Fundamental Models for
Composite Dark Matter and Higgs
Candidata
Relatore
Dott. Elena Vigiani
Prof. Alessandro Strumia
Ciclo XXIX
Overview 1
1 Introduction on Dark Matter 7
1.1 Evidences of Dark Matter . . . 7
1.1.1 Dark Matter as a non relativistic thermal relic . . . 9
1.2 Dark Matter searches . . . 12
1.2.1 Direct detection . . . 12
1.2.2 Dark Matter at LHC . . . 17
1.2.3 Indirect detection. . . 19
1.3 Dark Matter models . . . 19
2 Accidental Composite Dark Matter 21 2.1 Introduction. . . 21
2.2 Vectorlike Confinement models . . . 23
2.2.1 Accidental symmetries . . . 28
2.3 Techni-baryons as Dark Matter candidates. . . 30
2.3.1 SU(NTC) composite Dark Matter models . . . 32
2.3.2 SO(NTC) composite Dark Matter models . . . 36
2.4 Phenomenology of composite Dark Matter . . . 39
2.4.1 Direct detection of complex Dark Matter . . . 40
2.4.2 Direct detection of real Dark Matter . . . 42
2.4.3 Techni-pions at colliders . . . 47
2.4.4 Unification of SM gauge couplings . . . 48
3 Techni-pions as Dark Matter candidates 51 3.1 Introduction. . . 51
3.1.1 Models with accidentally stable techni-pions. . . 52
3.1.2 Less minimal models . . . 54
3.2 Techni-pions interactions and phenomenology . . . 56
3.2.1 Anomalous couplings to SM vectors . . . 56
3.2.2 Dark Matter interactions with techni-pions and gluons . . 59
3.3 Models with colored constituents . . . 62
3.3.1 A two species model example . . . 63
4.1.1 The Higgs as a composite Nambu-Goldstone boson . . . 74
4.1.2 Custodial symmetry . . . 75
4.1.3 Partial compositeness . . . 76
4.2 Fundamental theory of fermions and scalars . . . 78
4.2.1 Global symmetries . . . 80
4.2.2 Dynamical symmetry breaking . . . 82
4.3 Quartic couplings among scalars . . . 85
4.4 Successful models. . . 88
4.4.1 Model with SU(5)GUT fragments and Y = −1/2 . . . 93
4.4.2 Model with SU(5)GUT fragments and Y = +1/2 . . . 95
4.4.3 Model with minimal custodial symmetry and Y = 0 . . . . 96
4.4.4 Model with SU(5)GUT fragments and a scalar doublet . . . 96
4.4.5 Imperfect SO(NTC) model with minimal custodial symme-tries . . . 97
4.4.6 Model with a full family of scalars . . . 97
4.5 Higgs properties . . . 98
4.5.1 Yukawa couplings. . . 99
4.5.2 Higgs potential . . . 100
4.5.3 Flavor violations . . . 104
Conclusions 109 A Group theory for techni-baryons 113 B Effective chiral Lagrangian 115 B.1 Effective Lagrangian for techni-pions . . . 116
B.2 Including techni-baryons . . . 118
C Fundamental model of two composite Higgses 123 C.1 An issue with custodial symmetry . . . 124
This work could not be possible without my supervisor Alessandro Strumia and all my collaborators, especially Michele Redi. Moreover, I would not have been here at this point without the support of Stefania De Curtis who lovely introduced me to the theoretical physics research. Finally a special thanks goes to my friend and collaborator Andrea Tesi who always supported me during the hard work. Then, I am infinitely grateful to my parents and to my husband for their love and their continuos support.
Given the lack of evidences of new physics at the Large Hadron Collider (LHC), the role of Naturalness as a guiding principle for theories beyond the Standard Model may need to be reconsidered. The Standard Model (SM) is a very suc-cessful theory and LHC confirmed it, so far. At the same time, we know that it is affected by a list of problems that cannot be solved without postulating the existence of new physics. The more famous is the Higgs hierarchy problem: why is the Higgs mass so small with respect to the Planck scale? The presence of an elementary scalar in the SM introduces a non dynamically generated mass scale that we would expect to be zero, if protected by a symmetry, or of order of the Planck scale ΛPl. Since the mass of an elementary scalar is not protected by any
symmetry, the fact that Mh ≈ 125 GeV ΛPl represents the so called
hierar-chy problem. Naturalness requires new physics at a scale of order few hundreds GeV, at odds with the LHC results: maybe Naturalness is a valid principle but it is not perfectly realized in Nature. If it is the case, the new physics may be slightly above the energy scale explored by LHC and a modest fine tuning must be accepted. Even if the new physics that cuts off the quadratic divergences of the Higgs is slightly out of the reach of LHC, it can manifest itself someway at lower energies.
A simple and plausible way for extending the SM is that of adding new vec-torlike fermions with a mass scale <∼ TeV or greater, that are in real represen-tations of the SM gauge group. While adding new chiral fermions requires a lot of new particles for anomaly cancellation and their masses generation through electroweak symmetry breaking leads to significant corrections to precision elec-troweak and flavor observables, these problems are naturally avoided if vectorlike fermions are considered. In particular, since they can have a mass without cou-pling to the Higgs, they can be decoupled from the electroweak symmetry break-ing sector, thus avoidbreak-ing dangerous corrections to the SM precision observables. If these fermions are also charged under a new gauge strong interaction confining at a scale of order TeV, a composite sector analog to that of the QCD is generated. Since the new fermions are in vectorial representation of the SM, when conden-sates form, the SM symmetry is left unbroken. These features characterize the so called Vectorlike Confinement scenario and set the difference with respect to the Technicolor theories: Technicolor aimed to solve the hierarchy problem directly at the TeV scale, by means of fermions charged under the new confining force
and in chiral representations of the electroweak group, so that when condensates formed, the electroweak symmetry was broken. Technicolor theories, at least in their original realizations, are excluded by electroweak precision observables and flavor measurements.
In the Vectorlike Confinement scenario, the composite sector includes particles analog to the QCD baryons, with a mass scale of order or above the confinement scale, and states analog to the pions, with a significantly lower mass. The lat-ter can be accessible at LHC even if the other composite resonances cannot be produced. The Higgs can be a pion-like composite particle arising from the new confining theory, thus addressing the hierarchy problem: Vectorlike Confinement can provide a fundamental description of Composite Higgs models.
A typical feature of renormalizable gauge theories is the presence of accidental global symmetries and a striking success of the SM is that all the observed global symmetries arise accidentally. The proton is stable thanks to an accidental U(1)B
global symmetry that rotates, with an equal phase, the quark fields leading to the conservation of the baryon number. Cosmological observations suggest that in Nature at least another particle must be stable. Indeed, it is well know from decades that a big fraction of the Universe energy density is made by an unknown matter, that must be stable on cosmological scales.
A huge amount of experimental evidences show that two other ingredients, beside the usual baryonic matter, must fill the Universe: Dark Energy and Dark Matter. Dark Energy is responsible for the Universe’s accelerated expansion and, differently from matter, it does not cluster. Dark Matter behaves as usual matter from the gravitational point of view, but it does not emit, absorb nor diffuse light. The evidences for Dark Matter range from the galactic scales, where the presence of non-luminous matter is needed to explain the observed stellar dynamics, to cosmological scales, where it is necessary to correctly reproduce the structures formation mechanism. Including Dark Energy and Dark Matter, the cosmological model works very well and reproduces what we know about the Universe, its formation and evolution.
The presence of Dark Matter (DM) calls for new physics beyond the SM, that must provide particles stable on cosmological scales, with no electric charge, no color and almost no coupling to the Z vector boson. Since a weakly interacting particle can reproduce the thermal relic DM abundance in the Universe if its mass is around 100 GeV, a lot of attempts have been made for explaining DM in the context of natural extensions of the SM. As an example, in Supersymmetry, the DM candidate is provided by the lightest neutralino particle, stable thanks to the R-parity discrete symmetry. R-parity, under which the SM and SUSY particles have positive and negative parity respectively, is imposed to guarantee the stability of the proton and, as a bonus, gives rise to a DM candidate. However, the lesson from the SM makes particularly appealing the idea that DM might be stable thanks to an accidental symmetry of the Lagrangian that is supposed to extend the SM to higher energies.
describing the Vectorlike Confinement scenario: DM candidates can arise as com-posite particles, analog to QCD baryons or pions, with no electric charge, no color, nor hypercharge and accidentally stable thanks to a symmetry. Such DM can-didates can give rise to interesting phenomenology since they can have peculiar interactions with the SM gauge bosons, thanks to their composite nature.
In this Thesis, we will explore the Vectorlike Confinement scenario from two different points of view:
- Firstly, we will investigate how Dark Matter candidates automatically arise in the Vectorlike Confinement framework thanks to accidental symmetries. Under certain assumptions, we will perform a systematic search of all the renormalizable models giving rise to a good composite Dark Matter candi-date. We will do it by remaining agnostic about the Naturalness problem and assuming a SM Higgs.
- Then, we will consider the possibility that the Higgs emerges as a compos-ite pion-like particle. Aiming to reproduce the SM fermions masses in a fundamental way, we will extend the Vectorlike Confinement scenario by adding fundamental scalars charged under the SM gauge group and also under the new strong interaction.
In Chapter 1 we will shortly review what we know about DM and what are the main experimental strategies to detect it. In Chapter2we will introduce the Vectorlike Confinement scenario based on a non abelian new gauge interaction at which we will refer as techni-color (TC), being clear the differences with the old Technicolor theories. We will show how a DM candidate can arise automati-cally as the lightest composite techni-baryon stable thanks to an accidental U(1) global symmetry analog to the SM baryon number. This kind of candidates can reproduce all the DM thermal abundance in the Universe if their mass scale is around 100 TeV. Despite their large mass, they can give rise to interesting sig-natures in direct detection DM experiments. In particular, theories based on a new SU(NTC) gauge strong interaction give rise to complex DM candidates with
electric and magnetic dipole moments leading to a direct detection cross section with a special dependence on the DM relative velocity and on the DM/nucleus transferred momentum. The electric dipole moment can be generated by a CP-violating coupling between techni-baryons and techni-pions proportional to the θTC-angle of the new strong sector, in complete analogy with the neutron electric
dipole moment. Similar theories based on SO(NTC) gauge interactions provide
real DM candidates that in special cases can give rise to an interesting inelastic scattering phenomenology.
The possibility that DM candidates arise as composite techni-pions is inves-tigated in Chapter 3. The DM thermal relic abundance can be reproduced by a sub-TeV stable techni-pion and in this case the compositeness scale (the scale
at which confinement takes place) can be of order TeV and the techni-pions can be accessible at LHC. CP-violating interactions between techni-pions generated by the θTC-angle of the hidden sector and TCπTCπ scatterings contribute to the
DM annihilation cross section thus determining its thermal relic abundance in the Universe. Moreover, the unstable singlets taking part to the composite techni-pionic sector are coupled to the SM gauge bosons through the triangle anomaly, exactly as the π0 has an anomalous coupling to photons. In particular, if they
are made by colored constituents, they have an anomalous coupling to gluons and can be resonantly produced at LHC with a significant cross section, possibly giving rise to interesting signatures. Given that, it is interesting to investigate the possibility that DM can show up at collider not directly, but through some signature from the other unstable techni-pions, that share with the composite DM candidate a common origin.
In Chapter 4 we will change perspective and try to realize a fundamental theory predicting a composite Higgs and automatically generating the masses of the SM fermions. The idea that the Higgs may be a composite particle, so that its mass is protected from radiative corrections above the compositeness scale, is old and widely investigated in the literature. However, composite Higgs models have been mainly considered from an effective field theory point of view.
A composite Higgs is considered natural if it emerges as a pseudo-Nambu-Goldstone boson from a new confining strong dynamics, so that it acquires a potential and a mass because of the explicit breaking of the Goldstone symmetry: if this breaking is small, its mass can be significantly lower than the composite-ness scale. The dynamically generated Higgs potential triggers the electroweak symmetry breaking, exactly as in the SM.
The masses of the SM fermions are generated through the mechanism of partial compositeness. This consists in the fact that each elementary fermion is coupled linearly with a composite fermion so that the mass eigenstates are a linear combination of elementary and composite states. Since the composite fermions are coupled to the Higgs, the SM fermions get a mass thanks to their composite fraction so that the more a fermion is composite, the more it is massive.
A fundamental theory where the Higgs emerges as a composite pion-like par-ticle made by two fermions, can be realized in the framework of Vectorlike Con-finement, the main challenge being that of realizing the partial compositeness mechanism. The SM Yukawa couplings are generated by running down to low energy the couplings that parametrize the mixing between elementary and com-posite fermions. Simple dimensional considerations suggest that such a mech-anism is effective if the composite fermions are made by a fermion with mass dimension 3/2 and a composite scalar with effective mass dimension 1. Realizing this scenario with fundamental fermions requires to postulate unnaturally large anomalous dimensions for the composite fermionic operators, or to assume highly involved extra dimensional theories. The simplest option seems to be that of con-sidering a composite fermion made by a fundamental fermion and a fundamental
scalar, so that the linear mixing between elementary and composite fermions sim-ply comes from a Yukawa coupling in the renormalizable Lagrangian. The idea of introducing fundamental scalars charged under the new confining gauge inter-action is unconventional and rarely considered in the literature. Obviously this choice reopens the Naturalness problem and, since the theory is no more analog to QCD, the role of scalars in the non perturbative dynamics is unknown. The in-troduction of scalars poses many theoretical and practical problems, but in some sense it seems to be unavoidable. Thus, we will remain agnostic about the issue of Naturalness of fundamental scalars and we will try to explore what we can learn from a fundamental theory of confining fermions and scalars. Furthermore, since in Composite Higgs effective models ad hoc symmetries are postulated in order to make them phenomenologically viable, we will try to construct fundamental models where such symmetries appear accidentally.
This Thesis is based on the following papers :
O. Antipin, M. Redi, A. Strumia, E. Vigiani, “Accidental Composite Dark Matter”, JHEP 1507 (2015) 039 [arXiv:1503.08749].
M. Redi, A. Strumia, A. Tesi, E. Vigiani, “Di-photon resonance and Dark Matter as heavy pions”, JHEP 1605 (2016) 078 [arXiv:1602.07297]. The article has been expanded and decoupled from the now disappeared 750 GeV diphoton resonance.
F. Sannino, A. Strumia, A. Tesi and E. Vigiani, “Fundamental partial com-positeness”, JHEP 1611 (2016) 029 [arXiv:1607.01659].
Other works from the same author are :
S. De Curtis, M. Redi and E. Vigiani, “Non Minimal Terms in Composite Higgs Models and in QCD”, JHEP 1406 (2014) 071 [arXiv:1403.3116]. G. M. Pelaggi, A. Strumia and E. Vigiani, “Trinification can explain the di-photon and di-boson LHC anomalies”, JHEP 1603 (2016) 025 [arXiv:1512.07225]. G. M. Pelaggi, F. Sannino, A. Strumia and E. Vigiani, “Naturalness of asymptotically safe Higgs” [arXiv:1701.01453].
Introduction on Dark Matter
In this Chapter we shortly introduce what we know about Dark Matter and what are the main experimental strategies to detect it. The aim is not that of realizing a complete and detailed review on the topic, but that of giving the motivations and the ingredients necessary for the following two Chapters of the Thesis.
1.1
Evidences of Dark Matter
The SM describes only a small fraction of the Universe: many observations on astrophysical and cosmological scales indicate that about 25% of the Universe is filled by non-baryonic matter, that we know as Dark Matter (DM). Another huge fraction of the Universe content is made by something dark, that is the Dark Energy. This constitutes about 70% of the Universe and probably can be interpreted as a cosmological constant, that is a constant energy density perme-ating the Universe and responsible for its accelerated expansion. In this Thesis we will consider models that can provide DM candidates, while the problem of Dark Energy will not be addressed.
The existence of DM is strongly supported by a lot of astrophysical and cos-mological data collected in the last 80 years, for some reviews on the argument see [1] and references therein. The first hint of the presence of DM was presented in the 30’s by Fritz Zwicky [2], based on the measurements of the velocity disper-sion of galaxies in the Coma Cluster. He realized that these galaxies could’t be bound to the cluster by the gravitational attraction of the visible matter alone. This idea was rediscovered in the 70’s, with the measurements of the rotational velocity of disk galaxies that was expected to fall as 1/√r out of the visible disk but that instead showed a constant behavior [3]. This suggested that galaxies are made also by non luminous matter, that interacts gravitationally. Observations of strong and weak gravitational lensing [4] supported this idea. The most ex-traordinary example of weak lensing is the evidence of a mass discrepancy in the so called Bullet Cluster [5]: the gravitational potential of the system of two col-liding clusters of galaxies shows that the center of the total mass of the system is 8 σ away from the center of the baryonic mass. The explanation is that after the
collision the interacting gas was leaved behind while the DM passed through, thus giving an upper bound on the DM self interactions. While the galactic rotation curves can be explained also by a Modified Newtonian Dynamics (MOND) [6], it is very hard to explain the Bullet Cluster by a modification of classical gravity: it needs DM.
So we know that DM is stable or has a lifetime bigger than the age of the Universe, it interacts gravitationally, but it is not observed to interact electromag-netically. It can be neutral or have a small electric or magnetic dipole moment, or it can have a very small effective electric charge.
The small or absent interaction of DM with light has the consequence that DM, differently from baryonic matter, cannot cool during large structure forma-tion by radiating photons. DM begins to collapse into halos well before baryonic matter, that interacts with photons in the primordial plasma, so that at the time of recombination, baryonic matter falls into the potential wells created by the DM halos. This accelerates the galaxy formation, that in absence of DM would occur much later in the Universe than what is observed. The observed Large Scale Structure (LSS) [7] of luminous matter is consistent with the presence of a certain amount of DM that must be mostly dissipationless, even if a small part of it could be dissipative. Moreover, LSS suggests that DM must be non rela-tivistic, that is “cold”, or at most “warm”, but it cannot be totally “hot”. This classification depends on how relativistic is the DM when galactic size perturba-tions enter into the horizon, that happened at a temperature T ' keV. If DM is relativistic (hot) it would free-stream out of galactic sized regions so that only very large structure could form early: small structures as galaxies would derive from a fragmentation of super-clusters. This picture does not reproduce what we know about structure formations. Warm DM can still be compatible with LSS but cold DM is the preferred scenario.
Other strong evidences in favor of DM come from Cosmology, in particu-lar from the Cosmic Microwave Background (CMB) radiation firstly observed in 1964. This is like a picture of the Universe at the time of recombination, when neutral atoms formed and photons decoupled and started to propagate. This relic radiation carries a lot of informations on the structure of the Universe at that time [8]. CMB anisotropies give us informations about the matter distribution of the Universe and fits to the CMB power spectrum allow to fix a lot of parameters of the ΛCDM model, that is considered the best description of the present exper-imental data about the Universe. It is characterized by a non zero cosmological constant Λ that encodes for the presence of Dark Energy in the Universe, respon-sible for its accelerating expansion [9], and it is also characterized by a relevant amount of cold DM. The latest observations by the Planck Collaboration of the anisotropies in the CMB, combined with other observations [10], give:
Ωbh2 = 0.02230 ± 0.00014 ,
ΩDMh2 = 0.1188 ± 0.0010 ,
where h = 0.678 ± 0.009 is the present expansion rate of the Universe, that is the Hubble constant, in units of 100 km/Mpc · s and Ωi = ρi/ρic is the ratio between
the energy density today and the critical energy density necessary to make the Universe flat. The flatness of the Universe, that is measured to be very close to 1, is another open problem whose most popular solution is Inflation.
The small fraction of the Universe energy density made by baryonic matter is independently confirmed by the Big Bang Nucleosynthesis (BBN) that is quite well understood and based on SM physics. BBN predicts the abundances of the light elements in the early Universe in perfect agreement with observations. So, BBN too confirms that a huge fraction of the Universe is made by something dark.
The only particles of the SM that, in principle, can be DM candidates are neutrinos. However, since SM neutrinos are very light thermal relics, they are predicted to emerge from the early Universe as highly relativistic particles and thus represent an example of hot dark matter. As we discussed above, hot DM cannot account for the entire DM density, furthermore SM neutrinos are too light to be able to reproduce the DM abundance in the Universe. Thus the existence of DM necessary calls for new physics beyond the SM. The mass range in which we can find a DM candidate spans from 10−25eV to 1024kg:
��-�� ��-�� ��-�� �� ���� ���� �� ���� ���� ���� ���� ������ ���� ���� ����� ������ ����� ����� ���� ����������� ������� ���������� ����� ������ ������ �����-����� ������� ����� ��������� ��������� ν� ������� ������
see [1] for a review about the possible DM candidates. This makes evident how the hunt for DM is multi-pronged and interdisciplinary, interesting many sectors of physics, from astrophysics to cosmology and particle physics, going from high energy to low energy physics.
1.1.1 Dark Matter as a non relativistic thermal relic
Here and in the following we will focus on cold DM thermally produced, see [11] for a review about the possible DM production mechanisms. We assume that DM is produced via interactions with the thermal bath in the early Universe and then reaches chemical and thermal equilibrium with the bath. The density per comoving volume of non relativistic particles in equilibrium decreases exponen-tially with decreasing temperature: at a given time the reactions that keep DM in equilibrium with the thermal bath become ineffective. This happens when the annihilation rate becomes smaller than the Hubble expansion rate (equivalently the mean free path for DM collisions becomes longer than the Hubble radius), DM is said to freeze-out and its numerical density becomes constant.
The Boltzmann Transport Equation describes the evolution of the number density n of the DM particles (we assume that there is no particle/anti-particle
asymmetry and that DM particles and anti-particles can only be created and annihilated in pairs): dn dt = −3Hn − hσavri(n 2− n2 eq) . (1.2)
H = ˙a/a is the Hubble parameter representing the expansion rate of the Uni-verse and a is the scale factor of the UniUni-verse, t is the time and neqis the number
density at the equilibrium. The quantity hσavri is the product of the DM
annihi-lation cross section and the relative velocity of the system particle/anti-particle, averaged over the thermal distribution of momenta of the DM particles. The first term in the right-hand side of the equation above accounts for the Uni-verse expansion, while the n2 and n2eq terms account for the particle annihilation (DM DM → SM SM) and creation (SM SM → DM DM) respectively. Assum-ing that there is no entropy change in radiation plus matter neither durAssum-ing the decoupling nor after, this equation can be conveniently combined with the law of entropy conservation ds/dt = −3Hs, where s is the entropy density. Thus we can write an equation in terms of the dimensionless quantities Y = n/s and x = m/T : x Yeq dY dx = − Γa H Y2 Y2 eq − 1 , Γa= neqhσavri . (1.3)
This equation holds in the approximation s = S a3 ∼ T3, or equivalently a ∼ T−1,
that is well satisfied in most of the time. The equilibrium annihilation rate Γa
decreases with decreasing T faster than H and freeze-out happens when Γa<∼ H. In the limit Γa H, eq. (1.3) clearly says that the number of particles for
comoving volume becomes a constant with temperature and thus with time. The numerical solution of the Boltzmann equation, assuming that the Uni-verse is radiation dominated, is depicted in fig.1.1. Notice that for T < Tfo (after
freeze-out) the number density becomes constant and its value depends on the annihilation cross section. For a typical weakly interacting non relativistic parti-cle (WIMP), freeze-out occurs at xfo= MDM/Tfo ' 25 and the DM relic density
is estimated as ΩDMh2 0.1 ≈ 2.2 · 10−26cm3/s hσavri . (1.4)
The averaged annihilation cross section times velocity is usually approximated with a non relativistic expansion
hσavri = σa(0)+ hvr2iσ(1)a + O(v4r) , (1.5)
with σ(0)a and σa(1) corresponding to the s-wave and p-wave annihilation channels
respectively. Notice that the smaller is the annihilation cross section of the DM candidate, the bigger is its relic density. This can be easily understood since a WIMP with a stronger annihilation rate would remain in equilibrium with the bath longer and when the freeze-out happens, its equilibrium density would be
Figure 1.1: Typical evolution of the DM number density in the early Universe. Notice that the lower is the annihilation rate of the candidate, the higher is its relic density after freeze-out.
more suppressed by the Boltzmann factor. If the DM mass is the only relevant mass scale in the annihilation process, the cross section can be estimated by dimensional analysis σa≈ g4 16π2M2 DM . (1.6)
Considering only the s-wave term in the non relativistic expansion of hσavri, from
eq. (1.4) we get g4 16π2M2 DM ≈ 1.9 · 10 −9 GeV2 , (1.7)
so that for a typical weak value for the coupling g, the observed DM abundance is obtained for a WIMP mass in the 100 GeV ÷ 1 TeV range, that is the same in which new physics is expected according to the Naturalness paradigm. A new physics model that is able to make the Higgs natural can at the same time provide a DM candidate: this is known as the WIMP “miracle”. Nowadays, this has to face with the negative results of the new physics searches at LHC.
It is important to stress on the assumptions that we made above: DM is stable and with no particle/anti-particle asymmetry; it is in equilibrium with the thermal bath at high temperature and then it decouples when the Universe is radiation dominated; the entropy of radiation plus matter is unchanged during the decoupling and after. If some of these assumptions is no more satisfied, the relic density (1.4) would be different. We refer to [1,11] for a complete discussion about the argument.
1.2
Dark Matter searches
Focussing on non relativistic DM candidates, there are three kind of searches that can be used to detect DM, as represented in the scheme below:
χ SM χ SM direct detection indirect detection production at collider 1.2.1 Direct detection
Direct Detection experiments (Xenon, LUX and PandaX to make some examples) are based on underground detectors looking for small amounts of energy deposited in the detector in case of a scattering between DM and the atomic nuclei of the detector. The main challenges of this kind of experiments are related to the very small amount of energy expected to be deposited in the detector, of order 1 ÷ 100 keV and the small rate of expected events. The detectors must be made of large amounts of material and located underground, in order to be shielded from the cosmic rays. Thanks to the experimental community efforts, noise levels below 1 event per kg per year where achieved, making possible the search of DM-nuclei interactions.
Detectors located on the Earth are hit by a large flux of DM from the galactic halo, that can be estimated as
ρ0vr MDM ≈ 6.6 · 106 GeV MDM 1 cm2s, (1.8)
where we used the value ρ0 ≈ 0.3 GeV/cm3for the DM density in the galactic halo
and vr ≈ 220 km/s for the velocity of the DM wind in the Earth’s coordinates
system.1 In an elastic DM-nucleus scattering, the recoil energy of the nucleus is ER= q2 2MN = µ 2 Nvr2 MN (1 − cos θ) , (1.10) where q is the transferred momentum, MN is the nucleus mass and µN = MMDMMN
DM+MN
is the reduced mass of the system DM-nucleus and θ is the scattering angle in the
1More precisely, the DM velocity with respect to the Earth’s reference frame is given by
~
vr= ~vsun+ ~vearth(t) , vsun= (12 + 220 ± 50) km/s , vearth≈ 30 km/s , (1.9) where we considered the orbital velocity of the Sun around the center of the Galaxy and the orbital velocity of the Earth around the Sun. The latter depends on the period of the year, giving an interesting possibly observable annual modulation in the event rate of DM.
DM-nucleus center of mass frame. The maximal recoil energy can be estimated as: ERmax≈ 2 keV MDM 10 GeV 2 100 GeV MN for MDM MN, (1.11)
ERmax≈ 2 A keV for MDM MN, (1.12)
where in the second equation we used MN ' A GeV with A the nuclear mass
number. Notice that in the latter case, the deposited energy increases with A, but at the same time the DM flux decreases with the mass of the DM. According to the characteristic shape of the recoil energy, the mass of the DM particle can be determined most accurately in the special case MDM ≈ MN. Moreover, it
is convenient also to perform the same experiment with different nuclei species. The typical transferred momentum q ≈ µNvr is of order 1 ÷ 10 MeV, that is
much smaller than the inverse radius of most of the nuclei so that the nucleus interact with DM as a point-like particle. Decoherence effects will be important only for heavy nuclei and/or heavy DM particles and will be taken into account by a nuclear form factor.
The differential recoil rate in units of events per unit mass of the detector per keV of recoil energy per day is:
dR dER = NN |{z} exp. × Z vmaxr vmin r d~vrf (~vr, t) vrρ0 MDM | {z } astroph. × dσN dER | {z } particle ph. . (1.13)
The formula above is determined by inputs from multiple branches of physics: - Experimental physics enters through NN, that is the number of target nuclei
per unit mass of the detector, but also through efficiency functions not showed in eq. (1.13).
- Astrophysics and cosmology parametrize the DM velocity distribution in the galactic halo f (~vr, t), where vr is the DM velocity in the Earth frame.
The simplest distribution that we can consider is the Maxwell-Boltzmann distribution, in the galactic rest frame, with zero average and dispersion v0
that can be estimated as v0' 220 km/s. It is truncated as vesc ' 500 km/s
that is the escape velocity of the DM from our galaxy. The lower extremum of the integral can be determined from eq. (1.10)
vmin= 1 µN r MNEth 2 + δ √ 2MNEth , (1.14) where Eth is the minimum energy deposition that the detector can
dis-criminate. The parameter δ vanishes for an elastic scattering, while for an inelastic scattering it corresponds to the mass energy difference between the almost degenerate states that enters in the scattering [12]. Important signatures of direct detection signals come from the Earth’s motion through
the galaxy that induces a seasonal modulation of the event rate [13] and a forward-backward asymmetry. The differential event rate is expected to show a peak for small recoil energies in winter, for larger recoil energies in summer. Measuring the forward-backward asymmetry requires the abil-ity to measure the recoil direction besides the energy: new generation of experiments will be able to perform directional searches [14].
- Particle physics enters with the scattering differential cross section dσN/dER.
This is often parametrized in terms of independent (SI) and spin-dependent (SD) cross sections at zero momentum transfer and nuclear form factors depending on the recoil energy [1]
dσN dER = MN 2µ2Nv2 r σSIFSI2(ER) + σSDFSD2 (ER) , (1.15)
where FSI,SD(0) = 1. The SI and SD cross sections at zero momentum are
defined as σSI= µ2 N µ2 p Z + (A − Z) ·fn fp 2 σp, (1.16) σSD= 32µ2NG2F π J + 1 J [aphSpi + anhSni] 2, (1.17)
where σp and µp are the cross section and the reduced mass of the system
DM-proton, A and Z are the nuclear mass and atomic numbers respec-tively and J is the spin of the nucleus. GF is the Fermi constant, fp,n,
ap,n are effective couplings of the DM particle with the nucleons in the
SI and SD case respectively and depends on the characteristic of the DM. hSp,ni = hN |Sp,n|N i are the expectation values of the total proton and
neutron spin operators and depends on the nuclear model. For fn/fp = 1,
corresponding to an isospin conserving interaction, σSI = A2(µ2N/µ2p)σp,
thus increasing quadratically with the nuclear mass number. Even ignoring the details of the DM and nucleus nature, from eq. (1.16) and eq. (1.17) we can immediately notice that the SI cross section scales as A2, until decoher-ence effects enter into play trough the nuclear form factor, differently from the SD cross section. As a consequence, the SD cross section will be in general smaller than the SI so that experimental bounds on SD scattering processes are expected to be weaker that those on SI processes.
We can give a numerical example to have an idea of the typical value of the event rate. We assume a DM mass MDM≈ 100 GeV and a nuclear mass number
A = 100, so that MN ≈ MDM ≈ 100 GeV. The DM-nucleus scattering cross
section goes as σN ∼ 1/MDM2 and for a typical electroweak scale interaction is of
order σN ≈ 10−38cm2, resulting in a rate of events:
R ∼ 10 3N A A · kg × ρ0vr MDM × σN ≈ 0.13 events kg year (1.18)
Figure 1.2: Top: Direct detection bounds on DM-nucleon spin-independent cross section, taken from [15]. Bottom: Latest updates presented by the LUX Collab-oration [18] and also including the PandaX-II results [19].
where NA is the Avogadro number and we assumed ρ0 ≈ 0.3 GeV/cm3 and
vr ≈ 220 km/s.
Fig.1.2summarizes the direct detection bounds on the SI DM-nucleon cross section. In the top panel, the neutrino floor is also showed: along the curve, neutrino/nucleus scatterings induce an event rate comparable to that of non rel-ativistic DM. Thus, as the sensitivity of direct detection experiments increases, neutrinos from astrophysical sources, including the Sun, the atmosphere and Supernovae, will become an important background [15]. As an example, solar neutrinos starts to be a problem at σSI<∼ 10−46cm2. However, since they give recoil energies smaller than (4 ÷ 40) keV depending on the nucleus mass, if the DM is heavy enough a cut on the recoil energy allows to get rid of this back-ground. Another possible solution is that of using directional detectors that are able to discriminate events originating from the Sun. Thus, when the experimen-tal sensitivity would reach the neutrino floor, only with a better knowledge of the
Figure 1.3: Figures taken from [14] and showing the spin-dependent DM-nucleon cross sections limits. For the proton case, the new results from indirect DM searches by IceCube [16] are also shown. Notice that they give stronger limits on the spin-dependent DM-proton cross section than direct detection experiments.
neutrino background and thanks to the possibility to detect special signatures like seasonal modulation and forward-backward asymmetry, it would be possible to make progresses in DM direct searches. Fig.1.3 shows the current limits on the SD nucleon cross sections. Notice that the strongest limits on the SI DM-proton cross section come from the indirect detection searches by IceCube [16]. DM may be captured in the Sun, where its self-annihilation to SM particles can result in a flux of high energy neutrinos, that can be detected by IceCube. These indirect searches for DM are sensitive to the DM-proton scattering cross section, which initiates the capture process in the Sun. Under the assumption of equilib-rium between DM capture and annihilation in the Sun, limits on the annihilation rate can be converted into limits on the SI and SD DM-proton cross sections [17]. The limits on SI cross section are comparable but not competitive to the limits coming from the direct detection searches, while the limits on the SD DM-proton cross section are stronger than the ones coming from direct detection experi-ments. Comparing fig.1.2 with fig.1.3we can see that the SD bounds are much weaker than the SI ones, as we expected since the SI DM-nucleus cross section increases with the nuclear mass number.
Direct detection experiments give strong constraint on the nature and prop-erties of the DM. We already know that it must be electrically neutral, moreover from direct detection searches it turns out that DM candidates with non-zero hypercharge are excluded, apart from special cases. Indeed a DM particle with Y 6= 0 has vectorial interactions with the Z giving rise to SI elastic cross sec-tions [20] σSI= κ G2FMN2 2π Y 2(A + 4 sin2θ WZ)2, (1.19)
name operator type D1 mΛ3qχχ¯¯ qq scalar D5 Λ12χγ¯ µχ¯qγµq vector D8 Λ12χγ¯ µγ5χ¯qγµγ5q axial-vector D9 Λ12χσ¯ µνχ¯qσµνq tensor D11 4Λα33χχG¯ µνGµν scalar
Table 1.1: Some effective operators that couple a Dirac fermion DM χ to quarks and gluons, valid when the energy of the process is lower than Λ. For a complete list of the effective operators, also extended to the case of a Majorana DM particle or of a real or complex scalar DM, see [21].
mass and atomic number and θW is the Weinberg angle. The formula above
assumes that the DM mass is much bigger than the nuclear mass, that is MDM
MN. Taking MN ≈ 50 GeV as an example, we get the numerical estimation
σSI∼ κ Y210−38cm2 that is well above the present bounds. This bound can be
avoided in special cases such that of the Higgsino in supersymmetric models of DM. In that case the mixing with the Majorana gauginos can split its components by a small δm so that the lightest component becomes itself a Majorana fermion and cannot have a vectorial coupling to the Z. In Chapter 2 we will describe a similar mechanism.
1.2.2 Dark Matter at LHC
The typical signature of DM particles at LHC is represented by missing transverse energy: DM particles escape from the detectors bringing with them a significant amount of the energy involved in the scattering process. Avoiding strongly model dependent searches based on complicated decay chains, we can look for a process in which DM is pair produced besides a visible particle emitted by the initial or by the intermediate SM particles. The observable particle, that can be a photon (monophoton event), a gluon (monojet event) but also a vector boson or a Higgs, is necessary to detect the event. In various works, the collider bounds on this kind of events have been derived with an effective field theory approach. We assume that the DM particle, that can be either a fermion or a scalar, is coupled to quarks and gluons by effective operators such as those of table1.1, where DM is assumed to be a Dirac fermion. A complete list of the non renormalizable operators can be found in [21], both in the case of a Dirac or Majorana fermion DM and of a complex or real scalar DM. The DM pair production at LHC can be described by these effective operators only if the energy of the process is much lower than the cutoff Λ. This is not an obvious requirement for an high energy collider as LHC, so that the effective field theory description must be taken with care. If the process is due to the exchange of a mediator with mass M and coupling λ, such mediator is assumed to be much heavier than the exchanged momentum
Figure 1.4: Figure taken from [22] to which we refer for the details. LHC bounds on the operators D5 and D11 are compared with direct detection bounds on a SI
DM-nucleus scattering.
so that the propagator becomes λ2/M2 ∼ 1/Λ2. If the mediator of the process
is heavy respect to the exchanged momentum in the LHC process, that is M is greater than few hundreds GeV, it is surely heavier than the typical transferred momentum in the direct detection experiments. So it is interesting to compare the direct detection bounds with the LHC ones, being aware of their limited validity. An example of this kind of comparison is depicted in fig.1.4, where the monojet LHC bounds on two of the operators of table1.1 are shown, together with SI direct detection bounds [22]. As we can see from the plot, the collider bounds are significant only for low DM masses. In this region of the parameter space, the DM collider searches are complementary to the direct detection searches, that are ineffective for low DM masses because of the detector energy threshold. The complementarity of the different kind of DM searches is crucial but requires a careful comparison between data coming from different experiments. Referring again to fig.1.4, if the DM is a self-conjugate particle, the phase space of the LHC processes takes a factor 1/2 while the phase space for the direct detection processes is not affected by this factor. As a consequence, the LHC bounds for non self-conjugate DM particles are stronger by a factor of 2 in cross section. More importantly, while it is true that if the effective field theory approach is valid for LHC processes it is also valid for direct detection experiments, the reverse is not correct. Thus a plot like that of fig.1.4does not always make sense. In the case of light mediators, the effective field theory description breaks down and we need to specify the details of the model. An intermediate step is that of considering simplified models, see for example [23].
1.2.3 Indirect detection
Indirect searches experiments look for excesses in the flux of cosmic rays with respect to the flux expected from astrophysical background. DM in the galactic halo of our galaxy or other galaxies, as well as DM accumulated in the Sun and in the Earth can decay or annihilate into SM particles that then produce fluxes of energetic cosmic rays. The most dense regions, such as the galactic center or the center of the Sun, seem to be the most promising for indirect searches of DM. However, the detection of DM requires to understand, with a good accuracy, the astrophysical background and this is often more difficult in these regions. So it may happen that the most promising regions for indirect detection are not the denser ones. The most interesting SM particles to which look for are photons, neutrinos and stable anti-particles such as positrons and anti-protons. The latter are particularly interesting since the amount of anti-matter in the Universe is small and only few anti-particles are produced via astrophysical processes. These kind of searches are viable if DM is a thermal relic: the same reactions that allowed DM to be in equilibrium with the primordial plasma, make it possible to detect DM through its annihilations or decay products. We refer to [24] for a review about the status of indirect detection DM searches.
1.3
Dark Matter models
Given the lack of informations on the DM nature, proposing DM models is quite easy and thus it is important to find a way for selecting the “theoretically mo-tivated” models. Concretely, we can define as theoretically motivated, models entering in one or more of the following categories:
1. Models motivated by other issues, such as the Higgs hierarchy problem or the strong CP problem to make some examples;
2. Simple models, employing a minimal amount of extra degrees of freedom, such as one scalar or one fermion, or a minimal additional structure such as one symmetry;
3. Predictive models, where the DM properties are predicted in terms of a small number of parameters;
4. Nice models, providing an elegant and plausible answer to questions such as: why is DM stable?, why it is so dark?;
5. Gateways models, employing ingredients that tend to appear in more complex theories;
6. Models predicting novel signals, thus proposing a new strategy to de-tect DM.
Using this as a list of good intentions, we will try to construct theoretically moti-vated DM models. In the framework of Vectorlike Confinement, where a natural extension of the SM can be realized, we will construct simple renormalizable models based on a new confining gauge interaction from which DM arises as a composite particle made by charged fermions. The lightest composite particle are automatically neutral and stable because of an accidental global symmetry. Its composite nature makes it interacting with the SM gauge bosons in a peculiar way, giving an interesting phenomenology. Starting from these ingredients, we will construct models satisfying point 2 and point 4 of the list above.
Finally, a model can be good just because it can be easily tested by the experiments. This kind of models, that can be defined as “phenomenologically motivated”, will be considered in Chapter 3.
Accidental Composite Dark
Matter
This Chapter is based on [25], where in the context of Vectorlike Confinement sce-nario concrete extensions of the SM valid up to high energies are presented. We perform a systematic search of renormalizable models, compatible with SU(5)GUT
unification and valid up to the Planck scale, where DM candidates arise as com-posite states of a new confining gauge interaction, kept stable by accidental sym-metries. In particular we investigate the possibility that DM is made by composite states analog to the QCD baryons and stable thanks to an accidental symmetry, exactly as the proton is stable thanks to the baryon number conservation. Cosmol-ogy suggests that the DM thermal relic abundance can be reproduced if the mass scale of these states is around 100 TeV. We show that, despite its high mass, com-posite DM is characterized by an interesting phenomenology. SU(NTC) theories
can give rise to Dirac DM, with observable electric and magnetic dipole moments leading to a peculiar SI cross section. The electric dipole moment of the DM candidate arises exactly as the neutron electric dipole and it is proportional to a CP-violating coupling induced by the θTC-angle of the new strong sector. A
Ma-jorana DM can appear in SO(NTC) theories, with challenging SD cross section
or inelastic scatterings. Each model predicts also a set of lighter composite states, analog to the pions in QCD, possibly accessible at colliders.
2.1
Introduction
An interesting extension of the SM, still compatible with the lack of evidences at LHC, is the Vectorlike Confinement scenario [26]. This is one of the simplest and most plausible scenario for new physics beyond the SM, obtained by adding new fermions in vectorlike representations of the GSM gauge group, charged under
a new gauge force that we assume to be strong and to confine. Since the new fermions are in vectorial representations of the SM, when condensates form, the SM symmetries are left unbroken. Furthermore, since vectorlike fermions can have mass without coupling to the Higgs, the new physics is not related to the
electroweak symmetry breaking sector and it is not constrained by the electroweak precision observables. Assuming that the new fermions are connected with SM particles only via renormalizable SM gauge interactions, new physics accidentally realizes Minimal Flavor Violation (MFV) [27], meaning that the only sources of violations are the SM Yukawas. This clarifies the difference between Vectorlike Confinement and old Technicolor theories [28], where new fermions charged un-der a new strongly coupled gauge interaction were introduced with the purpose of generating the electroweak scale. In old Technicolor theories, the new fermions were in chiral representations of the SM gauge group that was broken at the confinement scale, putting these theories at odds with electroweak precision ob-servables and with flavor physics [29]. With abuse of language, we will refer to this new gauge force as techni-color (TC), being clear the difference between the two scenarios.
The framework of Vectorlike Confinement can be illustrated thanks to an analogy with the QED-QCD system, as presented in [26]. At the scale of about 100 MeV particle physics is made by leptons that interact via electromagnetic force. Leptons can be seen as the analog of all the SM particles and QED as the analog of the SM gauge interactions. Now let’s consider e+e− collisions at the GeV scale (as an analog of pp collisions at the LHC), where quarks are pair produced. Those are vectorlike particles under QED and feel a new strong in-teraction that is QCD, in analogy with the new vectorial fermions. Because of QCD confinement we do not see free quarks but pions as well as the ρ resonance. The mass of the first vector resonance is ≈ 770 MeV, while pions are much lighter since they are pseudo-Nambu-Goldstone bosons of the approximated chiral sym-metry breaking. In the same way, if the new techni-color confines at a scale >∼ TeV, the LHC phenomenology will be dominated by light techni-pions and by the techni-rho. Coming back to our e+e− accelerator, we see that some of the hadrons decay via electromagnetic interactions (as π0 → γγ) while other particles
(as π±) cannot decay directly via QED or QCD. As it is well known, the weak interactions with typical scale of order 100 GeV, mediate the decays π±→ µ±+ν.
At the collision energy scale of order GeV, the weak interactions correspond to effective non renormalizable interactions. In the same way we will see that some techni-pions are coupled to SM vectors, while other are stable because of acci-dental symmetries, that can be broken by non renormalizable operators. If these effective operators are neglected, the SM gauge interactions are the only bridge between the SM particles and the new vectorlike fermions.
The phenomenology of Vectorlike Confinement theories is very rich, in par-ticular it can give rise to good Dark Matter (DM) candidates with interesting signatures. A typical feature of renormalizable gauge theories is the presence of accidental symmetries possibly leading to stable particles that can be DM candi-dates. In the following we will pursue the idea that DM can be stable because of an accidental symmetry, exactly as the proton is stable because of baryon num-ber conservation. In particular this Chapter will be focused on the possibility
that DM is a techni-baryon: a techni-color singlet made by an anti-symmetric combination of NTC techni-quarks, with NTC the number of techni-colors.
2.2
Vectorlike Confinement models
To be concrete, we consider a SU(NTC) gauge theory (later extending the
discus-sion to SO(NTC))1 with NF Weyl fermions Qi transforming in the fundamental
of SU(NTC) plus their NF partners Qci in the anti-fundamental. In the Dirac
notation, they correspond to NF Dirac fermions with Qi playing the role of the
left-handed component and (Qci)†that of the right-handed one. In principle, the Vectorlike Confinement scenario can be extended including fundamental scalars charged under the TC gauge group: this unconventional possibility will be con-sidered in Chapter4. Here we will only consider fundamental fermions, whose strong dynamics is well known from QCD.
Neglecting mass terms and SM gauge interactions, the fundamental Lagrangian has an accidental non anomalous global symmetry SU(NF)L⊗ SU(NF)R⊗ U(1)V.
Assuming QCD-like dynamics, the strong interactions confine giving rise to con-densates hQiQcji ∼ ΛTCfTC2 δij, that spontaneously break the chiral global
sym-metry as
SU(NF)L⊗ SU(NF)R→ SU(NF)V (2.1)
at the scale ΛTC, giving rise to NF2 − 1 Nambu-Goldstone bosons (NGB). We
assume the usual large-NTC scaling [30]
ΛTC ∼ gTCfTC, gTC ∼
4π √
NTC
, (2.2)
where ΛTCis at the mass scale of the lightest vector techni-meson (the techni-rho)
and fTC is the NGB decay constant.
Now let’s consider the SM gauge interactions. The fermions in the fundamen-tal representation of the gauge group, that we will call techni-quarks (TCq), live in vectorial representations of the SM so that their condensates do not break the SM gauge group. We will use the following notation:
TCq : NS X i=1 QRi⊕ Qc Ri , (2.3)
where QRi = (NTC, Ri) is a Weyl fermion in the fundamental of the TC group
and in a generic representation of the SM and QcR
i is in the anti-fundamental of
TC and in the conjugate Rci SM representation. Here we will consider the sim-plest Ri representations that can be embedded in SU(5)GUT, listed and named in
table2.1. This is not the only possibility, but this choice might allow to include
1We do not consider Sp(N
TC) theories since they do not give rise to stable techni-baryons: the anti-symmetric combination of NTC techni-quarks decays into NTCtechni-mesons. While Sp(NTC) turns out to be not interesting for this Chapter, we will consider it in the following Chapters.
the theory in a Grand Unification scheme. Ns is the number of species with
mass below the confinement scale, and it is related to the techni-flavor number as NF=PNi=1s dim(Ri). Notice that for complex Ri we can construct two
inequiv-alent representations QRi = (NTC, Ri) and QRci = (NTC, R
c
i). Thus, including
also the mass terms and the SM gauge interactions, the fundamental Lagrangian can be written as2 L = LSM− 1 4g2TCG A µνG µν A + θTC 32π2G A µνG˜ µν A + Q † ii /DQi+ Qc †i i /DQci −miQciQi+ h.c. +LYuk, (2.4) DµQi = ∂µ+ igTCGµATA+ ig AaµTa Qi, (2.5)
where Gµ is the techni-gluon field, ˜Gµν = 12µναβGαβ and with Aµ we mean the
generic SM field that couples to QRi, depending on its quantum numbers, with
generic coupling g.
The SM Lagrangian also includes the Higgs: here we will remain agnostic about the explanation of the smallness of the electro-weak scale and assume a SM Higgs. This scenario can be seen as a low energy effective theory of a more complicated UV dynamics: an example of a fundamental theory giving rise to a composite Higgs, will be considered in Chapter4. In [31] a general study of composite DM is performed in the same framework we are considering, but adopting a specific point of view with respect to the Naturalness problem: any explicit mass term is allowed in the Lagrangian, power divergences are unphysical and all the masses arise via dimensional transmutation. In the absence of explicit mass terms, the resulting models can have the same number of free parameters as the SM, thus being very predictive (up to theoretical uncertainties in the new strong dynamics).
The topological term proportional to θTC is physical only if all the
techni-quark masses are different from zero and will play an important role in our dis-cussion below. Indeed, the θTC angle does not contribute significantly to the
neutron EDM, so that a sizable θTC<∼ 1 is allowed, see AppendixB for more details.
LYuk contains all the Yukawa couplings between the SM Higgs and the TCq
allowed by their quantum numbers. These couplings do not give problems with flavor since the theory always satisfies MFV [27]: in the limit of zero SM Yukawas, the SM has a global symmetry SU(3)3. On the other hand, the couplings be-tween the SM Higgs and the new fermions induce corrections to the electroweak precision observables, see [32]. However, this is not a phenomenological issue if the compositeness scale is sufficiently high and the couplings between the Higgs and the new fermions are small, as it would be the case in our models. Indeed, since vectorlike TCq can have mass without coupling to the Higgs, the Yukawa
2
SU(5)GUT SU(3)c SU(2)L U(1)Y name ∆b3/NTC ∆b2/NTC ∆bY/NTC 1 1 1 0 N 0 0 0 ¯ 5 ¯3 1 1/3 D 1/3 0 2/9 1 2 −1/2 L 0 1/3 1/3 10 ¯3 1 −2/3 U 1/3 0 8/9 1 1 1 E 0 0 2/3 3 2 1/6 Q 2/3 1 1/9 15 3 2 1/6 Q 2/3 1 1/9 1 3 1 T 0 4/3 2 6 1 −2/3 S 5/3 0 8/9 24 1 3 0 V 0 4/3 0 8 1 0 G 2 0 0 ¯ 3 2 5/6 X 2/3 1 25/9 1 1 0 N 0 0 0
Table 2.1: Techni-quarks are assumed to belong to fragments of SU(5)GUT
rep-resentations (plus their conjugates for complex reprep-resentations). We give the SM decomposition and list the contributions ∆bi to the SM β-function coefficients.
In the SU(NTC) models, since the techni-quarks are vectorial we have to multiply
the ∆bi by a factor of two. In the SO(NTC) models, the ∆bi must be multiplied
by a factor of two only for complex SM representations.
couplings above will have the only role of breaking some accidental global sym-metries and they can be small.
The spontaneous breaking of the chiral symmetry produces N2
F− 1 NGB, that
we will call techni-pions (TCπ). Those are QQc composite states transforming in the adjoint representation of the unbroken techni-flavor group SU(NF):
AdjSU(NF)⊕ 1 = "NS X i=1 Ri # ⊗ NS X j=1 Rcj , (2.6)
where the extra singlet is the analog of the η0in QCD. The TCq masses explicitly break the chiral symmetry so that the TCπ are not exact NGB and acquire a mass of order
m2π ∼ mQΛTC. (2.7)
TCπ carrying SM charges also receive quantum corrections by the SM gauge interactions, that explicitly break the techni-flavor symmetry, estimated as
∆gaugem2π ∼
3
4π α3C2(r3) + α2C2(r2) + αYY
2 Λ2
TC, (2.8)
where r3, r2 mean the color and weak representations which the TCπ belongs to
and Y is its hypercharge. C2(rN) is the quadratic Casimirs of the rN
represen-tation of SU(N ), equal to (N2− 1)/2N for the fundamental and to N for the adjoint. Electroweak symmetry breaking induces a further splitting between the
components of an electroweak multiplet, of order 100 MeV. In the chiral limit, we can argue that the lightest TCπ are those with the smallest SM charges, even if the introduction of the techni-quark masses can change this conclusion.
Given that each Ri⊗ Rci contains a singlet, any model predicts at least Ns
singlets, that we denote as η. Extra singlets exist if a fermion representation ap-pears with a multiplicity. The singlet associated with the generator proportional to the identity in techni-flavor space, analog to the η0in QCD, is anomalous under the SU(NTC) gauge interactions and acquires a large mass even in the limit of
massless TCq. This mass can be estimated in a large-N expansion as m2η0 ∼
NF
NTC
Λ2TC. (2.9)
Analogously to QCD baryons, techni-baryons (TCb) are techni-color singlets made of NTC TCq, so that their properties strongly depend on the details of
the model, in particular on NF and NTC. Their wave function must be
anti-symmetric in the exchange of TCq and, being a techni-color singlet, it must be also anti-symmetric in the exchange of techni-color quantum numbers. For the lightest TCb that have no orbital angular momentum, the spatial wave func-tion is symmetric, so that the states must be symmetric in spin × techni-flavor. Following [33], we can find the totally symmetric representations of spin and techni-flavor using the embedding SU(2)spin⊗ SU(NF) ⊂ SU(2NF). TCb are
composed of NTC fermions and transform as the completely symmetric
represen-tation made by NTC copies of 2NF representations. Using Young tableaux, we
here list some examples
NTC = 3, NF = 3 : = 56 of SU(6) ,
NTC = 3, NF = 4 : = 120 of SU(8) ,
NTC = 4, NF = 3 : = 126 of SU(6) .
The task is finding which tensor product of SU(2)spin and SU(NF)
representa-tions made with NTC boxes contains the completely symmetric representation
of SU(2NF). We realize that the completely symmetric representations above
appear when the Young tableaux for spin and techni-flavor are the same, since in this case any pair of SU(2)spin and SU(NF) indices transforms in the same
way under permutations. Simply listing the Young tableaux with NTC boxes of
SU(2)spin and SU(NF) we conclude that the TCb live in the following
represen-tations: NTC = 3, NF = 3 : ⊗ | {z } spin 1/2, 8 of SU(3)F ⊕ ⊗ | {z } spin 3/2, 10 of SU(3)F , NTC = 3, NF = 4 : ⊗ | {z } spin 1/2, 20 of SU(4)F ⊕ ⊗ | {z } spin 3/2, 2000of SU(4)F ,
NTC = 4, NF = 3 : ⊗ | {z } spin 0, 6 of SU(3)F ⊕ ⊗ | {z } spin 3, 15 of SU(3)F ⊕ ⊗ | {z } spin 2, 150of SU(3) F .
For an even (odd) number of boxes, the SU(2)spin always contains the spin 0
(spin 1/2) representation. Assuming that the lightest multiplet is the one with lowest spin, we conclude that the lightest TCb must belong to the representation of SU(NF) with two rows and NTC/2 boxes in each row for NTC even, (NTC+
1)/2 boxes in the first row and (NTC − 1)/2 in the second row for NTC odd.
Interestingly, TCb are fermions for NTC odd and bosons for NTC even. The case
NF = 1 is special since techni-flavor cannot be anti-symmetrized, so that TCb
have spin NTC/2.
Differently from techni-pions, TCb get a common mass from the new strong interaction that, assuming the large-N scaling, can be estimated as
mB∼ NTCΛTC, (2.10)
and the mass splitting between flavor multiplets is expected to be of order ΛTC.
Then the techni-flavor multiplets are split by the SM gauge interactions and the TCq masses (such as by Yukawa couplings if allowed). For TCπ, we ar-gued that in the chiral limit the lightest states are those with the smallest SM charges, the same is not rigorously proved for TCb. However, the electromagnetic splitting of baryons in QCD hints to the fact that the lightest states are those with the lowest SM charges and we will make this assumption in the following. The splitting between SM representations is of order ΛTC/100 or larger, then
electroweak symmetry breaking induces a small splitting (of order 100 MeV) be-tween the components of each electroweak multiplet. Again, a hierarchy in the techni-quark masses can change our conclusion.
Heavier TCb usually decay into a lighter TCb and a TCπ, but it could hap-pen that a higher spin TCb is stable due to an accidental symmetry and if it is the lightest state charged under this symmetry, it can be a DM candidate. This can happen if any of the TCq masses is comparable to ΛTC. In QCD, the
strangeness conserving decay Ω−(sss) → Ξ0(ssu) + K−(¯us) is kinematically for-bidden so that in absence of s-number violating weak interactions (analog to our non renormalizable interactions), the spin-3/2 Ω− baryon would be stable.
The decomposition of a techni-flavor multiplet of TCb under the SM group is non trivial (the technicalities are described in AppendixA) and requires group theory computations that we performed with the help of the Mathematica package Lie Art [34].
2.2.1 Accidental symmetries
The scenario presented above is characterized by accidental global symmetries that can give rise to stable DM candidates.
Species number: SM gauge interactions explicitly break the techni-flavor sym-metry, but leave unbroken the U(1)1⊗ · · · ⊗ U(1)NSspecies numbers, that rotate
each TCq with a different phase. As a consequence, the lightest TCπ with neat species number, that is QiQcj (i 6= j), is stable due to this accidental symmetry.
It cannot decay into SM particles unless some non renormalizable operator, such as a four fermion operator, breaks the species number. Analogously, in low en-ergy QCD, the π± would be stable if weak interactions were neglected, because of u-number and d-number symmetry. Also the lightest TCb made of different species can be stable if species number conserving decays into a TCb and a TCπ are kinematically forbidden.
Techni-baryon number: The Lagrangian is accidentally symmetric under a U(1)TB global symmetry that rotates all the techni-quarks with the same phase.
TCb are charged under this symmetry and the lightest one is thus stable. On the contrary, TCπ have zero techni-baryon number since they are QQc states. In some sense, as we will see, techni-baryon number is more robust than species number. As a consequence, it is particularly interesting to consider TCb as DM candidates.
G-parity3: In models where the techni-quarks belong to an SU(2)L
represen-tation with zero hypercharge, an analog of G-parity symmetry in QCD can be defined [36]. This symmetry acts on techni-quarks as
Q → eiπT2Qc, (2.11)
corresponding to a rotation around the second axis in the isospin space combined with charge conjugation. This transformation replaces any SU(2)L
representa-tion with its conjugate, that is equivalent to the original one since the SU(2) group is pseudo-real. G-parity acts on TCπ so that those with even (odd) isospin are even (odd) under G-parity, see [36] for details. SM states are left invariant under this new symmetry, so that the lightest G-odd TCπ is accidentally stable. This symmetry can be defined for any model where only real or pseudo-real rep-resentations are included, such as models with TCq in any SU(2)Lrepresentation
or in the adjoint of SU(3)cwith vanishing hypercharge. G-parity was introduced
in [36] for a SU(N ) gauge theory, but it can be easily extended to SO(N ) or Sp(N ) theories.
3In QCD G-parity allows to extend the concept of C-parity as charge conjugation eigenvalue to charged states, when isospin is a valid symmetry [35].
The symmetries above can be broken by various effects. When quantum num-bers allow for a Yukawa couplings QciHQj, both the species number symmetry
and G-parity are violated. In such a case otherwise stable states decay through the Higgs so that they cannot be DM candidates. Notice that Yukawa couplings preserve techni-baryon number. Species number can also be broken with ad-hoc model building, that is by adding new scalars with quantum numbers such that Yukawa couplings arise.
In a standard effective field theory approach, the symmetries above can be broken by non renormalizable operators arising from unspecified UV physics or from integrating out heavy degrees of freedom.4 G-parity and species number can be broken by dimension-5 operators such as
1 ΛQQ cHH , 1 ΛQ cσµνB µνQ , (2.12)
where G-parity acts as a Z2 symmetry on QQc and species number is broken by
QiQcj (with i 6= j) operators. The scale Λ represents the scale of the unknown UV
physics giving rise to these operators. Such operators would give a TCπ lifetime of order τπ ≈ 8πΛ2 m3 π ≈ 105s Λ ΛPl 2 1 TeV mπ 3 , (2.13)
that is much lower than the lifetime of the Universe (∼ 1017s). On the other hand, techni-baryon number violation requires dimension-6 (or higher) opera-tors, depending on quantum numbers. Suppose that we can write a dimension-6 operator that breaks techni-baryon number, of the form
1
Λ2QQQQ , (2.14)
the TCb turns out to be still stable on cosmological scales. More explicitly, assuming the typical mass of a TCb DM candidate to be of order 100 TeV, as we will see in section2.4, we get
τB≈ 8πΛ4 m5B ≈ 10 25s Λ ΛPl 4 100 TeV mB 5 , (2.15) that is compatible with the bounds from indirect detection searches [37]. From an effective field theory point of view, we can argue that techni-baryon number is more robust than species number and G-parity, in this sense TCb seems to be the most promising DM candidates.
In the following of this Chapter we will pursue this idea, scanning all possible phenomenologically allowed models that can give good TCb DM candidates. This will be done first in the framework of SU(NTC) Vectorlike Confinement models,
then the discussion will be extended to SO(NTC) models.
4Notice that higher dimensional operators violate flavor in general, but if the cutoff scale is sufficiently high, as it will be the case in our discussion, there are no phenomenological problems.
