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PHYSICAL REVIEW D 69, 063501 ~2004!
Antiprotons in cosmic rays from neutralino annihilation
F. Donato,1,* N. Fornengo,1,2,† D. Maurin,3,‡ P. Salati,4,§ and R. Taillet4,i
1Dipartimento di Fisica Teorica, Universita` di Torino, Istituto Nazionale di Fisica Nucleare, Sezione di Torino,Via P. Giuria 1, I–10125 Torino, Italy
2School of Physics, Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, Ko3Service d’Astrophysique, SAp CEA-Saclay, F-91191 Gif-sur-Yvette CEDEX, France
4Laboratoire de Physique The´orique LAPTH, Annecy-le-Vieux, 74941, France and Universite´ de Savoie, Chambe´ry, 73011, France~Received 10 June 2003; published 2 March 2004!
We calculate the antiproton flux due to relic neutralino annihilations, in a two-dimensional diffusion modelcompatible with stable and radioactive cosmic ray nuclei. We find that the uncertainty in the primary fluxinduced by the propagation parameters alone is about two orders of magnitude at low energies, and it is mainlydetermined by the lack of knowledge of the thickness of the diffusive halo. On the contrary, different darkmatter density profiles do not significantly alter the flux: a Novarro-Frenk-White distribution produces fluxeswhich are at most 20% higher than an isothermal sphere. The most conservative choice for propagationparameters and dark matter distribution normalization, together with current data on cosmic antiprotons, cannotlead to any definitive constraint on the supersymmetric parameter space, either in a low-energy effectiveminimal supersymmetric standard model, or in a minimal supergravity scheme. However, if the best choice forpropagation parameters—corresponding to a diffusive halo ofL54 kpc—is adopted, some supersymmetricconfigurations with the neutralino massmx&100 GeV should be considered as excluded. An enhancement fluxfactor—due for instance to a clumpy dark halo or a higher local dark matter density—would imply a moresevere cut on the supersymmetric parameters.
DOI: 10.1103/PhysRevD.69.063501 PACS number~s!: 95.35.1d, 96.40.2z, 98.35.Gi, 98.35.Pr
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I. INTRODUCTION
The recent Wilkinson Microwave Anisotropy Prob~WMAP! measurements of the cosmic microwave baground~CMB! anisotropies@1# point toward a flat universewith a fractionVL.0.7 of the closure density in the form oa negative pressure component—such as a cosmologicalstant or a scalar field—while the remainingVm.0.3 is mat-ter. These conclusions are independently reached fromdetermination of the relation between the luminosity distaand the redshift of supernovas of type Ia~SNIa! @2# on theone hand and from the large scale structure information frgalaxy and cluster surveys@3#. The WMAP values ofVm50.2760.04 andVB50.04460.004 indicate that most othe matter is nonbaryonic. The amount of baryonic maVB deduced from the CMB is in perfect agreement with tresults from primordial nucleosynthesis and observationsthe deuterium abundance in quasar absorption lines@4#.
The nature of this astronomical dark matter has been clenging physicists for several decades and is still unresolvThe favored candidate is a weakly interacting massive pticle. The so-called neutralino naturally arises in the framwork of supersymmetric theories as the lightest combinaof neutral Higgsinos and gauginos. A great deal of effort hbeen devoted to pinning down these evading species@5#.Experimental techniques@6–8# have been devised in order t
*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]§Electronic address: [email protected] Electronic address: [email protected]
0556-2821/2004/69~6!/063501~19!/$22.50 69 0635
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be sensitive to the recoil energy which a neutralino mayposit as it crosses a terrestrial detector. The annihilation ptons from the neutralinos that populate the Milky Way ha@9# or extragalactic systems@10# are under scrutiny. As amatter of fact, a gamma-ray excess has been recentlyported by HEGRA in the direction of the giant elliptical M8@11#. Antimatter cosmic-ray particles are also expected frneutralino annihilations inside our galaxy. A subtle featurethe positron spectrum has actually been measured byHEAT Collaboration@12# for energies beyond 7 GeV.
This work is devoted to cosmic-ray antiprotons whoenergy spectrum has already been measured with somecuracy. Much larger statistics will soon be collected by tAMS Collaboration on board the International Space Stat~ISS! by the BESS-Polar long duration balloon experimeand by the PAMELA satellite. Secondary antiprotons anaturally produced by the spallation of primary nucleimostly cosmic-ray protons and helions—on the diffuse gof the Milky Way ridge. If neutralinos pervade our galaxy,primary component adds up to that secondary distributiThe spectral distortion that ensues is expecteda priori in thelow-energy region for merely kinematic reasons@13#: unlikefor a neutralino annihilation, the center-of-mass frame ospallation event is not at rest with respect to the galaxy.principle, an excess of low-energy antiprotons is the sigture of an unconventional production—either neutralino anihilation or small black hole evaporation@14# for instance.However, because antiprotons undergo inelastic yet nonahilating collisions with the interstellar material, the highenergy particles tend to lose energy and to populate the lenergy tail of the spectrum, which consequently is muflatter @15# than previously estimated. This motivated thsearch for other cosmic-ray signatures such as antideute
©2004 The American Physical Society01-1
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
@16,17#. Antiproton production from primary cosmic-raspallations is the natural background to any unconventioexcess that would signal, for instance, the presence ofputative neutralinos. The detailed calculation of that secoary component@18# has required the determination of thpropagation-diffusion parameters that are consistent withB/C data@19#. By varying those parameters over the entrange allowed by the cosmic-ray nuclei measurements,theoretical uncertainty on the antiproton secondary fluxbeen found to be 9% from 100 MeV to 1 GeV. It reachemaximum of 24% at 10 GeV and decreases to 10% atGeV. This small scatter in the secondary antiproton spectis not surprising. Cosmic-ray nuclei such as LiBeB and sondary antiprotons are both manufactured in the saplace—the interstellar gas of the galactic disk—throughsame production mechanism—the spallation of primaries
The aim of this article is to calculate the supersymmecosmic-ray antiproton flux that arise from the diffusiopropagation parameter space and to estimate the unceties due to its spread. Since neutralinos annihilate all overMilky Way and are not confined to the disk alone, we antipate that the uncertainty in that primary component willmuch larger than for secondaries.
The discussion will be split into two main directionbrought to the fore by the structure of the equation descing the primary flux. Production and propagation maydisentangled in the limit where the energy does not changantiprotons travel. That is not strictly correct, as diffusireacceleration as well as adiabatic and Coulomb losseserate a diffusion in energy space that is discussed in Appdix. The elementary process of supersymmetric antiproproduction through neutralino annihilation is discussedSec. II, both in an effective minimal supersymmetric stadard model~MSSM! and in a supergravity-inspired modeThe two-zone propagation-diffusion model and the depdence of the primary antiproton fluxF p
SUSY on the propaga-tion parameters are described in Sec. III. The thicknessL ofthe magnetic halo is naively expected to be the dominsource of uncertainty forF p
SUSY as the larger the confinemenlayers, the larger the fiducial volume where neutralino anhilations take place, and the larger the supersymmetric aproton flux. Actually,L is combined with the diffusion coefficient K(E) and the galactic wind velocityVc in order to geta precise value for the B/C ratio and for the antiproton fluWe present the results for the primary flux in Sec. IV, whewe estimate the uncertainties induced by the spread ofdiffusion-propagation parameter space. We also brieflycuss the modifications ofF p
SUSY due to different choices inthe dark matter distribution function, in its normalizatioand in the core radius values. In Sec. V, the comparisonthe latest antiproton measurements with the antiproton flupredicted in different supersymmetric schemes will be dcussed as a function of the propagation-diffusion parameand of the neutralino galactic distribution. Conclusions aperspectives will be presented in Sec. VI.
II. THE NEUTRALINO-INDUCED ANTIPROTONS:THE SOURCE TERM
Antiprotons can be produced by self-annihilation of netralinos in the galactic halo. Dark matter neutralinos may
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considered almost at rest in the galactic frame since taverage velocity is of the order of 300 km s21. They aretherefore highly nonrelativistic. The production differentirate per unit volume and time is a function of space coornates (r ,z defined in the galactic rest frame! and antiprotonkinetic energyTp . It is defined as
qpSUSY
~r ,z,Tp!5^sannv&0g~Tp!S rx~r ,z!
mxD 2
, ~1!
where^sannv&0 denotes the average over the galactic velity distribution function of the neutralino pair annihilatiocross sectionsannmultiplied by the relative velocityv, mx isthe neutralino mass, andrx(r ,z) is the mass distributionfunction of neutralinos inside the galactic halo. Since reneutralinos behave as cold dark matter, their distributionto follow the matter density profilerDM(r ,z) of the galactichalo:
rx~r ,z!5jrDM~r ,z!, ~2!
wherej parameterizes the fact that the dark halo may nottotally made of relic neutralinos (j<1). This would be thecase when neutralinos are not responsible for the tamount of dark matter in the Universe, i.e., when their reabundanceVxh2 is much smaller than the measured valfor VDMh2. This is a situation that occurs in many supersymetric models. It is reasonable to assume thatj has no spacedependence and that it is related to the relative amounVxh2 with respect toVDMh2. We will assume the standardefinition
j5min~1,Vxh2/0.05!, ~3!
where we have considered that neutralinos withVxh2
,0.05 cannot be the dominant dark matter component.Finally, the second term in Eq.~1!, g(Tp), denotes the
antiproton differential spectrum per annihilation event, dfined as
g~Tp
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ANTIPROTONS IN COSMIC RAYS FROM NEUTRALINO . . . PHYSICAL REVIEW D69, 063501 ~2004!
The source termqpSUSY(r ,z,Tp) is therefore a combination
of astrophysical factors~the dark matter density profile of thgalactic halo! and of particle physics properties~the neu-tralino self-annihilation cross section and the hadronizatinto antiprotons of the neutralino annihilation products!. Theastrophysical and particle physics quantities are factoredand can be studied separately. With the definitions giabove, we can rewrite the antiproton source term as
qpSUSY
~r ,z,Tp!5Yg~Tp!rDM2 ~r ,z! ~5!
where we have defined the supersymmetric flux factorY as
Y5j2 ^sannv&0
mx2
, ~6!
which entirely depends on properties of supersymmemodels.
We move now to discuss each term separately.
A. The galactic distribution of dark matter
For most of our discussion, we will assume that the dmatter density distribution is described by a cored isothersphere. In terms of the radial distancer in the galactic planeand of the vertical coordinatez, the density profile is
rDM~r ,z!5r l
a21R(2
a21r 21z2, ~7!
wherea denotes the core radius of the dark halo andR( isthe distance of the Sun from the galactic center. We havea53.5 kpc and the IAU-recommended valueR(58.5 kpc.The valuer l for the total local dark matter density is detemined by taking into account the contribution given by tmatter density of Eq.~7! to the local rotational velocityv rot@21#. The value ofr l compatible with observations rangefrom 0.18 GeV cm23 ~for a low value of the rotational velocity, v rot5170 km s21, and a nonmaximal dark halo! to0.71 GeV cm23 ~for v rot5270 km s21 and a maximal darkhalo! @21#. The interval relative to the preferred value for throtational velocity (v rot5220 km s21) is 0.30 GeV cm23
&r l&0.47 GeV cm23 @21#. Our results will be presented for l50.3 GeV cm23. Since in the primary antiproton fluxr l
2
enters as a pure normalization factor, the fluxes obtaineddifferent values ofr l are easily rescaled. For instance, fr l50.47 GeV cm23, the antiproton fluxes would be a factoof 2.45 higher than the corresponding ones forr l50.3 GeV cm23.
We will come back to the topic of the dark matter densprofile at the end of the paper, in Sec. IV B.
B. Supersymmetric models
The existence of a relic particle in supersymmetric thries arises from the conservation of a symmetry,R parity,which prevents the lightest of all the superpartners fromcaying. The nature and the properties of this particle depon the way supersymmetry is broken. The neutralino can
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the dark matter candidate in models where supersymmetbroken through gravity-~or anomaly-!mediated mechanismsThe actual implementation of a specific supersymmescheme depends on a number of assumptions on the struof the model and on the relations among its parameters. Tinduces a large variability of the phenomenology of netralino dark matter. In this paper we will consider neutralidark matter in two different supersymmetric schemes: a loenergy effective-theory implementation of the minimal spersymmetric standard model~EMSSM! and a minimal su-pergravity model~MSUGRA!.
The EMSSM is defined as an implementation of supsymmetry directly at the electroweak scale, which is whthe phenomenology of neutralino dark matter is actuastudied. The large number of free parameters is reducedset of assumptions which are sufficient to shape the proties of the model at the electroweak scale. All the relevparameters, which set the mass scales and couplings othe supersymmetric particles~and of the Higgs sector! aretaken into account. The free parameters are the gaugino mparameterM2, the Higgs boson mixing parametersm, theratio of the two Higgs boson vacuum expectation valutanb, the mass of the pseudoscalar Higgs bosonmA , a com-mon soft scalar mass for the squarksmq , a common softscalar mass for the sleptonsml , and a common dimensionless trilinear parameterA for the third family (Ab5At[Amq andAt[Aml ; the trilinear parameters for the othefamilies are set equal to zero!. We assume the standard granunification relation between theU(1) and SU(2) gauginomass parameters:M155/3 tan2uWM2. The parameters willbe varied in the following intervals: 100 GeV<M2<1000 GeV, 100 GeV<umu<1000 GeV, 100 GeV<mA<1000 GeV, 100 GeV<mq ,ml <3000 GeV, 1<tanb<50, and23<A<3.
A different approach is to embed supersymmetry in apergravity scheme with boundary conditions at some critihigh-energy scale, such as the grand unification~GUT! scale,and keep the number of free parameters and assumpminimal. This is our MSUGRA. In this class of models wconsider gauge coupling constant unification at the Gscale. In addition, all the mass parameters in the supersmetric breaking sector are universal at the same GUT scThe low-energy sector of the model is obtained by evolvall the parameters through renormalization group equatifrom the GUT scale down to the electroweak scale: this pcess also induces the breaking of the electroweak symmin a radiative way. This model is very predictive, sincerelies on only very few free parameters, but at the same tit has a very constrained phenomenology at low energyalso appears to be quite sensitive to some standard mparameters, like the mass of the top and bottom quarksmtandmb) and the strong coupling constantas . In this class ofmodels there are four free parameters: the universal gaumass parameterM1/2 at the GUT scale, the universal soscalar mass parameter for both the sfermions and the Hbosonsm0 at the GUT scale, a common trilinear coupling fthe third family at the GUT scaleA0, and tanb. The param-eters will be varied in the following intervals: 50 Ge<M1/2<1000 GeV, 0<m0<3000 GeV, 1<tanb<50, and
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
23<A0<3. The standard model parametersmt , mb , andas are varied inside their 2s allowed ranges.
C. The supersymmetric flux factor Y
The flux factorY defined in Eq.~6! acts as a normalization factor for the antiproton flux and is a purely supersymetric term. It depends on the mass and couplings of ntralinos in the supersymmetric framework under study.Fig. 1 we show the flux factor as a function of the neutralimass for a scan of the EMSSM. Figure 2 reports the casethe MSUGRA scheme. We show the values ofY separatelyfor the case of comsologically [email protected]<Vxh2
<0.3, Fig. 1~a! and Figs. 2~a! and 2~c!# and subdominan@Vxh2,0.05, Fig. 1~b! and Figs. 2~b! and 2~d!# relic neu-tralinos. Among the cosmologically relevant ones, we ashow the configurations that yieldVxh2 inside the preferredrange for CDM, as determined by the combined WMA12DFGRS1Lyman-a analysis: 0.095<VCDMh2<0.131@1#. The results in the EMSSM show that the upper valuesY are around 10212 GeV24 for neutralino masses close tthe experimental lower bound~around 50 GeV! and thendecrease below 10214 GeV24 for mx;1 TeV.
In the case of dominant relic neutralinos, the intervalvalues forY is restricted, at all masses: in order to havalues ofVxh2 that fall in the cosmologically relevant rangthe annihilation cross section integrated from freeze-down to the present time must be inside the interva310211 GeV22&^sannv& int&2310210 GeV22. We recallthat the relic abundance depends on^sannv& int (Vxh2
}^sannv& int21). This cross section, due to the nonvanishi
temperature in the early Universe, may differ quite substtially from the zero-temperature cross section^sannv&0,which is instead relevant for the antiproton signal. Usuallcorrelation betweensannv& int and ^sannv&0 is present whenthe zero-temperaturesannv&0 is large; on the contrary, whe^sannv&0 is small, temperature corrections in the early U
FIG. 1. ~Color online! Scatter plot of the supersymmetric flufactorY[j2^sannv&0 /mx
2 as a function of the neutralino massmx ,calculated in the EMSSM.~a! refers to supersymmetric configurations with the neutralino as a dominant dark matter component~i.e.,0.05<Vxh2<0.3, and therefore a rescaling factorj51). The light~green! circles show the EMSSM configurations for which the netralino relic abundance lies in the preferred range for cold dmatter ~CDM!, as determined by the combined WMA12DFGRS1Lyman-a analysis: 0.095<VCDMh2<0.131 @1#. ~b!refers to the neutralino as a subdominant dark matter par(Vxh2,0.05).
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verse induce ^sannv& int to deviate, also sizably, from^sannv&0. This difference in the two cross sections is resposible for the band of values ofY shown in Fig. 1~a!.
When the neutralino relic abundance is low, such tneutralinos are not the dominant component of dark maY acquires an additional dependence onVxh2 through therescaling factorj2. This is shown in Fig. 1~b!. The effect ofj2 is obviously to reduceY: the lower the relic abundancethe smallerj and thus the flux factor. The lowest pointsFig. 1~b! are the ones with lower values ofVxh2. Theseconfigurations, even though they give a large^sannv&0 ~lowVxh2 has large values of sannv& int , and in this case^sannv& int}^sannv&0), nevertheless have a low flux factor bcause they are underabundant. This implies that largely sdominant relic neutralinos are likely to provide~almost! un-detectable antiproton fluxes. This is somewhat at variawith the case of direct detection: the difference arises frthe fact that the antiproton signal~as well as the other galactic signals! depends quadratically on the dark matter dens~and hence on the rescaling factor!, while for direct detectionthe dependence is linear and the suppression is much m@7#.
The situation of MSUGRA is shown in Fig. 2. In thicase, the largest values of the flux factorY are about anorder of magnitude lower that in the EMSSM:Y&10213 GeV24 for light neutralinos. This is a consequenc
-k
leFIG. 2. ~Color online! The same as in Fig. 1, calculated in th
MSUGRA scheme.~a! and ~c! refer to cosmologically dominanneutralinos (0.05<Vxh2<0.3); ~b! and ~d! to subdominant neu-tralinos (Vxh2,0.05). The upper row~a! and ~b! is obtained forthe universal soft scalar massm0 smaller than 1 TeV~for thesemodels, the neutralino is mostly a Bino state!; the lower row~c! and~d! refers to values ofm0 in excess of 1 TeV~in this case theneutralino may have a substantial Higgsino component!. The light~green! circles in~a! and~c! show the MSUGRA configurations fowhich the neutralino relic abundance lies in the preferred rangeCDM, as determined by the combined WMAP12DFGRS1Lyman-a analysis: 0.095<VCDMh2<0.131@1#.
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ANTIPROTONS IN COSMIC RAYS FROM NEUTRALINO . . . PHYSICAL REVIEW D69, 063501 ~2004!
of the properties of neutralinos in this constrained typemodel: neutralinos turn out to be mainly gauginos and thcouplings, especially to Higgs bosons, which require a miHiggsino-gaugino neutralino content, are in general smathan in some sectors of the EMSSM. The lower panelsFig. 2 show the situation in a sector of the MSUGRA schewhere the soft scalar masses are large:m0.1 TeV @22,23#.In this sector, the neutralino may acquire a nonvanishHiggsino component@22,23#, as a consequence of the raditive electroweak symmetry breaking, and their couplingsHiggs bosons are enhanced@22#: the consequence on the flufactor is in fact a mild enhancement, up to values ofYaround (3 –4)310213 GeV24, closer to the EMSSM uppevalues.
D. The differential antiproton spectrum g„Tp…
Let us move now to a discussion of differential spectraantiprotons which are produced by neutralino annihilatiThe capability of producing antiprotons depends on the psibility for neutralinos to produce quarks or gluons, eithdirectly or through decay of their annihilation producquarks and gluons will then hadronize and eventually pduce antiprotons. We have modeled the hadronizationcess by using thePYTHIA Monte Carlo model~MC! @24#. Theneutralino annihilation is calculated analytically as describin Ref. @25#. Neutralino annihilation occurs at rest in thgalactic frame, and the different final states that are otherefore depend on the neutralino mass. The annihilamay proceed through the following channels: production ofermion pair; production ofWW and ZZ; production of aHiggs boson pair; production of a Higgs boson together wa gauge boson~which can be theZ boson or theW dependingwhether the Higgs boson is neutral or charged!. Apart fromthe direct production of quarks or gluons, the decay chainthe annihilation products until a quark is produced is callated analytically. At this stage, the antiproton differentflux is obtained from the MC modeling. More details agiven in Appendix A.
A sample of p spectra for the four types of neutralinannihilation final states is shown in Fig. 3, for different vaues of the neutralino mass: panel~a! shows the spectra caculated for annihilation into a purebb state; panel~b! refersto annihilation into a pureZZ state; panel~c! refers to anannihilation into a Higgs boson pair, where the scalar Higboson has a mass ofmh5120 GeV, the pseudoscalar Higgboson mass ismA5200 GeV, for tanb510 and for a van-ishing value of the Higgs boson mixing parametera; panel~d! refers to annihilation into anhZ pair, for mh5120 GeV, tanb510, anda50. Figure 3 shows the dependence of the antiproton spectra on the production enefixed by the neutralino mass. For instance, in panel~a! theantiprotons are produced by the hadronization ofb quarksinjected at the energy given by the neutralino mass; in pa~b!, antiprotons are produced by quarks produced by thecay ofZ bosons in motion with respect to the neutralino rframe: a Lorentz boost on the hadronization spectra is thfore operative in shifting the fluxes to larger kinetic energi
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Spectra like the ones shown in Fig. 3 are used to calcuthe differential spectrag(Tp). However, as it is clear fromEq. ~4!, we also need to know the values of the branchratios of each neutralino annihilation final state. The braning ratios will weight the different differential spectra, likthe ones shown in Fig. 3. An example of branching ratiosneutralino annihilation is given in Fig. 4 for the EMSSMscheme, and in Fig. 5 for the MSUGRA models.
In the EMSSM, we notice that the annihilation in fermons may be sizable and dominant for masses lower thanGeV. The two-Higgs-boson final state is usually of the ord
of or lower than thef f final state, while the gauge bosofinal state may dominate, except for very large neutralmasses. The mixed gauge1Higgs boson final state tends tbe dominant at very large neutralino masses.
In the case of MSUGRA models, since the neutralitends to be a gaugino which couples effectively to fermio
through sfermion exchange, thef f final state usually domi-nates. A relevant production of final states other than ferons, especially gauge bosons, occurs in the large sfermmass regime (m0.1 TeV), where sfermion exchange is supressed by the large sfermion mass and at the same timHiggsino component for the neutralino arises: this facilitathe coupling to both Higgs and gauge bosons.
FIG. 3. Antiproton differential energy distribution for pure annihilation channels as a function of the reduced kinetic energyxp
[Tp /mx . ~a! refers to annihilation into abb pair, for neutralinomasses ofmx510,60,100,300,500,1000 GeV~from bottom to top!;~b! refers to annihilation into a ZZ pair, for mx
5100,300,500,1000 GeV~from top to bottom!; ~c! refers to anni-hilation into a scalar1 pseudoscalar Higgs boson pairhA, for mx
5300,500,1000 GeV ~from top to bottom!, and for mh
5120 GeV, mA5200 GeV, tanb510 ~ratio of Higgs bosonvacuum expectation values! anda50 ~Higgs boson mixing param-eter!; ~d! refers to annihilation into ahZ pair, for mx
5300,500,1000 GeV ~from top to bottom!, and for mh
5120 GeV, tanb510, anda50.
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
FIG. 4. ~Color online! Branching ratios for the neutralino selannihilation cross section in the EMSSM.~a! shows the amount othe branching ratio for the annihilation into a fermion-antifermi
final state (xx→ f f ). ~b!, ~c!, and~d! show the amount, relative to
the f f final state, of the annihilation into Higgs bosons, gaubosons, and the mixed Higgs boson–gauge boson final state.~red! points denote configuration with 0.05<Vxh2<0.3 ~dominantrelic neutralinos!. Light ~green! circles indicate configuration withVxh2,0.05 ~subdominant relic neutralinos!.
FIG. 5. ~Color online! The same as in Fig. 4, calculated in thMSUGRA scheme. Dark~red! points denote configuration with0.05<Vxh2<0.3 ~dominant relic neutralinos!. Light ~green! circlesindicate configuration withVxh2,0.05 ~subdominant relic neu-tralinos!. Crosses~in blue! indicate the MSUGRA configurationwith m0.1 TeV.
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The final result of this analysis is the calculation of reistic antiproton differential spectra for neutralino annihiltion. Some representative examples are shown in Fig. 6,different values of the neutralino mass. All the spectra reto neutralinos selected to haveVxh250.1 and large valuesof the flux factorY ~close to the upper values of Fig. 1, foeach mass!. All these spectra properly take into accountthe ingredients discussed in this section: the hadronizaspectra and the annihilation branching ratios. The spectfor mx5100 GeV is the reference spectrum for the analyof the astrophysical properties in the next sections. The sptra shown in Fig. 6 are selected from our sample ofEMSSM, except the one formx510 GeV, which refers to anEMSSM without the grand unification condition between tgaugino mass parametersM1 andM2 @26,27#. In this class ofmodels the neutralino can be as light as a few GeV@26,27#,in contrast to the standard EMSSM, where LEP constraimply a lower bound on the neutralino mass of aboutGeV. For completeness, we have therefore included alsorepresentative spectrum formx510 GeV, in order to illus-trate the effect of propagation on the primary flux from ligneutralinos. However, a complete study of the EMSSM wiout grand unification gaugino universality is beyond tscope of this paper.
Now that we have discussed the source term, we procto the second step of the calculation: the study of how th
ark
FIG. 6. ~Color online! Representative differential antiprotospectra per annihilation eventg(Tp) from neutralino self-annihilation, as a function of the antiproton kinetic energyTp . Thedifferent curves refer to different neutralino masses:mx510 @dotted~blue!#, 60 @short dashed~black!#, 100 @solid ~red!#, 300 @longdashed~green!#, and 500 GeV@dot-dashed~magenta!#. The spectraare selected from our sample of the EMSSM, except the onemx510 GeV which refers to an EMSSM without grand unificatiogaugino universality@26,27#. All the spectra refer to neutralinowith Vxh250.1 and large values of the flux factorY. The spectrumfor mx5100 GeV is the reference spectrum for the analysis ofastrophysical properties in the next sections.
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antiprotons diffuse and propagate in the galaxy and insolar system. The result of this analysis will be the interslar and top-of-atmosphere~TOA! fluxes of primary antipro-tons.
III. DIFFUSION AND PROPAGATION IN THE GALAXY
The propagation of cosmic rays in the galaxy has bconsidered in the framework of a two-zone diffusion modwhich has been described at length in Refs.@18,19,28#. Herewe only recall the main features of this model, and referthe above-mentioned papers for all the details and mottions. We also present in detail the quantitative dependeof the secondary and primary signals on the propagationrameters.
A. The framework
The disk of the galaxy is described as a thin disk of radR520 kpc, which contains the interstellar gas with a surfadensityS52hnISM with h5100 pc andnISM51 cm23. It isembedded in a thicker diffusion halo, supposed to havcylindrical shape with the same radiusR as the disk andheight L which is not well known. The matter densitymuch lower in the diffusion halo so that spallations~rateG[2hnISMs yielding the secondary species! of the chargednuclei occur only in the disk. Moreover, the standard souralso happen to be located in the disk.
The spatial diffusion of cosmic rays is assumed to ocuniformly in the whole~disk and halo! diffusion volume,with the same strength. The corresponding diffusion coecient has been defined asK(E)5K0b(R/1 GV)d, whereRstands for the particle rigidity andK0 andd are free param-eters of the model. We also consider the possibility thagalactic wind blows the particles away from the disk in thzvertical direction, with a constant speedVc . It induces anadiabatic dilution of the energy of the particles in the ddue to the sudden change inVc . Several other processemodify the antiproton energy distribution: ionization losswhen interacting with the neutral interstellar matter, or froCoulomb losses in a completely ionized plasma, dominaby scattering off the thermal electrons. To end with, minimreacceleration on random hydrodynamic waves, i.e., dision in momentum space, described by a coefficientKpp re-lated to the spatial diffusionK(E), is inevitable@29#. Thisprocess is assumed to occur only in the disk and is relatethe velocity of disturbances in the hydrodynamical plasVA , called the Alfven velocity. In summary, our diffusionmodel has five free parametersK0 ,d,L,Vc ,VA which de-scribe the minimal number of physical effects thoughthave some role in antiproton propagation.
The sets of diffusion parameters were constrained iprevious work@19# ~see also Ref.@30#! by analyzing stablenuclei ~mainly by fitting the boron to carbon ratio B/C!. Thevalues we obtained were also shown to be compatible wthe observed secondary antiprotons@18# and the flux of ra-dioactive isotopes@31#. However, in a first step we will disregard these constraints: in order to clarify which of tpropagation parameters are important if one wants to cpare any possible primary component to the backgro
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~secondaries!, we study the effect of each parameter on tsignal and the background. Only in a second step isadditional information on the constraints used to draw cclusions about the variation that will result in the primasupersymmetric signal~see Sec. IV A!.
B. Solutions for primary and secondary antiprotons
We are interested in the cosmic ray antiproton flux
F p~r ,z,E!5v p
4pNp~r ,z,E!. ~8!
It is related to the differential densityNp(r ,z,E)[dNp(r ,z,E)/dE which satisfies the steady-state diffusioequation. The general procedure for solving this equationwell as references for a detailed derivation are given in Apendix B. At variance with the solutions already presenelsewhere, it proves to be useful, as suggested by a studthe spatial origin of secondary and primary cosmic rays C~see Ref.@32# for what is ment by ‘‘spatial origin’’!, to in-troduce the quantities
r w[2K~E!
Vc, ~9!
r sp[K~E!
hG inel~E!. ~10!
Since many configurations ofK(E)5K0bR d andVc lead tothe samer w and r sp, these new parameters automaticaavoid a useless discussion about many degenerate valuthe diffusion coefficient and make the dependence onimportant parameters more evident in formulas~the physicalmeaning of these new parameters is explained below;also Ref.@32#!.
The solutions are given below discarding energy redisbutions ~see Appendix for the procedure to include them!.Energetics are not the dominant effects so it is interestingfocus on the analytical formulas obtained in that case.
(a) The primaries. Let us first inspect the primaries: thsource term is given byqp
SUSY(r ,z,Tp) described by Eq.~1!and discussed in detail in Sec. II. Primary antiprotonsproduced throughout the whole diffusive halo, which is ebedded in the dark matter halo. An advantage when enelosses and gains are discarded is that the solution canrecast as~see Appendix B!
Np,prim~r 5R( ,z50,E!5E sourceprim ~E!3Sastro
prim~R(,0,E!,
where theelementarysource term~spectrum from a pointsource! given by
E sourceprim ~E
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
Sastroprim~R(,0,E![(
i 51
`
P i~E,R(!H E0
R
J0~z i r /R!
3E2L
L
e2z/r wsinh@Si~L2z!/2#
sinh@SiL/2#
3w~r ,z!J dzrdr. ~12!
In the above equation,w(r ,z) is the effectivespatial distri-bution of the primary [email protected].,rDM
2 (r ,z) for supersym-metric particles andrDM(r ,z) for evaporating primordialblack holes~PBHs!#. We have defined
P i~E
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ANTIPROTONS IN COSMIC RAYS FROM NEUTRALINO . . . PHYSICAL REVIEW D69, 063501 ~2004!
dence~through R d). At sufficiently high energy~above afew tens of GeV!, r w ,r sp@1 andAi andSi become independent of E, so that the sole energy dependence 1/K(E) isfactored out ofP i(E,R(), i.e., of the Bessel sums. Asresult, the quantityS is insensitive to the choice ofK(E),whatever the value of the other parameters. There issubtlety left: the secondary elementary productionE source
sec (E), as defined above, in contrast to that of the pmary, is not fully elementary, because it does depend onvalue ofK(Ep) above 7 GeV. However, as we will see lateall propagation parameters are designed to have abousame K(E) at 100 GeV, so that the quantitNp, prim/Np, sec(r 5R(,0,E) is eventually not very sensitiveto this parameter.
1. The diffusive halo size L and the radius R of the galaxy
These parameters are related to the escape probafrom the confinement volume~the magnetic halo of the galaxy!. The largerL andR, the greater the probability for particles emitted in remote sources to reach us. Actually,side boundary plays almost no role for several reasons. Fescape is driven by the closest boundary, which is the onz56L asL is likely to be smaller thanR; second, the sourcedistribution is peaked near the galactic center and decreto very small values at large radii~see Ref.@32# for moredetails!. Hence, forL&5 kpc, settingR520 kpc or R5`leavesSandB unchanged. The enhancement of fluxes withLcan be seen in Fig. 7~we use here and in other figures thisothermal profile for the dark matter distribution!, showingS@L,r w5`,r sp5`# as a function ofL ~we limit the discus-sion to the supersymmetric case, but the reader can straforwardly extend to PBH’s!. For smallL, only the sourcesvery close to the solar neighborhood contribute and, asdark matter source distribution is normalized to 1, the supsymmetric and PBH cases yield the same value. AsL in-creases, escape is less efficient, and more sources~secondary
FIG. 7. This plot displays the quantitiesK(E)3Sastroprim and
K(E)3Bastrosec , see Eqs.~12! and ~18! ~left panel! andS defined by
Eq. ~19! ~right panel!, as a function of the propagation parameteL(r w5r sp5`, i.e., no wind, no spallations! for an isothermal profile.Two cases have been considered for the primary signal. The clabeled ‘‘supersymmetric’’ corresponds to an effective source teproportional tow(r ,z)5rDM
2 , whereas the curve PBH corresponto a source term proportional tow(r ,z)5rDM , as for primordialblack holes.
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or primary! effectively contribute to the signal. This enhancement is more important for primaries than for secoaries, as the effective number of sources increases, restively, as L3 ~volume distribution! and L2 ~surfacedistribution!. In the case of primaries, part of the enhancment is also due to the mere fact that the number of souwithin the diffusive box increases withL ~we recall that thesources from the dark halo to be propagated are thoseclosed inside the diffusive box; see Ref.@14#!. Both effectsare responsible for the evolution ofS. For L*5 kpc, nofurther significant enhancement is observed, as the bulkthe primary sources~the core radius of the dark matter ditribution! is then almost entirely enclosed in the diffusivhalo. To be quantitative,S is increased by a factor of 3 foL515 kpc compared toL51 kpc. Notice that the quantityK(E)3Sastro
prim plotted in the left panel of Figs. 7, 8 and 9 donot depend onK(E). To understand this property it is sufficient to look at the expression forSastro
prim in Eq. ~12!.
2. The galactic wind Vc through rw
At high energy~generally a few tens of GeV!, propaga-tion is dominated by diffusion. At low energy convectiomay become the most efficient process~parameterr w'1;see Refs.@32,33#! and it may compete withL for escape. Theeffect of convection is to blow the particles away from tdisk, leading to an effective size of the diffusive haloL!
;r w . There is a difference from the effect ofL as the de-
ve
FIG. 8. Same quantities as in Fig. 7, but as a function ofpropagation parameterr wind , and for three values ofL (r sp5`).
FIG. 9. Same quantities as in Fig. 7, but as a function ofpropagation parameterr sp, and for three values ofL (r w5`).
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
crease ofr w does not lead to a decrease of the numberprimary sources enclosed in the diffusive volume. Howevit turns out that the effect of the galactic wind is also moimportant for primaries than for secondaries, as the fluxexponentially decreased withz for particles created at heighz in the diffusive halo. This can be seen in Fig. 8~left panel!.We clearly see the competition betweenL andr w in the rightpanel. For largeL, the evolution ofS is completely driven byr w , so that we can compare the result to those of FigWhen wind is present, the sensitivity to a signal is mumore reduced than that we would obtain with a similarL~i.e., a factor;25 in S between the casesr w51 kpc andr w515 kpc, compared to a factor of 3 forL in the samerange!. For small r w , all primary curves converge to thsame value, independently ofL, because then the cosmrays become blind to this boundary, being convected awbefore having a chance to reach the top or bottom of the b
3. The relative rate of spallation through rsp
At low energy, particles can be destroyed more easbecause the probability of crossing the disk, and thus inacting with matter, increases relative to the escape~diffusiveor convective! probability. The dependence ofS on r sp isdisplayed in Fig. 9.
When r spal increases, we are sensitive to sources locafarther away, and as forL and r wind , the effect is more im-portant for primaries than for secondaries. However, thefect of r spal is milder. This is because the cutoff due to splations is less efficient than escape or convective windpreventing particles coming from faraway sources froreaching us.
D. A comment about secondaries fromGALPROP model
Among several other models that are used to desccosmic-ray propagation, the fully numerical approach impmented inGALPROP@34# has been widely used. Some resuobtained within this framework, in particular when studyinthe secondary antiproton spectrum, seem to differ~see, e.g.,Ref. @35#! from ours, obtained with a semianalytical modIn our paper we want to derive constraints on the supersmetric contribution which can be added to the secondone, when confronting with data. Therefore, we take theportunity of this specific work to briefly summarize and dcuss some of the differences between the two approachestheir results.
First, the approximation that may appear crucial is thatorder to find analytical expressions for the cosmic ray dsity, we have to use a simplified description of the matdistribution in the galaxy, whereas with a numerical aproach any distribution can be considered. However, thesults are not strongly affected by this hypothesis. Inframework of steady-state diffusion models,@32# has shownthat the stable nuclei detected in the solar neighborhood wemitted from sources located in a region large enoughthat, having sampled very different regions of the galacdisk, they are sensitive to a mean density. Moreover, inducing a radial dependence of the matter distribution dnot induce sizable difference in the results@36#. In relation to
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this first point, we have to emphasize that Ref.@35# actuallydoes not use a detailed description of the local~i.e., on ascale of a few hundreds of pc! gas distribution. As a resultthe authors cannot provide a reliable analysis of the radiotive species, which are very sensitive to the local structurethe interstellar medium@31#.
Second, the numerical approach is still costly in termscomputation time, and is less suited to the systematic stof different effects. For example, Ref.@34# using a pre-defined small valued50.3 for the diffusion coefficient spectral index, finds that the observed spectrum of B/C requismall values of the galactic wind. Indeed, a full scan of tparameter space, extended to a range of values ford, re-vealed that models with higher values ofd and with largervalues of the galactic wind, were actually preferred. This aother results have been thoroughly discussed in Ref.@30#,and also compared with different propagation models~suchas GALPROP!. This point is of great importance for thpresent work, as the theoretical uncertainties in the antipton flux are underestimated if some parameters are not vaover all their plausible values.
The last relevant difference is actually not related toastrophysical model but to the production cross sectionsparticular, those relevant for B/C have been recently upda~see references in Ref.@35#!, whereas we use a standard s~see, references, e.g., in Ref.@19#!. This is a possible way toexplain the discrepancy between the secondary antiproflux, but we estimate that this is unlikely. Indeed, the two sof cross sections differ mainly at low energy, for which thweight of experimental data is not the greatest. Usingupdated set should not change the propagation paramderived from B/C and used to propagate antiprotons; the firesults would essentially remain unaffected.
To conclude, we do not see any physically relevant diffence between the two approaches, and they are probequally valuable. There is still some work to be done froboth sides to understand the origin of the differences inresults, which may lie in the methods and interpretationthe results, more than in the models themselves.
IV. RESULTS AND UNCERTAINTIES FOR THE PRIMARYFLUXES
We now use all the ingredients previously discussed~aswell as all energy changes! to evaluate the primary interstelar flux. We try to quantify all the uncertainties that couhamper a clear selection or exclusion of supersymmetric cfigurations. They are substantially induced by the degenerin propagation parameters~see Sec. IV A! @19,30# and thechoice of a peculiar dark matter profile~see Sec. IV B!.
A. Primary fluxes and related uncertainties
The propagation parameters have been constrained banalysis of the observed boron to carbon~B/C! ratio, bymeans of axB/C
2 test over 26 data points and five free paraeters@19#. The bestxB/C
2 was found to be 25.5. A value of 4was considered quite conservative, corresponding roughl
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ANTIPROTONS IN COSMIC RAYS FROM NEUTRALINO . . . PHYSICAL REVIEW D69, 063501 ~2004!
the 4s confidence level on B/C data interpretation, whilexB/C
2 530 can be assigned to about the 2s confidence level@19#.
In Fig. 10 we present the result for the primary antiprotflux for our reference source term formx5100 GeV, whoseg(E) is plotted in Fig. 6. We plot the fluxes correspondingthe parameters providing the maximal and minimal fluxwhen all the astrophysical configurations are taken tocompatible with the analysis on stable nuclei, i.e.,xB/C
2
,40. For the same set of astrophysical parameters weplot the secondary antiproton flux. The variation of thetrophysical parameters induces a much larger uncertaintthe primary than on the secondary flux: in the first case,uncertainty reaches two orders of magnitude for energTp&1 GeV, while in the second case it never exceeds 2~notice that these uncertainties are smaller than the nucones; see Ref.@18#!. A thorough discussion about whycombination of parameters gives the same secondary fluskipped here, but the reader is referred to Ref.@30# for moredetails. The large variation in the primary signal can be
FIG. 10. The solid lines represent the antiproton flux formx5100 GeV neutralino and for maximal, median, and minimastrophysical configurations, forxB/C
2 <40. Dotted lines: the samebut for xB/C
2 <30. The dot-dashed band corresponds to the secoary flux as taken from Ref.@19# for all the configurations givingxB/C
2 <40.
06350
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derstood from the previous discussion: first, the exotic sigis more sensitive to astrophysical parameters than the sdard, as already underlined. Second, this has to be weigby the fact that the secondary flux has in its source termadditionalK(E). While many combinations ofK0 , d, L, andVc lead to the same secondary flux, it is not straightforwato decipher which ones lead to the maximum and minimprimary fluxes. DecreasingL and r w decreases the flux, buat the same time, to keep the fit to B/C good,K0 has to bedecreased also@19,30#, in turn increasing the flux@primariesdepend on 1/K(E)]. However, the first two parameters amore important~especially the wind effect! than the latter.We give in Table I the values for these parameters yieldthe maximum and minimum of the error band in both pmary and secondary fluxes. The resulting variation in Fig.can be read off from Figs. 7, 8, and 9~left panels! and TableI: a factor;2000 because ofr w andL, an additional factor&4 for r sp ~see Fig. 9, left panel! divided by a factor;50because of the value ofK(E), leading to a net scattering o;100. This is almost independent of the specific supersymetric configuration.
As emphasized before, energy redistributions relate a scific supersymmetric configuration@by means ofg(Tp)] tothe given propagation configuration. The effect on the resing antiproton flux is expected to be mild. We show in F11 the result of our analysis for the representative EMSspectra shown in Fig. 6, corresponding tomx
560,100,300,500 GeV and for the median astrophysicalrameters. The low-energy behavior of the fluxes is similarall the masses: this is a consequence of the propagatiothe source spectra, which reduces the intrinsic differencethe original fluxes at low kinetic energies. The high-enerbehavior of the fluxes reflects the fact that for higher netralino masses the phase space for antiproton productiolarger. Since neutralinos in the galaxy are highly nonrelaistic, their mass acts as an effective cutoff on the antiproproduction kinetic energy.
The effect of propagation on the primary antiproton sptrum may also be shown by the following function@13#:
CSUSYprop ~Tp!5
F p~R(,0,Tp!
Yg~Tp!, ~20!
where F p((,Tp) is the interstellar antiproton flux aftepropagation, normalized to supersymmetric elementary pduction term. The propagation functionCSUSY
prop (Tp) is a mea-sure of how the source fluxes are deformed by propaga
l
d-
roton1
TABLE I. Astrophysical parameters giving the maximal, median, and minimal supersymmetric antipflux and compatible with B/C analysis (xB/C
2 ,40). r w andr sp ~kpc! are also given for two kinetic energiesGeV and 10 GeV.
Case d K0 L Vc VA xB/C2 r w ~kpc! r sp ~kpc!
(kpc2/Myr) ~kpc! ~km/s! ~km/s! @1 GeV/10 GeV# @1 GeV/10 GeV#
max 0.46 0.0765 15 5 117.6 39.98 29.0/73.0 26.0/57.0med 0.70 0.0112 4 12 52.9 25.68 2.4/9.2 4.4/15.0min 0.85 0.0016 1 13.5 22.4 39.02 0.33/1.8 0.69/3.1
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
and diffusion before reaching the solar position in the galaand is shown in Fig. 12 for the same representative spectFig. 6. The energy dependence is steeper for low-masstralinos, and it becomes somewhat more symmetric arothe maximal values for neutralinos of increasing mass. Tsteep rise ofCSUSY
prop (Tp) near the end of the antiproton production phase space atTp5mx is due to reaccelerationwhile the source factorg(Tp) is rapidly vanishing, the propagated flux F p(R(,0,Tp) decreases in a much milder wabecause of reacceleration effects. This effect is more pnounced for the maximal astrophysical configuration, whVA is maximal, and it disappears ifVA is set to zero. Figure12 also shows that the maximal, median, and minimal setastrophysical parameters affect not only the absolute matude of the fluxes but also their energy dependence: thetortion of the original flux differs depending on the valuesthe propagation parameters, as discussed in the previoustions. In particular, the energy of maximal transfer for netralino masses above 60 GeV shifts from about 1–2 GeVthe maximal set to 5–6 GeV for the minimal set. Figureshows, at low kinetic energies, a hierarchy in the behavioCSUSY
prop (Tp) which follows the hierarchy of the neutralinmasses: the propagation function is larger at low energiesheavier neutralinos, i.e., for harder antiproton fluxes.
The propagation functionCSUSYprop (Tp) can be used directly
to estimate the propagation effects once the supersymmproduction termYg(Tp) is known.
B. Uncertainties related to the dark matter distribution
We have performed all the calculations assuming thatgalactic dark matter is distributed as an isothermal sph
FIG. 11. Interstellar primary fluxes calculated as a functionthe antiproton kinetic energy. The fluxes are calculated for thedian set of astrophysical parameters. Solid, long dashed, sdashed, and dotted lines correspond tomx560,100,300,500 GeV,respectively. The fluxes correspond to the representative differeantiproton spectra shown in Fig. 6.
06350
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with a core radiusa53.5 kpc and local dark matter densitr l50.3 GeV cm23. For this density profile, we estimatethat the antiproton propagation induces an uncertainty onprimary antiproton flux of about two orders of magnitudespecially at low kinetic energies.
Another source of uncertainty on the primary flux comfrom the shape of the dark matter density profile, whichonly poorly known, and from the allowed range of valuesr l for any given density distribution. We have already comented that for an isothermal spherical distribution the lodark matter density may range from 0.18 GeV cm23 to0.71 GeV cm23. Moreover, the dark matter distribution mabe quite different from a simple isothermal sphere~see, forinstance, Refs.@21,37–40# and references therein!: the colddark matter distribution could be nonspherically symmetrit can be singular at the galactic center, as suggested fnumerical simulations, or it can present a clumpy distributin addition to a smooth component. Since the shape ofgalactic halo enters asrDM
2 (r ,z) in the evaluation of the astrophysical part for the primary signalSastro
prim(R(,0,E), it is amain ingredient in the determination of the primary antiprton flux, and the uncertainties in the description ofrDM(r ,z)may sizably affect the predicted signal.
The uncertainty inr l , determined by a detailed modelinof the galactic component@39,40# and mainly due to thevalue of the local rotational velocity@21#, depends on theshape of the galactic halo. For the same isothermal sphthe range inr l may change the primary fluxes by a factthat ranges from 0.36 to 5.6: overall, even for the simchoice of an isothermal sphere, the antiproton flux has
FIG. 12. Propagation functionCSUSYprop of the primary supersym-
metric antiproton fluxes as a function of the antiproton kinetic eergy, calculated for the reference fluxes of Figs. 6 and 11. Dolines refer tomx510 GeV, short dashed tomx560 GeV, longdashed tomx5100 GeV, dot dashed tomx5300 GeV, and solid tomx5500 GeV. For each set of curves, the upper, medium,lower lines refer to the maximal, median, and minimal sets oftrophysical parameters.
fe-ort
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ANTIPROTONS IN COSMIC RAYS FROM NEUTRALINO . . . PHYSICAL REVIEW D69, 063501 ~2004!
TABLE II. Sensitivity to the core radius of an isothermal profile, and comparison of the Navarro-FWhite ~NFW! and isothermal profiles, for three representative propagation sets atTp51 GeV. These propa-gation parameters correspond to the minimum, median, and maximum primary fluxes compatible withanalysis. The reference valueSref
prim is for an isothermal halo whose core radius isa53.5 kpc. Notice that forhigher energies the results would be the same as those provided by the setL515 kpc ~purely diffusivetransport!.
L~kpc!, r w , r sp Sa52.5prim 2Sref
prim
Srefprim
Sa55prim2Sref
prim
Srefprim
SNFWprim 2Sref
prim
Srefprim
15, 28.66, 25.54 269.5% 123.9% 119%4, 2.38, 4.41 221.5% 19.9% ;0%1, 0.33, 0.69 ,1% ,0.2% ;0%
ratg
of
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nd
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ir-lethe
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are
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ofhe
uncertainty of a factor of about 15, on top of the two ordeof magnitude due to antiproton propagation. We anticipthat, among all the uncertainties due to the shape of thelactic halo, the uncertainty coming fromr l will turn out to bethe most relevant one~apart from, eventually, the presenceclose clumps!.
Independently of the normalizationr l , any given densityprofile could, in principle, modify the signal. In particuladistribution functions derived from numerical simulatioare singular toward the galactic center, where a very hneutralino annihilation rate would then occur. We could thexpect that such dark matter profiles would induce anhanced antiproton flux with respect to a nonsingular disbution. In this class of modified density profiles we can ainclude an isothermal sphere with different values of the cradius a. We expect that enlarging the core radius wouincrease the signal. We therefore estimated the modificaof the cosmic antiproton flux when different core radii adark matter profiles are used in the source term. The reence flux is obtained with our spherical isothermal distribtion, with core radiusa53.5 kpc. The results are shownTable II. Notice that we have used for all the profiles tnormalization r l50.3 GeV cm23, in order to extract thechange of the antiproton flux which is due entirely to tdifferent shapes of the halos. It is clear that each denprofile will have to be further implemented with its owvalue ofr l @21,39,40#. From Table II we notice, first of all,that for smallL and r w we are completely blind to whaoccurs near the galactic center. Only the very local properof the dark matter distribution are of some relevance forstudy. For a diffusive halo of 4 kpc, we varied the core radof the isothermal distribution from 2.5 to 5 kpc. With respeto our reference values of 3.5 kpc, smalla leads to a reduc-tion of the flux by about 20%, while large values ofa give a10% increase. ForL515 kpc—and all the other propagatioparameters modified consequently—a 2.5 kpc core radiusminishes the reference flux by 70% and a 5 kpc one pushesup by 25%. The uncertainty of a factor of 2 on the coradius of the isothermal distribution then reflects in a facof 4 indeterminacy of the primary antiproton flux. As for thsingular density profiles, Table II shows that a NFW@38#distribution function does not strongly modify the flux: whecompared to the isothermal case, the flux is increased bmore than about 20%, and this occurs when the diffushalo size is the largest. ForL&5 kpc, the difference betwee
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an isothermal profile and a NFW singular distribution isrelevant. This result clearly shows that it is very improbabfor an antiproton produced at the galactic center to reachEarth.
Finally, one can deal with halos which contain regionsenhanced density called clumps. In these subhalos, thetralino annihilation is more effective and the signal canincreased by some enhancement factor. However, assuggested by Ref.@41#, this enhancement is not propagatiodependent and simply acts on the antiproton flux as a nmalization factor. From the analysis of Ref.@41#, the averageenhancement is likely to be smaller than a factor of 5.detailed analysis of this point is beyond the scope ofpaper; however, the effects of such an enhancementbriefly discussed at the end of Sec. V.
In conclusion, we wish to remark that our choice ofisothermal sphere with a core radiusa53.5 kpc and localdark matter densityr l50.3 GeV cm23, together with thebest choice for the astrophysical parameters which govdiffusion and propagation in the galaxy, represent an optimchoice for the prediction of the antiproton signal. Our resuwill not be dramatically modified by a different choice fothe density profile, while a different choice for the local damatter density is easily taken into account as a normalizafactor.
V. TOP-OF-ATMOSPHERE FLUXES:COMPARISON WITH DATA AND RESULTS
FOR SUPERSYMMETRIC MODELS
Now that we have calculated the interstellar fluxes of atiprotons at the Sun’s position in the galaxy, we havefurther propagate them inside the heliosphere, wherecosmic-ray particles which eventually reach the Earthaffected by the presence of the solar wind. We modeleffect of solar modulation by adopting the force field aproximation of the full transport equation@42#. In this model,the top-of-atmosphere antiproton fluxF p
TOA is obtained as
F pTOA
~EpTOA
!
F pIS
~EpIS
!5S pTOA
pIS D 2
~21!
where E and p denote the total energies and momentainterstellar and TOA antiprotons, which are related by tenergy shift
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
EpIS
5EpIS
2f, ~22!
where the parameterf is determined by fits on cosmic radata. In our analysis, we will adopt the valuef5500 MVfor periods of minimal solar activity, corresponding to thyears around 1995–1998,f5700 andf51300 MV for atransient period and for the solar maximum, respectivwhich will be used for years 1999 and 2000.
Figure 13 shows the TOA antiproton fluxes for themx
5100 GeV reference configuration and for the maximal aminimal sets of astrophysical parameters. The figure shthat solar modulation has the effect of depleting the loenergy tail of the antiproton flux. The effect is clearly mopronounced for periods of strong solar activity, when tsolar wind is stronger.
Data on antiprotons at the Earth are now abundant, moafter the missions of the balloon-borne detector BESS. Texperiment has provided measurements at different perof solar activity @43–45#. It has now collected more tha2000 antiprotons between 200 MeV and 4 GeV. Data at sminimum were also taken by the AMS experiment on bothe shuttle@46# in an energy range similar to BESS, andthe CAPRICE balloon at higher energies, namely, betweeGeV and 40 GeV@47#. All the data at solar minimum areplotted in Fig. 14 along with the secondary reference fl~for details, see Ref.@18#! and our predictions for primaryfluxes at different neutralino masses in the EMSSM:mx
560,100,300,500 GeV and for the median set of astrophcal parameters. We notice that the primary flux from netralino annihilation is at most of the same order of magnitu
FIG. 13. Top-of-atmosphere antiproton fluxes as a functionthe antiproton kinetic energy for themx5100 GeV reference caseThe upper~lower! set of curves refers to the maximal~minimal! setof astrophysical parameters. Solid curves show the interstefluxes. Broken curves show the effect of solar modulation at difent periods of solar activity:f5500 MV ~long dashed!, f5700 MV ~short dashed!, andf51300 MV ~dotted!.
06350
y,
ds
-
e
tlyisds
ard
5
x
i--e
as the secondary flux, and this occurs for neutralino masclose to their current lower bound in the EMSSM, whicharoundmx.50 GeV. We recall that the representative supsymmetric configurations plotted in Fig. 14 refer to a larantiproton production for each mass~i.e., they correspond tolarge values of theY parameter shown in Fig. 1!. This indi-cates that the antiproton signal for neutralino dark matter whardly produce an excess over the secondary flux, forisothermal matter profile of the galactic halo and forr l50.3 GeV cm23. This occurs for the median~and best!choice of the astrophysical parameters which govern thefusion and propagation of antiprotons in the galaxy. Cleathe maximal set of astrophysical parameters, which produfluxes about one order of magnitude larger than the medset, may produce a large excess, for neutralino masses b100–200 GeV. This excess could then be used to constsupersymmetric models since the secondary flux is perfecompatible with the data. However, for setting constraintssupersymmetry in a conservative way, we should insteadthe set of astrophysical parameters which produces the mmal fluxes. In this case, the primary fluxes are lower thanones plotted in Fig. 14 by about one order of magnitudediscussed in the previous section. In conclusion, our analshows that, due to the large uncertainties in the primfluxes, the antiproton signal is not suitable at presentsettingconservativeconstraints on supersymmetric modeFor this we need a better knowledge of the astrophys
f
ar-
FIG. 14. Primary TOA antiproton fluxes as a function of thantiproton kinetic energy, for the representative spectra of Fig.the EMSSM. The solid line refers tomx560 GeV, the long dashedline to mx5100 GeV, the short dashed line tomx5300 GeV, andthe dotted line tomx5500 GeV. The astrophysical parameters crespond to the median choice. Solar modulation is calculated fperiod of minimal solar activity. The upper dot dashed curve corsponds to the antiproton secondary flux taken from Refs.@18,28#.Full circles show the BESS 1995–1997 data@43#; the open squaresshow the BESS 1998 data@44#; the stars show the AMS data@46#;and the empty circles show the CAPRICE data@47#.
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ANTIPROTONS IN COSMIC RAYS FROM NEUTRALINO . . . PHYSICAL REVIEW D69, 063501 ~2004!
parameters that govern the diffusion and propagation ofmary antiprotons in the galaxy.
Antiproton data are also available for periods of intensolar activity from the BESS detector. Figure 15 shows thdata together with the secondary flux and the primary flcalculated for the representativemx5100 GeV configura-tion. The astrophysical parameters are fixed at their medvalues. We see that at solar maximum also the secondaryis compatible with the data and the supersymmetric fluxremarkably smaller than the secondary one.
In our discussion so far, we have commented thatconser-vative and solid constraints on supersymmetric modelsquire the use of the minimal set of astrophysical parametThis attitude is needed in setting limits. However, the band most probable choice of astrophysical parameters ismedian one, and we will therefore adopt from here on tset of parameters for our analyses. It is likely that a sharping of the knowledge of the propagation parameters will leto a shrinking of the allowed uncertainty band aroundcentral~median! value.
In order to compare the experimental results with a fscan of the supersymmetric parameter space, we calcthe TOA antiproton fluxes in two different energy bins acompare our results with the excess which can be accomdated above the secondary flux in order not to enterconflict with the experimental data in that energy bin. Whave chosen a low-energy binTp50.23 GeV, and a high-energy oneTp537.5 GeV. As can be seen in Fig. 14, in thlow-energy bin the secondary flux is perfectly compatib
FIG. 15. Primary TOA antiproton fluxes at solar maximum fthe transient periods of solar activity of the years 1999 and 20The upper set of curves shows the antiproton secondary fluxeslower set of curves shows the primary antiproton fluxes obtainedthe representativemx5100 GeV case. The solar modulation paraeter is fixed at 700 MV~solid lines! and at 1300 MV~dotted lines!.The astrophysical parameters correspond to the median case.and full circles correspond to BESS 1999 and 2000, respecti@45#.
06350
i-
eex
nuxs
-s.thesn-de
late
o-o
with the data, therefore no excess is needed: this allows uset an upper bound on the possible amount of antiprotonprimary origin which can be accommodated:F p
TOA(Tp
50.23 GeV)&2.0931023 m22 s21 sr21 GeV21. This valueis obtained by taking into account the values and uncertties of both data and secondary flux atTp50.23 GeV. AtTp537.5 GeV, even though the data and the secondaryare statistically compatible, a possible excess may be accmodated, since the central value of the experimental pindicates a much larger flux as compared to the secondcomponent. In this case, we can define an interval of valfor a possible excess: 0.0431023&F p
TOA(Tp537.5 GeV)&1.8731023 in units of m22 s21 sr21 GeV21. We comparethese intervals with our calculations in the EMSSM andMSUGRA.
Figure 16 shows the scatter plot of the antiproton flcalculated atTp50.23 GeV for a generic scan of thEMSSM scheme. The supersymmetric fluxes are clealargest at low neutralino masses, and they decrease asneutralino mass increases mostly because the neutranumber density in the galaxy scales asmx
22 . A small fractionof configurations with masses below 100 GeV can prov
0.her
tarsly
FIG. 16. ~Color online! Antiproton flux at solar minimum fromneutralino annihilation calculated atTp50.23 GeV, as a function ofthe neutralino mass for a generic scan of the EMSSM. The flucalculated for a smooth halo described by an isothermal profile wcore radiusa53.5 kpc and for the median set of astrophysical prameters. Crosses~red! refer to cosmologically dominant neutralnos (0.05<Vxh2<0.3); dots~blue! refer to subdominant relic neutralinos (Vxh2,0.05); light circles~in green! show the EMSSMconfigurations for which the neutralino relic abundance lies inpreferred range for CDM, as determined by the combined WM12DFGRS1Lyman-a analysis: 0.095<VCDMh2<0.131 @1#. Theshaded region~yellow! denotes the amount of antiprotons, in exceof the secondary component@18#, which can be accommodated aTp50.23 GeV in order not to exceed the observed flux, as msured by BESS@43,44#. All the points of the scatter plot that liebelow the horizontal black line are compatible with observation
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
fluxes which could be potentially at the edge of producingexcess, but we recall here that for a safe exclusion of thconfigurations we should use the minimal set of astrophcal parameters, which provides a flux that is about one oof magnitude smaller. In any case, a reduction of the untainties on the primary flux calculation and a future reductof experimental errors may eventually allow one either tolimits to supersymmetry or to show a positive excess oftiprotons in this low-energy bin, a fact which could thenexplained as originated by neutralinos of masses belowGeV. Figure 17 shows the scatter plot of the antiproton flcalculated at Tp537.5 GeV for the same scan of thEMSSM. In this case we observe that all the supersymmeconfigurations are compatible with data, but there are nopersymmetric models that allow us to explain the discrancy between the data and the secondary flux as due texcess of supersymmetric origin.
The situation in the MSUGRA scheme is shown in Fig.with the flux of antiprotons atTp50.23 GeV. In this case, aalready observed in connection with the properties ofMSUGRA source term, the antiproton fluxes are smaller thin the EMSSM case. Nevertheless, a restricted fractionMSUGRA configuration is potentially explorable in the future, with a reduction of the experimental error of aboufactor of 2–3.
The fluxes we have shown so far all refer to a dark madensity distribution in the form of a cored isothermal spheClearly, a halo profile that is able to produce an overdenwith respect to the isothermal sphere would produce a laantiproton flux. We can parameterize the enhancement o
FIG. 17. Antiproton flux at solar minimum from neutralino anihilation calculated atTp537.5 GeV, as a function of the neutralino mass for a generic scan of the EMSSM. Notations are aFig. 16. The shaded region~in yellow! denotes the amount of antprotons which would be required atTp537.5 GeV in order to ex-plain the possible excess in the BESS data@43,44# over the second-ary component@18#. All the points of the scatter plot that lie belowthe upper horizontal black line are compatible with observation
06350
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matter density by a multiplicative factorh, which then entersash2 in the calculation of the antiproton primary flux, sincthe flux depends on the square of the matter density.origin of the overdensity may be due, for instance, to flatting of the galactic halo or to the presence of clumps. Itlatter case, the enhancement factor is likely to be smathan about 5, once the results of Ref.@41# are implementedwith our discussion on the antiproton diffusion region in tgalaxy. The enhancement factor may also be relateddifferent choice of the local dark matter density, which hbeen fixed atr l50.3 GeV cm23 in our analysis. We recalthat our cored isothermal sphere allows factors ofh up toh;(0.71/0.3)52.4 @21#. Clearly, a complete reanalysis othe propagation and diffusion properties will be requiredeach different choice of the halo shape: this reanalysisgive the amount of enhancement concerning the spehalo. In any case, regardless of how the enhancementh isobtained, we can discuss the effect of such an increasedby usingh as a normalization factor to show the amountenhancement which is required in order to interpret the aproton excess atTp537.5 GeV as due to neutralino darmatter, without exceeding the upper limit on the antiprotflux at Tp50.23 GeV. Figure 19 shows the correlation btween the EMSSM antiproton fluxes atTp50.23 GeV andTp537.5 GeV forh53 and 10. The supersymmetric configurations which could satisfy this requirement are the othat fall inside the shaded area in Fig. 19. We see thatpossible excess atTp537.5 GeV requires halo overdensitieof at least a factor of 2–3 and neutralino masses larger ta few hundreds of GeV. This last property is simply undstood on the basis of the fluxes shown in Fig. 14: the phspace cutoff atTp5mx implies that a light neutralino wouldneed a huge overdensity factor in order to match theserved antiproton flux atTp537.5 GeV, but this would pro-duce an exceedingly large flux atTp50.23 GeV. On the
in
FIG. 18. The same as in Fig. 16, for a generic scan ofMSUGRA scheme.
1-16
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ANTIPROTONS IN COSMIC RAYS FROM NEUTRALINO . . . PHYSICAL REVIEW D69, 063501 ~2004!
contrary, heavy neutralinos have a phase space cutomuch higher kinetic energies, and therefore a mild overdsity may enhance the flux atTp537.5 GeV without givingconflict at low kinetic energies.
VI. CONCLUSIONS AND PERSPECTIVES
We have calculated the flux of antiprotons producedrelic neutralino annihilations in the galactic halo in a detaidiffusion model constrained by analysis of stable and radactive nuclei. The source of antiprotons is studied both ilow-energy minimal supersymmetric standard model andsupergravity-inspired supersymmetric scheme. We findthe interstellar primary antiproton fluxes are affected bylarge uncertainty, which spans two orders of magnitudelow antiproton kinetic energies. This is at variance with tsecondary antiproton fluxes~whose uncertainty never exceeds 24%@19#! and it is mainly related to the fact that thsource of the primary flux is located inside the diffusive hawhose size is unknown. By adopting a conservative chofor the dark matter density profile and propagation paraeters, no supersymmetric configuration can be excludedthe basis of an excess over the existing data. Actually,data are quite well explained by the secondary contribualone. However, if we adopt the best values for the propation parameters~corresponding to a thickness of the diffusivhalo of 4 kpc!, a window of low-mass neutralino configurations provides fluxes which, once summed up to the secoary contribution, are in excess of the experimental measments. We have shown that the sensitivity to the antiprosignal is increased with the halo size and limited by stroconvection. An improved knowledge of the propagation p
FIG. 19. ~Color online! Correlation between the antiproton fluat Tp50.23 GeV andTp537.5 GeV shown in Figs. 16 and 17 fothe astrophysical enhancement parameterh53 and 10~overdensehalos! and for the median set of astrophysical parameters. Cir~green!, dots ~blue!, and crosses~red! denote configurations withneutralino masses in the ranges 50 GeV,mx,150 GeV,150 GeV,mx,300 GeV, and 300 GeV,mx,1 TeV, respec-tively. The horizontal line denotes the upper limit on the antiproflux at Tp50.23 GeV coming from the BESS data@43,44#, once thesecondary component@18# is taken into account. The rightmost vetical line denotes the corresponding upper limit atTp537.5 GeV.The shaded area indicates configurations which can explain thesible excess in the data@43,44# over the secondary component@18#at Tp537.5 GeV, without giving an excess at low kinetic energ(Tp50.23 GeV).
06350
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-aaataat
,e-
onena-
d-e-ng-
rameters will certainly help in reducing the uncertaintythe primary flux and consequently it could allow us to smore severe constraints on the supersymmetric paramspace.
The sensitivity of the primary antiproton flux to the shaof the dark matter density profile has also been investigaWe have shown that the shape of the dark matter dendistribution does not introduce large uncertainties. In partilar, we have demonstrated that a NFW distribution cancrease the primary antiproton flux by no more than 20% wrespect to the isothermal profile. Indeed, it is very improable to detect at the Earth antiprotons produced in the cenregions of the galaxy, where the two distributions diffmost.
In future years several balloon-borne experiments suchBESS, space-based detectors such as AMS, and satesuch as PAMELA will provide very abundant and accuradata on the antiproton flux. At the same time, new datacosmic-ray nuclei are expected and would lead to a beknowledge of the cosmic-ray propagation mechanisms.could thus expect a dramatic reduction of the uncertainaffecting the neutralino-induced antiproton flux, and mumore definite predictions for antiprotons of supersymmeorigin will then be possible. Much effort is also devotedother indirect neutralino searches, such as positrons andtideuterons in cosmic rays, gamma rays, and up-gomuons, as well as to direct searches in deep undergrolaboratories, thus giving the hope that a more constrainanalysis on the existence of relic neutralinos in the haloour galaxy will be viable.
ACKNOWLEDGMENTS
We warmly thank Professor A. Bottino for very usefdiscussions. N.F. wishes to acknowledge the warm hospity and support of the Korean Institute for Advanced Stu~KIAS!, where part of this work was done.
APPENDIX A: CALCULATION OF THE ANTIPROTONDIFFERENTIAL SPECTRUM PER ANNIHILATION
EVENT
The antiproton differential spectrum per annihilatioeventg(Tp) is calculated by following analytically the decachain of the neutralino annihilation products until a quarka gluonh is produced. The antiproton spectrum is then otained by a Monte Carlo modeling of the quark and gluhadronization~we make use of thePYTHIA package@24#!. Wehave produced thep differential distributions for anh5u,d,s,c,b gluon at various injection energies for eachh~the t quark is assumed to decay before hadronization antreated analytically, since in addition to its standard modecay intoW1b, it may have a supersymmetric decay inH1b). Whenever we need thep distribution for an injectionenergy different from the ones produced, we perform anterpolation. In order to obtain the antiproton differential dtribution in the neutralino rest frame we perform the necsary boosts on the MC spectra.
For instance, let us consider ap production from a neu-
s
os-
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DONATO et al. PHYSICAL REVIEW D 69, 063501 ~2004!
tralino decay chain of this type:
xx→A→a→h p. ~A1!
The antiproton differential spectrum per annihilation eveg(Tp) is then obtained by the product of the branching ratfor the production ofA, a, andh in the decay chain, with thedifferential distribution of antiprotons produced by the haronization of anh injected at an energyEprod ~defined in therest frame of thea decaying particle! double boosted to thexreference frame:
g~Tp!5BR~xx→A!BR~A→a!BR~a→h!
3F S dNph
dTpD
boost a→A
Gboost A→x
. ~A2!
The first boost transforms the spectrum from the rest fraof a ~in which h is injected with energyEprod) to the restframe of A. The second brings the distribution to the reframe ofx. Each boost is obtained by the following expresion:
g~Ep!51
2EE
28
E18 S dNp
h
dE8DU
Eprod
dE8
gbp8~A3!
where Ep5Tp1mp is the total antiproton energy,p8
5A(E821mp2), g andb are the Lorentz factors of the boos
and the interval of integration is defined by
E68 5minFEprod,gEpS 16bA12mp
2
Ep2 D G . ~A4!
APPENDIX B: SOLUTION OF THE DIFFUSIONEQUATION
In cylindrical geometry, the differential densityNp(r ,z,E)is given by
05H K~E!F ]2
]z21
1
r
]
]r S r]
]r D G2Vc
]
]zJ Np~r ,z,E!
1Q p~r ,z,E!22hd~z!G p~E!Np~r ,z,E!, ~B1!
where the energy losses have been omitted, for the sakclarity. The source term includes primary antiprotons, froexotic sources present in the dark halo, annihilating throu
l
tte
06350
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-
e
t-
of
h-
out the diffusive halo of the galaxy, and secondary antiptons, standardp and He CR’s spallating on the interstellagas in the thin disk, and may be written as
Q p~r ,z,E!5qp,prim~r ,z,E!12hd~z!qp,sec~r ,0,E!.~B2!
A convenient way to solve Eq.~B1! is to expand all thefunctions f (r ) @the densityN(r ) and the source distributionq(r )] that depend onr on the orthogonal set of Bessel funtions$J0(z ix)% i 51, . . . , ~wherez i are the zeros ofJ0). TheseBessel transforms are defined as
f ~r !5(i 51
`
f iJ0S z i
r
RD , ~B3!
with
f i52
J12~z i !
E0
1
r f ~rR!J0~z ir!dr. ~B4!
Using the Fourier-Bessel coefficientsNip(z,E) and
qip(z,E), there is no conceptual difficulty to extract solution
of Eq. ~B1!. We do not repeat the derivation, which can bebit lengthy. Solutions for primaries can be found in Ref.@14#,whereas the one for secondaries has been discussed in@18#.
1. Energy losses, tertiary component
Following the procedure described, e.g., in Ref.@19#, en-ergy losses and diffusive reacceleration lead to a differenequation onNi(z50,E):
AiNi~0,E!5Qi~E!22h]
]E H bloss~E!Ni~0,E!
2KEE
]Ni~0,E!
]E J , ~B5!
where bloss5bion1bCoul1badiab includes the three kinds oenergy losses. The exact forms for these terms may be foin Refs.@19,28#. The resolution of this equation proceedsdescribed in Appendixes A.2, A.3, and B of Ref.@18#, towhich we refer for further details. The source term also tainto account the so-called tertiary componentqi
ter(E), corre-
sponding to an inelastic but nonannihilating reaction ofp oninterstellar matter. This mechanism merely redistributestiprotons toward lower energies and tends to flatten thspectrum.
.
d
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