Asymmetric dark matter: residual annihilations and self-interactions

We thank the anonymous referees for their positive feedback on our paper and for their comments and suggestions. We address them below, explaining the related changes we implement in the revision version of our paper that we are submitting. The comments of the referees have allowed to us to improve the quality of our paper, and we hope the revised version can now be accepted for publication.

Reply to Report 1

1. The referee is right that direct detection (DD) constraints forbid early thermal equilibrium between the dark sector and the SM, in the region of parameter space interesting for Self-Interacting DM. We would like to observe that instead, for $m_V >$ few GeV, DD allows for $\epsilon \gtrsim 10^{-6}$, and therefore for early thermal equilibrium between the SM and the Dark sector (see e.g. footnote 8 of 1612.07295). That region is not interesting for SIDM, but it is interesting to study the impact of an asymmetry on some ID probes of this model, which is one of the purposes of our paper.
We would also like to observe that the above considerations assume that only the kinetic mixing is responsible for the communication between the two sectors.
It is possible that extra dynamics kept the two sectors in thermal equilibrium at large enough temperatures. Such dynamics could arise at an energy scale $\gg M_{p_D}, m_{\rm SM}$ and have no effect on the phenomenology we studied, and it would be somehow expected if one would want to link the origin of the baryon and dark matter asymmetries.

Following the referee's suggestion, we have added in Section 7 a synthesis of the above considerations, and a qualitative discussion of the effects, on our findings, of non thermal equilibrium.

2. We agree with the referee that the dark electrons may have various implications. Although we comment in our manuscript on some significant aspects, a detailed exploration of the effects of the dark electrons is beyond the scope of this work. With respect to the points raised by the referee:

2a. DM self-interactions due to the dark electrons.
Please see last paragraph of section 4. We have added a clarifying remark.

2b. Maximum dark photon mass.
We agree that for the efficient annihilation of the dark antielectrons, the dark photons have to be lighter than the dark electrons, and the inaccessible region of our parameter space grows. We have added a remark in the caption of fig. 1.
We note though that even if $m_{e_D}$ is smaller than $M_{p_D}$ by several orders of magnitude, the parameter space for self-interacting DM remains largely open or even unaffected.

2c. Dark electron freeze-out.
In our freeze-out computation, we have assumed that the thermal equilibrium between the SM and the dark sector ceases at temperatures larger than the dark proton mass (see comment above), and therefore we have taken into account the reheating of the dark sector due to $p_D$ becoming non-relativistic. This occurs before even the dark protons freeze-out, and affects the efficiency of their annihilation as much as that of the dark electrons.

The $e_D$ freeze-out may be further affected by the dark sector reheating due to $e_D$ becoming non-relativistic. This is a very mild effect, since the required cross-section scales as $\sigma_{\rm ann} v_{\rm rel} \propto T_D/T_{\rm SM} \propto g_{*,D}^{-1/3}$. This effect is far outweighed by the fact that the $e_D$ annihilation cross-section is larger than that of $p_D$, due to the smaller mass of the former. Quite generally, $r_\infty (e_D) \leqslant r_\infty (p_D)$.

3. We intended to write $v_{\rm rel} \lesssim 10^{-8}$ at the CMB epoch, we apologize for the typo, that we corrected. This value comes from the more detailed analysis in 1612.07295 (our ref [44], see Sections 3.1 and 4.3 therein). Dark electrons do not change the outcome of that analysis and we have added a footnote about this for completeness. Although, in presence of dark electrons the kinetic decoupling of the dark protons from the dark photons can be delayed, we have estimated that $v_{\rm rel}$ remains $< 10^{-8}$ at the CMB epoch.

4. We have added in Section 5 a comment as suggested by the referee, on the visualization of the CMB exclusion in the region where resonances are most dense.

5. Yes, in the calculation of the CMB constraints we have included the modification of the Sommerfeld enhancement factor due to unitarity restoration. We have added an explicit comment about this, again in Section 5.

6. The referee is right, that is a typo, we have now removed the extra $10^3$.

7. We agree with the referee, and we have added the encouragement suggested to the relevant sentence in Section 5. We also note that the general encouragement suggested by the referee, to present data in a way that it is amiable for reinterpretation, is also present in the Conclusions.

8. Finally, we have updated the plots after correcting an efficiency factor for the CDMSlite analysis, which resulted in slight changes to the direct detection constraints.

Reply to Report 2

Please see the response to Report 1 (item 2) for comments related to the dark electrons. With respect to the additional points raised:

If $m_{e_D} \approx M_{p_D}$, then the dark electrons should be included in the computation of the DM relic density and the estimation of the dark proton mass. This is a limiting case, which may arise due to a particular symmetry structure in the dark sector, and we shall not devote a specific analysis here. We have added a footnote about this in section 3.

The dissipation is a rather complex issue that depends sensitively of the energy splittings, interactions rates and the thermodynamic conditions. Even for massless mediators, it happens efficiently only in a very limited part of the parameter space. In our analysis, we focus on fairly massive mediators, $m_V > {\rm MeV}$, for which dissipation is expected to be negligible. We have added a comment in the end of section 4, and provided references to related studies.

The thermalisation of the two sectors is addressed by point 1 in the reply to the first referee report.