Quark Matter Induced Extensive Air Showers
QQuark Matter Induced Extensive Air Showers
Kyle Lawson
Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada ∗ (Dated: November 3, 2018)If the dark matter of our galaxy is composed of nuggets of quarks or antiquarks in a coloursuperconducting phase there will be a small but non-zero flux of these objects through the Earth’satmosphere. A nugget of quark matter will deposit only a small fraction of its kinetic energy inthe atmosphere and is likely to be undetectable. If however the impacting object is composed ofantiquarks the energy deposited can be quite large. In this case nuclear annihilations within thenugget will trigger an extensive air shower the particle content of which is similar to that producedby an ultrahigh energy cosmic ray. This paper gives a qualitative description of the basic propertiesof such a shower. Several distinctions from an air shower initiated by a single ultra high energynucleus will be described allowing these events to be distinguished from the cosmic ray background.The subtlety of these features may mean that some fraction of the high energy cosmic ray spectrummay in fact be due to this type of dark matter interaction.The estimated flux of dark matter nuggets and the energy deposited in the atmosphere are suchthat the Pierre Auger Observatory may prove an ideal facility to place constraints on the flux ofheavy quark matter objects. This paper attempts to highlight the best techniques to search for aquark matter signature through an extensive air shower signal. I. INTRODUCTIONA. quark matter as a dark matter candidate
It has been suggested that the dark matter may becomposed of macroscopically large, strongly interacting,composite objects comprised of the light quarks of thestandard model in a non-baryonic phase such as strangequark matter [1] or a colour superconducting phase [2].In the latter case the composite objects may be boundstates of either quarks or antiquarks which are stableover cosmological time scales. While strongly interact-ing these objects remain “dark” due to their large massto surface area ratio and the correspondingly low num-ber density required to explain the observed dark mattermass density. The total baryonic charge of the compos-ite object is the dominant uncertainty in this model as itdepends on the poorly understood physics of nugget for-mation (which occurs at the QCD phase transition.) Acombination of theoretical and observational constraintssuggest that the mean baryonic charge must exceed 10 [2]while the upper bound is dependent on the formationmodel and is not well constrained.A brief qualitative overview of the structure of a quarknugget is given in appendix A. Further, more precise,details of various phases of quark matter are available inthe reference given there.Previous works have studied the observational con-sequences of the presence of quark matter within thegalaxy. No contradictions are found with existing ob-servations, in fact the emission produced by these ob-jects may help to explain several anomalies in the galac-tic spectrum such as the strong 511keV line [3], [4], [5] ∗ E-mail: [email protected] the COMPTEL excess at 10 MeV [6], [7], the diffuse x-ray background [8], [9] and the WMAP “haze” [10], [11].Based on the simplest models of the dark matter distribu-tion and nugget interaction with the interstellar mediuma best fit to the galactic spectrum in this analysis is foundto favor a baryonic charge for the nuggets of B ∼ . B. high energy “cosmic rays” from quark matter
The cosmic ray spectrum is now observed to extend toenergies above 10 eV [12]. The incredibly small flux ofcosmic rays at these energies requires a correspondinglylarge detector to obtain useful statistics for these events.The aim of this work is to highlight the possibility thatthese detectors can also impose significant constraints onmassive composite dark matter candidates. Compositeobjects composed purely of matter will deposit only afraction of their kinetic energy in the atmosphere. Thesmall energy scales involved do not allow for substantialparticle generation and make direct detection unlikely.However, in the case of a nugget composed of antimatterthe dominant interactions between the atmosphere andantiquark matter will be strong force mediated matter-antimatter annihilations. The hadronic shower result-ing from these annihilations will be dominated by lightmesons and their decay products. The energy depositedby such an event will be considerably larger than thenugget’s kinetic energy and the resulting shower shouldbe readily observable. As in the case of a single ultra-high energy proton or ion a quark nugget impacting theearth’s atmosphere will be observable through the exten-sive air shower which develops around the primary par-ticle. However, in the model considered here the showeris driven not by the kinetic energy of the primary but bythe energy released in matter antimatter-annihilations.This makes these events fundamentally different than the a r X i v : . [ a s t r o - ph . H E ] M a y previously considered cases of highly accelerated dust orstrangelets [13]. Existing models of cosmic rays requirean accelerator capable of providing sufficient kinetic en-ergy for the primary particle to trigger an extensive airshower, in the present case no such accelerator is requiredas the shower is driven by energy released in nuclear an-nihilations. This allows a large air shower to developdespite the fact that the primary particle has a relativelysmall (galactic scale) velocity.This paper gives an overview of the process by which aquark nugget deposits energy in the atmosphere and theproperties of the resulting extensive air shower. As inthe case of an air shower initiated by a single ultrahighenergy cosmic ray these quark matter induced showersarise through a very large number of hadronic interac-tions which necessarily cascade down to similar final stateproducts. As such the particle content of the shower, asobserved at the earth’s surface will be quite similar tothat of a conventional shower. A detailed description ofthe resulting air shower would require large scale numer-ical simulations (similar to those conducted for proton ornuclei initiated showers) which are beyond the scope ofthis work. In an attempt to keep the physical picture asclear as possible the body of this work focuses on only themost essential features of the shower rather than micro-scopic details which may be strongly dependent on theprecise structure of the strong interactions at large den-sities. While a quark matter initiated shower is in manyways similar to a cosmic ray air shower there are alsoseveral critical differences in both the geometry and thetimescales involved. The final section of this work high-lights these differences and discusses potential techniquesfor the detection of quark matter induced air showers. II. TOTAL FLUX
The exact distribution of dark matter in the galaxyremains uncertain. Recent simulations indicate the pos-sibility of significant structure at subgalactic scales [14]which could significantly affect the flux of dark matterthrough the earth. In the interest of simplicity the fol-lowing analysis assumes a local density consistent witha smooth density profile and a velocity set by virialequilibrium. Under these assumptions the dark mat-ter density in the neighborhood of our solar system is ρ DM ≈ . . Assuming that the effective massof quarks in a colour superconductor is comparable tothat of hadronic quarks this mass density translates toa number density of nuggets approximately given by, n ∼ B − cm − . Where B is the total baryon numberof the nuggets. The number density can then be com-bined with the mean galactic velocity v g ∼ km/s toobtain a flux of nuggets at the earth’s surface. dNdA dt = nv g ≈ (10 km − yr − ) B − (1) Based on this order of magnitude estimation nuggets witha baryonic charge distribution near that favoured by fitsto the galactic spectrum will produce a flux compara-ble to that of cosmic rays near the GZK limit [15], [16].It is precisely this flux range that the Pierre Auger Ob-servatory [17] was designed to study and, consequently,it is also capable of constraining the presence of heavyquark matter in the cosmic ray spectrum. One mightalso consider looking for a quark nugget signal at largeunderground detectors however, as discussed in appendixC the larger surface area presented by Auger allows it toimpose much tighter constraints. III. ENERGETICS
This section gives an overview of the energy consid-erations related to a quark nugget induced air showerwithout focussing on the details of how this energy is de-posited in the atmosphere. While an antiquark nuggetcontains a large amount of antimatter very little of itactually annihilates as the nugget traverses the atmo-sphere. Instead the annihilation rate is limited by therate at which the nugget sweeps up atmospheric mat-ter which is dependent on the cross sectional area of thenugget and the atmospheric density. At the earth’s sur-face the integrated mass of atmospheric molecules is onthe order of 1 kg/cm while the nugget radius is generallyfound to be on the order of 10 − cm . For these values, ifall the atmospheric molecules striking the nugget annihi-late completely, the energy produced while crossing theatmosphere is,∆ E = 2 X at πR n = 10 eV (cid:18) R n − cm (cid:19) (2)This represents the total energy production from annihi-lations. The majority of this energy is thermalized withinthe nugget and will not take a readily observable form.It will also be shown that only a fraction of all moleculesincident on the nugget actually annihilate. Thus, the ex-pression (2) represents a maximum energy available tothe shower with the actual value likely to be several or-ders of magnitude smaller.For comparison the kinetic energy transferred to theatmosphere can be estimated by assuming that allmolecules in the atmosphere are accelerated from restto the typical nugget velocity of 200 km/s .∆ T = 12 X at πR n v n = 10 eV (cid:18) R n − cm (cid:19) (3)This is many orders of magnitude below the energy pro-duced by annihilations and represents only a minusculefraction of the total energy involved. Kinetic energytransfer may accelerate a large number of atmosphericmolecules but will be a purely elastic process producingneither new particles nor significant amounts of ioniza-tion. For this reason the following discussion will dealwith only the shower produced by antimatter nuggetsand the energy transferred by inelastic collisions will beignored. IV. SHOWER COMPONENTS
As stated above the quark matter induced shower willprimarily arise from the annihilation of atomic nucleiwithin the nugget. The main product of these annihila-tions will be light mesons (the exact composition of thesemesons depends on the form of quark matter realized inthe nuggets [18].) Given the relatively low momenta atwhich they are produced these strongly interacting modesare unlikely to escape across the quark matter surface.Instead, through a complex series of interactions, theywill loose energy to the lighter modes of the supercon-ductor. This process results in a collection of excitedelectromagnetically bound modes as well as thermalizingenergy within the nugget. The following sections give abrief overview of the particle content generated in theseinteractions.
A. electromagnetic shower
There are three primary mechanisms which will resultin the emission of energetic photons from the nugget.First annihilations within the nugget cascade from theinitial mesons down to the leptonic modes. As the light-est available energy carriers the positrons within thequark matter absorb the majority of this momentum. Apositron incident on the quark matter surface from withinthe nugget will rapidly decelerate within the strong elec-tric fields at the surface and remain bound to the nugget.This process leads to the emission of x-rays throughbremsstrahlung. A second radiation production mech-anism involves energetic electrons produced inside thenugget which annihilate with the positrons of the electro-sphere. These annihilations, as well as annihilations ofthe electrons of atmospheric molecules, produce gammarays with energies up to a few tens of MeV which will bereleased into the atmosphere. A final photon contribu-tion comes from thermal emission from the surface of theelectrosphere. As the nugget heats up due to the increas-ing rate of annihilations the surface can reach tempera-tures at the keV scale. This will result in the emissionof considerable amounts of thermal radiation. These en-ergetic photon components of the nugget emission spec-trum will generate an electromagnetic shower as the ion-ize the surrounding atmospheric molecules.
B. muons
As mentioned above the electrons and positrons pro-duced in the nugget are unlikely to be able to escape intothe atmosphere. Muons, because of their larger mass, lose energy less efficiently and are able to escape fromthe nugget’s surface. As such they are the dominantcharged particles deposited in the atmosphere. Initialmuon energies will be determined by the energy scale ofthe lightest hadronic modes of the colour superconduc-tor, typically around a few hundred MeV. After escapingthe nugget these muons lose energy to the surroundingatmosphere, generating fluorescence light in the process,until they decay into energetic electrons. The treatmentof muon energy loss to the surrounding atmosphere is de-scribed in appendix B and is important in determiningthe morphology of the resulting shower.The exact geometry of muon emission from the nuggetis a complex problem. At a basic level the majority of at-mospheric molecules first strike the nugget surface on thedownward directed face. The molecules will have rela-tively little time to migrate across the surface before theypenetrate into the quark matter and annihilate. As dis-cussed in [9] the combination of large penetration depthand the rapid energy loss from the jets produced by anni-hilations within the nugget favors the emission of muonsdirectly perpendicular to the quark matter surface abovethe point of annihilation. This argument, when combinedwith the preferential flux of atmospheric material alongthe axis of the nugget’s velocity implies preferential emis-sion in the forward direction. The simplest model wouldimply something like a cosine dependence but an exactestimate of this effect would depend on quite complicatedmaterial transport properties near the surface. In whatfollows it will simply be assumed that emission preferen-tially occurs from the forward directed face of the nugget.
V. NUGGET THERMODYNAMICS
Before proceeding to a more detailed description of aquark nugget induced air shower some basic thermody-namic properties of the nuggets must be introduced. Themajority of the energy deposited by nuclear annihilationsis thermalized within the nugget. The exact fraction,hereafter labeled f T , is dependent on the exact detailsof the quark matter and will not be calculated here. Asthe annihilations happen at low momenta the productsare likely to be emitted without a preferred direction andany energy moving deeper into the nugget will certainlybe thermalized. This basic geometric consideration sug-gests that 1 < f T < / A. thermodynamic equilibrium
This thermal energy is eventually radiated from thenugget’s surface at the point where the electrosphere be-comes transparent to thermal photons. This process wasdescribed in [11] where the emission spectrum was foundto be dEdt dA ≈ T α / π (cid:114) Tm e (4)implying a supression of thermal emission, with respectto blackbody, at low temperatures. The following analy-sis assumes that thermalization happens rapidly enoughthat the nugget remains near thermodynamic equilib-rium. Under this assumption the rate at which thermalenergy is deposited by annihilations will be equal to therate at which energy is radiated from the electrosphere.The accretion rate is set by the nugget’s velocity and thelocal atmospheric density and allows the nugget’s surfacetemperature to be determined at a given height. (cid:18) Tm e (cid:19) / = 3 πα / a b m e ρ at ( h ) v n f T = (cid:18) ρ at ( h )860 g/cm (cid:19) (cid:18) v n km/s (cid:19) f T (5)This estimation should remain valid as long as the tem-perature remains well below the electron mass (which istrue over the entire atmosphere.) This implies that thetemperature of a nugget near the earth’s surface will bearound 20 keV provided that all material in the nugget’spath is annihilated. B. molecular deflection
This section is devoted to determining the maximumrate at which matter can be deposited onto a quarkmatter surface. Intuitively as the flux of matter ontothe nugget’s surface increases so must the rate at whichthe resulting energy is transfered away from the surface.While the exact mechanism by which this energy transferoccurs may be quite complicated any plausible outwardtransfer of energy will exert a pressure on the incomingmatter and limit the rate at which it can be fed ontothe quark surface. This negative feedback suggests thatthere will be a density beyond which the annihilation ratesaturates. The following analysis attempts to be as gen-eral as possible to extract a generic scale at which matterannihilation rates reach a maximum.As demonstrated in [19] electron-positron annihilationsat low temperature are dominated by the formation of anintermediate positronium state. Positronium formationis a resonance process with a probability near one at lowmomenta but which falls off rapidly as the centre of massmomentum of the collision is increased. If the momen-tum is substantially larger that the positronium bindingenergy (2 m e α ) then the probability of forming a positro-nium bound state becomes negligible. This happens veryhigh in the atmosphere so that the primary annihilationchannel at relevant atmospheric densities is the direct e + e − → γ process described in [7]. At temperatures be-low the electron mass this process is actually less efficient than elastic scattering. In this case many positrons willscatter off of the incoming molecule before any of the elec-trons annihilate. The incoming molecules carry a kineticenergy T at = M at v , for a nitrogen molecule strikingthe nugget at 200 km/s this energy is a few keV. As thetemperature increases each positron scattering transfersmore energy until the energy transfer becomes sufficientto deflect the incident molecule. The exact temperatureat which this occurs is dependent on the exact details ofenergy transfer within the electrosphere and will not bedetermined here. Instead the following analysis will sim-ply assume that the temperature must be slightly abovethe kinetic energy of the incoming molecule. VI. FLUORESCENCE PROFILE
This section attempts to map the thermodynamic evo-lution described above onto a physical description of theresulting air shower. The atmospheric fluorescence yieldof a shower is determined primarily from the number ofcharged particles moving through the atmosphere at agiven point. These particles lose energy to the surround-ing atmosphere exciting nitrogen molecules which subse-quently radiate in the UV band.The fraction of muons per annihilated nucleon whichescape the nugget depends on the precise details of thequark matter surface and on the mass of the lightestmesons in the dense quark matter (the decay of thesebeing the primary muon production channel.) In vac-uum p ¯ p annihilations produce a large number of pions.The uncharged π s decay to photons while the chargedpions decay to muons. As such an annihilation in vac-uum typically yields between four and six muons. Thisshould be taken as the upper limit for total muon pro-duction per nucleon annihilated though only a fractionof these muons manage to escape the nugget. Thus, therate of muon production per annihilated nucleon, χ µ , hasa maximum possible value of order one while the actualvalue may be substantially lower. The uncertainty in χ µ is sufficient that the magnitude of the fluorescence yieldis only weakly constrained at the present level of analysis. A. geometry
In section V B it was argued that there must be a tem-perature at which the nuclear annihilation rate saturates.If this happens at a nugget surface temperature T max then this rate may be found from expression 4. dNdt = 323 R n α / T max m p (cid:114) T max m e ≈ × s − (cid:18) R n − cm (cid:19) (cid:18) T max keV (cid:19) / (6)Once this saturation point has been reached the decreasein the mean free path of an emitted particle with increas- nu m b e r o f c h a r g e d p a r t i c l e s FIG. 1: Muon content of a quark matter initiated shower as afunction of height. The curves are for saturation temperaturesof 10keV (solid), 15keV (dashed) and 20keV (dotted). ing atmospheric density implies that the flux of chargedparticles will decrease with atmospheric depth. The re-sulting shower profile, using the crude muon propagationmodel of B is shown in figure 1. It should be noted thatthe overall normalization of figure 1 is highly uncertainas it depends on both the muon production rate χ µ andthe mean energy with which muons escape the surface.Neither of these quantities are constrained beyond roughorder of magnitude estimates. Rather it is the overallgeometry of figure 1 that is relevant.The initial rise in muon flux is due to the increasingrate of nuclear annihilations with atmospheric density.The maximum charged particle number occurs near thepoint where the annihilation rate saturates and, as theatmospheric density increases beyond this point, its maineffect is to decrease the mean free path of a travelingmuon. This results in a more rapid loss of muons fromthe shower and thus a decrease in the integrated chargedparticle flux. B. timing
This basic shower geometry, growing to a maximumparticle content then decreasing rapidly beyond thatmaximum, is similar to that associated with an ultrahighenergy cosmic ray shower, however the fluorescence tim-ing will be substantially different. This difference arisesdue to the relatively small velocity of the nugget as com-pared to an ultra high energy cosmic ray. The later trav-els at the speed of light while the nuggets have typicalgalactic velocities, on the order of a few hundred kilome-ters per second, some three orders of magnitude slower.In both cases the secondary particles, produced inhadronic interactions, move outward at nearly the speedof light. As discussed in appendix B the charged particlesof a quark matter induced shower are generally confinedto a region within a few kilometers of the nugget due to their relatively small boost factors. The charged parti-cles spread through this volume over the course of tensof microseconds. However, the illuminated region of at-mospheric fluorescence will track with the nugget as itmoves through the atmosphere with the shower front ad-vancing quite slowly. The time scales for the progressof the nugget itself will be on the order of a tenth of asecond.The long duration of the atmospheric fluorescence andthe large photon multiplicity at any given time makethese events very difficult to observe above the variousbackgrounds. For this reason the fluorescence detector ofthe Pierre Auger Observatory is unlikely to trigger on aquark nugget air shower [20]. The difficulties inherent indetecting these fluorescence events likely favors searchesbased on surface detectors.
VII. LATERAL SURFACE PROFILE
When the shower reaches the earth’s surface it will betightly clustered around the nugget with only the high-est energy shower components able to travel far from theshower core. As with the fluorescence profile the exactdetails of the lateral profile are dependent on models ofmuon propagation through the atmosphere. Again theresults described here are based on the approximationsof appendix B which intends only to capture the mostgeneral features of the shower. As the majority of muonsare emitted at relatively low ( ∼ M eV ) energies theyare unable to travel far from the nugget in the denselower atmosphere. However, the shower also contains asmaller number of high energy muons able to travel alarger distance from the nugget. These higher energymuons produce an extended lateral distribution of par-ticles at the surface. An approximate lateral profile ofthe shower is plotted in 2. As with the fluorescence pro-file the total flux may be rescaled by slight changes inthe muon production rate and spectrum. The scalingof figure 2 is therefore less significant than the generalprofile shape. The essential feature of the radial surfaceprofile is a strong peak near the point where the nuggetstrikes the ground and an exponential drop off with ra-dial distance from this point. The controlling scale forthe exponential fall off is determined by the mean freepath of a muon averaged over the allowed initial energyscales as described in B. Numerically it is found that thisscale is in the range from a few hundred meters up to afew kilometers for the muon spectra given.
A. timescales
As with the fluorescence profile described above thesurface particle distribution is similar in geometry to thatof an air shower initiated by a single high energy cosmicray. But, once again, the timing signature will be verydifferent. In the case of a conventional air shower the F l u x FIG. 2: Particle flux per m as a function of distance fromthe shower core. The curves are for saturation temperaturesof 10keV (solid), 15keV (dashed) and 20keV (dotted). particles (primarily muons) arrive at the surface withina timescale of less than a microsecond. This is particu-larly true of the strongly beamed particles quite near theshower core while the arrival times of particles far fromthe shower core show considerably more scatter.In the case of a quark nugget initiated shower the timescale for particle arrival is determined by how long ittakes the nugget to pass through the region from whichthe emitted muons are able to reach the surface. Asdiscussed above the critical length scale for muon prop-agation is on the order of several hundred meters. Fora nugget moving at 200km/s this implies a shower dura-tion on the order of several milliseconds, several ordersof magnitude slower than the duration of an ultrahighenergy primary initiated shower.Near the shower core the difference in timing signaturesbetween an ultrahigh energy cosmic ray shower and aquark nugget shower will be very clear. However, in thecase of an off axis shower the situation is less clear. Atlarger radial distances the secondary particles of a cosmicray shower are less strongly beamed and have undergonea larger number of scatterings resulting in a longer showerduration. The opposite is true in the case of a quarknugget induced air shower. In this case it is only thehighest energy muons able to travel far from the showercore and the shower duration may be significantly shorterthan near the shower core. VIII. COMPARISON WITH CONVENTIONALSHOWERS
To this point emphasis has been placed on the simi-larities between the air shower induced by an antiquarknugget and one produced by a single ultra high energyprimary. There are however several important distin-guishing features between the two. The most importantof these arise from the much lower velocity of the primary particle. • A longer shower duration will be observable in atmo-spheric fluorescence producing an extended fluorescencetrack which lasts for a longer time. • This longer duration effect is likely also observablein the surface arrival times of secondary particles. De-pending on the timing cuts on the surface detector datait is likely that the muons associated with the shower willcontinue to arrive (with decreasing frequency) over timeson the order of microseconds. • The lower velocity of the primary particles will resultin a correlation between the arrival direction and the di-rection of earth’s motion with respect to the galaxy. Thiseffect produces both seasonal variation (similar to thatsearched for in the DAMA experiment [21]) as well asa correlation with the direction of somotion around thegalactic centre. • The arrival direction of quark nuggets is determinedby the local dark matter distribution and, as such, shouldshow no correlation with galactic or nearby intergalacticobjects. The presence of a quark matter component inthe cosmic ray spectrum would thus dilute any existingcorrelation with the source of typical ultrahigh energycosmic rays. • A distinguishing feature unrelated to the primaryparticle’s veloctiy is that shower evolution is dependenton the surface temperature of the nugget. As may beseen in equation 5 this scales with the atmospheric den-sity rather than the atmospheric depth of the shower.Conversely the evolution of a conventional shower is de-termined purely by the amount of atmospheric materialthrough which the shower has propagated. A possibleconsequence of this effect would be a larger apparentdepth of maximum for steeply inclined showers. However,without a detailed description of the thermal physics ofthe nuggets it is possible that the statistical variationin the saturation temperature may be large enough toobscure this effect. • A final distinguishing feature is observable in muonspectroscopy. In both cases the majority of particles willbe generated via the decay of pions with QCD scale ener-gies however an ultra high energy primary may producea number of muons with energies well above this scale.Conversely the QCD scale sets the highest energy avail-able to individual particles in a quark nugget initiatedshower. An analysis of the muon spectrum at the sur-face will thus show a high energy cut off around a GeVin the case of a quark nugget initiated shower while aconventional shower will show no such cut off.
IX. CONCLUSION AND DISCUSSION
The main purpose of this work has been to point outthat large surface area cosmic ray detectors are also wellsuited to search for the presence of dark matter in theform of quark nuggets. The impact of an antiquarknugget on the atmosphere will produce an extensive airshower consisting of a large number of secondary parti-cles observable through both their impact on surface de-tectors and the atmospheric fluorescence they generate.The resulting air shower is morphologically similar to onegenerated by a single ultra high energy primary particlein both the fluorescence profile and the lateral distribu-tion at the earth’s surface. It is therefore possible thatsome part of the high energy cosmic ray spectrum mayarise from the partial annihilation of dark matter in theform of heavy quark nuggets.The exact location of the shower maximum is depen-dent on rather complicated thermal physics in the elec-trosphere of the nuggets and, as such, cannot be explicitlyformulated in the preliminary treatment presented hereand will be the subject of future work. From this analysisit is only possible to argue that there must be an atmo-spheric density at which thermal pressure overcomes thekinetic energy of atmospheric molecules causing the anni-hilation rate to saturate. This effect leads to a nontrivialheight at which the shower will have a maximum particlecontent. In this context the observed break in the en-ergy spectrum 10 . eV [12] imposes limits on the totalparticle content and saturation temperature of the quark nugget.Finally it should be highlighted that additional workis needed on the atmospheric propagation of particleswithin this model. While the required simulations aresimplified by the absence of very high energy interac-tions (the properties of which are not well established)the injection of particles is dramatically different from aconventional shower. This requires a fundamentally dif-ferent formulation of the shower simulations from thosepresently employed. Without such simulations the ex-traction of statistical properties of the showers is notpossible. X. ACKNOWLEDGMENTS
Parts of this work were motivated by early discussionswith Brian Fick, I would also like to thank St´ephaneCoutu for many useful comments and Ariel Zhitnitskyfor his helpful discussions. This research was supportedin part by the Natural Sciences and Engineering ResearchCouncil of Canada. [1] E. Witten, Phys. Rev. D , 272 (1984).[2] A. R. Zhitnitsky, JCAP , 010 (2003), arXiv:hep-ph/0202161.[3] J. Kn¨odlseder et al. , Astron. Astrophys. , 513 (2005),arXiv:astro-ph/0506026.[4] A. Zhitnitsky, Phys. Rev. D , 103518 (2007),arXiv:astro-ph/0607361.[5] D. H. Oaknin and A. R. Zhitnitsky, Phys. Rev. Lett. ,101301 (2005), arXiv:hep-ph/0406146.[6] A. W. Strong, I. V. Moskalenko, and O. Reimer, Astro-phys. J. , 962 (2004), arXiv:astro-ph/0406254.[7] K. Lawson and A. R. Zhitnitsky, JCAP , 022 (2008),arXiv:0704.3064 [astro-ph].[8] M. P. Muno et al. , Astrophys. J. , 326 (2004),arXiv:astro-ph/0402087.[9] M. M. Forbes and A. R. Zhitnitsky, JCAP , 023(2008), arXiv:astro-ph/0611506.[10] D. P. Finkbeiner, Astrophys. J. , 186 (2004),arXiv:astro-ph/0311547.[11] M. M. Forbes and A. R. Zhitnitsky, Phys. Rev. D ,083505 (2008), 0802.3830.[12] The Pierre Auger, J. Abraham et al. , Phys. Lett. B685 ,239 (2010), 1002.1975.[13] J. Madsen, J.Phys. G , S833 (2005).[14] V. Springel et al. , Nature , 629 (2005), astro-ph/0504097.[15] K. Greisen, Phys. Rev. Lett. , 748 (1966).[16] G. T. Zatsepin and V. A. Kuzmin, JETP Lett. , 78(1966).[17] The Pierre Auger, B. Fick, Talk given at 28th Inter-national Cosmic Ray Conferences (ICRC 2003) (2003),astro-ph/0308512.[18] D. T. Son and M. A. Stephanov, Phys. Rev. D , 074012(2000), arXiv:hep-ph/9910491, Erratum: [22]. [19] M. M. Forbes, K. Lawson, and A. R. Zhitnitsky, Phys.Rev. D , 083510 (2010), 0910.4541.[20] A. Schmidt et al. , Nucl. Instrum. Meth. A601 , 347(2009).[21] P. Belli et al. , Phys. Rev. D , 023512 (2000), hep-ph/9903501.[22] D. T. Son and M. A. Stephanov, Phys. Rev. D ,059902(E) (2000), arXiv:hep-ph/0004095, Erratum to[18].[23] C. Alcock, E. Farhi, and A. Olinto, Astrophys. J. ,261 (1986).[24] J. Madsen, Phys. Rev. Lett. , 172003 (2001),arXiv:hep-ph/0108036.[25] M. Alford and K. Rajagopal, (2006), hep-ph/0606157.[26] D. H. Oaknin and A. Zhitnitsky, Phys. Rev. D , 023519(2005), arXiv:hep-ph/0309086.[27] Y. Fukuda et al. , Nucl. Instrum. Meth. A , 418(2003). Appendix A: Quark nugget structure
As alluded to above there are several possible phasesof quark matter from which the nuggets may be formed.Rather than performing detailed calculations within thecontext of a particular model this paper will rely onlyon general properties of quark matter. Reviews of theseideas are available in several previous works such as [23],[24], [25]. This section will present only the minimaldetails necessary for the discussion of the phenomenologyof quark nugget initiated air showers.The nuggets have a density at the nuclear scale andmay have a lower binding energy than the iron nucleus.If this is the case nuggets formed in the early universewill be stable over cosmological time scales.Of particular interest here is the proposal of [2] inwhich the nuggets may be composed of both matter andantimatter. The preferential formation of anti-nuggetshas been proposed as a mechanism for baryogenesis [26].In this model the formation of anti-nuggets is favored bya factor of 3:2 so that, beginning from a universe with nonet baryonic charge, antimatter is preferentially hiddenin the dark matter nuggets [26].At asymptotically large densities quark matter is com-posed of equal numbers of u,d and s quarks and is chargeneutral. However, the large s quark mass results in a de-pletion of s quarks in lower density quark matter. Evenif the bulk of the nugget is charge neutral the decreasingdensity near the quark surface results in a depletion ofs quarks and gives the quark matter a net charge, pos-itive in the case of a matter nugget and negative in thecase of an anti-nugget. To maintain charge neutrality thequark matter is surrounded by a layer of leptons. Theseleptons are only electromagnetically bound to the sur-face and extend beyond the quark surface. The exactstructure of this layer, known as the electrosphere, wasworked out in [19]. Near the quark matter surface theelectrons (or positrons) are tightly bound and at nucleardensities however the density falls off with distance downthe atomic scale. The presence of a large atomic densityshell of positrons surrounding the nugget will play a crit-ical role in interaction between the nugget and moleculesof the atmosphere. This layer also determines the ther-mal properties of the nugget as it is the point where thenugget first becomes transparent to low energy thermalphotons.
Appendix B: Muon propagation
This section gives a brief description of the approxima-tions made in describing the evolution of the air shower.While the model used is very simple it is intended only fordemonstrative purposes and highlights only the most ba-sic properties of the shower. As described above, the onlycharged particles capable of escaping the quark nuggetare muons. The main muon production channel is the de-cay of a meson-like excitation which will produce muonswith energies at the GeV scale. These muons rapidlylose energy in subsequent scatterings, primarily with thepositrons which are the lightest available modes. Energyloss will continue until the momentum of the muon is onthe same scale as the plasma frequency within the quarkmatter. This plasma frequency is generally found to beof the order ω p ∼ e Λ QCD ∼
10 MeV for a wide rangeof quark matter phases [23]. The muon energy spectrumwill therefore be peaked at this energy but may run upto the GeV scale for muons directly produced in annihi-lations near the surface. Energy loss scales exponentiallywith the depth at which the muon is produced thus the energy spectrum will be approximated as, dn µ dk = 1 ω p e ( ω p − k ) /ω p , m p > k > ω p (B1)where k is the muon momentum and m p is the protonmass. This will be taken as the initial spectrum formuons escaping the nugget.A muon traveling through the atmosphere will lose en-ergy scattering off the surrounding molecules. As theseare neutral on scales larger than a few times the Bohr ra-dius scattering requires the exchange of photons with anenergy above m e α . The cross-section for this processesin the limit where m µ >> m e α is given by, σ µ,e ≈ παm e v ≡ σ v − ≈ × − cm (cid:16) cv (cid:17) (B2)where v is the muon’s velocity. This translates to a scat-tering length of l s = 1 σ µ,e n at ( h ) (B3)where n at is the number density of the atmosphere. Scat-tering losses are dominated by events involving the lowestpossible intermediate energy photon, thus the muon willlose roughly m e α worth of energy on scattering. A muonwith initial kinetic energy T = E − m µ will then lose mostof its energy after T /m e α scatterings. Thus the stoppinglength for a muon of energy E and momentum p will be L s = E − m µ m e α l s = E − m µ m e α (cid:16) pE (cid:17) σ n at (B4)The other relevant length scale is the typical distancethat a muon travels before it decays L d = vγτ µ = pτ µ m µ (B5)where τ µ = 2 . × − s is the muon lifetime. Once themuon decays to an electron or positron it will rapidlybe lost in the electromagnetic component of the showerwhich this analysis makes no attempt to trace the evolu-tion of. A muon thus travels a distance L ( p ) = L d if L d < L s (B6) L s if L s < L d before its energy is dissipated into the electromagneticshower. Note that scattering is the dominant stoppingprocess for low momentum particles at large atmosphericdensities while decays dominate high in the atmosphereand for higher energy muons. The relevant quantity forwhat follows is actually the energy averaged length ob-tained by integrating over the muon spectrum (B1).¯ L = (cid:90) m p ω p dpω p L ( p ) e ( ω p − p ) /ω p (B7)Given the annihilation rate Γ an the total number of par-ticles produced at a given height may be estimated as N ( h ) = Γ an v n χ µ ¯ L (B8)This expression will be used in the context of (VI) inorder to track the evolution of the shower’s particle con-tent. Similar considerations can be used to approximatethe particle content at the earth’s surface. Under theassumption that particle emission from the nugget is pri-marily along the nugget’s direction of motion the numberof muons reaching an area of the surface will be dNdA = (cid:90) ∞ dh π ( h + b ) Γ an v n χ µ F (B9)where b is the distance from the shower core and F isthe fraction of initial muons which are able to propagatefar enough to reach the surface. Both loss mechanismsproduce an exponential extinguishing of the initial muonnumber, the characteristic length scale for this process isgiven by the energy averaged length in B7. F = exp (cid:34) − √ b + h ¯ L (cid:35) (B10)The integration of B9 with this expression for F is usedto approximate the surface flux of the shower. Appendix C: Underground detectors
This section briefly discusses the constraints imposedon quark nuggets based on underground detectors. While the muonic shower can be quite extensive in the atmo-sphere the higher density of rock strongly limits the rangeover which the muons can travel. This can be seen byreplacing the atmospheric density in B3 with the densityof surface rock. In this case the scattering length dropsby a factor of at least a thousand and the muons are ab-sorbed quite close to their production site. This is, ofcourse, precisely the reason why such experiments con-ducted under a large mass of shielding rock. Thus, theability to constrain the density of quark nuggets scalesalmost directly with detector area (or the effective crosssection presented by the cavity in which the detector islocated.) If we apply this to a relatively large detectorsuch as Super-Kamiokande [27] the effective detector sizeis limited to, at most on the order of 100m . In this case,even near the upper limit of the allowed flux (1/km /yr)one would expect an event rate of only ∼∼