On-shell effective theory for higher-spin dark matter

We thank the referee for the helpful comments and all the interesting points raised. Below, we address the referee's points sequentially, and detail the changes we have made in the draft to address their helpful feedback. For convenience we list the questions and respective answers below.

"Is equation 3.7 as written valid for spins $< 2$ ?"

The production amplitudes in Eqs.~(3.3)-(3.5) are valid as they stand also for $S=3/2$, and therefore the scaling in Eq.~(3.7) remains valid for $S=3/2$. For $S=1$ Eqs.~(3.3)-(3.5) are also valid, but with the convention that the first term involving $\sum_{k=1}^{2S-2}$ should be set to zero. As this first term dominates the scaling for generic $S$, Eq.~(3.7) is not valid for $S=1$, and one instead has $M_{\rm ann} \sim E^2/M_{\mathrm {Pl}}^2$, as follows from the second term involving $\sum_{k=0}^{2S-2}$ in Eqs. (3.3)-(3.5). Eqs.~(3.3)-(3.5) are not valid for $S<1$.

We added a comment below Eq.~(3.7) that the scaling is valid for $S \ge 3/2$ and commented on the scaling for $S=1$. We also added footnote 6 commenting on the (in)validity of Eqs.~(3.3)-(3.5) for $S\leq 1$.

" When this is possible, can the author compare their results with those of [Criado et al. 2010.02224] and [Alexander et al. 2010.15125] ?"

We added the next-to-last paragraph in Conclusions where we elaborate on the similarities and differences between our work and those of [Criado et al. 2010.02224] and [Alexander et al. 2010.15125]. We elaborate more about the differences between the models in the footnote of pag.2: "The main difference between [35,36] and our work is that neither uses the massive on-shell formalism of [15]. Moreover, Ref. [36] consider the case of DM production DURING inflation and we are considering the production AFTER inflation. Regarding [35], the leading DM-Higgs interaction is a 4-point contact term, which for us is a 3-point DM-Higgs-Higgs coupling. "

"I would suggest moving some of the bigger helicity amplitude equations (3.3-3.5, 4.2, 4.5-4.7) in some appendix."

We agree with the referee that the formulas could distract from the phenomenological results. However, we believe that equations (3.3)-(3.5) and (4.5)-(4.7) (now 4.4-4.6) are well justified to be in the main text. First, to our knowledge, it is the first time that the amplitudes for gravitational scattering of massive spin-$S$ particles on SM matter appear in the literature. Thus, these equations represent an important original contribution of our paper, and they are one of our main results. Second, these formulas also show explicitly what we advocate in the introduction: the massive spinor formalism is well suitable to obtain compact formulas for scattering amplitudes of particles with general spins (the analogous formulas obtained in the standard Lagrangian formalism would be lengthy and opaque). Last but not least, it is from these formulas that we could see in a transparent way the energy scaling of the amplitudes, which were then used for the dimensional analysis estimations. So, the formulas indeed improve our understanding of the physical processes relevant for higher-spin DM production.

On the other hand, we agree that Eq.~(4.2) from the previous version is not essential for the subsequent discussion. Therefore we moved it to Appendix C, as Eq.~(C.1). In the case, of equation 4.2, since the theory is defined in terms of the 3-point amplitudes we quoted all possibilities. This coupling is not particularly relevant to the phenomenology but since it is the same as in the dRGT gravity, we found worthy to write it.

Other changes:

• A coding mistake has been made when plotting the green region on Figure 6. The figure has been replaced, and now the green region matches the corresponding region in Figure 7. We also spotted a typo in the code related to the position of the violet region in Fig.~6 (Higgs-mediated pair production). After correcting this mistake, the violet region shifted to huge valus of $c_H$, which are not relevant withing the validity of the EFT. For this reason do not plot anymore the violet region where the Higgs-mediated pair production dominates, and changed the discussion below Eq.~(4.9). The same mistake was done in Figure 8, which has also been replaced. We apologize for these mistakes.
• On page 2 we added footnote 1 specifying that the no-go theorem for massless higher spins holds for Poincare-invariant, unitary, local QFTs in four spacetime dimensions (a loophole exists e.g. for QFTs in AdS background).

We thank the referee for the helpful comments and all the interesting points raised. Below, we address the referee's points sequentially, and detail the changes we have made in the draft to address their helpful feedback. For convenience we list the questions and respective answers below.


"What is the expected size of the coefficients associated to the contact terms? How do they scale in terms of coupling constants ($\hbar$ counting)? In this regard, does it make sense to consider $C_v^{(2)} \gg 1$ in Figure 4? A comment could be explicitly made (if it is not there already, apologies if I have missed it), that the overall normalization by $ M_{\rm Pl}^2$ could be counterbalanced by a large $C_v^{(2)} \sim ( M_{\rm Pl}/\Lambda)^2$. Same question/comment goes for $c_H$."

Below Eq.~(3.6) we added a short discussion about the scaling of the Wilson coefficients and the $\hbar$ counting. Indeed the Wilson coefficients are $\mathcal{O}(g_*^2 M_{\rm Pl}^2/\Lambda^2)$, so coefficients much larger than 1 are consistent with the effective theory assuming new physics producing these contact terms enters below the Planck scale.
We also added a comment on the scaling of $c_H$ below Eqs.~(4.1) and (4.16).


"Why does the estimate of $T_{\rm max}$ in Eq. (4.11) differs so much from the actual computation in Eq. (4.14)? This in turn makes the green region corresponding to $T_{\rm max}>T_{\rm EW}$ differ significantly from Figure 6 to Figure 7. Also, which of the two estimates in Eq. (4.11) is used for Figure 6?"

We thank the referee for this comment. Both Eq.~(4.11) (4.10 in the new version) and Eq.~(4.14) (now 4.13) are correct, however a coding mistake has been made when plotting the green region on Figure 6. The figure has been replaced, and now the green region matches the corresponding region in Figure 7.
When addressing this comment we also spotted a typo in the code related to the position of the violet region in Fig.~6 (Higgs-mediated pair production). After correcting this mistake, the violet region shifted to huge valus of $c_H$, which are not relevant withing the validity of the EFT. For this reason do not plot anymore the violet region where the Higgs-mediated pair production dominates, and changed the discussion below Eq.~(4.9). The same mistake was done in Figure 8, which has also been replaced.
We apologize for these mistakes.


"In the case that the DM candidate is not stable, is there any advantage from X having a high spin? I believe, from simple dimensional analysis, that the lifetime will not depend on the spin, although it could be nice to explicitly comment on this point."

Indeed, the decay width is not significantly affected by the spin of the DM candidate. We added a comment about it below Eq.~(4.18).
We do not see clear advantages for going to higher spins in this scenario.
This is precisely the reason why we only provide some simple dimensional analysis for $S>2$, without going through a detailed analysis.
One thing one could mention here is that, because $c_H\sim M_{\rm Pl} m^{S-2}/\Lambda^{S-1}$, it is more natural for $S>2$ to arrive at small $c_H$ and satisfy the DM decay constraints.
We added a comment about it below Eq.~(4.16).


"Are the authors aware of explicit UV completions where the lightest stable particle is of high spin? Any idea why this could be the case?"

This is a very good question and indeed it is not so common to find examples of dark sectors where the lightest particle has a large spin. However, there is no theoretical obstruction for such type of spectrum and one can find some examples where this is realized. As is very well known, in supergravity it is possible to have the gravitino (spin-3/2) as the lightest stable particle in broad regions of the parameter space. Spin-2 DM can appear e.g. in bigravity.
For $S>2$, arguably the most promising direction is $SU(N)$ gauge theories in the limit of large $N$. In particular, 1707.05380 studied a model where the DM particle is a dark baryon with higher spin. In this case one can argue that the lightest dark baryon corresponds to the totally symmetric spin configuration, therefore $S = N/2$. Finally, in nuclear physics there is a plethora of higher-spin ground states, but that's in the context of a non-relativistic EFT with $\Lambda \approx m$, so it's probably not directly relevant from the point of view of our paper.

We added a comment about the possible large-$N$ $SU(N)$ origin of our DM particle in the 2nd paragraph of Introduction.