1 Introduction

Baryon Destruction by Asymmetric Dark Matter

Hooman Davoudiasl, David E. Morrissey, Kris Sigurdson, Sean Tulin

Department of Physics, Brookhaven National Laboratory,

Upton, NY 11973, USA

Theory Group, TRIUMF,

4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada

Department of Physics and Astronomy, University of British Columbia,

email: hooman@bnl.gov, dmorri@triumf.ca, krs@physics.ubc.ca, tulin@triumf.ca

August 4, 2019

We investigate new and unusual signals that arise in theories where dark matter is asymmetric and carries a net antibaryon number, as may occur when the dark matter abundance is linked to the baryon abundance. Antibaryonic dark matter can cause induced nucleon decay by annihilating visible baryons through inelastic scattering. These processes lead to an effective nucleon lifetime of years in terrestrial nucleon decay experiments, if baryon number transfer between visible and dark sectors arises through new physics at the weak scale. The possibility of induced nucleon decay motivates a novel approach for direct detection of cosmic dark matter in nucleon decay experiments. Monojet searches (and related signatures) at hadron colliders also provide a complementary probe of weak-scale dark-matter–induced baryon number violation. Finally, we discuss the effects of baryon-destroying dark matter on stellar systems and show that it can be consistent with existing observations.

## 1 Introduction

Cosmological observations indicate that about 4.6% of the energy density of the Universe consists of baryonic matter, while 23% is dark matter (DM) [1]. Neither of these results can be explained with our current understanding of elementary particles, the standard model (SM). Cosmology therefore requires new fundamental physics, and it is important to find ways to detect such new physics experimentally.

In the majority of new-physics scenarios, the generation of baryons and DM occurs through unrelated mechanisms, offering no explanation for the similar magnitudes of their cosmological densities. The most thoroughly studied scenarios involve baryon production from CP-violating non-equilibrium processes during the electroweak phase transition, from decays of right-handed neutrinos, or from the coherent evolution of scalar fields [2], while the DM relic density is determined by thermal freeze out when a non-relativistic stable species falls out of equilibrium [3]. In this context, there is no reason to expect similar cosmological densities of baryons and DM.

This apparent coincidence may instead be a clue that both types of matter have a common origin. Several models of asymmetric dark matter (ADM) have been proposed along these lines where the DM density carries a net (approximately) conserved global charge shared by the SM [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], such as baryon number . These models generally fall into two classes depending on how the charge asymmetry is created:

1. An initial charge asymmetry, generated in either the visible or DM sector, is partitioned between the two sectors by chemical equilibration through a transfer operator [4, 9]. These charges are separately “frozen in” once the transfer operator goes out of equilibrium.

2. Non-equilibrium dynamics generate equal and opposite charge asymmetries in the visible and DM sectors, without any net overall charge asymmetry [10, 11, 12, 13, 14, 15]. In order to avoid washout, transfer operators must always be out of equilibrium once the asymmetries are created. We term this process hylogenesis (“hyle” = matter) [15].

Subsequently, in both cases, DM particles and antiparticles are assumed to annihilate efficiently, leaving only a remnant asymmetric component determined by the charge density.

In the present work, we investigate novel experimental signatures from hylogenesis scenarios where DM carries  [15]. Here, the lowest-dimensional, gauge-invariant transfer operator is given by

 Leff∼1Λ3uiRdjRdkRΨRΦ+h.c. (1)

where label generation. In this case, DM has two components, a fermion/scalar pair with total baryon number . Stability of both and requires . This operator is inactive cosmologically and does not wash out the baryon-DM asymmetry for provided the asymmetry is created at relatively low temperatures, below about a GeV. Our results may also be applicable to other ADM scenarios in which the same transfer operator appears with one or both of making up the DM.

In hylogenesis, the Universe is net -symmetric, and therefore the baryon asymmetry carried by DM is equal and opposite to that in visible baryons. Specifically, in the hylogensis scenario of Ref. [15], this has two important consequences:

• The number densities of satisfy . Therefore, cosmological observations imply . Together with the DM stability requirement, we have GeV.

• There exist many scenarios in which ADM is coupled to the SM via the “neutron portal” operator  [4], generally falling into the first class of “chemical equilibration” scenarios. In this case, DM is very often baryonic (), while for hylogenesis scenarios the DM must be antibaryonic ().111 In supersymmetric models, light superpartners can play an important role in chemical equilibration [16], potentially affecting the sign of in equilibration ADM scenarios. Also, for other applications of the neutron portal operator to baryogenesis, see Ref. [17].

In Ref. [15], we presented a specific realization for hylogenesis where the operator in Eq. (1) arises by integrating out heavy Dirac fermion mediatiors . Out-of-equilibrium decays of during reheating generate equal-and-opposite dark and visible baryon asymmetries. Furthermore, in this scenario are charged under an additional hidden gauge symmetry, that couples to the SM via kinetic mixing with hypercharge, to facilitate annihilation of the symmetric DM densities. More details about this realization are given in Appendix A.

Hylogenesis models with the operator of Eq. (1) have an interesting and unique signature: antibaryonic DM particles can annihilate visible baryonic matter, termed induced nucleon decay (IND). IND is a novel and unusual DM signal. These events are and , where is a nucleon and is a meson. Since the DM states are invisible, this mimics nucleon decay, with an effective lifetime dependent on the local DM density. This signal offers the new and exciting possibility of searching for DM in nucleon decay searches in deep underground detectors such as SuperKamiokande [18].

In Section 2, we compute the rates and kinematics of IND, and discuss the implications of IND for existing nucleon decay searches. Our main conclusions are: (i) due to different kinematics of IND, compared to standard nucleon decay, existing bounds do not apply over most of the region of parameter space of our DM model, and (ii) the effective nucleon lifetime can be around years, if the new physics scale in Eq. (1) is the weak scale.

Hadron colliders can probe (anti)baryonic DM scenarios through direct production of the weak-scale mediators transfering between visible and dark sectors, discussed in Section 3. The operator of Eq. (1) can give rise to observable signatures in the form of monojets (or jets and missing energy). Within our hylogenesis model, if IND were detected in nucleon decay searches, monojet signals are inescapable and should be observed at the Large Hadron Collider (LHC).

IND processes can potentially be relevant in stars. Since the IND rate scales with the DM density, capture and accretion of DM in stars is important, possibly leading to modification of stellar evolution due to baryon destruction and energy injection. In Section 4, we consider these effects in neutron stars, white dwarfs, and main-sequence stars. Our conclusions are summarized in Section 5.

## 2 Dark Matter Detection in Nucleon Decay Searches

Dark matter particles can annihilate nucleons , producing an energetic meson through inelastic scattering

 ΦN⟶¯ΨM,ΨN⟶Φ†M, (2)

shown in Figure 1. We restrict our attention to single meson final states, although multi-meson events are allowed and may have comparable rates. In general, both down-scattering and up-scattering can occur, defined as whether the heavier or lighter DM particle, respectively, is in the initial state; however, the latter is kinematically forbidden if . Since neither the initial DM nor final anti-DM particles are observed directly, these processes mimic standard nucleon decay events [19], with an undetected final state neutrino (or antineutrino ).

The observable meson energy for each case in Eq. (2) is, respectively,

 EM=(mN+mΦ)2+m2M−m2Ψ2(mN+mΦ),EM=(mN+mΨ)2+m2M−m2Φ2(mN+mΨ), (3)

neglecting the initial kinetic energy of the DM particles ().

Although hylogenesis (as a baryogenesis mechanism) works for any quark flavor, the most interesting signatures arise if the IND operators involve quarks only. Restricting our attention to the lightest mesons (, , ), the final state meson can have much more kinetic energy in IND than for standard nucleon decay (SND), summarized in Table. 1. For fixed masses and , the meson is either monochromatic or bichromatic (up to Fermi motion), depending on whether up-scattering is forbidden or allowed. The range of momenta corresponds to the allowed range GeV, provided that are stable and account for the observed .

### 2.1 Nucleon decay searches

Existing searches have been optimized for meson momenta MeV, while for IND mesons are typically much more energetic, with momenta GeV. Here, we briefly summarize existing SND search strategies and how they might be adapted for IND searches.

: The Super-Kamiokande experiment, a water Čerenkov detector, provides the strongest limits on these channels. For , they have three searches: (i) , giving three Čerenkov rings, (ii) with a prompt (from ), and (iii) a mono-energetic from , with no prompt . All three searches assume, as is the case for in SND, that the is emitted below Čerenkov threshold () and comes to rest before decaying. For IND, we estimate that an fraction of ’s do come to rest before decaying. However, except for up-scattering events close to kinematic threshold, the from IND has , adding an extra ring to the event topology. Furthermore, this additional radiation may make finding the prompt in search method (ii) more difficult. For , they have two searches: (iv) , giving four -like rings, and (v) , giving two -like rings. Both searches assume MeV, thereby excluding IND events (again, except for up-scattering near kinematic threshold). One difficulty in search (iv) is identifying all four -like rings. For IND, this may be more difficult as the rings would be more overlapping due to relativistic beaming. On the other hand, search (v) is promising for IND. In SND this mode suffers from a small efficiency that the are both above Čerenkov threshold. With greater energetics in IND, the efficiency may be much larger.

: The best limit is provided by the Soudan 2 experiment, an iron tracking calorimeter. Nucleon decay event candidates were required to have a single track, with ionization consistent with mass or , initial momentum MeV, and visible endpoint decays ). Their simulations showed that a originating from within an iron nucleus loses on average half its initial momentum. At higher , IND events may be more visible due to reduced background from atmospheric neutrinos. However, it is unknown to us what is the average momentum deposition in iron of the at much higher energy, and whether this can lead to fragmentation of the parent nucleus into exotic nuclear states.

: The best limits on these modes come from the IMB-3 experiment, a water Čerenkov detector.222We note that the IMB-3 experiment found an excess in events with total energy MeV (20 events vs. 6.1 expected background) [22]. Nearly all these events had between Čerenkov rings, large missing momenta ( MeV), and large invariant masses ( MeV). In IND, a large missing momentum would be expected, while a large invariant mass could arise through heavy meson or multi-meson final states. The channel may be more difficult at higher energies: due to decreased separation angle of the two photons (from relativistic beaming), they can appear as a single electron-like track. The channel will have greater photon separation and may be more promising. We find that the IND rates into and final states are comparable and are sensitive to the same underlying IND operator (shown below).

### 2.2 Effective nucleon lifetime from IND

An effective IND lifetime can be defined as the inverse scattering rate per target nucleon, , with local DM number density and IND scattering cross section . Numerically, we have

 τ−1N≈(1032yrs)−1×(ρDM0.3GeV/cm3)((σv)IND10−39cm3/s). (4)

Next, we compute using chiral perturbation theory. We perform an expansion in powers of , where MeV is the pion decay constant, and truncate at leading order. Since for IND we expect , our calculations should be regarded as order-of-magnitude estimates at best. Our analysis closely follows SND rate computations in Ref. [23].

There are four effective interactions that are relevant for IND processes with single meson final states. These are given by , with operators (given in two-component spinor notation)

 O1 =ϵαβγΦ(uαRdβR)(dγRΨR) (5) O2 =1√6ϵαβγΦ[(dαRsβR)(uγRΨR)+(sαRuβR)(dγRΨR)−2(uαRdβR)(sγRΨR)] (6) O3 =1√2ϵαβγΦ[(dαRsβR)(uγRΨR)−(sαRuβR)(dγRΨR)] (7)

where are color indeces, and the coefficients have mass dimension .333A fourth operator is relevant only for multi-kaon final states. A fifth operator vanishes by a Fierz identity. The linear combinations have been chosen such that have strong isospin , respectively. Here, it is useful to write where

 c≡⎛⎜ ⎜ ⎜ ⎜ ⎜⎝c2√6+c3√2000c2√6−c3√200c1−√23c2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,Oij≡12ϵαβγϵjkℓ(qαRkqβRℓ)(qγiRΨR)Φ, (8)

with . Under chiral symmetry transformations, the right-handed quark fields transform in the representation, (where ), while the IND operator transforms in the representation, . If we treat as a spurion in the representation, is invariant under chiral transformations. (The DM fields are chiral singlets.)

The IND interactions of baryon and (pseudo-Goldstone) meson fields are determined by the chiral transformation properties of the spurion . Following the conventions of Ref. [23], the only invariant operator is

 LIND=βTr[cξ†(BRΨR)Φξ], (9)

where . The meson and baryon fields are

 M=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝η√6+π0√2π+K+π−η√6−π0√2K0K−¯K0−√23η⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,B=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Λ0√6+Σ0√2Σ+pΣ−Λ0√6−Σ0√2nΞ−Ξ0−√23Λ0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (10)

Eq. (9) is invariant since the quantity is in the representation [23]. The unknown overall coefficient has been computed using lattice methods [24].

The Feynman diagrams for IND are given in Fig. 1. It is straight-forward to derive the Feynman rules for the interactions of baryons, mesons, and DM by expanding the matrix expressions in Eq. (9) and working to linear order in . From these, we compute the matrix elements and cross sections for IND processes. We assume that only one coefficient is non-zero at a time. modes depend only on , while modes are governed by .

Our numerical results are shown in Figs. 2 and 3, for the five decay modes we consider. We plot the velocity-weighted cross sections for each mode, as a function of , for the allowed range (as required by DM stability). In Fig. 2, the solid lines show the total cross sections for the channels and (left) and (right). All three modes arise from the same IND operator , and we have fixed . The individual cross sections for () are shown by the dotted (dashed) curves. Although the individual rates vanish where kinematically forbidden, the total rate is always non-vanishing. For , both up- and down-scattering rates are non-zero, and IND is bichromatic. Otherwise, only down-scattering is allowed, and IND is monochromatic.

In Fig. 3, we show the strange IND channels (left) and (right). Here, there are two relevant operators . The solid lines denote the total IND cross sections induced by each independently, taking , for . The individual cross sections for () are shown by the dotted (dashed) curves. Again, the kaons can be monochromatic or bichromatic, depending on whether up-scattering is kinematically allowed.

Existing nucleon decay bounds do apply in select regions of parameter space for up-scattering close to threshold, where the meson momentum is reduced. We illustrate these regions in Fig. 3, shown in grey, for the case of Super-Kamiokande. For , this region corresponds to : in this case, the is below Čerenkov threshold and the event topology is identical to . For , this region corresponds to a kinematic window MeV, as in their search. We emphasize that Super-Kamiokande bounds constrain only the up-scattering IND rate, which can be suppressed compared to down-scattering in these parameter regions.

The total IND rate depends sensitively on the unknown mass scale . The total IND cross sections and nucleon lifetimes for all channels are comparable, scaling as

 (σv)IND≈10−39cm3/s×(ΛIND1TeV)−6,τN≈1032yr×(ΛIND1TeV)6(ρDM0.3GeV/cm3). (11)

As we show in Sec. 3, the collider bound on this scale is GeV. Therefore, can in principle be as low as years. It is likely that such a short lifetime would be excluded from nucleon decay searches, but no dedicated IND search has yet been performed. We also note that future nucleon decay experiments are envisioned to have markedly better reach for . For example, a water Čerenkov detector with kton year of exposure can reach  yr [25, 26]; a similar capability is expected to be achieved with a liquid Argon detector with kton year of exposure [27]. For kaon final states, liquid Argon technology is expected to provide improved efficiency due to better imaging capabilities [27].

Lastly, we note that the IND cross sections satisfy certain relations, as a consequence of strong isospin symmetry:

 (σv)p→π+IND=2(σv)n→π0IND,(σv)p→K+IND=(σv)n→K0IND. (12)

The latter relation holds only if modes are dominated by either or , as assumed in Fig. 3. If neither operator is negligible, then . In this case, both modes are complementary and can be used to disentangle the underlying IND operator structure.

## 3 Collider Signals from Hylogenesis

ADM scenarios rely on transfer operators to connect global charge between the visible and dark sectors, and these operators can be probed at high-energy colliders. In the specific hylogenesis model of Ref. [15], is mediated between the two sectors by heavy Dirac fermions (), through interactions of the form

 −L⊃∑a=1,2λijkaM2(uiRdjR)(X†a,LdkR)+ζa(Xa,LΨL+Xa,RΨR)Φ+h.c. (13)

where label generation, color indices are implicitly contracted antisymmetrically, and other fermion contractions are also possible. Integrating out generates operators of the form of Eq. (1).

The particles can be produced at high-energy hadron colliders through the operator of Eq. (13). With decays , this gives rise to events involving missing energy and one or more jets. In this section, we investigate the sensitivity of the Tevatron and the LHC to such events and we derive a corresponding bound on the heavy mass scale suppressing the neutron portal operator. Related studies in the context of WIMP and other dark matter candidates can be found in Refs. [28, 29, 30, 31, 32, 33].

To be concrete, we will focus on the lighter state , with the specific interaction

 −L ⊃ λM2(X†LsR)(uRdR)+ζXΨΦ+h.c., (14)

We expect other flavour structures and fermion contractions to give qualitatively similar results. The operator of Eq. (14) can give rise to processes of the form

 q(p1)q′(p2)→¯q′′(p3)¯Ψ(p4)Φ†(p5) (15)

through either a real or off-shell , where quarks. The corresponding summed and averaged squared matrix element can take two possible forms, depending on how the fermions are contracted. They are:

 |M|2=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩23∣∣λζM2∣∣2∣∣∣1q2−m2x+iΓxmx∣∣∣2(p1⋅p2)[2(p3⋅q)(p4⋅q)−(q2−m2X)(p3⋅p4)] ;     s-% like23∣∣λζM2∣∣2∣∣∣mxq2−m2x+iΓxmx∣∣∣2(p1⋅p3)[2(p2⋅q)(p4⋅q)−(q2−m2X)(p2⋅p4)] ;     t-like (16)

Here, is the momentum carried by the intermediate state, and is the width of the state which we assume decays mainly into . The -like form correponds to the case where both initial state fermions are contracted together in the underlying operator while the -like form corresponds to a contraction between initial and final state quarks.

At the Tevatron and the LHC, we find that the full cross section derived from the matrix elements of Eq. (16) is frequently dominated by the pole in the intermediate propagator. This corresponds to the production of an on-shell state whose mass is not much smaller than the higher-dimensional operator scale , and corresponds to a large momentum transfer.444 This is a necessary condition for hylogenesis to create a sufficiently large asymmetry. In this case the higher-dimensional operator structures we are using become unreliable and the full dynamics of the unknown ultraviolet completion becomes relevant. Without specifying the underlying theory, we can still parametrize the generic behaviour in a reasonable way as follows. For -like contractions, whose structure would arise most naturally from a boson in the -channel, we make the replacement

 λM2→λ^s−M2+i√^sΓ, (17)

where is the parton-level Mandelstam variable and is the decay width of the mediator. In the absence of an underlying theory, we parametrize this quantity as and consider the values . For -like contractions, we make the replacement

 λM2→λ^t−M2, (18)

where is the parton-level Mandelstam variable. Since is negative, we do not bother adding a width term.

We apply these matrix elements to compute the leading-order (LO) monojet production cross sections at the Tevatron by convolving with CTEQ6.1M parton distribution functions [34] and integrating over phase space. To match the most stringent Tevatron monojet search bounds, we impose a cut of and on the outgoing jet. Following Ref. [30], we also apply a flat efficiency factor of to connect our parton-level cross section to the full hadronic jet reconstruction at the Tevatron detectors. The cross sections computed in this way (after applying cuts and the efficiency factor) are shown in Fig. 4 for the parameter values and . We also set and . This figure shows a significant resonant enhancement when the intermediate state is narrow. A similar enhancement was seen in Ref. [32]. The resonant enhancement only becomes fully operational at due to the cut on jet . The horizontal dashed line in Fig. 4 represents the current Tevatron exclusion limit on the net monojet cross section of (after cuts) [30], based on the CDF analysis of Refs. [35, 36] which uses the same set of jet cuts as applied to our signal estimates. This limit translates into a lower bound of for the operator of Eq. (14), depending on the mass of the state and the width of the unspecified intermediate state.

Monojet signals can also be detected at the LHC. Due to the expectation of a significant amount of associated QCD radiation, Ref. [37] investigated the reach of an inclusive search for a hard jet plus missing energy search at ATLAS with no veto on additional hard jets. To match this analysis, we compute the inclusive leading-order parton-level cross section at the LHC with subject to the cuts and on the outgoing jet. We also rescale the cross section by a conservative acceptance/efficiency factor of  [30]. The corresponding cross sections are shown in Fig. 5 for several values of with and . We also set and . Based on the background analysis of Ref. [37], the study in Ref. [30] estimated a net SM background production rate after the applied cuts of . Applying a simple measure on the detection significance, this leads to a sensitivity to monojet cross sections as small as () with () of data at . This is shown by the dotted lines in Fig. 5. The resulting LHC reach for the operator of Eq. (14) lies in the range . This is competitive with the sensitivity to this operator from searches for IND in existing nucleon decay experiments.

Additional quark operators beyond that given in Eq. (14) and considered above are expected to yield qualitatively similar collider rates and signatures in most cases. An interesting further possibility are quark operators involving top or bottom quarks. With a bottom quark in the final state, the monojet signal could be augmented with a -tag. In the case of a top quark in the final state, the signal would be a hard single top quark with large missing energy. While the search reach in monobottom and monotop channels is likely greater than for the light quark channels, the operators involving third-generation quarks do not correlate directly with IND processes.

The hylogenesis model in Ref. [15] also contains a hidden gauge symmetry that couples to , is spontaneously broken at the GeV-scale, and couples to the SM only through kinetic mixing with hypercharge. Direct pair production of DM particles via the corresponding vector boson with an associated jet can be another source of monojet signals [30, 31]. However, since this vector is relatively light, the existing bounds from the Tevatron and the expected reach of the LHC are both much weaker than the limits from DM direct detection through elastic scattering mediated by the .

## 4 IND and Stellar Evolution

DM can be captured in stars by elastic scattering with the nuclei they contain. Once captured, ordinary self-annihilating DM will collect in the middle of the star and annihilate with other DM particles there, releasing energy [38, 39, 40]. Self-annihilation is not possible for hylogenic dark matter (hDM) due to its conserved global charge. Instead, it can potentially destroy baryons within the star via IND processes such as . If it remains within the star, the anti-hDM reaction product can then annihilate with a particle already captured in the stellar core. The net result of this chain is the destruction of a baryon and the release of energy, both from the meson decay and the annihilation step.

In this section we investigate the effects of IND processes on several varieties of stellar species including neutron stars, white dwarfs, and main-sequence stars. To be concrete, we will concentrate on the specific model of hylogenesis presented in Ref. [15] and described in Appendix A with IND mediated by the operator of Eq. (1). We will assume a fiducial spin-independent proton scattering cross section of (and ) for both and induced by their coupling to the kinetically-mixed vector boson,555For DM masses below , this value is consistent with existing direct detection searches [41, 42, 43]. as well as an -anti- annihilation cross section of . We shall consider two cases for the IND cross section: a large value of , and a small value of . The general behaviour for other IND rates will lie somewhere between these two extremes.

### 4.1 Stellar Capture and Annihilation

Relic and particles will be captured in stars by scattering with nuclei to energies below the local escape velocity. Once captured, an hDM particle will undergo further scatterings, thermalize with the baryons in the star, and collect within the stellar core. This occurs quickly relative to the lifetimes of the stars we consider here for our large fiducial value of the proton scattering cross section. Once they thermalize, the DM particles are largely confined to a core of radius [39, 40, 44]

 ri,th=(9Tc4πGρcmi)1/2, (19)

where , is the mean temperature and is the mean (baryon) density in the stellar core.

The evolution of the total numbers of , , , and within a star is described the following system of equations:

 dNΨdt = CΨ−AΨNΨN¯Ψ−BΨNΨ (20) dN¯Ψdt = CΨ−AΨNΨN¯Ψ+ϵ¯ΨBΦNΦ (21) dNΦdt = CΦ−AΦNΦNΦ†−BΦNΦ (22) dNΦ†dt = CΨ−AΦNΦNΦ†+ϵΦ†BΨNΨ (23)

Here, the coefficients are the hDM capture rates, the coefficients describe hDM-anti-hDM annihilation, and the coefficients describe IND. A general expression for is given in Ref. [40], while the and coefficients are given to a good approximation by

 Ai ≃ (σv)i,ann/(4πr3i,th/3), (24) Bi ≃ (σv)i,IND(ρc/mn), (25)

where is the mass of a nucleon. The terms appearing in Eqs. (21) and (23) are the probabilities for the anti-hDM products of IND to be captured by the host star after they are created. In certain regimes additional processes can influence the evolution of the stellar populations of hDM and anti-hDM such as evaporation [39, 45] and direct annihilation to baryons (e.g. ). We will discuss these effects when they may be relevant.

### 4.2 Neutron Stars

Neutron stars are very dense objects supported by the Fermi degeneracy pressure of their neutrons. Despite their name, they also contain a significant mass fraction of protons and heavier nuclei, and the nuclear state of their cores is not fully understood [46]. Typical neutron star parameters are mass , with  kg the solar mass, radius , core temperature , and core baryon density  [44, 47, 48, 49].

The rate of capture of or upon a typical neutron star (including general relativistic corrections) is [47, 48, 49]

 Ci≃2.5×1025s−1(ρDMGeV/cm3)(5GeVmΨ+mΦ)(220km/s¯v)f, (26)

where is the local DM energy density, is the local DM velocity dispersion, and

 f=min{1,(xpσp+xnσn)/(2×10−45cm2)}, (27)

with and being the proton and neutron mass fractions. The factor accounts for the saturation of the cross section at the cross-sectional area of the star. This saturation sets in when the star becomes optically thick to DM – when a DM particle impinging upon the star is likely to scatter multiple times with nucleons in the star. For the fiducial nucleon scattering cross section we are using and assuming a proton mass fraction of , we find that neutron stars are optically thick to both and . This leads to as well as . We also find

 ri,th≃(140cm)(Tc105K)1/2(3GeVmi)1/2(1.4×1018kg/m3ρc)1/2, (28)

implying and for the large (small) IND rate ().

Consider first the case of a large IND rate. Using the evolution equations of Eqs. (2023) and assuming negligible initial hDM densities, we find that the numbers of hDM and anti-hDM particles within a typical neutron star reach a steady state. This behaviour is shown in Fig. 6 for the fiducial cross sections given above and , . The steady-state populations in the limit of are approximated well by the analytic expressions

 NΨ≃CBΨ(1+ξ), N¯Ψ≃BΨAΨ(1+ξ1+ξ−1), NΦ≃CBΦ(1+ξ−1), NΦ†≃BΦAΦ(1+ξ−11+ξ), (29)

with .666 For , we find: . A steady state is also attained when one of the IND rates vanishes due to kinematic suppression. For example, with and , we find

 NΨ=CBΨ,   N¯Ψ=0,   NΦ,Φ†=±C2BΨ+√(C2BΨ)2+CAΦ, (30)

where in the last equality the plus sign corresponds to and the minus sign to . The solution for and is identical but with and interchanged in the expressions above. In both cases, the steady-state particle populations are on the order of and (when they are non-zero) for the fiducial input values listed above.

The time needed to reach this steady state from a negligible initial dark matter density in the star is on the order of

 τss∼max{B−1,BCA}. (31)

where the first value corresponds to IND balancing capture and the second to annihilation balancing production of anti-hDM by IND. For the fiducial cross sections we are considering, we find that annihilation takes longer to balance, and leads to . This is ultra short relative to the lifetime of a typical neutron star.

Having reached a steady state, the main combined effect of the IND and annihilation processes is to inject energy into the host neutron star with rate . In this respect, hDM in the steady state limit has the same effect on neutron stars as ordinary self-annihilating DM. Energy injection by DM can interfere with and halt the cooling of old neutron stars, and therefore the observation of a very cool, old neutron star in a region of large DM density could put significant constraints on a wide variety of DM scenarios [44, 48, 49]. However, this effect is too small to be observed using existing observations, and appears to be challenging to probe in the near future [44, 48, 49]. With hDM, baryons within the neutron star are also destroyed by IND, but the number is negligible compared to the total of within a typical neutron star over the lifetime of the Universe unless the local DM density approaches an enormous value of .

Consider next the case of a vanishingly small IND cross section. The populations of of and hDM particles will now build up within the neutron star with rate given by Eq. (26), corresponding to populations of about over the lifetime of the Universe. We can compare this number to the populations required for DM particles to begin self-gravitating and to form a black hole. Self-gravitation begins when [44, 50, 51]

 (32)

This is much greater than the steady-state populations found above for the case of a large IND rate, and also greater than the numbers collected for a smaller IND rates for local hDM densities below .

Self-gravitating DM particles will form a black hole that can potentially destroy the host star unless they are stabilized in some way [44, 47, 50, 51, 52]. For non-interacting fermions there is a degeneracy pressure that must be overcome. To do so, the total number of fermions must exceed

 Ni≳Nfcrit≡(√8πM% Plmi)3≃6×1055(3GeVmi)3, (33)

with the reduced Planck mass. In the case of non-self-interacting bosons, there is still a zero-point pressure which can be overcome if the number of bosons exceeds

 Ni≳Nbcrit≡(√8πM% Plmi)2≃2×1037(3GeVmi)2. (34)

A bosonic black hole can also arise even before bulk self-gravitation sets in through the formation of a Bose-Einstein condensate [50, 51].

We argue that black holes are not likely to form in the specific theory of hylogenesis discussed in Ref. [15] and Appendix A, in which and carry equal and opposite charges under a spontaneously broken gauge symmetry, until . At distances much smaller than the inverse vector mass , the effects of breaking the can be neglected and the vector boson mediates a repulsive force between particles with like-sign charges. This induces a pressure among particles that prevents them from collapsing into a black hole provided and (where is the coupling of [50]. Instead, under these mild assumptions the formation of a black hole requires an approximately charge-neutral collection of hDM particles and therefore roughly equal numbers of and states. A necessary condition for this to occur is to overcome the fermion degeneracy pressure. Therefore we do not expect this specific theory of hylogenesis to lead to the formation of hDM black holes within neutron stars unless the local hDM density around the host star exceeds .

Between the two extremes of the large and small IND rates considered above, no new obvious observational bounds arise. With a small but non-zero rate for IND, the and populations may grow large enough to form some amount of anti-hDM that will subsequently annihilate away. Large hDM densities can also lead to direct annihilation of hDM to antibaryons, , and similarly for anti-hDM. This can be accounted for by adding terms of the form to the evolution equations for and , with and is the volume occupied by the hDM, whose evolution we must also keep track of. Note that appears here rather than because this effect starts to compete with standard IND only after the hDM population begins to self-gravitate, when . We defer an analysis of the effects of self-gravitation and self-annihilation of hDM on the structure of neutron stars to a future work.

### 4.3 White Dwarfs

We can perform a similar analysis for the effects of hDM on white dwarfs. A typical white dwarf consists primarily of carbon and oxygen and is supported by the degeneracy pressure of the electrons it contains. Typical white dwarf parameters are , , ,  [53]. We will approximate the internal structure of a white dwarf as consisting entirely of carbon with a uniform density [44].

The capture rate of DM through elastic scattering with nuclei in a white dwarf is approximated well by the expression [40, 44, 54, 55, 56]

 Ci≃(83π)1/2(ρi¯vmi)(3v2esc2¯v2)σeff, (35)

with being the local velocity dispersion, the escape velocity at the surface of the star, and the the effective cross section is defined to be

 σeff≡min{πR2,  σSIp∑kMmnxkAk[fpZk+(Ak−Zk)fn]2f2p(mrkmrp)2}, (36)

where the sum runs over all nuclear species in the star with mass fraction and atomic mass (number) (), is the spin-independent nucleon scattering cross section off protons, and are the relative coupling strengths of the DM to protons and neutrons, and () is the reduced mass of the DM-nucleus (DM-proton) system. The saturation of the effective cross section at corresponds to the star becoming optically thick.

We find that a typical white dwarf is indeed optically thick for the fiducial scattering cross section of (with , ). This gives

 CΨ = CΦ≡C ≃ (6×1027s−1)(R0.01R⊙)(M0.7M⊙)(ρhDM