Phenomenology of magnetic black holes with electroweak-symmetric coronas

Magnetically charged black holes (MBHs) are interesting solutions of the Standard Model and general relativity. They may possess a “hairy” electroweak-symmetric corona outside the event horizon, which speeds up their Hawking radiation and leads them to become nearly extremal on short timescales. Their masses could range from the Planck scale up to the Earth mass. We study various methods to search for primordially produced MBHs and estimate the upper limits on their abundance. We revisit the Parker bound on magnetic monopoles and show that it can be extended by several orders of magnitude using the large-scale coherent magnetic fields in Andromeda. This sets a mass-independent constraint that MBHs have an abundance less than 4 × 10−4 times that of dark matter. MBHs can also be captured in astrophysical systems like the Sun, the Earth, or neutron stars. There, they can become non-extremal either from merging with an oppositely charged MBH or absorbing nucleons. The resulting Hawking radiation can be detected as neutri- nos, photons, or heat. High-energy neutrino searches in particular can set a stronger bound than the Parker bound for some MBH masses, down to an abundance 10−7 of dark matter.


Introduction
With the discovery of the Higgs boson in 2012 [1,2], the complete particle content of the Standard Model (SM) of particle physics is confirmed. Although remaining puzzles such as neutrino mass, dark matter, and the baryon asymmetry may require physics beyond the SM, it is also important to know all possible states of matter in the SM. For ordinary life, the electromagnetic interaction provides a rich ensemble of both stable and metastable atomic states. Quantum chromodynamics (QCD) interactions are responsible for hadronic states and atomic nuclei. In high-temperature and/or high-density environments, exotic states also exist based on the QCD interaction, such as quark matter or colorsuperconducting matter [3], which could exist in the cores of neutron stars. For the SM electroweak (EW) interactions, less attention has been paid to the possible exotic states JHEP10(2020)210 with the exception of objects similar to the 't Hooft-Polyakov monopole [4,5]. Due to the non-compactness of the U(1) Y symmetry, there is no localized, finite-energy monopole solution in the weak sector (see refs. [6,7] for new interactions beyond the SM that facilitate a finite-energy solution and ref. [8] for a metastable monopole-antimonopole state connected by a string).
With the help of the gravitational interaction, more states based on the SM interactions could exist. One interesting example is the black hole with EW hair, discovered by Lee and Weinberg in ref. [9]. This magnetically charged black hole can have spherical nontopological monopole hair around a Reissner-Nordström (RN) black hole (BH) at the core. For a large magnetic charge Q, similar but much heavier objects exist. As recently pointed out by Maldacena in ref. [10], EW symmetry can be unbroken inside a non-spherical coronalike region surrounding the BH. The restoration of EW symmetry in a large magnetic field was pointed out a while ago in refs. [11][12][13][14][15]. Other objects with macroscopic EW-symmetric (EWS) regions have been studied in [16][17][18].
This magnetic (nearly-)extremal black hole (MeBH) with an EWS corona serves as a novel and interesting state just from the SM plus general relativity. Near extremality, this new state is stable against Hawking radiation [19]. For a large enough Q, the evaporation of magnetic charge via Schwinger production of magnetic monopoles like grand unified theory (GUT) monopoles or splitting into two smaller MeBHs is also suppressed. As a result, this MeBH with an EWS corona is stable on cosmological time scales, even at masses below 10 15 g where uncharged BHs would evaporate on timescales shorter than the age of the Universe. Leaving aside the early universe production of primordial MeBHs (PMBHs) (see refs. [20,21] for early universe production of primordial eBHs and refs. [22,23] for recent reviews of primordial uncharged BH production), the immediate questions are how abundant MeBHs can be in the current Universe and how to detect them, which will be the main topics of this paper.
From a phenomenological point of view, this MeBH has some similarities with the GUT monopole (see [24,25] for recent reviews). Both are heavy objects with magnetic charges, but MeBHs have a fixed, smaller charge-to-mass ratio and larger masses. Additionally, MeBHs could have a large-radius corona with unbroken EW symmetry, which will increase their interaction rates with ordinary matter. For instance, when we consider the capture rate of MeBHs by astrophysical objects, both the finite size and large magnetic charge of MeBHs will be taken into consideration. Furthermore, given the BH in the core, Hawking radiation also plays a role when either two MeBHs with opposite charges merge or one MeBH absorbs ordinary matter to become a non-extremal BH [21]. This Hawking radiation provides a unique signal not found in ordinary monopoles. As emphasized in ref. [10], an interesting two-dimensional (2d) Hawking radiation mode could happen when the Hawking temperature is above the mass of the lightest electrically charged particle, i.e., the electron. PMBHs thus also differ from Schwarzschild BHs, which have four-dimensional (4d) Hawking radiation. This paper is organized as follows. We first discuss some properties of MeBHs with an EWS corona in section 2. In section 3, we apply the Parker limits to MeBHs from both the Milky Way and Andromeda galaxies. We find that the large-scale coherent magnetic JHEP10(2020)210 fields in Andromeda provide a much stronger bound on monopoles and MeBHs than the traditional Parker bound from the Milky Way. The neutrino signals from captured PMBHs inside the Sun are discussed in section 4; the Earth heat and neutrinos are worked out in section 5; the photon signals from the PMBHs captured by neutron stars and white dwarfs are studied in section 6. For some MeBH masses, several of these signals can provide even stronger constraints than the Andromeda Parker bound. We discuss other constraints like gravitational lensing and direct detection searches, then summarize the various constraints on the fraction of PMBHs as dark matter in section 7. In appendix A, the formalism for 2d massless fermion modes in a PMBH magnetic field background and a qualitative understanding of the neutrino 2d modes are provided. In appendix B, stopping of a finitesized PMBH in various astrophysical objects is worked out.

Extremal magnetic black holes
A charged black hole is described by the Reissner-Nordström metric, Here, f (r) = 1 − 2M G/r + πQ 2 G/(e 2 r 2 ); G = 1/M 2 pl with M pl = 1.22 × 10 19 GeV; e is the electric gauge coupling defined as e = √ 4πα with α ≈ 1/137 as the fine-structure constant; Q is an integer and h Q ≡ Q h with h = 2π/e ≈ 68.5 e ≈ 21 is the magnetic charge times the coupling. Q = 1 is the minimal charge, manifesting the Dirac quantization condition e h = 2π. The magnetic field is B = Qhr/(4πr 2 ). For a RN magnetic extremal black hole (eBH), one has Note that the repulsive magnetic force between two same-sign extremal BHs is exactly equal to the attractive gravitational force.
Ref. [9] demonstrated the existence of another magnetically-charged BH solution, which appears when the event horizon radius is less than of order the monopole radius. By "monopole radius," we mean the radius that an 't Hooft-Polyakov monopole would have if it was admissible in the symmetries of the SM, i.e. R M m −1 W when Q = 2 with m W the W boson mass. This leads to a hairy BH with a cloud of EW fields outside the horizon, distinct from the RN solution.
The magnetic field near the event horizon of an eBH is where we have used R eBH ∼ R RN eBH . For a smaller Q, the magnetic field is larger, owing to the shrinking event horizon radius. As studied in refs. [11][12][13][14][15] [21,26,27]) Within this radius, the EW symmetry is unbroken. Analogously, there is also a QCD corona for the PMBH with radius where Λ QCD ∼ 1 GeV. This radius is larger than R EW . Within this radius, QCD may be in a different phase from the vacuum phase without a magnetic field [28,29].
Ref. [9] demonstrated the existence of a spherically symmetric solution for Q = 2 (corresponding to q = 1 in [9]) for this new class of magnetically charged BHs with hair. The hairy BH's mass is at least the summation of c W M RN eBH and the monopole mass M M . Here, c W = cos θ W ≈ 0.88, θ W is the Weinberg angle of the SM, and M M 4π m W /e 2 is the spherically symmetric monopole mass (again, assuming such a monopole was admissible in the SM symmetry group). The factor of c W appears because the EW symmetry is restored near the event horizon, so the BH carries magnetic hypercharge 2πQ/g Y = c W 2πQ/e, with g Y the hypercharge coupling constant. Its mass is bounded from above by requiring the mass not be larger than that of a BH with radius R EW . For a large Q, the corona boundary is anticipated to be non-spherical, and the mass M * • must be above c W M RN eBH plus the non-spherical Q-charged monopole mass M M (Q) [9]. The shape has not been worked out in detail, but may be expected to contain spiky features where vortex strings end on monopoles [10], which we denoted using subscript " * • ".
We now give a more precise estimate for the mass. Including the contributions from both the hypercharged BH mass and the positive vacuum energy of the unbroken EW symmetry, m 2 h v 2 /8, the EWS-corona BH mass is estimated to be Here, we have ignored the energy contributions from the transition boundary from symmetry-unbroken to broken regions as well as the non-sphericity of the corona configuration. We anticipate that those corrections are small in the limit of 1 Q Q max . The second term, which comes from the energy density of the corona, is only important when Q However, it is easy to see that M tot MeBH (2 Q) > 2 M tot MeBH (Q) due to the presence of the second term, so energetically it is preferable for an MeBH with a large charge to split into smaller MeBHs. Although the large-charged MeBH is metastable, its lifetime can be longer than the age of the Universe for Q Q min 10 6 given the existence of a GUT monopole with mass M GUT M ∼ 10 17 GeV [10]. This is a stronger condition than in (2.5). This metastability is in agreement with the weak gravity conjecture [30]: the non-gravitational interaction is stronger than the gravitational one. The range of viable charges Q min Q Q max corresponds to a mass range For reference, the mass of the Earth is M ⊕ = 6.0 × 10 27 g = 3.4 × 10 51 GeV.

Non-extremal magnetic black holes
Non-extremal BHs are also relevant for phenomenology. They appear, e.g., after mergers of oppositely charged PMBH or absorption of baryons by PMBHs. For these cases, the BH mass M BH > M * • , so the BH has a non-zero Hawking temperature given by (2.10) Here and elsewhere, M * • is taken to mean only the mass contribution from the BH and not the corona as in (2.8).
For a non-extremal PMBH with an EWS corona, the Hawking radiation inside the corona is effectively made up of 2d modes (see appendix A) leading to a radiated power [10,31] (2.11) Here, g * counts the number of left-and right-handed 2d modes using the hypercharges of chiral fermions. For instance, g * = |Q| for q L , L , d R , e R (the left handed quark, lepton doublets, right handed down quark, and electron of the SM) and g * = 2|Q| for u R (the right handed up quark). In the high-temperature limit, the total g * = 6|Q| for one family of SM fermions and g * = 18|Q| for three families. We emphasize that the 2d Hawking radiation only applies to fermions here (for spin-zero particles, the 2d modes are massive with a mass proportional to qeB(R eBH ), with q the particle's charge; for spin-one particles, the magnetic flux generates a negative mass and induces gauge boson condensation), so no photon modes with a large multiplicity |Q| are anticipated. Furthermore, not all of those When T (M BH , M * • ) m e , the previous 2d radiation is suppressed. The region within the EWS corona will be heated to the Hawking temperature of the BH. Both thermal photon and neutrino modes are stored in this region. As a result, the 4d blackbody radiation on the boundary of the EWS corona could be important and has radiated power with g * = 2 for photon and g * = 3 × 2 × h , for the merger product to have an EWS corona becomes So, unless Q 1 is infinitesimally close to Q 2 (or both are near Q max ), this condition can be easily satisfied and the PMBH merger product also has an EWS corona. In the limit of Q 1 − Q 2 Q 1 + Q 2 ≡ 2Q and hence M BH M * • , this condition becomes Thus, the produced non-extremal RN BH also has an EWS corona (we do not consider the situation that the charge distribution of PMBHs is exactly monochromatic with a delta function). Using (2.10), the Hawking temperature is charged particles can be produced from the 2d Hawking radiation. The heavy charged particles have various decay channels which generate neutrinos that can escape the Sun or the Earth's core and potentially be observed by neutrino telescopes. In addition, the radiated particles other than the neutrinos may heat up astrophysical bodies like the Earth, neutron stars, or white dwarfs, potentially providing a bound. Also note that the non-extremal RN BH can quickly 2d Hawking radiate to become (nearly) extremal. Using (2.11), the evaporation time scale is where we have chosen g * = 2|Q| and M BH = 2M * • . As emphasized in ref. [10], this 2d Hawking radiation time scale is much shorter than the 4d one, which scales like M 3 * • /M 4 pl [21]. PMBHs can also facilitate baryon number violation (BNV). As discussed in appendix A, baryons that enter an EWS corona become 2d modes and can be easily captured by the PMBH. For example, if R EW 1 GeV −1 corresponding to Q 10 4 , then baryon bound states are expected [17]. The PMBH can then reemit energy as Hawking radiation, which need not have the same baryon number. There is also the possibility that the EWS corona can mediate baryon number violation [32,33]. We do not consider that here because the extended sphaleron configuration is also relevant for the baryon-violating process, though the sphaleron energy is reduced [33]. Note that these processes are different from the Callan-Rubakov process, which applies to GUT monopoles [34][35][36][37][38]. Another interesting possibility, left for future work, is that the BNV process could facilitate baryogenesis.
For the case of PMBH absorption of baryons, the resulting BH mass is close to the extremal mass. In the limit of M BH − M * • M * • , the Hawking temperature is PMBHs with larger masses must absorb many baryons before reemitting via 2d modes. This may occur, e.g., in dense environments like stars. Using the 2d radiation in (2.11) with g * = 2|Q|, the fast 2d evaporation time scale is After this time scale, the BH follows the slow 4d evaporation process.

Parker limits from Milky Way and Andromeda galaxies
The Parker bound arises from the requirement that domains of coherent magnetic field are not drained by magnetic monopoles [39]. If a monopole transits such a domain, it will be JHEP10(2020)210 accelerated by the magnetic field and drain its energy. Thus, the energy loss to monopoles must be slower than the time it takes for the fields to be regenerated. To simplify our discussion, we will ignore the subleading second term in (2.8) for Q Q max and take Compared to a GUT monopole with Q = 1, a PMBH has a much larger mass-to-charge ratio. We now compare the PMBH flux to the various Parker-type bounds, updated to include charge dependence where necessary.
Assuming that PMBHs account for a faction f * • of all dark matter energy density and has an averaged speed v, the flux is For the local dark matter density in our solar system, we use ρ local ≈ 0.4 GeV cm −3 [40] and virial velocity v ≈ 10 −3 [41].
We follow the treatment of ref. [42], but include the Q-dependence in h Q and M * • .
First, a monopole can be accelerated in a coherent magnetic field to reach a speed v mag min 1, where 21 = c /(10 21 cm) is the coherence length of the magnetic field and B 3 = B/(3 × 10 −6 gauss) is the magnetic field strength in our Milky Way galaxy [43]. This velocity is less than the virial velocity of our galaxy, around 10 −3 . Thus, the PMBHs can remain bound in our galaxy and explain DM. They could also have a larger velocity and not be bound, thus unable to explain DM, but the flux bound presented below turns out to be the same. The Parker bound is set by requiring the mean rate of energy gained by PMBHs times the regeneration time t reg of the field by dynamo action to be smaller than the energy stored in the magnetic field, or with ∆E The magnetic-field-independent constraint on the PMBH flux is where v −3 = v/(10 −3 ) and t 15 = t reg /(10 15 s). Combined with (3.1), the constraint on the PMBH fraction from coherent fields in the Milky Way is independent of the PMBH mass and given by

JHEP10(2020)210
Thus, at present there is no constraint from magnetic field domains in our galaxy, regardless of PMBH mass and charge. To strengthen the bound in (3.5), one could look for systems with larger coherent magnetic field domains 21 > 1, longer times to regenerate the magnetic fields t 15 > 1, smaller virial velocities v −3 < 1 (although note if v < v mag , then PMBHs would not be bound to the galaxy so could not be DM, and a different constraint would apply [42]), or larger enhancements to the local DM density ρ 0.4 > 1.
We identify the Andromeda galaxy as an example of a system with larger coherent magnetic domains that take a correspondingly longer time to regenerate. Andromeda has an approximately azimuthal magnetic field around its whole circumference, measured between radii of 6 and 14 kpc [44]. This implies c ∼ 10 kpc ⇒ 21 ∼ 30 and t reg ∼ 10 Gyr ⇒ t 15 ∼ 300 [45]. The density of DM for Andromeda is very similar to the Milky Way [46,47], so we keep ρ 0.4 ≈ 1 and v −3 ≈ 1. Using these values in (3.5), we constrain the PMBH fraction in Andromeda to be Although there is a large uncertainty for c and t reg used in the Parker limit, the above limit suggests PMBHs cannot account for all dark matter in our Universe. While the above bounds come from galactic magnetic fields, intracluster magnetic fields were considered in ref. [48], although the bound is somewhat less secure as stated in their paper. Because of the smaller intracluster dark matter density ≈ 1.5 × 10 −6 GeV cm −3 [49], the constraint is much weaker than (3.6). Finally, the bound in [42] was extended in [50], which observed that the much smaller seed magnetic fields early in our galaxy's formation must also survive. Otherwise, there would be no fields today. The limit from the progenitor field reduces to the ordinary Parker bound when M * • /Q > 10 17 GeV, which is always satisfied for PMBHs.

Cosmic rays: solar neutrinos
PMBHs can be captured by the Sun, drift into the core of the Sun, and merge to produce non-extremal RN BHs that Hawking radiate energetic neutrinos. This high energy particle signal from merging extremal BHs was pointed out in a different context in [21]. In addition, PMBHs that are captured but have not yet annihilated can mediate BNV processes, which could also be detected by the energetic particles they emit. Measuring the neutrino flux from the Sun's direction by a large-volume neutrino detector can therefore constrain the PMBH fraction f * • of dark matter.
Before giving the detailed calculations, we provide a brief overview of the capture process, which will be applicable for the Sun, the Earth, neutron stars, and white dwarfs. First, the Sun captures PMBHs with a rate dependent on the flux and the strength of interactions between PMBHs and the stellar medium. The PMBHs will drift to the core of the Sun, and the time it takes should be short compared to other timescales in the problem. There, they may merge with oppositely-charged PMBHs and "annihilate" via Hawking radiation. The capture and annihilation rates often reach an equilibrium, so that the annihilation rate saturates to the capture rate. However, magnetic fields in the core JHEP10(2020)210 may separate oppositely-charged PMBHs, preventing them from annihilating. These can lead to a build-up of PMBHs, and these PMBHs can mediate BNV processes.

PMBH capture by the Sun
The capture rate of PMBHs by the Sun is estimated to be where R = 7.0 × 10 10 cm is the solar radius; v esc = 2 × 10 −3 is the solar escape speed on the solar surface; v = 10 −3 is the averaged dark matter speed; ∈ [0, 1] is the capture For a small radius PMBH, its stopping power by the solar plasma is similar to the GUT monopole case [51][52][53][54][55] except for a factor of Q 2 enhancement. Using the results of [55], any PMBH with Q > 30 or M * • 2 × 10 21 GeV is stopped. Following the analysis of ref. [52], in appendix B, we also take into account finite-size effects and demonstrate that our Sun can easily stop any large-radius PMBH above this minimal charge once it enters the Sun. We therefore choose = 1.
The captured PMBH will be slowed down by interacting with the plasma, thermalize with the medium, and drift into the core region of the Sun. Oppositely charged PMBHs "annihilate" or merge into a non-extremal RN BH. The annihilation or merger rate is related to the number density distribution of PMBHs.
We begin by presenting the usual calculation of solar capture and annihilation, relevant for DM with no self interactions aside from annihilations [56]. After that, we will include the effects of the magnetic fields of the Sun and PMBHs, which qualitatively and quantitatively change the results. For the non-interacting case, the PMBH radial distribution at the core follows a Maxwell-Boltzmann distribution ∝ e −φ(r)/(k B T ) where φ(r) is the gravitational potential. This can be written as a Gaussian with a characteristic radius of R: with n 0 as the PMBH number density at the center of the Sun. The annihilation or merger rate is estimated to be where a geometric cross section with the BH event horizon radius is used. The radius R can be estimated using the balance of the gravitational potential and kinetic energy where ρ c is the mass density in the core. If T is similar to the solar core temperature without PMBHs, T = T c = 1.5 × 10 7 K. If the gravitational potential is dominated by the solar plasma and using ρ c = ρ p ≈ 50 g/cm 3 [57], the radius is [41]

JHEP10(2020)210
with the proton mass m p = 0.938 GeV. The annihilation rate is given by The time evolution of the PMBHs in the Sun is given byṄ * where all PMBHs are assumed to have charges of equal magnitude. The solution to this which has an equilibration time So, for f * • = 1 and M * • > 6 × 10 23 GeV, τ eq is shorter than the age of the Sun t = 4.6 × 10 9 yr = 1.45 × 10 17 s. For τ eq < t , the annihilation rate is determined by the capture rate and given by The time between mergers Γ −1 A is longer than τ BH in (2.15) for M * • 10 39 GeV, so one could approximately treat the 2d Hawking radiation as occurring instantaneously below this mass. In other words, for low enough PMBH mass, PMBH mergers and annihilations can be thought of interchangeably. We will see that all masses for which constraints can be placed satisfy this approximation. The total number of PMBHs inside the Sun is then So far, the PMBHs were assumed to make their way to the center of the Sun nearly instantaneously. However, if the drag forces are large enough, it may take a long time for PMBHs to make their way from the surface to the center of the Sun. An estimate for this drift velocity can be obtained by setting the stopping force in appendix B equal to the gravitational attraction: Thus, using n e = 10 24 cm −3 and v th = 0.058 corresponding to T = 10 7 K, the drift time (for a vertical moving PMBHs) is (4.10) Thus, PMBHs with mass M * • 2 × 10 38 GeV will take shorter than t to settle to the center of the Sun. This sets an upper limit on the masses that can be probed by PMBH merger signals.
The above analysis has ignored the magnetic field in the core of Sun. A magnetic field could separate the locations of positively and negatively charged PMBHs, as emphasized in JHEP10(2020)210 ref. [53]. This would prevent oppositely charged PMBHs from merging unless the number of captured PMBHs is above a critical value. On the other hand, the attractive forces between oppositely charged monopoles can increase the annihilation rate well above that in (4.5). As we now demonstrate, the attractive forces are more important for the Sun when f * • is not too small, while for tiny f * • the Sun's magnetic field gives the dominant effect. In either case, τ eq in (4.6) will not be meaningful.
Choosing a constant B and assuming the +Q and −Q PMBHs are separated by a distance z from the center, the force balance equation is where the last term is the attractive force of 1 2 N * • PMBHs on one oppositely charged PMBH. We have assumed that the PMBHs are near-extremal so that the gravitational and magnetic forces are equal (they actually differ slightly, by a factor of c W , which we neglect). If the first two terms are dominant, the magnetic and gravitational forces can be balanced with a separation distance where B 100 = B/(100 gauss) (the precise value of magnetic field in the solar core region is still unknown [58]) and ρ c = ρ p ≈ 50 g/cm 3 . Compared to R th in (4.4), the magnetic force is more important than the thermal pressure and can potentially separate oppositely charged PMBHs and reduce the annihilation rate C A (as shown in figure 1). To consistently ignore the attractive force between PMBHs in the third term, the total number of PMBHs is required to be below a critical value

JHEP10(2020)210
Above this critical number, the third term in (4.11) reduces the distribution radius, and equilibrium is quickly reached between capture and annihilation with Starting from time zero, the captured PMBHs settle down in the solar core with a separation distance of z B . The amount of time to reach N crit * • is given by which is independent of PMBH mass and dramatically smaller than the age of the Sun Note that when 10 36 GeV M * • 2 × 10 38 GeV, there are only O(1) PMBHs within the distribution radius to annihilate. Also, for a heavy mass, the capture rate in (4.1) is small. The capture/annihilation process is discrete: the Sun waits a long time to capture the next PMBH, which quickly drifts into the core and annihilates with the existing one.

Solar neutrinos from PMBH annihilation
For the radiated charged fermions following a PMBH merger, the thermally averaged energy for 2d Hawking radiation is E f ≈ 1.19 T BH , with T BH given in (2.14). Ignoring order one factors, the number of charged particles from one PMBH annihilation event is N f ≈ M * • /T BH . Charged particles like µ ± , π ± , and K ± can decay into neutrinos directly or via cascade. The total number of high energy neutrinos from each annihilation event is where η ν is a factor to take into account the average number of high energy neutrinos from charged particle decays, e.g., η ν ≈ 2 for a muon. This assumes that the particles do not thermalize or significantly slow down in the solar plasma before decaying to neutrinos, which is true for prompt-decaying particles like the τ lepton, as well as the charm and bottom hadrons. The neutrino energy is The neutrinos generated from PMBH annihilation plus 2d Hawking radiation can propagate outside the Sun and reach the Earth to be observed by neutrino telescopes.
IceCube has performed a search for dark matter annihilations inside the Sun [59], which can be recast as a search for PMBHs. There are lower energy cuts to select neutrinos: E cut ν = 10 GeV for DeepCore selection and 100 GeV for IceCube selection. Requiring T BH > E cut ν in (2.14), the maximum mass that can be probed is (4.16) For a heavy PMBH, the annihilation rate could be so suppressed such that the separation time from one event to another event could be longer than the operation time, t exp , of the experiment. This sets another upper limit on the mass that can be probed For a given experiment, the combined upper limit on the testable mass is (4.18) The generated neutrino flux is where the distance from the Earth to the Sun is d ⊕ = 1.5 × 10 13 cm. The Sun is opaque for neutrinos with E ν 100 GeV [60], so the high energy neutrinos generated at the solar core will be converted to charged leptons from the charged-current interaction or to lower-energy neutrinos from the neutral-current interaction. A detailed analysis requires a numerical study of particle production, decay, and interaction. In our simplified recast of IceCube limits, we take the initial neutrino energy E ν ∼ T BH . When T BH below 100 GeV, we will take η ν = 1. For T BH 100 GeV, we assume that the neutrinos with energy below or around 100 GeV have a multiplicity factor proportional the neutrino energy, η ν ∼ (T BH /100 GeV), resulting from the cascades of higher energy particles.
For the IceCube searches [59], t exp = 532 day. The effective area can be read from the left panel of figure 4 of ref. [59], which can be approximated by for the acceptance area for the DeepCore selection. We also note that the search in ref. [59] is for muon neutrinos, which generate tracks in the detector with a better pointing ability.
Here, we absorb the additional neutrino flavor dependence and also neutrino oscillation effects into the factor η ν . Requiring the number of signal events I ν × A eff × t exp 100 (see figure 6 of [59] for the observed number of events and table 3  The second line comes from M max,t in (4.17). This limit would apply if the drift time were instantaneous, which is a good approximation for M * • 2 × 10 38 GeV from (4.10).
Thus, solar neutrino searches cannot probe merging PMBHs to higher masses than this, regardless of experiment time or energy threshold.

Baryon number violation process
Captured PMBHs absorb baryons in the Sun, increasing their mass above extremality and leading to 2d or 4d Hawking radiation depending on the temperature increase by the incoming flux of baryons. For a PMBH captured within the Sun, we can approximate the rate of change of the mass, where Θ(x) is the heavyside function and v p th is the proton thermal velocity. Here, the first term on the right side is energy deposited by the proton flux, while the second and third terms are power radiated in 2d or 4d modes, respectively, depending on the temperature. As the PMBH absorbs protons, the temperature increases and reaches equilibrium between the first and third terms on the r.h.s. of (4.23), giving For the Sun, T eq BH ≈ 2 × 10 −5 GeV ≈ 2 × 10 8 K, which is below m e , so the 2d radiation is suppressed. This gives the total power radiated via BNV by surviving PMBHs as (4.25) The above power is dramatically smaller than the observed solar luminosity L = 3.8 × 10 33 erg s −1 . The corresponding neutrino flux is also much below the solar neutrino flux [62]. Therefore, proton decay does not constrain the PMBH abundance.

Earth heat and neutrinos
As discussed in [63,64], if a significant fraction of DM is captured as it passes through the Earth and then annihilates efficiently to SM particles other than neutrinos inside the Earth, the heat so generated would surpass measurements of the internal heat of the Earth. This

JHEP10(2020)210
can be used to set a stringent constraint on DM candidates with scattering cross sections that are large enough to be captured by the Earth. As discussed in appendix B.1, the Earth can efficiently capture PMBHs for Q 1900 or M * • 1.2 × 10 23 GeV. Similar to (4.1), the capture rate is estimated to be where R ⊕ = 6.4 × 10 8 cm. The magnetic field in the Earth core is around 25 gauss [65], while the density is around 10 g/cm 3 [66]. Using (4.13), the critical number to overcome the separation of PMBHs due to the Earth's magnetic field is The corresponding time to reach this critical number is M * • 1 × 10 37 GeV. However, unlike for particle DM, the rate of mergers must be considered for heavy PMBHs. As a rough estimate, to set a bound we require the annihilation rate (which equals the capture rate in equilibrium) to be faster than the diffusion rate of heat from the core to the surface. The thermal diffusion timescale of the inner core alone is estimated as τ κ = (1.4±0.7)×10 9 yr = (4.4±2.2)×10 16 s [68], which is comparable to the age of the Earth. Demanding C cap τ −1 κ requires M * • 7 × 10 41 f * • , comparable to the bound from t drift < t ⊕ when f * • = 1. Thus, we do not expect a dramatic reduction in sensitivity even when the PMBH merger/annihilation events are very rare. In fact, this long diffusion timescale enables us to set bounds for much larger masses, rather JHEP10(2020)210 than being limited by how long we have been taking detailed measurements of Earth's heat flux.
In addition to Earth heating, the neutrino flux from PMBH mergers can be used to set a limit. We recast the results in the IceCube search for particulate DM χ [69]. Specifically, they set a mass-dependent bound on the rate of χ particle annihilations Γ A,χ going to both hard and soft channels. We expect post-merger PMBHs with a temperature T BH given by (2.14) will emit neutrinos with a similar energy spectrum as particulate DM whose mass m χ ∼ T BH in both the hard (χχ → W + W − ) and soft (χχ → bb) channels. The only difference is that the neutrino multiplicity is enhanced by a factor ∼ η ν M * • /T BH . As before, η ν is a neutrino multiplicity parameter, with η ν ∼ 1 for the hard component and At smaller masses, the post-merger temperature is higher. Thus, one could use IceCube to search for higher-energy neutrinos above the maximum mass of 10 TeV used in the DM search. However, the Earth is opaque to neutrinos with energies higher than 40 TeV [70]. Thus, such a search would be ineffective.
Alternatively, one could try to estimate the soft component of neutrinos coming from these lower-mass, higher-temperature PMBH mergers. Using a similar approximation as the solar case, we may expect a neutrino multiplicity enhancement of η ν ∼ T BH /E ν , with the neutrino energy E ν 40 TeV T BH . Then, the average rate of neutrino production at energy E ν is independent of mass in these approximations. This quantity should be compared against the IceCube bound Γ A,χ < 1.47 × 10 10 s −1 for m χ = 10 TeV in the soft channel [69]. Thus, there is an approximate bound f * • 10 −11 for M * • (2 × 10 32 GeV)f * • [from (4.17) and using the Earth capture rate in (5.1)]. The exact bound on f * • depends sensitively on the details of the soft neutrino multiplicity, which requires detailed numerical study of the Hawking radiation and its interactions and showering within the Earth, beyond the scope of this work. However, this assumed an instantaneous τ crit , neglecting Earth's magnetic field. Taking the value in (5.3) into account, the Earth neutrino bound is reduced to

JHEP10(2020)210
The low-mass cutoff is from the minimum stopping mass in the Earth, and the high-mass behavior is from requiring at least one annihilation during the observation period of the search. Because this is much weaker than the maximum bound with instantaneous τ crit , we expect it to be fairly robust against the neutrino multiplicity approximations we employed.
Earth heating and neutrino production can also occur from the BNV process. The temperature of PMBHs due to proton absorption can be calculated using (4.24). With v p th = (3 T core /m p ) 1/2 , ρ p = 12.2 g/cm 3 and T core ≈ 5700 K, T eq BH = 5.4 keV = 6.3 × 10 7 K. We can evaluate the power radiated by the proton decay as L BNV = N crit * • P 4 (T = 5.4 keV) = 1.2 × 10 6 W , (5.8) which is less than the internal heat of Earth and thus does not give any constraints. There are no direct experimental measurements for geoneutrinos at this low energy.

PMBH capture
PMBHs can get captured by other large bodies outside the solar system, such as neutron stars and white dwarfs. The calculation proceeds similarly to solar capture, though in the case of neutron stars, we include a relativistic correction factor, where R = R 10 × 10 km is the radius of the neutron star, v esc ≈ 0.47 for a neutron star and v = 10 −3 . In appendix B, we show that a MeBH will be stopped if it enters a neutron star or white dwarf, so = 1. The number of captured PMBHs per neutron star is then where τ NS = τ 10 × 10 10 yr is the age of the neutron star. The number of captured PMBHs per white dwarf (with R ≈ 7000 km and v esc ≈ 0.02) is Note that in the above analysis, we neglected the strong magnetic field of the neutron star. The B field of old neutron star is less than 10 8 gauss, which induces a force of order F EM Q h B. The gravitational force close to the surface of the neutron star is of order Neglecting relativistic corrections, which are O(1), for simplicity, the ratio of the forces is for a surface magnetic field B = B 8 × 10 8 gauss, the approximate value for an old neutron star. Old neutron stars are more prevalent and will have captured many more PMBHs than young neutron stars, so they are the most relevant for setting bounds. The magnetic force is much smaller than the gravitational force and can therefore be neglected to first approximation for the calculation of the encounter rate. JHEP10(2020)210

PMBH distribution
Like the Sun, magnetic fields inside neutron stars can separate PMBHs and prevent them from merging. The analysis of this possibility is complicated in this scenario by the unknown exotic phase of matter in the core of the neutron star. We begin by analyzing that case. At a depth of order few km below the crust, a neutron star is expected to be a proton superconductor [71,72] (see ref. [73] for a recent review), in which protons form Cooper pairs. This region forms the outer core. Magnetic fields in this region are confined to flux tubes at early times, which diffuse outward until the core is free of magnetic field. The diffusion time is somewhat uncertain, but can be less than the typical age of an old neutron star [74].
The nature of the inner core is not definitively known. For example, the environment could be a pion superconductor in which the pions acquire an isospin-breaking expectation value [75]. It could also be a color superconductor, which would break SU(3) c × U(1) EM → SU(2) × U(1), with the unbroken U(1) [76].
We are most concerned, as we will see, with the phase at a distance of around a kilometer from the center, very deep in the neutron star. There are essentially two qualitatively different scenarios. If the neutron star is a proton or pion superconductor at a distance of around a kilometer from the center, then the magnetic field lines of the PMBHs are confined to quantized flux tubes which create a large outward force that can balance the gravitational pull toward the core of the neutron star. Otherwise, if the magnetic field lines are not confined to flux tubes, there is a weak outward magnetic force (similar to the case of the Sun in figure 1), and the PMBH and anti-PMBHs collapse unimpeded toward the core.
As a PMBH enters a superconductor, its magnetic field is confined to flux tubes. In the case of a proton superconductor, the flux of each tube is Φ = π e , (6.5) which is half of the fundamental Dirac charge. The factor of half is due to the fact that it is charge-two Cooper pairs of protons that break the electromagnetic gauge group. For a pion superconductor, where EM charge is broken only by one unit, the flux per tube is twice that. There must then be 2 Q (Q) flux tubes per PMBH for a proton (pion) superconductor. For simplicity, we focus on the proton superconductor case, though the pion superconductor case is not qualitatively different. The typical size of these flux tubes is given by the London penetration depth λ = m p e 2 n p 1/2 ∼ 10 −12 cm , (6.6) for n p m p ≈ 4 × 10 14 g/cm 3 . The typical magnetic field inside the flux tube is then given by much larger than the surface magnetic field. A flux tube has an enormous tension force that would like to minimize the length of the flux tube. This force is given by [74] F T ∼ B 2 Φ π λ 2 ln (λ/ξ) ∼ 10 4 N , (6.8) where ξ ∼ few × 10 −13 cm is the Bardeen-Cooper-Schrieffer (BCS) correlation length for each flux tube. For GUT monopoles, the tension force is sufficient to eject monopoles inside the superconducting core when it forms or to prevent monopoles from entering except along flux tubes [74]. For PMBHs, however, the charge-to-mass ratio is lower and the gravitational force allows the PMBHs to penetrate into the superconducting region. The flux tubes are approximately radially outward going from the entry point of the PMBH. This configuration is qualitatively illustrated in figure 2. For a sufficient tension force, this allows for a stable shell of hanging PMBHs, also illustrated in figure 2. The radius of the shell is given by balancing the tension force with the neutron star gravitational force, neglecting for the moment self-gravitation contributions of the PMBH population,

JHEP10(2020)210
with ρ c ≈ m p n p ≈ 4 × 10 14 g/cm 3 . The energy of the flux tubes can be lowered if outward-going flux tubes from PMBHs merge with inward going flux tubes from anti-PMBHs, leading to a tension force that pulls JHEP10(2020)210 the PMBH and anti-PMBH together to annihilate. The dominant effect that initially prevents this from happening is that the drift of the flux tubes is very slow. 1 The drift velocity can be obtained by balancing the flux tube tension against the force due to impinging electrons and other elements of the degenerate fluid in the neutron star (see refs. [74,77] for estimation of the drift velocity).
The flux tubes from a given PMBH are grouped into "bundles" stretching roughly vertically from the PMBH (see figure 2 for an illustration). The spread of the flux tubes within a bundle is set by balancing the tension that restores the flux tubes toward the vertical with the pressure due to collisions of electrons in the plasma with the flux tube. The angular scale for this spread can be obtained by comparing the energy required to displace the flux tube from vertical by an angle θ with the temperature. In the limit of θ 1, the ratio is F T R c θ 2 /(2 T ) ∼ 1 for T 10 6 K and R c10 = R c /(10 km) the superconducting core radius in units of 10 km. Additionally, the flux tubes will not pack more tightly than ∼ λ [78], so θ T √ Qλ/R c . Putting these together, When the separation angular distance from one bundle to another bundle is larger than θ T , the tension force is more important and one can treat the whole bundle as one composite object; otherwise, the thermal pressure is more important. This condition can be translated into We can estimate the total encounter rate of two bundles using Γ enc = N * • n 2 σ v * • , with the 2d number density as (the second term from when the tubes are tightly packed is never relevant for P). The merger or annihilation rate is (6.12) The equilibrium time for the annihilation and capture rates is which is shorter than the age of neutron stars for M * 10 . The second term inside the minimum is never important when comparing τ eq to the age of 1 There is also a potential barrier given by the added tension of a configuration where the two flux tubes are merged rather than radially outward, but this potential will be negligible in the regime of small separation we consider.

JHEP10(2020)210
neutron stars τ NS ∼ 10 10 yr, so we can henceforth neglect it. The number of PMBHs in a neutron star is Note that this number of N * • satisfies the condition in (6.11).
One might also worry that the increased number of PMBHs will cause the stable shell of "hanging" PMBHs to shrink and destabilize due to the self-gravitation of the PMBHs. This happens when the flux tube tension is not sufficient to balance the combined radial gravitational force of the PMBHs and the neutron star. This occurs for a critical number , which is larger than the maximal captured number for the whole neutron star age. Therefore, this effect is also negligible.
If the inner core has unconfined magnetic fields, as is the case for a color superconducting phase, then the magnetic fields are drastically weaker, of order the surface magnetic field. This is analogous to monopole separation in the uniform field of the Sun, see figure 1. The magnetic fields are then insufficient to sustain a stable separation of PMBHs and anti-PMBHs, and annihilation proceeds uninhibited until it equilibrates with the capture rate once a critical number of captured PMBHs is reached. The critical number is given by (4.13), The annihilation rate is given by (4.7) and (6.1), For a white dwarf, the situation is simpler as they are not expected to be superconducting. Similar to the case for the Sun or a color superconducting neutron star core, annihilation quickly equilibrates with capture at N crit * • , with taking the white dwarf magnetic field to be B 6 × 10 6 gauss and the density to be ρ c ∼ 10 6 g/cm 3 . The annihilation rate is then

Constraints
In the scenarios that we consider, there are two potential sources of heat generation in the star: annihilation and BNV. The luminosity generated by annihilation which has equilibrated with capture is given simply by BNV has a cross section that is at least given by the EW radius R EW as argued above. Thus, the luminosity generated by nucleon decay is given by 20)

JHEP10(2020)210
where v is the mean speed of nucleons in the core of the star. For neutron stars, this is given by v ≈ 3 v F /4, where v F ∼ 0.2 is the Fermi velocity. For white dwarfs, the protons at the core are not expected to be degenerate, so the average velocity is set by the temperature, v = 2 2 T /(π m p ) ≈ 5 × 10 −4 , with T /m p ∼ 10 −7 . We now consider three scenarios: a neutron star with magnetic fields confined to flux tubes in the core, a neutron star without flux tubes, and a white dwarf. In the first case, the dominant luminosity comes from BNV. Using the abundance of captured PMBHs in (6.14), the luminosity due to BNV is The luminosity can be compared either to that of individual stars, the total from all stars, or to diffuse emissions. There are advantages and disadvantages to each approach as outlined in ref. [79]. All of these bounds are on the order of L γ < 10 32 erg s −1 per neutron star [79][80][81]. Assuming a fraction r = L tot /L γ of the total luminosity to photon luminosity, a bound is obtained after expanding P: (BNV) (6.22) Here, the limit in the last line comes from requiring at least one captured PMBH for all ∼ 10 8 neutron stars in our galaxy [see (6.2)], which is valid for the diffuse emission limits. For proton superconductors, r ∼ 1 [79,82,83]. If the core is instead a pion condensate, r could be around 10 3 or 10 4 [79,82,83], leading to a reduced bound. If the core is not superconducing or if it is a color superconductor, then annihilation equilibrates with capture at a lower captured PMBH abundance. The luminosity from annihilation is still subdominant compared to the luminosity from BNV, which is [using (6.15)] L NS Although BNV dominates annihilation luminosity, it is still unlikely to contribute a relevant bound, even if all of the power generated is emitted as photons. 2 For white dwarfs, the constraint determination proceeds similarly. The luminosity due to annihilation is given by L WD ann = 1.6 × 10 25 erg s −1 f * • , while the luminosity due to JHEP10(2020)210 BNV is given by L WD BNV = 3.3 × 10 23 erg s −1 f * • . The power from the dimmest nearby white dwarfs is around 2 × 10 29 erg s −1 [85], which is a few orders of magnitude larger than that generated by PMBHs such that no relevant constraint is set.

Discussion and conclusions
The constraints on PMBHs are summarized in figure 7. The Parker bound due to the coherent magnetic fields in M31/Andromeda comes from eq. (3.6). The bounds from solar neutrinos come from eq. (4.21) for IceCube (IC) and (4.22) for Super-K (SK). The Earth neutrino bound from IceCube is in (5.7), while the Earth heating bound is (5.5). The merger bounds are limited on the left by requiring the stopping length be smaller than the size of the bodies, and on the right by requiring at least one merger during the relevant timescale for the bound. We have approximated the left boundaries as sudden cutoffs at Q stop,min in table 1, but a more thorough investigation could account for the nonzero probability to capture smaller-charge PMBHs. The bounds from BNV processes in neutron stars are shown assuming either r = 1 (for a proton superconductor) or r = 10 4 (for a pion superconductor), setting R 10 = 1 and R c10 = 0.7 in (6.22). The left-most portion of the bounds becomes weaker at smaller masses because PMBHs annihilate. In the center, where the bound is a horizontal line, PMBHs do not annihilate efficiently. To the right, as the mass increases some neutron stars do not capture any PMBHs, decreasing the BNV luminosity. If neutron star cores do not confine magnetic fields to flux tubes, such as if they have color superconducting cores, neutron stars provide no bound. Thus, this bound is quite tentative, depending on the details of neutron star cores. Mass and charge are related πM pl /e ≈ 5.1 M pl , with Q min from above eq. (2.9), which assumes the existence of a GUT monopole with mass M M = 10 17 GeV, and Q max from (2.4), above which there is no EWS corona.
At higher masses, PMBHs can be constrained by gravitational lensing of stars. Subaru/HSC set limits using stars in Andromeda [86] for BH masses starting from M * • 10 23 g [87]. Other microlensing studies from Kepler [88], MACHO [89], EROS [90], and OGLE [91] cover the remainder of the possible PMBH masses up to Q max . These bounds will not be sensitive to the presence of the EWS corona [92,93], especially because the corona is inside the Einstein radius. Eventually, microlensing of X-ray pulsars [94] and femtolensing [95] or lensing parallax [96] of gamma ray bursts could set bounds down to M * • 10 17 g.
Taking the more conservative limit gives f * • < 1 × 10 3 M 26 , setting a constraint on masses M * • < 1 × 10 23 GeV, a bit stronger than the MACRO constraint. White dwarf destruction may also set a constraint, potentially adding a secondary constraint to the Parker bound between the annihilation to neutrino constraints and the lensing constraints. However, a detailed hydrodynamical simulation taking into account the effects of the EWS corona would be required [100], beyond the scope of this work.
We have not discussed how PMBHs form, instead focusing on the phenomenology if they are present. One example of a formation mechanism is that they may start as ordinary primordial BHs, then absorb on average N randomly charged monopoles leaving them with a typical charge ∼ √ N , then Hawking radiate until they approach near extremality [20,21]. In such a scenario, bounds from the effects of their evaporation on Big Bang nucleosynthesis, the cosmic microwave background, and gamma rays could set additional constraints [101]. Note, these constraints may be modified if the BHs obtain a large enough charge to form a corona before Hawking radiating to near-extremal. If, on the other hand, PMBHs are born extremal or near extremal, these constraints are relaxed.

JHEP10(2020)210
If PMBHs are indeed primordial, then they can form binaries in the early Universe that merge today, giving high energy neutrinos and gamma rays throughout the sky. An estimate of this signal is given in [21], but more detailed numerical work is needed, particularly on binary disruption (see, e.g., [102,103]). Alternatively, binaries may form in galactic halos, but the merger rate from this population of binaries is smaller than the merger rate of primordially-formed binaries [104].
To conclude, PMBHs are interesting long-lived objects that require no new physics beyond the SM and general relativity. We have outlined many search strategies and shown the PMBH abundance is already relatively constrained compared to dark matter. Nevertheless, they remain an interesting target for future searches. In particular, PMBH mergers or baryon number violating processes offer the possibility to detect Hawking radiation. Furthermore, this Hawking radiation would be emitted as 2d modes from the electroweak-symmetric corona, whose spectrum may be differentiated from ordinary 4d Hawking radiation. If a signal is observed, this distinction could provide strong evidence for the PMBH interpretation.

Acknowledgments
The work of YB, MK, and NO is supported by the U.S. Department of Energy under the contract DE-SC-0017647. The work of JB is supported by PITT PACC.

A Dirac equations and 2d modes
In this section, we follow ref. [105] to discuss the solutions to the Dirac equation in a background BH geometry and magnetic field. Rather than only considering the massless case in [105], we also keep the fermion mass in our discussion. For a general metric in spherical coordinates, and A φ = Q 2 cos θ. For extremal BHs, the above metric is related to the one in the ordinary spherical coordinate in (2.1) by Choosing the gamma matrices in the spinor representation [106], the four-dimensional spinors can be written as a tensor product of two dimensional spinors The Dirac operator in the bi-spinor representation is

JHEP10(2020)210
Using (A.12) for q Y q,m−1/2 and q Y q,m+1/2 , one has the following relation As a consistency check, we can compare the result in (A.5) with η 1,2 given by (A.10) and (A.11) with the results of ref. [109], which discusses a similar problem neglecting the curvature of spacetime due to the monopole. The solution using the gamma matrices in the spinor basis via (A.3) is with d 1,2 as normalization factor. The solution in ref. [109] is calculated based on the Dirac basis for the gamma matrix with where we have multiplied an additional factor of i for the Dirac matrices in ref. [109] to match the metric convention in ref. [105]. For the "type (3)" solution in ref. [109], one has with f (r) = (2/π) 1/2 sin (Er + δ 3 )/(Er) and g(r) = −i (2/π) 1/2 cos (Er + δ 3 )/(Er). Here, j = q − 1/2 with q > 0 and m = −j, · · · , j.
The second unitary matrix is a rotation of the tangent space relative to the coordinate space. In [105], the tangent space is chosen in the convention such that the x-axis aligns with the coordinate radial direction. This rotation acts on spinors as

JHEP10(2020)210
where c = cos θ 2 , s = sin θ 2 , and e ± = e ± i φ/2 . The two solutions in (A. 16) and (A.18) are related by Using the relation in (A.15), the above equation becomes One can explicitly see that the θ-and φ-dependences on the both sides of the equation come from the same function η 1 and the r-dependence is also identical. After demonstrating the consistency of solutions in the literature, one can ask which complimentary states must be added to the 2d modes to form a complete basis. One can use the other two types of solutions in (A.18) and perform a transformation via (A.21) to obtain the correponding solutions in the basis choice of (A. 16). Doing so, one could derive the equation of motion for the radial functions by including the curved spacetime and obtain the solutions for a Dirac fermion in the MeBH background.
To develop a qualitative understanding of the 2d neutrino modes, note that the neutrino has a non-zero hypercharge inside the EWS corona and zero electric charge outside. One could model this situation by choosing a step-function-like A φ = Q 2 cos θ Θ(R EW − r) (assuming the boundary condition is spherical, which is only approximately true for a large Q). One can explicitely check that a linear combination of solutions for Q > 0 and Q < 0 2d modes can have a standing wave solution inside R EW with vanishing wave-function outside. This type of "particle-hole state" has been discussed for the bosonization [110], and we leave the more detailed calculation to future exploration.
Finally, we also comment on the situation for the massive fermion case. For a non-zero mass m χ = 0, eq. (A.7) cannot be solved analytically. Defining which can be combined and become Using the relation between x and r in (A.2), we convert the above equation into

JHEP10(2020)210
with f (r) = (1 − R e /r) 2 and f (r) ≡ df (r)/dr. Note that when r → ∞, the equation provides a simple plane wave solution, ψ 1 (r) = e ikr , with the normal dispersion relation Close to the event horizon or r → R e , the mass term is less important. So, effectively, one could treat the system as a particle with a locationdependent mass, or equivalently with some "attractive potential". For E < m, bound state solutions are anticipated. For instance, when a nucleon scatters off the EWS corona region, its mass is reduced and the bound-state-mediated scattering can increase the scattering cross section to the geometric one [16].

B Stopping of a finite-sized PMBH by a plasma
We modify the treatment of monopole stopping in a plasma in [111] (see also [51]) to account for a finite-sized monopole. Since the properties of the plasma inside the EW radius R EW are uncertain, we conservatively assume that modes at impact parameter less than R EW do not contribute to stopping. To do so, we approximate the PMBH as a charged spherical shell R. Any interactions with impact parameter smaller than this are neglected. It is simple to verify that the stopping power by particles with impact parameter smaller than R that are absorbed by the BH is insufficient to stop the PMBH.
Note that the treatment in [111] neglects non-linearities in the plasma response, however non-linearities are important for the Sun. Thus, this calculation does not give an accurate result for the stopping power in the Sun. Still, it gives the correct proportionality of the stopping power on the parameters in the problem. As we will see, the stopping power monotonically increases with Q even when finite-size effects are included, and the minimum Q for which PMBHs are stopped in astrophysical bodies like the Sun is well within the point-particle regime. This leads us to conclude that finite-size effects do not affect whether or not PMBHs are stopped.
The charge and current densities for a PMBH with radius R, velocity V, and magnetic charge h Q can be taken as, ρ(r, t) = h Q 4πR 2 δ(|r − Vt| − R) , J(r, t) = Vρ(r, t) . (B.1) We take the Fourier transform of the magnetic charge density to obtain ρ(k, ω) = 2 π h Q δ(k · V − ω) j 0 (k R) , where j 0 (x) = sin x/x is a spherical Bessel function, which goes to 1 for x 1 and falls off as 1/x, up to oscillation, at large x.
The magnetic field in Fourier space is where all quantities on the right are assumed Fourier transformed. The transverse plasma dispersion T is given in [111]. In the limit z = ω/(k v th ) = k · V/(k v th ) 1 with v th the thermal velocity of the plasma particles,

JHEP10(2020)210
with the squared plasma frequency ω 2 p = 4πn e e 2 /m e for electrons with number density n e and mass m e . Then, the PMBH power loss is dW dt = d 3 r J(r, t) · B(r, t) (B.5) where the last line is in the limit V 1. The characteristic attenuation length is The last equality gives the value for the Sun with V = 10 −3 . The integral is truncated at finite k max to avoid a logarithmic divergence when R = 0. This corresponds to the assumption of a linear plasma response in T , which neglects short-range interactions. It is taken as related to the distance where the electrostatic and thermal energies are equal: k max ∼ 4π n L 2 D = T 2 /e 2 ≈ (2 × 10 −9 cm) −1 , with n the number density of particles in the plasma and L D the Debye length. However, the result is only logarithmically dependent on this. Then, the R = 0 result is To generalize to finite radius, we must now account for the j 0 (kR) term. We approximate the integral in (B.7) by taking j 0 (kR) = 1 for k ∈ [0, 1/R) and j 0 (kR) ∼ 1/(kR) for k ∈ [1/R, ∞) (ignoring the sine dependence). For R l, the result is the same as (B.9) but with k max replaced by k max = min(k max , 1/R), while for R l and k max 1/R The radius is the EWS corona radius, (B.11) Thus, the stopping length L S ∼ 1 2 M * • V 2 (dW/dx) −1 ∝ QV 3 (dW/dt) −1 is monotonic with Note L S is monotonically decreasing as Q increases regardless of how k max , R, and are related. Finally, note that in the point-like MeBH case, the stopping length is smaller than the solar radius for Q 30 (see below). For this charge, the point-like approximation is valid. Since the stopping length monotonically decreases with Q even after accounting for finite size effects, all extremal magnetic BHs above this charge are stopped.

B.1 Stopping in other materials
To a rough approximation, the stopping power in other materials like conductors, insulators, or degenerate gases is very similar the stopping power of plasmas considered in the previous section, but with the thermal velocity of electrons v th replaced by the Fermi velocity v F [112]. In other words, from the point-like approximation in (B.9), This can be used for objects like neutron stars and the Earth. The minimum charge for a PMBH to be captured while traveling through a body is then estimated by requiring where v ≈ 10 −3 is the velocity of the PMBH far from the body, v esc is the body's escape velocity, and R is the radius of the body. If this condition is satisfied, then the PMBH becomes gravitationally bound to the body and will, either during the first crossing or over the course of further crossings, stop inside the body. The resulting minimal charge is Q stop,min ∼ c W v 2 v F m e M pl 8 π 3/2 e n e R (v 2 + v 2 esc ) 1/2 , (B.14) which assumes that once the PMBH is gravitationally bounded, it will be quickly captured, usually during its first pass through the object (this is true for the Sun, the Earth and neutron stars, which provide non-trivial constraints). We provide relevant quantities in table 1. For stars, the proton-to-nucleon ratio (equivalent to the electron-to-nucleon ratio) is Z/A ∼ 1, while for the Earth Z/A ∼ 1/2. For the Sun, we use the more precise calculation of the stopping power in [53] (though with a factor of two larger stopping power [55]), which gives Q stop,min ∼ 30, compared to Q stop,min ∼ 390 using (B.14) and the quantities in table 1.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.