Two generic mechanisms trap DM inside the neutron star: accumulation during the star lifetime and gravitational capture at the star formation. These mechanisms are cumulative. Potentially, they may provide comparable amounts of trapped DM.

In the next sections, we consider accumulation of DM particles from the ambient galactic halo [27,28]. Far away from the neutron star, the distribution of DM velocities is nearly Maxwellian with small dispersion v¯2, e.g., v¯∼10-3 in the Milky Way (MW). But the particles acquire semirelativistic speeds ∼0.5 as they fall into the neutron star. With masses in the WIMP range, they lose energies of order mn—the neutron’s rest mass—in collisions with neutrons. This is much larger than the asymptotic energies of the particles; thus, most of them bind gravitationally to the neutron star after the first collision. Besides, the momentum transfer in their collisions marginally exceeds the Fermi momentum of neutrons, so the degeneracy of the latter does not play a crucial role. Neglecting the degeneracy is a crude approximation that overestimates the capture rate [29–31], but in view of comparable astrophysical uncertainties, we will use it for simplicity. Finally, we will ignore general relativity effects which enhance the capture rate by an order 1 number; see Ref. [32].

Under these assumptions, the capture rate, i.e., the number of trapped DM particles per unit time, takes the form [27,28] dNdt≈24πGρDMmv¯M*R*f.(3)Here, ρDM is ambient DM density, M* and R* represent the mass and the radius of the neutron star, and we introduced the probability f≤1 for the DM particle to scatter during one pass through the star. The value of f is proportional to the DM-neutron cross section σ, f=σ/σcritif  σ<σcrit,(4)and f=1 otherwise. Here, the proportionality coefficient (“critical” cross section [27]) depends on the neutron star parameters, σcrit∼R*2mn/M*∼10-45  cm2,where we used M*∼1.5  M⊙ and R*∼10  km in the estimate. Notably, σcrit is comparable to the upper limits on the DM-nucleon cross section coming from the direct detection experiments [33–37].

Since dN/dt∝m-1, the accumulation rate of dark mass does not directly depend on m. Integrating Eq. (3), we obtain the total DM mass inside the 10 Gyr old neutron star, MW:  Mtot∼7×1042  GeVf∼7×10-15  M⊙f,(5)dwarf:  Mtot∼8×1046  GeVf∼7×10-11  M⊙f.(6)The above two options differ by the parameters of the ambient DM distribution: v¯≈220  km/s, ρDM≈0.3  GeV/cm3 in the MW and v¯∼7  km/s, ρDM∼100  GeV/cm3 in the densest dwarf galaxies [38,39]. Besides, taking f=1 turns the above estimates into model-independent upper limits on the total DM mass that can be captured by the neutron star in the respective environments.

Another possibility is a gravitational trapping of the DM during formation of the neutron star [40,41]. The latter objects are created in the supernova collapses of ordinary stars with masses above ∼9  M⊙, which, in turn, are born in giant molecular clouds. Originally, as the baryonic gas contracts adiabatically into a proca star, a low-velocity fraction of the ambient DM gets trapped by its gravitational well. Eventually, this DM ends up in the center of a heavy progenitor star; estimates of Refs. [40,41] show that the captured DM mass is close to Eqs. (5) and (6) within an order of magnitude. Later, a fraction of the star DM is inherited by the neutron star in the course of supernova collapse. That last process has never been considered in the literature, but our rough estimate suggests that the respective suppression of Mtot lies between 1 and 10-3.

In the rest of this paper, we will use Eqs. (5) and (6) and ignore the second, “gravitational capture” mechanism despite the fact that it seems less sensitive to nongravitational DM interactions. Indeed, that additional mechanism cannot dramatically change the amount of captured DM even in the best case. Besides, it depends on the multistage neutron star formation which is subject to large astrophysical uncertainties. Finally, it does in fact rely on the DM-DM and DM-neutron interactions, albeit in a subtle and indirect way: the nongravitational couplings are needed to thermalize the DM particles inside the progenitor star and mix them in the phase space, or none would lose enough energy to get into the neutron star.

Once gravitationally bound, the DM particles settle on star-crossing orbits and continue to lose energies in repeating collisions with neutrons until a thermal equilibrium is reached. Their kinetic energies reduce to the neutron star temperature T∼105  K, and their orbits shrink to the “thermal” radius, rth≈(9T8πGρcm)1/2∼20  cm(100  GeVm)1/2,(7)where we substituted the density ρc∼1015  g/cm3 of the neutron star core. The particles continuously arrive to the central DM cloud during the entire lifetime of the neutron star. The rate of this process is given by Eq. (3).

The characteristic time of this thermalization process was estimated in Refs. [14,42]. For almost any model within our scenario, it is much shorter than the age of the Universe [14]. Indeed, we have already mentioned that BH formation requires a large number of DM particles inside the neutron star [18]. To capture all of them, one usually assumes the largest possible DM-neutron cross section σ∼σcrit, and even that may be insufficient. With these interactions, the DM particles equilibrate quickly, since every star crossing in the beginning of the process leads to scattering. On the other hand, we will be able to consider very small DM-neutron cross sections once the new mechanism for reducing the required multiplicity to Eq. (2) is invoked. In that case, a necessity to thermalize the DM imposes the strongest constraint on its interactions with neutrons.

As the DM particles continue to accumulate, the multiplicity N of the central cloud grows at a fixed radius (7). Eventually, the mean distance between the particles rthN-1/3 drops below the size (mT)-1/2 of their wave functions. This happens at BEC:  N≳0.2T3Mpl3ρc-3/2∼1036,(8)i.e., significantly before the maximal number of particles (5), (6) is reached. At this point, the particle wave functions start to overlap and Bose-Einstein condensate forms in the thermal cloud [43–46]. Since then, most of the thermalized DM particles occupy the lowest level in the combined DM and neutron star gravitational potential, with the thermal energy being carried by a few remaining particles.

Thanks to large occupation numbers (8), the Bose-Einstein condensate at the lowest level can be described by a classical DM field ϕ(x). From the very beginning, it forms a nonrotating [47] self-bound soliton, which is almost detached from the neutron star surroundings. Indeed, even at multiplicity (8), the gravitational field of this object can be estimated to exceed the neutron star’s, and with time, the soliton mass grows. We will call this soliton a Bose star [48,49] or a Q-ball [50–52] if it is mostly bound by self-gravity or by attractive self-interactions, respectively. In either case, the properties of the soliton can be determined by solving the stationary classical field equations, where a single free parameter is the total number of accumulated DM particles N.

With the soliton formation, the final (model-dependent and nonlinear) stage of DM evolution begins. As N increases from Eq. (8) to the maximal values in Eqs. (5) and (6), the soliton becomes heavier. It collapses gravitationally if the condition for collapse—the hoop conjecture—gets satisfied prior to accumulating the maximal amount of DM. If, to the contrary, the soliton size exceeds its Schwarzschild radius even for the largest N in Eq. (6), the black hole does not form.

Let us finish section by commenting on DM thermalization in white dwarfs—second-best compact objects accumulating DM. Their cores are significantly more dilute, ρc≲108  g/cm3, and have higher typical temperatures T≳106  K as compared to the neutron stars; see Ref. [53]. Thus, Bose-Einstein condensation of dark matter particles in their centers would require larger DM multiplicity N≳1049; see Eq. (8). But in fact, the DM inside the thermal radius becomes self-gravitating before that, at N≳1046(m/100  GeV)-5/2. It collapses gravithermally and forms a compact object where the subsequent DM cooling and (possibly) Bose-Einstein condensation occur. This process deserves a separate study, which is outside of the scope of this paper.

As we stressed in the Introduction, interactions generically detain the gravitational collapse until the number of DM particles becomes parametrically larger than in Eq. (2), and this is often too large to be accumulated by the neutron star during the Universe’s lifetime. To gain a more quantitative understanding of the numbers, we start with the simplest scalar DM described by a single complex field, L=|∇μϕ|2-V(|ϕ|).(9)We will try to optimize its scalar potential V in a way that minimizes the dark matter multiplicity required for collapse. We consider minimal coupling to gravity and ignore interactions with the visible matter—those will be added later. Importantly, we also assume that the typical scale(s) Λ in the potential V are essentially sub-Planckian, Λ≪Mpl.(10)This separates our model from the effects of quantum gravity.

Suppose a compact, stationary, and stable solitonic configuration of the scalar field—a Q-ball or a Bose star—is formed inside the neutron star. Let us compute its mass M and radius R as functions of the global charge N. By conservation law, the latter quantity counts the number of dark matter particles appended to the Bose-Einstein condensate. We recall, first, that a generic stationary solution in the model (9) has the form ϕ=φ(x)e-iωt.(11)Here, we introduced the real-valued soliton profile φ(x) and the energy ω of particles inside it. Second, the gravitational field of the soliton is expected to be small, gμν≈ημν, except for the critical case when gravitational collapse is about to happen. In this case, the flat-space Noether charge can be used, N≈i∫d3x(ϕ*∂0ϕ-ϕ∂0ϕ*)∼ωφ02R3,(12)where the last equality is a crude estimate in terms of the soliton size R and the field φ0∼φ(0) in its center. Since every particle inside the soliton has energy ω, the total mass of this object is of order M∼ωN.

Recall, however, that the soliton parameters φ0 and R are related by the field equations which involve the scalar potential V(φ0). We therefore consider two options. First, one can assume that the potential grows almost quadratically at weak fields, V∼m2|ϕ|2, and then flattens out beyond some scale Λatt like V∝|ϕ|α with α<2; see Fig. 1(a). This case corresponds to particle attraction inside the solitonic core, as the energy per unit charge is smaller than the particle mass. At strong fields, we approximate V≈m2|ϕ|αΛatt2-αat  |ϕ|≳Λatt,(13)where 0<α<2. It is precisely the scalar self-interaction that holds the soliton—the Coleman Q-ball [51]—together, since gravity is weaker: Λatt≪Mpl according to Eq. (10). In the field equation, the self-attraction is balanced by the kinetic pressure ∂i2ϕ∼V′∼ω2ϕ, or φ0/R2∼m2φ0α-1Λatt2-α∼ω2φ0;(14)see the Appendix for details. This gives ω∼R-1, N∼(RΛatt)2(Rm)4/(2-α),andM∼N/R,(15)where we parametrize the soliton configurations with their sizes R.

The soliton collapses gravitationally when its compactness Rg/R=2M/(Mpl2R) becomes of order 1, i.e., at masses above critical, Mcr≈RMpl2/2∼ωNcr.(16)Substituting this relation into Eq. (15), we obtain the number of DM particles needed for collapse, Ncr∼Mpl2m2(MplΛatt)2-αfor  0<α<2.(17)At the critical point, the field inside the soliton is Planckian: φ0,cr∼Mpl; cf. Eqs. (14), (15), and (17).

Notably, the critical multiplicity (17) is minimal at α=2 when the “free bosonic” expression (2) is recovered. The other values of α<2 are less advantageous, as Λatt is parametrically below the Planck scale. One obtains almost the “fermionic” multiplicity (1) in the case V∝|ϕ| and even larger Ncr for the ϕ-independent potential. Thus, contrary to naive expectations, particle attraction obstructs collapse, the reason being that the energy per particle ω becomes much smaller than m.

One can push the above “attractive” option to the extreme assuming that the scalar potential V(ϕ) decreases at strong fields, e.g., V∝-|ϕ|α. However, that would destabilize the soliton, making its field evolve toward lower V; cf. Refs. [54,55]. With no new positive terms in the potential to stop the process, the region with V(ϕ)<0 would be eventually reached [56]; then, the soliton turns into an expanding and Universe—destroying bubble of true vacuum [57]. Even if the bubble can be somehow forced to collapse gravitationally, the region in its center still breaks the positivity conditions, so a naked singularity may appear instead of a black hole. If, alternatively, the potential starts growing at larger fields, again, the field stops rolling at that point, thus bringing us back to the two options in Fig. 1.

Equation (17) hints at the possibility that the second option of repulsive self-interactions with α>2 may be more interesting; see Fig. 1(b). In this case, the only attractive force is gravity. The respective soliton is called a Bose star [48,49], since it is bound by gravitational attraction compensating interaction pressure of Bose particles inside it. Note that the repulsive energy gives a subdominant contribution to the mass of the subcritical object because its opponent—the gravitational energy—remains small until the rim of collapse. We therefore keep two terms in the potential, V≈m2|ϕ|2+m2|ϕ|αΛrep2-α,(18)where now α>2, the field ϕ is arbitrary, and Λrep satisfies Eq. (10). Performing the estimates similar to the ones before [58], we find out that the Bose star collapses gravitationally only at α>8/3; see the Appendix for details. In this case, the soliton field indeed gets stuck in the region of subdominant self-interactions φ0≲Λrep until the collapse, at which point φ0,cr∼Λrep and the soliton charge equals Ncr∼Mpl3Λrepm2for  α>8/3.(19)Notably, this critical multiplicity is independent of α; see also Refs. [58,59]. At Λrep∼m, the expression (19) reproduces the fermionic result (1). The way to decrease the multiplicity is to increase Λrep suppressing the self-repulsion. At Λrep∼Mpl, that force is as weak as gravity, the field inside the critical Bose star is Planckian, and we obtain the free bosonic formula (2), again.

The remaining region 2<α<8/3 is the worst of them all, since in that case the Bose-Einstein condensate does not clump under self-gravity. Indeed, an estimate of the Appendix shows that the self-repulsion |ϕ|α with α in this range is stronger than gravity at large distances. It makes the DM condensate spread over the entire volume available inside the neutron star gravitational field.

We conclude that the critical particle number Ncr is smallest in the restricted class of models with long, Planckian-size valleys |ϕ|≲Mpl and almost quadratic potentials V∝|ϕ|2 at their bottoms. Any interaction impedes collapse and sharply increases the critical multiplicity. In particular, self-repulsion becomes strong, almost equivalent to the Fermi pressure at large fields—hence the “fermionic” result (19). Self-attraction does not help, either; it provides negative binding energy and lowers the soliton mass, which is bad for collapse. The respective critical particle number is also parametrically larger; see Eq. (17).

In the optimal model with exactly quadratic potential, the kinetic pressure inside the Bose star is compensated by the gravitational attraction [48,60,61]. The respective object becomes gravitationally unstable at φ0,cr∼Mpl. It has the critical multiplicity [60] Ncr≈0.653Mpl2/m2for  α=2.(20)This result agrees with the estimate in Eq. (2). Nonrelativistic approximation remains valid during most of the Bose star growth and gets marginally broken at the rim of collapse.

How far can the DM model deviate from the free bosonic theory? To find out, we require that the critical multiplicity for collapse Ncr does not exceed the maximal amount (6) of captured DM: mNcr≲Mtotdwarf.

In the attractive case, this condition bounds from below the scale |ϕ|∼Λatt at which the scalar potential flattens out in Fig. 1(a): Λatt≳Mpl(2×10-9  GeVfm)1/(2-α);(21)see Eqs. (6) and (17), and recall that 0<α<2. Typically, Λatt is very large. Indeed, since the critical Q-ball has φ0∼Mpl, the model with flat potential should be trustable, i.e., renormalizable and weakly coupled, all the way up to the Planckian scale. The only manifestly renormalizable flat potential is V=const; it appears [62], e.g., in the celebrated Friedberg-Lee-Sirlin model [50]. In this case, Λatt≳5×1013  GeV(mf/100  GeV)-1/2,α=0.(22)Soon we will see that large scales (21), (22) are problematic because multiloop corrections to the scalar potential become relevant at φ0≪Λatt. They generate interaction pressure and prevent collapse.

The scale Λatt can be substantially lowered in a specific class of renormalizable multifield models where the potentials at the bottoms of curved valleys have α≈2. We will consider this option in Sec. IV.

In the opposite, self-repulsive case, the inequality mNcr≲Mtotdwarf also provides a large scale, Λrep≳2×108  GeV(mf/100  GeV)-1,(23)following from Eqs. (6) and (19). This condition strongly suppresses all repulsive self-interactions at fields |ϕ|≲Λrep; see Eq. (18). For example, in the λ4|ϕ|4/4 case, λ4≡(2m/Λrep)2≲7×10-13f2(m/100  GeV)4.(24)It is worth remarking that the renormalizability of the repulsive potential is not required, since the Bose star field φ0≲Λrep remains parametrically below the natural cutoff of the theory (18) prior to collapse.

Now, recall that our dark matter should interact with the visible sector, and strongly enough, in order to be captured by the neutron star. The respective couplings should be renormalizable and stay under control at strong fields—hence, their forms are constrained by the Standard Model symmetries. Let us demonstrate that, generically, loop corrections from these interactions break the desired properties of the dark matter potential: lift its flat parts and generate unacceptably large repulsive vertices, as was first pointed out in Ref. [18].

Couple, e.g., the field ϕ to the Higgs doublet H(x) by deforming the potential of the latter to V=λH(H†H-v22-y|ϕ|22λH)2+m2|ϕ|2;(25)cf. Refs. [63,64]. Here, v≈246  GeV and λH≈0.13 are the standard Higgs parameters, and the new constant y regulates its interactions with the dark sector. Notably, the model is weakly coupled at y≪(4πλH)1/2∼1.

The physical Higgs field h(x) is defined as H†H=(v+h)2/2. It interacts with neutrons n via the effective Yukawa vertex Vhnn≈2mnhn¯n/(9v), which is known up to light quark contributions giving a factor 1÷2 [65,66]; mn≈GeV is the neutron mass. This means that the dark matter also scatters off neutrons. The respective diagram is shown in Fig. 2, and the cross section at nonrelativistic momenta equals¹

Hadronic form factors and strong nucleon interactions may suppress this cross section by an additional factor of up to 10-3; see Refs. [31,67]. However, that would only sharpen the arguments of this section.

On the other hand, the same coupling (25) generates DM pressure at strong fields. In fact, we have already tuned this potential to cancel DM self-interactions at the classical level. Namely, at large fixed φ0=|ϕ|, the Higgs field adjusts itself to minimize the first term in Eq. (25), H†H≡12(h+v)2=v22+y|ϕ|22λH.(27)As a consequence, the tree potential is quadratic at the bottom of this potential valley, V≈m2|ϕ|2. Nevertheless, the self-coupling reappears, again, once loop contributions from the visible matter are included. As an illustration, let us first ignore all fields except for the Higgs boson and ϕ itself. Then, their one-loop effective potential [68] at large |ϕ|∝h along the valley (27) equals ΔV1-loop≈λ4,eff|ϕ|4/4,(28)where λ4,eff=y2L/(2π2)+O(y3/λH) is a correction to the |ϕ|4 self-coupling, L=ln(|ϕ|/Λren), and Λren∼m is a renormalization scale. Note that a proper calculation of the effective potential at strong fields includes renormalization group resummation of the leading logs [69,70]. However, that usually introduces an order-1 factor in λ4,eff which does not affect our estimates.

We see that Eq. (28) brings in the |ϕ|4 repulsion even if it was absent before. Moreover, since λ4,eff logarithmically depends on the field, the new contribution cannot be canceled by any renormalizable counterterms even if an arbitrarily precise fine-tuning is allowed. An obvious way out is to make the one-loop repulsion small, so that it does not preclude black hole formation. Requiring λ4,eff to satisfy Eq. (24) with σ given by Eq. (26), we obtain the inequality y≳400L1/2>400,(29)which cannot be satisfied in a weakly coupled model with y≪1. Moreover, even this unacceptable lower limit can be achieved only if m≳PeV. We conclude that either the coupling constant y is too small for accumulating the required amount of ϕ-particles or the interaction pressure caused by the same constant prevents the ϕ-condensate from collapsing.

As an alternative, one may try to couple ϕ to fermions that generate negative terms in the effective potential. In the model (25), this amounts to recalling that every massive Standard Model field gives a potential to Higgs and therefore produces a vertex |ϕ|4∝h4 along the valley (27). The leading contribution is negative and comes from the top quark Yukawa coupling yt≈1. We obtain δλ4,eff=-3y2yt2L/(8π2λH2). This definitely destabilizes the valley [71,72] unless even larger positive |ϕ|4 terms are introduced, which, however, returns us to the no-go estimates given above.

Somewhat more elegantly, one may organize a (partial) cancellation between the fermionic and bosonic loops. But that would mean upgrading the Standard Model to (Next to) Minimal Supersymmetric Model (MSSM). In that case, the inequality mNcr≲Mtotdwarf imposes strong constraints on the supersymmetry-breaking operators that detune the cancellation [18]. In Sec. IV, we consider more economic and general possibility.

To sum up, loop corrections from the visible sector are dangerous and generic and cannot be avoided. Thus, we need a special mechanism to tame them.

Let us summarize the properties of dark matter model needed for black hole formation inside the neutron star: (a)

The scalar potential of the model should include a long valley parametrized by the complex scalar ϕ. The valley should extend to large fields: to |ϕ|∼Mpl in the case of attractive self–interactions (13) or, in the repulsive case (18), at least to the scale |ϕ|∼Λrep given in Eq. (23). The model should remain weakly coupled at these fields, i.e., be renormalizable or have a sufficiently high cutoff.

We have already demonstrated that the condition (c) is usually violated by dark matter interactions with the visible sector in one-field DM models. We now turn to models where this can be avoided.

Start from the arbitrary model for the DM field ϕ, and add the second complex scalar χ in such a way that ϕ and χ have global charges 1 and 2, respectively. The coupling between the two is then chosen in a specific renormalizable and positive-definite form, V=λ|ϕ2-Λχ|2+m2|ϕ|2+λ′|ϕ|4/4,(30)where the last two terms represent the original ϕ potential. This model is invariant under phase rotations ϕ→eiθϕ and χ→e2iθχ. In the vacuum ϕ=χ=0, all fields are massive: mϕ=m and mχ=Λλ1/2. We will assume that the first term in the potential is the largest: m≪Λλ1/2andλ′≪λ.(31)This means, in particular, that the χ-particles are heavy and do not change cosmology.

The trick of the new model is to make the potential valley bend in the ϕ−χ space. Indeed, the low-energy field configurations are expected to minimize the largest term in Eq. (30), i.e., satisfy χ=ϕ2/Λ.(32)The other terms of V create a small potential along this valley. The same is true, in particular, for the stationary nonrelativistic soliton, ϕ=φ(x)e-iωt,χ=χ˜(x)e-2iωt,(33)which has real profiles φ, χ˜ satisfying Eq. (32).

Let us explicitly demonstrate that the valley extends in the ϕ direction at small fields, then takes a turn at ϕ∼χ∼Λ and goes along χ. To this end, we introduce a combination ρ(x) of the solitonic profiles which has a canonical kinetic term along the valley: (∂iρ)2=(∂iφ)2+(∂iχ˜)2. Using Eq. (32) and integrating, we find ρ(φ)=φ21+4φ2Λ2+Λ4arcsinh(2φΛ).(34)The classical potential V(ρ) at the bottom of the valley is obtained by inverting Eq. (34) and substituting φ(ρ) into the last two terms of Eq. (30).

At small fields, we get ρ≈φ; hence, the valley stretches in the ϕ direction, indeed. Taylor series expansion in ρ/Λ≪1 then gives a nonlinear valley potential, V(ρ)=m2ρ2+λρρ4/4+O(ρ6/Λ4)(35)with λρ=λ′-16m2/(3Λ2). We have already discussed in Sec. III that the ρ4 term cannot be large positive, or it would stop the soliton from growing. Requiring λ′/4=β(m/Λ)2,β≲1,(36)we ensure that the effective coupling is attractive at small fields: λρ<0. This is not dangerous for vacuum stability, since the overall potential of our model is explicitly positive definite.

The region ϕ≫Λ is entirely different. Expression (34) gives ρ≈φ2/Λ meaning that the valley runs along χ˜≈ρ. At the same time, the potential at the bottom of the valley is quadratic: V(ρ)=μ2ρ2+(smaller terms), where we denoted μ=mβ1/2≲m,(37)and used Eqs. (30) and (36). So, quartic self-interaction in this part of the valley is absent, and the effective ρ mass is smaller than m.

Now, we see how the bent valley (32) works. Originally, the model (30) includes the term λ′|ϕ|4 creating pressure. The same term, however, reduces to a mass |ϕ|4∝|χ|2 once the valley takes a turn. Moreover, this property is valid even at the quantum level.

Indeed, an explicit one-loop calculation [68] gives quartic effective potential at large fields: ΔV1-loop≈λ4,eff|ϕ|4/4+O(ϕ2) with λ4,eff=2π2(2λ2+λλ′+5λ′2/32)ln|ϕ|Λren.(38)Thus, quantum fluctuations of ϕ and χ somewhat increase the constant λ′→λ′+λ4,eff in front of |ϕ|4∝|χ|2 but do not prevent the Bose star from growing if Eq. (36) remains satisfied: λ′+λ4,eff<4(m/Λ)2.(39)The latter inequality is easily met if Λ is not too far away from m.

One may wonder why the dangerous terms ΔV=c1|ϕχ|2+c2|χ|4 are not generated at the one-loop level. If present, they would create an undesired pressure inside the strong-field soliton. However, the auxiliary field χ enters the interaction potential in the combination Λχ, where Λ has mass dimension 1. On dimensional grounds, c2∼c12∼(Λ/Λcutoff)4, where Λcutoff is a cutoff for loops. Thus, the multiloop diagrams for the dangerous terms converge; the constants c1, c2 do not depend logarithmically on the fields; and one can tune them to zero at ϕ,χ≫Λ. Barring this fine-tuning, the quantum corrections are harmless in the model (30).

Let us visualize the growth of the Bose-Einstein condensate in the model of this section. Initially, it forms a Bose star held by the gravitational forces. This object becomes denser with time due to continuous inflow of dark matter particles. At ρ∼m2/(|λρ|Mpl)≪Λ, the λρρ4 self-attraction overcomes the kinetic pressure, and the Bose star collapses as a bosenova [54,73]. This means that its central part starts squeezing in a particular self-similar fashion [54,55] developing strong fields. The squeeze-in halts when the field in the center reaches ρ∼Λ and the valley potential stops being attractive; cf. Refs. [55,74].

This is the moment when a droplet of much denser condensate with ρ∼χ≳Λ appears in the Bose star center. At first, it has the field strength χ∼Λ. However, the droplet is expected to grow in density toward χ≫Λ as more χ-particles join in. Indeed, once it is there, a condensation process ϕϕ→χ with energy release 2m-μ becomes allowed. Even if one assumes that this direct process is ineffective dynamically, the condensation should continue in another, recurrent way. Without direct transmutation into χ, the ϕ particles would create another dilute Bose star around the χ-droplet, and that star would collapse, again, feeding the droplet. In any case, all global charge should be eventually transported into the dense χ-cloud and may never leave it due to large binding energy 2m-μ of the χ particles; cf. Eq. (37).

Notably, we expect that the central χ-cloud quickly thermalizes into a dense and noninteracting Bose star with χ≫Λ. Indeed, the χ-particles effectively scatter with self-coupling ≳λ′ at the cloud boundaries. In there, χ∼Λ, and hence the particle number density is huge: n∼μΛ2. A conservative estimate constrains the relaxation time from the above [43,45,75], τχ∼(ξσϕϕvχnf)-1≪40  yrvχ2(100  GeV)3μΛ2,(40)where ξ≫(Λ/Mpl)2 is the time fraction spent by the particles at the cloud periphery, σϕϕ≳λ′2/(64πμ2) is their cross section, f∝n/(μvχ)3 is the phase-space density, vχ is velocity, and we used Eqs. (36) and (37). Even the most conservative choices of parameters give small τχ compared to the lifetime of the Universe—thus, the relaxation is effective, indeed.

Now, we recall that the self-interactions are absent at χ≫Λ. This means that the respective Bose star will grow pressureless until it collapses gravitationally into a black hole. The latter event requires the critical charge Ncr≈1.3Mpl2/(βm2),(41)where the numerical coefficient is larger by a factor 2/β than in Eq. (20) because χ∼ρ has global charge 2 and mass μ=mβ1/2.

We thus constructed a model with almost an optimal critical multiplicity for the collapse of Bose-Einstein condensate into a black hole. In the next section, we will see that interaction with the visible sector and hence DM capture can be added to this model without disrupting the picture.

We couple the DM field to the Higgs doublet in the same way as before—by adding the first term in Eq. (25) to the scalar potential (30). This makes the ϕ-particles scatter off neutrons with the cross section σ∝y2 in Eq. (26). Importantly, we leave χ to interact only with ϕ. This maintains the desired properties of the strong-field Bose stars in our model.

Now, the Higgs field changes along the potential valley (32) according to Eq. (27). As a consequence, a nonzero Higgs profile h(x) is generated inside every stationary soliton. A combination of the three profiles—φ, χ˜, and h—with the canonical kinetic term is ρ(φ)=∫0φdφ′1+4φ′2Λ2+y2φ′2λH(mH2+2yφ′2),(42)where the last contribution comes from the Higgs field.

Notably, with interactions (25) included, the effect of the bent valley remains essentially the same. Indeed, ρ is still proportional to φ and χ˜ at ρ≪Λ and ρ≫Λ, respectively. In the weak-field regime ρ≲Λ, the valley potential is nonlinear, Eq. (35), and has almost the same quartic constant λρ as before. We therefore again impose the condition (36) to make λρ negative at weak fields. Once the valley turns at ρ≫Λ, the potential V(ρ) becomes quadratic with mass (37), as guaranteed by the mechanism of the previous section.

All these nice properties remain valid because χ does not couple directly to the Standard Model fields and still enters the overall potential in the combination Λχ with dimensionful Λ. This makes the nonlinear interactions vanish in the strong-field region, and they cannot be generated by loops. We check the latter property by computing the one-loop effective potential in the three-field model of ϕ, χ, and H at the bottom of the valley (27), (32). The result at large fields is given by Eq. (28), where the constant λ4,eff equals Eq. (38) plus the Higgs contribution δλ4,eff=y2π2λH(λ+λH2+y2+3λ′8+y28λH)ln|ϕ|Λren.Similarly, the other Standard Model fields are expected to generate ΔV1-loop∝h4∝|ϕ|4. Regardless of the value of λ4,eff, the loop corrections safely change the mass |ϕ|4∝|χ|2 in the strong-field region. The value of μ then can be made positive by adjusting λ′. For simplicity, we assume below that the loop corrections are small compared to λ′.

Without the interaction pressure, the strong-field Bose star collapses gravitationally at almost optimal critical charge (41). This value cannot exceed the total number of captured dark matter particles, Mtot(dwarf)/m>Ncr. One obtains the inequality y>10-7(m100  GeV)1/2β-1/2(43)using Eqs. (6), (26), and (41). This constraint is relatively mild and can be easily satisfied.

Indeed, let us show that a viable parametric region for our model can be selected. One starts by specifying m within the WIMP mass range and β≲1. The coupling constant y is then chosen to satisfy Eq. (43) together with the constraints from the DM detection experiments. Below, we will argue that a narrower region, 2×10-6≲y(m100  GeV)-1/2≲2×10-5β-1/2,(44)is better for phenomenology. The next parameters are λ and λ′≪λ. In the simplest case, one takes λ∼y/λH1/2 and λ′≳λ2, thus suppressing the running of λ′: λ4,eff, δλ4,eff≲λ′. Finally, the scale Λ is given by Eq. (36). We conclude that all the constraints can be satisfied if the hierarchy λ′∼λ2∼y2/λH,Λ∼m/λ,β≲1(45)is valid and y belongs to the interval (44).

So far, we relied on fine-tuning to kill the |ϕχ|2 and |χ|4 terms in the potential. If added, these interactions should be extremely weak—say, the operator λ2′|χ|4/4 should satisfy the analog of Eq. (24): λ2′≲2×10-13f2β3(m/100  GeV)4.Can we allow stronger interactions at |χ|≳Λ?

The answer is yes, if extra auxiliary scalars are introduced. Namely, let us add the |χ|4 term along with the new field χ2 that has a global charge 4. The new terms in the potential are V2=λ2|χ2-Λ2χ2|2+λ2′|χ|4/4,(46)where λ2′=4β2(μ/Λ2)2 and β2≲1; cf. Eq. (36). Now, the valley takes a second turn at χ∼χ2∼Λ2, before the |χ|4 interaction becomes relevant.

One can continue this procedure and arrive at the “clockworklike” model with n fields χi, a hierarchy of scales Λ≪Λ2≪…≪Λn, masses at strong fields μi=μi-1βi1/2, and many coupling constants λi′≡(2μi/Λi)2, where βi≲1. The Bose star made of the last field χn collapses gravitationally at the critical charge Ncr=(Mpl/m)2∏i=1n(2/βi).(47)If n∼O(1), this multiplicity is not exceedingly large.

Of course, the field χ can be made interacting in other, old-fashioned ways—say, by supersymmetry. To this end, one upgrades this field to the supersymmetric sector with a flat direction [76] which represents the |χ|≳Λ part of the valley. The flat direction should be lifted by soft terms giving the mass to the condensate. Then, the superpotential will be protected from quantum corrections as long as its fields couple to the visible matter via the dimensionful constant Λ.