Direct search constraints on very heavy dark skyrmions

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 gV2MS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_V^2M_S$$\end{document} of Skyrme coupling gV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_V$$\end{document} and skyrmion mass MS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_S$$\end{document}. We also find that extrapolation of the direct search constraints from XENON1T to very high masses constrains the combination gwh/gV4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{wh}/g_V^4$$\end{document} as a function of MS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_S$$\end{document}, where gwh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{wh}$$\end{document} is the Higgs portal coupling of the dark skyrmions.


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]. 1 See also [2,3] for a possible relaxation of the mass bound. a e-mail: madeline.berezowski@usask.ca b e-mail: rainer.dick@usask.ca 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 nonthermal 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 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. sector without subsequent thermalization and thermal freezeout.
In particular, a chiral G = SU L (2) × SU R (2) → H = SU V (2) phase transition in the dark sector will create skyrmions from the non-trivial homotopy group π 3 (G/H ) = Z which arises from the mappings of compactified R 3 = S 3 into S 3 [16][17][18]20,21]. The compactification and the mapping arise from the boundary condition lim |x|→∞ U (x) = 1 on the fields U (x), which map R 3 → S 3 for fixed time t, Here the σ i are the Pauli matrices of a dark SU (2) isospin symmetry and the vectorŵ = w/|w| is the unit vector for the triplet w i (x) of dark mesons. The winding number W = d 3 x W 0 of the mapping S 3 → S 3 is related to the conserved current (with the convention 0123 = −1) We will denote 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.
The spherically symmetric skyrmions are stable with skyrmion number W = ±1 if w S (0) = ±π f w [19,21,22]. It is therefore natural that skyrmions are discussed as dark matter candidates. Electroweak scale dark skyrmions from a non-standard Higgs sector are constrained to masses up to about 10 TeV [23][24][25][26][27][28]. If the coupling of the skyrmions to baryons is strong enough to thermalize the skyrmions after the phase transition, then their relic abundance is determined by thermal freeze-out [23,[26][27][28]. However, if the skyrmions do not thermalize after the phase transition, then it is the phase transition itself which determines any relic skyrmion abundance. Such a scenario was discussed by Campbell et al. for electroweak skyrmions [24]. Furthermore, Murayama and Shu had pointed out that topological dark matter with masses up to 10 PeV could be generated through the Kibble-Zurek mechanism [12], and indirect signals from very heavy dark skyrmions through Higgs or neutrino portals were discussed in [29].
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 Γ H if the reverse process to SS 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.

Dark skyrmions from chiral phase transitions
We parametrize chiral symmetry breaking in a dark sector with a Lagrangian The fourth order term in the Skyrme Lagrangian stabilizes skyrmions as topological solitons against Derrick's theorem [16][17][18]20,21]. The coupling constant g V is usually written as 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]. The skyrmions (3) can be thought of as coherent excitations of the dark meson fields w(x), and the Skyrme Hamiltonian defines the mass for the static skyrmion solutions through the energy functional with the result [18,21] Their effective extension can be estimated as L S 1.2/ (g V f w ) [58]. The representation (6) for the Skyrme Hamiltonian uses the convention that summation over lower pairs of 4-indices is defined as Euclidean, e.g.
Just like for ordinary pions in chiral perturbation theory, the residual implies a standard spin-1 rotation representation on the dark mesons, Here (L i ) jk = i jk are the standard SO(3) generators. Adding a mass term for the Goldstone bosons, respects the unbroken symmetry and is compatible with the effective Goldstone boson interpretation of the w triplet if m w M S . Following Ref. [29] we also include a Higgs coupling for the dark mesons, The exact nature of chiral phase transitions for different chemical potentials and different numbers of underlying fermion flavors is still a matter of investigation. The nonthermal abundance of topological defects from a second order phase transition has been estimated by Muramaya and Shu [12], while Campbell et al. provided an estimate for a first order transition [24].
The finite correlation length ξ from quenching during a second order phase transitions implies that anti-skyrmions and skyrmions with different centers will form with a density of order ξ −3 during the phase transition. The Kibble-Zurek based estimate of Murayama and Shu for the abundance of particle-like topological defects from cosmological phase transitions therefore applies, Here n S denotes the sum of skyrmion and anti-skyrmion densities, and a critical exponent ν 2/3 was assumed [12]. On the other hand, Campbell et al. assume a first order phase transition in an approximately scale-invariant model with chiral symmetry breaking and a pseudo-dilaton χ . They find an estimate for the skyrmion abundance depending on the ratio of dilaton mass to vacuum expectation value m χ /v χ [24]. We will find that the relic dark skyrmion abundance decouples from the order of the phase transition and the initial skyrmion abundance if Γ (T c ) H (T c ) while thermal skyrmion creation is kinematically suppressed.
Numerical simulations indicate that skyrmions annihilate with their classical cross section σ SS = π L 2 S into a number 2 < N w < M S /m w of w particles [59,60]. The simulations for hadronic skyrmions actually yield 2 < N w 7, which is in remarkable agreement with observations of lowenergy p p annihilations. These annihilations produce 5 ± 1 pions, with pion multiplicities 3 ≤ n π ≤ 7 in 99% of measurements [61]. The w-bosons then annihilate into baryons with annihilation cross sections for collision energies which are much larger than the top mass, √ s w ≥ 2m w m t [29], and Their leading order annihilation cross section into Standard Model states is therefore Both quenching and bubble nucleation delay completion of a phase transition, such that topological defects would be born into a heat bath with lower temperature T 1 than their mass, T 1 T c ∼ M S . Furthermore, we would generically need collision of N w ∼ M S /m w 2 w particles to produce an SS pair, such that these processes are phase-space suppressed with a factor exp(− N w E w /T ). Both of these effects suppress the thermal creation of the dark skyrmions from baryonic matter. Therefore, similar to magnetic monopoles [62], we need to solve the balance equation with initial conditions given by Eqs. (12) or (13), respectively, to determine the relic skyrmion density today. We encounter a factor 1/2 in the skyrmion annihilation rate in equation (17) because n S =ñ S +ñ S 2ñ S is the sum of skyrmion and anti-skyrmion densities, and the skyrmion annihilation rate is The temperature-dependence of vσ SS depends on how efficiently the dark mesons can couple the skyrmions to the primordial heat bath. The possible range of temperature dependences can be parametrized by with 0 ≤ β ≤ 1 [4].
Evolving the skyrmion density from the time t c of creation to the epoch of radiation-matter equality yields for t eq t c Annihilation is negligible at later stages, and we find a relic skyrmion density where z c is the redshift at the epoch of the chiral phase transition in the dark sector. This can be expressed in terms of entropy ratios The annihilation term in the denominator in Eq. (21) is related to the ratio of the skyrmion reaction rate and the expansion rate at the phase transition, This will be larger than 1 at t c , and we find Time and temperature at the phase transition are related by radiation domination, Hadronic physics indicates 5 x c ≡ M S /T c 10, and therefore we will adopt this range in evaluations of Eq. (26). We can also use a non-relativistic approximation σ v c π L 2 S √ 3/x c , β = 1/2. Furthermore, all radiative components are thermalized, g * S (T c ) = g * (T c ), and we assume that they are dominated by Standard Model degrees of freedom, g * (T c ) = 112. This assumption is in agreement with the proposal that the phase transition which generates the dark skyrmions is the last phase transition in the early universe before electroweak symmetry breaking. This yields We use current particle data group values [63] for the entropy density s 0 = 2891.2 cm −3 and the critical density cr h −2 = 1.05371 × 10 −5 GeV/cm 3 to constrain the skyrmion param- Fig. 1. This shows that dark skyrmions can be PeV to multi-PeV scale dark matter for skyrmion couplings g 2 V 1.

Constraints on the heavy dark mesons
We are particularly interested in the very heavy dark skyrmions as the dominant dark matter component, and therefore presume that the heavy w mesons do not contribute to Ω C DM at a significant level. I.e. besides 1 TeV ≤ m w 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 < 4π for the perturbative calculation of the Higgs portal annihilation and recoil cross sections. The corresponding domains where g wh < 4π and Ω w < 0.01Ω C DM , 0.01Ω C DM < Ω w < 0.1Ω C DM , and 0.1Ω C DM < Ω w < Ω C DM , 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 Ω w < 0.1Ω C DM .
In addition to thermally created heavy dark mesons, annihilation of relic very heavy dark skyrmions will also generate heavy dark mesons, which will then annihilate into Standard Model states following the balance equation Energy conservation implies an estimate N w / s −1 w 8M 2 S / N w , and detailed balance for those non-thermally created w mesons yields This implies for the non-thermal dark w component the estimate According to Figs. 1 and 2, m w /g 2 V M S ∼ 10 −3 while g wh is of order 1. Furthermore, we certainly have v SS 1 for the very heavy relic skyrmions, i.e. we can safely assume that the very heavy dark skyrmions are the dominant dark matter component in the model for the parameter ranges identified in Figs. 1 and 2. The "dark mediator bottleneck" that was considered as a generic possibility in Ref. [29] therefore does not materialize in this model. 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.
To calculate the expected nucleon recoil cross section from virtual Higgs exchange, we consider the skyrmion as a coherent superposition of the w mesons, i.e. we use the coherent state Here ω 2 (k) = k 2 + m 2 w and w S (k) is the skyrmion in k space, The state (31,32) has the property to yield the skyrmion as the expecation value of the w meson operator i.e.

S(t)|w(x)|S(t)
The state (31) and operator (34) are the interaction picture state and operator, respectively, with H 0 = d 3 khω(k)a + (k) · a(k). The Schrödinger picture skyrmion state |S ≡ |S(0) is of course timeindependent for the static skyrmion solution. The timedependent factor exp[iω(k)t] in ζ (k, t) arises from the shift of the factor from the interaction picture operator a + (k, t) = a + (k) exp[iω(k)t] into the amplitude ζ (k).
The expectation value of the w meson number in the skyrmion state is This yields for the dark skyrmion-nucleon recoil cross section the estimate with the nucleon recoil cross section of the w particles (for m w m N ) We use the abundance weighted average nucleon mass in stable or long lived Xenon isotopes m N = 930.6 MeV in evaluations of Eqs. (39,40). The coupling constant g h N in Eq. (40) is the effective Higgs-nucleon coupling [64] with a modern value g h N v h = 289 MeV [65]. We use the semi-analytic approximation a = 0.9451 [19], for the skyrmion profile in numerical evaluations. This yields Fig. 3 for the skyrmion profile in k space. The w meson number is shown in Fig. 4. On the other hand, we expect that n w M S /m w . Fitting this expectation to the result (38) yields estimates n w 47/gV 2 and m w 1.3g V f w .
We can now collect the unknown heavy dark sector parameters in the nucleon recoil cross section in a single parameter, and for future comparisons with collider searches it is useful to express this in terms of the lighter w particles, This yields the skyrmion-nucleon recoil cross sections in Fig. 5.
On the other hand, for comparisons with direct dark matter searches it is more useful to express this in terms of the actual dark matter mass M S , Fig. 6 The skyrmion-nucleon recoil cross section for 100 TeV ≤ μ S ≡ g 4 V M S /g wh ≤ 100 PeV This yields the skyrmion-nucleon recoil cross sections in Fig. 6. Extrapolation of the XENON1T 1-year constraints to high masses implies σ SN 8.3 × 10 −46 cm 2 M S [TeV] × F(M S ) [66], where F(M S ) = |S(1 TeV)/S(M S )| 2 is the corresponding ratio of Helm structure factors squared. Our result (43) then constrains the coupling constant η ≡ g wh /g 4 V as a function of dark skyrmion mass. This is displayed in Fig. 7.
The constraints yield g V 1.8 × g wh for M S = 100 TeV and g V 0.32 × g 1/4 wh 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.
The improved sensitivities of next-generation and nextto-next generation direct search experiments to g wh /g 4 V are displayed in Fig. 8. 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.

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 nonthermal 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 Ω S ≤ Ω CDM then constrains the parameter g 2 V 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 η = g wh /g 4 V as a function of skyrmion mass M S , as shown in Fig. 7.
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