# Direct search constraints on very heavy dark skyrmions

## Abstract

In the standard dark matter creation scenario, dark matter arises from freeze-out due to decoupling from the thermal heat bath in the early universe. On the other hand, topological solitons can also emerge during phase transitions through the Kibble–Zurek mechanism or through bubble nucleation. In particular, Murayama and Shu found that the Kibble–Zurek mechanism can produce topological defects up to about 10 PeV, and Bramante et al. had recently pointed out that direct search constraints can be extrapolated to very large masses. Motivated by these observations, we examine direct search constraints for PeV scale dark skyrmions with a Higgs portal coupling to baryons. We find abundance constraints on the combination \(g_V^2M_S\) of Skyrme coupling \(g_V\) and skyrmion mass \(M_S\). We also find that extrapolation of the direct search constraints from XENON1T to very high masses constrains the combination \(g_{wh}/g_V^4\) as a function of \(M_S\), where \(g_{wh}\) is the Higgs portal coupling of the dark skyrmions.

## 1 Introduction

The dark matter puzzle motivated numerous investigations of the question how dark matter may have been produced in the early universe. Dark matter up to a mass^{1} of about 100 TeV [1] can be produced in freeze-out from a thermal heat bath if the dark matter interaction rate with baryons becomes suppressed by the cosmic expansion [4]. Dark matter can also arise in the form of topological defects in phase transitions [5], or due to freeze-in if the interactions were too weak to ever establish thermal equilibrium in the dark sector [6]. Generation during or near the end of inflation is another possibility [7, 8].

Numerous other proposals have been discussed in the literature, but here we would like to focus on direct search constraints on very heavy dark matter from the Kibble–Zurek mechanism [5, 9, 10, 11]. This is motivated by the observation of Murayama and Shu that this mechanism can produce non-thermal PeV scale dark matter [12], and by the recent observation of Bramante et al. that direct search constraints can be extrapolated to very high mass values [13]. We assume that the dark matter can interact with the baryonic sector through a Higgs portal coupling for the calculation of the pertinent nucleon recoil cross section. We use a skyrmion scenario where the very heavy skyrmions are coherent states of heavy “mesons”, and it is the heavy mesons which couple to baryons through the Higgs portal. The ensuing separation of mass scales ensures that the very heavy dark matter can be non-thermally created as topological defects while the mesons have strong enough coupling to baryons to prevent the mesons’ freeze-out^{2} into a sizable dark matter component. This sets the present model apart from standard thermal Higgs portal models and implies different constraints from direct search experiments.

Higgs portal dark matter from thermal freeze-out has been extensively discussed and is well understood, see e.g. [14, 15] and references there. However, the mechanism of dark matter generation determines what (if any) relation exists between dark matter mass and coupling strength to baryons, thus also determining the relation between dark matter mass and nucleon recoil cross section from the theoretical side of matching to direct search constraints. We are therefore interested in the constraints on topological Higgs portal dark matter which arises through a phase transition in the dark sector without subsequent thermalization and thermal freeze-out.

*U*(

*x*), which map \({\mathbb {R}}^3\rightarrow S^3\) for fixed time

*t*,

*SU*(2) isospin symmetry and the vector \(\hat{\varvec{w}}=\varvec{w}/|\varvec{w}|\) is the unit vector for the triplet \(w_i(x)\) of dark mesons.

*W*as skyrmion number, since the standard designation as “baryon number” is obviously inappropriate: The skyrmions in the dark sector are not baryons, neither in the hadronic sense nor in the cosmologist’s definition of baryonic matter.

In the present paper we are interested in direct search constraints on very heavy dark skyrmions, since Bramante et al. had pointed out that limits from existing direct search experiments can principally be extrapolated to very high masses [13]. The non-thermalized dark skyrmions are not constrained by the standard indirect [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] and direct [14, 15, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56] search analyses for Higgs portal dark matter since the different creation mechanism changes the relation between mass and couplings. However, we can adopt estimates on the ratio \(x_c=M_S/T_c\) from hadronic physics. This allows us to constrain the skyrmion parameter \(g_V^2 M_S\) from the requirement \(\varOmega _S\le \varOmega _{\mathrm {CDM}}\). Direct search constraints from XENON1T then allow us to constrain a combination of the skyrmion coupling and the skyrmion’s Higgs portal coupling as a function of skyrmion mass.

We will review some pertinent aspects of dark skyrmions in Sect. 2 and confirm that skyrmions are good candidates for PeV scale dark matter through a new analysis of their current abundance which takes into account that the Boltzmann equation erases initial conditions for \(\varGamma \gtrsim H\) if the reverse process to \(S{\overline{S}}\) annihilation is kinematically suppressed. Section 3 discusses constraints on the heavy *w* mesons which arise from the requirement that the relic very heavy dark skyrmions, but not the heavy *w* mesons, are the dominant dark matter component. The direct search constraints on PeV scale dark skyrmions are then discussed in Sect. 4. Section 5 summarizes our conclusions.

## 2 Dark skyrmions from chiral phase transitions

*e*, but we avoid this notation. The designation \(g_V\) is motivated by the observation that the skyrmion coupling can arise from vector dominance due to a hidden local \(SU_V(2)\) invariance [18, 57].

*SU*(2) transformation (1), i.e. under parity \(\mathrm {P}\), \(\mathrm {P}\circ U(x)\circ \mathrm {P}=U^{-1}(x)\). Just like for ordinary pions in chiral perturbation theory, the residual \(SU_V(2)\) transformation

*SO*(3) generators. Adding a mass term for the Goldstone bosons,

*w*-bosons then annihilate into baryons with annihilation cross sections for collision energies which are much larger than the top mass, \(\sqrt{s_w}\ge 2m_w\gg m_t\) [29],

This shows that dark skyrmions can be PeV to multi-PeV scale dark matter for skyrmion couplings \(g_V^2\lesssim 1\).

## 3 Constraints on the heavy dark mesons

*w*mesons do not contribute to \(\varOmega _{CDM}\) at a significant level. I.e. besides \(1\,\mathrm {TeV}\le m_w\ll M_S\), we also assume that \(g_{wh}\) is sufficiently large to prevent thermal freeze-out of the dark mesons at a level where they would contribute a significant fraction to the dark matter abundance.

^{3}On the other hand, we still assume \(g_{wh}<4\pi \) for the perturbative calculation of the Higgs portal annihilation and recoil cross sections. The corresponding domains where \(g_{wh}<4\pi \) and \(\varOmega _w<0.01\varOmega _{CDM}\), \(0.01\varOmega _{CDM}<\varOmega _w<0.1\varOmega _{CDM}\), and \(0.1\varOmega _{CDM}<\varOmega _w<\varOmega _{CDM}\), respectively, are indicated in Fig. 2. The small blips near \(m_w=4.2\) TeV occur as a consequence of the fact that the top quarks do not contribute to the effective number \(g_*(T_f)\) of relativistic degrees of freedom at the freeze-out temperature \(T_f\) any more if the thermal dark matter mass drops below that value.

Since we are interested in observational implications of the very heavy dark skyrmions as the dominant dark matter component, we assume that \(\varOmega _w<0.1\varOmega _{CDM}\).

*w*mesons yields

*w*component the estimate

^{4}

## 4 Direct signals from the dark skyrmions

Bramante et al. have pointed out that existing exclusion limits from direct searches can be extrapolated to very high masses [13], and we are particularly interested in direct search constraints on PeV scale dark skyrmions.

*k*space.

On the other hand, we expect that \(\langle n_w\rangle \simeq M_S/m_w\). Fitting this expectation to the result (38) yields estimates \(\langle n_w\rangle \simeq 47/gV^2\) and \(m_w\simeq 1.3g_Vf_w\).

The constraints yield \(g_V\gtrsim 1.8\times g_{wh}^{1/4}\) for \(M_S=100\) TeV and \(g_V\gtrsim 0.32\times g_{wh}^{1/4}\) for \(M_S=10\) PeV, i.e. an underlying gauge theory to induce the very heavy dark skyrmion model through the BKUYY mechanism [57] could require non-perturbative coupling strength \(g_V\) to satisfy direct search constraints at the lower end of the non-thermal dark matter mass range, but very weak gauge coupling \(g_V\) would be still possible for higher dark skyrmion masses.

For actual parameters for the next-generation Xenon based experiments, we used XENONnT [67], but the anticipated sensitivities and time scales for LZ [68] and PandaX-4T [69] are comparable. For the DARWIN experiment see [70]. The neutrino floor is too close to the DARWIN limit in the high mass region to resolve on the logarithmic scales used in Fig. 8.

## 5 Conclusions

In the future, multi-tonne scale direct search experiments with long exposures will provide stronger constraints on very heavy dark matter, and this will eventually also probe the non-thermal dark matter mass range beyond the Griest-Kamionkowski bound around 100 TeV. Indeed, extrapolation from current direct search results already implies constraints for nucleon recoil cross sections at very high masses. It is therefore of interest to develop theoretical models and techniques for nucleon recoils from very heavy dark matter. We find that very heavy dark skyrmions from an early chiral phase transition provide an interesting avenue to theoretical descriptions of dark matter in the PeV mass range. Their non-thermal creation implies that the parameter \(x_c=M_S/T_c\) is not determined from a freeze-out condition, but we can use estimates for \(x_c\) from hadronic skyrmions. The requirement \(\varOmega _S\le \varOmega _{\mathrm {CDM}}\) then constrains the parameter \(g_V^2 M_S\) as a function of \(x_c\), as shown in Fig. 1. We could also derive an estimate for the nucleon recoil cross section of the dark skyrmions through Higgs exchange (43). Comparison with direct search constraints then limits the parameter \(\eta =g_{wh}/g_V^4\) as a function of skyrmion mass \(M_S\), as shown in Fig. 7.

## Footnotes

- 1.
- 2.
The designation “freeze-out” is also extensively used in the literature to describe the formation of topological defects during quenching during a phase transition, as described by the Kibble–Zurek mechanism. Here we will use “freeze-out” only to refer to the standard cosmological freeze-out from the primordial heat bath, but not to the inhomogeneous formation of local ground states during a phase transition.

- 3.
- 4.
What has been missed in Ref. [29] was the energy conservation estimate for \(\langle N_w\rangle /\langle s_w^{-1}\rangle \) and the fact that contrary to \(n_w/n_S\), the relevant ratio \(\varOmega _w/\varOmega _S\) comes with an extra power of \(m_w/M_S\).

## Notes

### Acknowledgements

This work was supported by a University of Saskatchewan President’s NSERC grant.

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