The PAMELA excess from neutralino annihilation in the NMSSM

25 Pages·2009·0.44 MB·English

FERMILAB-PUB-09-115-T
The PAMELA excess from neutralino annihilation in the NMSSM
Yang Baia, Marcela Carenaa,b, and Joseph Lykkena
aTheoretical Physics Department, Fermilab, Batavia, Illinois 60510,
bEnrico Fermi Institute, Univ. of Chicago, 5640 Ellis Ave., Chicago, IL 60637
We examine whether the cosmic ray positron excess observed by PAMELA can be explained
by neutralino annihilation in the Next-to-Minimal Supersymmetric Standard Model (NMSSM).
The main dark matter annihilation products are the lightest CP-even scalar h1 plus the lightest
CP-oddscalar a1,with thea1 decayingintotwomuons. Theenergeticpositronsneededtoexplain
PAMELAarethusobtainedintheNMSSMsimplyfromkinematics. Therequiredlargeannihilation
9
0 cross section is obtained from an s-channel resonance with the heavier CP-odd scalar a2. Various
experiments constrain the PAMELA-favored NMSSM parameter space, including collider searches
0
2 for a light a1. These constraints point to a unique corner of the NMSSM parameter space, having
a lightest neutralino mass around 160 GeV and a very light pseudoscalar mass less than a GeV.

p A simple parameterized formula for the charge-dependent solar modulation effects reconciles the
e discrepancy between the PAMELA data and the estimated background at lower energies. We
S
also discuss the electron and gamma ray spectra from the Fermi LAT observations, and point out
4 the discrepancy between the NMSSM predictions and Fermi LAT preliminary results and possible
resolution. An NMSSM explanation of PAMELA makes three striking and uniquely correlated

] predictions: therise in thePAMELA positron spectrum will turnoverat around 70 GeV,thedark
h matter particle mass is less than the top quark mass, and a light sub-GeV pseudoscalar will be
p
discovered at colliders.
-
p
PACSnumbers: 12.60.Jv,95.35.+d
e
h
[
I. INTRODUCTION can be explained if dark matter particles first an-

2 nihilate into some intermediate particles, which are
v so light that their decays to hadrons are kinemat-
4 Recently, the PAMELA collaboration has ob-
ically forbidden [5], or couple dominantly to Stan-
6 served an anomalous positron abundance in cos-
dard Model (SM) leptons [6]. To explain the large
9 mic radiation [1]. The positron over electron frac-
2 tion turns over and appears to rise at energies from dark matter annihilation cross section in the galac-
. tic halo while remaining consistent with a thermal
5 10 GeV to 100 GeV. However, from the same de-
dark matter relic abundance, one can introduce an
0 tector, no obvious antiproton excess is seen for the
attractive force between two dark matter particles
9 same energy range [2]. Many suggestions have been
0 and use the Sommerfeld enhancement to boost the
made to explain the positron excess at PAMELA.
: annihilation cross section in the galactic halo [7].
v Among different approaches, dark matter annihila-
This scenario suggestsinteresting signatures [8] and
i tion is especially interesting and could imply future
X can be tested at the LHC [9]. A second approach is
signals in dark matter direct detection experiments
r toconsidernon-thermalrelics. Otherlong-livedpar-
and/or at the Large Hadron Collider (LHC).
a
ticles candecay to the darkmatter particlesand in-
Inrecentmodel-independentstudies,darkmatter
creasethe darkmatterrelicabundanceinthe galac-
particlesarerequiredbothtoannihilatedominantly
tic halo [10].
to leptons and to have a much larger annihilation
rate in the galactic halo than would be implied by Instead of constructing a dark matter model ad
a traditional thermal relic estimate [3] [4]. The lack hoc to explain the cosmic ray observations, in this
ofantiprotonexcessinthe PAMELAexperiment[2] paper we ask whether there is an existing well-
2
motivated model that naturally contains the nec- the final state can dominantly be a + h with the
1 1
essary ingredients to explain the positron excess of formerdecayingtoleptons. Thekinematicshelpsus
PAMELA. The most developed framework to ad- to obtain hard leptons in the final state of the dark
dress the naturalness problem of the SM is the matter annihilations.
MinimalSupersymmetricStandardModel(MSSM), In Section II, we develop the notation by deriv-
whichprovidesthe lightestsuperpartner(LSP)pro- ing the spectrum and interactions in the NMSSM,
tected by R-parity as the dark matter candidate. and show two sets of representative model points
In general, the annihilation products in the MSSM allowed by current experimental constraints. We
contain not only leptons but also a large fraction of calculate the dark matter annihilation cross section
hadrons. This makes it difficult for the MSSM to in Section III and positron excesses from neutralino
explainthe positronexcess atPAMELA,andatthe annihilation in Section IV. In Section V, we con-
same time be consistent with the antiproton spec- sider the constraints on the model parameter space
trumatPAMELA[11]. Fromthe theoreticside,the fromthePAMELAantiprotonspectrum. Wediscuss
MSSM suffers the µ-problem, which can be solved the gamma-ray spectrum in Section VI and point
elegantly by introducing a new gauge singlet chiral out a discrepancy with recent preliminary Fermi
supermultiplet, as proposed in the Next-to-Minimal LAT results and comment on a possible resolution.
Supersymmetric Standard Model [12]. A recent ex- In Section VII we demonstrate the consistency of
plorationoftheNMSSMparameterspaceshowsthat our model points with direct constraints from LEP,
thelightestCP-oddparticlea (mainlyfromthesin- Tevatron, CLEO, B-factories and the magnetic mo-
1
glet component) can be naturally lighter than 2m ment of the muon. Finally, we discuss dark matter
b
and mainly decay into two τ’s [13]. One should no- direct detection and conclude in Section VIII.
tice that the mass of a in the NMSSM is protected
1
bytheU(1) symmetryandcanevenbelighterthan
R
1 GeV if the soft terms associated with the singlet II. SPECTRUM AND INTERACTIONS IN
are small. In this case the a will decay mainly into THE NMSSM
1
two muons for a mass of a few hundred MeV.
To describe the NMSSM model, wefollow the no-
Therefore, if the dark matter candidate in the
tation in the Ref. [15]. The superpotential in the
NMSSMcanannihilatemostlyintoa ’s,wecanhave
1 NMSSM is
a leptonic final state in the annihilation products
κ
simply from kinematics. The NMSSM provides the W = λSˆHˆ Hˆ + Sˆ3, (1)
u d
necessary ingredients to make this happen. Notice 3
that the dark matter candidate in the NMSSM is
and the soft supersymmetry-breaking terms are
the LSP neutralino (for existing studies for light
neutralino dark matter in the NMSSM, see [14]). κ
V = λA SH H + A S3 + h.c.. (2)
If its mass happens to be around half that of the λ u d 3 κ
heavier CP-odd scalar a mass, a large dark mat-
2
ter annihilation cross section is obtained through Here, hatted capital letters denote superfields, and
the s-channel resonance effects with a . To have unhatted capital letters the corresponding scalar
2
leptons dominant in the final state, we should have components. The minimization of the scalar po-
a large branching ratio of a to a plus h . This tential determines their vacuum expectation values
2 1 1
will naturally happen, provided that the dark mat- (VEVs): hu Hu , hd Hd and s S . The
≡ h i ≡ h i ≡ h i
ter LSP mass is less than the top quark mass, and electroweak scale v = h2 + h2 = 174 GeV. Since
u d
thus that the decay of a to tt¯is kinematically for- µ = λs, there are four new parameters in the
2 eff
p
bidden. Since the CP-odd scalar coupling to other NMSSM, which we take to be real: λ, A , κ and
λ
fermions is proportional to their Yukawa couplings, A . With sign conventions for the fields, λ and
κ
3
tanβ h /h are positive, while A , κ, A and and with its mass approximated as
u d λ κ
≡
µ can have either sign.
eff g2v2(M + µ sin2β)
whTichherisespexoinsttasnaeouZs3lysbyrmokmenetaryndfionrduthceesNaMdoSmSaMin, mχ = M1 + 1 2(M12 µ2) . (8)
1 −
wall problem. One can introduce higher dimension
Hereafter, we use a simple notationχ to replace the
operators to explicitly break this discrete symme-
usual notation χ˜0 for the lightest neutralino.
tryandperhapscircumventthisproblem[16]. Since 1
those breaking operators have small effects on
3
Z
theanalysisperformedinthispaper,wewillneglect
B. Higgs sector at tree level
them from now on.
The charged Higgs H± = cosβHu± + sinβHd±
has a mass
A. Neutralinos
2 g2
M2 = λs(A + κs) + ( 2 λ2)v2.(9)
There are five neutralinos in the NMSSM: the H± λ sin2β 2 −
U(1) gaugino λ , the neutral SU(2) gaugino λ ,
Y 1 W 2
the Higgsinos ψu0 and ψd0 and the singlino ψs. In For κ and λ of order of unity, the mass of H± is
the basis ψ0 =( iλ , iλ ,ψ0,ψ0,ψ ), we have the generically (µ).
neutralino mass−mat1ri−x 2 u d s ExpandinOgaroundtheHiggsfieldsVEVs,theneu-
tral scalar fields are defined as
1
= (ψ0)T (ψ0) + h.c., (3)
L −2 M0 H0 = h + HuR + iHuI ,
where u u √2
M1 0 g1√h2u −g√1h2d 0 Hd0 = hd + HdR √+2iHdI ,
 0 M2 −g2√h2u g√2h2d 0  SR + iSI
S = s + . (10)
M0 = g1√h2u −g2√h2u 0 −µ −λhd .(4) √2
 −g√1h2d g√2h2d −µ 0 −λhu  For the three CP-even neutral states, we can diag-
 0 0 λh λh 2κs  onalize their 3 3 mass matrix by an orthogonal
 − d − u  matrix S to o×btain the mass eigenstates (ordered
  ij
Toobtainthe neededdarkmatter annihilationrate, in mass): h = S (H ,H ,S ) , with masses
i ij uR dR R j
we consider the case that the LSP, the lightest neu- denoted by m . For v s, we write the lightest
tralino, is mainly made of the bino with mixings CP-even Higghsias ≪
with Higgsinos. For g v,λv M < µ < M ,
i 1 2
a moderate tanβ > 1 and µ≪< 0|, th|e li|gh|test|neu|- h1 cosα cosθSHuR + sinα cosθSHdR
≈
tralino is approximately sinθ S , (11)
S R
−
χ iλ (sinα cosβ sinα sinβ)ψ0
≈ − 1 − 1 − 2 u with
(sinα cosβ + sinα sinβ)ψ0, (5)
− 2 1 d π λM2 sin2β sin4β
α β Z ,
with ≈ 2 − − 4κµ2
1 √2g vµ cos2β λ2v κ
α = arctan 1 ,(6) θ = 1 sin2β + (v3/s3)(.12)
1 2 µ2 M2 + (λ2 g2/2)v2 S −2κ2s − λ O
− 1 − 1 (cid:16) (cid:17)
α = 1 arctan√2g1v(M1 + µ sin2β), (7) Thesingletcomponentofh1 issmallandsuppressed
2 2 M2 µ2 by v/s.
1 −
4
There are three CP-odd pseudoscalar fields, one maining 2 2 mass matrix in the (A˜,S ) basis with
I
×
of which is a massless Goldstone mode eaten by the A˜ cosβH + sinβH , is
uI dI
≡
Z boson. Dropping the Goldstone mode, the re-
λsh2u+h2d(A + κs) λ h2 + h2(A 2κs)
= huhd λ u d λ − . (13)
odd
M λ h2 + h2(A 2κs) 4λκhph + λA huhd 3kA s 
u d λ − u d λ s − κ
 
We introduce a mixing anglpe θ to diagonalize the above matrix:
A
2s(A + κs) v2(A 2κs)2 sin22β
tanθ = λ , cos2θ = λ − , (14)
A −v(Aλ − 2κs) sin2β A 4s2(Aλ + κs)2 + v2(Aλ − 2κs)2 sin22β
andarriveatthephysicalCP-oddstatesa (ordered and
i
in mass)
s
tanθ ,
a1 = cosθA(cosβHuI + sinβHdI) + sinθASI, A ≈ v sin2β
a2 = sinθA(cosβHuI + sinβHdI) + cosθASI. v2 sin22β v2 sin22β
− cos2θ ,(19)
(15) A ≈ v2 sin22β + s2 ≈ s2
ThelightestCP-oddparticlea ismainlycomposed
1
for v s.
ofthesingletfieldwhencosθA 0,andisadoublet ≪
→
fieldotherwise. Since we are lookingfor a verylight
scalar, we observe from the determinant of
odd
M
that this occurs if Aκ and Aλ are small. This is C. Interactions and decay modes
technicallynaturalsinceA ,A 0isasymmetry-
κ λ
→
enhancing limit.
Sinceweareinterestedindarkmatterannihilation
TheheavierCP-oddscalarmassisapproximately
through s-channel a exchange, we list the relevant
2
M2 = 2λs(Aλ + κs) vertices associated with CP-odd scalars in this sec-
a2 sin2β tion. For v s, the couplings of CP-odd scalars to
≪
λv2(A 2κs)2 sin2β fermions are
λ
+ − . (16)
2s(A + κs)
λ m sinθ
a t tc : i t A ,
with the condition Aλ + κs > 0. 2 L R − √2v tanβ
proFxoirmsamtealflorAmλualaned: Aκ, we have the following ap- a2bLbcR : imb ta√n2βvsinθA ,
M2 3s(3λAλv2 sin2β − 2κAκs2),(17) a t tc : i mt cosθA ,
a1 ≈ 2(s2 + v2 sin22β) 1 L R − √2v tanβ
2κλs2 m tanβ cosθ
M2 + 2κλv2 sin2β, (18) a b bc : i b A . (20)
a2 ≈ sin2β 1 L R √2v
5
Thecouplingsofa anda tothelightestneutralino with α and α defined in Eqs. (6–7). Therefore,
2 1 1 2
are the coupling of a χχ increases as one increases the
2
Higgsinocomponentsofthe LSP.Theircouplingsto
iga2χχ ≡ ig1 sinθA(cos2β sinα1 − sin2β sinα2), gauge bosons are
(21)
ig ig cosθ (cos2β sinα sin2β sinα ),
a1χχ ≡ − 1 A 1 − 2
(22)
ig sinθ
a (p)H+(p)W : 2 A (p p) , (23)
2 ′ µ− − 2 − ′ µ
ig cosθ
a (p)H+(p)W : 2 A (p p) , (24)
1 ′ µ− 2 − ′ µ
ig ig M2 sinθ sin4β
a (p)h (p)Z : sinθ cos(α+β)(p p) Z A (p p) , (25)
2 1 ′ µ −√2 A ′− µ ≈−√2 2M2 ′− µ
a2
ig ig M2 cosθ sin4β
a1(p)h1(p′)Zµ : √2 cosθA cos(α+β)(p′ − p)µ ≈ √2 Z 2MA2 (p′ − p)µ. (26)
a2
The couplings among a , a and h depend on Thus for v/s 1 the decay h 2a is suppressed
2 1 1 1 1
the diagonalizationof the CP-even scalar mass ma- infavorofh ≪ b¯b. Notethissu→ppressionisdirectly
1
→
trix, which may have significant one-loop contribu- connected to the small values of A and A . For
λ κ
tions. For simplicity, we use the tree-level results in large values of A and A , the h 2a decay
λ κ 1 1
→
Eq. (11) to obtain analytic formulae. We arrive at would be the dominant one [17].
the following dimensional couplings to the leading Similarly for a , if its mass is below twice the top
2
power in v/s [17] quark mass, the leading two decay channels are
κλ
a1a1h1 : wa1a1h1 =O(v3/s2), (27) Γ(a2 → h1a1) ≈ 32π Ma2 sin2β, (31)
a a h : w = √2κµ+ (λ2v2/s).(28) 2
2 1 1 a2a1h1 − O Γ(a b + ¯b) 3Ma2 mb tanβ sinθA (3.2)
2
→ ≈ 8π √2v
Here the approximation is valid for small values of (cid:18) (cid:19)
A and A . We also need the main decay channels The bosonic decay channel can be dominant for
λ κ
of h1 and a2. For the CP-even particle h1 with a modest values for κ and λ. However, if the a2 mass
mass below 2M , it mainly decays into 2 a ’s or 2 exceedstwicethetopquarkmass,thedecaychannel
W 1
b’s with the decay widths calculated as following: into tt¯opens:
2
3M m sinθ
Γ(h1 2a1) = 1 (v6/s4), (29) Γ(a2 → t + t¯) ≈ 8πa2 √2tv tanAβ
→ 32πMh1 O (cid:18) (cid:19)
Γ(h b + ¯b) 3Mh1 mb 2 . (30) 1 4m2t , (33)
1 → ≈ 8π (cid:18)√2v(cid:19) ×s − Ma22
6
and this would become the dominant decay for a caystothreepionsaresuppressedbythethree-body
2
assuming κ,λ<1. phase space. In the following, we will approximate
a 2µ as 100% for case 1. In the second case,
1
→
because the couplings of a to fermions are propor-
1
D. Spectrum from numerical calculations tional to fermion masses, we anticipate that a de-
1
cays mainly to 2 τ’s. We do not consider the case
Inthis sectionwefind NMSSMmodelpoints that that a1 mainly decays into two electrons because of
canprovidetheneutralinoasaDMcandidatetoex- stringent constraints from the beam-dump experi-
plainPAMELA. There are two relevantpossibilities ment at CERN [18].
depending on the mass of a1: To find the interesting parts of the parameter
space,weusetheprogramNMHDECAY[15]fornu-
1. The massofthe lightestCP-oddparticlea is
1 merical checks. There are many experimental con-
in the range (2m ,1 GeV).
µ straints considered in NMHDECAY such as various
Higgs searches at LEP, b sγ and Υ(1S) a γ.
2. The massofthe lightestCP-oddparticlea is 1
1 → →
The model points presented in this paper pass all
in the range (2m ,2m ).
τ b
of the constraints embedded in NMHDECAY. Fur-
In the first case a mainly decays to two muons thermore, we will discuss updated constraints on
1
because decays to mesons are kinematically sup- Υ(3S) a γ µ+µ γ from BaBar and searches
1 −
→ →
pressed. Because a is CP-odd, decays to two pi- for dimuon resonances at LEP and Tevatron in sec-
1
ons are forbidden due to CP symmetry, while de- tion VII.
tanβ λ κ Aλ Aκ µeff M1 M2
3.1 0.24 0.194 -0.05 -0.273 -190 178.5 200
mχ Ma1 Ma2 Mh1 Mh2 MH± mχ± Γa2
161.8 0.81 320.1 114.7 297.6 325.4 175.6 0.22
Br(h1 →b¯b)=78.3% Br(h1 →ττ¯)=8.1% Br(h1 →a1a1)=1.0%
Br(a1 →µ+µ−)≈100%
Br(a2 →a1h1)=68.0% Br(a2→b¯b)=22.5% Br(a2 →Zh1)=5.4%
χ = −0.595(−iλ1) + 0.347(−iλ2) + 0.599(ψu0) + 0.404(ψd0) − 0.061(ψs) cosθA =0.12
TABLE I: µ-favored model point. Masses are in GeV.
Forbothcases,inordertoisolatethedarkmatter masses for sleptons, 1 TeV for squarks, 1 TeV for
discussion, we choose the less relevant soft terms to gluinoand 2.5TeVforalltheA-termsinthequark
−
be heavy. For example, we choose 500 GeV soft and lepton sectors. We have used the updated top
7
quark mass m = 173.1 0.6 1.1 GeV [19], which hadrons being disfavored by the null antiproton ex-
t
± ±
hasasignificantcorrelationtotheHiggsbosonmass cessatPAMELA.Itisintriguingthatacombination
for small tanβ as considered in this paper. of NMSSM and PAMELA results forces us to have
For the muon-favored case, we choose values for a dark matter mass below around 170 GeV.
otherparametersintheNMSSMasinTableI,which ItistechnicallynaturaltohaveM below1GeV
a1
also shows the relevant spectrum and branching ra- for tiny values of A and A as reported here, since
λ κ
tios of the light scalars. The default lower limit on the U(1) symmetryprotectsitsmass(onecanalso
R
the a mass is 1 GeV in NMHDECAY. One has to use the U(1) symmetry to obtain a light pseu-
1 PQ
change the code file named mhiggs.f to obtain an doscalar, see [20] for example). Notice that the
a mass below 1 GeV. As can be seen from Table I, branching ratio of h a a is below 1.2% to sat-
1 1 1 1
→
if a can be produced fromχχ annihilation through isfy the current null results of searches of a in the
2 1
the resonance effect, the final products of the DM channel h a a 4µ atD0 (see Section VIIB).
1 1 1
→ →
annihilation mainly contain a +h . The a decays For the tau-favored case, we list the values of
1 1 1
into two muons to provide the positrons needed to model parameters, spectrum of particles and inter-
explain the excess at PAMELA. The dark matter esting branching ratios of light scalars in Table II.
mass is mainly controlled by the parameter M in The current direct searches only impose mild con-
1
thismodel,whichischosentohavethelightestneu- straints on the model parameters. Therefore, we
tralino mass below the top quark mass. Otherwise, choose one representative point in the parameter
a willdecayintott¯withasignificantbranchingra- space to have h a a 4τ as the main decay
2 1 1 1
→ →
tio, and the neutralino annihilation produces a lim- channel of h , and hence to have six τ’s in the final
1
ited amount of positrons and lots of hadrons, the state of dark matter annihilations.
tanβ λ κ Aλ Aκ µeff M1 M2
2.0 0.519 0.458 -14.83 -3.6 -200.0 162.25 1000
mχ Ma1 Ma2 Mh1 Mh3 MH± mχ± Γa2
161.7 7.8 322.0 123.0 296.9 304.7 208.0 0.79
Br(h1 →a1a1)=92.2% Br(h1 →b¯b)=7.6%
Br(a1 →τ+τ−)=85.0% Br(a1 →gg)=7.7% Br(a1→cc¯)=5.4%
Br(a2 →a1h1)=95.5% Br(a2→b¯b)=2.8% Br(a2 →a1h2)=0.7%
χ = 0.968(−iλ1) + 0.003(−iλ2) − 0.236(ψu0) − 0.080(ψd0) + 0.043(ψs)
TABLE II: τ-favored model point. The masses are in
GeV.
III. DARK MATTER ANNIHILATION terannihilationproducts. Thisgoalcanbeachieved
CROSS SECTION
Our first goal is to look for parameter space in
the NMSSM having leptons as the main dark mat-
8
if the lightest CP-odd scalar is an intermediate an- two cases. The annihilation rate is
nihilation product and subsequently decays to two
taus or two muons from kinematic constraints. The σ(χχ a2 a1h1)vdm
→ →
othergoalistohavealargedarkmatterannihilation g2 w2 M2
crosssectiontoexplainthesizeofthePAMELAex- = a2χχ a1a2h1 1 h1 ,
cess. Theannihilationcrosssectioncanbeenhanced 16π (Ma22 − 4m2χ)2 + Γ2a2Ma22 − 4m2χ!
through s-channel resonance effects. Assuming the (cid:2) (cid:3) (34)
lightest neutralino is the dark matter candidate in
the NMSSM, the heavier CP-odd scalar a is the up to (v2 /c2) where the average dark matter
only particle which can play this role. T2he CP- speed iOs vddmm/c 10−3 in the galactic halo. Here
∼
evenscalarsare ruledout by the CP symmetry, be- the massofa1 isneglectedandthecouplingwa1a2h1
cause the initial state with two identical Majorana hasmassdimensionone. Theco-annihilationeffects
fermions is CP-odd. The lighter CP-odd scalar a can be neglected here, because other superpartner
1
is far below the necessary mass region for the reso- masses are much larger than the LSP mass. For
nance effect. the parameter points in Table I and Table II, we
daHrkavminagtteerstaanbnliishhieladtiχonχc→hanan2el→, thXe masaitnhaenmniahiin- hanavde3σ01(χpχbc→, reas1phec1t)ivvdemly.apAplsrooxfirmomateTlyab1l3e5Ipabn·dc
·
lationproductsareequivalenttothedecayproducts Table II, the width Γa2 of a2 is below 1 GeV, so
of a . The main dark matter annihilation products the width part in the denominator of Eq. (34) can
2
are shown in Fig. 1 for the two cases exemplified by be neglected for |Ma2 − 2mχ| > 1 GeV consid-
Table I and Table II. ered here. Therefore, a large dark matter annihila-
tioncrosssectioncaneasilybeobtainedforthecase
of m < m . On the contrary, if m > m , the
b χ t χ t
χ h1 decay channel of a2 → tt¯ is open and the decay
¯b width of a2 is of order 10 GeV. Then the dark mat-
terannihilationcrosssectionislimitedbythe width
a2 part in Eq. (34), generically below 10 pbc and not
χ a1 µ− large enough to explain the PAMELA da·ta. Thus
wehavetwo reasonstobelievethatthelightestneu-
µ+
tralino mass is below the top quark mass: one is
τ− tohavedominantlyleptonicannihilationfinalstates
a1
τ+ andtheotheroneistohavealargeannihilationcross
χ h1 τ− section.
a1 Our annihilation rates are by far larger than the
a2 τ+ necessary one ( 1 pbc) to satisfy the dark matter
χ a1 τ− thermal relic de∼nsity. ·One possible explanation for
τ+ this discrepancy is that the dark matter is nonther-
mal. For example, other long-livedparticles like the
FIG. 1: The Feynman diagram of the dominant neu- gravitinoormodulicandecayintotheLSPatalater
tralino annihilation channel. The upper panel is for the time [10]. As argued in [10], a long-lived modulus
muon-favored case, while the lower panel is for the tau- fieldwithalifetimeshorterthanonesecondcannat-
favored case. urallyappearintheanomaly-mediatedSUSYbreak-
ing model. The main dark matter relic abundance
To calculate the dark matter annihilation cross will be determined by the moduli annihilation cross
section,wewillconcentrateonthemainannihilation section, which in principle can be smaller than the
channel χχ a a h , which is true for both onefortheneutralinoandprovidetheobserveddark
2 1 1
→ →
9
matter relic density at the current time. thermallyaveragedannihilationcrosssectionandto
a good approximation can be replaced by the for-
mula in Eq. (34); dN /dE is the energy spec-
e+ e+
IV. POSITRON EXCESSES FROM trum of positrons; ρ(x) is the dark matter distri-
NEUTRALINO ANNIHILATION bution inside the Milky Way halo. For the dark
matterdistributionweuseeithertheNavarro,Frenk
We now calculate the positron spectrum out and White (NFW) profile [21] or the cored isother-
of dark matter annihilation and compare it with mal (ISO) profile [22]; the use of other profiles may
the PAMELA results to determine the PAMELA- change discussions in this paper, especially for the
favored parameter space in the NMSSM. We gamma ray spectrum. The NFW and ISO profiles
first discuss the source term of primary positrons are parametrized as:
from DM annihilations, propagation of cosmic ray
positrons and then the positron fluxes measured at
PAMELA. We also consider the charge-dependent
solarmodulationonthe positronfractionspectrum.
2
r r + r
s
ρNFW(r) = ρ ⊙ ⊙ ,
A. The source term for primary positrons ⊙ (cid:16) r (cid:17) (cid:18) rs + r (cid:19)
r2 + r2
For the two cases we are considering in this pa- ρISO(r) = ρ⊙ (cid:18)rss2 + r⊙2(cid:19) , (36)
with r = 20 kpc (NFW), 5 kpc (ISO) is the
per, the source term for primary positrons can be s
radius of the central core; r = 8.5 kpc is the
generally written as: galactocentric distance of the⊙solar system; ρ =
1 ρ(x) 2 dN 0.3 GeV cm−3 is the solar neighborhood DM⊙den-
q(x,E) = σv e+ . (35) sity.
2h i(cid:18) mχ (cid:19) dEe+
Here the overall factor 1/2 is from the Majorana The positron energy spectrum function for the µ
propertyoftheneutralinoDMcandidate; σv isthe case can be generally expressed as
h i
dN dN
e+ = dE dE a1 (E E ) (E E )
dEe+ Z a1 µ+ dEa1 P a1 → µ+ P µ+ → e+
dN
+ 2 dE dE dE h1 (E E ) (E E ) (E E ). (37)
h1 a1 µ+ dE P h1 → a1 P a1 → µ+ P µ+ → e+
Z h1
Here (E E )denotestheprobabilityofaparti- favored case. Because positrons and electrons from
i j
P →
cle i with energy E decaying into a particle j with b decays are relatively soft, including them makes
i
energy E ; the factor of 2 in the expression is be- only a slight change in the spectrum below 10 GeV,
j
causeh decaysinto2a ’s. Weneglectthepositrons where the background is anyway dominant.
1 1
and electrons from the b quark decays although the
h mainly decays into two b quarks in the muon-
1
From two χ’s annihilating into a and h , and
1 1
10
neglecting the mass of a , we have
1
EΤ+=10GeV
ddNEaa11 = Br(a2 → h1a1)δ Ea1 − (mχ − 4Mmh21χ)! , Ntot 0.1 NEΤu+m=.3F00ittGedeV
(cid:144) 0.01
E
dN M2 d
dEhh11 = Br(a2 → h1a1)δ Eh1 − (mχ + 4mhχ1 )! . (cid:144)dN 0.001
(38)
10-4
Sincetheemissionofmuonsinthea1 restframeand 0.0 0.2 0.4 0.6 0.8 1.0
theemissionofa ’sintheh restframeareisotropic,
1 1 E (cid:144)E
we have e+ Τ+
1 EΤ+=10GeV
P(Ea1 →Eµ+) ≈ Ea1 H(Ea1 − Eµ+), 0.01 EΤ+=300GeV
Br(h1 a1a1) Ntot 0.001 Num.Fitted
(E E ) → , (39)
P h1 → a1 ≈ Eh1 (cid:144)dE 10-4
(cid:144)
with (x)istheheavy-sidefunctionandthemasses N
of µ+Hand a1 are neglected in the approximation d 10-5
formula. Neglecting the muon polarization, the
positron energy probability in muon decay has the 10-6
0.0 0.2 0.4 0.6 0.8 1.0
following analytic form [4]
Ee-(cid:144)EΤ+
1
(E E ) = 5 9x2 + 4x3
P µ+ → e+ 3Eµ+ − FIG.2: Upperpanel: thepositronenergyspectrumfrom
h(E(cid:2) E ), (cid:3)(40) inclusivetaudecays. Theredandtheblackpointsarefor
× µ+ − e+ 10 GeV and 300 GeV τ+ energy, respectively. The blue
with x E /E . Here the functions (E line is from a fittedanalytic function in thetext. Lower
≡ e+ µ+ P i →
E ) are normalized such that dE (E E ) = panel: similar as theupperpanel, but for electrons. On
j j i j
Br(i j). P → average, 1.16 positrons and 0.16 electrons are generated
Th→e τ-favored case is more cRomplicated than the from one τ+ decay.
µ-case, because other than leptonic channels τ can
also decay to various charged mesons, which decay
eventuallytoelectronsandpositrons. Inordertoob-
tainthe electronenergydependent probabilityfrom
τ decays, we use PYTHIA [23], which calls the pro-
gram TAUOLA [24], to simulate the inclusive elec-
tron/positronenergyspectrumfrombothdirectand
indirect τ decays. As can be seen from Fig. 2, once for Ee−/Eτ+ >0.5 can be understood from the fact
that only high multiplicity final states can contain
theenergyofthe(unpolarized)τ ismuchlargerthan
an e with a different charge from τ+.
its mass, the fractional electron and positron en- −
ergy spectra are independent of the energy of the
τ. The large fluctuations for Ee−/Eτ+ > 0.5 in
the lower panel of Fig. 2 are due to limited Monte For these e+ and e spectra from inclusive τ+
−
Carlo statistics. The tiny fraction of e out of τ+ decays, the following fitted functions provide good
−