The Electrosphere of Macroscopic "Quark Nuclei": A Source for Diffuse MeV Emissions from Dark Matter
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The Electrosphere of Macroscopic “Quark Nuclei”:A Source for Diffuse MeV Emissions from Dark Matter.
Michael McNeil Forbes ∗ Institute for Nuclear Theory, University of Washington, Box 351550, Seattle, Washington, 98195-1550, USA
Kyle Lawson and Ariel R. Zhitnitsky
Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada (Dated: August 20, 2018)Using a Thomas-Fermi model, we calculate the structure of the electrosphere of the quark antimat-ter nuggets postulated to comprise much of the dark matter. This provides a single self-consistentdensity profile from ultrarelativistic densities to the nonrelativistic Boltzmann regime that use topresent microscopically justified calculations of several properties of the nuggets, including theirnet charge, and the ratio of MeV to 511 keV emissions from electron annihilation. We find that thecalculated parameters agree with previous phenomenological estimates based on the observationalsupposition that the nuggets are a source of several unexplained diffuse emissions from the Galaxy.As no phenomenological parameters are required to describe these observations, the calculation pro-vides another nontrivial verification of the dark-matter proposal. The structure of the electrosphereis quite general and will also be valid at the surface of strange-quark stars, should they exist.
PACS numbers: 95.35.+d, 52.27.Aj, 78.70.Bj, 98.70.Rz,
CONTENTS
I. Introduction 1II. Dark Matter as Dense Quark Nuggets 2III. Core Structure 4IV. Electrosphere Structure 5A. Thomas-Fermi Model 51. Boundary Conditions 6B. Profiles 7C. Nugget Charge Equilibrium 8V. Diffuse Galactic Emissions 9A. Observations 101. The 511 keV Line 102. Diffuse MeV scale emission 103. Comparison 10B. Annihilation rates 111. Positronium Formation 112. Direct-annihilation Rates 11C. Spectrum and Branching Fraction 12D. Normalization to the 511 keV line 14VI. Conclusion 14Acknowledgments 15A. Density Functional Theory 151. Thomas-Fermi Approximation. 16 ∗ E-mail: [email protected]
2. Analytic Solutions 16a. Ultrarelativistic Regime 16b. Boltzmann Regime 173. Numerical Solutions 17B. Debye Screening In the Electrosphere 17References 19
I. INTRODUCTION
In this paper we explore some details of a testable andwell-constrained model for dark matter [1–5] in theform of quark matter as antimatter nuggets. In par-ticular, we focus on physics of the “electrosphere” sur-rounding these nuggets: It is from here that observableemissions emanate, allowing for the direct detection ofthese dark-antimatter nuggets.We first provide a brief review of our proposal inSec. II, then describe the structure of the electrosphereof the nuggets using a Thomas-Fermi model in Sec. IV.This allows us to calculate the charge of the nuggets,and to discuss how they maintain charge equilibriumwith the environment. We then apply these results tothe calculation of emissions from electron annihilationin Sec. V, computing some of the phenomenological pa-rameters introduced in [6] required to explain currentobservations. The values computed in the present pa-per are consistent with these phenomenologically moti-vated values, providing further validation of our modelfor dark matter. The present results concerning the den-sity profile of the electrosphere may also play an im-portant role in the study of the surface of quark stars,should they exist.
II. DARK MATTER AS DENSE QUARK NUGGETS
Two of the outstanding cosmological mysteries – thenatures of dark matter and baryogenesis – might be ex-plained by the idea that dark matter consists of com-pact composite objects ( cco s) [1–5] similar to Witten’sstrangelets [7]. The basic idea is that these cco s –nuggets of dense matter and antimatter – form at thesame qcd phase transition as conventional baryons(neutrons and protons), providing a natural explana-tion for the similar scales Ω dm ≈ Ω B . Baryogenesisproceeds through a charge separation mechanism: bothmatter and antimatter nuggets form, but the natural cp violation of the so-called θ term in qcd – whichwas of order unity θ ∼ qcd phase transi-tion – drives the formation of more antimatter nuggetsthan matter nuggets, resulting in the leftover baryonicmatter that forms visible matter today (see [2] for de-tails). Note, it is crucial for our mechanism that cp vio-lation be able to drive charge separation: though not yetproven, this idea may already have found experimen-tal support through the Relativistic Heavy Ion Collider( rhic ) at Brookhaven [13], where charge separation ef-fects seem to have been observed [14, 15]The mechanism requires no fundamental baryonasymmetry to explain the observed matter/antimatterasymmetry. Together with the observed relation Ω dm ≈ Ω B (see [16] for a review) we have B universe = = B nugget + B visible − ¯ B antinugget (1a) B dark-matter = B nugget + ¯ B antinugget ≈ B visible (1b)where B universe is the overall asymmetry – the totalnumber of baryons minus the number of antibaryonsin the Universe – and B dark-matter is the total numberof baryons plus antibaryons hidden in the dark-matter If θ is nonzero, one must confront the so-called strong cp problemwhereby some mechanism must be found to make the effective θ parameter extremely small today in accordance with measurements.This problem remains one of the most outstanding puzzles of theStandard Model, and one of the most natural resolutions is to in-troduce an axion field. (See the original papers [8–10], and recentreviews [11].) Axion domain walls associated with this field (orultimately, whatever mechanism resolves the strong cp problem)play an important role in forming these nuggets, and may play inimportant role in their ultimate stability. See [1, 2, 12] for details. Note that we use the term “baryon” to refer in general to anythingcarrying U B ( ) baryonic charge. This includes conventional coloursinglet hadrons such as protons and neutrons, but also includestheir constituents – i.e. the quarks – in other phases such as strangequark matter. nuggets. The dark matter comprises a baryon chargeof B nugget contained in matter nuggets, and an an-tibaryonic charge of ¯ B antinugget contained in antimatternuggets. The remaining unconfined charge of B visible isthe residual “visible” baryon excess that forms the reg-ular matter in our Universe today. Solving Eq. (1) givesthe approximate ratios ¯ B antinugget : B nugget : B visible ≃ | B | ≈ – 10 ,so they have an extremely tiny number density. Thisexplains why they have not been directly observed onearth. The local number density of dark-matter parti-cles with these masses is small enough that interactionswith detectors are exceedingly rare and fall within allknown detector and seismic constraints [3]. (See also[17, 18] and references therein.)2. The nugget cores are a few times nuclear density ρ ∼
10 GeV/fm , and thus have a size R ∼ − –10 − cm.Their interaction cross section is thus small σ / M ≈ π R / M = − –10 − cm /g: well below the typicalastrophysical and cosmological limits, which are on theorder of σ / M < /g. Dark-matter–dark-matter in-teractions between these nuggets are thus negligible.3. They have a large binding energy such that the bary-onic matter in the nuggets is not available to participatein big bang nucleosynthesis ( bbn ) at T ≈ ∆ ≈
100 MeV, andcritical temperature T c ∼ ∆ / √ ≈
60 MeV, as this scaleprovides a natural explanation for the observed photonto baryon ratio n B / n γ ∼ − [2], which requires aformation temperature of T form =
41 MeV [19]. Thus, on large scales, the nuggets are sufficientlydilute that they behave as standard collisionless colddark matter ( ccdm ). When the number densities ofboth dark and visible matter become sufficiently high,however, dark-antimatter–visible-matter collisions mayrelease significant radiation and energy. In particular,antimatter nuggets provide a site at which interstellarbaryonic matter – mostly hydrogen – can annihilate,producing emissions that should be observable fromthe core of our Galaxy of calculable spectra and energy. If the average nugget size ends up in the lower range, then thePierre Auger observatory may provide an ideal venue for searchingfor these dark-matter candidates. At temperatures below the gap, incident baryons with energies be-low the gap would Andreev reflect rather than become incorpo-rated into the nugget.
These emissions are not only consistent with currentobservations, but naturally explain several mysteriousdiffuse emissions observed from the core of our Galaxy,with frequencies ranging over 12 orders of magnitude.Although somewhat unconventional, this idea natu-rally explains several coincidences, is consistent withall known cosmological constraints, and makes testablepredictions. Furthermore, this idea is almost entirelyrooted in conventional and well-established physics. Inparticular, there are no “free parameters” that can be– or need to be – “tuned” to explain observations: Inprinciple, everything is calculable from well-establishedproperties of qcd and qed . In practice, fully calculat-ing the properties of these nuggets requires solving thefermion many-body problem at strong coupling, so wehave generally resorted to “fitting” a handful of phe-nomenological parameters from observations. In thispaper we examine the qed physics of the electrosphere,providing a microscopic basis for some of these parame-ters. Once these parameters are determined, the modelmakes unambiguous predictions about other processesranging over more than 10 orders of magnitude in scale.The basic picture involves the antimatter nuggets –compact cores of nuclear or strange-quark matter (seeSec. III) surrounded by a positron cloud with a profileas calculated in Sec. IV. Incident matter will annihilateon these nuggets producing radiation at a rate propor-tional to the annihilation rate, thus scaling as the prod-uct ρ v ( ~ r ) ρ dm ( ~ r ) of the local visible and dark-matterdensities. This will be greatest in the core of the Galaxy.To date, we have considered five independent observa-tions of diffuse radiation from the core of our Galaxy:1. S pi / integral (the spi instrument on the integral satel-lite) observes 511 keV photons from positronium decaythat is difficult to explain with conventional astrophys-ical positron sources [20–22]. Dark-antimatter nuggetswould provide an unlimited source of positrons as sug-gested in [23, 24].2. C omptel / crgo (the comptel instrument on the Comp-ton Gamma Ray Observatory ( crgo ) satellite) detectsa puzzling excess of 1–20 MeV γ -ray radiation. It wasshown in [6] that the direct e + e − annihilation spectrumcould nicely explain this deficit, but the annihilationrates were crudely estimated in terms of some phe-nomenological parameters. In this paper we providea microscopic calculation of these parameters (Eqs. (19)and (21)), thereby validating this prediction.3. Chandra (the
Chandra x-ray observatory) observes a dif-fuse keV x -ray emission that greatly exceeds the en-ergy from identified sources [25]. Visible-matter/dark-antimatter annihilation would provide this energy. Itwas shown in [4] that the intensity of this emission isconsistent with the 511 keV emission if the rate of pro-ton annihilation is slightly suppressed relative to therate of electron annihilation. In Sec. IV C we describe the microscopic nature of this suppression.4. E gret / crgo (the Energetic Gamma Ray ExperimentTelescope aboard the crgo satellite) detects MeV to GeVgamma rays, constraining antimatter annihilation rates.It was shown in [4] that these constraints are consistentwith the rates inferred from the other emissions.5. W map (the Wilkinson Microwave Anisotropy Probe)has detected an excess of GHz microwave radiation– dubbed the “ wmap haze” – from the inner 20 ◦ core of our Galaxy [26–29]. Annihilation energy notimmediately released by the above mechanisms willthermalize, and subsequently be released as thermalbremsstrahlung emission at the eV scale. In [5] it wasshown that the predicted emission from the antimatternuggets is consistent with, and could completely ex-plain, the observed wmap haze.These emissions arise from the following mechanism:Neutral hydrogen from the interstellar medium ( ism )will easily penetrate into the electrosphere, providing asource of electrons and protons.The first and simplest process is the annihilation ofthe electrons through positronium formation, produc-ing 511 keV photons as discussed in [23, 24]. Note thatthis mechanism predicts that, within the environmentof the electrosphere, virtually all of the low-energyemission should be characterized by a positronium de-cay spectrum, including 25% as a sharp 511 keV linefrom the decay of the singlet state and the remaining75% as the broad three photon continuum resultingfrom the triplet state. As emphasized in [30], simplypostulating a dark-matter source of positrons does notsuffice to explain the observed spectrum characterizedby 94 ±
4% positronium annihilation: The positronsmust annihilate in the appropriate cool environment asprovided by the electrosphere of the nuggets.Our proposal can thus easily explain the observed511 keV radiation. If we assume that this process is thedominant source, then we can use this to normalize theintensities of the other emissions. A remarkable featureof this proposal is that it then predicts the correct inten-sity for all of the other observations, even though theyspan many orders of magnitude in frequency.The second process is direct annihilation of theelectrons on the positrons in the electrosphere. Asdiscussed in appendix B, the electrons are stronglyscreened by the positron background, and some frac-tion can penetrate deep within the electrosphere. Therethey can directly annihilate with high-momentumpositrons in the Fermi sea producing radiation up to10 MeV or so. This process was originally discussedin [6] where the ratio of direct MeV annihilation to511 keV annihilation was characterized by several phe-nomenological parameters chosen to fit the observa-tions. In Sec. V we put this prediction on solid groundand show that these parameter fits agree with the micro-scopic calculation based on the electrosphere structure,which depends only on qed .The other radiation originates from the energy de-posited by annihilation of the incident protons in thecore of the nuggets. In [4] we argue that the protonswill annihilate just inside the surface of the core, releas-ing some 2 GeV of energy. Occasionally this processwill release GeV photons – the rate of which is consis-tent with the egret / crgo constraints – but most of theenergy will be transferred to strongly interacting com-ponents, and ultimately about half will scatter downinto ∼ Chandra observations which cannot resolve thethermal falloff, but future analysis might be able to dis-tinguish between the two.We shall shed some light on the interaction betweenthe protons and the core in Sec. IV C, but cannot yetperform the required many-body analysis to place all ofthis on a strong footing as this would require a practicalsolution to outstanding problems of high-density qcd .The remaining energy will thermalize within thenuggets, until an equilibrium temperature of about T ∼ qed – extends to very low frequencies, and theintensity of the emission in the microwave band is justenough to explain the wmap haze [5]. III. CORE STRUCTURE
A full accounting of this nugget proposal requires aproper description of the high-density phase found inthe core of the nuggets. Unfortunately, a quantitativeunderstanding of this phase requires a practical solu-tion to the notoriously difficult problem of high-density qcd . The density of the core will be within several or-ders of nuclear matter density ∼ −
100 GeV/fm . Thisis not high enough for the asymptotic freedom of qcd may be used to solve the problem perturbatively, so onemust resort to nonperturbative techniques such as thelattice formulation of qcd . Unfortunately, at finite den-sity, the presence of the infamous sign problem rendersthis approach exponentially expensive and it remains a famously intractable problem. As such, we cannot ex-actly quantify the nature of the core and must constrainits properties from other observations; we list the im-portant properties in this section.Fortunately, the observable emissions discussed inthis paper result primarily from the calculable physicalprocesses in the electrosphere of the nuggets, and arethus largely insensitive to the exact nature of the core.Nevertheless, the core structure must be addressed, andit is possible that future developments concerning theproperties of high-density qcd could rule out the feasi-bility of our nugget proposal.The first problem concerns the stability of the core.All evidence suggests that, in the absence of an externalpotential (such as the gravitational well of a neutronstar), nuclear matter will fragment into small nucleiwith a baryon number no larger than a few hundred.This suggests one of two possibilities: • The first possibility suggested by Witten [7] is that aphase of strange-quark matter [31] becomes stable athigh density. In this case, the nuggets are simplystrangelets and antistrangelets and the novel featurehere is that the domain walls associated with strong CP violation provide the required mechanism to con-dense enough matter to catalyze the formation of thestrangelets before they evaporate. (For a brief review,see [32] and references therein.)Although strangelets have not yet been observed, thepossibility of stable strange-quark mater has not yetbeen ruled out (see for example [33]). This must be care-fully reconciled with future astrophysical observationsand constraints as it is conceivable that this possibilitymight be ruled out in the future. If the nuggets are aform of strange-quark matter, one must also considerthe possibility of mixed phases as suggested in [34] (seefor example [35, 36] and references therein). This wouldmost likely have to be ruled out energetically to preventthe nuggets from fragmenting. • The second possibility is that the domain walls responsi-ble for forming the nuggets at the qcd phase transitionbecome an integral part, providing a surface tensionthat holds the nuggets together, even in the absence ofabsolutely stable strange-quark matter. The stability ofthis possibility has been discussed in detail in [1] andwe shall not repeat these arguments here. In this case,the core may be something more akin to dense nuclearmatter as might be found in the core of a neutron star.As discussed above, in order to explain the observa-tions, the following core properties are crucial to ourproposal. This provides some insight into the requirednature of the core:1. The nuggets must be stable. Stable strange-quark mat-ter would offer a nice explanation. Otherwise, the struc-ture of the core – especially the surface – must be con-sidered in more detail. This is a complicated problemthat we do not presently know how to solve.2. As discussed above, the formation of the objects muststop at T =
41 MeV to explain the observed photonto baryon ratio. This could be naturally explained bythe order 100 MeV pairing gaps expected in colour-superconducting strange-quark matter. If the proposalturns out to be correct, then, this would provide aprecise measurement of the pairing properties of high-density qcd . (The exact relationship between the pair-ing gap and the formation temperature will be quitenontrivial and require a detailed model of the forma-tion dynamics.) This favours a model of the core withstrong pairing correlations.3. In order to explain the Chandra data, our picture of theemission mechanism requires that the proton annihila-tions occur somewhat within the core, not immediatelyon the surface where there would be copious pion pro-duction. The simplest explanation for this would be thepresence of strong correlations in the core, delaying theannihilation until the protons penetrate a few hundredfm or so. Again, if the core is a colour superconductor,then the pairing correlations could explain this, but adetailed calculation is needed to make sure.An argument supporting the presence of such correla-tions in a dense environment is the observation that theannihilation cross section of an antiproton on a heavynuclei is many times smaller than the vacuum p ¯ p and n ¯ n annihilation rates suggest. Indeed, it is expected thatantiproton/nuclei bound states may persist with a life-time much longer than the vacuum annihilation ratespredicts (see [37, 38]). This annihilation suppressionshould become even stronger with increasing density.Thus, the hypothesis of strange-quark matter simplifiesthe picture quite a bit, but we do not yet see a way to di-rectly link this hypothesis with the existence or stabilityof the nuggets. IV. ELECTROSPHERE STRUCTURE
Here we discuss the density profile of the electron cloudsurrounding extremely heavy macroscopic “nuclei”. Wepresent a type of Thomas-Fermi analysis including thefull relativistic electron equation of state required tomodel the relativistic regime close to the nugget core.We consider here the limit when the temperature ismuch smaller than the mass T ≪
511 MeV. The solu-tion for higher temperatures follows from similar tech-niques with fewer complications.The observable properties of the antimatter nuggetsdiscussed so far [1–6] depend on the existence of a non-relativistic “Boltzmann” regime with a density depen-dence n ( r ) ∼ ( r − r B ) − (see appendix 1 of Ref. [5]for details). This region plays an important role in ex- plaining the wmap haze [5] as well as in the analysis ofthe diffuse 511 keV emissions [2]. The techniques pre-viously used, however, were not sufficient to connectthis nonrelativistic “Boltzmann” regime to the relativis-tic regime through a self-consistent solution determinedby parameters T , µ , m e .The main point of this section is to put the exis-tence of a sizable Boltzmann regime on a strong foot-ing, and to calculate some of the previously estimatedphenomenological parameters that depend sensitivelyon density profile. These may now be explicitly com-puted from qed using a justified Thomas-Fermi approx-imation to reliably account for the many-body physics.We show that a sizable Boltzmann regime exists for allbut the smallest nuggets, which are ruled out by lackof terrestrial detection observation. We also addressthe question of how the nuggets achieve charge equilib-rium (Sec. IV C), discussing briefly the charge-exchangemechanism, and determining the overall charge of thenuggets (see Table I). A. Thomas-Fermi Model
To model the density profile of the electrosphere, weuse a Thomas-Fermi model for a Coulomb gas ofpositrons. This is derivable from a density functionaltheory (see Appendix A) after neglecting the exchangecontribution, which is suppressed by the weak coupling α . The electrostatic potential φ ( ~ r ) must satisfy the Pois-son equation ∇ φ ( ~ r ) = − π en ( ~ r ) . (2)where en ( ~ r ) is the charge density. Outside of the nuggetcore, we express everything in terms of the local effec-tive chemical potential µ ( ~ r ) = − e φ ( ~ r ) , (3)and express the local charge density through the func-tion q ( ~ r ) = en [ µ ( ~ r )] where n [ µ ] contains all of the in-formation about the equation of state.As with the Thomas-Fermi model of an atom, the self-consistent solution will be determined by the chargedensity of the core (“nucleus”). We shall simply imple-ment this as a boundary condition at the nugget coreboundary at radius r = R , and thus only consider theregion r > R . The resulting solution may thus be ex-pressed in terms of the equations ∇ µ ( ~ r ) = πα n [ µ ( ~ r )] , ǫ ~ p = q k ~ p k + m , (4a) n [ µ ] = Z d ~ p ( π ) (cid:20) + e ( ǫ ~ p − µ ) / T − + e ( ǫ ~ p + µ ) / T (cid:21) , (4b)with the appropriate boundary conditions at r = R and r = ∞ . In (4b) we have explicitly included both particleand antiparticle contributions, as well as the spin de-generacy factor, and have used modified Planck unitswhere ¯ h = c = πǫ = e = α and energy,momentum, inverse distance, and inverse time are ex-pressed in eV.We assume spherical symmetry so that we may write ∇ µ ( r ) = r dd r r dd r µ ( r ) = µ ′′ ( r ) + µ ′ ( r ) / r . (5)Close to the surface of the nuggets, the radius issufficiently large compared with the relevant lengthscales that the curvature term 2 µ ′ ( r ) / r may be ne-glected. We shall call the resulting approximation “one-dimensional”. The resulting profile does not dependon the size of the nuggets and remains valid until theelectrosphere extends to a distance comparable to theradius of the nugget, at which point the full three-dimensional form will cut off the density. When ap-plied to the context of strange stars [39–42], the one-dimensional approximation will be completely suffi-cient. See also [43] where a Thomas-Fermi calculationis used to determine the charge distribution inside astrange-quark nugget.To help reason about the three-dimensional equation,we note that (4a) may be expressed as µ ′′ ( x ) = x − πα n [ µ ( x )] (6)where x = r . A nice property of this transformationis that µ ( x ) must be a convex function if the chargedensity has the same sign everywhere.
1. Boundary Conditions
The physical boundary condition at the origin followsfrom smoothness of the potential. By combining thiswith a model of the charge distribution in the core andan appropriate long-distance boundary-condition, onecould in principle model the entire distribution of elec-trons throughout the nugget.In the nugget core, however, beta-equilibrium essen-tially establishes a chemical potential on the order of 10– 100 MeV [39] that depends slightly on the exact equa-tion of state for the quark-matter phase (which is notknown). Thus, we may simply take the boundary con-dition as µ ( R ) = µ R ≈
25 MeV. Our results are not verysensitive to the exact value, though if less that 20 MeV,then this acts as a cutoff for the direct e + e − emissiondiscussed in Sec. V A 2.The formal difficulty in this problem is properly for-mulating the long-distance boundary conditions. At T =
0, the large-distance boundary condition for non-relativistic systems is clear: n ( r → ∞ ) =
0. From (6) wesee that µ ( x ) is linear with slope µ ′ ( x ) = − eQ where Q is the overall charge of the system (ions with a defi-ciency of electrons/positrons are permitted in the the-ory; see for example [44]): eQ ( r ) = Z r d˜ r πα n ( ˜ r ) ˜ r = Z r d˜ r (cid:16) ˜ r µ ′ ( ˜ r ) (cid:17) ′ = r µ ′ ( r ) .At finite temperature, however, this type of boundary-condition is not appropriate. Instead, one must con-sider how equilibrium is established.In a true vacuum, a finite temperature nugget will“radiate” the loosely bound outer electrosphere untilthe electrostatic potential is comparable to the temper-ature eQ / r ∗ ∼ T . At this radius, radiation becomesexponentially suppressed. Suppose that the density atthis radius is n ( r ∗ ) . We can estimate an upper boundfor the evaporation rate as 4 π r ∗ vn where v ∼ √ T / m . This needs to be compensated by rate of charge de-position from the surrounding plasma which can be es-timated as 4 π R v n + , ism n n + , ism where v n + , ism ∼ − c is the typical relative speed of the nuggets and chargedcomponents in the Inter-Stellar Medium ( ism ) and 4 π R is approximately the cross section for annihilation onthe core (see Sec. IV C). The density n n + , ism of chargedcomponents in the core of the Galaxy is typically 10 − to 10 − of the total density n ism ∼ − . Thus, oncethe density falls below n < n rad ∼ R r r mT v n + , ism n ism − (7)charge equilibrium can easily be established. This al-lows us to formulate the long-distance boundary condi-tions by picking a charge Q and outer radius r ∗ suchthat eQ / r ∗ ∼ T , and n ( r ∗ ) ∼ n rad , establishing boththe typical charge of the nuggets as well as the outerboundary condition for the differential equation (4). Note that the Thomas Fermi approximation is nottrust-worthy in these regimes of extremely low density,but the boundary condition suffices to provide an es-timate of the charge of the nuggets: were they to beless highly charged, then the density at r ∗ would besufficiently high that evaporation would increase thecharge; were they more highly charged, then the evap-oration rate would be exponentially suppressed, allow-ing charge to accumulate. The Boltzmann averaged velocity is lower h v i = p T / ( π m ) , andcutting off the integral properly will lower this even more, so thisgives a conservative upper bound on the evaporation rate. Strictly speaking, at finite T , the equations do not have a formalsolution with a precise total charge because there is always somedensity for µ >
0. Practically, once m − µ ≪ T , the density becomesexponentially small and the charge is effectively fixed. Thus, for the antimatter nuggets, the density profile isdetermined by system (4) and the boundary conditions µ ( R ) = µ R ∼
25 MeV, (8a) eQ / r ∗ = T ∼ n ( r ∗ ) = n rad ( r ∗ ) . (8b) B. Profiles
In principle, one should also allow the temperature tovary. However, tThe rate of radiation in the Boltzmannregime is suppressed by almost six orders of magnitudewith respect to the black-body rate; meanwhile, the den-sity of the plasma n B ∼ ( mT ) is quite large [5]. Thus,the abundance of excitations ensures that the thermalconductivity is high enough to maintain an essentiallyconstant temperature throughout the electrosphere.One can now numerically solve the system (4)and (8). We shall consider six cases: total (anti)baryoncharge B ∈ { , 10 , 10 } and quark-matter densi-ties n core ∈ {
1, 100 } fm − . The resulting profiles areplotted in Fig. 1. As a reference, we also plot the “one-dimensional” approximation obtained numerically byneglecting the curvature term in (5) as well as twoanalytic approximations: One for the ultrarelativisticregime [39, 40, 46] where n [ µ ] ≈ µ /3 π : n ur ( z ) ≈ µ R π ( + z / z ur ) , z ur = µ − R r π α , (9)and one for the nonrelativistic Boltzmann regime [5] B n core
R Q
100 fm − − cm 5 × e − × − cm 2 × e
100 fm − × − cm 10 e − − cm 4 × e
100 fm − × − cm 10 e − − cm 4 × e TABLE I. Antimatter nugget properties: the core radius R andtotal electric charge Q for nuggets with two different assumedcore densities n core and three different antibaryon charges B .These correspond to the profiles shown in Fig. 1. Note that the ultrarelativistic approximation employed in [45] dis-cussing nonrelativistic physics not only overestimates the densityby 3 orders of magnitude in the Boltzmann regime, but also hasa different z -dependence [(9) verses (10)]: The ultrarelativistic ap-proximation is not valid for µ < m where most of the relevantphysical processes take place. For T =
0, see appendix A 2 a. where n [ µ ] ∝ e µ / T : n B ( z ) = T πα ( z + z B ) . (10)Since we have the full density profiles, we fit z B here FIG. 1. Density profile of antimatter nuggets (bottom plot isa zoom). The thick solid line is the one-dimensional approxi-mation neglecting the curvature of the nugget. Six profiles areshown descending from this in pairs of thin solid and dashedcurves. From left to right, each pair has fixed baryon charge B = (red), 10 (black) and 10 (blue) respectively. Thesolid curves represent nuclear density cores while the dashedcurves represent 100 times nuclear density. The light shaded(yellow) regions correspond to the nonrelativistic Boltzmannregime where the Boltzmann approximation (10) (dash-dotted(cyan) line) is valid. Only the B = profiles visibly de-part from the one-dimensional approximation in this regime.The two upper curves in the top plot use the scale on theright and are the annihilation rates Γ Ps (right (green) curve)and Γ dir (left (blue) curve) normalized to the saturated value Γ Ps ( µ = ∞ ) = vq / ( π m α ) (12). These curves comprisea range of cutoffs q ∼ m α such the positronium annihilationrates (12) vary by 10%, and a range of incoming velocities10 − c < v < − c . The scaling of the abscissa in the up-per figure is ln ( z + z ur ) where z ur ≈ × − so that the T = so that the this approximation matches the numericalsolution at n B = ( mT /2 π ) , (a slightly better approx-imation than used in [5]).All of our previous estimates about physical proper-ties of the antimatter nuggets – emission of radiation,temperature etc. – have been based on these approxima-tions using the region indicated in Fig. 1, it is clear thatthey work very well, as long as these regimes exist . Thepotential problem is that the nuggets could be highlycharged or sufficiently small that the one-dimensionalBoltzmann regime fails to exist with the density rapidlyfalling through the relevant density scales. We can seefrom the results in Fig. 1 that the nonrelativistic re-gion exists for all but the smallest nuggets: As longas B > – as required by current detector con-straints [4] – then the one-dimensional approximationsuffices to calculate the physical properties.The annihilation rates discussed in Sec. V have alsobeen included in Fig. 1 to show that these effects onlystart once the densities have reached the atomic den-sity scale, which is higher than the Boltzmann regime.Thus these results are also insensitive to the size of thenugget and the domain-wall approximation suffices. Itis clear that one must have a proper characterization ofthe entire density profile from nonrelativistic throughto ultrarelativistic regimes in order to properly calculatethe emissions: one cannot use simple analytic forms.We now discuss how charge equilibrium is estab-lished, and then apply our results to fix the relative nor-malization between the diffuse 1 – 20 MeV emissionsand the 511 keV emissions from the core of our Galaxy.As we shall show, the relative normalization is nowfirmly rooted in conventional physics, and in agreementwith the previous phenomenological estimate [6], pro-viding another validation of our theory for dark matter. C. Nugget Charge Equilibrium
In order to determine the effective charge of the nuggetswe must consider how equilibrium is obtained. Forthe matter nuggets, equilibrium is established throughessentially static equilibrium with the surrounding ism plasma, however, for the antimatter nuggets, nosuch static equilibrium can be achieved. Instead, onemust consider the dynamics of the following charge-exchange processes:1. Deposition of charge via interaction and annihilation ofneutral ism components (primarily neutral hydrogen).2. Deposition of charge via interaction and annihilation ofionized ism components (electrons, protons, and ions).3. Evaporation of positrons from the antinugget’s surface.First we consider impinging neutral atoms ormolecules with velocity v n , ism ∼ − c . Even if the antinugget is charged, these are neutral and will stillpenetrate into the electrosphere. Here the electronswill annihilate or be ionized leaving a positively chargenucleus with energy T ism ∼ m n v n , ism . As the elec-trosphere consists of positrons, this charge cannot bescreened, so the nucleus will accelerate to the core. Atthe core the nucleus either will penetrate and annihi-late, resulting in the diffused x-ray emissions discussedin [4] and providing the heat to fuel the microwaveemissions [5], or will bounce off of the surface due tothe sharp quark-matter interface. ¯ Budu ¯ Bddu ¯ Budu ¯ Budu
FIG. 2. Charge-exchange process (top diagram): A chargedincoming proton (upper left) exchanges an up quark u for adown quark d with the antimatter nugget ¯ B , reflecting as aneutron (upper right). This is Zweig suppressed relative tothe simple reflection process illustrated below, however, theneutron can escape from the system whereas the reflected pro-ton will be trapped by the electric fields. This will enhance theoverall rate of charge-exchange reactions because the protonwill continue to react again and again until either the chargeexchange occurs, or it eventually annihilates. In order to ex-plain the relative intensities of the observed 511 keV and dif-fuse x-ray emissions from the core of our Galaxy [4], the ratioof the charge-exchange rate to the annihilation rate must be f ∼ It is well known from quantum mechanics that any sufficientlysharp transition has a high probability of reflection of low energyparticles.
Note that, if the nuggets are sufficiently charged suchthat kinetic energy of the incoming nucleus at the pointof ionization is smaller than the electrostatic energy, T ism ∼ m n v n , ism ∼ eV ≪ eQr ∼ keV ∼ K, (11)then the charged nucleus will be unable to escape andwill return to the core to either annihilate or undergoa charge-exchange reaction to become neutral, afterwhich it may leave unimpeded by the electric field. Fornuggets with B ∼ , 10 , and 10 respectively, thiscritical charge is Q / e ≫
1, 10 , and 10 respectively. Asshown in Table I, the charge established through evapo-ration greatly exceeds this critical charge: The chargednucleus will not be able to escape the antinugget unlessit undergoes a charge-exchange reaction (see Fig. 2).For a single proton, such a charge-exchange reactionconsists of an up quark being replaced by a down quarkat the surface of the nugget, converting the proton to aneutron which can then escape. This process involvesthe exchange of a quark-antiquark pair and is a stronginteraction process, but suppressed by the Zweig ( ozi )rule. The overall ratio of charge-exchange to annihila-tion reactions is amplified by a finite probability of re-flection from the core boundary: this results in multiplebounces before annihilation.A better understanding of the details concerning theinteraction between the proton and the quark-matterboundary is required to quantitatively predict the ratiosof the rate of charge-exchange to the rate of annihilationand thus to confirm or rule out the suppression factor f ∼ ∼
10 keV x-ray emis-sions seen from the core of the Galaxy [4]. This requiresan estimate of the rates of reflection, annihilation, andcharge-exchange; the elasticity of collisions; and the en-ergy loss through scattering.We emphasize that the corresponding calculationsdo not require any new physics: everything is rootedfirmly in qed and qcd . They do, however, require solv-ing the many-body physics of the strong interaction in-cluding the incident nucleons and the quark antimatterinterface (the phase of which may be quite complicated).A full calculation is thus very difficult, requiring insightinto high-density qcd : estimates can probably be madeusing standard models of nuclear matter for the core,but are beyond the scope of the present paper.In any case, the nucleus will certainly deposit itscharge on the antinugget, and one may neglect the in-teractions with neutral ism components for the purposeof establishing charge neutrality.Instead, we must consider the charged components.Using the same argument (11) one can see that theCoulomb barrier of the charged nuggets will be highenough to prevent electrons from reaching the electro-sphere in all but the very hot ionized medium ( vhim ), which occupies only a small fraction of the ism in thecore (see [47] for example). Protons however, will beable to reach the core to annihilate with a cross sec-tion ∼ π R (this is not substantially affected by thecharge). Thus, positive charge can be deposited at arate ∼ π R v n + , ism n n + , ism where n n + , ism is the densityof the ionized components in the ism , which is typically10 − to 10 − of the total density n ism ∼ − This rate of positive charge accumulation must matchthe evaporation rate of the positrons from the elec-trosphere (see Eq. (7)), giving the boundary condi-tions (8b), and the resulting charges summarized in Ta-ble I. (These are consistent with the estimates in [18]and the universal upper bound [48].)Note that the rate of evaporation is much less thanthe rate of proton annihilation and carries only ∼ V. DIFFUSE GALACTIC EMISSIONS
Previous discussions of quark (anti)nugget dark matterconsidered the electrosphere in only two limiting cases:the inner high-density ultrarelativistic regime and theouter low-density Boltzmann regime. While this anal-ysis was sufficient to qualitatively discuss the differentcomponents of the spectrum, it did not allow for theircomparison in any level of detail. In particular, com-paring the relative strengths of the 511 keV line emis-sion (emitted entirely from the nonrelativistic regime)with that of the MeV continuum, (emitted from therelativistic regime) required introducing a phenomeno-logical parameter χ ≈ χ is supported bya purely microscopic calculation rooted firmly in qed and well-understood many-body physics. The agree-ment between the calculated value of χ and the phe-nomenological value required to fit the observationsprovides another important and nontrivial test of ourdark-matter proposal. In addition, using the same nu-merical solutions of the previous section, we computethe spectrum in the few MeV region which could not becalculated in [6] with only the ultrarelativistic approxi-mation for the density profile.0 A. Observations
1. The 511 keV Line S pi / integral has detected a strong 511 keV signal fromthe Galactic bulge [49]. A spectral analysis shows thisto be consistent with low-momentum e + e − annihila-tion through a positronium intermediate state: Aboutone quarter of positronium decays emit two 511 keVphotons, while the remaining three-quarters will de-cay to a three photon state producing a continuumbelow 511 keV. Both emissions have been seen by spi / integral with the predicted ratios.The 511 keV line is strongly correlated with the Galac-tic centre with roughly 80% of the observed flux comingfrom a circle of half angle 6 ◦ . There also appears to be asmall asymmetry in the distribution oriented along theGalactic disk [50].The measured flux from the Galactic bulge is foundto be d Φ /d Ω ≃ − s − sr − [51]. Afteraccounting for all known Galactic positronium sourcesthe 511 keV line seems too strong to be explained bystandard astrophysical processes. These processes, aswe understand them, seem incapable of producing asufficient number of low-momentum positrons. Severalprevious attempts have been made to account for thispositron excess. Suggestions have included both modi-fications to the understood spectra of astrophysical ob-jects or positrons which arise as a final state of someform of dark-matter annihilation. At this time there isno conclusive evidence for any of these proposals.If we associate the observed 511 keV line with theannihilation of low momentum positrons in the elec-trosphere of an antiquark nugget, these properties arenaturally explained. The strong peak at the Galacticcentre and extension into the disk must arise becausethe intensity follows the distribution ρ v ( ~ r ) ρ dm ( ~ r ) of vis-ible and dark-matter densities. This profile is unique todark-matter models in which the observed emission isdue to matter–dark matter interactions and should becontrasted with the smoother ∼ ρ dm ( ~ r ) profile for pro-posals based on self-annihilating dark-matter particles,or the ∼ ρ dm ( ~ r ) profile for decaying dark-matter pro-posals. The distribution ρ v ( ~ r ) ρ dm ( ~ r ) obviously impliesthat the predicted emission will be asymmetric, withextension into the disk from the Galactic center as ittracks the visible matter. There appears to be evidencefor an asymmetry of this form [50]. In our proposal,no synchrotron emission will occur as the positrons aresimply an integral part of the nuggets rather than be-ing produced at high energies with only the low-energycomponents exposed for emission. Contrast this withmany other dark matter based proposals where the rel-atively high-energy positrons produced from decayingor annihilating dark-matter particles produce strong synchrotron emission. These emissions are typically inconflict with the strong observational constraints [21].
2. Diffuse MeV scale emission
The other component of the Galactic spectrum we dis-cuss here is the diffuse continuum emission in the 1 –30 MeV range observed by comptelcrgo . The interpre-tation of the spectrum in this range is more complicatedthan that of the 511 keV line as several different astro-physical processes contribute.The conventional explanation of these diffuse emis-sions is that of gamma rays produced by the scatteringof cosmic rays off of the interstellar medium, and whiledetailed studies of cosmic ray processes provide a goodfit to the observational data over a wide energy range(from 20 MeV up to 100 GeV), the predicted spectrumfalls short of observations by roughly a factor of 2 in the1 – 20 MeV range [52].Background subtraction is difficult, however, and es-pecially obscures the spatial distribution of the MeV ex-cess. However, the excess seems to be confined to theinner Galaxy ( l = ◦ – 30 ◦ , | b | = ◦ – 5 ◦ ), [52] with anegligible excess from outside of the Galactic centre.
3. Comparison
As our model predicts both of these components to havea common source, a comparison of these emissions pro-vides a stringent tests of the theory. In particular, themorphologies, spectra, and relative intensities of bothemissions must be strongly related.The inferred spatial distribution of these emissionsis consistent: both are concentrated in the core of theGalaxy. Unfortunately, this is not a very stringent testdue to the poor spatial resolutions of the present ob-servations. The prediction remains firm: If the mor-phology of the observations can be improved, then afull subtraction of known astrophysical sources shouldyield a diffuse MeV continuum with spatial morphol-ogy identical to that of the 511 keV.Present observations, however, do allow us to test theintensity and spectrum predicted by the quark nuggetdark-matter model. The model predicts the intensityto be proportional to the 511 keV flux with calculablecoefficient of proportionality. Therefore, the integral data may be used to fix the total diffuse emission flux,removing the uncertainties associated with the line ofsight averaging which is the same for both emissions.The resulting spectrum and intensity were previouslydiscussed in [24] and [6], but the 511 keV line emissionwas estimated using the low-density Boltzmann approx-imation, while the MeV emissions were estimated using1the high-density ultrarelativistic approximation. Theproportionality factor linking these two was treated as aphenomenological parameter, χ , which required a valueof χ ≈ B. Annihilation rates
As incident electrons enter the electrosphere ofpositrons, the dominant annihilation process is throughpositronium formation with the subsequent annihila-tion producing the 511 keV emission. As the remainingelectrons penetrate more deeply, they will encounter ahigher density of more energetic positrons: Direct anni-hilation eventually becomes the dominant process. Themaximum photon energy, ∼
10 MeV, is determined bythe Fermi energy of the electrosphere at the deepestdepth of electron penetration. We stress that this scale isnot introduced in order to explain the comptel / crgo gamma-ray excess: Our model necessarily produces astrong emission signature at precisely this energy.In the following section we integrate the rates forthese two processes over the complete density profile,allowing us to predict the relative intensities and spec-tral properties of the emissions.
1. Positronium Formation
In principle, we only need to know the e + e − annihila-tion cross section at all centre of mass momenta. Athigh energies, this is perturbative and one can use thestandard qed result for direct e + e − → γ emission. Atlow energies, however, one encounters a strong reso-nance due to the presence of a positronium state thegreatly enhances the emission. This resonance rendersa perturbative treatment invalid and must be dealt withspecially, thus we consider these as two separate pro-cesses. Our presentation here will be abbreviated: de-tails may be found in [24].While the exact positronium formation rate –summed over all excited states – is not well established,it is clear that the rate will fall off rapidly as the centre ofmass momentum moves away from resonance. A sim-ple estimate suggests that for momenta p > m α the for-mation rate falls as ∼ p − . To estimate the rate, we thusmake a cutoff at q ≈ m α as the upper limit for positro-nium formation: for large center of mass momenta weuse the perturbative direct-annihilation approximation.The scale here is set by the Bohr radius a b = ( m α ) − for the positronium bound state: If a low-momentum e + e − pair pass within this distance, then the probabilityof forming a bound state becomes large, with a naturalcross section σ Ps ∼ π a b . The corresponding rate is Γ Ps = Z p . q v σ Ps n ( p ) d p ( π ) ∼ vm α ( n ( p ) π p F . q q π p F & q .(12)where n ( p ) is the momentum distribution of thepositrons in the electrosphere, and v ∼ α is the incom-ing electron velocity. The second expression representsthe two limits (valid at low temperature): a) of low den-sity when the Fermi momentum p F . q and all statesparticipate, resulting in a factor of the total density n ,and b) of high density where the integral is saturatedby q /3 π ≈ ( m α ) /3 π .As discussed in Sec. IV C, and in more detail in Ap-pendix B, electrons will be able to penetrate the chargedantimatter nuggets in spite of the strong electric field.Initially the electrons are bound in neutral atoms. Onceionized, the density is sufficiently high that the chargeis efficiently screened with a Debye screening length λ D that is much smaller than the typical de Broglie wave-length λ = ¯ h / p of electrons. Thus, the electric fields– although quite strong in the nugget’s electrosphere– will not appreciably effect the motion. The bindingof electrons in neutral atoms complicates the analysisslightly, but the binding energy – on the eV scale – willnot significantly alter the qualitative nature of our esti-mates. For a precision test of the emission properties,this will need to be accounted for. Here we includebands comprising ±
10% relative variation in the over-all positronium annihilation rate to show the sensitivityto this uncertainty.
2. Direct-annihilation Rates
While positronium formation is strongly favoured atlow densities due to its resonance nature, deep withinthe electrosphere the rapidly growing density of statesat large momentum values result in a cross sectioncharacterized by the perturbative direct-annihilationprocess. Conceptually one can imagine an incidentGalactic electron first moving through a Fermi gas ofpositrons with roughly atomic density with a relativelylarge probability of annihilation through positroniumto 511 keV photons. A small fraction survive to pen-etrate to the inner high0density region where directannihilation dominates. The surviving electrons thenannihilate with a high-energy positron near to the in-ner quark-matter surface releasing high-energy pho-tons. The spectrum of these annihilation events willbe a broad continuum with an upper cutoff at an en-ergy scale set by the Fermi energy of the positron gas at2
FIG. 3. Electron survival fraction, n e ( r ) / n ∞ (16), of an in-coming electron with velocities v = c [leftmost light gray(red) band], v = c [middle (green) band] and v = c [rightmost dark gray (blue) band] from left to right, respec-tively. The thickness of the bands includes a ±
10% variationin the positronium annihilation rate (12). The local Fermi mo-mentum p F is shown along the top and with vertical dottedlines including the cutoff scale q ≈ the maximum penetration depth of the electrons. Thespectral density for direct e + e − annihilation at a givenchemical potential was calculated in [6]:d I ( ω , µ ) d ω d t = Z d n p ( µ ) v p ( µ ) d σ ( p , ω ) d ω (13) = Z d p ( π ) + e ( µ − E p ) / T pE p d σ ( p , ω ) d ω d σ ( p , ω ) d ω = πα mp " − ( m + E p )( m + E p )( m + E p − ω ) − + ω ( m + E p )( m + E p ) − ( m ω ) ( m + E p ) ( m + E p − ω ) ,where E p = p p + m is the energy of the positron inthe rest frame of the incident (slow-moving) electronand ω is the energy of the produced photons. The anni-hilation rate at a given density, Γ dir ( n µ ) , is obtained byintegrating over allowed final state photon momentum.This was previously done in the T → C. Spectrum and Branching Fraction
To determine the full annihilation spectrum we first de-termine the fraction of incident electrons that can pene- trate to a given radius r in the electrosphere (see Fig. 3.We then integrate the emissions over all regions.Consider an incident beam of electrons with density n ∞ and velocity v . As they enter the electrosphere ofpositrons, the electrons will annihilate. The survivalfraction n e ( r ) / n ∞ will thus decrease with a rate pro-portional to Γ ( r ) = Γ Ps ( r ) + Γ dir ( r ) which depends cru-cially on the local density profile n [ µ ( r )] calculated inSec. IV B:d n e ( r ) d t = v − d n e ( r ) d r = − Γ ( r ) v − n e ( r ) . (15)Integrating (15), we obtain the survival fraction: n e ( r ) n ∞ = exp (cid:18) − Z ∞ r d r v − Γ ( r ) (cid:19) . (16)This is shown in Fig. 3. One can clearly see that in theouter electrosphere, positronium formation – indepen-dent of v – dominates the annihilation. Once the densityis sufficiently high ( p F & v of the incident particle.The initial velocity v ∼ − c is determined by thelocal relative velocity of the nuggets with the surround-ing ism . This will depend on the temperature of the ism , but the positronium annihilation rate is insensitiveto this. As we mentioned previously, most of the elec-trons in the interstellar medium are bound in neutralatoms, either as neutral hydrogen HI, or in molecularform H . (The ionized hydrogen HII represents a verysmall mass fraction of interstellar medium.) These neu-tral atoms and molecules will have no difficulty enter-ing the electrosphere.The remaining bound electrons that do not annihilatethrough positronium will ionize once they reach denserregions, and will acquire a new velocity set by a combi-nation of the initial velocity v ∼ − c and the atomicvelocity v ∼ α ∼ − c imparted to the electrons asthey are ionized from the neutral atoms: The latter willtypically dominate the velocity scale. This only occursin sufficiently dense regions where the Debye screen-ing discussed in Appendix B becomes efficient. Hence,the electric fields will not significantly alter the motionof the electrons after ionization. The direct-annihilationprocess depends on the final velocity v ; as the dominantcontribution comes from ionization, this will remain rel-atively insensitive to the ism . During ionization, somefraction (roughly half) of the electrons will move awayfrom the core, but a significant portion will travel withthis velocity v toward the denser regions.Two other features of Fig. 3 should be noted. Firstis the value of the survival fraction χ ∼ p F = µ R ∼
100 MeV or so – virtually no Galactic electronspenetrate deeply enough to annihilate at these energies.For nonrelativistic electrons the maximum energy scalefor emission is quite generally set at the ∼
20 MeV scale,depending slightly on v .Having established the survival fraction as a functionof height it is now possible to work out the spectraldensity that will arise from the e + e − annihilations. Ata given height we can express the number of annihila-tions through a particular channel as,d n d r = n e ( r ) v − Γ ( r ) . (17)Integrating this expression over all heights will thengive the total fraction of annihilation events proceedingvia positronium f , and via direct annihilation f MeV .(The numerical values are given for q = m α and arequite insensitive to v .) f = Z ∞ R d r n e ( r ) n ∞ v − Γ Ps ( r ) ≈ f MeV = Z ∞ R d r n e ( r ) n ∞ v − Γ dir ( r ) ≈ vn ∞ . As such, the ratio of 511 keVphotons to MeV continuum emission is independent ofthe relative densities (though it will show some depen-dence on the local electron velocity distribution). Nu-merically, with a ±
10% variation in the positronium an-nihilation rate, we find χ = f MeV f ≈ − v . This ratiowas introduced as a purely phenomenological param-eter in [6] to explain the observations. Here we havecalculated from purely microscopic considerations that,for a wide range of nugget parameters, the requiredvalue χ ∼ v − ):d I total d ω = Z ∞ R d r v − n e ( r ) n ∞ d I ( µ ( r )) d ω d t . (20) FIG. 4. Spectral density (scaled by ω to compare with [52]of photons emitted by an electron annihilating on antiquarknuggets with incoming velocities v = c [uppermost (red)band], v = c [middle (green) band] and v = c [low-est (blue) band to the lower left] from right to left, respec-tively, including the cosmic ray background determined in[52] (dotted line). The thickness of the bands includes a ± comptel data points. Note: This spectrum should still beinterpreted as a qualitative effect - a detailed calculation ofthe ionization and hence distribution of the velocity v mustbe performed to yield a quantitative prediction. The generalstructure and magnitude, however, can be trusted as these de-pend on the overall density profile which we have carefullymodeled. (Compare with Fig. 5, for example, which uses onlythe ultrarelativistic density profile: The resulting intensity is2 orders of magnitude too large.) The resulting spectrum is shown in Fig. 4 and is sensi-tive to both the incoming velocity v (which will dependon the local environment of the antimatter nugget) andthe overall normalization of the positronium annihila-tion rate (12). (The latter is fixed in principle, but re-quires a difficult in-medium calculation to determineprecisely.) These two parameters are rather orthogonal.The velocity v determines the maximum depth of pene-tration, and hence the maximum energy of the emittedphotons (as set by the highest chemical potential at theannihilation point): If the electron velocity is relativelylow ( v <
100 km/s ≈ c ), almost all annihilationshappen immediately and the MeV continuum will fallrapidly beyond 5 MeV. As the velocity increases the elec-trons are able to penetrate deeper toward the quark sur-face and annihilate with larger energies. In contrast, thedetails of the positronium annihilation do not alter thespectral shape, but do alter the overall normalization.As already mentioned earlier, the parameter χ ∼ ∆ ω ≃ ω is due to the increasing density of statesas a function of depth. Above ∼
10 MeV, the emissionbecomes suppressed by the inability of Galactic elec-trons to penetrate to depths where the positrons havethis energy. While the exact details may vary with pa-rameters, such as the local electron velocity and preciserate of positronium formation, the general spectral fea-tures are inescapable consequences of our model, allow-ing it to be tested by future, more precise, observations.
D. Normalization to the 511 keV line
To obtain the observed spectrum, one needs to aver-age (20) along the line of site over the varying matterand dark-matter density and velocity distributions. Nei-ther of these is known very well, so to check whether ornot the prediction is significant, we fix the average rateof electron annihilation with the observed 511 keV linewhich our model predicts to be produced by the sameprocess. As the intensity of the 511 keV line (resultingfrom the two-photon decay of positronium in the S state) has been measured by spi / integral along theline of sight toward the core of the Galaxy (see [53] fora review), we can use this to fix the normalization of theMeV spectrum along the same line of sight. The totalintensity is given in terms of Eq. (18a):d Φ d Ω = C f ∼ s sr , (21)where the factor of 4 accounts for the three-quartersof the annihilation events that decay via the S chan-nel (also measured, but not included in the line emis-sion). This fixes the normalization constant C ≈ − s − sr − . The predicted contributionto the MeV continuum must have the same morphol-ogy and, thus, an integrated intensity along the sameline of sight must have the same normalization factor.This normalization has been used in Fig. 4 to comparethe predicted spectrum with the unaccounted for excessemission detected by comptel ([52]).The exact shape of the spectrum will be an averageof the components shown over the velocity distributionof the incident matter. Our process of normalization isunable to remove this ambiguity because the predicted 511 keV spectral properties are insensitive to this. It isevident, however, that MeV emission from dark-matternuggets could easily provide a substantial contributionto the observed MeV excess. Note also, that the ex-cess emission can extend only to 20 MeV or so. This iscompletely consistent with the more sophisticated back-ground estimates discussed in [54] which can fit almostall aspects of the observed spectrum except the excessbetween 1 and 30 MeV predicted by our proposal. VI. CONCLUSION
Solving the relativistic Thomas-Fermi equations, we de-termined the charge and structure of the positron elec-trosphere of quark antimatter nuggets that we postu-late could comprise the missing dark-matter in our Uni-verse. We found the structure of the electrosphere tobe insensitive to the size of the nuggets, as long as theyare large enough to be consistent with current terrestrialbased detector limits, and hence, can make unambigu-ous predictions about electron annihilation processes.To test the dark-matter postulate further, we usedthe structure of this electrosphere to calculate the an-nihilation spectrum for incident electronic matter. Themodel predicts two distinct components: a 511 keVemission line from decay through a positronium inter-mediate and an MeV continuum emission from direct-annihilation processes deep within the electrosphere.By fixing the general normalization to the measured 511keV line intensity seen from the core of the Galaxy, ourmodel makes a definite prediction about the intensityand spectrum of the MeV continuum spectrum withoutany additional adjustable model parameters: Our pre-dictions are based on well-established physics.As discussed in [30], a difficulty with most otherdark-matter explanations for the 511 keV emission isto explain the large ∼ A priori , there is no reason to expect that the predictedMeV spectrum should correspond to observations: typi-cally two uncorrelated emissions are separated by manyorders of magnitude. We find that the phenomenolog-ical parameter χ (19) required to explain the relativenormalization of MeV emissions arises naturally fromour microscopic calculation. This is highly nontriv-ial because it requires a delicate balance between thetwo annihilation processes from the semirelativistic re-gion of densities that is sensitive to the semirelativis-tic self-consistent structure of the electrosphere outsidethe range of validity of the analytic ultrarelativistic andnonrelativistic regimes. (See Fig. 1 and 3).If the predicted emission were several orders of mag-5nitude too large, the observations would have ruledout our proposal. If the predicted emissions were toosmall, the proposal would not have been ruled out, butwould have been much less interesting. Instead, weare left with the intriguing possibility that both the 511keV spectrum and much of the MeV continuum emis-sion arise from the annihilation of electrons on dark-antimatter nuggets. While not a smoking gun – at leastuntil the density and velocity distributions of matterand dark-matter are much better understood – this pro-vides another highly nontrivial test of the proposal that, a priori , could have ruled it out.Both the formal calculations and the resulting struc-ture presented here – spanning density regimes from ul-trarelativistic to nonrelativistic – are similar to those rel-evant to electrospheres surrounding strange-quark starsshould they exist. Therefore, our results may proveuseful for studying quark star physics. In particular,problems such as bremsstrahlung emission from quarkstars originally analyzed in [55] (and corrected in [56])that uses only ultrarelativistic profile functions. The re-sults of this work can be used to generalize the corre-sponding analysis for the entire range of allowed tem-peratures and chemical potentials. Another problemwhich can be analyzed using the results of the presentwork is the study of the emission of energetic electronsproduced from the interior of quark stars. As advo-cated in [57], these electrons may be responsible forneutron star kicks, helical and toroidal magnetic fields,and other important properties that are observed in anumber of pulsars, but are presently unexplained.Finally, we would like to emphasize that this mech-anism demonstrates that dark matter may arise from within the standard model at the qcd scale, and thatexotic new physics is not required. Indeed, this is nat-urally suggested by the “cosmic coincidence” of almostequal amounts of dark and visible contributions to thetotal density Ω tot = ( ) of our Universe [16]: Ω dark-energy : Ω dark-matter : Ω visible ≈
17 : 5 : 1.The dominant baryon contribution to the visible portion Ω visible ≈ Ω B has an obvious relation to qcd throughthe nucleon mass m n ∝ Λ qcd (the actual quark massesarising from the Higgs mechanism contribute only asmall fraction to m n ). Thus, a qcd origin for the darkcomponents would provide a natural solution to theextraordinary “fine-tuning” problem typically requiredby exotic high-energy physics proposals. Our proposalhere solves the matter portion of this coincidence. Fora proposal addressing the energy coincidence we referthe reader to [58] and references therein. The axion is another dark-matter candidate arising from the qcd scale, but with fewer observational consequences. The idea concerns the anomaly that solves the famous axial U ( ) A ACKNOWLEDGMENTS
M.M.F. would like to thank George Bertsch, Au-rel Bulgac, Sanjay Reddy, and Rishi Sharma for use-ful discussions, and was supported by the LDRD pro-gram at Los Alamos National Laboratory, and the U.S.Department of Energy under grant number DE-FG02-00ER41132. K.L and A.R.Z. were supported in part bythe Natural Sciences and Engineering Research Councilof Canada.
Note in proof:
While this paper was in preparation, areview [62] was submitted which mentions the mech-anism discussed here, but dismisses it based on thearguments of [45]. This work makes it clear that theassumptions used in [45] are invalid (see footnote 7)and addresses the issues raised therein. The argu-ments of [45] were previously addressed in [5] (at theend of appendix 1) where it was emphasized that [45]incorrectly applies a relativistic formula to the non-relativistic regime.
Appendix A: Density Functional Theory
We start with a Density Functional Theory ( dft ) formu-lated in terms of the thermodynamic potential: Ω = ∑ i f i Z d ~ r ψ † i ( ~ r )( ǫ − i ¯ h ~ ∇ − µ ) ψ i ( ~ r )++ Z d ~ r V ext ( ~ r ) n ( ~ r ) + Z d ~ r ε xc h n ( ~ r ) i n ( ~ r )++ π e Z d ~ r d ~ r ′ n ( ~ r ) n ( ~ r ′ ) k ~ r − ~ r ′ k − T ∑ i f i ln f i , (A1a)where ǫ p = p p + m is the relativistic energy of theelectrons, ε xc ( n ) is the exchange energy. The density is n ( ~ r ) = ∑ i f i ψ † i ( ~ r ) ψ i ( ~ r ) , (A1b)where we have explicitly included the spin degeneracy,and used relativistic units where ¯ h = c = e = α .We vary the potential with respect to the occupationnumbers f i and the wave functions ψ i subject to the con-straints that the wave functions be normalized, Z d ~ r ψ † i ( ~ r ) ψ j ( ~ r ) = δ ij , (A1c) problem, giving rise to an η ′ mass that remains finite, even in thechiral limit. Under some plausible and testable assumptions aboutthe topology of our Universe, the anomaly demands that the cosmo-logical vacuum energy depend on the Hubble constant H and qcd parameters as ρ de ∼ Hm q h ¯ qq i / m η ′ ≈ ( × − eV ) – tantalisinglyclose to the value ρ de = [ ( ) × − eV ] observed today [16]. This yields the Kohn-Sham equations forthe electronic wave functions: h ǫ ( − i ¯ h ~ ∇ ) − µ + V eff ( ~ r ) i ψ i ( ~ r ) = E i ψ i ( ~ r ) , f i = + e E i / T .where (we take e to be positive here) V eff ( ~ r ) = e φ ( ~ r ) + V ext ( ~ r ) + ε xc [ n ( ~ r )] + n ( ~ r ) ε ′ xc [ n ( ~ r )] and φ ( ~ r ) = π e Z d ~ r ′ n ( ~ r + ~ r ′ ) k ~ r k is the electrostatic potential obeying Poisson’s equation, ∇ φ ( ~ r ) = − π en ( ~ r ) .
1. Thomas-Fermi Approximation.
If the effective potential V eff ( ~ r ) varies sufficientlyslowly, then it is a good approximation to replace itlocally with a constant potential. The Kohn-Sham equa-tions thus become diagonal in momentum space, i ≡ k , ψ k ∝ e ikr and may be explicitly solved: E ~ k ( ~ r ) = ǫ ~ k − µ + V eff ( ~ r ) , n ( ~ r ) = Z d ~ k ( π ) (cid:18) + e E ~ k ( ~ r ) / T − + e − E ~ k ( ~ r ) / T (cid:19) ,where we have also explicitly included the contribu-tion from the antiparticles. We piece these homoge-neous solutions together at each point ~ r and find theself-consistent solution that satisfies Poisson’s equation.This approximation is a relativistic generalization of theThomas-Fermi approximation.In principle, the dft method is exact [59], however,the correct form for ε xc is not known and could beextremely complicated. Various successful approxima-tions exist but for our purposes we may simply ne-glect this. The weak electromagnetic coupling constant α ∼ α n ∼ α µ /3 π & m , which corresponds to µ &
50 MeV.Near the nugget core, many-body correlations maybecome quantitatively important. For example, the ef-fective mass of the electrons in the gas is increased This is most easily realized by introducing the single-body densitymatrix ρ = − Cρ T C where C is the charge-conjugation matrix. by about 20% when µ ≈
25 MeV and doubles when µ ≈
100 MeV (see, for example, [60]). These effects,however, will not change the qualitative structure, andcan be quite easily taken into account if higher accuracyis required. We also note that, formally, the Thomas-Fermi approximation is only valid for sufficiently highdensities. However, it gives correct energies within fac-tors of order unity for small nuclei [44] and is known towork substantially better for large nuclei [61]. As longas we do not attempt to use it in the extremely low-density tails, it should give an accurate description.
2. Analytic Solutions
There are several analytic solutions available if we con-sider the one-dimensional approximation, neglectingthe curvature term 2 µ ′ ( r ) / r in (5), which is valid closeto the nugget where z = r − R ≪ R , the distance fromthe nugget core, is less than the radius of the nugget. a. Ultrarelativistic Regime The first is the ultrarelativistic approximation where µ and/or T are much larger than m and the limit m → n [ µ , T ] m ≪ µ , T = µ π + µ T z ur : µ ( z , T ) = T π √ h T q απ ( z + z T ) i , (A6) z ur = T r απ sinh − T π √ µ R ! . (A7) In this limit, then integrals have a closed form: n [ µ , T ] m ≪ µ , T == π Z ∞ d p (cid:20) p + e ( p − µ ) / T − p + e ( p + µ ) / T (cid:21) == T π Z ∞ d x (cid:20) x + e x − µ / T − x + e x + µ / T (cid:21) == − T π Γ ( ) h Li ( − e µ / T ) − Li ( − e − µ / T ) i (A3)where Li s ( z ) is the PolylogarithmLi s ( z ) = ∞ ∑ k = z k k s . (A4) T ∼ ≪ m so we can also take the T → n ur ( z ) ≈ µ R π ( + z / z ur ) , z ur ≈ µ − R r π α . (A8)This solution persists until µ ≈ m , which occurs at adistance z B ≈ z ur (cid:16) µ R m − (cid:17) ≈ m − r π α . (A9) b. Boltzmann Regime Once the chemical potential is small enough thate µ / T ≪ e m / T , we may neglect the degeneracy in thesystem, and write n [ µ ] | exp ( µ / T ) ≪ exp ( m / T ) ≈ n e µ / T . (A10)This occurs for a density of about n B ≈ (cid:18) mT π (cid:19) (A11)and lower. If the one-dimensional approximation is stillvalid, then another analytic solution may be found: n ( z ) = n B (cid:16) + z − z B z α (cid:17) , z α = s T πα n B (A12)from which we obtained (10). The shift z B must be de-termined from the numerical profile at the point where n ( z B ) = n B . Note that z α ≪ z B , so this regime is validand persists until z ∼ R , at which point the one-dimensional approximation breaks down. It is in thisBoltzmann regime where most of the important radia-tive processes take place [5].
3. Numerical Solutions
The main technical challenge in finding the numericalsolution is to deal effectively with the large range ofscales: T ∼ ≪ m ∼
500 keV ≪ µ R ∼
25 MeV. Forexample, the density distribution (4b) is prone to round-off error, but the integral can easily be rearranged togive a form that is manifestly positive: n [ µ ] = π Z ∞ d p p sinh (cid:0) µ T (cid:1) cosh (cid:18) √ p + m T (cid:19) + cosh (cid:0) µ T (cid:1) . (A13)The derivative may also be safely computed. Let A = p p + m T , B = µ T , (A14) to simplify the expressions. These can each be com-puted without any round-off error. The first derivativepresents no further difficulties:˙ n [ µ ] = T π Z ∞ d p p + cosh A cosh B ( cosh A + cosh B ) . (A15)The differential equation is numerically simplified if wechange variables to logarithmic quantities. We wouldalso like to capture the relevant physical characteristicsof the solution, known from the asymptotic regimes.Close to the nugget, we have µ = µ R / ( + z / z ) , weintroduce an abscissa logarithmic in the denominator a = ln (cid:18) + zz (cid:19) = ln (cid:18) + r − Rz (cid:19) . (A16)The dependent variable should be logarithmic in thechemical potential, so we introduce b = − ln ( µ / µ ) . Wethus introduce the following change of variables: r = R + z ( e a − ) , µ = µ e − b , (A17a) a = ln (cid:18) + r − Rz (cid:19) , b = − ln µµ . (A17b)With the appropriate choice of scales z describing thetypical length scale at the wall r = R and µ ∼ µ R , theseform quite a smooth parametrization. The resulting sys-tem is¨ b ( a ) = ˙ b ( a ) (cid:18) ˙ b ( a ) + R − z − z e a R − z + z e a (cid:19) + − πα z e a n [ µ ] µ . (A18)It is imperative to include a full numerical solutionto the profile in order to obtain the proper emissionspectrum. Using only the ultrarelativistic approxima-tion (A8) produces a spectrum (Fig. 5) 2 orders of mag-nitude too large, in direct contradiction with the obser-vations [52]. The actual prediction depends sensitivelyon a subtle – but completely model-independent – bal-ance between the ultrarelativistic, relativistic, and non-relativistic regimes. The consistency between the pre-dicted spectrum shown in Fig 4 and the observations isa highly nontrivial test of the theory. Appendix B: Debye Screening In the Electrosphere
Here we briefly discuss the plasma properties insideof the nugget’s electrosphere. The main point is thatthe Debye screening length λ D is much smaller thanthe typical de Broglie wavelength λ = ¯ h / p of elec-trons. Thus, the electric fields – although quite strong inthe nugget’s electrosphere – will not appreciably effect8 FIG. 5. Incorrect spectral density obtained by using a purelyultrarelativistic approximation for the profile (A8). Note thatthis is two orders of magnitude larger than the spectrum ob-tained from the full numerical solution (Fig. 4) and, if cor-rect, would have easily ruled out our proposal. This servesto demonstrate the highly nontrivial nature of the predictedspectrum shown in Fig. 4 that is consistent with the observa-tions [52]. the motion of electrons within the electrosphere as thecharge is almost completely screened on a scale ∼ λ D .This screening effect was completely neglected in [45],which led the authors to erroneously conclude thatall electrons will be repelled before direct annihilationcan proceed. The overall charge (see Table I) will re-pel incident electrons at long distances as discussed inSec. IV C, but neutral hydrogen will easily penetrate theelectrosphere, at which point the screening becomes ef-fective, allowing the electrons to penetrate deeply andannihilate as discussed in Sec. V.To estimate the screening of an electron, we solve thePoisson equation with a δ ( ~ r − ~ r ) function describingthe electron at position ~ r (see also 2), ∇ φ ( ~ r ) = − π e h n ( ~ r ) − δ ( ~ r − ~ r ) i (B1)where φ ( ~ r ) is the electrostatic potential and n ( ~ r ) isthe density of positrons. As before, we exchange φ for µ , the effective chemical potential (3). Now let n ( ~ r ) and µ ( ~ r ) be the solutions discussed in Sec. IV Bwithout the potential δ ( ~ r − ~ r ) . We may then de-scribe the screening cloud by µ ( ~ r ) = µ ( ~ r ) + δ µ ( ~ r ) and n ( ~ r ) = n ( ~ r ) + δ n ( ~ r ) where ∇ δ µ ( ~ r ) = π e h δ n ( ~ r ) − δ ( ~ r − ~ r ) i ≈ πα (cid:20) ∂ n [ µ ] ∂µ δ µ ( ~ r ) − δ ( ~ r − ~ r ) (cid:21) (B2)where n [ µ ] is given by (4b). If the density is sufficientlylarge compared to the screening cloud deviations, we may take ∂ n / ∂µ to be a constant, in which case we maysolve (B2) analytically with the boundary conditions:lim ~ r → ~ r δ µ ( ~ r ) = πα k ~ r − ~ r k , lim ~ r → ∞ δ µ ( ~ r ) =
0. (B3)This gives the standard Debye screening solution δ µ ( ~ r ) = − πα k ~ r − ~ r k exp (cid:18) − k ~ r − ~ r k λ D (cid:19) (B4)where the Debye screening length is λ − D = πα ∂ n [ µ ] ∂µ . (B5)On distances larger than λ D , the charge of the electronis effectively screened. In particular, we can neglect theinfluence of the external electric field on motion of theelectron if λ D is small compared to the de Broglie wave-length λ = ¯ h / p of the electrons:1 ≪ λλ D ∼ mv s πα ∂ n [ µ ] ∂µ . (B6)In the ultrarelativistic limit (A5), one has ∂ n / ∂µ ≈ µ / π + T /3 whereas in the Boltzmann limit (A10),one has ∂ n / ∂µ ≈ n / T .The relevant electron velocity scale in our problemis v ∼ − c ( T ∼ n ≫ m v T πα ∼ n B (B7) n B = ( mT /2 π ) is the typical density in the Boltz-mann regime. The typical electric fields in this regimeare E ∼ ∇ φ ∼ eT /2 z α , which yield an ionization po-tential of V ionize ∼ eEa ∼ α Ta /2 z α ∼ ≪
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