Antideuteron fluxes from dark matter annihilation in diffusion models

18 Pages·2013·0.6 MB·English

DFTT 06/2007
Antideuteron fluxes from dark matter annihilation in diffusion models
F. Donato∗
Dipartimento di Fisica Teorica, Universit`a di Torino
Istituto Nazionale di Fisica Nucleare, via P. Giuria 1, I–10125 Torino, Italy
N. Fornengo†
Dipartimento di Fisica Teorica, Universit`a di Torino
Istituto Nazionale di Fisica Nucleare, via P. Giuria 1, I–10125 Torino, Italy
D. Maurin‡
Laboratoire de Physique Nucl´eaire et Hautes Energies, CNRS-IN2P3/Universit´e Paris VI et VII,
8
0 4 Place Jussieu, Tour 33, 75252 Paris Cedex 05, France
0 (Dated: June10, 2013)
2
Antideuteronsare among themost promising galactic cosmic ray-related targets for dark matter
r
a indirectdetection. Currentlyonlyupperlimitsexistontheflux,butthedevelopmentofnewexperi-
M ments,suchasGAPSandAMS–02,providesexcitingperspectivesforapositivemeasurementinthe
near future. In this Paper, we present a novel and updated calculation of both the secondary and

8 primary d fluxes. Weemploy a two–zone diffusion model which successfully reproduces cosmic–ray
1 nucleardata and theobserved antiproton flux. Wereview thenuclearand astrophysical uncertain-

ties and provide an up to date secondary (i.e. background) antideuteron flux. The primary (i.e.
]
h signal) contribution iscalculated for genericWIMPsannihilating inthegalactic halo: weexplicitly
p considerandquantifythevarioussourcesofuncertaintyinthetheoreticalevaluations. Propagation
- uncertainties,asisthecaseofantiprotons,aresizeable. Nevertheless,antideuteronsofferanexciting
p
target for indirect dark matter detection for low and intermediate mass WIMP dark matter. We
e
h thenshowthereachingcapabilitiesofthefutureexperimentsforneutralinodarkmatterinavariety
[ of supersymmetric models.

1
PACSnumbers: 95.35.+d,98.35.Gi,11.30.Pb,12.60.Jv,95.30.Cq
v
0
4
I. INTRODUCTION nos, gamma rays, positrons and antiprotons are already
6
2 available at a sensitivity level allowing some inspection
. on possible exotic contributions. In the seminal Paper
3 Theidentificationandtheunderstandingofthenature
0 ofdarkmatter(DM)isoneofthedeepestopenproblems, [1], it was proposed to look for cosmic antideuterons (d)
8 as a possible indirect signature for galactic dark mat-
together with the solution to the dark energy mystery,
0
ter. It was shown that the antideuteron spectra deriv-
: in the fundamental physics research. Many experimen-
v ing from DM annihilation is expected to be much flat-
tal efforts devoted to the detection of the astronomical
i
X dark matter in the halo of our and nearby galaxies have ter than the standard astrophysical component at low
kinetic energies, T < 2-3 GeV/n. This argument mo-
r been carried out in underground laboratories, in large- d
a tivated the proposal∼of a new space-borne experiment
area surface telescopes as well as in space. In the near
[2,3,4]lookingforcosmicantimatter(antiprotonandan-
future, the LHC willprovideus with invaluableinforma-
tideuteron) and having the potential to discriminate be-
tion on particle physics extending beyond the Standard
tween standard and exotic components for a wide range
Model, thus probing a wide class of theoretical models
of DM models. AMS–02 has also interesting capabilities
hosting the most viable DM candidates.
of looking for cosmic antideuterons [5]. Antideuterons
The indirect dark matter detection is based on the
have not been measured so far, and the present experi-
search for anomalous components due to the annihila-
mental upper limit [6] is still far from the expectations
tion of DM pairs in the galactic halo, in addition to the
on the secondary antideuteron flux which are produced
standard astrophysical production of neutrinos, gamma
by spallation of cosmic rays on the interstellar medium
raysandlightantimatterincosmicrays. Dataonneutri-
[7,8],butinfactperspectivesforthenearfuturearevery
encouraging.
InthepresentPaperweupdateandimproveourcalcu-
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] lationoftheantideuteronprimaryfluxinafulltwo–zone
‡Electronicaddress: [email protected] diffusion model, consistent with a number of indepen-
2
dent cosmic ray (CR) measurements, and explicitly esti- turbulent magnetic fields of the Galaxy. As well as be-
matetheuncertaintieswhichaffectthesignaldetermina- ing responsible for spatial diffusion, the Alfv´enic waves
tion. In addition, we also provide a new determination also lead to energy drift and reacceleration. A minimal
of the secondary component. In Sect. II, the framework reaccelerationschemeiswell-motivated[12]andallowsto
andingredientsforthesolutionofthetwo-zonetransport calculate the f and s coefficients. Similar albeit more
o o
equationarerecalled. InSect.III, the coalescencemodel empiricalforms have alsobeen used [13, 14]. In allthese
for the nuclear fusion process is discussed both for sec- models,thestrengthofthereaccelerationismediatedvia
ondary and primary d production. Sect. IV is dedicated the Alfv´enic speed V of the scatterers.
a
to the secondary d flux and to the possible uncertain-
ties affecting its evaluation. In Sect. V the production
of antideuterons from DM particles is detailed and re- A. The two-zone disk-halo model
sults on the propagated fluxes are presented, together
with the estimation of the uncertainties due to propa- A full numerical treatment is generally required to
gation, to the dark matter halo profile and to the DM solvethetransportequation,asdescribed,e.g.,inStrong
annihilation final states. Finally, Sect. VI demonstrates and Moskalenko [15]. However, analytical (or semi-
that antideuterons are probably one of the most power- analytical) solutions may be derived assuming a simpli-
fuldarkmatterindirectdetectionchannel,andshowsthe fieddescriptionofthespatialdependenceofsomeparam-
optimistic potentials of next-to-come balloon and space eters in Eq. (1). The two-zone diffusion model [16, 17],
based missions. We finally draw our conclusions in Sect. basedon the description of the Galaxy as a thin gaseous
VII. diskembeddedinathickdiffusivehalo,provedtobesuc-
cessfulinreproducingthenuclear[14,18,19],antiproton
[20] and radioactive isotopes data [19]. It also allows to
II. THE PROPAGATION MODEL
treat contributions from dark matter (or other exotic)
sourceslocatedin the diffusive halo [21, 22, 23], whichis
Cosmic ray fluxes are determined by the transport
the aim of this Paper. We remind below the salient fea-
equationasgiven,e.g.,inBerezinskiietal.[9]. Ifsteady-
tures of this model, which has been extensively detailed
state is assumed, the transport equation for any nuclear
in Refs. [14, 24, 25].
species can be rewritten in terms of the cosmic ray dif-
a. Geometry(L,Randh). TheGalaxyisdefinedas
ferential density dn(~r)/dE N(~r) as:
≡ a cylinder with a diffusive halo of half-height z =L and
∂ ∂N(~r) radius r = R. The halo thickness L is a free parameter
~ K~N(~r) V~ N(~r) f N(~r)+s
c o o of the model. The interstellar (IS) gas and the nuclei
−∇ ∇ − −∂E − ∂E
(cid:20) (cid:21)
h i accelerators are contained in a thin-disk of half-height
=Q (~r) n(~r)vσ N(~r). (1)
source ine
− h L. The two parameters h and R are set to 100 pc
≪
The l.h.s describes the spatial diffusion (K) and convec- and 20 kpc, respectively.
tion (Vc), and the energy transport (first and second or- b. Diffusion coefficient (K0 and δ). Diffusion arises
der terms). The r.h.s corresponds to the primary, sec- becausechargedparticlesinteractwiththegalacticmag-
ondary and tertiary (only for antinuclei) source terms, netic field inhomogeneities. The diffusion coefficient
and the sink (spallative destruction) for the considered K(~r,E) is related to the power spectrum of these inho-
species. The most general form for the diffusion coeffi- mogeneities, which is poorly known. Several analytical
cientisK(E,~r). Energygainandlossesdependon(E,~r) forms for K have been assumed in the literature. We
as well. The first order term f (E,~r) corresponds to consider here the standard rigidity ( = pc/Ze) depen-
o R
the sum of four contributions: ionization, Coulomb and dent form K(E) = βK0 δ, where the normalization
×R
adiabatic losses, and first order reacceleration. Ioniza- K0 is expressedin units of kpc2 Myr−1. The same diffu-
tion losses take place in the neutral interstellar medium sioncoefficientisassumedthroughouttheGalaxy,i.e. in
(ISM), while the Coulomb ones in the completely ion- the disk and in the halo. K0 and δ are free parameters
ized plasma, dominated by scattering off the thermal of the model.
electrons [10, 11]. The spatial dependence of these two c. Galactic wind and Alfv´enic speed (V and V ).
c a
terms is encoded in the distribution of the neutral and The convective wind is assumed to be of constant mag-
ionized gas. Adiabatic losses are due to the expanding nitude directed outwards perpendicular to the Galactic
wind and their spatial dependence is related to the gra- plane V~ =V ~e . The reacceleration strength, mediated
c c z
dient of V~ . The last contribution to f (E,~r) has the by the Alfv´en velocity V , is confined to the thin disk.
c o a
same origin as the second-order term s (E,~r). Those The first and second order terms f and s in Eq. (1)
o o o
come from the scattering of the chargedparticles off the follow the formulation given in Ref. [14].
3
The cross sections in Eq. (3) refer to inelastic non-
case δ K0 L Vc VA χ2B/C annihilating processes and are detailed in the Appendix.
(kpc2/Myr) (kpc) (km/s) (km/s) Thetertiarymechanismdoesnotactuallycreatenewan-
max 0.46 0.0765 15 5 117.6 39.98 tideuterons. It merely states that the number of antin-
med 0.70 0.0112 4 12 52.9 25.68 uclei observed at energy E has to take into account the
min 0.85 0.0016 1 13.5 22.4 39.02 redistribution of those with energy E′ > E (first term,
positive contribution), minus the total number of d re-
distributedto lowerenergies(secondterm, negativecon-
TABLEI:Transport parametersprovidingthemaximal, me-
tribution). The tertiary contribution is treated as a cor-
dian and minimal primary antideuteron flux and compatible
rective factor and dealt with iteratively: the equilibrium
with B/C analysis (χ2 <40) [14, 24].
B/C spectrum N(0)(~r,E) is first calculated with Q(0) 0,
ter ≡
then Q(1) calculated with N(0)(~r,E) in Eq. (3) to ob-
ter
The free parameters of this propagation model—the tain N(1)(~r,E), etc. For antideuterons, due to the small
diffusive halo size L, the normalizationK andthe slope crosssection—asshownin Ref. [8]—only one iterationis
0
δ of the diffusion coefficient, the value of the constant necessaryto convergetothe solution(comparedtoafew
galactic wind V and the level of reacceleration through iterations for antiprotons [20]).
c
the Alfv´enic speed Va—were constrained from the study Hence, whether secondaryor primary(or a mixture of
of the B/C ratio in Refs. [14, 24]. In this Paper we take both) sources are considered, three cross sections always
advantageoftheparametersfoundinRef.[14]andlisted enter the calculation: the differential non-annihilating
in Table I. Actually, when fitting to existing B/C data, inelastic cross section dσnon−ann/dE, the total non-
a strong degeneracy of the transport parameters is ob- annihilating inelastic cross section σnon−ann and the to-
served, meaning that many sets of these five parameters talannihilating inelastic crosssectionσann [appearingin
are acceptable and lead to the same B/C ratio, but also the r.h.s. of Eq. (1)]. Considering the ISM as a mix-
to the same secondary (standard) antiproton flux [20]. ture of H and He, the full calculation requires six cross
This degeneracy is broken for sources located in the dif- sections. Our calculations, based on the parameteriza-
fusive halo, leading to large astrophysical uncertainties tions discussed at length in Duperray et al. [8], are dis-
for the relevantfluxes [21, 26, 27]. The same conclusions cussed in the Appendix where, in particular, we update
are obtained here for antideuterons. We will come back dσnon−ann/dE. Note however,that even if some of these
to this point in Secs. IV and V. cross sections are slightly modified, the impact on the
propagated spectra do not change the conclusions found
in Ref. [8].
B. The case of antideuterons
In the two-zone diffusion/convection/reacceleration
model, it is possible to extract semi-analytical solutions
Once the astrophysicalframeworkfor the transportof
of Eq. (1), based on Bessel expansions of the transport
(anti)nucleiisset,thecalculationoftheantideuteronflux
equation. We do not wish to repeat the various steps of
rests onthe d specificities regardingthe sourceterm and
this derivation, nor to rewrite the complete form of the
its nuclear interactions [r.h.s. of Eq. (1)]. The source
solutions, which have already been given in several pa-
term for antinuclei is usually cast in separate contribu-
pers. The solution for the so-called antideuteron stan-
tions:
dard source (of secondary origin in the galactic disk)
and the numerical procedure to treat reacceleration is
Q (~r,E)=Q (~r,E)+Q (~r,E)+Q (~r,E).
source prim sec ter
detailed in Ref. [20]. The solution for an exotic source
(2)
distributed in the whole diffusive halo of the Galaxy can
Among these three terms, only primary and secondary
befoundinRef.[21,26]. Actuallythesetwopapersrefer
are true sources. They will be discussed in Secs. IV
to p, but formally, the solutions apply to d as well.
and V.
The tertiary term was emphasized in Ref. [28] to de-
scribe the process corresponding to the non-annihilating
III. ANTIDEUTERON PRODUCTION
interaction of a CR antinucleus with an atom of the IS
gas. In a medium of constant density n (v is the CR
velocity): The production of cosmic antideuterons is based on
the fusion process of a p and n pair. One of the sim-
+∞ dσnon−ann
Q (~r,E) = nv′ d¯H→d¯X E′ E N(~r,E′)dE′ plestbutpowerfultreatmentofthefusionoftwoormore
ter
ZE dE { → } nucleons is based on the so–called coalescence model
nv σnon−ann E N(~r,E). (3) which, despite its simplicity, is able to reproduce re-
− d¯H→d¯X { }
4
markably well the available data on light nuclei and second antinucleon is reduced by twice the energy car-
antinuclei production in different kinds of collisions. In ried away by the first antinucleon. Instead, in Duperray
the coalescence model, the momentum distribution of et al. [8] the threshold production was phenomenologi-
the (anti)deuteron is proportional to the product of the cally taken into account through an A+2 phase space
(anti)proton and (anti)neutron momentum distribution factor. The latter description seems more appropriate
[29, 30]. That function depends on the difference ∆ as,while preservingthe correctasymptotic properties,it
~k
between (anti)nucleon momenta. It is strongly peaked does not favor any mechanism for the pair production
around∆ ~0 (compare the minimum energy to form a [34].
~k ≃
d,i.e. 4m ,withthe bindingenergy 2.2MeV),sothat The coalescence momentum p is linked to the mea-
p 0
∼
sured coalescence factor B (hereafter simply B ):
A=2 2
~kp¯≃~kn¯ ≃ ~k2d¯ . (4) d3σR dσR −2
Theddensityinmomentumspaceisthuswrittenasthep B2 ≡ σiRnel·Ed¯ d~kd¯d¯ · Ep¯d~kp¯p¯! , (8)
densitytimestheprobabilitytofindann¯ withinasphere
so that
of radius p around~k (see, e.g. Ref. [31]):
0 p¯
−1/3
γdNd¯ = 4πp3 γdNp¯ γdNn¯ . (5) p0 = B1 · mm2d¯· 43π . (9)
d~kd¯ 3 0· d~kp¯ · d~kn¯ (cid:18) 2 p¯ (cid:19)
The B coefficienthas been measuredfor proton-proton,
2
The coalescence momentum p is a free parameter con-
0 proton-nucleus and heavy ion collisions (see a summary
strained by data on hadronic production. Note that the
and references in Refs. [8, 35]). More recently, several
coalescence model has been refined to account for heavy
other channels have also been measured at high energy:
nucleireactions[see, e.g.32,33], but as it is notrelevant
photo-production [35], DIS production [36] and e+e−
for this study, we will stick to the simple Eq. (5).
productionattheZ [37]andΥ(1S)[38]resonances. The
The number d R of particles X produced in a single
reaction R and wNhXich momenta are~k can be expressed e+e− channelis ofparticularinterestforthe DMannihi-
X lation reactions.
as a function of the total available energy √s, the inclu-
sive(i.e. totalinelasticorreactioncrosssection)andthe
differential cross section: A. Hadronic production
1
d R = d3σ (√s,~k ). (6)
NX σR X X For the hadronic processes, the coalescence momen-
inel
tum can directly be fitted to data. However, different
For instance, in our specific case X are the antinucleons assumptions regarding the set of data to retain can lead
andantideuteronscreatedinthepp,pHeandHeHereac- to differentvalues of p . Note that many recentdata are
0
tionsbetweenp-HeCRsandH-HeintheISM.Assuming available for A+A systems, in addition to pp and pA
the usual equality between the unmeasured n and the reactions. However,themechanismsatplayinheavyion
measured p cross sections, and combining the two previ- collisionsarenotnecessarilythoseoflightersystems(see,
ous expressions Eqs. (5) and (6) we get: e.g., discussionin Sect. II.A of Ref. [8] and the results of
Ref. [39]), so that these reactions are discarded in the
2
Ed¯dd3~kσd¯dR¯ = σiR1nel · 43πp30· mmd2p¯¯· Ep¯ddσ~kp¯Rp¯! . (7) resCthoafroduornnaentaleytsiasl.. [7] used pp data from Refs. [40, 41,
42]andpAcollisionsfromRefs. [42](seeFig.1inRef.[7])
The hypothesis of factorization of the probabilities is to derive a coalescence momentum p = 58 MeV. Based
0
fairly well established from experiments at high energies onkinematicalrelevanceofthemeasuredreactions,these
[see, e.g., 8]. For spallation reactions, however, the bulk authorsdisfavoredtheppdatafromRef.[41],butunder-
of the antiproton production takes place for an energy lined that a value p = 75 MeV, compatible with their
0
√s 10GeV,whichturnsouttobeofthesameorderof whole set of data, would merely provide twice as more
∼
magnitude as the antideuteron mass. Pure factorization antideuterons.
should break in that case as a result of energy conser- In Duperray et al. [8], a larger set of data is used,
vation. Two ansatz have been used in order to correct including many pA reactions (see their Tab. I and ref-
that effect for this regime: in Ref. [1, 7] it was assumed erences therein). The approach is more sound since a
that,whilethefirstantinucleonisproducedwith√s,the χ2 analysis on the momentum distribution of the frag-
center of mass energy availablefor the production of the mentswasperformed,takingalsointoaccountthe phase
5
space. This leads to an estimate of p = 79 MeV. Note
0
that at variancewith the choice of Chardonnetet al. [7],
Duperray et al. discarded the data from Ref. [42] be-
cause they give a poor χ2 value compared to all other
data. It is thus not surprising that these authors end up
withavalueclosetop =75MeVquotedinChardonnet
0
etal.[7]. InthepresentPaper,wetakedirectlythecross
sections derived in Duperray et al. [8] using the value
p =79 MeV.
0
B. Weak production
At LEP energies, (anti)deuteron production occurs
through e+e− annihilations into qq¯ pairs, a mechanism
similartothedproductioninDMannihilationreactions.
Based on theoretical arguments, it has been argued [31]
that the antideuteron yields in e+e− reactionsshould be
FIG. 1: Contribution of all nuclear channels to the d sec-
smallerthaninhadronicreactions. However,theALEPH
ondaryflux. Dashedlines,fromtoptobottomreferto: p+H,
Collaboration[37]hasfoundthatthis theoreticalpredic-
p+He, He+H, He+He. Dotted lines, from top to bottom
tion (see Fig. 5 in ALEPH paper) underestimates their stand for: p+H, p+He. Solid line: sum of all the compo-
measureddinclusivecrosssection. Theyderive(seetheir nents.
Fig. 6) a value B =3.3 0.4 0.1 10−3 GeV2 at the
2
± ± ×
Z resonance,which translates into p =71.8 3.6 MeV,
0
very close to the p = 79 MeV derived for th±e hadronic areevaluatedusingthepfluxcalculatedinthesamerun.
0
production. Hence, in the remaining of the Paper, the Theproductioncrosssectionsforthesespecific processes
value of p = 79 MeV will be retained for both the pro- are those given in Ref. [8].
0
cesses of hadronic and electroweak origin. The different contributions to the total secondary an-
tideuteron flux, calculated for the best fit propagation
configuration (the “med” one in Table I), i.e. K =
0
IV. SECONDARY ANTIDEUTERONS 0.0112 kpc2 Myr−1, L = 4 kpc, Vc = 10.5 km s−1 and
V = 52.1 km s−1, are shown in Fig. 1. As expected,
a
thedominantproductionchannelistheonefromp-pcol-
Secondary antideuterons are produced in the galactic
lisions, followed by the one from cosmic protons on IS
diskfromthecollisionsofcosmicprotonsandheliumnu-
helium (p-He). As shown in Ref. [8], the p+H channel
clei over the ISM. We evaluate here the d propagated
is dominant at low energies,and negligible beyond a few
fluxes as well as the nuclear and propagation uncertain-
GeV/n. The effect of energy losses, reacceleration and
ties, similarly to what was done for p in Ref. [20].
tertiaries add up to replenish the low energy tail. The
maximum of the flux reaches the value of 2 10−7 parti-
·
A. Median flux cles (m2 s sr GeV/n)−1 at 3-4 GeV/n. At 100 MeV/n it
is decreased by an order of magnitude, thus preserving
an interesting window for possible exotic contributions
The secondary d flux is the sum of the six con-
characterizedby a flatter spectrum.
tributions corresponding to p, He and p cosmic ray
fluxes impinging on H and He IS gas (other reactions
are negligible [8]). The p and He fluxes were fitted
on BESS [43] and AMS [44, 45, 46] high energy data B. Propagation uncertainties
with a power law spectrum (see details in Ref. [20])
Φ(T) = N(T/GeV/n)−γ. The best fit corresponds to For a determination of the propagation uncertainties,
N = 13249 m−2 s−1 sr−1 (GeV/n)−1 and γ = 2.72, we follow the same approach as in Ref. [20]. We calcu-
p p
andN =721m−2s−1sr−1(GeV/n)−1andγ =2.74. late the secondary antideuteron flux for all the propa-
He He
The uncertaintyonthese twofluxesissmallandleadsto gation parameter combinations providing an acceptable
negligible uncertainties inthe p andd spectrum. Contri- fit to stable nuclei [14]. The resulting envelope for the
butions to the d flux from p¯+H and p¯+He reactions secondary antideuteron flux is presented in Fig. 2. The
6
as the value ofp depends on the choicefor dσR. Hence,
0 p¯
to be very conservative and to keep a simple approach,
we have spanned all hadronic production cross sections
in the range +100% around their reference value. If we
−50%
wished to translate this into an uncertainty on the coa-
lescencemomentum,thiswouldleadtotheeffectiverange
p =79+26 MeV. Finally, in order to estimate the maxi-
0 −13
malfluxwiththemostconservativeattitude(thehighest
is the secondary flux the lowest is the chance to outline
an exotic contribution), the non-annihilating cross sec-
tion was doubled, as its value is probably only a lower
limit(seeAppendix). Ontheotherhand,toevaluatethe
minimal flux, we switched off the p¯+H(He) d¯+X
→
contributions, which intensity remains very uncertain.
The dotted lines in Fig. 2 take into account the sum
of all the possible uncertainties of nuclear source, as de-
scribed above. At the lowest energies the flux is uncer-
tainby almostoneorderofmagnitude,at100GeV/nby
FIG.2: Dominantuncertaintiesontheinterstellar secondary
afactorof4. Wehavecheckedthatthesolarwindmildly
d flux. Solid lines: propagation uncertainty band. Dotted
decreasestheISfluxatlowenergiesbutleavestheuncer-
lines: nuclear uncertainty band.
tainty magnitude unchanged. It is obvious from Fig. 2
thatthe uncertainties onnuclearand hadroniccrosssec-
solidlinesdelimittheuncertaintybandduetothedegen- tions (dashed lines) are more important than the ones
eracy of the propagation parameters: at energies below coming from the propagation models (solid lines). We
1–2 GeV/n, the uncertainty is 40-50 % around the aver- emphasize once more our conservative attitude in esti-
ageflux,whileat10GeV/nitdecreasesto 15%. This mating the nuclear band. However,if no dedicated cam-
∼
behavior is analogous to that obtained for p [20] and is paigns of measurements for these cross sections will be
easily understood. The degenerate transportparameters carriedoutinthefuture,theseuncertaintiesarenotlikely
combine to give the same grammage in order to repro- to be significantly reduced.
duce the B/C ratio. Indeed, the grammage crossed by
C to produce the secondary species B is also crossed by
V. PRIMARY ANTIDEUTERONS
p and He to produce the secondary p and d. In short,
a similar propagationhistory associatedwith a wellcon-
strained B/C ratio explains the small uncertainty. With The source term for primary d to be cast into Eq. (1)
better measurement of B/C expected soon, e.g. from is:
PAMELA [47] or TRACER [48], this uncertainty will 2
further decrease and could become negligible. qprim(r,z,E)=ηξ2 σ v dNd¯ ρDM(r,z) , (10)
d¯ h ann i0 dEd¯ (cid:18) mχ (cid:19)
where σ v is the thermal average of annihilation
ann 0
C. Nuclear uncertainty cross sehction itimes the WIMP velocity, dNd¯/dEd¯ is the
sourcespectrum, ρ (r,z)is the distribution ofthe DM
DM
The possible nuclear uncertainty can arise from two in the Galaxy and m is the WIMP mass. The quantity
χ
different sources. The first one is directly related to σ v depends on the particle physics model. If not
ann 0
h i
the elementary production process dσR. It was found in differently stated, we fix its value to 2.3 10−26 cm3 s−1,
p¯
·
Ref.[20]thatthismaybecastintoa 25%inthepprop- which corresponds to a thermal CDM relic able to ex-
±
agated flux, so that it should be translated to a rough plain the observed amount of cosmological dark matter
50% in the d flux. Second, there is the uncertainty [49, 50, 51]. This will be our reference value for most of
±
onthe coalescencemomentum p . Using anindependent the analysis. The coefficient η depends on the particle
0
model (i.e. different from the coalescence scheme) for beingornotself–conjugate: forinstance,fora fermionit
d production, Ref. [8] found that, conservatively, the d is 1/2 or 1/4 depending on whether the WIMP is a Ma-
background was certainly no more than twice the flux joranaora Diracparticle. Inthe followingwe willadopt
calculated with p =79 MeV. η =1/2. The quantity ξ parameterizes the fact that the
0
To some extent, these two uncertainties are correlated dark halo may not be totally made of the species under
7
scrutiny (e.g a neutralino or a sneutrino) when this can- Applying the factorization–coalescencescheme discussed
didate possesses a relic abundance which does not allow above leads to the antideuteron differential multiplicity
it to be the dominant DM component (see e.g [52] or 2
e[5r3e]n)c.eIvnaltuheisfocrasheσaρnχnv=i0ξcρlDeaMrlywiotnheξh<as1ξ.=F1o.rTohuerDreMf- ddNEdd¯¯ =(cid:18)34kpd30¯(cid:19)·(cid:18)mm2pd¯¯(cid:19)·XF,hBχ(Fh)(ddNEpp¯¯h (cid:18)Ep¯= E2d¯(cid:19)) .
candidate may then be identified with a neutralino [21] (13)
or a sneutrino [54] in various supersymmetric schemes, Weassume,asdiscussedinSect.III,thatthesamevalue
butforthepurposesofourdiscussionitdoesnotneedto of the coalescence momentum p =79 MeV holds as for
0
be specified. We in fact wish to maintain the discussion hadronic reactions.
atthemostgenerallevel: wejustneedtospecifythefinal The evaluation of the differential antiprotonspectrum
state particles produced in the DM annihilation process dNh/dE follows the treatment of Ref. [21]. We refer
p¯ p¯
andtheensuingenergyspectra. Thefinal–stateparticles to this paper for the details of the p spectra from all
all belong to the Standard Model, and this allows us to theannihilationchannels. Theresultingdsourcespectra
perform our discussion on a totally general basis. We from different final states are not directly shown here.
willattheendspecifyourcandidatetobetheneutralino Instead,wewillprovideexamplesofpropagateddspectra
and discuss experimental capabilities in the framework for the various final states in Fig. 3.
of some specific supersymmetric scheme.
Below, we briefly recall the main steps for the calcu-
B. Dark matter halo profile
lation of the source term, before focusing on the propa-
gation of these antideuterons in the Galaxy (Sect. VC),
which is one of the main novelty in this Paper. The distribution of DM inside galaxies is a very de-
bated issue (see e.g. Ref. [58] for a brief highlight on re-
cent results and relevant references). Different analyses
A. Antideuteron source spectrum of rotational curves observed for several types of galax-
iesstronglyfavouracoreddarkmatterdistribution,flat-
The production of antideuterons from the pair- tenedtowardsthecentralregions(Ref.[59]andreferences
annihilation of dark matter particles in the halo of our therein). Onthe otherside,manycollisionlesscosmolog-
Galaxy was proposedin [1]. The interestin this possible ical N-body simulations in Λ-CDM models are now in
DM detection channel has been the physics case for the good agreement among themselves [60], but for the very
proposaloftheGAPSexperiment[2,3,4]andithasalso central regions some resolution issues remain open. It
been considered in Refs. [55, 56, 57]. has been recently stressed that asymptotic slopes may
As previously discussed (see Sect. III) the production not be reached at all at small scales [61, 62, 63, 64, 65].
of a d relies on the availability of a p – n pair in a single However,itisnotclearwhetherthecentralcuspissteep-
DM annihilation. In the case of a WIMP pair annihila- enedorflattenedwhenthebaryonicdistributionistaken
tion, the differential multiplicity for antiproton produc- into account(e.g. [66, 67]). For definiteness,we consider
tion may be expressed as a generic dark matter distribution:
dNp¯ = B(F) dNp¯h . (11) ρ ρ (r)=ρ r⊙ γ 1 + (r⊙/a)α (β−γ)/α ,
dEp¯ χh dEp¯ χ ≡ CDM ⊙ r 1 + (r/a)α
XF,h n o (cid:26) (cid:27) (14)
The annihilation into a quark or a gluon h is realized
wherer =8kpcisthedistanceoftheSolarSystemfrom
⊙
through the various final states F with branching ratios
thegalacticcenter. Thesphericalpseudo-isothermaland
B(F). Quarksorgluonsmayinfactbe directly produced
χh cored DM profile with (α,β,γ)=(2,2,0) will be the ref-
whenaWIMPpairannihilatesortheymayalternatively
erence in our calculations. The total local—Solar Sys-
result from the intermediate production of Higgs bosons
tem—CDMdensityhasbeensetequaltoρ =0.42GeV
⊙
or gauge bosons. Each quark or gluon h then generates cm−3, the core radius to a=4 kpc. This value and the
jets whose subsequent fragmentation and hadronization
total local density may be varied in large intervals by
yield an antiproton energy spectrum dNh/dE .
p¯ p¯ maintaining good agreement with observations. The an-
As in Ref. [1], we assume that the probability to form
tideuteron flux is very sensitive to the local distribution
an antiproton (or an antineutron) with momentum ~k
p¯ ofdark matter ρ , since it appearssquaredin the deter-
(~k ), is essentially isotropic: ⊙
n¯ minationoftheflux,whileitislesssensitivetothechosen
dN darkmatter distributionfunction(aswasalreadyunder-
p¯(χ+χ p¯+...) = 4πk E (√s=2m,E ).
dE → p¯ p¯Fp¯ p¯ linedinRef.[21]forp). Forcompletenessandforcompar-
p¯
(12) ison, we also consider some of the profiles obtained from
8
FIG. 3: Antideuteron flux for a WIMP mass m =100 GeV FIG. 4: Antideuteron flux for WIMPs of m =50 GeV. Dot-
χ χ
annihilatingintodifferentfinalstates: solid(black)linerefers ted(black)linesrefer totheinterstellar flux,solid (red)lines
to¯bb, dotted (red) to u¯u, short-dashed (blue) to WW, long- standforthetop–of–atmosphereflux,modulatedatsolarmin-
dashed (green) line to ZZ. The dot-dashed (magenta) refers imum. Foreachsetofcurves,thethreelinesrefertothemax-
to t¯t and m =200 GeV. The annihilation cross section (here imal,medianandminimalpropagationconfigurationsdefined
χ
and in thefollowing figures) is fixedat thevalue: hσannvi0 = in Table I.
2.3·10−26 cm3 s−1.
ently stated,TOAfluxes correspondto a solarminimum
Λ-CDM simulations: i) a standard NFW profile having
activity with modulation potential φ=0.5 GV. For the
(α,β,γ)=(1,3,1),witha=21.746kpc[68],ii)thesteeper
reference propagationconfigurations,we refer the reader
DMS-1.2 (1,3,1.2) profile with a = 32.62 kpc [69], iii)
to Table I.
and the modified NFW profile with an logarithmic slope
(hereafter N04), with a = 26.4 kpc [61] (similar to the
Einasto profile [63]). Scale radii for NFW and DMS-1.2
profiles are taken from Ref. [70], while the parameters
1. Fluxes for various annihilation states
for the N04 DM density distribution are the same as in
Ref. [71]. All these profiles are normalized to ρ = 0.42
⊙
GeV cm−3, in orderto isolatethe effect of the localden- Figure 3 displays the d flux from a WIMP of mass
sity, which can be easily rescaled in the flux evaluation. mχ =100GeV.Eachcurvecorrespondstodifferentpure
Wedonotincludeanyboostfactorduetohalosubstruc- (i.e. with BR=1) annihilation final states: ¯bb, u¯u, WW,
tures. This conservative attitude is corroborated by the ZZ and t¯t (for which mχ = 200 GeV). The aim of the
resultsofLavalleetal.[58],whereithasbeenshownthat figure is to show the effect on the observable flux of the
the boostfactoris typicallycloseto unity: only forsome different χ-χ annihilation final states from which the d
extreme and unlikely configuration it can reach a factor originate. The transportparameters are the “med” ones
of 10. of Tab. I (providing the best fit to B/C data) and the
fluxes are not modulated (IS spectra). In the low energy
part of the spectrum – around and below 1 GeV/n – it
C. Primary antideuteron flux and uncertainties turnsoutthatthesefluxesshowquitesimilarshapesand
comparable normalization when varying the final state.
In the present Section, we show our results for the This energetic range is the one in which a primary flux
propagatedantideuteronflux fromDMannihilation. We might emerge from the secondary counterpart. In addi-
follow the prescriptions detailed in the previous Sections tion, as we will also discuss at the end ofour Paper,it is
forthe productionandthe propagationofantideuterons. the window explorable by experiments in a near future.
Top–of–atmosphere (TOA) fluxes are derived from the For these reasons, we will adopt the antideuteron yield
interstellar(IS)onestreatingthe effectofthe solarmod- fromanannihilationintoapure¯bbfinalstateasasimple
ulation with the force field approximation. If not differ- but representative case for our discussions.
9
FIG. 5: Uncertainty due to propagation models on the an- FIG. 6: Effect of changing the DM halo density profile, for
tideuteron(blacksolidlines)andantiproton(reddottedlines) a m =50 GeV WIMP and for the “max” (solid) and “med”
χ
interstellar fluxes. The WIMP mass has been fixed at the (dotted)configurationsof TableI. Theeffect isshown asthe
value m =50 GeV. For each set of curves, the three lines relative change in the IS antideuteron flux as compared with
χ
refer to the maximal, median and minimal propagation con- the reference case of a cored isothermal profile. The lower
figurations defined in Table I. (black)linesrefertotheNFWprofile[68],themedian (blue)
linestoacuspyprofilewith1.2slope[69]andtheupper(red)
ones to theN04 profile of Ref. [61].
2. Propagation uncertainties
certainty on the primaries are, at high energy, the range
Fig.4showstheuncertaintiesontheprimarydfluxdue
allowedfor the halo size L (see Tab I), whereas,at lower
to propagation parameters. The three curves (dashed
energy,this uncertainty is further increasedby the effect
line: IS fluxes; solid lines: TOA fluxes) correspond to
of the galactic wind (2K/V becomes smaller than L).
the maximum, median and minimal set of propagation c
Whencomparingindetailsthepandthedfluxesfrom
parameters as gathered in Table I. The band between
Fig. 5, the following conclusions can be drawn. First,
the upper andlowercurveestimates the uncertaintydue
the antiproton fluxes are a factor of 104 higher than the
to propagation. At the lowest energies of hundreds of
antideuteron ones, as expected from the fusion process
MeV/n the total uncertainty reaches almost 2 orders of
into d. Then, at high energies the difference between
magnitude, while at energies above 1 GeV/n it is about
the two fluxes is to be ascribed to their source spectra,
a factor of 30. The figure refers to a WIMP mass of 50
which for antideuterons is the square of the antiproton
GeV but the results are insensitive to this parameter,as
one. Thiseffect,addedtothedifferentweightofdestruc-
well as from the solar modulation. The magnitude of
tion cross sections, is visible also in the lowerenergy tail
the propagation uncertainty is similar to the one affect-
ofthespectrum. Thedestructionoftheantideuteronnu-
ing the primary antiprotons [21], as explained in Fig. 5.
clei on the ISM alters the flux by a factor of two, while
This behavior is drastically different from that observed
the antiproton one is modified by a mere 20–25%.
onthesecondaries(seeFig.2). Indeed,theirpropagation
history is very different. Whereas secondaries originate
fromstandardsourcesin the thin disk ofthe Galaxy,ex-
otic primaries are produced in all the diffusive halo of 3. Dark Matter Halo profile uncertainty
the Galaxy. As shown in Ref. [72], these primary antin-
uclei do not suffer large energy losses, reacceleration or The effect of changing the DM distribution function
tertiary redistribution as they rarely cross the thin disk. ρ (r,z) on the d flux is demonstrated in Fig. 6. We
DM
Most of them arrive at Earth—substantially unshifted only modify the shape of the density distribution (as
in energy—from an effective diffusion cylinder of height discussed in Sect. VB), while keeping frozen the local
L∗ = min(L,2K/V ) and radius of a few L∗ centered on DM density to ρ =0.42 GeV/cm3. The DM mass is
c ⊙
the observer[73]. Hence, the parameters driving the un- m =50 GeV, but as explained in the previous Section,
χ
10
FIG. 7: Interstellar and Top–Of–Atmosphere (TOA) an- FIG. 8: TOA fluxes for primary (solid lines) and secondary
tideuteron fluxes. The dashed (blue) line shows the primary (dashed line) antideuterons for the median propagation pa-
flux for mχ=50 GeV and hσannvi0 = 2.3·10−26 cm3 s−1, rameters. Fromtoptobottom,thesolidlinesrefertoWIMPs
the (red) dotted line denotes the secondary component and with mass m =50, 100, 500 GeV.
χ
the(black)solidlinestandsforthetotal(signal+background)
flux. Propagation model is the median one in Table I.
VI. POTENTIAL FOR DETECTION: RESULTS
AND DISCUSSION
the source term mostly factors out, so that these con-
clusions hold for any neutralino mass. We plot the ratio We now turn to the determination of the totalflux we
(φd¯−φrd¯ef)/φrd¯ef whereφrd¯ef isthereferencefluxcalculated can expect from the standard astrophysical source (see
with the cored isothermal profile and φd¯corresponds to Sect. IV) added to a possible contribution from DM an-
theNFW,DMS–1.2andN04profiles(seeSect.VB). The nihilation. InFig.7,weshowtheISandTOA(solarmin-
two classes of curves correspond to the maximal (upper, imum) secondary and primary d fluxes, and their sum.
solid) and median (lower, dotted) propagation parame- The primary flux is for a WIMP with mass m = 50
χ
ters. The difference on the fluxes calculated with the GeV, annihilation cross section σ v = 2.3 10−26
ann 0
h i ·
minimal set of propagation parameters (not shown) is cm3 s−1, an isothermal profile and ρ =0.42 GeV/cm3,
⊙
negligible. as in the previous figures. The discrepancy between pri-
The increasing steepness of the profile in central re- maryandsecondaryfluxforTd¯<2GeV/nisstriking. A
gions of the Galaxy is responsible for an increasing of signal from DM annihilation as∼the one in our example
the d flux which is more relevant for higher diffusive ha- would definitely increase by a large amount the number
los. In the case L = 15 kpc, the d obtained with a 1.2 of expected antideuterons in the lowestenergy bins with
cuspy profile [69] is a factor of 2 higher than the cored respect to the purely secondary flux. At 100 MeV/n the
one, while the NFW [68] halo gives fluxes 30-40%higher expecteddfluxfromacosmologicallydominantDMpar-
than the isothermalone depending on energy. The high- ticle of 50 GeV mass is two orders of magnitude larger
estfluxisobtainedwiththelog-slopeNFW-likeprofileof than the secondary d flux calculated within the same
Navarroetal.[61],whichpredicts,amongtheconsidered propagationmodel. One has to remindthat the primary
DM profiles, the highest DM density in a wide radialin- flux scales as m2: this means that, in the low energy
χ
tervalaroundtheSolarSystem,althoughitisflatterthan sector, the signal can overwhelm the background up to
the DMS–1.2 andNFW profilesin the centralkpc ofthe masses of the order of few hundreds of GeV. This figure
Galaxy. The flux obtained with the median parameters demonstrates that the search for cosmic antideuterons
(L = 4 kpc) is less significantly modified by a change is definitely one of the most powerful indirect detection
in the halo profile. Indeed, a charged particle produced meansfortheDMannihilationinthehaloofourGalaxy.
around the galactic center can more easily reach the So- Thediscriminationpowerbetweensignalandbackground
larSystemwhenthemagneticdiffusivehaloislargerand canbeashighasfewordersofmagnitude. Amajorlimit
when it is more energetic. to this kind ofexperimentalinspectionmay residein the