Corso di Laurea Magistrale in Fisica Teorica
COLORED DARK MATTER
Tesi di Laurea Magistrale in Fisica Teorica
Relatore:
Chiar.mo Prof.
Alessandro Strumia
Candidato:
Valerio De Luca
25 Giugno 2018
Anno Accademico 2017/2018
Introduction 5
1 Introduction to Dark Matter 8
1.1 A brief compendium of cosmology . . . 8
1.2 What is Dark Matter? . . . 12
1.3 Dark Matter detection . . . 18
2 The model 28 2.1 Confinement . . . 31
2.2 Eigenvalues in a linear plus Coulombian potential . . . 31
2.3 Decay rates of excited bound states . . . 33
2.4 Cross section for formation of a loose QQ bound state . . . 37
2.5 Cross section for formation of a un-breakableQQ bound state . . 38
3 Cosmological relic densities 43 3.1 Q decoupling at T ∼ MQ/25 . . . 43
3.2 Q recoupling at T >∼ ΛQCD . . . 60
3.3 Chromodark-synthesis at T ∼ ΛQCD . . . 61
3.4 Q redecoupling at T <∼ ΛQCD . . . 61
3.5 Nucleodark-synthesis . . . 65
4 Signals of relic hybrid hadrons 68 4.1 Direct detection of hybrid hadrons . . . 68
4.2 Searches for accumulated hybrid hadrons . . . 70
4.3 Abundance of hybrid hadrons in the Earth . . . 72
4.4 Abundance of hybrid hadrons in meteorites. . . 73 3
4 CONTENTS
5 Dark matter signals 75
5.1 Direct detection of DM . . . 75 5.2 Indirect detection of DM . . . 80 5.3 Collider signals of DM . . . 84 6 Conclusions 87 A Boltzmann equations 90 B Toy redecoupling 94 Bibliography 96
Many models of particle Dark Matter (DM) have been proposed; one common feature is that DM is a new neutral and uncolored particle. We challenge this view [1]: can DM be instead colored or charged, and be dominantly present today in the form of neutral bound states kept together by ordinary electromagnetic or strong interactions analogously to hydrogen or neutrons? The answer is no for electric binding: two charged particles with mass M me form a negligible amount of neutral bound states, when their thermal
relic abundance matches the DM cosmological abundance.
On the other hand, colored particles necessarily form hadronic bound states. We add to the Standard Model (SM) a new stable heavy colored particleQ, for simplicity neutral. Q could be a heavy quark in the 3 ⊕ ¯3 representation of SU(3)c, or a ‘Dirac gluino’ in
the 8⊕ 8 representation, such that Q annihilates with ¯Q, but not with itself. We dub this neutral quark as quorn. Perturbative annihilations and recombination between Q and ¯Q leave a thermal relic density of order ΩQh2 ∼ 0.1 MQ/7 TeV. After the quantum
chromo-dynamics (QCD) phase transition at temperature T <∼ ΛQCD ≈ 0.27 GeV colored
particles bind in hadrons. Subsequent annihilations among hadrons reduce their relic abundance, increasing the value of MQ such that DM has the observed cosmological
abundance, ΩDMh2 ∼ 0.1 for MQ ≈ 12.5 TeV.
The quorn-onlyum hadrons made of Q only (QQ if Q ∼ 8, and QQQ if Q ∼ 3) are acceptable DM candidates, as they have a small Bohr-like radius a ∼ 1/α3MQ.
This scenario is believed to be excluded because it predicts other hybrid hadrons where Q binds with SM quarks q or gluons g. Such hybrids, Qqq, QQq, Q¯q (if Q ∼ 3) and Qg, Qq¯q0 (if Q ∼ 8), which could be charged and are subject to strong bounds, have size of order 1/ΛQCD and thereby undergo scatterings with cross sections of order
σQCD ∼ 1/Λ2QCD, such as Qg + Qg → QQ + gg, that lead to the formation of a loose
Q-onlyum bound state. These loose bound states, with binding energy comparable to those of light quarks, EB ∼ ΛQCD, will decay fast enough to unbreakable deepQQ bound
6 CHAPTER 0. INTRODUCTION states with much bigger binding energy EB ∼ α23MQ.
The hybrids’ cosmological abundance must be orders of magnitude smaller than the DM abundance ΩDM ≈ 0.1, while naively one might expect that cosmological evolution
results into Ωhybrid ΩDM, given that quarks and gluons are much more abundant than
quorns Q.
We will show that the cosmological evolution is characterized by several phases, from the usual decoupling of free Q at T ∼ MQ/25, to recoupling at T >∼ ΛQCD, to
T ∼ ΛQCD, to redecoupling at T <∼ ΛQCD and to nucleosynthesis at T ∼ 0.1 MeV, and
it gives Ωhybrid ∼ 10−4ΩDM, such that this scenario is allowed. This is not surprising,
taking into account that quorn-onlyum has a binding energy EB ∼ α23MQ ∼ 200 GeV
much larger than hybrids, EB ∼ ΛQCD. Quorn-onlyum thereby is the ground state,
reached by the universe if it has enough time to thermalise. This depends on two main factors:
i) quorns are much rarer than quarks and gluons: nQ ∼ 10−14nq,g when the DM
abundance is reproduced;
ii) QCD interactions are much faster than the Hubble rate H ∼ T2/MPl: a loose bound
state with a σQCD cross section recombines N ∼ nq,gσQCD/H ∼ MPl/ΛQCD ∼ 1019
times in a Hubble time at temperature T ∼ ΛQCD.
Since 1019 is much bigger than 1014, chromodark-synthesis cosmologically results into
quorn-onlyum plus traces of hybrids. This is analogous to Big Bang Nucleo-synthesis, that leads to the formation of deeply bounded Helium plus traces of deuterium and tritium.
The cosmological DM abundance is reproduced for MQ ≈ 12.5 TeV, compatibly with
direct searches (the Rayleigh cross section, suppressed by 1/M6
Q, is close to present
bounds), indirect searches (enhanced by QQ + ¯Q ¯Q → Q ¯Q + Q ¯Q recombination), and with collider searches (where Q manifests as tracks, pair produced via QCD). Hybrid hadrons, made of Q and of SM quarks and gluons, have large QCD cross sections, and do not reach underground detectors. Their cosmological abundance is 104 times smaller than DM, such that their unusual signals seem compatible with bounds. Those in the Earth and stars sank to their centers; the Earth crust and meteorites later accumu-late a secondary abundance, although their present abundance depends on nuclear and geological properties that we cannot compute from first principles.
This thesis is organised as follows: In chapter 1 we present a summary of cosmology and focus on the dark matter scenario. In chapter2we define the model and summarize the main features of its QCD interactions. In chapter3 we discuss how cosmology leads to dominant formation ofQ-onlyum hadrons. In chapter 4we show that the abundance of hybrids is small enough to be compatible with bounds. In chapter 5 we show that Q-onlyum DM is compatible with bounds. A summary of our results is given in the conclusions in chapter 6.
Chapter 1
Introduction to Dark Matter
The relationship between particle physics and cosmology has pointed out several unan-swered questions in the modern physics puzzle.
In what follows we shall present a short summary of the basic principles and ideas on which cosmology is based, focusing on the spacetime geometry and on the thermody-namics of the early universe. Afterwards, we focus on the dark matter scenario, showing what we know about its properties, and finally we describe the possible ways of detecting it [2–4].
In this work we will use the natural units ~ = c = kB = 1, where kB is the Boltzmann
constant, and the following signature for the metric: ηµν = diag(−1, 1, 1, 1).
1.1
A brief compendium of cosmology
The Universe is observed to be, to a good approximation, spatially isotropic and homo-geneous, on large-enough scales. The most general space-time metric with isotropic and homogeneous spatial sections is the Robertson-Walker metric:
ds2 = dt2− a(t)2 dr2 1− kr2 + r 2dΩ (1.1) where the curvature k is normalized to +1 (‘closed’, or positive-curvature), 0 (‘flat’) or −1 (‘open’, or negative-curvature), and where a(t) is a dimensionless scale factor. The dynamics of this scale factor is dictated by the fundamental equations of General Relativity, the Einstein’s equations:
Rµν −
1
2gµνR− Λgµν =−8πGTµν (1.2)
where G is the Newton’s gravitational constant, Rµν is the Ricci’s tensor, with trace R,
gµν is the metric tensor, Tµν is the matter’s energy-momentum tensor and Λ is the
cosmo-logical constant or the vacuum energy. In cosmology, matter and energy are customarily modeled as perfect fluids with energy density ρ and pressure p, i.e.
Tνµ= (ρ + p)uµuν + pδνµ = diag(−ρ, p, p, p) (1.3)
with 4-velocity uµ= (1, 0, 0, 0) in a comoving coordinate system, and such that it holds
the equation of state p = ωρ. If our system is composed by several fluids, then the equa-tions of state are valid for each fluid, but often not for the whole system. Conservation of energy implies that, for constant ω,
ρ∝ a−3(1+ω). (1.4) The most relevant fluids in cosmology are non-relativistic matter (ω = 0, pressure-less fluid), relativistic matter or radiation (ω = 13), and also vacuum energy (ω =−1).
From the Einstein’s equations we can derive the Friedmann equations: ¨ a a =−4πG(ρ + 3p) + Λ 3 ˙a a 2 = 8πGρ 3 − k a2 + Λ 3 (1.5)
Introducing two useful quantities like the Hubble parameter H ≡ ˙a/a and the critical density ρcrit ≡ 3H
2
8πG, whose present values are H0 = h · 100 km/s ·Mpc and ρcrit,0 =
1.878· 10−29h2· g/cm3, factoring out h = 0.6774± 0.0046, we can define the ratio of any species i to the critical density as Ωi ≡ ρi/ρcrit, such that the second Friedmann equation
can be recast as:
ΩΛ+ Ωm+ Ωr+ Ωk = 1 (1.6)
where Ωk ≡ −Hk2a2, Ωm accounts for non-relativistic matter (mostly dark matter, with a
small contribute from baryonic matter), Ωr accounts mainly for photons and neutrinos
and ΩΛ refers to the unknown dark energy. We can see that the sign of k, and so the
‘geometry’ of the universe, is determinated by whether Ω = P
i=Λ,m,rΩi is less, equal
or greater than one; measurements from the Cosmic Microwave Background (CMB) indicate that now the universe is observationally consistent with being flat, being Ωk =
10 CHAPTER 1. INTRODUCTION TO DARK MATTER by the Planck satellite, it has been possible to extract the values of the cosmological parameters [5]:
ΩΛh2 = 0.3171± 0.0072, Ωmh2 = 0.14170± 0.00097 (1.7)
where the radiation ratio plays a negligible role, Ωrh2 ∼ 4.15 · 10−5. It is clear that the
present composition of the universe is dominated by dark energy; by contrast, in the early Universe, radiation dominated over the other species.
We shall now explore the very early Universe, where particles were in thermal equi-librium in a hot and dense plasma. It is useful to work with the statistical mechanics’ language, introducing the number density n, the energy density ρ and the pressure p of a particle species with mass m as an integral of the distribution function in the phase-space with momenta k: n = g (2π)3 Z f (~k)d3k, ρ = g (2π)3 Z E(k)f (~k)d3k, p = g (2π)3 Z |~k|2 3E(k)f (~k)d 3k (1.8) where E = q
|~k|2+ m2 is the relativistic dispersion relation, g is the number of internal
degrees of freedom and f (~k) = exp((E−µ)/T )±11 is the phase-space distribution function of the species in kinetic equilibrium, with +1 for fermions (Fermi-Dirac) and −1 for bosons (Bose-Einstein), and where µ is the chemical potential; in our computations we will consider T µ, neglecting it. In the non-relativistic limit, T m, we have that those physical quantities reduce to:
n = g mT 2π 3/2 e−m/T ρ = m· n p = n· T ρ (1.9)
while in the relativistic limit, T m, we have: n = ( (ζ(3)/π2)gT3 for bosons (7/8)(ζ(3)/π2)gT3 for fermions ρ = ( (π2/30)gT4 for bosons (7/8)(π2/30)gT4 for fermions p = ρ 3 for both (1.10)
The energy density of all relativistic species in the thermal bath can be written as: ρR= π2 30g∗(T )T 4, g ∗(T )≡ X bosons i gi Ti T 4 +7 8 X fermions i gi Ti T 4 (1.11) where g∗(T ) is an effective number of relativistic degrees of freedom that accounts for the
possibility that species i have thermal distributions with a different temperature Ti from
that of photons T ; in the Standard Model we have g∗(T 300 GeV) & 106.75. From
the first Friedmann equation we can therefore write the expansion rate of the universe as: H(T ) = ˙a a = r 4π3g ∗(T ) 45 T2 MPl . (1.12)
In the early Universe the entropy per co-moving volume is conserved as long as thermal equilibrium holds, and the entropy density can be written as:
s = ρ + p T = 2π2 45 g∗s(T )T 3 , g∗s(T )≡ X bosons i gi Ti T 3 +7 8 X fermions i gi Ti T 3 . (1.13) Entropy conservation implies that:
d(sa3)
dt = 0 ⇒ s∼ a
−3
⇒ T ∼ 1/a. (1.14)
From this behaviour it is easy to see that the universe’s expansion implies the decrease of its temperature.
Let us now focus on the realization of ‘thermal equilibrium’: we can say that a particle species is kept in thermal equilibrium if it is part of a reaction characterized by an interaction rate per particle Γ bigger than the Hubble expansion rate, Γ & H. If this reaction is a decay, then Γ is its decay width; by contrast, if it is a scattering reaction, then it can be recast as Γ = nσv, where n is its number density, σ is the reaction cross section and v is the relative species velocity.
As the universe cools, the previous condition can no longer be satisfied, with the consequence that the reaction is too slow to keep the particle species in thermal equi-librium; when this happens the particle species undergoes a thermal decoupling. If the particle species freezes out while it is relativistic, then it is a hot relic; by contrast, if it decouples in a non-relativistic regime, then it is called a cold relic. Of course, a hot relic can become a cold one cooling when the temperature gets smaller than its mass.
In cosmology all the most relevant phenomena, such as thermal decoupling, are the result of non-equilibrium physics, that can be studied through the formalism of the Boltzmann equations, see appendixA to get the details.
12 CHAPTER 1. INTRODUCTION TO DARK MATTER
1.2
What is Dark Matter?
Considerations of cosmological nucleosynthesis and astrophysical observations ranging from galactic to cosmological scales lead to the conclusion that most of the mass in the universe is not in the form of ordinary baryonic matter, i.e. atomic nuclei and electrons; these observations can be explained satisfactorily postulating the existence of a new form of matter which is usually referred to as ‘dark matter’. At the moment, all the evidences for dark matter are due to the fact that it interacts through the force of gravity. We shall briefly review the main observations and then present a summary of how they constrain the properties of dark matter.
1.2.1 Observational evidences
The first strong evidence for the existence of such particles comes from observations of clusters of galaxies: in fact, galaxies are not randomly distributed through the Universe but tend to aggregate in clusters with a typical scale of Rcluster ∼ 10 Mpc. In 1933
the astronomer Fritz Zwicky [6] tried to estimate the mass of the Coma cluster by first counting the number of galaxies and converting their total luminosity into a mass of order Mlum ∼ 1045 g, and later looking at its velocity dispersion, finding that extra
matter was necessary to keep them together. Zwicky’s determination was based on the virial theorem, assuming equilibrium: in a toy model with N 1 objects of mass m at equal distance r interacting through gravity, the virial theorem allows to determine their total mass mN : Nmv 2 2 = 1 2 N2 2 Gm2 r ⇒ mN = 2rv2 G . (1.15)
Therefore, measuring the mean square velocity of the galaxies, its mass can be inferred, and it showed out to be of order Mtot ∼ 1047 g, resulting into a factor of 100 bigger
than the luminous mass observed; the simplest interpretation is that there exists some invisible ‘missing mass’, that he firstly called ‘dunkel materie’, that did not shown up.
A further indication, obtained in 1970 by Freeman [7], is shown from observations of the rotation curves of spiral galaxies, that were made using the Doppler shift of the 21 cm hyperfine transition of the neutral hydrogen, which permits to measure the velocity vc of the gas circling around the galaxies as a function of the distance r from the galaxy’s
axis. For a spherical distribution, classical mechanics gives: v2c(r) = GM (r)
Figure 1.1: The rotation curves for the galaxies M31, M101, and M81 (solid lines) obtained by Roberts and Rots in 1973. The rotation curve of the Milky Way Galaxy was included by the authors for comparison. From [8].
where M (r) is the total mass enclosed in a sphere of radius r. For distances bigger than the galaxy disk’s radius, all the visible mass should be concentrated in the disk and it should be therefore constant; for this reason we should obtain the known ‘Keplerian’ fall-off behaviour, vc∼ r−1/2. However, as it is clear from figure 1.1, see [8], observations
give constant velocity at large distances, i.e. M (r) ∝ r, implying flat rotation curves. Hence we are lead to conclude that a vast amount of matter in spiral galaxies is not visible and it is not confined in the galactic disk, giving another hint for the presence of dark matter in the galactic environment.
From the knowledge of the rotation curves of the galaxy it is possible to reconstruct the density profile of matter inside a galaxy [9]. The result, for spiral galaxies, is that while luminous matter is distributed on a disk, dark matter forms a spherical halo that extends on scales of order Rhalo ∼ 100 kpc. These results are supported also by numerical
14 CHAPTER 1. INTRODUCTION TO DARK MATTER
Figure 1.2: The velocity distribution found with the Via Lactea simulation (solid red), with the 68% scatter (light green) and the maximum and minimum ranges (dark green), compared with the Maxwell-Boltzmann best-fit. From [11].
must be surrounded by an halo of dark matter, with a central spike and many sub-haloes [10], following an Einasto profile:
ρ(r) = ρ−2· exp −α2 r r−2 α − 1 (1.17) where r−2 is the radius at which dlnρdlnr =−2 and α ∼ 0.17 from simulations on Milky-Way
sized halos. The Galactic dark matter velocity distribution can be computed through a best-fit on the Via Lactea simulation realised by [11], with a Maxwell-Boltzmann distribution truncated at the galaxy’s escape velocity:
f (~v) = 1 Nesc 3 2πσ2 3/2 e2σ23~v2 |~v| < vesc 0 |~v| ≥ vesc (1.18)
where σ ' (279 ± 33) km/s is the r.m.s velocity dispersion [12], vesc ' (544 ± 64) km/s
is the escape velocity [13] and Nesc = erf(z)− 2π−1/2ze−z
2
, with z ≡ vesc/(p2/3σ). Fig.
1.2 compares the simulation with the functional form of eq. (1.18), and we can observe high-velocity ‘tails’ and ‘clumps’ in velocity space.
Today the most striking evidence for the presence of dark matter on the length scales of galaxy clusters comes from the observations of a pair of merging clusters known as the ‘bullet cluster’ (located 3.7 Gyr away, with a catalog name 1E0657-558) [14]
Figure 1.3: Bullet cluster. The merging of a pair of clusters of galaxies, with the col-ored map representing the X-ray image of the hot baryonic gas. This is displaced from the distribution of the total mass reconstructed through weak lensing, shown with green contours. The white bar corresponds to the length of 200 kpc. From [14].
and of similar systems. Most of the baryonic mass in the bullet cluster is in the form of hot gas whose distribution can be traced through its X-ray emissions. The mass density profile reconstructed through weak gravitational lensing indicates two distinct massive sub-structures and appears to be displaced with respect to the distribution of hot gas, with visible matter and dark matter spatially separated. The interpretation is the following: in the past each of the two objects was an ordinary system, with the visible and dark matter mixed together. Then these two objects collided 150 million years ago and visible matter interacts significantly with itself, so that the hot gas from the two clusters experienced a collisional shock wave; dark matter, on the other hand, experienced negligible collisions with itself and with normal matter, such that the DM components of the two systems simply passed through each other. This lead to the present separation between the two components, visible in fig. 1.3. The fact that the two DM halos did not slow down compared to the collision-less trajectories implies an upper bound, obtained by hydrodynamical simulations, on the DM self-interaction cross section, assuming point interactions [15]:
σχχ Mχ . 1 cm2 g = 1.8 mb GeV = 4580 GeV −3 (1.19)
where Mχ indicates the dark matter mass. For comparison, we recall that 50 mb is a
typical QCD cross section, for example for the pp scattering. Self-interactions can also affect the shape of halos, generally making them rounder.
16 CHAPTER 1. INTRODUCTION TO DARK MATTER
Figure 1.4: Planck 2015 temperature power spectrum. The best-fit base ΛCDM theo-retical spectrum fitted to the Planck TT+lowP likelihood is plotted in the upper panel. Residuals with respect to this model are shown in the lower panel. The error bars show ±1σ uncertainties. From [5].
Furthermore without dark matter it is almost impossible to explain the timely for-mation of non-linear structures in the universe: given a pattern of density perturbations shared between baryons and photons in the very early universe, up to their decoupling, we can measure the size of these inhomogeneities by measuring angular anisotropies in the temperature fluctuations of the CMB, since baryonic density fluctuations δρ/ρ can be computed from CMB temperature fluctuations δT /T , giving δρ/ρ . 10−5 at recom-bination. In the linear regime, such inhomogeneities grow as the factor scale a, that in turn has grown from recombination by a redshift factor (1 + zrec) ∼ 1100. With such
values, not enough time has passed for observed structures (which have δρ/ρ 1) to form, if there are only baryons. The solution to this problem lies on the existence of an other particle species that decoupled from the thermal plasma much earlier than re-combination and created seeds of inhomogeneity like gravitational potential wells much deeper than baryonic ones, being δρ/ρ|DM 10−4, from which it is not able to stream
out. This can happen only if this species is electrically neutral and cold (non relativistic at decoupling), and DM reflects well these properties. Therefore baryons fall into those
pre-existing gravitational potential wells seeded by dark matter, accounting for structure formation.
Finally, we should point out how cosmological observations on scales r > 100 Mpc strengthen the evidence for dark matter and clarify unequivocally its non-baryonic na-ture. Modern calculations predict the relative abundance of primordial light elements synthesized at Big Bang Nucleosynthesis (BBN) in term of the ratio of nucleons to pho-tons: η = nb/nγ; in particular it has been possible to measure the deuterium/hydrogen
ratio in very early intergalactic matter by observing deuterium as well as hydrogen ab-sorption lines in the spectra of quasistellar objects, due to abab-sorption in intervening intergalactic clouds of large redshift, giving (D/H)p = (2.53± 0.04) · 105, from which it
is inferred η = (6.0± 0.1) · 10−10 [16]. From this value and the knowledge of the number density of photons, which can be obtained integrating the black body spectrum describ-ing CMB radiation, it is possible to derive the present day baryon density parameter Ωbh2 ∼ 0.022. The comparision of this value with that obtained by visible matter (i.e.
stars) Ωlumh2 ∼ 0.0012 [17] suggests that a large portion of baryonic matter is given by
intergalactic plasma.
More accurate measurements come from the best-fits of the CMB power spectrum observed by the Planck satellite [5], like that of figure1.4, giving the following values for the cosmological parameters:
Ωbh2 = 0.02226± 0.00023, Ωmh2 = 0.14170± 0.00097. (1.20)
As we can see, the baryonic matter density cannot account for all the cold matter density, and such a result gives a strong evidence that the vast majority of non-relativistic matter in the Universe is non-baryonic dark matter, with cosmological abundance:
ΩDMh2 = 0.1186± 0.0020. (1.21)
The agreement of all this different independent considerations is impressive and strengthen the hypothesis that all these observations can be explained in terms of a new form of non-baryonic matter, whose elementary nature, interactions and properties are still largely unknown.
1.2.2 Summary
On the back of these evidences, the idea that some kind of dark particle pervades the Universe is now universally accepted. The scope of this section is to review the main properties that such a particle should have:
18 CHAPTER 1. INTRODUCTION TO DARK MATTER ◦ Non-baryonic, since Big Bang Nucleosynthesis and CMB data provide a determi-nation of the density parameter for baryonic matter and dark matter separately.
◦ Dark, i.e. it is not observed to interact with light of any frequency. This implies that it should be neutral under the electromagnetic interactions, i.e. have very small electromagnetic coupling; constraints on its charge are derived in [18]. An important consequence of this fact is that dark matter cannot cool radiating photons. Therefore it will not collapse to the center of galaxies, radiating away their energy electromagnetically as baryons do, and this is what allows for the existence of DM halos.
◦ Stable, or it should have a lifetime τ τtoday, otherwise it would have decayed
before the present time.
◦ Cold: we know that DM density perturbations in the early Universe plant the seeds for structure formation and thus determine the major features of the large scale structures of the Universe. Numerical simulations on the cosmological formation of large scale structures suggest that dark matter should be cold, meaning that it has to be non-relativistic already at the epoch of structure formation [19].
◦ Collisionless, since its self interactions are bounded by the upper limit σχχ/Mχ ≤
1 cm2 g−1.
◦ Classical, in the sense that there should exist some constraints on its quantum nature not ‘interfering’ with observation. For example, the de Broglie wavelength of the DM particle must be smaller than galactic scales to have a coherent (in the quantum sense) dark matter halo.
◦ Fluid: it should be sufficiently fine on galactic scale. The main flaw of discrete DM is that it will produce a time dependent gravitational potential which may disrupt bound system like globular clusters if Mχ> 103M .
◦ Its relic abundance must be: ΩDMh2 = 0.1186± 0.0020, as discussed previously.
◦ Dark matter has attractive gravitational interactions, necessary to explain the ob-served rotational curves of stars in spiral galaxies, the formation of the primordial grav-itational wells and the excess in galaxies gravgrav-itational lensing.
1.3
Dark Matter detection
After this brief review of all the most important features about a dark matter candidate, it is now crucial to highline the most common ways through which dark matter can be detected. In particular, there are three main avenues of investigation: direct detection,
indirect detection and detection at colliders.
1.3.1 Direct Detection
Direct detection experiments aim at detecting the recoil event produced by the scatter-ings of galactic Dark Matter particles hitting atomic nuclei or electrons of a opportunely super-shielded and closely monitored underground detector made of ultrapure semicon-ductors, noble gasses, pristine crystals, etc, to avoid background noises. These searches try to answer the following questions: what is the rate for dark matter scattering off of nuclei? What is the typical energy deposition we need to measure? Which dark matter particle mass can we hope to detect with a detector with a certain given energy threshold?
We will focus now on the elastic scatterings of a DM particle of mass Mχ on nuclei
of mass mN at rest in the lab frame; moving in the center-of-mass frame, the energy
transferred from the dark matter particle to the nucleus detector, known as recoil energy, is given by: ER=
µχN2
mN v
2(1− cosθ) = q2/2m
N, where µχN = MχmN/(Mχ+ mN) is the
reduced mass of the DM-recoiling target nucleus system, θ is the scattering angle in the center-of-mass frame, v is the relative velocity and q is the exchanged momentum between DM and nucleus. At a given velocity v, the differential recoil energy is dER =
(dcosθ)(µ2
χN/mN)v2, and so the corresponding differential detector event rate R is given
by: dR dER = 2πmN µ2 χNv2 dR dΩ (1.22)
where we assumed no azimutal angular dependence. Given that the differential rate is expressed as dR/dΩ = nχNNv(dσχN/dΩ), then we have:
dR dER = 2πρχNNmN Mχµ2χNv dσχN dΩ (1.23)
where NN is the number of target nuclei and nχ = ρχ/Mχ is the dark matter number
density. In the center-of-mass frame the differential scattering cross section can be ex-pressed as a function of the modulus squared of the amplitude of the relevant process, giving: dσχN dΩ = µ2 χN M2 χm2N 1 64π2|M| 2. (1.24)
As it is clear from the previous equation, all the physics behind our DM model comes from the differential cross section: in particular the DM scattering cross section off on nuclei
20 CHAPTER 1. INTRODUCTION TO DARK MATTER σχN scales with that off on nucleons σχN depending on the type of interaction; if the
dark matter couples to the nucleon number A, the scattering is coherent on the nucleus and nucleon amplitudes add coherently, such that we have σχN ≡ σSIA2 ≈ σχNA2,
otherwise dark matter couples to nucleon spin and spins add incoherently inside nuclei, σχN ≈ σχN. The former case is known as spin-indipendent direct detection, while the
latter as spin-dipendent, where we have neglected velocity suppressed interactions. In the spin-indipendent approximation the amplitude of the process can be written as:
M = FN(q)(Zfp+ (A− Z)fn) (1.25)
where fp,fn are the SI couplings with protons and neutrons in an effective field theory of
DM interacting with non-relativistic nucleons, such as L =P
i,N =p,nf i
NONi ; it’s
interest-ing to see that tipically DM scatters differently on protons and neutrons, and that the two terms feel an interference. FN(q) = 3e−q
2s2/2
[sin(qr)− qrcos(qr)]/(qr)3, with s' 1
fm, r =√R2− 5s2 and R = 1.2A1/3 fm, is a Helm nucleus form factor that accounts for
a possible decoherence for nonzero q, and reproduces the coherent enhancement A2 for
small momenta exchanges, FN(0) = 1.
We can recast the spin-indipendent cross section for DM scattering on nucleons as: σSI = σχN|q=0 µ2χN µ2 χNA2 = Zfp+ (A− Z)fn A 2µ2 χp(n) µ2 χN σχp(n) (1.26)
where the second equality converted it into the SI cross section for DM scattering on a single proton (neutron), σχp(n). Evaluating the first expression, we have:
σSI= 1 16π µ2 χN M2 χm2N |Zfp+ (A− Z)fn|2 A2 . (1.27)
Therefore assuming no angular dipendence in the cross section, i.e. dσ/dΩ = σ/4π, we can express our rate for recoil energy as:
dR dER = σSI Mχµ2χN · A 2N NmN · ρχ 2v · FN( p 2mNER) 2 . (1.28)
At this point, all the dark matter particle physics feeds into the first term of this equation, while the second is only dependent on the target considered; the third term accounts for astrophysics and the fourth for nuclear physics. Making an average over the relative velocity according to the dark matter halo velocity probability distribution function f (v) shown in eq. (1.18), we finally obtain the scattering rate for recoil energy:
dR dER = σSI Mχµ2χN · A2N NmN · Z ∞ vmin dvρχ 2vf (v)· FN( p 2mNER) 2 (1.29)
Figure 1.5: Dark matter direct detection diagrams. Typical values for the DM scattering cross sections from tree level Z exchange (left), Higgs exchange (middle), and 1 loop electroweak correction (right). From [21].
where vmin =
q
mNER/2µ2χN is the minimal attainable velocity for the energy spectrum
that ranges from ER = 0 to ER= 2µ2χNv2/mN. In more complex scenarios we can also
add to this equation a DM form factor|Fχ(q, v)|2, strongly dependent on the model built.
Experiments can tag the scattering event and reconstruct ER by observing at least
one of the following three end-products: heat (phonons), ionization and scintillation. However, there are two necessary conditions to have a statistically significant detection: i) having enough events and ii) having a large enough signal-to-noise. However, other than suppressing background, one can use other handles to distinguish a signal event from a background event, such as the seasonal modulation due to the Earth motion around the Sun, i.e. a∼ 60 km/s effect, which translates into a ∼ 10% signal ‘sinusoidal’ modulation with a maximum in June and a minimum in December [20].
In fig. 1.5 it is possible to see examples of Feynman diagrams relevant for direct detection, with a general behaviour of their SI cross section (the vector effect vanishes if the DM candidate is a real particle, e.g. a Majorana fermion).
The historic progress of the bounds on the spin-indipendent cross section is shown in fig. 1.6, taking the DM mass to be Mχ ≈ MZ, for example. DM particles of this
mass that would interact strongly with visible matter were immediately excluded by the mere fact that NaI scintillation crystals in other experiments do not glow in the dark [22]. In the figure we also denote the typical values for several DM scattering cross
22 CHAPTER 1. INTRODUCTION TO DARK MATTER
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Figure 1.6: History of direct detection bounds for spin-independent DM-nucleon scatter-ing, setting the DM mass to equal the Z boson mass. Only the most stringent bounds (blue) at any given moment are shown, along with a claimed DAMA signal. Projected sensitivities are presented by green circles. From [21].
sections, from tree level Z exchange, Higgs exchange and from scattering induced through 1-loop electroweak diagram (see also fig. 1.5). The most sensitive experiment at present, Xenon1T, sets a bound on the DM/nucleon cross section down to σSI . 10−46 cm2 [23].
This experiment, located at the Laboratori Nazionali del Gran Sasso (LNGS) in Italy, is designed primarly for detecting nuclear recoils from weak interacting massive particles (WIMPs) scattering on Xenon nuclei.
In fig. 1.7we show the bounds on the spin-indipendent cross section for direct detec-tion of dark matter. The maximal sensitivity is for Mχ ≈ mN ∼ 100 GeV, but where
the irreducible background of coherent neutrino/nucleus scatterings could represent a problem. For heavier DM, nχ = ρχ/Mχ gets smaller and the bound on σSI weaker;
by contrast, for lighter DM the recoil energy is small, and the signal goes below back-grounds. The dominant background for light dark matter Mχ . 10 GeV is given by solar
neutrinos, while the atmosferic and diffuse supernova background neutrinos can become important for heavier DM masses. A powerful strategy to discriminate against neutrino background could be to do measurements with different nuclei.
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Figure 1.7: Bounds on the spin-independent cross section for direct detection of Dark Matter. From [21].
the recoil energy spectrum can differ significantly from the neutrino background, giving an extra handle to reject against it.
1.3.2 Indirect Detection
The fundamental particle nature of dark matter can be probed indirectly, i.e. without directly detecting the particle itself, with the detection of the particle debris produced or affected by dark matter as an elementary particle. The main processes are:
• the pair-annihilation of dark matter particles χ into Standard Model particles, usually known as ‘messengers’: χ + χ→ SM;
• the decay of dark matter particles into SM particles: χ → SM.
It is important to note that none of the processes listed above is bound to necessarily oc-cur in any particle dark matter model: for example the first one can be highly suppressed or not occur at all if the coupling of the dark matter sector to the SM is suppressed; by constrast, the second one does not occur if dark matter is absolutely stable.
24 CHAPTER 1. INTRODUCTION TO DARK MATTER These SM particles can be detected by searching for an excess in cosmic rays, collected on Earth, with respect to the presumed astrophysical contribution. Promising sources are generically the regions where DM is expected to be densest, such as the center of our Galaxy, its inner halo, nearby galaxies dominated by it, the center of the Sun and the Earth, etc. However, some of these regions (notably the Galactic Center) are also the most complicated from the point of view of the underlying astrophysics; in fact it is not always possible to reconstruct the charged rays arrival directions, since we do not know how they propagate, and we have not understood yet which and how much are the background sources relevant for the energy spectrum of those particles. The best detection opportunities might thus come from selecting targets which are not necessarily the richest in DM but rather for which the signal over background ratio is most favorable. This then also depends on which species of cosmic ray one is looking for: in general terms, the SM messengers that we hope to detect are high energy photons (γ-rays), low energy photons (X-rays, radio waves), electrons, neutrinos, and maybe even some antiparticles like positrons or anti-protons, since they come towards us in straight lines.
Let us compute, for example, the rate of photons seen at Earth with a detector of exposure surface dA and volume dV from an annihilation of two DM particles at a distance r. The number of photons from this annihilation that reach the detector is given by: Nγ = nγ· 1 2n 2 χhσvidtdV · dA 4πr2 (1.30)
where the first term nγ is the number of photons produced in an annihilation process,
the second term accounts for the number of annihilation processes, depending on the DM number density nχ = ρχ/Mχ, the exposure time dt and the averaged annihilation
cross sectionhσvi, with a factor 1/2 introduced not to double count identical pairs, and the last term accounts for the effective amount of photons that reaches the detector over a sphere of radius r centered around the annihilation site. Therefore the event rate for photon energy is given by:
dNγ dEγ = dAdt·hσvi M2 χ dnγ dEγ · ρ2χdV 8πr2 (1.31)
where we have assumed that redshifting is irrilevant, and where we have factorized the dependence on the detector properties, on the particle physics and on astrophysics, respectively. Integrating over the distance r, we obtain the observed differential rate
for photon energy, exposure time and surface and solid angle dΩ: dNγ dEγ dA dt dΩ = hσvi M2 χ dnγ dEγ · J, J = 1 8π Z ∞ 0 ρ2χ(r)dr (1.32) where we introduced the astrophysical J factor describing the DM density distribution as a function of the distance r from the observer. For a DM particle decaying with a long lifetime τχ τtoday, this expression reduces to:
dNγ dEγ dA dt dΩ = 1 Mχτχ dnγ dEγ · 1 4π Z ∞ 0 ρχ(r)dr (1.33)
where here nγ represents the number of photons emitted in a decay.
With this example we highlined how the key ingredients to understand indirect de-tection are the production rates of the relevant messengers, the energy scale of these SM particles, set by the mass of the dark matter particle, and the annihilation products; all these ingredients permit us to make suitable quantitative and qualitative predictions, that can be probed as long as we have a large enough number counts N = φχ· Aeff· texp,
where φχ indicates the relevant dark matter-induced event rate, such as the flux of a
certain type of SM particle in the appropriate angular region, Aeff is an effective area
and texp is the relevant ‘exposure time’, and a large enough signal-to-noise.
Finally, it is interesting to see that dark matter could also have some possible effects on astrophysical objects, including solar physics, neutron star capture, supernova and star cooling, protostars and planets warming, covering an amazing variety of possibilities.
1.3.3 Detection at Colliders
Searching for Dark Matter can also be realized through particle colliders, where Standard Model particles make collisions, and if DM particles interact sufficiently strongly with ordinary matter, they could be one of the products of those collisions. Even if such a production is possible in principle, the unknown DM mass means that we do not know whether the center-of-mass energy available in present experiments is sufficient for the production to be kinematically allowed [24].
A key difference between direct or indirect dark matter searches and collider searches is that whatever particle is produced at colliders, it will be in general unknown whether such particle has anything to do with the Galactic/cosmological dark matter particle, and it will even be unclear whether the particle is, at all, long-lived enough to be the dark matter candidate. Moreover, given that DM must be a stable neutral particle,
26 CHAPTER 1. INTRODUCTION TO DARK MATTER
Figure 1.8: Artistic view of the Dark Matter theory space. From [25].
thanks for example to a Z2 symmetry, at colliders it can be produced only in pairs (the
energy of the collider must be bigger than√s > 2M ) and, once produced, these particles would be invisible to any detector in the vicinity of the collision point and would hence reveal their presence only via an apparent imbalance in the total transverse momentum (so-called missing transverse momentum), which would be associated with the energy carried away by it. Such minimal signal of DM is detectable only if some other visible particle is also produced during the collision, otherwise the event goes unnoticed. Surely, this condition to be transverse should be valid only for hadronic colliders, since otherwise it is not necessary.
Nevertheless, searches for DM particles at the Large Hadron Collider (LHC) are a thriving research field, because the SM backgrounds and their differential distributions are now understood to sufficient accuracy that even small distortions in the missing energy spectrum (in particular in its high-energy tail) may be observable and can be used to constrain DM models that predict these distortions.
There are, in particular, two distinct possibilities as to how make predictions to search for dark matter at colliders:
• a top-down approach, where a ‘complete’ theory is given, and distinct, specific search strategies are considered at colliders;
approximated either with contact interactions (EFT) or through simplified models. One can construct a large number of qualitatively different Dark Matter models. Col-lectively these models populate the ‘theory space’ of all possible realizations of physics beyond the SM with a particle that is a viable DM candidate. The members of this theory space fall into three distinct classes (see fig. 1.8):
◦ First of all, if DM is the only accessible state to our experiments, then effective field theory (EFT) allows us to describe the DM-SM interactions mediated by all kine-matically inaccessible particles. The DM-EFT approach can be used to derive stringent bounds on the ‘new-physics’ scale that suppresses the higher dimensional operators, that can be compared with the limits following from direct and indirect DM searches. How-ever, the signal is so suppressed that the EFT approach is not really useful: in particular, measurements from LHC with √s = 13 TeV give strong bounds on those effective oper-ators.
◦ The large energies accessible at the LHC lead toward simplified DM models, char-acterized by the most important state mediating the DM interactions with both SM and itself. They are able to describe correctly the full kinematics of DM production at the LHC, because they resolve the EFT contact interactions into single-particle s-channel or t-channel exchanges.
◦ Complete DM models can close the gap given by the fact that simplified models are likely to miss important correlations between observables, by adding more particles to the SM. The classical example is the Minimal Supersymmetric SM, in which each SM particle gets its own superpartner and the DM candidate, the neutralino, is a weakly interacting massive particle. Phenomenological models in this class have of order 20 parameters, leading to varied visions of DM.
Chapter 2
The model
In our work we will focus on Composite Dark Matter models, i.e. models in which the Dark Matter candidate is composed by other elementary particles. The first possibility is that those constituents are part of the Standard Model of particle physics and that their interactions are those described, for example, by the Quantum Chromo-Dynamics (QCD). However, we can also build a model where a new particle is added to the SM, interacting with the SM forces, or we can also introduce a new force, like a new strong force, to describe new and old particles’ interactions.
Within the SM, QCD could give rise to Dark Matter as ‘strangelets’ made of many uds quarks [26] or as ‘sexaquark’ uuddss [27], that is a neutral and a flavour-singlet, such that it does not couple to photons, pions and most other mesons; it is probably more compact than ordinary baryons and its mass would make it difficult to distinguish kinematically from the neutron. However there is no experimental nor lattice evidence that such objects exist.
We thereby consider the following extension of the SM:
L = LSM+ ¯Q(i /D− MQ)Q. (2.1)
The only new ingredient is Q: a Dirac fermion with quantum numbers (8, 1)0 under
SU(3)c⊗ SU(2)L⊗ U(1)Y, i.e. a neutral color octet. We choose a neutral component
instead of a charged one because charged components form a negligible amount of bound states when their thermal abundance matches that of dark matter [28]. In fact, a neutral bound state composed by charged components is equivalent to a neutron, composed by charged quarks; however such a bound state would have a non-zero weak charge that would make it interact weakly (this is the reason for which a neutron undergoes weak
interactions like n→ p + e−+ ¯νe). For this reason we decided to focus only on a neutral
component, with only free parameter represented by its mass MQ.
Like in Minimal Dark Matter models [29], Q is automatically stable, as no renor-malizable interactions with SM particles allow its decay, which can first arise due to dimension-6 effective operators such asQDDU and QLDQ where Q(L) is the SM quark (lepton) doublet, and U (D) is the right-handed SM up-type (down-type) quark. For example, the effective lagrangian associated with the first operator is:
Leff = gQDDU
Λ2 ≡
QDDU ˜
Λ2 (2.2)
introducing an effective coupling g and an energy scale Λ. Its decay rate is given by: Γeff ∼ g2MQ MQ Λ 4 ∼ MQ MQ ˜ Λ 4 . H (2.3)
that is cosmologically negligible if it is smaller than the inverse of the age of the universe, i.e. smaller than the Hubble rate H. This happens if such operators are suppressed by the Planck scale, Λ∼ MPl.
After confinementQ forms bound states. For MQ ΛQCD/α3states made byQ-only
are Coulombian. TheQ ¯Q bound states are unstable: Q and ¯Q annihilate into gluons and quarks. No such annihilation arises inQQ bound states as we assumed that Q carries an unbroken U(1)DB dark baryon number that enforces the Dirac structure such thatQQ is
stable. This U(1)DB symmetry is imposed in our model building, since we want that the
Majorana mass associated to our quorn Q is zero, allowing it to annihilate with its anti-particle ¯Q, but not with itself. The DM candidate is the quorn-onlyum QQ ground state, neutral, color-less and with spin-0. As we will see, ifQQ is a thermal relic, the observed cosmological DM abundance is reproduced for MQ ∼ 12.5 TeV. This mass is large enough
that Q does not form QCD condensates. The QQ potential in the color-singlet channel is V (r) = −3α3/r, so the binding energy is EB = 9α23MQ/4n2 ≈ 200 GeV/n2, which is
bigger than ΛQCD up to n∼ 20. We adopt the value ΛQCD ≈ 0.27 GeV.
Since Q follows the adjoint representation of SU(3)c, then it can couple with SM
particles like gluons and quarks, creating hybrid hadrons like Qg and Qq¯q0, with non exotic quantum numbers, i.e. with quantum numbers that are not possible for simple q ¯q systems. We expect that the isospin singlet Qg is lighter than Qq¯q0 (isospin 3⊕ 1) by an amount of order ΛQCD, which accounts for the relative motion of q and ¯q0, where q, q0 =
{u, d}. A lattice computation is needed to safely establish who is lighter. Assuming that Qq¯q0 is heavier, then its neutral component
30 CHAPTER 2. THE MODEL 1/ΛQCD. The slightly heavier components Qu ¯d and Qd¯u with electric charges ±1 have
a lifetime of order v4/Λ5QCD, given from the lifetime of the neutral component 1/ΛQCD
multiplied by a weak decay suppression v4/E4 ∼ v4/Λ4
QCD, where v ∼ 246 GeV is the
vacuum expectation value for the electroweak interactions.
The above DM model has possible extra motivations: in Supersymmetry, a gluino is the hypothetical supersymmetric partner of a gluon, interacting via the strong force as a color octet, and in models that conserve R-parity gluinos decay via the strong interaction to a squark and a quark, where we remember that R-parity is aZ2 symmetry
acting on Minimal Supersymmetric Standard Model (MSSM) fields that forbids all the renormalizable couplings in the theory that do not conserve baryon and lepton number. The fermion Q appears as a ‘Dirac gluino’ in some N = 2 supersymmetric models [30], i.e. models in which there are N = 2 irreducible real spinor representations, where fermion superpartners, called sfermions, can mediate its decay if R-parity is broken.
Alternatively, the heavy quarksQ could be identified with those introduced in KSVZ axion models [31]; in such a case our U(1)DB symmetry would be related to the
Peccei-Quinn symmetry.
Moreover, there is a 3-loop RGE effect which arises even if the heavy quarkQ has only strong interactions and no electroweak interactions: in fact, this heavy quark interacts with gluons, that interact with the top quark, that in turn interacts with the Higgs, such that corrections to the Higgs mass squared are proportional to MQ2:
δmh2 ∼ g34y2t (4π)6M 2 Qln M2 Q µ2 (2.4) where µ describes the energy scale, g3 =√4πα3 is the SM strong coupling and yt is the
coupling with the top quark; these corrections are comparable to its measured value for MQ ≈ 10 TeV [32].
Other assignments of quantum numbers of Q are possible. A scalar would give a similar physics. A fermionicQ ∼ (3 ⊕ ¯3, 1)0 under SU(3)c⊗ SU(2)L⊗ U(1)Y would give
theQQQ baryon as a viable DM candidate. As the gauge quantum numbers of a neutral color triplet are exotic, theQQq, Qqq and Q¯q hadrons containing light quarks would have fractional charges. Fractionally charged hadrons are subject to stronger experimental bounds [28]. A Q ∼ (3, 2, 1/6) = (Qu,Qd), with the same quantum numbers of SM
left-handed quarks Q, would give as lightest state the neutral DM candidate QuQdQd.
This is excluded by direct detection mediated at tree level by a Z boson, being a weak doublet with hypercharge Y 6= 0. Allowing for an additional confining group, a Q ∼ 8
can be build out of Q ∼ 3 obtaining double composite Dark Matter.
2.1
Confinement
QCD confinement happens in cosmology through a smooth crossover. Working in Cor-nell parametrization [33], the QCD potential between two quarks in the F undamental representation of SU(Nc) at temperature T in the singlet configuration is approximated
as Vq ¯q(r) ≈ −αF eff/r + σFr, where σF is the string tension in this representation. In
the perturbative limit one has αF eff = CFα3, where CF = (Nc2 − 1)/2Nc = 4/3 is the
quadratic Casimir for Nc= 3 and α3 is renormalized around 1/r. In our computation we
will work with energy scales around ΛQCD, so with distances of order r∼ 1/ΛQCD, and
lattice simulations find αF eff = 0.4 and σF ≈ (0.45 GeV)2 [34]. The potential between
two adjoints is similarly approximated by a Coulombian term plus a flux tube: VQQ(r)≈ −
αeff
r + σr (2.5)
where the color Coulomb force is always attractive in the color singlet channel. Pertur-bation theory implies VQQ/CA ≈ Vq ¯q/CF [35] where CA = Nc = 3. Thereby αeff ≈ 3α3
and σ ≈ 9σF/4 ≈ (0.67 GeV)2 at zero temperature, as verified on the lattice [36].
At finite temperature the Coulombian force gets screened by the Debye mass and the string appears only below the critical temperature of the transition to deconfinement Tc ≈ 155 MeV as σ(T ) ≈ σ(0)p1 − T2/Tc2 [34].
2.2
Eigenvalues in a linear plus Coulombian potential
We will need the binding energies of a non-relativisticQQ hadron. We thereby consider the Hamiltonian H = ~p2/2µ + V (r) in 3 dimensions that describes its motion around the center of mass, with reduced mass 2µ' MQ. The potential is given by eq. (2.5). As usual,
wave-functions are decomposed in partial waves as ψ(r, θ, φ) = P
˜
n,`,mRn`˜ (r)Y`m(θ, φ)
where ˜n is the principal quantum number. For each ` = 0, 1, 2, . . . we define as ˜n = 1 the state with lowest energy, so that ˜n = 1, 2, 3, . . .. The radial wave function R˜n`(r)
has ˜n− 1 nodes. Differently from the hydrogen atom, there are no free states: angular momentum ` is not restricted to ` < ˜n. In order to match with the Coloumbian limit in its usual notation we define n≡ ˜n + ` such that, at given `, only n ≥ ` + 1 is allowed.
The reduced wave function u˜n`(r) = rRn`˜ (r) obeys the Schroedinger equation in one
32 CHAPTER 2. THE MODEL -�� -� -� -� -��� -��� -��� � ��� ��� ��� � �� �� �� �� � � �� �� �� �� ������� ������ � = � - ℓ ������� ������ ℓ ��=�� ���� σ = (���� ���)�� α���=�
Figure 2.1: Binding energies E˜n` in GeV for aQQ system. States with En`˜ <−0.2 GeV
(in green) are well approximated by the Coulombian limit. Increasing MQ leads to a larger
number of Coulombian states and to a deeper ground state. QQ states are cosmologically mostly produced in the region with larger ` of the band E ∼ ΛQCD.
the kinetic barrier is suppressed for heavy states with large reduced mass µ. Rescaling arguments imply that energy eigenvalues have the form:
En`˜ = αeff2 µ×f(ε, ˜n, `), where ε ≡ σ 4α3 effµ2 = 10−8 σ GeV2 10 TeV MQ 2 1 αeff 3 . (2.6) Since our potential is a sum of two potentials, −αeff/r and σr, then we can apply the
potential-envelope method to construct general upper and lower bounds on its eigenval-ues; in particular from [37] we extract the approximation valid at leading order in ε 1, using the sum approximation for these bounds:
En`˜ = α2 effµ 2 − 1 n2 + 8εn(` + 1.37) +· · · . (2.7)
The first term is Coulombian. The second term accounts for the linear potential, and becomes relevant at large n, `, being ε very small. In particular, assuming ` ' n 1, Coulombian states with negative binding energy, En`˜ < 0, exist up to ` <
α3/4eff MQ1/2/(8σ)1/4. The ground state has binding energy EB = −E10 ∼ 200 GeV for
MQ ∼ 10 TeV.
In the opposite limit where the linear force dominates and the Coulomb-like force can be neglected, all energy levels are positive and states with higher ` have higher
energy [37] En`˜ ≈ 3σ2/3 (2µ)1/3 0.897˜n + ` 2 − 0.209 2/3 (2.8) such that thermalisation lowers `. In this limit the dependence on σ, µ and the ground state energy can also be computed variationally with the Hellmann-Feynman theorem, assuming a trial wave-function ψ(r) = e−r/rc/r3/2
c , such that the typical size is rc ∼
(µσ)−1/3. Fig.2.1 shows the binding energies for relevant values of the parameters, and it is easy to see that the states near the continuum threshold are dominated by the linear term.
We next discuss a bound state BQ made of a heavy Q and a gluon. It cannot
be described by non-relativistic quantum mechanics, for the presence of the relativistic gluon. Nevertheless, its binding energy can roughly be obtained by eq. (2.8) taking a small reduced mass µ∼√σ. One then expects that such state is in its ground state at T <∼ ΛQCD, so after confinement, and that its mass is MBQ = MQ+O(ΛQCD).
2.3
Decay rates of excited bound states
Energy losses due to quantum decay of aQQ state with n, ` 1 into deeper states can be approximated with classical abelian Larmor radiation, with the subsequent emission of pions. This holds in dipole approximation, where a state can only decay to states with `0 = `± 1, according to the angular momenta selection rules.
To see this, we consider an hydrogen-like system with potential V = −α/r and reduced mass µ = MQ/2, neglecting at first the linear part of the potential. Assuming
a circular orbit as in [38], since it determines the minimum radius for a given angular momentum, one gets the Larmor emitted power:
WLarmorcirc = 2αa
2
3 (2.9)
where a = α/µr2 identifies the acceleration as a function of the orbital radius r.
Let us now consider a generic state. Classically, a generic elliptic orbit is parameter-ized by its energy E and by its angular momentum ` ≤ `circ, where `circ =pα2µ/2E is
the value corresponding to a circular orbit. The Larmor radiation power, averaged over the orbit, is:
hWLarmori = WLarmorcirc
3− (`/`circ)2
2(`/`circ)5
34 CHAPTER 2. THE MODEL Due to the larger acceleration at the point of minimal distance, the radiated energy for ` `circ is much larger than in the circular case: this is why e¯e colliders are built
circular.
From the following quantum numerical computation we observe that the quantum decay rates are fully reproduced, in the classical limit, by those decay rates described by Larmor radiation.
Let us summarize the known quantum results for the decay rates of an hydrogen-like system in dipole tree-level approximation [39]. We denote the initial state as (n, `), and the final states as (n0, `0). Their energy gap is:
∆E(n, n0) = α 2µ 2 1 n2 − 1 n02 . (2.11)
At quantum level, a circular orbit corresponds to a state with maximal angular mo-mentum ` = `circ = n. Converting the orbital radius into n2 times the Bohr radius as
r = rn= n2/αµ, then the Larmor emitted power for circular orbits becomes:
WLarmorcirc = 2αa
2
3 =
2µ2α7
3n8 . (2.12)
Similarly, the binding energy is E =−α/2r = −α2µ/2n2. In dipole approximation such
a state decays only to n0 = `0 = n− 1, emitting a soft photon with energy ∆ELarmor =
|En− En−1| ' α2µ/n3, such that the decay rate is:
ΓcircLarmor = W circ Larmor |∆ELarmor| = 2 3 α n 5 µ. (2.13)
For a general non-circular orbit the spontaneous emission rate, in dipole approximation, is given by: Γ(n, `→ n0, `0) = 4α 3 ∆E3 2` + 1 X m,m0 |hn0 , `0, m0| ~r |n, `, mi|2. (2.14) where we made a sum over all possible z-components of the states’ angular momenta, m and m0, because of rotational invariance. Selection rules imply ∆` =±1, and the matrix elements are X m0 |hn0 , `− 1, m0| ~r |n, `, mi|2 = ` + 1 2` + 1 1 (αµ)2 Rnn, `0, `−1 2 (2.15) X m0 |hn0, `, m0| ~r |n, ` − 1, mi|2 = ` 2` + 1 1 (αµ)2 Rnn, `−10, ` 2 (2.16)
Figure 2.2: Total decay rate (Left) and energy loss rate (Right) from an initial Coulom-bian state (n, `) and summed over all available lower-energy states.
where we factorized the radial integral Rnn,`0,`0 =
Z ∞
0
dr r3Rn`Rn0`0 (2.17) with Rn`(r) the radial part of the hydrogen wave-function. These integrals can we
rewritten as a function of the Hypergeometric2F1 function, 2F1, as
Rnn`0,`−1 = (−1) n0−` 4(2`− 1)! s (n0+ `− 1)!(n + `)! (n0− `)!(n − ` − 1)! (4nn0)`+1(n− n0)n+n0−2`−2 (n + n0)n+n0 × 2F1 − n + ` + 1, −n0+ `, 2`,− 4nn 0 (n− n0)2 − (2.18) + n− n 0 n + n0 2 2F1 − n + ` − 1, −n0+ `, 2`,− 4nn 0 (n− n0)2 .
A similar formula can be obtained for Rnn,`−10,` by the interchange of the indices n and n0. The total decay rate and energy loss rate from an initial state (n, `) is obtained by summing over all available lower-energy states with n0 < n, and their behaviour is shown in fig. 2.2, where we computed numerically those rates for MQ = 12.5 TeV and
for α = 3α3 ≈ 0.3. These results can be approximated as:
Γn` ' 2α5µ 3n3`2, Wn`' 2α7µ2 3n8 3− (`/n)2 2(`/n)5 (2.19)
36 CHAPTER 2. THE MODEL
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Figure 2.3: Energy loss rate from an initial state (n, `) summed over all available lower-energy states for a Coulombian + linear bound state.
that reproduces perfectly the results for circular orbits in the limit n ' ` and matches with those for non-circular orbits for n, ` 1.
In the quantum computation the enhancement at small ` < n appears after summing over the available final states with small n0 ≥ ` − 1 which allows for energy jumps |En− En0| larger than in the circular case.
In the opposite limit where the linear part of the potential dominates over the Coulom-bian part, energy losses of highly excited states are again well approximated by classical Larmor radiation, which in this regime does not depend on the shape of the orbit, given that the force does not depend on the radius: in fact we have a = σ/µ and therefore WLarmor = 2αa2/3 = 8ασ2/3MQ2, that is negligibly small for our bound states. This is
confirmed by numerical quantum computations.
In general, for a bound state composed by two quorns Q with a potential VQQ(r) ≈
−α/r + σr, the energy loss rate from an initial state (n, `), summed over all available lower-energy states, is computed solving the equation of motion with such a potential to find its acceleration and using the Larmor expression (2.9). The energy loss rate in GeV2 with both terms in the potential is shown in fig. 2.3 for M
Q = 12.5 TeV and for
lower-energy states, being Coulombian, and a slight difference for higher-energy states, where the contribution of the linear term in the potential makes it bigger.
2.4
Cross section for formation of a loose
QQ bound state
We here estimate the cross section σtot(BQ + BQ → BQQ+ X) for the formation of a
loose bound state containing two heavy quarks Q, starting from two bound states BQ
containing one Q, with the emission of some kind of particles X. This bound state BQQ
is loose in the sense that it can be easily destroyed by the thermal plasma, and it won’t represent a good candidate for our stable particle unless it reaches a deeper state losing energy.
Just as in heavy quark effective theory, let us picture these BQ =Qg as consisting of
a Q and a gluon kept together by a flux tube with binding energy ∼ ΛQCD and length
` ∼ 1/ΛQCD, and it is clear that the following geometrical picture emerges from the
strong interactions. The cross section is σtot ≈ π`2℘ at energies E ∼ MQv2<∼ ΛQCD such
that there is not enough energy for breaking the QCD flux tubes, being suppressed by large Nc, and the recombination probability of two flux tubes is ℘ ∼ 1, like in string
models. Independently from the above geometric picture, the size of the bound state is of order 1/ΛQCD, and thereby one expects a cross section of order σQCD= c/Λ2QCD, with
c ≈ π in the geometric picture. In the following we will consider c = {1, π, 4π}, since we do not know the exact value for this constant, but only its general behaviour. For example the measured pp cross section corresponds to c≈ 10.
While this expectation is solid at energies of order ΛQCD, at lower temperatures the
cross section might be drastically suppressed if the residual van der Waals-like force has a repulsive component, which prevents the particles to come close enough. We will ignore this possibility, which would result into a higher abundance of hybrid relics BQ.
More in general, processes that only require a small energy exchange E can have large cross sections of order 1/E2. The authors of [40] propose a quantum mechanical model where processes analogous to σ(BQ + BQ → BQQ+ X) are computed in terms of cross
sections suppressed by 1/MQ. This large suppression seems to derive from their arbitrary
assumption that the cross section should be dominated by an s-channel resonance, which proceeds through an off-shell bound state. However we won’t consider such a suppression for our prospective computation.
38 CHAPTER 2. THE MODEL
2.5
Cross section for formation of a un-breakable
QQ bound
state
We can finally compute the quantity of interest for us: the thermally averaged cross section σfall(T ) for collisions between two Qg states which produce an unbreakable QQ
hadron. The issue is whether those loose bound states discussed in the previous section can survive in the thermal bath long enough to radiate away energy and angular mo-mentum and become unbreakable or, by contrast, get broken by it. This happens when they radiate more energy than ∼ T , i.e. the amount of energy that the system can gain colliding with the thermal bath in the time ∆t before the next collision, such that it becomes un-breakable and later falls down to its deep ground state.
In view of the previous discussion, we proceed as follows. A large total cross section σQCD∼ π/Λ2QCD∼ πb2 needs a large impact parameter b∼ 1/ΛQCD, and thereby theQQ
state is produced with large angular momentum `∼ MQvb; in particular, in the thermal
bath at temperature T one has v ∼ pT /MQ, and thereby ` ∼ pMQ/ΛQCD ∼ 200 at
T ∼ ΛQCD.
The issue is whether a bound state with large ` gets broken or radiates enough energy becoming un-breakable [38]. As discussed in section 2.3, energy losses are well approximated by classical Larmor radiation, and it is crucial to take into account that non-circular orbits radiate much more than circular orbits because of the enhancement at small ` < n. The QQ potential is given by eq. (2.5), with a large αeff ≈ 3α3(¯µ)
renormalized at ¯µ∼ 1/r ∼ ΛQCD, i.e. where it is expected to bind, and since we are near
the critical temperature for confinement, we cannot have a precise value for the coupling constant renormalization scale.
The cross section for falling into an un-breakable QQ bound state is therefore com-puted as follows. We simulate classical collisions between two hybrid bound states, numerically solving the classical equation of motion for the QQ system:
µ¨~x(t) =− ~x |~x|V
0
(|~x|) (2.20)
starting from an initial relative distance b and an orthogonal relative velocity v. From the solution ~x(t) we compute the radiated energy ∆E by integrating the radiated power WLarmor ∼ 2αeff~x¨2/3 for a time ∆t, where ∆t is the average time at temperature T
between two collisions of such a bound state with the thermal plasma, dominated by pions after confinement, that we estimate as ∆t∼ 1/nπvπσQCD, where σQCD = c/Λ2QCD
