The WIMP Miracle and Neutralino

Young-kyoung BaeKorea Advanced Institute of Science and Technology(KAIST), South Korea

(Dated: November 4, 2018)

Since Fritz Zwicky had inferred the dark matter, there were many candidates what it is, theMACHO, axion, etc. The weakly interacting massive particle(WIMP) is one of the most promisingcandidates of dark matter. In this article, I will introduce the ’freeze out’ process of dark matter,the WIMP miracle and a neutralino which is the best motivated WIMP candidate.

I. INTRODUCTION

Identifying the nature of dark matter(DM) is one of themost important unsolved problem in physics. Since F.Zwicky inferred its existence[1] and V. C. Rubin providedthe rotation curve which gives the strong indication ofDM[2], there were many trials to determine what it is andhow to detect it. Through their research, we can expectsome properties of DM particles[3]: 1) Non-relativistic(The Milky Way’s DM halo, 〈v〉 ∼ 200km/s), 2) Non-baryonic (neither electric nor (preferably) color charges),3) Stable (the life-time of DM particle must exceed theage of Universe.) By using these properties, it is possibleto get the lower bound mass of DM particle without anyspecific model. If the DM is an ultra-light scalar(fuzzy),then there is no limit of occupation number. So we canestimate the lower bound mass of a scalar DM particle byusing the uncertainty principle, mscalar ≥ 10−22eV. (Forthe tightest bounds, we used halos surrounding dwarfgalaxies.[4]) If the DM particle is a fermionic particle,it has different lower bound mass which is much morestringent due to the Pauli exclusion, mferm ≥ 0.7keV[4].

These calculated range of DM mass suggests the de-tection range, but its range is too wide to detect theparticles. So, many people focused on the early Uni-verse, because DM relics are considered to be producedat that time. There are two ways(at least) to describethis process: 1) DM particles are generated in thermalequilibrium, e.g. WIMP. 2) the DM relics produced out-side the thermal equilibrium, e.g. axion. (Actually, thenon-thermal WIMP scenario also exists. But in general,WIMP is treated as thermal relics.)

In this article, we will focus on the weakly interact-ing massive particles(WIMPs), which is the most oftenconsidered candidate for cold DM(CDM). Therefore, atfirst, we will review the thermal history of the Universeon section 2 to know the thermal equilibrium process.And then, section 3 describes what ’the WIMP miracle’is. Section 4 gives the detection strategies of WIMPsbriefly. Finally, section 5 will explain a neutralino, whichis one of the most promising candidates of CDM.

II. THERMAL HISTORY

To understand the thermal history of the Universe, wehave to consider two things, the rate of interaction Γ

and the rate of expansion H. For a process of the form1+2 3+4, then the interaction rate of particle 1 is Γ1 =n2〈σv〉, where n2 is the density of particle 2 and 〈σv〉 isthe velocity-averaged cross section. The expansion rate,H, can be expressed H ∼ √ρ/Mpl at the early Universeby using the Friedmann equation.

These two rates determine the evolution of particles.When Γ H, then the time scale of particle interac-tion is much smaller than the expansion time scale. Sountil the expansion rate becomes relevant, particles arein thermal equilibrium. As the Universe cools, the inter-action rate decreases faster than the expansion rate. AtΓ ∼ H, the particles decouple from the thermal bath andthe ’freeze-out’ occurs. On the DM particle case, afterthat time, the relic density remains frozen. Let’s explorethis history in detail.

In the early Universe, particles are relativistic in ther-mal equilibrium. So the distribution function can bewritten as

f(p) =1

e(E(p)−µ)/T ± 1

By using this form, we can get the number density andthe energy density at the relativistic limit, n ∼ T 3 andρ ∼ T 4. These densities are used to calculate the ratioof two rates, Γ, H.

The thermal history of the early Universe can be di-vided into two parts by the electroweak symmetry break-ing, which occurs at T ∼ 100 GeV. Since the gaugebosons receive masses after the electroweak symmetrybreaking by the Higgs mechanism, the interaction rate ofparticles associated with the weak force become different.When T ≥ 100GeV, the cross section for the strong andthe electroweak interaction can be estimated, σ ∼ α2/T 2

where α is the generalized structure constant. By usingthis, we can get the ratio of two rates

Γ

H∼ α2Mpl

T∼ 1016GeV

T

where we have used α ∼ 0.01. According to this formula,we can verify that the interaction rate is much larger thanthe expansion rate at 100GeV ≤ T ≤ 1016GeV. However,when T ≤ 100GeV, the cross section for the electroweakinteraction becomes different because the gauge bosonsreceive masses by the Higgs mechanism, σ ∼ G2

FT2 where

GF is the Fermi’s constant. Therefore, the ratio also

2

becomes different

Γ

H∼ G2

FMplT3 ∼

( T

1MeV

)3According to this, we can verify that particles which in-teract through the weak interaction decouple around 1MeV. This thermal history is important for DM particlebecause we can get the relic abundance of DM by usingthis history.

III. WHAT IS THE WIMP MIRACLE?

To describe the WIMP scenario, let’s assume that2 → 2 interaction and the Cold DM(CDM). Before the’freeze-out’, Γ > H, the interaction is inelastic scatteringχχ↔ XX where χ is the DM particle and X is a Stan-dard Model(SM) particle. To evaluate the DM numberdensity, we follow the evolution of this scattering usingthe Boltzmann equation.

dnχdt

+ 3Hnχ = gχ

∫C[fχ]

d3pχ(2π)3

where C is the collision operator. The right-side thermfor DM particle is then

gχ

∫C[fχ]

d3pχ(2π)3

= −∑spins

∫[f2χ(1± fX)2|Mχχ→XX |2

−[f2X(1± fχ)2|MXX→χχ|2]

×(2π)4δ4(2pχ − 2pX)d2Πχd2ΠX

where gi and fi are the spin degrees of freedom andphase-space densities for particle i, and Mx→y is the ma-trix element for the reaction x→ y[5]. To make the man-ageable form, let’s suppose the following assumptions: 1)Kinetic equilibrium is maintained. 2) The temperature ofeach species satisfies Ti Ei−µi where µi is the chemi-cal potential. Then we can ignore this term: (1±fi) ∼ 1.3) The SM particles are in thermal equilibrium with thephoton bath. By using these assumptions, we can reducethe collision term

gχ

∫C[fχ]

d3pχ(2π)3

= −∫

(σvMφl)χχd2nχ

+

∫(σvMφl)XXd

2nX

where the Mφoller velocity is defined as

(vMφl)ij =

√(pi · pj)2 − (mimj)2

EiEj

for ij → kl process. Therefore, the Boltzmann equationcan be written as

dnχdt

+ 3Hnχ = 〈σvMφl〉χχ(n2eq − n2χ)

FIG. 1. The DM number density Y (the solid curve) as a func-tion of x. The dashed line is the equilibrium expectation.[6]

where neq is the number density of DM particle in thechemical equilibrium with X particle. Since the DM num-ber density decrease as the Universe expands, we canchange this form by defining Y = nχ/s where s is thetotal entropy density of the Universe.

dY

dx= −xs〈σvMφl〉

H(m)(Y 2 − Y 2

eq)

where x = m/T and m is the DM mass. To simplifythis equation, we have to consider the thermal historyof the DM. When Γ H, the thermal equilibrium ismaintained. But when Γ H, it is very hard to be thechemical equilibrium because the time scale of interactionis too small to meet each other. Therefore, We can guessthe shape of Y as illustrated in Fig. 1. So we can knowthat Y (x < xf ) ' Yeq(x) and Y (x > xf ) ' Yeq(xf )where xf is the value at the freeze-out. By using thisequation and [5], we can conclude the relic abundance of

DM today, Ytoday ' xfλ where λ =

〈σvMφl〉0s0H(m) .(Of course,

it can be changed by another particle physics model.)Finally, we can calculate the DM density today.

Ωχh2 =

mstodayYtodayρcrit

h2 ' 0.1(0.01

α

)2( m

100GeV

)2taking xf ∼ 10 and 〈σvMφl〉 ∼ α2/m2. If we assumea weakly interaction DM particle(WIMP), we can usethat α ∼ 0.01 and m ∼ 100GeV. Then the DM densitywhich we calculate has the correct abundance comparedto the value measured by Planck, 0.1200±0.0012[7]. Thiscoincidence is called the ’WIMP miracle’.

However, there is no reason that the DM must have aweakly interaction. Actually, the important thing is theratio of α and m, not their value. So it is possible to getother range of mass while keeping the ratio α/m fixed.The WIMPless DM models[8] and the Forbidden DM[9]are examples of these scenarios. Furthermore, if we break2 → 2 interaction assumption and assume 3 → 2 inter-action, then a strongly interaction is also possible.(theSIMP miracle[10]. But most of them are excluded byrecent searches.)

3

FIG. 2. Neutralino relic density from a scan over 19-dimensional SUGRA model parameter space. The measuredrelic abundance is indicated by the green dashed line.[11]

IV. NEUTRALINO

The lightest neutralino(simply, a neutralino) is a hy-pothetical particle of supersymmetric(SUSY) theoriesand the most promising candidate of WIMP for DM.This particle is composed of two electrical neutral gaug-ino(wino and bino) and higgsinos(the superpartners ofneutral Higgs bosons). So a neutralino is a Majoranafermionic mass eigenstate, χ0

1, χ02, χ

03 and χ0

4. We can es-timate the upper bound mass of neutralino based on ’neu-tralness’ for electroweak physics, mχ ≤ 1TeV. However,prior to the discovery of Higgs bosons, this SUSY modelcould not predicts the correct measured abundance ofCDM. In Fig. 2, a scan of 19-dimension parameter space(Now, much of these parameter has been excluded bysparticle searches and the Higgs mass at LHC) with thechargino mass to lie above 103.5 GeV(LEP2 searches)and mχ < 500 GeV, the expected relic abundance of thelightest supersymmetric particle(LSP) is illustrated. Abino-like LSP predict higher relic density and a wino andhiggsino predict lower density than the measured Ωχh

2.In order to match the measured abundance, specific neu-tralino types or annihilation mechanism are required: 1)well-tempered neutralino(the mix of bino-wino-higgsinofor the LSP) 2) co-annihilations 3) resonance annihila-tion, etc.

But after the discovery of the Higgs bosons and theshift of the SUSY breaking scale to the TeV range, inthe multi-TeV region of superparter masses there existsa higgsino with the right value of relic density and itsmass is 1TeV. This framework is called the Minimal Su-persymmetric Standard Model(MSSM), phenomenologi-cal SUSY scenarios. So new LUX and XENON100 startto detect the higgsino region 1TeV and have excludedthe mixed neutralino region and the well-tempered neu-tralinos regions. It is shown in Fig.3.

FIG. 3. Direct detection search limits on the spin-independentWIMP-nucleon cross section.[3]

V. DETECTION STRATEGIES OF THE WIMP

In this section, we will review the well-known strategiesto detect WIMPs.a. Direct detection Direct detection is the way to

detect DM passing through the Earth and scattering offa nucleon. If WIMPs interact with a particle in a un-derground detector (inelastic scattering), the recoil en-ergy is ER ' 50keV (mχ/100GeV )2(100GeV/mN ) wheremN is nucleus mass. Since the expected WIMPs mass is10GeV 1TeV, we can know that the target recoil energyis 1keV 100keV. To get the lower threshold recoil en-ergy of detector, we have to use the high nucleon mass.So higher mass nuclei are preferred for this search, fromGe to Xe.(LUX, XENON100, etc.) The interaction rateis also important. This rate is highly related to the massof DM and the cross section for DM-nucleus interaction.Fig. 3 shows many direct detection searches limits on thespin-independent. Additional, direct detection must con-sider the annual modulation since the ’wind’ of DM andan effect of ’gravitational effect’ is different each month.b. Indirect detection The goal of this search tech-

nique is to detect the products of DM annihilation(elasticscattering) using a satellite or ground-based telescope onEarth. This is very rare event after the freeze-out, butit is possible in regions of high DM density. When itoccurs, there are many possibilities of products such asphotons or other SM particles. The indirect detectionobserves photons either products of DM annihilation orsecondary products of SM particles.

4

[1] F. Zwicky. Die rotverschiebung von extragalaktischennebeln. Helvetica Physica Acta, pages 110–127, 1933.

[2] N. Thonnard V.C. Rubin, W.K. Ford. Rotational prop-erties of 21 sc galaxies with a large range of luminositiesand radii, from ngc 4605 /r = 4kpc/ to ugc 2885 /r = 122kpc/. Astrophysical Journal, Part 1, 238:471–487, 1980.

[3] Jihn E. Kimc Leszek Roszkowski Howard Baer a, Ki-Young Choi b. Dark matter production in the early uni-verse: Beyond the thermal wimp paradigm. Physics Re-ports, pages 1–60, 2015.

[4] D. Baumann. Cosmology, part iii mathematical tripos.[5] G. Gelmini P. Gondolo. Cosmic abundance of stable par-

ticles: Improved analysis. Nuclear Physics B, 360:145–179, 1991.

[6] M. Lisanti. Lectures on dark matter physics. Theo-retical Advanced Study Institute in Elementary ParticlePhysics: New Frontiers in Fields and Strings (TASI 2015

), 2015(2016).[7] Planck Collaboration: P. A. R. Ade et. al. Planck 2018

results. vi. cosmological parameters.[8] J. L. Feng and J. Kumar. Dark-matter particles

withoutweak-scale masses orweak interactions. PhysicsReview Letter, 2008.

[9] K. Griest and D. Seckel. Three exceptions in the cal-culation of relic abundances. Physics Review D, 43(10),1991.

[10] T. Volansky Y. Hochberg, E. Kuflik and J. G. Wacker.Mechanism for thermal relic dark matter of strongly in-teracting massive particles. Physics Review Letter, 113,2014.

[11] A. D. Boxa H. Baera and H. Summya. Neutralino versusaxion/axino cold dark matter in the 19 parameter sugramodel. Journal of High Energy Physics, 2010.