A$'. In addition, I was confused by the subsequent sentence: why is $A\to ah$ enhanced at low $M_H\,$? Perhaps the authors mean $A\to aH$, or I am misinterpreting this sentence.

4) In the Conclusions appears the following sentence, `If future $WW$ measurements can be made less model-dependent, for example by including all relevant kinematic cuts ...', which I believe refers to a discussion presented by three of the authors in the Les Houches 2019 report [9], pages 85-88. A reader of the present manuscript is likely to get confused, unless they have already read the Les Houches report. Similarly for footnote 6. These discussions should be slightly expanded to explain these aspects, so the present paper can be read without necessarily consulting [9].

5) A couple of suggestions concerning the Introduction: I think it would be useful to mention explicitly an important reason for interest in pseudoscalar mediation (the fact that single $a$ exchange between DM and quarks only gives rise to suppressed spin-dependent cross section, so direct detection bounds are easily satisfied), and to cite 1404.3716 as an important early study of the 2HD + a + DM model.

6) A few typos: page 3, exotics Higgs bosons $\to$ exotic Higgs bosons; page 6, CP even $\to$ CP-even; on page 11, `... those that couple to top quarks' could be replaced by '... those that involve top quarks' for clarity.