Composite Dark Matter from Strongly Interacting Chiral Dynamics

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FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI Corso di Laurea Magistrale in Fisica Teorica

Composite Dark Matter from Strongly-Interacting Chiral

Dynamics

Tesi di Laurea Magistrale in Fisica Teorica

Relatore:

Prof. Roberto Contino Relatore Interno:

Prof. Enore Guadagnini

Candidato: Filippo Revello

19/09/2018

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Introduction

The nature of dark matter, whose existence has been inferred indirectly from a large variety of as-trophysical and cosmological observations, is arguably one of the greatest mysteries in fundamental physics today. So far, many alternatives have been suggested as a possible solution, ranging from modified theories of gravity to beyond the Standard Model physics or primordial black holes, but un-fortunately none of them has received any compelling empirical justification. Amongst them, particle dark matter seems particularly well motivated from a theoretical point of view, since it could also help to shed some light on the open issues in the Standard Model, such as the strong CP problem, grand unification or supersymmetry.

An intriguing proposal is that dark matter may be realized as a composite state of a strongly interacting dynamics, coming from a non abelian-gauge theory which confines in the infrared. This idea is particularly appealing, since it can account for many of the properties of dark matter in a conceptually economical way, mimicking some already well-known mechanisms in particle physics. For example, adopting an effective field theory point of view, we see that neglecting the non-renormalizable interactions at low energies can enforce some accidental symmetries in our description, which may naturally give rise to the stability of the lightest particle in each multiplet. This is exactly what happens in the Standard Model for baryon number, forbidding proton decay. Since all of these theories do not aim to address electroweak symmetry breaking and related issues, an important fact to keep in mind is that the dark condensation must not break the Standard Model gauge group, and this is what distinguishes them, for example, from Technicolor or Composite Higgs. Up to now, most of the work has focused on Vector-like Confinement 1 _{scenarios, which were first considered in the context} of LHC Phenomenology [1] and later applied to dark matter model building. Their structure provides significant simplifications with respect to the general case, since anomaly cancellation constraints are trivially satisfied and the real representation2 _{enables one to conjecture the existence of a global} symmetry breaking pattern in the dark sector [2] which preserves the electroweak group. In [3], all possible models consistent with a SU(5) unification of SM gauge forces have been classified, showing that many of them are compatible with current bounds and may exhibit characteristic detection

1_{A model is called Vector-like if, for any set of fields transforming under a certain representation of the gauge group,}

a corresponding set of fields transforming under the conjugate representation is also present (or alternatively, the model can be constructed out of Dirac fermions only)

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signatures. It is then relevant to ask if the same framework can be successfully applied to chiral models, and whether new phenomenological features can be obtained this way. By chiral we mean that the new fermions transform in a complex representation of the gauge group, with the immediate effect that mass terms are forbidden by gauge invariance. This hypothesis appears to possess some elegant features, and to emerge naturally in the context of physics beyond the Standard Model:

• Since mass terms for the dark fermions are not allowed, there is no need for any new, artificial scale. The only relevant one is generated dynamically through dimensional transmutation, and it is what determines the spectrum of the composite states.

• Chiral theories play an important role in our understanding of particle physics, with the Standard Model itself being the most obvious example.

• If the Standard Model interactions unify in a single gauge group at high energies, the resulting theory must be chiral. The prototypical GUT is the famous SU(5) [4], where each generation of SM matter fields is embedded into a ¯5 and a 10. While it is certainly possible to add vector-like fermions transforming in fragments of chiral GUT representations, one must then introduce an additional mechanism to explain why the other members of the multiplets have acquired a much higher mass. This is analogous to the doublet-triplet splitting problem of Grand Unified Theories (See [5] for a review).

For these reasons, chiral models are often thought to be more fundamental; an instructive example can be found in the vector-like QED-QCD system, which finds the explanation of its structure and mass scales in a chiral UV completion, the SM. Chiral models 3 _{have been considered in [}₆_,₇_{], where the} possibility of a dark, non-anomalous U(1) spontaneously broken by a Higgs mechanism is examined, and also in the context of the mirror world scenario [8]. In addition, some algorithmic methods to construct chiral, anomaly-free extensions of the Standard Model have been devised in [9,10].

An interesting example was recently considered in [11,12]4_{, where the authors introduced a composite} chiral model with four dark fermions charged under the gauge group SU(N)DC⊗ U(1)D5. The theory

confines in the infrared, breaking U(1)D spontaneously (thus giving the dark photon a mass) and

forming dark pions and dark baryons, which are bound states exactly analogous to those of QCD. Due to the existence of a global, accidental U(1)V symmetry in the renormalizable lagrangian, the

two dark pions are stable and provide a viable dark matter candidate. Interactions with the Standard Model are solely achieved by means of a mixing between the dark and the standard photon [13], which is expected to happen on quite general grounds. The purpose of the thesis is to search for possible extensions of this framework, and examine their phenomenological implications. A possible general-ization involves the assignment of charges under the SM gauge groups, adding new portals between the dark sector and SM; this could potentially be very relevant for collider experiments, and also for direct detection. In fact, there are no previous examples of a composite chiral model where the dark

3_{But not necessarily composite} 4

From now onwards, we will refer to it as the CHN model, from the initials of the authors

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constituents are not singlets with respect to the SM.

More specifically, we identify a new scenario in which the dark fermions also transform as repre-sentations of SU(2)L, resulting in a non-trivial dark sector characterized by a rich and distinctive

phenomenology. Since the model exhibits an approximate SU(4)L⊗SU(4)R⊗U(1)Bsymmetry

(spon-taneously broken to SU(4)V ⊗ U(1)B), the low energy spectrum of the theory contains 15 Pseudo

Nambu-Goldstone bosons (NGBs), one of which is eaten by the dark photon due to the spontaneous breaking of U(1)D induced by condensation in the dark sector. In addition to the U(1)V symmetry

described above, which is still present, we point out the existence of a discrete symmetry analogous to charge conjugation, CD, with important implications for processes involving a dark photon.

The model satisfies all the necessary requirements to provide viable dark matter candidates: • The cancellation of gauge anomalies is still guaranteed

• Landau poles below the GUT scale are avoided for a reasonable range in the choices of the dark charge eD and the number of dark colours NDC

• The symmetry breaking pattern is under control, preserving the weak gauge group SU(2)L

• A calculation of the pion effective potential shows that the explicit breaking induced by the gauge interactions is sufficient for each of them to obtain a mass, which can be calculated under the assumption of QCD-like sum rules.

More specifically, we identify a new scenario in which the dark fermions also transform as repre-sentations of SU(2)L, resulting in a non trivial dark sector characterized by a rich and distinctive

phenomenology. Since the model exhibits an approximate SU(4)L⊗SU(4)R⊗U(1)Bsymmetry

(spon-taneously broken to SU(4)V ⊗ U(1)B), the low energy spectrum of the theory contains 15 Pseudo

Nambu-Goldstone bosons (NGBs), one of which is eaten by the dark photon due to the spontaneous breaking of U(1)D induced by condensation in the dark sector. In addition to the U(1)V symmetry

described above, which is still present, we point out the existence of a discrete symmetry analogous to charge conjugation, CD, with important implications for processes involving a dark photon.

The model satisifies all the necessary requirements to provide viable dark matter candidates: • The cancellation of gauge anomalies is still guaranteed

• Landau poles below the GUT scale are avoided for a reasonable range in the choices of the dark charge eD and the number of dark colours NDC

• The symmetry breaking pattern is under control, preserving the weak gauge group SU(2)L

• A calculation of the pion effective potential shows that the explicit breaking induced by the gauge interactions is sufficient for each of them to obtain a mass, which can be calculated under the assumption of QCD-like sum rules.

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We write down an appropriate chiral lagrangian for the dark pions containing all the interactions between themselves and with the gauge bosons, thus making the model very predictive. An inspection of these interactions enables one to conclude that only the lightest states charged under U(1)D are

stable and electromagnetically neutral, and to analyze all the decay channels in detail. We also discuss the possibility of direct detection and analyze the question of collider searches. In particular, we calculate the cross section for the production of electroweak-charged NGBs at the LHC and use it to set bounds on the parameters of the dark sector by comparing with actual data. As regards Cosmology and Astrophysics, we derive constraints assuming the correct relic abundance is obtained through a standard thermal freeze-out proceeding via the annihilation channel

DM DM → γDγD (1)

and examine the possible bounds coming from indirect detection. The material is organized as follows:

• Chapter 1 provides an introduction to the problem of Dark Matter, including observational evidence for its existence and possible strategies for discovery.

• Chapter 2 deals with Composite Dark Matter, with a particular focus on chiral models.

• Chapter 3 describes the construction of our model, and its analysis from a theoretical point of view.

• Chapter 4 deals with the generic phenomenological features of the model, with an emphasis on astrophysical and cosmological aspects

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Chapter 1

The Dark Matter paradigm

Over the last 40 years, a diverse and vast amount of astrophysical evidence has accumulated to point towards the existence of an unidentified (and otherwise undetectable) form of matter in the universe. Up to now, this hypothetical form of matter, named Dark Matter (DM) because of its supposed inability to interact with light, has only been observed to interact gravitationally and over very large distances, ranging from galactic to cosmological scales. Furthermore, its origin and nature are still a mystery, and very little is known about its microscopic properties. Although many different explanations, such as modified theories of gravity (e.g. Modified Newtonian Dynamics (MOND) [14], and its attempted covariant formulations [15]), have been proposed to account for the observed anomalies, most of these seem to be rather ad hoc hypotheses and usually succeed in explaining only a few of the phenomenological manifestations commonly attributed to the presence of DM. Therefore, the ”new particle” explanation still remains the most accredited as of today, and an intense effort has been devoted to unveil its fundamental nature, from both the theoretical and experimental sides. Historically, this interpretation has also been supported by the fact that some of the most well motivated extensions to the Standard Model of Particle Physics naturally contain dark matter candidates, even though many of them have now been excluded by data. The aim of this chapter is to give a brief review on the observations which have lead to suggest its existence, along with some the implications these may carry for the structure of DM on a fundamental level. Useful reviews can be found in [16–20], both on general and selected topics.

1.1 Astrophysical tests

From a historical perspective, the first to suggest1 _{the presence of an invisible form of matter was} the Swiss astronomer F. Zwicky [21], who in 1933 came up with a simple way to estimate the mass content of a given galaxy cluster by observing the motion of its internal components. Using just the virial theorem from Classical Mechanics, he was able to argue that the total mass of a cluster may be

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calculated with the expression

Mtot = 5R totv2

3GN (1.1)

where v is the velocity of an individual galaxy inside the cluster, and the double bar indicates a double average, both over time and galaxy velocity. Rtot is the visible radius of the galaxy. He was then able

to derive the mass-to-light ratio for the Coma cluster, which is defined as the quotient between the total mass contained in given volume and its luminosity:

Υ ≡ Mtot

Ltot (1.2)

and is expressed in solar units Υ. Although his estimates were influenced by a number of imprecise

inputs, his result of Υcoma ∼ 500 is quite close to the presently measured values, which asymptote

at around Υcoma ∼ 400 as the cluster size increases. Of course, this by itself does not establish the

existence of DM, since we now know that the vast majority of the baryonic matter in the universe is not contained inside stars. However, the baryon mass density can be measured independently, and it turns out there is a sizeable mass fraction which cannot be attributed to known forms of matter in the universe. With the aid of more modern and accurate techniques, similar methods are used today to estimate the matter content of the universe. In [22], for example, cosmological large-scale simulations for the mass-to-light ratios are compared with observations to derive

Ωm= 0.16 ± 0.05 (1.3)

Although there are more precise ways to measure this quantity (see below), it is reassuring to find that very different methods all give compatible results.

The first unambiguous type of evidence for the existence of DM came from the observations of galactic rotation curves through the Doppler shift of the hydrogen 21 cm line, enabled by the advent of the first radio telescopes in the early 70’s. For the first time, it was possible to extract indirect information on the internal mass distribution of spiral galaxies up to large radii by mapping the velocity of the interstellar gas, whereas optical instruments could only allow to perform studies of the inner, more luminous regions. It is a well-known result in Classical Mechanics that the velocity of a particle orbiting around outside a spherically symmetric body obeys

v(r) =

s

GNM(r)

r (1.4)

where M(r) is the total mass contained in a spherical shell of radius R. Assuming a homogeneous distribution with density ρ inside a galaxy of radius r, one would therefore expect to observe the following behaviour:    v(r) =q4πGNρ 3 r if r ≤ R v(r) = q GM r if r > R (1.5) These assumptions are somewhat oversimplifying, but the linear behaviour at small r and the so-called Keplerian fall-off (∝ r−1

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cilindrical symmetry is present. However, the experimental results only show the former of these two characteristic, and the velocity dependence actually asymptotes to a constant value at large radii, as shown in Fig. 1.1. Assuming the laws of gravitation to be valid on galactic scales, one is therefore

Figure 1.1: Rotation curves for 7 galaxies of different Hubble type, clearly showing the flat behaviour at large distances. Image taken from [23]

forced to admit the existence of dark halos of gravitationally-interacting only matter which extend well beyond above the visible galaxy radius, with a density2

ρ(r) = v

2

c

4πGNr2 (1.6)

On larger scales, another technique which can be used to probe the matter content of a given system is that of gravitational lensing. As predicted by GR more than a hundred years ago, strong gravita-tional fields can distort the trajectories of light, and thus the images of distant sources registered by telescopes on Earth can be distorted by the presence of massive objects on the line of sight, such as cluster of galaxies. Most importantly, this method is sensitive to the total mass of the system under exam, and it can be used to make a comparison with other astrophysical observations which are only able to account for the visible mass. A spectacular example of this multi-technique approach is the famous Bullet Cluster (1E0657-558), which consists of a pair of merging galaxy clusters (hence the name). Through the study of the thermal X-ray emission of the interstellar gas, it is possible to dis-entangle the distribution of the hot baryon components from that of the whole clusters, reconstructed through gravitational lensing. The result, depicted in Fig. 1.2, clearly shows a displacement between the centres of the two mass distributions, suggesting that the two might have experienced different interactions during the collision. Unlike the thermal gas, the dark halos exhibit an almost collisionless

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Figure 1.2: Superposition of the X-ray image of the Bullet Cluster (1E0657-558) with the gravitational potential profiles derived through weak gravitational lensing. The offset between the galaxy mass distribution (whose predominant component is assumed to be dark matter) with respect to that of baryonic matter is particularly evident.

behaviour, which also allows one to set an upper bound on the dark matter self interactions through hydrodynamical simulations [24–26]: σsi M . 1gcm −2_{= 1.8} mb GeV = 4580 GeV3 (1.7)

Crucially, this observation is particularly difficult to justify with modified theories of gravity, in which the opposite behaviour would be expected.

1.2 Cosmology

Strong indications for the existence of dark matter also come from the study of cosmology and large-scale structure formation. The currently accepted ΛCDM cosmological model relies on the existence

of a cold, non-baryonic form of matter in order to be able to fit all the observations, allowing for the most precise determination of its relic density. To this end, the finest probe available today is the Cosmic Microwave Background (CMB), an almost uniform blackbody radiation composed of photons which decoupled from ordinary matter at the time of recombination3 _{and have been free-streaming} in the universe since then. Because the baryon-photon fluid before recombination was very strongly coupled through Thomson scattering, the CMB temperature inhomogeneities carry an imprint of the matter density perturbations in the early universe.

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The amplitudes and the position of peaks in the anisotropy power spectrum of the CMB are sensitive to the baryonic and total matter densities in the universe at recombination separately, and these can be easily evolved to the present day value with

Ωb/mρcrit= ρb/ma3 (1.8)

The most precise measurements are those performed by the Planck collaboration [27], with the result Ωbh2 = 0.02225 ± 0.00023 ΩDMh2 = 0.1198 ± 0.0015 (1.9)

where the dimensionless Hubble parameter h, defined as

H0= h km s−1Mpc−1 (1.10)

is introduced to eliminate uncertainties coming from the determination of the Hubble constant. The latest measurement from Planck gives

h= 0.677 ± 0.04 (1.11)

so in practice h2 _' _{0.5. Up to a very good approximation, the dark matter relic density is simply}

obtained as the difference between the matter and baryonic ones, since all other SM particles give a negligible contribution.

ΩDM ≡Ωm−Ωb (1.12)

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As a confirmation of these results, compatible values for Ωb and Ωm have been determined

inde-pendently at different length scales and at different epochs in the cosmic evolution, even though with much larger errors. For Ωb, these include:

• X-ray observations of interstellar gas inside galaxies point towards Ωbh2∼0.02

• Luminosity measurements of distant quasars allow one to compute the line of sight column density by looking at how much light is absorbed on the way to Earth, which also gives Ωbh2 ∼

0.02

• Comparing the deuterium production predicted by the theory of Big Bang Nucleosynthesis (BBN) with the observed abundance. With this technique, the authors of [28] were able to derive

Ωbh2= 0.0214 ± 0.002 (1.13)

As regards the total matter density, the following techniques can also be used • Gravitational lensing

• Galaxy surveys directly probe the large-scale structure of the universe, measuring the matter power spectrum. The 2dF Galaxy Redshift Survey [29] obtained

Ωm = 0.231 ± 0.021 Ωb/Ωm= 0.185 ± 0.046 (1.14)

while the Sloan Digital Sky Survey (SDSS) [30] found

Ωm= 0.286 ± 0.018 (1.15)

• Mass-to-light ratio measurements, as described in the previous section

1.3 Non-gravitational Dark Matter searches

On the experimental side, a wide range of unsuccessful dark matter searches has been carried out in the last decades, imposing strong bounds on the accessible parameter space for many candidate theories. As a result, there is now a wide range of constraints which any new model must be able to evade in order to be considered as a viable alternative.

They can be schematically divided into three categories, as shown in Fig. 1.4: • Direct detection in nuclear experiments

• Indirect astrophysical observations • Collider searches

In the following, we will present a brief review of all three methods, discussing advantages and weak-nesses.

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Figure 1.4: Schematic illustration of the different kinds of particle dark matter searches. For each of these, the direction of the arrows show the prototypical process under study.

Direct detection

Since our galaxy itself is expected to be immersed in DM halo, a promising strategy is to look for interactions between the dark matter wind and detectors placed on Earth. The most common tech-nique involves searching for elastic scattering between WIMPS and target atomic nuclei, through the measurement of their recoil energy spectra. The main limitations in sensitivity are provided by the small DM density, radioactive backgrounds and by the maximal attainable recoil energy which, for a fixed energy threshold Eth, implies a lower bound on the minimum detectable dark matter mass.

The expected differential rate both depends on the microscopic characteristics of the DM candidate (nucleon-DM cross section, mass, etc) and on the knowledge of its macroscopic astrophysical proper-ties, such as the density ρ and the velocity distribution f(v):

dN dER = NN ρ mDM Z vmax √ (mNEth)/(2µ2) dσ dER vf(v) dv (1.16)

It is interesting to note that some of these dependences can be used to distinguish a genuine signal from systematic or backgrounds effects, for example exploiting the diurnal and annual modulations of the dark matter wind velocities with respect to earth, which directly reflect into the event rates predicted by Eq. 1.16. As regards the cross sections, two distinct behaviours can be found in the non-relativistic regime. If the nucleon scattering amplitudes add coherently, the cross section is said to be spin-independent and

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Figure 1.5: Spin-dependent and spin-independent WIMP-nucleon cross section bounds from various experiments, taken from [31]. The closed contours show the claims of positive discoveries, although being still highly controversial. At the bottom of the figure the neutrino floor is also shown.

This effect can be used both to increase the expected event rate and to discriminate against background events by using targets with a different atomic number A. On the other hand, if DM couples to nucleon spin the amplitudes add incoherently, and

σDM −N ' σDM −n (1.18)

where σDM −n also contains a J(J + 1) spin factor. This last case is known as spin-dependent in the

literature. Finally, an effect worth mentioning is the so-called neutrino floor, an irreducible source of background due to the scattering between nuclei and cosmic neutrinos, which is however far from present day sensitivities. A combined plot showing the bounds in the σ − MDM plane from all the

main experiments is presented in Fig. 1.5.

Indirect detection

These studies concern the possibility of inferring the existence of dark matter through astrophysical effects which are non-gravitational in nature, mainly through the emission of visible SM particles which can be detected by telescopes. On a microscopic level, the two typical processes which can give measurable effects are

• Dark matter decays into the SM, which can occur if DM is not absolutely stable but only long lived.

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• Pair annihilations into SM particles. In the simplest thermal freeze-out scenario, this is the same process determining the DM relic abundance.

In spite of the smallness of the individual decay/annihilation rates, it is possible to observe sizeable signals by observing regions where the DM density is thought to be particularly high, such as galaxy clusters or in the centers of galaxies (including our own). On the downside, there are also large sources of astrophysical uncertainties involved in the determination of the expected rates, even in the simplest circumstances. As an example, let us consider neutral4 _{particles (essentially photons and neutrinos);} they travel in straight lines (in flat spacetime) and can be easily traced backed to their sources, making them the ideal probe. However, they are still influenced by

• The absorption factor between source and observer, determined by the optical depth τ(z, E) along the line of sight

• The local dark matter density ρ(θ, ϕ, z), whose unknown distribution introduces intrinsic uncer-tainties

• The redshift z of the source, if extragalactic

In formulas, the expected SM differential fluxes per unit energy for annihilation and decay can be calculated as dNann dE dA dt = Z _dΩ 4π Z dz e−τ (z,E) _dN0 dE0 E0_=E(1+z) ρ(θ, ϕ, z)2 1 H(z)(1 + z)3 hσvi 2m2 DM (1.19) and dNdec dE dA dt = Z _dΩ 4π Z dz e−τ (z,E) _dN0 dE0 E0_=E(1+z) ρ(θ, ϕ, z) 1 H(z)(1 + z)3 ΓDM mDM (1.20)

In the case of cosmic rays, the situation is even more difficult since their propagation inside galaxies is governed by a diffusion equation dependent on many different effects (convection, diffusive reaccela-ration, decays and fragmentations just to name a few). Moreover, all the possible signals described in this section suffer from the presence of large backgrounds. For photons, background processes can vary from neutral pion decays, pulsar synchrotron radiation, thermal gas emission and the CMB depending on the energy range. As a final side remark, it is interesting to note how some observable effect may be generated on a macroscopic level, such as in supernova/star cooling [32] or even in the physics of the solar system [33].

Collider searches

Accelerators are used to hunt for dark matter production through their coupling to SM particles, especially at the LHC. The fundamental reason to believe in the existence of a large enough matter-DM interaction enabling a measurable signal is that the most accredited matter-DM production method in the

4

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Figure 1.6: Schematic illustration of the main indirect detection experiments and their energy reach. Reading from top to bottom, neutrino telescopes,cosmic-ray detectors, photon telescopes in the radio and microwave bands, and photon telescopes in the hard UV, X-ray and gamma-ray bands. Image adapted from [20]

early universe, thermal freeze-out, requires DM to have an annihilation cross section into SM particles of the order

hσannvi '3 × 10.26cm3/s (1.21)

It is therefore reasonable to imagine that the inverse process could take place with a sizeable strength, with two SM particles fusing into a DM particle-antiparticle pair in a collision, as illustrated in Fig.

1.7. With respect to the other methods discussed above, collider searches have many advantages, which include the ability to control, simulate and measure all backgrounds and systematics, the large achievable luminosities and the possibility to measure kinematic distributions of the outgoing particles. Since DM is expected to interact very weakly within the detectors, the most promising avenues for

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discovery include missing energy measurements5 _{and/or eventual decay products. In the case of} missing energy searches, the DM particles have to be produced in conjunction with other SM states to be reconstructed, and for this reason one usually talks of Mono-X 6 _{searches, where X is usually} taken to be a jet or an electroweak gauge boson. Another important aspect to keep in mind for LHC searches is the trigger; since only a small fraction of the events can be recorded, trigger algorithms are employed to discriminate whether an event has any chance of being interesting, rejecting all others. This is particularly challenging, since the LHC has many different scopes other than DM detection, and it is difficult to optimize the algorithms for all the different signatures of putative New Physics Models. As concerns our discussion, it is worth mentioning that all events (above a certain energy) containing either leptons, photons or missing energy are usually kept, but we refer to the appropriate literature for a detailed description of all the experimental cuts.

Figure 1.7: Depiction of the connection between thermal freeze-out and collider production of a DM candidate. Picture taken from [19]

For a wide class of models, the most stringent constraints come from the LHC, which can reach the highest center of mass energies √s= 14 TeV. In order to extract quantitative information about the unknown DM particle properties two complementary approaches can be adopted, either top-down or bottom-up. As regards the latter, the minimal hypothesis is to assume the DM candidate to be the only accessible state in the dark sector, whose low energy interactions can be parametrized by an Effective Field Theory (EFT) where all the irrelevant degrees of freedom have been integrated out. A study of all the possible low dimensional operators can be used to derive constraints on the candidate mass and the new physics scale [34,35]. Increasing the degree of sophistication, we encounter a possibility commonly referred to as Simplified Models [36–39], which also assumes the existence of a new mediator between the SM and DM sectors. In spite of its generality, the bottom-up approach may be too simplified7 _{to account for all the phenomenological features of DM, and in fact there exist} some complete models which cannot be mapped to these simple ones. As a result, full models should be considered to exhaust the range of possibilities and to obtain more accurate (but also more specific) bounds. They can also be better motivated from a theoretical perspective, since UV-complete models can aim to address other open questions in Particle Physics; examples include Supersymmetry, extra dimensions (related to the hierarchy problem) or axions (strong CP problem).

5

The relevant quantity for hadron colliders is actually the missing transverse energyET, since the longitudinal

components of the partons’ initial momenta are unknown. See Chapter 5 for a discussion.

6

Adding more final states usually lowers the cross section

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Chapter 2

Composite Dark Matter and BSM Physics

2.1 The Standard Model as an Effective Field Theory

The Standard Model of Particle Physics (SM) has provided a description of the microscopic world with an unprecedented degree of accuracy and has received countless experimental confirmations, resisting numerous attempts to test the limit of its validity. From a mathematical point of view, it is formulated in the language of Quantum Field Theory, based on the gauge group

GSM= SU(3)c⊗ SU(2)EW⊗ U(1)Y (2.1)

The elementary constituents are the gauge bosons, the fermionic matter fields and the scalar Higgs field. which together account for the properties of all the particles that are observed in Nature. The fermionic matter fields can be further divided into leptons, singlets under SU(3)c, and quarks, which

lie in the fundamental of SU(3) and cannot be observed as free particles, since they confine into hadrons. They all fill irreducible representations of GSM, with the quantum numbers shown in table

2.1.

Field SU(3)c SU(2)W U(1)Y

QL 3 2 +1₆ uR 3 1 +2₃ dR 3 1 −1₃ LL 1 2 −1₂ eR 1 1 −1 νR 1 1 0 H 1 2 +1 2

Table 2.1: Standard model particle content, classified as irreducible representations of GSM. Each

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In addition, the Electroweak sector undergoes spontaneous symmetry breaking

SU(2)EW ⊗ U(1)Y → U(1)EM (2.2)

and three gauge bosons (W±_,_{Z) acquire mass, while the photon remains massless. While many}

alternatives were proposed to account for this mechanism, the 2012 discovery of the Higgs boson by the ATLAS and CMS experiments [40,41] established it as due to the destabilization of the vacuum by the Higgs scalar potential, which is the simplest, explicit realization of this idea. Incidentally, this specific model also provides a way to give mass to the fermions, through the Yukawa couplings to the Higgs. Then, the Standard Model lagrangian LSM is uniquely determined as the most general

structure including the above particles and compatible with the following principles: • Lorentz Invariance

• Locality

• Gauge invariance • Renormalizability

L_SM = L_gauge+ L_{f ermions}+ L_{Y ukawa}+ L_Higgs (2.3) The first three are universally recognized as fundamental principles of Physics, but the nature of the last one needs to be clarified. From a modern perspective, renormalizability per se is not seen as fundamental property of Nature, but rather as a consequence of the UV suppression of the higher-order, non-renormalizable operators. The Standard Model lagrangian, from an EFT point of view, is the renormalizable part of the infinite series

L= L_SM+ X

i,n≤4

C_i(n)O_i(n), [O_i(n)] = n (2.4) where irrelevant operators, appearing for n ≥ 4, are suppressed by the supposed scale of New Physics ΛUV as

C_i(n)∼ 1 ΛUVn−4

(2.5) Then, a natural explanation for the spectacular accuracy achieved by SM computations is the assump-tion that the scale ΛUV be much higher than the highest possible energies available in present-day

experiments. This viewpoint is supported by the fact that there are a number of experimental hints pointing towards extensions of the theory:

• Experiments on neutrino oscillations have shown that at least two of them have non zero masses, which cannot be accounted for in the renormalizable Standard Model lagrangian (assuming right handed neutrinos don’t exist).

• The existence of dark matter seems now well established, from a large variety of indirect obser-vations. See Chapter 1 for a short review.

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• There is a large asymmetry in the cosmological distribution of matter (mostly baryons) over that of anti-matter, quantified by the experimentally measured photon-to baryon ratio

nB− n_B¯

nγ

∼10−10 (2.6)

Although any models have been proposed to address the issue of baryogenesis in the early universe, the large majority of them requires physics beyond the SM.

Furthermore, there are also some pressing theoretical problems still in need of a solution: • The coefficient of the (CP-violating) theta term

LCP =

ϑ 32π2ε

µνρσ_G

µνGρσ (2.7)

in QCD is constrained to be extremely small, by measurements of the neutron electric dipole moment. There is currently no universally accepted explanation for this fact, which is known as the Strong CP problem.

• The observed value of the Higgs mass requires an extraordinary tuning in the parameters of the high energy theory, since it is quadratically sensitive to any scale of new physics. This is known as the Hierarchy Problem.

• The measured value of the cosmological constant is unnaturally small, much lower (by a factor O(10120_{)) than what would be expected on the basis of naive, zero point energy estimates. This}

can also be thought of as a hierarchy problem.

• Gravity can be introduced in the same framework, but the resulting theory is non-renormalizable: a UV completion is necessary.

• The apparent unification of gauge couplings at high energies has induced speculations on Grand Unified Theory (GUT) scenarios.

• An explanation of the pattern of the Yukawa couplings to the fermions is still lacking.

2.2 Composite Dark Matter

2.2.1 Preliminaries

For the sake of the discussion, it will be useful to give a short overview on the concept of real and complex representations, along with some of the implications these carry for model building. Given a representation r of a group G, it is said to be real if it is unitarily equivalent to its complex conjugate r∗. This means that there exists a unitary transformation U such that

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At the level of the infinitesimal generators, this translates to

(i Ta₎∗_{= U iT}a_U† _(2.9)

Taking the complex conjugate of 2.8and reinserting it into the same formula then gives

U U∗ = ±1 or equivalently U = ±UT (2.10)

In this case r is said to strictly real or pseudo-real respectively. If2.8is not satisfied, the representation is said to be complex. For simplicity, we adopt a notation in which all spinors are left handed, which can be achieved by charge-conjugating all the right-handed ones:

ψ_Li ψj_R ! → ψ i L iσ2ψRj∗ ! (2.11)

Thus, a given generator is written as

Ta= " tL_a 0 0 −tR a∗ # (2.12) With these conventions, a model is said to be real or complex (also chiral) depending on the represen-tation of the gauge group in which the matter fields lie. An important subclass of the first category is that of vector-like models, in which the representation is reducible

r= r1⊕ r2⊕ ... ⊕ rN (2.13)

and for every irreducible component ri its complex conjugate ri∗ is also present. This is equivalent to

saying that the model can be constructed out of Dirac spinors only, which are obtained by pairing the conjugate Weyl fermions. An important point is that for chiral models, gauge invariance forbids the existence of a fermionic mass term. Due to the anti-commuting nature of Grassmann variables, the most general mass term for a set of Weyl fermions is

Lmass= −εαβχmαMmnχnβ (2.14)

Where M is a symmetric matrix, greek indices run on the spinor components and latin ones on the irreducible representations. Requiring that 2.14be invariant leads to

M T_aT = −TaM (2.15)

which is exactly the reality condition above. Thus, M must be zero.

2.2.2 The role of accidental symmetries

An interesting consequence of the EFT approach is the emergence of global, accidental symmetries, which are only exact when a finite number of operators in the expansion is considered. These are actually quite powerful when trying to constrain Physics beyond the SM, since New Physics effects

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are expected to violate them on general grounds. A well-known example is baryon number conservation in the Standard Model, a U(1) global symmetry of the renormalizable lagrangian1 _{which prevents the} decay of the proton, since it is the lightest particle with B = +1. More precisely, it consists in a rotation of each quark by a common phase θ, and each anti-quark by the opposite phase −θ:

U(1)B : qi → e

iθ

3 ¯q_i→ e −iθ

3 (2.16)

Note that the explanation is different (and less robust) from that of other stable particles in the SM, such as the electron; in this case what is forbidding the decay is a gauge symmetry, which must be exact for consistency reasons. On the contrary, baryon number is accidental and can already be violated by operators in the SM2 _{with a dimension of six or higher [}₄₄_–₄₆_{],thus giving rise to a small} decay rate. It is noteworthy to mention that such effects also appear in many well-motivated extensions of the SM, most notably Grand Unified Theories (GUTs), and thus have attracted a significant interest in the literature.

On the experimental side, data accumulated for almost 20 years at the Super-Kamiokande exper-iment [47] imply

τ(p → e+π0) ≤ 1.6 × 1034yrs and τ(p → µ+π0) ≤ 7.7 × 1033yrs (2.17) at the 90 % confidence level, allowing to set bounds on the supposed scale of New Physics. Similarly, under the assumption of massless neutrinos, lepton number is also conserved at the renormalizable level, preventing the decay

µ±→ e±γ (2.18)

and similar phenomena. Other examples include isospin, parity and the chiral symmetry in the strong interactions.

It is therefore natural to explore the possibility of a similar mechanism in order to explain the stability of dark matter particles, trying to extend those principles that have made the Standard Model so successful, and it is this idea that leads us to the notion of composite dark matter.

2.2.3 Composite models

Composite models have had a long tradition in particle physics, from the experimentally established theory of the strong interactions (QCD) to more modern and exotic hypotheses such as Technicolor or Composite Higgs theories. Their distinguishing feature is postulating the existence of a confining dark colour gauge group GDC which gives rise to a spectrum of bound states in the infrared, held together

by the new strong interactions. In the context of dark matter, some of these hadrons may be stable due to the existence of accidental symmetries, and thus emerge as possible dark matter candidates.

1

There is a small caveat here, since there are non-perturbative effects at the renormalizable level which break U (1)B,

as it is anomalous with respect to the weak interactions. However, these baryon number violations are exponentially suppressed and there is a selection rule forbidding proton decay [42,43].

2

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To guide one through the plethora of models that can be formulated this way, we provide a brief list of requirements which necessarily have to be satisfied for a theory to be considered plausible:

(i) GDC⊗ GSM must be anomaly-free.

(ii) The quark condensation in the dark sector must leave the electroweak vacuum invariant. If this were not the case, a most striking consequence would be a contribution to the electroweak gauge boson masses of order (see Chapter 3 for a discussion)

δmW,Z ∼ gDfπD '

gDΛDC

4π (2.19)

and phenomenologically one usually finds ΛDC has to be above or at least comparable to the

electroweak scale. More in general, even smaller confinement scales would give measurable effects on electroweak precision observables. This is exactly what happened in Technicolor theories [48,49], a popular alternative to the Higgs mechanism, where one tried to describe electroweak symmetry breaking as a consequence of SSB in a strongly interacting, hidden sector. The condition is particularly tricky to examine, as in many cases the exact symmetry breaking pattern is not known.

(iii) No massless bound states should be obtained in the spectrum, since these are excluded for phenomenological reasons.

(iv) No Landau poles should appear for any (weak) coupling below the ultraviolet cutoff of the theory. This can be taken, for example, at the GUT scale ΛGUT ∼ 1015/1016GeV or the Planck scale

Mpl∼1019GeV

(v) The lightest states transforming under a given accidental symmetry must not be charged with respect to GSM (or at least the electroweak SU(2)L⊗U(1)Y, since there can be some exceptions

as regards SU(3)c [50])

Vector-like confinement

Up to now, most of the effort has been concentrated on the study of vector-like models, since they allow for a much simpler treatment. Condition (i), for instance, is automatically satisfied because all of the anomalies cancel out pair by pair between complex conjugate representations (see below for a more rigorous analysis), and the presence of mass terms for dark quarks eliminates the problems of massless bound states altogether. In addition, it is possible to employ the Vafa-Witten [51] theorem, which states that vectorial symmetries cannot be broken spontaneously in the absence of a theta term3 _{in a} vector-like theory, to infer the correct symmetry breaking pattern. If quark masses are much smaller than the confinement scale, a model with F Dirac fermions transforming in a complex representation enjoys a global, approximate symmetry

SU(F)L⊗ SU(F)R⊗ U(1)V (2.20)

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and assuming the theory preserves the maximal unbroken subgroup the pattern is QCD-like:

SU(F)L⊗ SU(F)R⊗ U(1)V→ SU(F)V⊗ U(1)V (2.21)

This result has been confirmed independently by lattice simulations. A class of theories known as Vec-torlike Confinement were first introduced as a viable and phenomenologically rich scenario for New Physics at the LHC in [1], where the possibility that some of the accidentally long-lived states could be proposed as dark matter was already highlighted. Their set-up includes a certain number F of hyper-flavors - fermions transforming in the fundamental representation of the dark gauge group SU(N)DC

but also charged with respect to the SM. The choice of complex representations under SU(N) guar-antees that the approximate symmetry breaking pattern is the one described above, preserving all of the SM gauge symmetries since these are vectorial too. In addition, the vector-like character of the theory with respect to the weak group, together with the generic absence of renormalizable interac-tions between hyperquarks and the Higgs (the Yukawas’), enables one to evade electroweak precision tests: once the hypersector has been integrated out, all the effects on EWPT observables come from non-renormalizable interactions containing at least one Higgs field H, which means the leading contri-butions come from diagrams two loops beyond the SM ones. As regards the accidental symmetries, at the renormalizable level it is possible to associate a conserved U(1) - dubbed species number - rotating all flavors in a given representation of GSM simultaneously, ensuring the stability of various particles.

Real models and Accidental Composite Dark Matter

Under the hypothesis of maximal flavour subgroup alignment [2,52], it is also possible to analyze the global symmetry breaking pattern for strictly real (SR) and pseudo-real (PSR) irreducible represen-tations. For 2N Weyl fermions the un-gauged lagrangians exhibit the respective symmetries:

(SR) : SU(2N) (PSR) : SO(2N) (2.22)

Assuming the condensate to be a fermionic bilinear of the form

hΩ| ψiΣijψj|Ωi (2.23)

it is possible to constrain the matrix Σ with the help of the relation 2.8. The most general way to satisfy gauge invariance is by using

Σ = U−1 _(2.24)

meaning Σ has to be (anti-)symmetric for a (P)SR representation. Then, to preserve the largest possible group one can make the ansatz that Σ is the identity matrix or a N × N symplectic matrix respectively, implying the following patterns:

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Numerical studies have been conducted to simulate some special cases on the lattice, always finding agreement with this hypothesis. In [53], for example, the distribution of the Dirac operator eigenvalues is used to study the symmetry breaking patterns for some exotic real and pseudo-real representations of SU(2), SU(3) and SU(4). An exhaustive analysis of dark models based on real and pseudo-real representations was performed in [3], under the name of Accidental Composite Dark Matter. In addition to the requirements listed above, the authors assume the following:

• The dark colour group is either SU(N) or SO(N) • There are no fundamental scalars

• Only representations with up to two indices are considered

• Compatibility with SU(5) unification is advocated, sothe new fermions must fill fragments of SU(5) representations

A classification of all the acceptable models satisfying the above criteria is presented, alongside a study of the relevant phenomenology. They find various ”golden class”4 _{models for both SU(N) or SO(N),} where the dark matter candidates are dark baryons and dark pions respectively. In the case of a thermal freeze-out, the correct relic abundance is obtained for MDM ∼100 TeV in both classes, while

a DM mass of ∼ 3 TeV is necessary in the first case if the production mechanism proceeds through a dark baryon asymmetry.

2.3 Chiral Composite Dark Matter

The study of chiral composite models has received less attention in the literature with respect to its vector-like counterpart, because of the intrinsic difficulties in dealing with representations which are neither real nor pseudo-real. In this second part of the chapter, we will review the main obstructions and show how they are overcome in the CHN model [11,12]. However, not only there are no funda-mental reasons to think this possibility should have been disfavoured by nature, but there also some compelling motivations to justify its plausibility. As a first point, gauge invariance forbids the exis-tence of mass terms for fermions, making the model fully natural and more predictive, as it contains less free parameters. The only energy scale, ΛDC, emerges as a product of dimensional transmutation

when the dark colour coupling becomes strong and the dark quarks confine into hadrons, eliminating the need to introduce new mass scales by hand. Furthermore, the Standard Model itself is the proto-type of a chiral theory (it suffices to think that there are no right-handed partners for the left-handed neutrinos 5_{), so it appears as a natural road to pursue in BSM model building. Fermion masses are}

4

Which are defined as those models where all the desirable properties depend only on the form of the renormalizable lagrangian, and no assumptions on the structure of non-renormalizable interactions are needed

5

The hypothesis of a right-handed sterile neutrino is a very interesting dark matter candidate in its own right, but has not insofar been proven [54].

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forbidden by gauge invariance, and they only arise because of the Yukawa couplings to the Higgs, L_yuk = −λd_ijQ¯i_LHdj_R− iλu_ijQ¯i_Lσ2H∗dj_R− −λeij¯LiLHe

j

R (2.26)

after spontaneous symmetry breaking gives a vev H = √1 2 0 v ! (2.27)

This does not happen in the scenarios under consideration since the introduction of scalar fields is avoided, both for naturalness concerns and for minimality reasons. Finally, complex representations appear ubiquitously in the context of Grand Unified Theories, which are an attempt to unify the indi-vidual constituents of GSM in a single, larger group GGU T. Their attractiveness lies in an explanation

of the SM hypercharge assignments (and thus of charge quantization) and in the unification of the gauge couplings, eliminating the number of parameters in the theory. Well known examples are SU(5) unification [4], in which the SM matter content is organized into two irreducible representations (a ¯5 and a 10), and SO(10) [55], where, assuming the existence of the right handed neutrinos, only one spinorial representation is necessary. A slightly different example is the Pati-Salam model [56], based on the semi-simple group SU(2) ⊗ SU(2) ⊗ SU(4), which aims to explain the similarities between the lepton and quark generations. Although the simplest realizations of this idea turned out to be ex-cluded, more sophisticated models have been built to address their problems, such as supersymmetric GUTs. Therefore, it is not unreasonable to assume the existence of a dark sector which is unified with the SM at very high energies, and this where chirality comes into play. Now, since each representation of GGU T can be decomposed under GSM as

R= R1⊕ R2⊕ ... ⊕ RN (2.28)

It follows that if R is a real representation, for every Ri appearing in2.3 also its complex conjugate

R_i∗ is also present, and the particle content is also real under GSM. Since this low energy structure is

not observed (the SM is chiral), one of the following must be true: (i) GGU T is chiral

(ii) There is an unknown mechanism to give the unseen ”mirror” fermions a mass much larger than their SM counterparts, and for this reason they are not observed. Although some models have been constructed, this seems quite unlikely, since the standard and mirror fermions are exactly equivalent from the point of view of gauge symmetry.

Assuming the first to be true, it then follows that an eventual dark sector transforming under GGU T

will, in general, be chiral as well.

Finally, let us briefly mention the possibility of chiral but not necessarily composite theories, which can still feature accidental dark matter candidates. The minimal possibility is to postulate the existence of a new, non-anomalous U(1) gauge symmetry, which is spontaneously broken by a dark Higgs mechanism [7,6]. In [6] a new U(1)ν, under which some new ”neutrino-like” SM singlets are charged,

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is introduced, leading to the construction of an explicit model which accounts for both an accidentally stable dark matter candidate and small neutrino masses. Although born out of a different context, another related idea is that of the mirror world scenario [8], where the existence of a copy of the Standard Model is assumed. These mirror particles can interact with ordinary matter only through the Higgs sector (and gravitationally), and dark baryons can serve as dark matter candidates because of their stability.

2.3.1 Global symmetry breaking patterns

For chiral representations, there are fewer analytical and numerical tools to analyze the symmetry breaking patterns induced by dark quark condensation, and no kind of observational evidence is available. In turn, this makes it difficult to tell whether these break the electroweak group, apart from some fortunate cases. As a first step, the results discussed in the previous section can be used to derive the following lemma: if the (globally complex) representation is real under GDC, then the SM

gauge group cannot be preserved. The reason is that, assuming the SSB pattern described in 2.2.3, if the bilinear operator 2.23 is invariant under SM-only gauge transformations, then it is possible to write a mass terms for the dark quarks, but this is absurd. One could then try to reason by analogy and apply some of the criteria derived for real representations, but there is an intrinsic difficulty due to the fact that, by definition, there are no gauge-invariant fermionic bilinears: the emergence of a condensate is invariably linked to the breaking of the gauge group G to one of its subgroups G0. Depending on whether the matter content is still chiral under G0, this process can repeat itself until the representations are real and then finally confinement occurs, as discussed in 2.2.3. This conjecture, known as tumbling, was first hypothesized by Dimopolous, Raby and Susskind in [57], suggesting that it might help to explain the observed patterns in quark and lepton masses. This is because the successive breakings can give rise to a hierarchy of exponentially separated scales, due to the logarithmic running of the gauge coupling. However, this still leaves open the question of which condensate is actually selected at every step of this procedure. A possibility is the so-called Maximally Attractive Channel (MAC) criterium [57], according to which the condensate occurs in the channel where quarks experience the strongest Coulombian attraction due to single gluon exchange. One should then minimize the potential

V (r) = g 2 2r[C (r0) 2 − C (r1) 2 − C (r2) 2 ] (2.29)

where C2 is the quadratic Casimir corresponding to each representation: r0 of the bilinear, r1 and r2 of

the two fermions. It is somehow surprising that the same pattern, for a wide range of models, can be predicted through the use of istanton techniques [58]. Another possibility is that different condensates form simultaneously and at the same scale [58], occurring in the maximally attractive channel for each irreducible representation. As regards the computational side, chiral fermions are notoriously difficult to put on the lattice [59], and no applicable results have been obtained so far. To conclude, in [60,61], Renormalization Group techniques are used to give a constraint on the number of light degrees of

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freedom in the IR and gain knowledge about spontaneous symmetry breaking.

As a possible caveat to the picture outlined above, one can consider the case of a product dark gauge group

G_DM = GS

DM⊗ GDMW (2.30)

which features a strongly interacting (and confining) subgroup GS

DM alongside a ”weak” component

GW

DM. Then, if the matter fields are globally complex but real under GDMS , it is possible to neglect

the weak gauging and assume the symmetry breaking pattern to be that of the corresponding real representation. Note that it is impossible for GW

DM to be a subset of GSM, since representations cannot

be real under the full dark gauge group. This is precisely the assumption made in [11,12], and we are also going to rely on it in the construction of our model.

2.3.2 Anomalies

Throughout the rest of the chapter, we will heavily rely on the concept of anomalies, a fundamental theoretical tool in the study of QFT’s. Here, we review some of the main properties and explore the consequences for our model building goals. We will see how chiral theories have to satisfy very stringent conditions (the so-called gauge anomaly cancellations) to be fully consistent at the quantum level, and how the study of anomalies can be used to relate properties of the IR and UV behaviour of confining gauge theories through the ’t Hooft anomaly matching.

Let us consider a theory whose classical action Scl is left invariant under a continuous symmetry

group, which is not broken spontaneously by the dynamics of the system. Such a symmetry is said to be linearly realized, and the equations of motion imply the corresponding Noether current is conserved at the classical level:

∂µJµ(x) = 0 (2.31)

However, the generalization of this relation is not guaranteed to hold when quantum corrections are taken into account, and in the general case

hΩ| ∂µJµ(x) |Ωi = A(x) (2.32)

which provides a definition of the anomaly A(x) associated to that particular symmetry.

The axial anomaly

Perhaps the simplest explanation of how a relation such as 2.32can emerge is using the path integral formulation, as first realized by Fujikawa in [62,63]. We shall examine the case of a theory with chiral fermions transforming under an arbitrary gauge group G, which we embed in a single, reducible representation ψi for convenience. The key point of this analysis resides in the fact that the fermionic

measure in the path integral is not left invariant by an arbitrary chiral, global transformation, in-troducing anomalous contributions in the evaluation of correlation functions which give rise to the

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anomaly. In the presence of background gauge fields, the generating functional of the theory can be written as Z[η, ¯η] = Z [D ¯ψ][Dψ]eiScl+i R ¯ ηψ+ ¯ψη _(2.33)

It is possible perform the local version of a global, axial symmetry

ψ0= U ψ (2.34)

where

U = eiα(x)T γ5 (2.35)

and T is hermitian, so that U is a pseudo-unitary matrix acting with the opposite sign on left and right-handed fields. Since this is just a change of variables, the generating functional must be left invariant: Z [D ¯ψ][Dψ]eiScl+i R ¯ ηψ+ ¯ψη ₌Z _{[D ¯}_ψ0_][Dψ0_]eiS0cl+i R ¯ ηψ0+ ¯ψ0η _(2.36)

However, under an axial transformation the measure picks up a phase [D ¯ψ][Dψ] → det(U)det( ¯U)[D ¯ψ][Dψ] ≡ ei

R

d4_xA

U(x) (2.37)

where we have defined

Umn(x, y) ≡ δ(4)(x − y)U(x)mn U¯mn(x, y) ≡ [γ0U(x)†γ0]mnδ(4)(x − y) (2.38)

Some of these expressions are ill-defined, and must be regularized using a suitable gauge-invariant prescription. The result is

AU(x) = − 1

16π2ε

µνρσ_Ga

µνGbρσT r[TaTbT] (2.39)

Then, expanding 2.36at first order in the transformation parameter α(x), one finally obtains

hΩ| ∂_µJ_Uµ(x) |Ωi = AU(x) (2.40)

for the axial current

J_Uµ(x) = −i ¯ψγ5T ψ (2.41)

From a historical perspective, this result turned out to be of great phenomenological relevance, enabling the calculation of the π0 decay width from the electromagnetic anomaly of the U(1)A symmetry in

QCD, in perfect agreement with the experimental value.

Direct calculation and gauge anomalies

Although very clear from a conceptual point of view, the path integral is not the most general method to compute anomalies. In particular, when dealing with generic chiral anomalies (such as those concerning gauge symmetries) it is more convenient to calculate them directly as Feynman diagrams, as originally done by Adler [64], Bell and Jackiw [65]. The LSZ formula and the Dyson-Schwinger

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equations enable one to calculate the expectation value of a current’s divergence through the ”triangle” diagrams (see Fig. 2.1) representing correlation functions between three currents

Γµνρ

abc(x, y, z) ≡ hΩ| J µ

a(x)Jbν(y)Jcρ(z) |Ωi (2.42)

with Jµ _{defined as in Eq.} _2.41_{. Although this quantity can be shown to be convergent, its spatial}

Figure 2.1: One-loop triangle diagrams contributing to a gauge anomaly

derivative ∂µΓµνρ_abc(x, y, z), which is the fundamental ingredient, diverges, and must be properly

regu-larized. What emerges from the calculation is that it is impossible to find a regulator preserving all three current conservations if the fully symmetric symbol

Dabc≡ 1₂T r[{Ta, Tb}, Tc] (2.43)

is different from zero. More precisely, the anomaly associated to a given current is hΩ| ∂_µJ_cµ|Ωi = −Dabc

32π2ε

µνρσ_Ga

µνGbρσ (2.44)

In the case of a global symmetry, this means the theory cannot be regulated without spoiling either gauge invariance or the symmetry under consideration, and so the latter must be sacrificed; thus2.44

reduces to 2.40. For a gauge symmetry, instead, the anomaly appears as a violation of the Ward-Takahashi identities (or their non-abelian Slavnov-Taylor correspective), signaling an inconsistency in its renormalization procedure: an infinite number of counterterms are then required to renormalize the theory. In addition, gauge symmetry, which is just a redundancy in our description, is a crucial ingredient in allowing us to establish the equivalence between the unitary and covariant gauges, in which unitarity and Lorentz-invariance respectively are manifest [66–69]. This brings us to the notion of gauge anomaly cancellation, which any QFT must satisfy in order to provide a consistent and physically acceptable formulation:

X

r

T r[{T_a(r), T_b(r)}, T_c(r)] = 0 (2.45) where the sum is on all the (left-handed) representations. For real representations (and vector-like theories as a special case), this condition is automatically satisfied, since the existence of a similarity

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transformation such that

(iTa)∗ = S(iTa)S−1 (2.46)

implies

Dabc= −Dabc (2.47)

However, in the chiral case the condition becomes non-trivial, and imposes very strong conditions in the space of possible theories.

Gravitational anomalies

If one also takes gravity into account, there is also an anomaly associated to the expectation value of a generic matter current Jµ_{= ¯}_{ψT γ}µ_ψ_{in the presence of an external gravitational field. A computation}

of the relevant triangle diagram with two gravitons gives [70]

∂µJµ∝ T r[T ]εαβγδRαβρσRρσγδ (2.48)

Not only does this vanish for real or pseudo-real representations, but also for simple Lie algebras, such as SU(N) for N > 1.

Gauge anomaly cancellation in the Standard Model

A prominent example is the SM itself, since its gauge group admits complex representations, and the cancellation of its gauge anomalies computed with the experimentally measured charges provides a strong consistency check on its structure. We briefly outline how the computation is carried out, referring to Tab. 2.1for the SM charges.

• For any anomaly containing a single SU(2) or SU(3) factor,

Dabc∝ T r[Ta] = 0 (2.49)

• SU(3)3

The representation is vector-like with respect ot SU(3), so the anomaly vanishes. • SU(2)3

SU(2) only has real or pseudo-real representations. • U(1)3 X r Y_r3= 6 −1 6 3_{+ 3} 2 3 3_{+ 3} −1 3 3_{+ 2} 1 2 3_{+ (−1)}3_{= 0} _(2.50) • SU(3)2_⊗ _U(1) X ¯ 3,3 Yr= −1 6 − 1 6 + 2 3− 1 3 = 0 (2.51)

Composite Dark Matter from Strongly Interacting Chiral Dynamics