Universit´e Libre de Bruxelles Facult´e des Sciences

(1)

Universit´e Libre de Bruxelles

Facult´e des Sciences

Matter asymmetry and gauge unification

Nicolas Cosme Service de Physique Th´eorique

Septembre 2004

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences

(2)

Acknowledgments - Remerciements

Bien avant le commencement de cette thèse, je croyais na¨ıvement que cette période serait remplie d’une douce bohème. C’était bien entendu sans compter sur le caractère nécessairement personnel dont est empreint ce type de travail. Au delà de l’apparente inaction contemplative et de la relative liberté dont jouit le téméraire prétendant au titre de docteur, il est question en réalité d’une traversée intérieure; au tréfonds des sentiers inconnus de la science.

C’est avec plaisir que j’ai partagé ces moments avec plusieurs personnes qui de près ou de loin ont concouru à l’accomplissement de ce travail.

En effet, est-ce pour m’avoir permis de me divertir ou pour m’avoir empêché de travailler que je me sens redevable au département de physique? Sûrement pour les deux, mais définitivement la saveur de ce “quasi-lustre” n’aurait pas été si goûteuse sans le bonheur d’avoir pu m’évader des contrées désertes de la recherche par les échanges entre personnes (enseignants ou étudiants).

Avec mes compagnons de voyages, Yves, Laura, Paola, ce sont des instants en dehors du monde que nous avons partagés, du désespoir à l’euphorie tout deux aussi déraisonnés. Pour leur aide et leur amitié, je les remercie jusqu’à la lie.

Pour m’avoir offert, en passionné, de découvrir les voies secrètes de la profession de chercheur, et pour m’avoir laissé mon autonomie et ma liberté dans le travail tout en m’y apportant son sou- tient, je remercie Jean-Marie Frère de tout coeur.

Michel Tytgat m’a également fourni une aide précieuse spécialement par ces conseils pleins de motivations et de pondérations. De même, Malcolm Fairbairn obtient pour moi la palme de l’écoute et de l’attention aux autres.

D’un point de vue personnel, on ne serait rien sans les personnes qui nous entourent dans notre quotidien. C’est pour cela que je tiens à remercier ma famille et particulièrement mon frère pour sa relecture active de ce qui suit.

Mais c’est à toi Annabelle que je dois le plus. Et pour ne pas que les mots me trahissent, je laisse ici s’exprimer l’un de nos auto-portraits que je préfère.

i

(3)

Introduction

That there is matter in the Universe is quite obvious. What may be less so, is why! Or, more precisely, what element of fundamental interactions leads to what will appear below as an unbalance between matter and antimatter? A look in this direction leads straight to the heart of the nature of matter, of its elementary constituents, and of their interactions.

One particularly striking point is the existence of ”antiparticles”, partners of our ordinary particles, appearing in high energy collisions. Nothing indeed in our daily surroundings seems to hint at their existence, to the exception of surreptitious phenomena, instant outbursts quickly resorbed by the predominance of matter.

There is indeed the problem: matter and antimatter annihilate each other. This is inherent to the way they are introduced, as a requirement of relativity when combined with electrodynamics in the framework of field theory. The (near) perfect symmetry is observed to amazing precisions in high energy phenomena.

If the mirror relating matter to antimatter is so perfect, why has the first taken over to such an extent that the latter is next to absent in our world ? Can the explanation be found in the evolution of the Universe? The ideally symmetrical description of particles and antiparticles must therefore yield under a more thorough examination, and the behaviour of particles and antiparticles must indeed be distinguishable.

Nevertheless, despite the existing evidence, the violation of matter-antimatter symmetry calls for highly subtle mechanisms. The violation is furthermore not direct, and appears deeply related to one specificity of electroweak gauge interactions, dealing with their inequivalent treatment of spinor (or fermionic) particles of different chiralities. For this reason, a true difference between particles and antiparticles will be observed in phenomena which violate the more complete CP symmetry (resulting from the combination of charge reversal C and spatial parity reversal P).

This combined symmetry (CP) is however the intrinsic symmetry of (pure) gauge interactions. Its breaking must be found in the still rather obscure sector of mass and scalar interactions.

Violations of matter-antimatter symmetry in this sense have indeed been observed explicitly both in the K and B mesons systems, but remains tenuous in terms of absolute rates (even if some CP-breaking branching ratios appear large).

CP violation therefore appears as a privileged path in understanding the still poorly described part of fundamental interactions, since it cannot stem from the pure (well-known) gauge sector. It addresses directly the sector of scalar particles and Yukawa interactions, responsible notably for

(7)

Introduction 2

the apparition of particle masses. Such interactions escape the predicted universality of gauge interactions, and are therefore responsible for most of the remaining free parameters of the Standard model (in the quark sector alone, 6 masses, 3 mixing angles and one phase...).

In a first part of this thesis, we accept the (usual) introduction of CP violation as a phenomenological description, through complex Yukawa couplings, and study how these microscopic physics considerations may lead to the present overwhelming domination of matter over antimatter, through the mechanism known as ”leptogenesis”.

Recent experimental results have indeed shown that neutrinos, tenuous particles which only take part in weak interactions, possess a mass and mixing structure leading to the phenomenon of

”oscillations” between flavours. The existence of such masses motivates the addition of ”sterile”

(often called right-handed) neutrinos, appearing at energies much higher than the already known particles. This extension is otherwise naturally motivated when usual gauge interactions (both strong and electroweak) are unified in a simple structure, which later breaks to provide the different scales.

The basic leptogenesis scenario assumes that the desintegration of such heavy neutrinos gen- erates a tiny early leptonic asymmetry, later converted into the currently flagrant domination of matter over antimatter. A successful description depends heavily on the values of the Yukawa couplings between these heavy neutrinos and the currently observed light ones. It is therefore related to the values of the mass spectrum of the observed light neutrinos, but also depends critically on the gauge interactions of the heavy neutrinos, which are intrinsically related to the unification pattern of electroweak and strong interactions.

In the second part of this thesis, we focus more precisely on the nature of CP violation as such, and explore some possible paths to a fundamental understanding.

As already mentioned, in 4 (or, rather 3+1) dimensions, CP violation seems rooted, in the cur- rent description, in the ill-understood sector of Yukawa couplings, which lack the strict constraints imposed on gauge couplings. As a result, all these (complex) couplings are introduced ”by hand”

as phenomenological or ad-hoc parameters. A really unified theory of fundamental interactions should put an end to such arbitrariness and most likely relate the Yukawa couplings to the gauge sector. In such a context, CP violation could only appear by some symmetry breaking mechanism.

In this direction, we study the possible extension of our physical description to involve extra dimensions, and examine how CP violation may result from gauge interactions through the mechanism of dimensional reduction (compactification of the extra dimensions) related to the gauge breaking. Such introduction of extra dimensions finds some motivation in the attempted unification of gravity with other fundamental forces through string theory, but also in more phenomenological considerations, like a possible understanding of the different scales assigned to these different forces. We will finally examine how this requirement of CP violation through the dimensional reduction mechanism severely constrains the choice of unification groups.

(8)

Part I

Matter asymmetry and leptogenesis

(9)

Chapter 1 Matter-antimatter asymmetry

Our day to day experience of the world since early childhood make us confident regarding the question of existence of matter in the universe. Nevertheless, it is worth asking the question and then face the diluted character of the answer.

Indeed, there is matter in the universe but, as we will see in this chapter, several independent observations reveal us that reality is quite different from our immediate perception: on average in the universe, you will find one poor baryon for more than a billion of photons.

Despite this depressing conclusion, we can however appreciate the value of this very small amount of matter from the point of view of physics.

It accounts for the creation of light chemical elements in the early universe, as we will see below through nucleosynthesis. Moreover, consider the standard cosmology model where the be- ginning of the universe is a thermal bath of an equal number of particles and anti-particles. These particles and anti-particles annihilate with each other and will not be recreated since the temperature is falling with the expansion. Fortunately, annihilation stops after the nucleon formation when the expansion rate of the universe becomes faster than the annihilation rate, resulting in a freeze-out of the interactions.

But the freeze-out temperature T_{f reeze}₋_out∼20MeV enables only a remaining nucleon density of

n_B n_γ =n_B_¯

n_γ ' m_N T_f₋_o

!3/2

Exph

−m_N/T_f₋_oi

'10⁻¹⁸, (1.1)

(with m_N the nucleon mass), that is hundred million times smaller than what is observed (10⁻¹⁰).

Therefore, we should not be too sad with the amount of matter we are observing and the com- parison between the two numbers (10⁻¹⁰ and 10⁻¹⁸) clearly requires an additional phenomenon in the above-mentioned description.

Beside the question of matter is also the question of existence of antimatter (section 1.1).

We will address below the observational evidences of the remaining matter amount in the universe, in particular predictions of big bang nucleosynthesis and observations of anisotropies in the cosmic microwave background. We also quickly review the relevant physics behind these phenomena.

(10)

1.1 Antimatter? 5

The agreement between the baryon to photon ratio extracted from nucleosynthesis analysis on the light chemical elements (section 1.2) and the inferred value from the cosmic microwave background (section 1.3) is really surprising since these two results arise from different types of measurements and different physical processes occurring at two separate periods of the universe.

Both results indicate an earlier origin for this matter density, that is back before nuclei formation.

A candidate for this origin is discussed in this work, i.e. leptogenesis: the creation of a leptonic asymmetry induced by the out-of-equilibrium decay of heavy Majorana neutrinos (chapter 4).

1.1 Antimatter?

Let us begin with the more awkward problem of the present existence of primeval antimatter, that is antiparticles produced in the early universe. This last precision is not useless since actually all reported detections of antimatter can be deduced to come from a matter source: e.g. on earth, positrons from nuclear decays, anti-particles in accelerators, etc.

Of more interest is the observed anti-protons cosmic flux, with an anti-protons to protons ratio of the order of 10⁻⁴. The measurements are however consistently reproduced by calculation of the production of secondary antiprotons from galactic cosmic ray interactions with interstellar gas nuclei ( e.g. p+p→3p+p) [1], excluding therefore a primordial origin.¯

Constraints on persisting primordial antimatter can also be found through upper limits obtained by the Alpha Magnetic Spectrometer (AMS) detector [2] on the cosmic flux ratio of anti-helium to helium ranging between 10⁻⁵and 10⁻⁶depending on the energy.

These results are direct evidences against the existence of significant primeval antimatter in our galaxy but, even in this case, the possibility of distinct regions made of antimatter still persists, giving rise to a globally symmetric universe.

One can invoke for instance a separation mechanism to avoid excessive nucleon-antinucleon annihilation in the description given above. In order to get enough baryon density, n_B/n_γ = n_B_¯/n_γ∼10⁻¹⁰, such a mechanism had to operate at T>40MeV . The resulting causally connected region would then contain only about 10⁻⁷ solar masses which is not acceptable with respect to our visible universe. The argument is however not valid with inflation but this scenario poses other serious drawbacks as e.g. dilution of the baryon density due to an exponential expansion.

In any case, however such a separation mechanism takes place, we expect annihilation at the boundaries of separate matter and antimatter regions leading to a γ-ray background radiation.

According to the observation and using a phenomenological analysis, A. Cohen et al. [3] estimated the size of our matter island to be of the order of 1Gpc that is roughly the size of the visible universe.¹

So, even though we cannot fully reject the existence of primeval antimatter, it is very unlikely that it still exists.

1pc=3.3 light year=3.0910²⁴cm

(11)

1.2 Big Bang nucleosynthesis 6

1.2 Big Bang nucleosynthesis

The origin of the light nuclear elements has long been a question mark for physicists. Indeed, no present astrophysical process is known to account for their observed abundances. In particular, the

4He mass fraction produced in regions with significant stellar production is only about 5% while observed values in galactic and extragalactic clouds of ionised hydrogen gas range around 25% of the total mass. Moreover, even the very small abundance of deuterium (D/H∼10⁻⁵) is very hard to account for, since deuterium is easily destroyed by astrophysical processes due to its very weak binding energy.

The idea of primordial nucleosynthesis, i.e. synthesis of nuclei in a primeval hot bath of particles, dates to 1946 by Gamow [5]. It now accounts for observations on abundances of the light nuclear elements (D, ³He, ⁷Li, ⁴He) while determining fundamental physical quantities as the baryon to photon ratioη=n_B/n_γ and the number of light neutrino families.

The number density of a non-relativistic nuclear species A is given, according to Boltzmann statistic by

n_A=g_A m_AT

2π 3/2

Exp

µ_A−m_A T

, (1.2)

withµ_Athe chemical potential. Using chemical equilibrium for the production of nuclei with mass number A and charge Z out of Z protons and A−Z neutrons, and moving to the mass fraction to the total nucleon density, we can describe the amount of a nuclear species A(Z)at equilibrium by

X_A= n_AA

m_n+m_p+∑_i(m_AA)_i ∝g_AA^5/2 T

m_N ^3(A₂⁻¹⁾

η^A⁻¹X_p^ZX_n^A⁻^ZExp(B_A/T). (1.3) The quantity depends on the nuclei binding energy B_A=Zmp+ (A−Z)m_n−m_A, respectively the proton and neutron mass fractions X_p and X_n, and of most interest to us here the baryon to photon rationη.

The formation of nuclei is tightly connected to the evolution of interactions between their constituents, that is protons and neutrons. The equilibrium between the latter is ensured by weak interactions:

n ↔ p+e⁻+ν¯e

νe+n ↔ p+e⁻

e⁺+n ↔ p+ν¯e, (1.4)

from which one can deduce their relative abundance at equilibrium:

n p

eq

=Exp(−Q/T), (1.5)

for Q=mn−mp, the mass difference∼1.293MeV .

Once weak interactions rates became smaller than the expansion rate of the universe, at a temperature ∼0.8MeV , this ratio freezed out at a value ∼1/6, up to residual interactions as neutron decay. It is also at that time that neutrinos decoupled from the plasma and followed their

(12)

own evolution giving rise to a fossil neutrino gas much like the well-known cosmic microwave background constituted by photons decoupled at the time of neutral atoms recombination.

On the other hand, the equilibrium nuclear abundances were no longer limited by their binding energy through the exponential factor Exp(B_A/T), but nevertheless remained very small due to the entropy factor η^A⁻¹ which indicates the low probability for baryons to meet and the high possibility of nuclei photo-dissociation. This effect delayed for a while the nuclei production which eventually occurred at temperature around 0.1MeV .

The precise determination of the light nuclei abundances relies on a full computation of their evolution taking into account the different production reactions. As already mentioned, the final result strongly depends on the baryon to photon ratio ( see Figure 1.1) for which we will discuss below some relevant effects.

Indeed for increasing η, we observe a reduction of D and ³He abundances coupled with an increase of⁴He. This comes from the fact that deuterium and³He constitute the fuel for⁴He pro- duction whose rate behaves asΓ∼X_2,3(ηn_γ)hσ|v|i. So for higherη, the production of⁴He stops later resulting in lower D and³He abundances left unburnt and therefore a higher⁴He production.

Incidentally, ⁷Li has two different production processes: ⁴He(³H,γ)→⁷Li forη<3×10⁻¹⁰ and⁴He(³He,γ)→⁷Be→⁷Li+β forη>3×10⁻¹⁰. This implies the particular form of the⁷Li abundance as a function ofη.

The main two other parameters of nucleosynthesis are the neutron life-time and the number of relativistic degrees of freedom (which gives essentially the number of light neutrino families).

Their increase induces an earlier freeze-out of weak interactions, and thus a higher ⁴He mass fraction.

We now quickly summarise the astrophysical observations of light elements abundances which lead to an estimate of the baryon to photon ratio.

The important point is that while nucleosynthesis predicts primordial abundances, one can only observe abundances at much later epochs. The ejected remains of stellar nucleosynthesis can then alter light element abundances but also produce heavy elements such as C, N, O and Fe. In order to get accurate results, one therefore observes astrophysical sites with low metal abundances and measures both the light element abundances and the metalicity. The primordial abundances is then inferred for zero metalicity [6]:

• ⁴He is observed in clouds of ionised hydrogen (H II region). The observed mass fraction is given by [6]: Y_P=0.238±0.002±0.005, with respectively the statistical and systematic errors.

• ⁷Li observations have been carried out for hot, metal-poor stars belonging to the halo pop- ulation (Pop II) of our galaxy, i.e. the oldest ones. These observations have long shown that ⁷Li does not significantly vary in such stars having metalicities <1/30 of solar. The extrapolation to zero metalicity gives the value: Li/H|p= (1.23±0.06^+0.68+0.56₋_0.32 )×10⁻¹⁰, where there remains serious uncertainties.

• Deuterium has long provided an upper bound on the matter density through observations on local interstellar medium since astrophysical processes are not able to create it and even

(13)

3___He H ___D

H 0.23 0.22 0.24 0.25 0.26

Baryon-to-photon ratio η₁₀ 10⁻⁴

10−3

10⁻⁵

10⁻⁹

10⁻¹⁰

2 3 4 5 6 7 8 910

1 2 5

Ω_Bh²

0.01 0.02 0.03

0.005

3He/H _p D/H _p 4He

7Li/H _p Y_p

Figure 1.1: Primordial abundances of⁴He, D,³He and⁷Li predicted by big bang nucleosynthesis as a function of the baryon to photon ratioη=n_B/n_γ=η₁₀10⁻¹⁰. Boxes indicates the observations with 2σ statistical errors for small boxes and 2σ statistical and systematic errors for larger boxes [6].

more likely destroy it. For example, such observation gives D/H = (1.5±0.1)×10⁻⁵ requiringη≤9×10⁻¹⁰.

More recently, observations on high-redshift low-metalicity quasar absorption systems gave the primeval deuterium abundance to be D/H= (3.0±0.4)×10⁻⁵[6]. The associated error is however increased in Figure 1.1 due to lower values reported in different systems.

All these observations combined with the big bang nucleosynthesis predictions are ultimately used to find the baryon to photon ratio. The overlap in theη ranges spanned by the error boxes in Figure 1.1 indicates overall concordance which holds for 2.6×10⁻¹⁰≤η≤6.2×10⁻¹⁰.

Incidentally, combining all light elements observations in a likelihood analysis, S. Burles et al [7] found:

η= (5.5±0.5)×10⁻¹⁰. (1.6) This type of analysis has long been the most accurate way to determinate the matter density in the universe. However, an independent and really competitive cross-check is now provided by analysis of anisotropies in the cosmic microwave background as recalled in the next section.

(14)

1.3 The cosmic microwave background 9

1.3 The cosmic microwave background

Recent precision measurements on the slight fluctuations in the cosmic microwave background (CMB) made by the now famous Wilkinson Microwave Anisotropy Probe (WMAP) have enabled us to discriminate between cosmological models and the deduction of the relevant parameters, including the baryon density of interest here.

Before going to the results, and in order to understand the interest in these fluctuations, we will first briefly review the context in which the CMB takes place [8], [9].

The cosmic microwave background is a prediction of big bang nucleosynthesis which states that a large fraction of the light chemical elements (D, He, Li) are produced in a primeval thermal bath of particles at a high temperature, as seen in the previous section.

Before electrons and nuclei form electrically neutral atoms, photons were tightly coupled to the bath in thermal equilibrium according to a black body distribution. Once neutral atoms got formed at a temperature of a few eV ’s, photons ceased interacting and decoupled from the other particles, encoding the possible structure of this primeval bath.

The cosmic microwave background has been measured in temperature and polarisation from all directions in the sky and has shown a remarkable isotropy up to one part in 10⁵. Although these measurements confirm big bang nucleosynthesis predictions, they give rise at the same time to the well-known horizon problem: How can different regions, not connected causally at the time of decoupling, reach such a level of isotropy? Indeed, the causal horizon at the last scattering was roughly 300 000 light years which corresponds nowadays to about one degree on the sky.

A possible explanation for this nearly perfect isotropy might be found in a global phenomenon predicting also the flatness of the universe and potentially accounting for large-scale structures:

This candidate is a period of inflation, which is an exponential expansion of the universe due for instance to a slowly rolling scalar field along its potential [10].

While it implies a global homogeneity of the universe, inflation also accounts for fluctuations and predicts a scale-invariant fluctuations spectrum.

How does inflation create fluctuations? Actually, they all find their origin in quantum fluctuations that are exponentially magnified by the expansion and then get stuck once their scale is larger than the causal horizon.

Once the horizon increased enough to contain the fluctuation, the latter starts to oscillate in the plasma of matter and photons. The observed anisotropies in the cosmic microwave background are the imprint of these oscillations at the time of recombination, i.e. when photons ceased interacting through Thompson scatterings with electrons and nuclei. We refer to these observations as the surface of last-scattering even though the quoted surface has actually a finite width.

The description of temperature fluctuationsΘ(n) =ˆ ∆T/T with respect to the black-body tem- perature T in a given direction in the sky ˆn is decomposed in multipole moments:

Θ_lm= Z

d ˆn Y_lm^∗(n)ˆ Θ(n),ˆ (1.7) where Y_lmare the usual spherical harmonics.

(15)

Assuming a Gaussian distribution (predicted by inflation), the multipole distribution has zero mean and is fully characterised by its variance, i.e. the power spectrum:

hΘ^∗_lmΘ_l0m⁰i=δ_ll0 δ_mm0C_l. (1.8) Since for small angular scales, where we can neglect curvature, spherical harmonics reduce to usual Fourier functions, the l number may be identified to a wavenumber implying that large multipole moments correspond to small scale fluctuations.

The resulting diagram shows the power spectrum of fluctuations as a function of the multipole moment l, as shown in Figure 1.2 for observations from WMAP [11]. We observe a succession of peaks and troughs corresponding to fluctuations of decreasing scale known as acoustic peaks. For instance, the first peak corresponds to the largest fluctuation entering in the horizon and caught in its first compression by recombination.

The theoretical construction of this peaks structure depends on several cosmological parameters which are associated to several subtle effects that will not be summarised here. We will just mention the main effects of matter density in the evolution of acoustic waves in the photon-baryon plasma that filled the universe before recombination.

The first obvious effect of baryons in the primeval plasma is the modification of the sound

Figure 1.2: Power spectrum of the cosmic microwave background observed by the WMAP exper- iment. The curve shows the best fit according to a basic cosmological model: a flat universe with radiation, baryons, cold dark matter and cosmological constant, and a power-law power spectrum of adiabatic primordial fluctuations [11]. The grey dots are the unbinned data.

(16)

Ω_Λ 0.2 0.4 0.6 0.8 20

40 60 80 100

∆T(µK)

Ω_tot 0.2 0.4 0.6 0.8 1.0

Ω_bh2

10 0.02 0.04 0.06

100 1000

20 40 60 80 100

l

∆T(µK)

10 100 1000

l

Ωmh2 0.1 0.2 0.3 0.4 0.5

(a) Curvature (b) Dark Energy

(c) Baryons (d) Matter

Figure 1.3: Variation of the acoustic peaks depending on cosmological parameters.

speed by the baryon-photon momentum ratio R= p_b+ρ_b

/ p_γ+ργ

=3ρ_b/4ργ:² c_s= 1

p3(1+R).

Adding baryons therefore decreases the sound horizon. The consequence is that oscillating fluc- tuations are displaced to smaller scales and conversely acoustic peaks translated to larger l.

Moreover baryons act on oscillations as a mass on a spring in a constant gravitational field. It lowers the equilibrium point. The symmetry of the oscillations is broken since baryons enhance only the compressional phase through the extra gravity provided by baryons in overdense regions.

The consequence is an enhancement for odd-numbered peaks.

Finally, we also mention the damping of oscillations for large multipole moment l due to the imperfect coupling of photons to the plasma at short distances. Indeed, photons have a mean free path corresponding roughly toλ∼1/p

neσ_TH (neelectron density,σ_T Thompson scattering cross-section and H Hubble parameter) which limits the coupling of fluctuations to the photons bath for smaller wavelengths. This is observed in the power spectrum through smaller peak am- plitudes for large l.

The precise determination of the cosmological parameters depends on several assumptions regarding the cosmological model and on the data sets coming from different independent observations. For instance, even in the simplest (most constrained) cosmological model with assumed vanishing curvature, perfect power law spectrum, negligible neutrino density, no gravitational waves and w=−1 in the equation of stateρ=wp; the basic cosmological parametersΩm(matter

2paandρabeing respectively the pressure and density of the a species.

(17)

density) and H (Hubble parameter) are only given with errors of 50% and 20% respectively for the only WMAP data [13].

Including informations on the power spectrum from the galaxies survey for instance (SDSS) [12], the errors are dramatically reduced to 10 and 5%, but depending crucially on the assumption of perfectly flat space.

Conversely, once WMAP and SDSS are combined, the constraints on the three parameters (Ω_Bh², Ω_DMh² and n_s) are quite robust regarding the choice of theoretical priors on the other parameters [13].³The baryon fraction to the critical density is evaluated to

Ω_bh²=0.023±0.001. (1.9)

In order to relateΩ_bh²to the value ofη=n_b/n_γ in big-bang nucleosynthesis, one have to know quantities as the present photon temperature and the average mass per baryon number. Assuming further that the expansion has been adiabatic since BBN,η and the baryon density are related by Ω_bh²= (3.650±0.004)×10⁷η[7]. Therefore the above value forΩ_bh²translates to :

η=n_b

n_γ = (6.3±0.3)×10⁻¹⁰. (1.10)

3H=100 h km s⁻¹M pc⁻¹, the Hubble constant.

(18)

Chapter 2 Baryogenesis

As seen in the previous chapter, all observations point to an asymmetric universe filled in by baryons and no anti baryons. A priori, we cannot reject the possibility that this asymmetry is an initial condition even though it would be an unsightly one. But if it is the case, we would therefore have to accept an unexplained new parameter of the order of one over a billion in the cosmological model.

Conversely, we will explore here ways to generate dynamically such an asymmetry in the evolution of the universe.

2.1 Sakharov ’s three conditions

Already in 1967, A.D. Sakharov [14] identified three conditions allowing the generation of a baryon asymmetry in a usual field theory conserving CPT .¹

These are

1. Baryon number violation, 2. C and CP violation,

3. Departure from thermal equilibrium.

The first condition is obvious since we want to start from a symmetric universe, but is also quite constrained by the well-established stability of the proton and the unobserved neutron-antineutron oscillation.

In the second place, C symmetry ensures the processes i→ f and ¯i→ ¯f have the same probabil- ity. Therefore, a symmetric universe would produce an equal amount of f and ¯f having opposite baryon numbers. The consequence is a vanishing net baryon number. Furthermore, CP invari- ance implies that the rates of the processes i(x)→ f(x)and its CP conjugate ¯i(−x)→ f¯(−x)are equal. Then, even though a baryon asymmetry could be created locally, integrating over all space directions and quantum numbers will completely wash out the effect.

1C: charge conjugate; P : space parity; T : time reversal; to be discussed further in sections 5.2 and 7.1.

(19)

2.2 Baryogenesis in the standard model 14

Finally, let us suppose that a certain species X with mass m_X is in thermal equilibrium. The phase space distribution is given by the Fermi-Dirac or Bose-Einstein statistic :

f_X(p) = (Exp[(E_X(p)−µ_X)/T]±1)⁻¹, (2.1) (with E_X(p) =q

p²+m²_X) according to whether the particle is a fermion (+) or a boson (−). The condition of chemical equilibrium of a given process X+A→B+C requires a relation between chemical potentials:

µ_X+µ_A=µ_B+µ_C. (2.2)

Then, considering the annihilation process : X ¯X→γγ, we get the relation:

µ_X+µ_X_¯ =2µ_γ =0. (2.3)

The baryon number violating process X X→X ¯¯X implies µ_X =µX¯ =0 which combined with the CPT requirement m_X =m_X_¯ ensures an equal density (2.1) for baryons and anti-baryons, i.e. no asymmetry.

A simple rule to estimate the departure from equilibrium is to compare the relevant particle interaction rateΓto the expansion rate of the universe H. The particle density decouples since Γ<<H.

In general, the Sakharov conditions are sufficient but not necessary, and one can avoid these conditions for different exotic theories [15], [16].

2.2 Baryogenesis in the standard model

What is the status of baryogenesis in the standard model (SM)? Which of the Sakharov conditions can be fulfilled by the standard model? Actually, even if it is not straightforward, all three conditions can be realized in principle. Nevertheless, and we will understand the reason below, according to the present experimental status of the standard model, it fails to account for a baryon asymmetry of the right order. The origin of this failure is to be found in the very weak match of Sakharov conditions within the experimental standard model parameters.

Let us first come back to whether or not Sakharov conditions can be met in the standard model.

B+L anomaly

At the classical level, baryon number (as lepton number, assuming no Majorana masses) is a global symmetry of strong and electroweak interactions. It results in the conservation of the baryonic and leptonic currents, i.e.:

∂µJ_B^µ=0, ∂µJ_L^µ=0, (2.4)

with J_B^µ=∑q1

3qγ¯ ^µq and J_L^µ=∑_l¯lγ^µl (since quarks carries a baryon number of one third).

At the quantum level, however, these currents are no longer conserved because only left-handed fermions are gauged in SU(2)_L and that the U(1)_Y hypercharge is different for left and right components. This gives rise to a chiral anomaly [17]. We thus obtain:

∂µJ_B^µ=∂µJ_L^µ= N_F

32π² −g²TrW_µνW˜ ^µν+g⁰²B_µνB^µν

, (2.5)

(20)

2.2 Baryogenesis in the standard model 15

where W_µν, B_µν denote respectively the SU(2)_L and U(1)_Y field strength tensors and g, g⁰ their respective gauge couplings; N_F being the number of fermion families.²

We can rephrase here the B and L non conservation in more interesting terms for further con- sideration as a conservation of B−L and a violation of the orthogonal combination B+L.

Integrating (2.5) over space-time, we can deduce that the variation in the baryon (or lepton) number is quantified by

∆B=∆L=N_F ∆N_CS, (2.6)

where N_CS, the Chern-Simons number, is the topological charge of a pure gauge vacuum config- uration in the non-abelian SU(2)_Lgroup. The transition between two different vacua is therefore accompanied by B+L violation.

At zero temperature, such a transition occurs only through quantum tunnel effect, whose rate is extremely small. On the other hand, at higher temperature, a classical solution interpolating two ground states whose Chern-Simons numbers differs by one unit, can be identified as an unstable solution of the equations of motions of gauge and scalar fields.

This solution, called sphaleron, is characterised by the saddle point at the top of the potential barrier between the two vacua and has energy (see [18] for a review):

E_sph=2m_W α_W B

m_φ m_W

, (2.7)

where m_φ is the Brout-Englert-Higgs(BEH) boson mass, B _m

φ m_W

has been evaluated numerically to 1.56 to 2.72 as m_φ/m_W varies from zero to infinity (α_W = ₄^g_π²). The height of the barrier in the electroweak theory is therefore of the order 10 TeV . At high temperature, the system may then jump to the sphaleron solution and after fall down to another vacuum with a different B+L number. Therefore, The transition occurs without tunneling.

The corresponding transition rates depend on whether one consider the broken or unbroken phase of electroweak theory. Indeed, in the first case, when temperature lies below the electroweak phase transition temperature T<T_EW ∼100GeV , the sphaleron rate behaves as:

Γ^sph∝Exp

−E_sph/T

, (2.8)

i.e. is Boltzmann suppressed.

On the other hand, in the unbroken phase, the transition rate behaves as :

Γ ∝(α_WT)⁴. (2.9)

Equilibrium

Eventually, we compare this rate to the expansion rate of the universe in order to check the departure from equilibrium.

In the unbroken phase, we find that these processes are in thermal equilibrium for tempera- tures T_EW ∼100 GeV <T .10¹²GeV . This conclusion is a strong constraint on baryogenesis

2W˜_µν =¹₂ε_µνρσW^ρσ.

Universit´e Libre de Bruxelles Facult´e des Sciences