Photon spectra from WIMP annihilation

21 Pages·2011·1.86 MB·English

PHYSICAL REVIEW D 83, 083507 (2011)
Photon spectra fromWIMP annihilation
J.A.R. Cembranos,1,2,*A. de la Cruz-Dombriz,2,†A. Dobado,2,‡ R.A. Lineros,3,x and A.L. Maroto2,k
1William I.Fine Theoretical Physics Institute,University ofMinnesota,Minneapolis, Minnesota55455,USAandSchool ofPhysics
and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455,USA
2Departamento de F´ısica Teo´rica I, Universidad Complutense de Madrid, E-28040 Madrid, Spain
3INFN sezione di Torino, I-10122 Torino,Italy andDipartimento di Fisica Teorica, Universita` di Torino, I-10122 Torino,Italy
(Received 15 October 2010; published 11 April 2011)
IfthepresentdarkmatterintheUniverseannihilatesintostandardmodelparticles,itmustcontributeto
the fluxes of cosmic rays that are detected on the Earth and, in particular, to the observed gamma-ray
fluxes.Themagnitudeofsuchacontributiondependsontheparticulardarkmattercandidate,butcertain
features oftheproducedphotonspectramaybeanalyzedinarather model-independentfashion.Inthis
work we provide the complete photon spectra coming from WIMP annihilation into standard model
particle-antiparticle pairs obtained by extensive Monte Carlo simulations. We present results for each
individualannihilationchannelandprovideanalyticalfittingformulasforthedifferentspectraforawide
range of WIMP masses.
DOI: 10.1103/PhysRevD.83.083507 PACS numbers: 95.35.+d, 98.80.Cq
(WIMPs) are stable, two of them may annihilate into
I. INTRODUCTION
ordinarymattersuchasquarks,leptonsandgaugebosons.
According to present observations of large scale struc- Their annihilation in different places (galactic halo, Sun,
tures, cosmic microwave background anisotropies and Earth, etc.) produce cosmic rays to be discriminated
light nuclei abundances, the most important component through distinctive signatures from the background. After
ofmatterintheUniversecannotbeaccommodatedwithin WIMPsannihilationacascadeprocesswouldoccur.Inthe
the standard model (SM) of elementary particles. Indeed, end the potentially observable stable particles would be
dark matter (DM) cannot be made of any of the known neutrinos,gammarays,positrons,andantimatter(antipro-
particles, and this is one of the most appealing arguments tons, antihelium, antideuterions, etc.), which may be ob-
fortheexistenceofnewphysics.IndeedDMisarequired served through different devices. Neutrinos and gamma
componentnotonlyoncosmologicalscales,butalsofora rays have the advantage of maintaining their original di-
satisfactory description of rotational speeds of galaxies,
rection thanks to their null electric charges. On the con-
orbitalvelocitiesofgalaxiesinclusters,gravitationallens-
trary,chargedparticles’searches,suchasthoseofpositrons
ing of background objects by galaxy clusters, such as the
andotherantimatterparticles,arehinderedbypropagation
bulletcluster,andthetemperaturedistributionofhotgasin
trajectories.
galaxies and clusters of galaxies. The experimental deter-
The detection of such indirect signals would not con-
mination of the DM nature will require the interplay of
stituteconclusiveevidence forDMsincetheuncertainties
collider experiments [1] and astrophysical observations.
in the specific DM interactions, DM densities, and back-
These searches use to be classified in direct or indirect
grounds from other sources are not fully understood yet.
searches(see[2,3]fordifferentalternatives).Nevertheless,
Nevertheless, this work precisely focuses on this kind of
nongravitational evidence of its existence and a concrete
detection as an indirect method to get information about
understandingofitsnaturestillremainelusive.Concerning
the DM nature, abundance, and properties.
directsearches,theelasticscatteringofDMparticlesfrom
Photon fluxes in specific DM models are usually ob-
nuclei should lead directly to observable nuclear recoil
tained by software packages such as DARKSUSY and
signatures. Although the number of DM particles which
MICROMEGASbasedonPYTHIAMonteCarloeventgenera-
passes through the Earth each second is quite large, the
tor.Ingeneral,foraparticularsupersymmetrymodelanda
weak interactions between DM and the standard matter
given WIMP mass, the total photon spectrum is obtained
make DM direct detection extremely difficult.
fromtheadditionofthecontributionsfromdifferentchan-
Ontheotherhand,DMmightbedetectedindirectly,by
nels.Inthis sense,itwouldbeinterestingtohaveafitting
observing their annihilation products into SM particles.
functionfortheshapeofthespectracorrespondingtoeach
Thus, even if weakly interacting massive particles
individualannihilationchanneland,inaddition,determine
the dependence of such spectra on the WIMP mass in a
*[email protected] model-independent way.Previous attemptsof analytically
†dombriz@fis.ucm.es
capturing the photon spectra for different annihilation
‡dobado@fis.ucm.es
[email protected] channels were presented in [4,5] but only for some chan-
kmaroto@fis.ucm.es nels and two particular WIMP masses. These efforts did
1550-7998=2011=83(8)=083507(21) 083507-1 (cid:1)2011 American Physical Society
J.A. R. CEMBRANOS et al. PHYSICAL REVIEW D 83, 083507 (2011)
not cover the whole photon energy range since the pro- decaychainsandnonperturbativeeffectsrelatedtoQCDis
posed parametrization for the gamma-ray flux was given ahardtasktobeaccomplishedbyanyanalyticalapproach.
by a single exponential. More recent attempts [6] also The second piece in (1) is a line-of-sight integration
studied two WIMP masses (500 GeV and 1 TeV) for through the DM density distribution. We will discuss
quarks, W,Z, gluon, and (cid:1) lepton channels. each of these pieces separately.
The obtained fitting function could be used to obtain
photon fluxes for alternative dark matter candidates for
A. Particle physics model
which software packages have not been developed. On
Althoughannihilationcrosssectionsarenotknown,they
theotherhand,theinformationaboutchannelcontribution
are restricted by collider constraints and direct detection.
and mass dependence can bevery useful in order to iden-
In addition, the thermal relic density in the range
tify certain gamma-ray features as signals of specific
WIMP candidates. (cid:3)CDMh2 ¼0:1123(cid:3)0:0035 which is determined by fit-
ting the standard (cid:4)CDM model to the WMAP7 data
Thepaperisorganizedasfollows:inSec.II,webriefly
(WilkinsonMicrowaveAnisotropyProberesultsfor7years
review the standard procedure for the calculation of
of observations) [7], the latest measurements from the
gamma-ray fluxes from WIMP pair annihilations. In
Baryonacousticoscillationsinthedistributionofgalaxies
Sec. III, we comment on several aspects of detectors and
backgrounds. Section IV is then devoted to the details of [8] and the Hubble constant (H0) measurement [9], does
not allow an arbitrary contribution from the DM gamma-
specificsimulationsperformedwithPYTHIA.InSec.V,we
rayfluxes.
introducethefittingformulasthatwillbeusedtodescribe
As already mentioned, the annihilation of WIMPs is
the spectra and in Sec. VI the results for the simulations,
closely related to SM particle production. The time scale
the fitted parameters, and their dependence on the WIMP
ofanannihilationprocessisshorterthantypicalastrophys-
mass are presented. Then, in Sec. VII we provide some
icalscales.Thisfactimpliesthatonlystableorverylong-
informationabouttheperformednumericalcodesobtained
livedparticlessurvivetotheWIMPannihilationsandmay
from our results and available online. Section VIII is then
thereforebe observed by detectors.
devoted to the main conclusions of the work. Finally, an
FormostoftheDMcandidates,theproductionofmono-
Appendix is provided to illustrate the obtained results for
some studied annihilation channels. energeticphotonsisverysuppressed.Themainreasonfor
suchasuppressioncomesfromthefactthatDMisneutral.
Thus, it is usually assumed that the gamma-ray signal
II. GAMMA-RAY FLUX FROM DM
comes fundamentally from secondary photons originated
ANNIHILATION
inthecascadeofdecaysofgaugebosonsandjetsproduced
Let us denote the DM mass by M and its thermal fromWIMPannihilations.Theseannihilationswouldpro-
averaged annihilation cross section into two SM particles duce in the end a broad energy distribution of photons,
(labeled bythesubindexi)byh(cid:2)ivi.Thenthe(cid:3)-rayflux which would be difficult to be distinguished from the
from all possible annihilation channels is givenby background. However, the directional dependence of the
gamma-ray intensity coming from these annihilations is
d(cid:1)DM 1 X dNi
(cid:3) ¼ h(cid:2)vi (cid:3) mainly localized in pointlike sources as will be discussed
dE 4(cid:4)M2 i dE inthefollowingsection.Thisfactcouldthereforeprovidea
(cid:3) |fflfflfflfflfflfflfflfflfflfflfflfflfflfflifflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl(cid:3)}
distinctive signature.
Particlemodeldependent In conclusion, for a particular DM candidate, a unique
Z Z
1
annihilationchannelmaydominate,butingeneral,theyall
(cid:1) d(cid:3) (cid:5)2½rðsÞ(cid:2)ds; (1)
(cid:2)|fflffl(cid:3)fflfflfflfflfflfflfflfflffl(cid:2)fflfflffl(cid:3)fflfflfflfflfflfflfflfflfflfflffl{zl:ffloffl:fflsfflffl:fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} contribute. All those channels contributions produce a
broad energy gamma-ray flux, whose maximum consti-
Darkmatterdensitydependent
tutes a potential signature for its detection. Typically, this
where (cid:5) is the DM density as a function of distance from peak is centered at an energy that is 1 order of magnitude
its center r, which depends on the heliocentric distance s. lower than the mass of the DM candidate.
Theintegralisperformedalongtheline-of-sight(l.o.s.)to On the other hand, a different strategy can be followed
the target and averaged overthe detector solid angle (cid:2)(cid:3). by taking into account the fact that the cosmic-ray back-
The first piece of the right-hand side in (1) depends on ground is suppressed at high energies. Primary photons
theparticularparticlephysicsmodelforDMannihilations. coming from the Weicksa¨cker-Williams radiation domi-
Inparticular,theself-annihilationcrosssectionsaremainly natethespectrumatenergiesclosetothemassoftheDM
described by the theory explaining the WIMP physics, candidateandtheirsignatureispotentiallyobservableasa
whereasthenumberofphotonsproducedineachdecaying cutoff[10].Thisapproachhastheadvantageofbeingless
channel perenergy interval involves decays and/orhadro- sensitive to electroweak corrections which may be impor-
nization of unstable products, for instance quarks and tant if the mass of the DM candidate is larger than the
gauge bosons. Consequently, the detailed study of these electroweak scale [11].
083507-2
PHOTONSPECTRA FROM WIMPANNIHILATION PHYSICAL REVIEW D 83, 083507 (2011)
B. DMdensity directionality resolution. Typical reference sizes for the solid angle are
(cid:2)(cid:3)¼10(cid:4)5 sr for ACTs and FERMI and (cid:2)(cid:3)¼10(cid:4)3 sr
The line-of-sightintegration can be obtained from
for EGRET. In any case, the mentioned values only
Z
: 1
hJi ¼ JðcÞd(cid:3) illustrate the general situation, since the solid angle for
(cid:2)(cid:3) (cid:2)(cid:3) (cid:2)(cid:3) each experiment depends on the energy. In particular, for
2(cid:4) Z(cid:6) FERMI, the 0.1(cid:5) point spread function is only valid
¼ maxd(cid:6)sin(cid:6)
(cid:2)(cid:3) for energies of about 100 GeV [17].
0
Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi There are different main sources of background for the
s
(cid:1) maxds(cid:5)2ð s2þs20(cid:4)2ss0cos(cid:6)Þ; (2) signalunderconsideration:hadronic,cosmic-rayelectrons,
smin localizedastrophysicalsources,andthediffuse(cid:3)rays.The
where latter is negligible for ACTs, but only the last two are
Z present for satellite experiments like FERMI or EGRET.
JðcÞ¼ ds(cid:5)2ðrÞ: (3) For heavy WIMPs, the produced high-energy gamma
l:o:s: photonscouldbeintherange 30GeV–10TeV,detectable
by ACTs such as HESS, VERITAS, or MAGIC. On the
The angled brackets denote the averaging over the solid
angle(cid:2)(cid:3),andsminandsmaxaretheloweranqdffiffiffiuffiffiffipffiffipffiffiffieffiffirffiffiffilffiiffiffimffiffiffiffiiffitffiffis contrary,forlighterWIMPs,thephotonfluxeswouldbein
the range detectable by space-based gamma-ray observa-
of the line-of-sight integration: s0cos(cid:6)(cid:3) r2t (cid:4)s20sin2(cid:6). tories [18] such as EGRET, FERMI, or AMS, with better
In this formula s0 is theheliocentric distance and rt is the sensitivities around 30 MeV–300 GeV.
tidal radius.
Traditionally,theGalacticcenterhasattractedtheatten-
IV. MONTE CARLOSPECTRAGENERATION:
tion of this type of directional analysis since standard
TECHNICALITIES
cusped Navarro-Frenk-White halos predict the existence
of a very important amount of DM in that direction [12]. In this section, we explicitly specify how gamma rays
However, this assumption is in contradiction with a sub- spectrahavebeengenerated.Wehaveusedawidelyknown
stantial body of astrophysical evidences [13], and a core particle physics software, PYTHIA (version 6.418) [19], to
profile is not sensitive to standard DM candidates. On the obtain the results we are about to present. In a first ap-
contrary, cusped profiles are not excluded for the local proximation, the WIMP annihilation is described by two
group dwarf spheroidals that constitute interesting targets separated processes: The first one describes the annihila-
since theyare muchmore dominated by DM. In this way, tion of WIMP particles and its output which are particle-
directional analysis towards Canis Major, Draco, and antiparticle SM pairs. The details are contained in the
Sagittarius or Segue 1 [14] are more promising. theory describing the WIMP physics. The second process
In any case, galaxy clusters are also promising targets considers the evolution (decays and/or hadronization) of
[15]. Another alternative strategy takes advantage of the the SM unstable products, for instance, quarks and gauge
largefieldofviewofFERMI,thatmaybesensitivetothe bosons. Unfortunately, a first-principle description of this
continuum photon flux coming from DM annihilation at latter step is too complex due to chain decays and non-
moderate latitudes (jbj>10(cid:5)) [12]. Other proposed tar- perturbative QCD effects.
gets,asthelargemagellaniccloud[16],arelessinteresting As we mentioned above, in this work we have used
sincetheircentralpartsaredominatedbybaryonicmatter. PYTHIA to generate the photon energy spectra starting
from pairs of SM particles, where each pair respects
WIMP annihilation quantum numbers like neutral charge
III.DETECTORS AND BACKGROUNDS
andcolorsinglet.Aswillbedescribedbelow,wewillallow
(cid:6)maxinEq.(2)istheangleoverwhichweaverage,andis forfinalstateradiationfromchargedparticlestocontribute
boundedfrombelowbytheexperimentalresolutionofthe to the photon spectra. Because of the expected velocity
particular detector: dispersion of DM, we expect most of the annihilations to
Z happen quasistatically. This fact offers the center of mass
(cid:6)
(cid:2)(cid:3)¼2(cid:4) maxd(cid:6)sin(cid:6)¼2(cid:4)ð1(cid:4)cosð(cid:6)maxÞÞ: (4) (CM) frame as the most suitable frame to produce the
0
photonspectra.Hence,theprocessisdescribedbythetotal
The quoted point spread function widths for the various energy:
experiments are typically: 0.4(cid:5) (EGRET), 0.1(cid:5)
ECM ’2M; (5)
(CANGAROO-III, FERMI, HESS, MAGIC, and
VERITAS). EGRET and FERMI are satellite detectors where M is the mass of the WIMP particle. Therefore, by
withlowenergythresholds(about100MeV),high-energy considering different CM energies for the SM particles
resolution ((cid:6)15%) but only moderate angular precision. pairs in each WIMP annihilation process we are indeed
The others are atmospheric Cerenkov telescopes (ACTs) studyingdifferentWIMPmasses.Theproceduretoobtain
with higher thresholds ((cid:7)100 GeV) but better angular the photon spectra is thus straightforward, except for the
083507-3
J.A. R. CEMBRANOS et al. PHYSICAL REVIEW D 83, 083507 (2011)
particular case of the t quark. For any given pair of SM TABLE I. Total number of photons—in 107 units—generated
particleswhichareproducedintheWIMPannihilation,we from WþW(cid:4), ZZ, and tt(cid:5)channels for different WIMP masses.
countthenumberofphotonsineachbinofenergyandthen
Channel\Mass
normalize them to the total number of simulated pair
(GeV) 100 125 150 200 250 350 500 1000
collisions. The bins which we have considered in the x
variable, x(cid:8)E(cid:3)=M, are: ½10(cid:4)5;10(cid:4)3(cid:2), [10(cid:4)3, 0.2], [0.2, WZZþW(cid:4) 50..2412 6(cid:9).(cid:9)0(cid:9)1 12..9911 16.48.95 (cid:9)(cid:9)(cid:9)(cid:9)(cid:9)(cid:9) 17.48.23 22..8911 22..8052
0.5], [0.5, 0.8], and [0.8, 1.0]. Nevertheless, for some tt(cid:5) (cid:9)(cid:9)(cid:9) (cid:9)(cid:9)(cid:9) (cid:9)(cid:9)(cid:9) 0.70 0.86 0.32 2.81 1.41
studied channels more precision was needed in some
particular energy intervals and additional bins were
considered.
TABLE II. Totalnumberofphotons—in107 units—generated
The number of required photons in each bin was fixed
from (cid:1)þ(cid:1)(cid:4) and (cid:7)þ(cid:7)(cid:4) channels for different WIMP masses.
apriori.Nevertheless,thisnumberwaschangedsometime
The required number of photons per bin was fixed to the same
inordertoprovidesuitablestatisticsofproducedphotons.
figureforbothchannelsineachbin.Thischoicewasstatistically
For instance, for the high-energy bins many collisions are justified according to the obtained simulated plots.
required to get a significant number of photons, whereas
for low-intermediate energy, many photons are usually Channel\Mass
(GeV) 25 50 100 200 500 1000 104 5(cid:1)104
produced even for a small number of collisions. The total
number of photons corresponding to the different gener- (cid:1)þ(cid:1)(cid:4) 2.25 2.25 2.23 1.07 2.81 2.33 8.41 7.80
ated pairs in terms of the WIMP mass are presented in (cid:7)þ(cid:7)(cid:4) 2.25 2.25 2.23 1.07 2.81 2.33 8.41 7.80
Tables I, II, and III.1 When these results are normalized
with respect to the number of WIMP annihilations, one
gets the number of photons per WIMP annihilation for
TABLE III. Total number of photons—in 107 units—gener-
each channel and WIMP mass. At the end of the paper, ated from uu(cid:5), dd(cid:5), ss(cid:5), cc(cid:5), and bb(cid:5) channels for different WIMP
Fig. 9 inthe Appendix illustrates these results.
masses.
The SM particle pairs decays generated are W and Z
gauge bosons; (cid:1) and (cid:7) leptons; and u, d, s, c, b and t Channel\
quarks.Foreachannihilationchannelwehavestudiedthe Mass (GeV) 50 100 200 500 1000 2000 5000 70008000
gamma-ray spectra produced for different WIMP masses. uu(cid:5) 2.05 11.9 2.422.81 3.82 10.8 5.91 (cid:9)(cid:9)(cid:9) 2.11
The result of the simulations were fitted to analytical dd(cid:5) 1.04 1.962.422.81 2.81 2.81 2.31 (cid:9)(cid:9)(cid:9) (cid:9)(cid:9)(cid:9)
expressions as is described in the following section. ss(cid:5) 15.3 2.00 1.972.81 9.82 2.71 2.71 11.0 (cid:9)(cid:9)(cid:9)
cc(cid:5) 2.41 1.9916.82.81 2.81 3.8112.0 (cid:9)(cid:9)(cid:9) 3.00
A. Finalstate radiation bb(cid:5) 11.7 1.912.622.61 8.81 2.20 3.81 (cid:9)(cid:9)(cid:9) 1.70
If the final state in the annihilation process contains
charged particles, there is a finite probability of emission
of an additional photon. This is discussed in detail in energy photons from WIMP annihilation, its contribution
[20]. In principle there are two types of contributions: is relevant only in models and at energies where the lines
that coming from photons directly radiated from the contribution is dominant over the secondary photons. For
external legs, which is the final state radiation we have those reasons and since the aim of the present work is to
considered in the work, and that coming from virtual provide model-independent results for photon spectra,
particles exchanged in the WIMP annihilation process. only final state radiation was included in our simulations.
The first kind of contribution can be described for rela-
tivistic final states by means of a universal Weizsa¨cker- B. The case for tquarkdecay
Williams term fundamentally independent from the par-
The decay of top quark is not explicitly included in the
ticle physics model [20]. On the other hand, radiation
from virtual particles only takes place in certain DM PYTHIApackage.Wehaveapproximatedthisprocessbyits
models and is only relevant, in particular, cases, for dominantSMdecay,i.e.,each(anti)topdecaysintoWþð(cid:4)Þ
instance, when the virtual particle mass is almost degen- and(anti)bottom.Inordertomaintainanynonperturbative
erate with the WIMP mass. Even in these cases, it has effect, we work on an initial four-particle state composed
been shown [21] that although this effect has to be by Wþb coming from the top and W(cid:4)b(cid:5) from antitop,
which keeps all kinematics and color properties from the
included for the complete evaluation of fluxes of high-
original pair. Starting from this configuration, we have
forced decays and hadronization processes to evolve as
1The raise in the total number of photons for certain mass
PYTHIAdoesand therefore,thegammaraysspectra corre-
valuesinthesetablesisduetothefactthatinordertogetgood
enoughstatisticsincertainchannelsand/orWIMPmassesaddi- sponding to this channel have also been included in our
tional bins and computation timewas required. analysis.
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PHOTONSPECTRA FROM WIMPANNIHILATION PHYSICAL REVIEW D 83, 083507 (2011)
V. ANALYTICAL FITS TO PYTHIA B. W and Z bosons
SIMULATION SPECTRA FortheW andZgaugebosons,theparametrizationused
In this section we present the fitting functions used for tofit the MonteCarlo simulation is
(cid:3) (cid:4)(cid:5) (cid:6)
the different channels. According to the PYTHIA simula- dN c ln½pðj(cid:4)xÞ(cid:2) q
tions described in the previous section, three different x1:5 dx(cid:3) ¼a1exp (cid:4)b1xn1 (cid:4)xd1 lnp : (8)
1
parametrizations were required in order tofit all available
data from the studied channels. The first one for quarks This expression differs from the expression (6) in the
(exceptthetop)andleptons.Then,asecondoneforgauge absence of the additive logarithmic contribution.
bosons W and Z and a third one forthe top. Nonetheless, this contribution acquires a multiplicative
behavior. The exponential contribution is also quite sim-
A. Quarksand leptons plifiedwithonlyonepositiveandonenegativepowerlaws.
For quarks (except the top), (cid:1) and (cid:7) leptons, the most Moreover,a1,n1,and qparameters appeartobe indepen-
dent of the WIMP mass M as will be seen in the corre-
general formula needed to reproduce the behavior of the
spondingsection.Therestofparameters,i.e.,b1,c1,d1,p,
differential number of photons per photon energy may be
and j, are WIMP mass dependent and will be determined
written as:
for each WIMP mass and for the W and Z separately. In
(cid:3) (cid:4)
x1:5dN(cid:3) ¼a exp (cid:4)b xn (cid:4)b xn (cid:4) c1 þ c2 bothcaseswehavecoveredaWIMPmassrangefrom100
dx 1 1 1 2 2 xd xd to104 GeV.Nonetheless,atmasseshigherthan1000GeV,
1 2
we have observed no significant change in the photon
x2(cid:4)2xþ2
þqx1:5ln½pð1(cid:4)xÞ(cid:2) : (6) spectra for both particles.
x
Inthis formula,thelogarithmicterm takesintoaccount C. t quark
the final state radiation through the Weizsa¨cker-Williams
Finally, for the top, the required parametrization turned
expression [20,22]. Nevertheless, initial radiation is re-
outto be
movedfromourMonteCarlosimulationsinordertoavoid (cid:3) (cid:4)(cid:5) (cid:6)
wrongly counting their possible contributions. x1:5dN(cid:3) ¼a exp (cid:4)b xn (cid:4) c1 (cid:4) c2 ln½pð1(cid:4)xlÞ(cid:2) q:
Strictly speaking, the p parameter in the Weizsa¨cker- dx 1 1 1 xd1 xd2 lnp
Williams term in the previous formula is ðM=mparticleÞ2 (9)
wheremparticleisthemassofthechargedparticlethatemits
LikewisethepreviouscaseforWandZbosons,gamma-
radiation.Howeverinourcase,itwillbeafreeparameter
ray spectra parametrization for the top is quite different
to be fitted since the radiation comes from many possible
fromthatgivenbyexpression(6).Thistime,theexponen-
chargedparticles,whichareproducedalongthedecayand
tial contribution is more complicated than the one in ex-
hadronization processes. Therefore we are encapsulating
pression (8), with one positive and two negative power
all the bremsstrahlung effects in a single Weizsa¨cker-
laws.Again,theadditivelogarithmiccontributionisabsent
Williams–like term.
Concerning the (cid:7) lepton, the expression above (6) be- but it acquires a multiplicative behavior. Notice the expo-
nent l in the logarithmic argument, which is required to
comessimplersincetheexponentialcontributionisabsent.
The (cid:7)(cid:4) decays in e(cid:4)(cid:8)(cid:5)e(cid:8)(cid:7) with a branching ratio of (cid:6)1 provide correct fits for this particle.
The covered WIMP mass range for the top case was
and therefore the only contribution in addition to its own
from200to105 GeV.Nevertheless,atmasseshigherthan
bremsstrahlung, is providedby the radiation comingfrom
1000GeVwehaveobservedagainthatthereisnosignifi-
the electron. The total gamma rays flux is thus well fitted
cant change in thegamma-ray spectra.
by
dN x2(cid:4)2xþ2
x1:5 (cid:3) ¼qx1:5ln½pð1(cid:4)xlÞ(cid:2) ; (7) VI. RESULTS FROM PYTHIA SIMULATION
dx x
In this section we present the results of our fit of the
wherethelparameterinthelogarithmisneededinorderto parameters given by expressions (6), (8), and (9) after
fit the simulations as will be seen in the corresponding having performed the PYTHIA simulations described in
sections. Sec.IV.Foreachstudiedchannel,wehaveconsideredthe
Let us mention at this stage that for the gamma rays possibilityofparametersdependingontheWIMPmass.
obtained from electron-positron pairs, the only contribu- Oncetheparametersinexpressions(6),(8),and(9)have
tion is that coming from bremsstrahlung. Therefore, the been determined for each channel and different WIMP
previous expression (7) is also valid with q¼(cid:9)QED=(cid:4), masses, it is possible to study their evolution with the
p¼ðM=me(cid:4)Þ2, and l(cid:8)1. This choice of the parameters WIMP mass M. Some parameters in expressions are
corresponds of course to the well-known Weizsa¨cker- WIMP mass independent and take values that depend on
Williams formula. the studied channel. The rest are WIMP mass dependent.
083507-5
J.A. R. CEMBRANOS et al. PHYSICAL REVIEW D 83, 083507 (2011)
TABLE IV. W boson: b1, c1, d1, p, and j parameters corre- TABLE VI. Z boson: b1, c1, d1, p, and j parameters corre-
sponding to (8) in the WþW(cid:4) channel for different WIMP sponding to (8) in the ZZ channel for different WIMP masses.
masses. Mass independent parameters in (8) for this channel Mass independent parameters in (8) for this channel are pre-
are presented at the bottom of the table. sented at the bottom of the table.
WIMP mass (GeV) b1 c1 d1 p j WIMP mass (GeV) b1 c1 d1 p j
100 9.48 0.651 0.292 973 0.790 100 10.3 0.498 0.323 7010 0.702
150 8.87 0.808 0.261 783 0.919 125 9.74 0.612 0.294 4220 0.836
200 8.64 0.882 0.250 684 0.955 150 9.49 0.675 0.280 3850 0.894
350 8.56 0.907 0.245 593 0.991 200 9.28 0.734 0.268 3630 0.943
500 8.51 0.917 0.244 560 0.996 350 9.02 0.800 0.257 3380 0.978
1000 8.45 0.931 0.242 535 1.000 500 8.95 0.813 0.255 3260 0.988
a1 ¼25:8; n1 ¼0:510; q¼3:00 1000 8.91 0.819 0.254 3140 0.997
a1 ¼25:8; n1 ¼0:5; q¼3:87
ForsomechannelsandinsomerangeofWIMPmasses,
we observed that this dependence was given by a simple Some of these results are presented in Fig. 1 in the
power law. In fact, for a given channel (i) and a generic Appendix for four WIMP masses: 100, 200, 350, and
mass-dependentP parameter,asimplepower-lawscaling 1000 GeV. Besides, mass-dependent parameters b1, c1,
behavior would correspondto an expression like d1,p,and j were presented intheAppendix in Fig. 2.
P ðiÞðMÞ¼mPðiÞMnPðiÞ; (10) depCeonndceenrtnpinagramtheetersscgaivlienngbybeehxapvreiossrionof(10th),esweeombatasisn-
withmPðiÞ andnPðiÞ constantvaluestobedeterminedforthe thatb1,c1,andjparametersscalewithasimplepowerlaw
different studied channels. Values of mPðiÞ and nPðiÞ and ofMathighmasses.Infact,b1andc1parametersfollowa
theirrangeofvalidityarepresentedforeachstudiedchan- twopower-lawbehavioratlowmasses.Forthed1parame-
nel inthe following. ter,wefindthatthesumoftwopowerlawscoversthishigh
massesinterval,whereasasimplepower-lawatlowmasses
A. W boson is obeyed. Parameter p scales with two power laws in the
whole studied mass interval. These results are shown in
As commented above, the correct parametrization for
Table V.
theW bosonsimulationswasgivenbyexpression(8).For
tch1i,sdb1o,spo,na,ntdhejrewahroesefivvaelumeassasr-ededpeetnaidleendtinpaTraabmleetIeVrs.:Tbh1e, B. Z boson
mass independent parameters are a1 ¼25:8, n1 ¼0:51 FortheZbosonthecorrectparametrizationisagainthe
and q¼3:00. The mass range considered for this boson one given by expression (8). For this boson there are five
is 100 to 105 GeV. In fact, from M ¼1000 GeV, the mass-dependentparameters:b1,c1,d1,p,andjwhichare
photon spectrum does not change. The parameters ob- detailed in Table VI. The mass independent parameters
tained fit the enegy spectra from x¼2(cid:1)10(cid:4)4 until the are a1 ¼25:8, n1 ¼0:5, and q¼3:87. The studied
end of the allowed interval. It can be seen that for low WIMP mass range for this boson was from 100 to
masses the spectrum does not end at x¼1 but at smaller 105 GeV. However, above M ¼1000 GeV the energy
energies (e.g. x’0:78 for M ¼100 GeV) and as masses spectrum does not change as can be seen from our
get higher, the energy tail approaches x¼1. simulations.
TABLE V. Parameters corresponding to (10) for the W boson. It can be seen that the p
parameter follows two different power laws depending on the WIMP mass interval. For the
remainingmass-dependentparametersthereisauniquepower-lawbehaviorintheWIMPmass
interval 350(cid:10)M(cid:10)1000.
Parameter WIMP mass interval (GeV) Fitting power law(s)
b 100(cid:10)M(cid:10)200 0:0433M0:765þ46:4M(cid:4)0:382
1
200<M(cid:10)1000 9:29M(cid:4)0:0139
c 100(cid:10)M(cid:10)200 (cid:4)27M(cid:4)0:240þ35:0M(cid:4)0:0643(cid:4)16:5
1
200<M(cid:10)1000 0:743M0:0331
d 100(cid:10)M(cid:10)240 2:64(cid:9)10(cid:4)4M1:03þ2:28M(cid:4)0:470
1
240<M(cid:10)1000 0:265M(cid:4)0:0137
p 200(cid:10)M(cid:10)1000 105M(cid:4)1:13þ285M0:0794
j 385(cid:10)M(cid:10)1000 0:943M0:00852
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PHOTONSPECTRA FROM WIMPANNIHILATION PHYSICAL REVIEW D 83, 083507 (2011)
The chosen parameters values fit the photon spectra TABLE IX. Parameterscorresponding to (10) for the t quark.
from x¼5(cid:1)10(cid:4)4 until the end of the allowed interval. Itcanbeseenthatallmass-dependentparametersforthetquark
As for the W case, it can be seen that for low masses the followasimplepower-lawscalingbehavioratintermediateand
spectrumdoesnotendatx¼1butatsmallerenergies(e.g. high WIMP masses. Parameters b1 and n1 follow the simple
powerlawforM>300 GeVwhereastherestoftheparameters
x’0:7 for M ¼100 GeV) and as masses get higher, the
presented do so in the interval 200(cid:10)M(cid:10)1000.
high-energy tail approaches x¼1.
Concerning the power-law scaling of the parameters WIMP mass
with M, we obtained that parameters b1, c1, d1, and j Parameter interval (GeV) Fitting power law(s)
follow a simple power-law behavior for high WIMP b 200(cid:10)M(cid:10)350 9:32M0:0507þ11:0(cid:9)106M(cid:4)2:91
masses.Parameterpfollowsatwosumpower-lawbehav- 1 350<M(cid:10)1000 16:4M(cid:4)0:0400
ior for masses higher than 170 GeV. Concerning the d1 n 200(cid:10)M<300 21:4M(cid:4)0:818þ0:00867M0:589
1
parameter, the whole accessible WIMP mass interval is 300(cid:10)M(cid:10)1000 0:559M(cid:4)0:0379
covered by different either one or two power laws. These c 200(cid:10)M(cid:10)1000 8910M(cid:4)3:23
2
results can be seen in Table VII. p 200(cid:10)M<1000 5:78(cid:1)10(cid:4)5M1:89
q 200(cid:10)M(cid:10)1000 0:133M0:488
C. t quark l 200(cid:10)M(cid:10)1000 21:9M(cid:4)0:302
Forthetop,therearesixmass-dependentparameters:b1,
n1, c2, p q, and l which are detailed in Table VIII. The
mass independent parameters are a1 ¼290, c1 ¼1:61, SomeoftheseresultsarepresentedgraphicallyinFig.3
d1 ¼0:19,andd2 ¼0:845.Themassrangeforthisquark (in the Appendix), for four WIMP masses: 200, 250, 500,
is from 200 to 105 GeV. Nevertheless, from 1000 GeV and 1000 GeV. Also in the Appendix, mass-dependent
onwards, the photon spectra do not change as was proven parameters b1, n1, c2, p, q,and l are plotted in Fig. 4.
byconsideringseveralhighermasses.Thechosenparame- Concerning the scaling behavior of the c2, p, q, and l
ters fit the spectra from x¼10(cid:4)4 until the end of the parameters, they obey a simple power law in the whole
allowed interval. Again for low masses, the spectra do accessible WIMP mass range. Nevertheless, forb1 and c1
not end at x¼1 but at smaller energies (e.g. x’0:7 for parameters the simple power-law behavior starts from
m¼200 GeV)and,asmassesgethigher,thespectraltail masses bigger than 350 GeV. These results can be seen
approaches x¼1. inTable IX.
TABLE VII. Parameters corresponding to (10) for the Z boson. It can be seen that all mass-
dependent parameters for the Z boson follow a simple power-law scaling at intermediate and
high masses.
Parameter WIMP mass interval (GeV) Fitting power law(s)
b 500(cid:10)M(cid:10)1000 9:36M(cid:4)0:00710
1
c 465(cid:10)M(cid:10)1000 0:765M0:00980
1
d 100(cid:10)M(cid:10)191 0:00999M0:530þ21:5M(cid:4)1:01
1
191<M(cid:10)360 2:02(cid:1)10(cid:4)9M2:56þ0:491M(cid:4)0:115
360<M(cid:10)1000 0:272M(cid:4)0:00990
p 170(cid:10)M(cid:10)1000 8550M(cid:4)0:166þ0:476M0:984
j 350(cid:10)M(cid:10)1000 0:884M0:0175
TABLE VIII. tquark:b1,n1,c2,p,q,andlparameterscorrespondingto(9)inthett(cid:5)channel
fordifferentWIMPmasses.Massindependentparametersin(9)forthischannelarepresentedat
the bottom of the table.
WIMP mass (GeV) b1 n1 c2 p q l
200 14.4 0.477 3:34(cid:1)10(cid:4)4 1.34 1.76 4.42
250 13.5 0.457 1:54(cid:1)10(cid:4)4 1.95 1.96 4.14
350 13.0 0.448 5:99(cid:1)10(cid:4)5 3.78 2.32 3.74
500 12.8 0.442 1:69(cid:1)10(cid:4)5 7.40 2.75 3.36
1000 12.4 0.436 1:80(cid:1)10(cid:4)6 30.0 3.85 2.72
a1 ¼290; c1 ¼1:61; d1 ¼0:19; d2 ¼0:845
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J.A. R. CEMBRANOS et al. PHYSICAL REVIEW D 83, 083507 (2011)
TABLE X. (cid:1)lepton:n1andpparameterscorrespondingto(6) TABLE XII. (cid:7) lepton: Parameters corresponding to (7) for
inthe(cid:1)þ(cid:1)(cid:4) channelfordifferentWIMPmasses.Massindepen- different WIMP masses. All parameters in expression (7) are
dent parameters in (6) for this channel are presented at the WIMP mass dependent.
bottom of the table.
WIMP mass (GeV) p q l
WIMP mass (GeV) n1 p 25 9510 3:37(cid:1)10(cid:4)3 0.787
25 10.1 221 50 23600 3:40(cid:1)10(cid:4)3 0.642
50 10.0 767 100 54600 3:45(cid:1)10(cid:4)3 0.579
100 9.91 2520 200 1:12(cid:1)105 3:50(cid:1)10(cid:4)3 0.548
200 9.80 8660 500 2:54(cid:1)105 3:61(cid:1)10(cid:4)3 0.523
500 9.67 4:01(cid:1)104 1000 4:13(cid:1)105 3:70(cid:1)10(cid:4)3 0.511
1000 9.57 1:35(cid:1)105 5000 1:18(cid:1)106 3:91(cid:1)10(cid:4)3 0.484
104 9.25 4:80(cid:1)106 104 1:84(cid:1)106 4:00(cid:1)10(cid:4)3 0.474
5(cid:9)104 9.14 5:44(cid:1)107 5(cid:1)104 5:29(cid:1)106 4:23(cid:1)10(cid:4)3 0.454
a1 ¼14:7; b1 ¼5:40; b2 ¼5:31; n2 ¼1:40; c1 ¼2:54; d1 ¼
0:295; c2 ¼0:373; d2 ¼0:470; q¼0:00260
For this particle, it is worth mentioning the increasing
D.Leptons and quarks contribution of the logarithmic term in (6) as the WIMP
massincreases.Thisfactcanbeseeninthepresentedplots
For the rest of the quarks and leptons, the parametriza-
fromx¼0:5onwards.Asaconsequence,thevaluesofthe
tion given in (6) is completely valid. Now we present
resultsfor(cid:1)and(cid:7)leptonsandallquarksexceptforthetop. pparameter increase as the WIMP massesincrease.
2.(cid:7) lepton
1. (cid:1) lepton
For the (cid:1) lepton, there are only two mass-dependent Forthe(cid:7)particleandaccordingtoexpression(7),there
parameters in the spectra fitting function (6): n1 and p. are only three mass-dependent parameters: q, p and l.
These values are presented in Table XII. In this case, the
The remaining parameters are mass independent for this
50pa::34r17ti,0cn,le2aan¼ndd1q:t4h¼0e,irc01:v0a¼0l2u62e0:s5.4aTr,edh1eas1¼e¼r0e:1s2u49l:57ts,,cba2r1e¼¼p0r5:e3:s47e03n,,tedbd22 ¼¼in c1o0nT4shGiedeeVsrce.adlinragngoef tfhoer pWIpMarPammetaesrsewsitihs tfhroemW2IM5 Ptom5a(cid:1)ss
shows two well differentiated regimes, with different
Table X.
asymptotic power laws: one from M ¼25 GeV to
In this case, the WIMP mass interval considered
ranges from 25 to 5(cid:1)104 GeV. For masses higher than M ¼100 GeV, and another from M ¼750 GeV to M ¼
5(cid:1)104 GeV, the spectra do not seem to change, within 5(cid:1)104 GeV. On the other hand, q and l parameters
present a sum of two power laws. These results are pre-
the statistical uncertainties, with respect to that corre-
sponding to 5(cid:1)104 GeV. sented inTable XIII.
simTphlee np1owpearralmawetefrorscMale<s 5w(cid:1)ith10th4eGWeVI.MFPormtahsesoatshear WIAMsPfomrtahses(cid:1)inlcerpetaosne,s.thIenfltuhxisocfapshe,ottohnesqinpcraeraasmeestaesr tihne-
mass-dependent p parameter, the power-law behavior is creases as the WIMP masses do so, instead of the p
parameter as was the case forthe (cid:1).
validintwo separated intervalswith aninflectionpointin
thebehavioratM ¼1000 GeV.Theseresultscanbeseen
3. u quark
in Table XI.
SomeoftheseresultsarepresentedgraphicallyinFig.5 The mass independent parameters are a1 ¼5:58, b2 ¼
(in the Appendix), for four WIMP masses: 25, 100, 1000, 5:50,c1 ¼0:315,c2 ¼0:0(therefored2 isirrelevant)and
and 5(cid:1)104 GeV. Also in the Appendix, mass-dependent q¼9:30(cid:1)10(cid:4)4.Themass-dependentparametersareb1,
parameters n1 and p are presented in Fig. 6. n1,n2,d1,andp.TheseresultsarepresentedinTableXIV.
TABLE XI. Parameters for expression (10) for the (cid:1) lepton. It can be seen that n1 and p
parameters follow power-lawbehaviors.
Parameter WIMP mass interval (GeV) Fitting power law(s)
n 25(cid:10)M<104 10:6M(cid:4)0:0148
1
104 (cid:10)M(cid:10)5(cid:1)104 (cid:4)7:00M(cid:4)1:99þ179M(cid:4)0:763þ9:09
p 25(cid:10)M<1000 0:773M1:75
1000(cid:10)M(cid:10)5(cid:1)104 3:07M1:55
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PHOTONSPECTRA FROM WIMPANNIHILATION PHYSICAL REVIEW D 83, 083507 (2011)
TABLE XIII. Parameters for expression (10) for the (cid:7) lepton. It can be seen that the p
parameterfollowssimplepowerlawsintwodifferentWIMPmassregimes.Nevertheless,forthe
q and l parameters a sum of two power laws accounts for the WIMP mass dependence in the
whole studiedWIMP mass interval.
Parameter WIMP mass interval (GeV) Fitting power law(s)
p 25(cid:10)M(cid:10)100 176M1:25
750(cid:10)M(cid:10)5(cid:1)104 4530M0:653
q 25(cid:10)M<5(cid:1)104 0:00230M(cid:4)0:911þ0:00291M0:0348
l 25(cid:10)M<5(cid:1)104 0:626M(cid:4)0:0300þ16:4M(cid:4)1:34
The analyzed mass range for this quark is from 50 to x¼2(cid:1)10(cid:4)3 for M ¼50 GeV whereas for the rest of
8000 GeV. massesthe fits work very well until x¼5(cid:1)10(cid:4)4.
Thespectraofthetwohigheststudiedmasses(5000and Thescalingofb1withMisgivenbyasimplepower-law
8000 GeV) clearly differ in the low energy interval. inthewholemassinterval,whereasn1,n2,andc1followa
Therefore,noconclusioncanbemadeabouttheexistence sumoftwopower-lawbehaviorinthewholestudiedmass.
of an asymptotic high masses limit in the spectral shapes. Finally,thepparameterpresentsapower-lawbehaviorfor
Concerning the mass evolution of the parameters for this M>50 GeV.These results can be seen in Table XVII.
quark,weobservesimplepower-lawbehaviorsforbothb1
and p parameters in thewhole studied WIMP mass range 5.s quark
interval. On the other hand, n1, n2, and d1 parameters are For the s quark, there are just four mass-dependent
fitted by a sum of two power laws in the studied range. parameters b1, n2, d1, and p. The mass independent pa-
f5Toh(cid:1)rest1he0er(cid:4)ep4saurulatnsmticelatnethrbseeteusnerdnenooiufnttThtaoeblfiaeltlXothwVe.edTspheeenccetrhragoysferoinnmvtearlxvuae¼ls. bra2m¼ete6r:s50f,ocr1th¼is0p:a3r3ti5c,lce2in¼(60):0a(rde2ai1s¼irre4l:e8v3a,nnt1as¼fo2r:0th3e,
Nevertheless, for some masses, the fit also applies for
lower energies, i.e., lower x values, up to 10(cid:4)4. TABLE XV. Parameterscorrespondingto(10)foruquark.b1
and p parameters follow a simple power-law behavior in the
4. d quark whole studied WIMP mass interval. n1, n2 and d1 parameters
follow a sum of two power lawsin thewhole mass interval.
For this channel there are five mass-dependent parame-
ters:b1,n1n2,c1,andp.Themassindependentparameters WIMP mass
are a1 ¼5:20, b2 ¼5:10, d1 ¼0:410, c2 ¼0:0260, Parameter interval (GeV) Fitting power law(s)
d2 ¼0:570, and q¼1:40(cid:1)10(cid:4)4. All parameters in this b 50(cid:10)M(cid:10)8000 2:96M0:0506
case are presented in Table XVI. The mass range studied 1
n 50(cid:10)M(cid:10)8000 2:91M(cid:4)0:351þ1:90M0:0172
1
for this quark was from 50 to5000 GeV. n 50(cid:10)M(cid:10)8000 0:0587M0:146þ0:848M(cid:4)0:145
2
In this channel, no conclusion can be drawn about the d 50(cid:10)M(cid:10)8000 0:317M(cid:4)0:0300þ0:403M(cid:4)0:351
1
existence of an asymptotic high mass limit in the spectral p 50(cid:10)M(cid:10)8000 4:74M0:839
shape. The chosen parameters provide good fits from
TspAoBndLiEngXtIoV.expureqsusiaornk:(6b)1,wnh1e,nn2a,pdp1li,eadndtopthpearuau(cid:5)mcehtearnsnceolrfroe-r TspAoBndLiEngXtVoI.expdreqsusaiorkn:b(61),nw1,henn2,acp1,plaineddptopadrda(cid:5)mcehtaenrsneclorfroe-r
differentWIMPmasses.Massindependentparametersin(6)for
different WIMP masses. Mass independent parameters in (6)
this channel are presented at the bottom of the table.
for this channel are presented at the bottom of the table.
WIMP mass (GeV) b1 n1 n2 d1 p WIMP mass (GeV) b1 n1 n2 c1 p
50 3.60 2.77 0.585 0.383 129
50 4.09 2.69 0.561 0.327 17.7
100 3.75 2.64 0.551 0.355 225
100 4.24 2.47 0.522 0.293 66.2
200 3.88 2.54 0.521 0.332 409 200 4.39 2.34 0.480 0.258 166
500 4.04 2.44 0.490 0.308 856 500 4.56 2.28 0.448 0.228 483
1000 4.18 2.40 0.472 0.293 1540 1000 4.75 2.25 0.426 0.212 1270
2000 4.34 2.39 0.463 0.281 2800 2000 4.91 2.24 0.409 0.200 3130
5000 4.55 2.38 0.450 0.266 6000 5000 5.11 2.23 0.394 0.187 10200
8000 4.67 2.34 0.448 0.259 8900 a1 ¼5:20; b2 ¼5:10; d1 ¼0:410; c2 ¼0:0260; d2 ¼0:570;
a1 ¼5:58; b2 ¼5:50; c1 ¼0:315; c2 ¼0:0; q¼9:30(cid:1)10(cid:4)4. q¼1:40(cid:1)10(cid:4)4
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J.A. R. CEMBRANOS et al. PHYSICAL REVIEW D 83, 083507 (2011)
TABLE XVII. Parameters corresponding to (10) for the d TABLE XIX. Parameters corresponding to (10) for the s
quark.Ascanbeseen,theb1 parameterfollowsasimplepower quark. As can be seen, the b1 parameter follows a simple
law in the whole accessible mass interval. n1, n2, and c1 power-law behavior for masses higher than 1000 GeV. The n2
parameters follow a sum of two power laws in the whole and d1 parameters follow the sum of two power laws for the
accessible mass interval. Finally, the p parameter follows a whole studied WIMP mass interval. Finally, the p parameter
power lawfor M>50 GeV. presentstwopowerlaws:oneformassessmallerthan1000GeV
and another for masses higher than 1000 GeV.
WIMP mass
Parameter interval (GeV) Fitting power law(s) WIMP mass
b 50(cid:10)M(cid:10)5000 3:39M0:0485 Parameter interval (GeV) Fitting power law(s)
1
n 50(cid:10)M(cid:10)5000 21:8M(cid:4)0:993þ2:25M(cid:4)0:00113 b M>1000 4:54M0:0339
1 1
n 50(cid:10)M(cid:10)5000 0:848M(cid:4)0219þ0:161M0:0573 n 50(cid:10)M(cid:10)7000 3:68M(cid:4)1:01þ0:744M(cid:4)0:0352
2 2
c 50(cid:10)M(cid:10)5000 0:722M(cid:4)0:270þ0:0544M0:0874 d 50(cid:10)M(cid:10)7000 0:621M(cid:4)0:674þ0:414M(cid:4)0:0588
1 1
p 50<M(cid:10)5000 0:168M1:29 p 50(cid:10)M(cid:10)100 4:01M0:981
100<M(cid:10)7000 12:8M0:732
u quark), and q¼2:40(cid:1)10(cid:4)4. All these parameters are
detailed in Table XVIII. The studied mass range for this TABLE XX. c quark: b1, n1, c1, d1, and p parameters corre-
quark is between 50 and 7000 GeV. spondingtoexpression(6)inthecc(cid:5)channelfordifferentWIMP
As in the d quark case, no conclusion can be drawn masses.Massindependentparametersin(6)forthischannelare
abouttheexistenceofanasymptotichighmasslimitinthe presented at the bottom of the table.
spectral shape. The scaling with M of the parameters for
this quark is a simple power law for the b1 parameter for WIMP mass (GeV) b1 n1 c1 d1 p
masseshigherthan1000GeV,thesumoftwopowerlaws 50 5.93 2.35 0.239 0.428 210
forn2andd1parametersinthewholestudiedWIMPmass, 100 5.48 2.08 0.283 0.374 379
and two power laws for the p parameter: one for masses 200 4.98 1.86 0.330 0.330 673
smallerthan1000GeVandanotherformasseshigherthan 500 4.50 1.65 0.378 0.288 1230
1000 GeV. These results are shown in Table XIX. 1000 4.00 1.50 0.406 0.264 2110
2000 3.70 1.35 0.432 0.245 4050
6.c quark 5000 3.27 1.17 0.470 0.221 8080
As for the d quark, there are five mass-dependent pa- 8000 3.08 1.11 0.494 0.208 12000
rameters. In this case b1, n1, c1, d1, and p which are a1 ¼5:58; b2 ¼7:90; n2 ¼0:686; c2 ¼0:0; q¼9:00(cid:1)10(cid:4)4
presented in Table XX. The mass independent parameters
are a1 ¼5:58, b2 ¼7:9, n2 ¼0:686, c2 ¼0:0 (therefore
d2 is irrelevant),and q¼9:00(cid:9)10(cid:4)4. spectral shape. Higher masses simulations would be thus
Likewise the u quark, the studied mass rangewas from required also inthis case.
50to8000GeVandagainnoconclusioncanbemadeabout Thescalingofb1 andn1 withMshowsasimplepower-
the existence of an asymptotic high mass limit for the law behavior in the considered range. For c1 and p, the
singlepower-lawevolutionisonlyvalidformassesabove
200 GeV. Finally, the d1 parameter follows a sum of two
power laws in the studied mass range. These results are
TABLE XVIII. s quark: b1, n2, d1, and p parameters corre-
shown in Table XXI.
sponding to expression (6) when applied to ss(cid:5) channel for
different WIMP masses. Mass independent parameters in (6)
for this channel are presented at the bottom of the table.
TABLE XXI. The parameters corresponding to (10) for the c
WIMP mass (GeV) b1 n2 d1 p quark.Itcanbeseenthatthemass-dependentparametersfollow
apower-lawbehaviorforintermediate andhighWIMPmasses.
50 4.78 0.719 0.367 186 In particular the d1 parameter follows a sum of two power-law
100 5.31 0.669 0.332 409 behavior in hewhole accessible WIMP mass range.
200 5.43 0.648 0.315 605
500 5.60 0.612 0.290 1180 WIMP mass
Parameter interval (GeV) Fitting power law(s)
1000 5.73 0.592 0.276 1980
2000 5.87 0.575 0.263 3320 b 50(cid:10)M(cid:10)8000 9:90M(cid:4)0:130
1
5000 6.07 0.557 0.249 6500 n 50(cid:10)M(cid:10)8000 4:14M(cid:4)0:148
1
7000 6.12 0.548 0.244 7570 c1 500(cid:10)M(cid:10)8000 0:210M0:0951
a1 ¼4:83; n1 ¼2:03; b2 ¼6:50; c1 ¼0:335; c2 ¼0:0; d1 50(cid:10)M(cid:10)8000 1:50M(cid:4)0:632þ0:479M(cid:4)0:0942
q¼2:40(cid:1)10(cid:4)4 p 200<M(cid:10)8000 8:11M0:812
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