Semi-annihilation of Dark Matter

0.52 MB·English

by  Francesco D'Eramo

Semi-annihilation of Dark Matter
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Francesco D’ERAMO
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MassachusettsInstituteofTechnology
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J E-mail: [email protected]

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] Thesemi-annihilationreactiontakestheschematicformψiψj →ψkφ, whereψi arestabledark
h matter particles and φ is an unstable state. Such reactions are allowed when dark matter is sta-
p bilized by a larger symmetry than just Z . They lead to non-trivial dark matter dynamics in the
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p earlyuniverse,andthethermalproductionoftherelicparticlescanbecompletelycontrolledby
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h semi-annihilations. This process might also take place today in the Milky Way, enriching the
[ (semi-)annihilationfinalstatespectrumobservedinindirectdetectionexperiments.

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IdentificationofDarkMatter2010
July26-302010
UniversityofMontpellier2,Montpellier,France
Speaker.
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(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Semi-annihilationofDarkMatter FrancescoD’ERAMO
1. Introducingsemi-annihilation
The evidence for dark matter is one of the best motivations for physics beyond the Standard
Model(SM)[1]. Awell-motivateddarkmattercandidateareso-calledWeaklyInteractingMassive
Particles(WIMPs),whoseproductionintheearlyuniverseisthermal[2,3]. Thethermalfreeze-out
isverypredictive,thusitisimportanttoknowhowtocorrectlycomputetherelicdensity,especially
whenthereliccomputationcannotbereducedtothestandardcase(seee.g. [4]).
Thethermalabundancecanbedramaticallyaffectedbythepresenceofthe“semi-annihilation”
reaction. Ifthedarkmatteriscomposedofmorethanonestablecomponentψ,ittakestheform
i
ψψ ψ φ, (1.1)
i j k
→
where φ is a SM state or a new particle which decays to the SM. We see that unlike ordinary
annihilation where the total dark matter number changes by two units, in semi-annihilation the
total dark matter number changes by only one unit, as shown in figure 1. As long as the triangle
inequalitym <m +m issatisfied(aswellasitscrossedversions),thesemi-annihilationreaction
k i j
ψψ ψ φ can take place without making any relic particle unstable. Such reactions take place
i j k
→
oncewerelaxsomeassumptionsaboutthesymmetrystructureofWIMPinteractions. Forexample
the dark matter can be stabilized by a Z symmetry [5], composed of non-Abelian gauge bosons
3
[6],ormoregenerallystabilizedby“baryon”and/or“flavor”symmetries,asinQCD-liketheories.
Westudytwoexplicittoyexamplesofsuchmodels[7]. Wefindthattocorrectlycomputetherelic
abundance,semi-annihilationmustbeincluded,anditcandominateoverstandardannihilation. We
alsoexploretheeffectsofsemi-annihilationsonindirectdetectionexperiments.
We introduce semi-annihilation with a simple case. We assume dark matter to be composed
of a complex scalar χ, which is stabilized by a Z symmetry. The χ particle interacts with a real
3
scalarφ,whicheventuallydecaystoSMstates. The χ particlesarekeptinthermalequilibriumin
theearlyuniversebythereactions χχ¯ φφ and χχ χ¯φ. Whilewefocusonthecaseofafield
→ →
φ “portal”totheSM[8],inmoregeneraldarkmatterscenariosφ couldbeaSMfielditself[9].
Single-component dark matter with a Z symmetry is not representative of models in which
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semi-annihilation is relevant. The second model we consider is a minimal multi-component dark
matter model having semi-annihilations among its allowed reactions.1 In this toy system, dark
matter is composed of two stable components, a complex scalar field χ and a vector-like fermion
bandbc. WeimposeaU(1)globalsymmetryunderwhichb,bc,and χ havecharges+1, 1,and
−
2, respectively. Wealsointroducearealscalarportalφ field, whichweassumetobeinthermal
−
equilibrium in our calculation and decays to SM states. Fermion number conservation always
guarantees that b is stable, whereas stability for χ requires the triangle inequality rule m 2m
χ b
≤
(isoscelestriangle). Themodelanditssymmetriesaresummarizedintable1.
The remainder is structured as follows. In Section 2 we present a study of the relic density
in the two toy models we consider. In Section 3 we explore the effects of semi-annihilations on
indirectdetectionexperiments,andconcludeinSection4.
1Amorerealisticconstruction,whereasupersymmetricgaugetheorywithNf =Nc+1isconsidered,canbefound
in[7].Here,weprefertofocusonasimplermodelavoidingunnecessarycomplicationscausedbyhavingsuperpartners.
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Semi-annihilationofDarkMatter FrancescoD’ERAMO
ψ φ ψ φ
DM SM DM SM
ψ φ ψ ψ
DM SM DM DM
(a) (b)
Figure 1: Reactions that keep the dark matter particles in thermal equilibrium in the early universe: (a)
annihilation;(b)semi-annihilation. Thefieldφ decaystoSMstates.
SM
Fields Spin U(1)charge
b Weylleft +1
bc Weylleft 1
−
χ complexscalar 2
−
φ realscalar 0
Table1: Fieldcontentandsymmetriesofaminimalmulti-componentmodelwithsemi-annihilation.
2. Implicationsfortherelicdensity
Thenumberdensityevolutionofχ inthemodelwithZ symmetryisdescribedby2
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dn (cid:104) (cid:105) 1 (cid:104) (cid:105)
χ +3Hn = σv n2 neq2 σv n2 n neq . (2.1)
dt χ −(cid:104) (cid:105)χχ¯→φφ χ− χ −2(cid:104) (cid:105)χχ→χ¯φ χ− χ χ
In the σv =0 limit we recover the standard case. The above Boltzmann equation can be
χχ χ¯φ
(cid:104) (cid:105) →
solvedsemi-analyticallyinanalogywiththemethodin[3],bysolvingtheequationintworegimes,
early and late times, and then matching the two solutions at the freeze-out point. The freeze-out
temperaturex ,wherewedefinex=m /T,isfoundbysolvingtheequation
f χ
(cid:20) (cid:21) (cid:20) (cid:21)
g m M 1 c+1 σv
χ χ Pl χχ χ¯φ
xf =log 0.038c(c+2) σv χχ¯ φφ +log 1+ (cid:104) (cid:105) → , (2.2)
(cid:104) (cid:105) → √g xf 2 c+2 σv χχ¯ φφ
∗ (cid:104) (cid:105) →
where c = √2 1 for s-wave amplitudes [3]. The first term on the right-hand side of (2.2) is
−
the solution we usually have in the Lee-Weinberg scenario. Thus we see that the effect of semi-
annihilation is to shift the freeze-out temperature by only a small logarithmic amount. The mass
densityoftheχ particletodayisgivenby
1.07 109GeV 1 (cid:90) ∞dx(cid:20) 1 (cid:21)
Ωχh2=2× √g×MPlJ(xf)− , J(xf)= xf x2 (cid:104)σv(cid:105)χχ¯→φφ+2(cid:104)σv(cid:105)χχ→χ¯φ . (2.3)
∗
In single component dark matter models, the conventional assumption is that we have only an-
nihilation (namely σv =0), thus once we fix the dark matter mass m the cross section
χχ χ¯φ χ
(cid:104) (cid:105) →
2WecorrecttheBoltzmannequationin[7]withthe1/2factorinthesemi-annihilationterm.WethankBrianBatell.
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Semi-annihilationofDarkMatter FrancescoD’ERAMO
Κ(cid:61)0, Ε(cid:61)0, (cid:87)h2(cid:62)100 Κ(cid:61)4Π, Ε(cid:61)0, (cid:87)h2(cid:62)100 Κ(cid:61)0, Ε(cid:61)5, (cid:87)h2(cid:62)0.1
10(cid:45)7 10(cid:45)7 10(cid:45)7
10(cid:45)9 10(cid:45)9 10(cid:45)9
Y10(cid:45)11 Y10(cid:45)11 Y10(cid:45)11
10(cid:45)13 10(cid:45)13 10(cid:45)13
10(cid:45)15 10(cid:45)15 10(cid:45)15
15 20 30 50 70 15 20 30 50 70 15 20 30 50 70
x x x
(a) (b) (c)
Figure 2: Y vs. x evolution in the bbχ model for the b (blue lines) and the χ (green lines). We define
Y =n/s,wherei=b,χ andsistheentropydensity. Weplottheequilibriumvalues(dashedlines)andthe
i i
numericalsolutions(solidlines). Wefixm =0.8TeV,m = 1TeV,α =2andβ =0.01. Thevaluesofκ,
χ b
ε andΩ h2 areshownineachplot. Here, χ particleswouldbeoverproduced(a), butbecauseofphase
DM
space suppression κ (b χ conversion) is ineffective (b). However, ε (semi-annihilation) is never phase
↔
spacesuppressedandcanrestorethedesiredrelicdensity(c).
σv is uniquely determined by the relic density measurement. However, if we allow also
χχ¯ φφ
(cid:104) (cid:105) →
semi-annihilations,awiderregionofparameterspacegivesthecorrectrelicdensity,thusthether-
malproductionoftherelicparticlescanalsobecompletelycontrolledbysemi-annihilations.
In models which involve more than one stable dark matter component a semi-analytical so-
lution for the relic density is simply not possible, and one must resort to numerically solving the
complete set of coupled Boltzmann equations. In particular, the simplifying assumptions made
whenanalyzingco-annihilatingmodels[4]arenotapplicableforsemi-annihilation[7]. Asacon-
sequence, the dark matter dynamics in the presence of semi-annihilation are far more varied than
indecoupledmulti-componentscenarios.
In our second example model, the relic abundance of b and χ is described by a system of
twocoupledBoltzmannequations. WecannowwritedownandsolvenumericallytheBoltzmann
equationsfortherelicabundanceofbandχ andseehowsemi-annihilationsaffectthedarkmatter
relicdensity. Oncewefixthedarkmattermassesandlimitourconsiderationtos-waveannihilation,
therearefourfreeparameters,whichwechoosetobethefollowings-wavematrixelements
M =α, M =β, M =κ, M =ε. (2.4)
bb¯ φφ χχ¯ φφ bb¯ χχ¯ χb φb¯
→ → → →
Thefirsttwoamplitudescomefromthestandardannihilationtolightparticlesinthefinalstate,the
thirdfromtheprocesswheretwodarkmatterparticlesofonespeciesareentirelyconvertedtotwo
darkmatterparticlesoftheotherspecies. Thisprocessisphasespacesuppressedfortypicalkinetic
energiesatfreeze-out. Thelastcorrespondstothesemi-annihilationprocess,andweareinterested
toseehowtherelicabundancechangeswiththeparameterε.
In our numerical study we fix the dark matter masses to be m = 0.8TeV and m = 1TeV
χ b
for concreteness, and we vary the matrix elements amplitudes. We fix the values of α and β
(annihilationamplitudes)andstudyhowtherelicdensityisaffectedonceweturnontheparameters
κ (b χ conversion)orε (semi-annihilation). Themaindifferencebetweensemi-annihilationand
↔
speciesconversionisthattheformerisneverphasespacesuppressed. Thisisbeautifullyillustrated
in the case when pure annihilations would give a lot of χ and very few b, as shown in figure 2(a)
forα =2andβ =0.01. Thesemi-annihilationprocessχb b¯φ isstillabletodestroyχ particles
→
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Semi-annihilationofDarkMatter FrancescoD’ERAMO
e+
e+
φ e
ψ SM − φ e
DM ψ SM −
DM
e+
ψDM φSM ψ ψ
DM DM
e
−
(a) (b)
Figure3: Twodifferentcontributionstoindirectdetectionspectra: (a)annihilation; (b)semi-annihilation.
Thefieldφ isaportalfield,butinmoregeneraldarkmatterscenariosφ couldbeaSMfielditself.
SM
andbringtherelicdensitybacktotheobservedvalue,figure2(c)forε =5. However,thereaction
χχ¯ bb¯ is powerless, even if we badly break perturbation theory for κ =4π as in figure 2(b).
→
Thisreactionisforbiddenatzerokineticenergy,andatfreeze-outthethermalenergyisverysmall
compared with the mass splitting between the two particles. Thus, semi-annihilation is a truly
uniquespecieschanginginteractionthataffectsearlyuniversecosmology.
3. Implicationsforindirectdetection
Wenowexploretheimplicationsofsemi-annihilationsfordarkmatterdetectionexperiments.
Sincesemi-annihilationdoesnotgiveanynewcontributionstodirectdetectionrates,wefocuson
indirect detection only. There has been much recent interest on indirect detection of dark matter,
motivated by observations suggesting the presence of a new primary source of galactic electrons
andpositrons[10]. Theseanomaliesmightbeevidencefordarkmatterannihilationordecayinour
galaxy.
Ingeneral,semi-annihilationdoesnotleadtothe“boostfactor”necessarytoexplainthemag-
nitude of these candidate dark matter indirect signals. However the presence of such a reaction
can still have considerable implications for indirect searches. Semi-annihilations are additional
channels to produce light particles in dark matter interactions in our galaxy, as shown in figure
3, which have no analog in standard multi-component scenarios. Thus the predicted spectrum is
enrichedwithrespecttothestandardscenarios. Inaddition,ourexamplesmodelscanaccountfor
the fact that dark matter (semi-)annihilates preferably into leptons; if we assume a portal φ field
massm 2GeVthedecayofφ toantiprotonsiskinematicallyforbidden[8].
φ
≤
We present here the results for the bbχ model.3 In this case, there are four reactions to pro-
duceφ particlesinthefinalstate,theordinaryannihilationsofthetwocomponentsandtwosemi-
annihilations. Thusthespectrumisthismulti-componentcaseisquiterich,sincethefourreactions
yeld four monochromatic φ lines. We present numerical results for the φ spectrum for the same
values of the dark matter masses chosen in Section 2, namely m =0.8TeV and m =1TeV. To
χ b
simplify the following discussion we perform our analysis for κ =0. Two example spectra are
3ForthespectraintheZ modelseereference[7].
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Semi-annihilationofDarkMatter FrancescoD’ERAMO
Α(cid:61)1.5, Β(cid:61)0.75, Ε(cid:61)0.4 Α(cid:61)0.5, Β(cid:61)0.25, Ε(cid:61)2.1
1.0 1.0
0.8 0.8
0.6 0.6
al al
n n
g g
si 0.4 si 0.4
0.2 0.2
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
E(cid:144)m E(cid:144)m
b b
(a) (b)
Figure4: Spectraofφ particlesinthebbχ modelforthermalproductionmostlycontrolledby: (a)annihi-
lations,(b)semi-annihilations. Thevaluesofthematrixelementamplitudesareshownineachplot.
shown in figure 4, where two opposite cases are considered. In the first case, figure 4(a), the
semi-annihilationcontributionistakentobeverysmall,thusthepureannihilationlinesdominate.
When the thermal production is mostly controlled by semi-annihilations, as in figure 4(b), the
semi-annihilationlinesoverwhelmthestandardcontributions.
One expects the indirect detection spectra in models with more than two stable components
to have an even richer structure of φ lines. Of course, once convolved with the φ decays, the
spectrum is less distinct [7]. More detailed study is necessary to know whether such a spectrum
canbemeasuredinγ raystelescopes.
4. Conclusions
WerelaxedsomeassumptionsaboutthesymmetrystructureofWIMPinteractionsbyallowing
relic particles to “semi-annihilate”. We studied two explicit examples where semi-annihilation is
present: asinglespeciesdarkmattermodelwithaZ symmetryandamultiplespeciesdarkmatter
3
model with “baryon” and “flavor” symmetries. We saw that the semi-annihilation process can
haveaconsiderableeffectonthedarkmatterrelicabundance,andthedarkmatterdynamicsinthe
presenceofsemi-annihilationarefarmorevariedthanindecoupledmulti-componentscenarios.
The inclusion of semi-annihilation has interesting implications for indirect detection experi-
ments. The overall integrated flux is not very affected by semi-annihilation. However, the final
statespectruminsemi-annihilatingmodelsisfarricherthaninstandardsscenariosbecauseofthe
differingkinematicsbetweensemi-annihilationandordinaryannihilation.
Inthelanguageof[4],semi-annihilationisinsomewaysthe“fourthexception”inthecalcu-
lationofdarkmatterthermalrelicabundances. Wefinditintriguingthatunlikethetraditionalthree
exceptions(co-annihilation,annihilationbelowamassthreshold,andannihilationnearapole),this
fourth exception not only affects dark matter interactions in the early universe, but also leaves an
imprinttodayviatheindirectdetectionspectrum. Weexpectthatthereareawidevarietyofmulti-
component dark matter models with species changing interactions, motivating further studies of
thesemi-annihilationprocess.
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Semi-annihilationofDarkMatter FrancescoD’ERAMO
Acknowledgments
WethankJesseThalerforcollaboration. ThisworkissupportedbyU.S.DepartmentofEnergy
(D.O.E.)undercooperativeresearchagreementDE-FG0205ER41360.
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