Leptocolor symmetric completions of Composite Higgs theories

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UNIVERSIT

A DI PISA

Dipartimento di Fisica

Corso di Laurea Magistrale in Fisica

Tesi di Laurea Magistrale

Leptocolor symmetric completions

of Composite Higgs theories

Candidato

Marco Costa

Relatore

Prof. Dario Buttazzo

Relatore interno

Prof. Alessandro Strumia

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5 UV sector characterization 73 5.1 Leptocolor only . . . 74 5.1.1 H = SO(n) . . . 75 5.1.2 H = Sp(2n) . . . 76 5.1.3 H = SU (n)V . . . 77 5.2 Complete models . . . 78 5.2.1 p = 1 . . . 79 5.2.2 p = 2 . . . 83 5.2.3 p = 3 . . . 87 5.2.4 p = 4 . . . 90 6 Chiral patterns 92 6.1 General strategy . . . 92 6.1.1 Condensation consequences . . . 95 6.2 Chiral patterns . . . 98 6.2.1 Electroweak coset . . . 98 6.2.2 Leptocolor coset . . . 99

6.3 Class I basic models . . . 100

6.3.1 p = 1 . . . 100

6.3.2 p = 2 . . . 101

6.3.3 p = 3 . . . 105

6.4 Class I non-basic models . . . 105

6.4.1 p = 1 . . . 105

6.4.2 p = 2, vector-like sector . . . 106

6.4.3 p = 2, multiple copies . . . 111

6.4.4 p = 3, multiple copies . . . 111

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CONTENTS v 7 Composite-SM mixing 114 7.1 NGB couplings . . . 114 7.1.1 Higgs couplings . . . 116 7.1.2 Colored NGB couplings . . . 118 7.2 NGB gauge interaction . . . 120 7.3 Hyperfermion masses . . . 122 7.4 Potential estimate . . . 123 7.4.1 Higgs potential . . . 123 7.4.2 Colored NGB potential . . . 124 7.5 Next-to-minimal models . . . 125 Conclusions 126 A Group theory 129 A.1 Generators and Lie algebras . . . 129

A.2 Dynkin index . . . 130

A.3 Quadratic Casimir . . . 131

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Introduction

One of the pillars of our understanding of Nature is the Standard Model (SM) of particle physics. It encodes three of the four different fundamental interactions: the electromagnetic force, the weak nuclear force and the strong nuclear force. One of the key ingredients in the formulation of the SM is the unification of the weak and electromagnetic forces in the electroweak interaction.

The different behaviour of the two forces in the low energy regime can be explained thanks to the Higgs mechanism: the Higgs field acquires a vacuum expectation value and gives mass to the weak gauge bosons W±, Z0_{, while keeping the photon massless.}

After several experimental efforts, the Higgs boson, a scalar excitation around the Higgs vacuum with mass mh = 125 GeV, was finally discovered in 2012 at LHC.

Despite these astounding successes, the SM is not a theory of everything: it has several cosmological shortcomings, like the explanation of the origin of dark matter, and it cannot describe correctly gravitational interactions.

To address gravitational effects, a new theory must replace the SM around the Planck scale MP L ' 1019GeV, which is the energy at which gravitational interactions

become relevant. Excluding these problems, there are no strong evidences suggesting the existence of new physics beyond the SM at scales lower than MP L.

In the best case scenario (or worst, depending on the point of view), the SM is correct from above the electroweak scale up to the Planck scale.

If this is true, the SM can be seen as an effective field theory correctly describing low energy phenomena below MP L. However in this case large quantum fluctuations

of SM fields with energy right below the cut-off MP L give a contribution to the Higgs

mass proportional to MP L. Therefore mh is expected to be around MP L. Without

invoking a very precise cancellation between the SM radiative corrections and the ones of the fields above MP L, the physical mass of the Higgs mh is expected to be

much heavier than its observed value.

To address this hierarchy problem, several authors considered the possibility of a composite nature of the Higgs boson. Thanks to compositeness, the quantum corrections to the Higgs mass do not run up to MP L: other composite particles will

contribute to the radiative corrections, effectively lowering the cutoff of the SM from MP L down to the compositeness scale ΛN P. What is the value of this scale?

The naturalness argument suggests that new physics should appear at a natural scale, such that contributions of quantum fluctuations to the mass mh are not much

larger than its observed value. A simple calculation gives that the natural scale for the new physics stabilizing the Higgs mass should be around (1 ÷ 10) TeV. This gives a criterion to estimate the compositeness scale ΛN P if we identify it with the

naturalness scale.

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After a first wave of works focused mostly on the phenomenological implications of the compositeness hypothesis, several authors started to study possible new dynamics beyond the SM responsible for the Higgs formation. We focus on extensions of the SM with the following characteristics:

a strongly interacting gauge group GHC, called hypercolor group.

a set of Weyl left-handed fermions, called hyperfermions, charged under GHC

and displaying a global flavor symmetry group G.

The SM, which is the low energy limit of the full theory, exhibits an approximate global symmetry group, the custodial symmetry:

Gcus= SU (3)c× SU (2)L× SU (2)R× U (1)B−L.

We want to extend this symmetry also to the hyperfermionic sector in order to preserve it. In particular, we require that the full Gcus is a symmetry of the hyperfermions, so

G must be large enough to contain Gcus. The hypercolor interaction is much stronger

than the SM ones: the effects of the SM gauging can be neglected when determining the GHC dynamics. At the scale ΛHC ≡ ΛN P ≈ (1 ÷ 10) TeV the interaction GHC

confines and the hyperfermions are bound together to form resonances, among which there is the Higgs boson. Moreover hyperfermionic bilinear condensates are formed, spontaneously breaking the flavor group G to a subgroup H, which again must contain Gcus since it’s a symmetry of the infrared theory. On general grounds, the

mass of GHC bound states is expected to be around the compositeness scale ΛHC.

In composite Higgs theories (CH) the Higgs is not a generic bound state, but a light pseudo-Nambu-Goldstone boson associated to the spontaneous symmetry breaking G/H (also known as coset ). In this way the Higgs can be naturally lighter than the typical resonance. Thanks to this mechanism, the bounds coming from searches (both direct and indirect) for the heavy resonances can be bypassed. The peculiarity of the Higgs is not only its lightness compared to ΛHC, but also its characteristic

mexican hat potential that is crucial to achieve electroweak symmetry breaking. In modern composite Higgs theories the potential for the Higgs is generated via radiative corrections coming from terms that explicitly break the flavor symmetry G of the hyperfermionic sector. These usually are direct mixing terms with the SM elementary fermions and couplings with the SM gauge bosons. Indeed G is only a symmetry of the hyperfermionic sector, and in general not of the SM fields: any mixing between the two sectors will spoil G invariance.

In previous works [50] the possible new fermionic sectors that could be the completion of a composite Higgs theory were classified. The classification consists in finding all the possible gauge groups GHC and the respective hyperfermionic content

with flavor group G. The classification is important because it gives an idea of the various possibilities to get a dynamically motivated CH theory. In general different sectors have a different phenomenology depending on GHC and its hyperfermionic

content, like for example different Nambu-Goldstone spectra.

The purpose of this thesis is to critically re-examine such classification under two points of view. The first is to consider sectors that have a larger symmetry group than Gcus. Remarkably the SM fermions, with the addition of a right-handed

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3

neutrino NR, are organized in approximate multiplets of the famous Pati-Salam

group GP S:

GP S= SU (4)LC× SU (2)R× SU (2)L' SO(6) × SO(4) .

Actually the SM fermions can be organized in a complete 16-dimensional multiplet of a grand unified SO(10) group. We will focus on the Pati-Salam setup only. In this scenario, the SU (3)cand the U (1)B−Lare unified in a global leptocolor group SU (4)LC:

the leptons are seen as the fourth color together with the standard three colors of QCD. It’s natural to extend the GP S global symmetry to the hyperfermions. Additionally,

we require that our models dynamically break via GHC condensates the leptocolor

symmetry to the usual SU (3)c× U (1)B−L. This has as a consequence the appearance

in the spectrum of light Pati-Salam scalar leptoquarks, and eventually of vector resonances with the same quantum numbers. We mention that leptoquark models based on GP S have recently garnered much attention in relation to experimental

anomalies in B meson decays. Although it’s worthwhile to investigate this connection, we will not focus on it: we will simply limit ourselves to the study of the possible UV sectors displaying a spontaneously broken leptocolor symmetry. We proceed at first by studying simple vector-like models that exhibit the desired leptocolor dynamical breaking pattern, forgetting the problem of having the Higgs among the NGBs. Vector-like theories are defined to have a GHC content that allows the construction

of massive Dirac spinors by pairing a fermion with its conjugate. If a theory is not vector-like is called chiral. The vector-like scenario is easier to study than the chiral one, since there are many strong results on the behaviour of such theories, like the Vafa-Witten theorem. After having found the basic blocks that achieve leptocolor breaking, we combine these with the electroweak counterparts that contain the Higgs among the Goldstone bosons. In this way we can give a classification of minimal CH theories satisfying our requests.

The second point we want to address is a review of the results of [50]. Indeed some interesting class of theories, such as models with a chiral GHC content, are completely

neglected by the authors. Although the behaviour of strongly interacting chiral gauge theories is still theoretically unknown, we will employ heuristic arguments to understand some of their aspects. This will lead us to find a set of interesting dynamics for composite Higgs theories that escaped the previous classification.

We stress that models considered in our analysis only satisfy the minimal necessary conditions to be a composite Higgs theory with an enlarged UV leptocolor symmetry. Not all models presented are phenomenologically viable: this is something to be checked case by case, eventually in future works.

The thesis is organized as follows. In chapter 1 we introduce the Higgs boson and discuss in detail the naturalness problem related to its light mass. Then we motivate in depth the goals of our work.

Chapter 2 is meant to be a review of the key ideas and results in strongly interacting gauge theories that are crucial to understand their infrared behaviour. It can be skipped by readers familiar with such concepts.

In chapter 3 we review the framework of composite Higgs theories, with particular attention on how a realistic electroweak symmetry breaking can be achieved in this class of models.

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Chapter 4 is devoted to introduce the Pati-Salam symmetry and the role of the leptocolor symmetry.

The original content is presented in chapters 5, 6, and partly in chapter 7. In chapter 5 we discuss the minimal vector-like scenarios in which a breaking of the leptocolor symmetry SU (4)LC to SU (3)c× U (1)B−L is achieved. Then we give a full

classification of vector-like models that contain both a composte Higgs Goldstone and a dynamically spontaneously broken leptocolor symmetry.

In chapter 6 we explore chiral models. We will find possible composite Higgs dynamics that realize symmetry breaking patterns not allowed in vector-like models.

Finally in chapter 7 we briefly discuss the mixing of the UV sector with the elementary fields, and see if EWSB can be achieved without breaking the SM SU (3)c

color symmetry. We will find that this is in general possible due to the stronger color interaction and the possibility to independently give positive masses to the EW and colored Goldstone bosons.

In the conclusions we recap the results obtained and give an outlook of possible future directions of this work.

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Chapter 1 Motivations

1.1 The Standard Model

Our current best understanding of high energy phenomena taking place in particle colliders is captured in the Standard Model (SM) of particle physics. The SM is a simple and elegant theory describing a vast range of phenomena, from pion decay to nucleon scattering. From a mathematical point of view, the SM is a quantum Yang-Mills theory based on the gauge group:

GSM = SU (3)c× SU (2)L× U (1)Y . (1.1)

This gauge group is responsible for different interactions depending on its subgroups: The color SU(3)c: its gauge bosons are the gluons. It describes nuclear strong

forces. The subset of the SM describing only this force is called quantum chromodynamics (QCD).

The electroweak (EW) SU(2) × U(1)Y sector: it’s responsible for the weak

nuclear forces, and, thanks to the Higgs mechanism, for the emergence of electromagnetism at lower energy.

In this work we will focus mostly on the EW sector, although in chapter 2 we will give a brief overview of some properties of QCD useful to describe beyond the Standard Model physics (BSM). To write the SM Lagrangian, we have to specify its matter content. At first, we will consider a single family of fermions. We specify their representations under the GSM and Lorentz group SO(3, 1). The colored triplet

Field SO(3, 1) SU (3)c SU (2)L U (1)Y qL (1/2, 0) 3 2 1/6 uR (0, 1/2) 3 1 2/3 dR (0, 1/2) 3 1 −1/3 lL (1/2, 0) 1 2 −1/2 eR (0, 1/2) 1 1 −1

Table 1.1: Fermionic field content of a single family of the Standard Model.

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elements are simply the three different color of quarks. Instead, the SU (2)L doublet

components are labelled in the following way: qL= uL dL , lL = νL eL . (1.2)

The fields contained in table 1.1 form a family or generation. There are in total three families that are an exact copy of the ones in table 1.1. The family index is also called flavor index. The only differences between the three families are the masses of the fermions: the second family is heavier than the first, and the third is even heavier. A remarkable property of the SM is that it maximally violates parity: indeed only left-handed fields interact with the EW SU (2)L gauge bosons.

The right-handed fermions are singlets under such interaction. Moreover there is no right-handed partner of the left-handed neutrino, although in chapter 4 we will study extension of the SM including it. Now that the fermionic sector is defined, we can write the Lagrangian containing the fermionic and gauge bosons kinetic terms:

Lkin = − 1 4G iµν_Gi µν− 1 4W aµν_Wa µν − 1 4B µν_B µν+ + X ψL=qL,lL ¯ ψLi /DψL + X ψR=uR,dR,eR ¯ ψRi /DψR (1.3)

where D is the gauge-covariant derivative

Dµ= ∂µ− igcGiµTi− igwWµaTa− ig 0

Y Bµ. (1.4)

A diagonal sum on the flavor index is left implicit. A single flavor family of the SM enjoys an additional, accidental, non-anomalous U (1) symmetry: the difference between baryon and lepton numbers B − L.

qL→ eiθ/3qL, uR→ eiθ/3uR, dR→ eiθ/3dR

lL→ e−iθlL, eR → e−iθeR .

(1.5) This simmetry is important since in section 1.2.4 it will be used to give to the fermions the right hypercharge in a more general setting. For simplicity we will sometimes use the rescaled X generator:

X = 3(B − L) . (1.6) One could think to add bare masses for the fermions in equation (1.3). However this is mathematically not possible. Indeed the SM is a chiral model : the left-handed fields are in different representations than the right-handed ones. For example, there is no right-handed partner with the same quantum numbers of the left-handed, SU (2)Ldoublet lL. The same holds for the colored qL. This implies that the fermions

cannot be combined together to form a Dirac spinor, and thus that a gauge-invariant mass term cannot be written for such fermions. However, fermions do have a mass: for example the electron has a mass of approximately 0.5 MeV. Another problem with the Lagrangian in equation (1.3) is that it describes a total of 12 massless gauge bosons: 8 gluons and 4 EW gauge bosons. It can be shown that the strongly

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1.2. THE HIGGS FIELD 7

interacting gluons are screened and found only in bound states (such as glueballs and other resonances). Therefore the fact that there is no Coulomb-like potential at low energies for QCD is not a problem. We will briefly cover this phenomenon in chapter 2. However, out of the remaining 4 gauge bosons, only the photon is experimentally observed as massless. The other three gauge bosons, the W±, Z0_,

instead have masses of MW = 80 GeV, MZ = 91 GeV. It can be argued that this

problem can easily be fixed by putting in the Lagrangian a bare mass term for the vector bosons: Lm = m2 2 W µa_Wa µ . (1.7)

However a bare mass term for the non-abelian vectors like in equation (1.7) usually implies a non-renormalizabilty of the theory, or problems with perturbative unitarity [92, 94]. In order to address both the fermion and vector boson masses, the scalar Higgs field must be introduced.

1.2 The Higgs field

The Higgs field H is a fundamental complex scalar field that plays two fundamental roles in the SM:

1. It breaks spontaneously the EW group [111] via the Higgs mechanism [46, 69]: SU (2)L× U (1)Y → U (1)Q ,

where U (1)Q is the electromagnetic interaction.

2. It gives mass to the fermions of the SM.

Additionally, after spontaneous symmetry breaking the field can be reparametrized to describe the interactions of a massive, electrically neutral scalar excitation, the famous Higgs boson. The charges of the Higgs field H under GSM are:

H = (1, 2)1/2 . (1.8)

Thus its components can be written in terms of 4 real components as [113]:

H = h2+ ih1 h4+ ih3

. (1.9)

We can write the most general renormalizable Lagrangian describing the Higgs-gauge interactions: LHiggs = (DµH) † DµH − V H†H . (1.10)

The potential for the Higgs contains at most quartic terms at the renormalizable level. The sign of the coefficients is actually crucial to achieve spontaneous symmetry breaking (reviewed in chapter 2):

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The positive sign of the quartic is necessary to have a potential bounded from below. The negative sign of the “mass” term gives to the potential in equation (1.11) the characteristic mexican-hat shape. This allows the Higgs field to develop a non-zero vacuum expectation value. We can use the polar decomposition to rewrite the Higgs field as: H = eiφav T a 0 σ . (1.12)

The minimum of the potential can be calculated taking the derivatives of equation and setting the field H constant in space-time in (1.11):

σ2 = H†H = µ 2 2λ ≡ v2 2 ' 1 2(246 GeV) 2 (1.13) where v is known as Fermi scale or electroweak scale. Now we can expand the Higgs field around its vacuum expectation value (vev):

H = eiφavT a   0 v + h √ 2   . (1.14)

The radial excitation is the Higgs boson. We can employ the gauge invariance of the Higgs Lagrangian to go to the unitary gauge:

H → H0 = e−iφaf T a H =   0 v + h √ 2   . (1.15)

In the unitary gauge the φ angular modes disappear explicitly from the Lagrangian. Only the radial h mode is still present. This has several consequences on the masses of gauge bosons, fermions and interactions of h.

1.2.1 The Higgs boson

The Lagrangian in equation (1.10) when expressed in unitary gauge describes the self interaction of the Higgs boson h with itself. In particular it predicts its mass as a function of the vev of the field H.

m2_h = 2µ2 = 2λv2 . (1.16) The self interaction terms can be read from the potential in unitary gauge:

Lh = m2 h 2v h 3₊ m2h 8v2h 4 _. _(1.17)

The existence of this massive excitation was first proved experimentally at LHC in 2012 [27, 29]. The prediction of a single massive scalar excitation is a peculiar characteristic of the Higgs model. An example of theories of electroweak symmetry breaking not exhibiting this excitation are original technicolor theories (TC) [104]. Such models achieve the spontaneous breaking of the EW group via a dynamical

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condensation1 [104] of a strong interaction called technicolor. In their most na¨ıve formulation, TC theories do not predict any additional massive bosonic excitations like h: all the Nambu-Goldstone bosons are eaten by the gauge vectors. This is of course excluded experimentally.

1.2.2 Gauge bosons

Remarkably in the unitary gauge it becomes apparent that the vector bosons acquire a mass. To show this, consider for simplicity only the electroweak Lagrangian, based on the SM gauge subgroup SU (2)L× U (1)Y:

LEW = − 1 4W µνa W_µνa − 1 4B µν Bµν+ LHiggs . (1.18)

The mass matrix in the Wa

µ, Bµ basis is: M2 = v 2 4     g_w2 0 0 0 0 g2 w 0 0 0 0 g2 w gwg0 0 0 gwg0 g0 2     (1.19)

After diagonalization, we obtain three massive vector fields and only one massless gauge field, corresponding to the photon:

M_W2 = v 2_g2 w 4 , M 2 Z = (g2 w+ g 0 2_{) v}2 4 ≡ M2 W cos2_θ w , M_A2 = 0 , (1.20) where we defined the Weinberg angle θw. The coupling of the massless photon, or

the electric charge value, is related to the EW couplings via θw:

e ≡ gwsin θw = g0cos θw . (1.21)

By retaining the terms with h we can find the trilinear and quadrilinear interaction terms between the Higgs and vector bosons:

LV h = 2 M_W2 v W + W−h +M 2 W v2 W + W−h2+M 2 Z v Z 2 h +1 2 M_Z2 v2 Z 2 h2 . (1.22)

1.2.3 Fermions

As we already remarked, in a chiral gauge theory like the SM, chiral fermions cannot form gauge-invariant bilinear. This implies that no Dirac mass term can be written in such a theory. However, quarks and leptons certainly have a mass. Indeed, the Higgs boson can be used to build a fermionic bilinear after setting it to its vev. In addition to the kinetic term and self-interaction in Lagrangian (1.10), we can write Yukawa couplings: gauge invariant terms built with two fermions and the Higgs field. In particular, we can write:

LYuk = λdq¯LHdR+ λuq¯LHcuR+ λl¯lLHeR+ h.c. , (1.23)

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where we defined the conjugate Higgs field Hc ≡ iσ2H∗. In unitary gauge, the

Yukawas contain a mass term for the SM fermions: LYuk ⊇ λuu¯LuR

v √

2 . (1.24)

It’s clear that the mass of a given fermion is proportional to the Yukawa coupling and the Higgs vev. The most relevant Yukawa interaction is the one with the top quark, which is the most massive fermion in the SM.

mt= λt

v √

2 ' 174 GeV . (1.25) The other terms in the Yukawa Lagrangian are terms describing trilinear coupling between fermions and the Higgs boson:

Lψh =

m_u

v u¯LuRh

+ (d ↔ u) + (e ↔ u) . (1.26) As expected, the trilinear coupling is proportional to the mass of the fermions. We discussed only the interaction of the Higgs field with a single SM family to illustrate concisely how the Higgs mechanism works. When there is more than one family, the Yukawa couplings λu,d,l become matrices in flavor space. There is the need to

diagonalize these Yukawa matrices to obtain the mass eigenstates. In general the mass eigenstates are not eigenstates of the weak interactions: there is a mixing matrix VCKM called the Cabibbo-Kobayashi-Maskawa (CKM) matrix This causes mixing

between the generations. In particular in the SM the charged weak interactions mediated by the W± bosons can cause the decay from a generation to the lighter ones. Since we are not interested in flavor in the present work, we will not consider this aspect further.

1.2.4 Custodial symmetry

The shape of the Higgs potential V H†H was determined by imposing renor-malizability and gauge invariance under SU (2)L× U (1)Y. However V is accidentally

invariant under a bigger symmetry group. To see this, notice that the argument of the potential is:

H†H = h2₁+ h2₂+ h2₃+ h2₄ . (1.27) So the potential V is a function of the square norm of a 4 dimensional real vector. Since the symmetry group of V is the symmetry group of its argument, it’s clear that V is invariant under SO(4). This is called custodial symmetry, for reasons that will become apparent later. Notice that locally the following isomorphism holds:

SO(4) ' SU (2) × SU (2) . (1.28) The two SU (2) factors can be identified as:

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where SU (2)L is the same gauged SU (2)L, rotating the two complex components

h2+ ih1 and h4+ ih3 together. The action of SU (2)L× SU (2)R on the Higgs field is

more clear if we build the following 2 × 2 matrix: H = √1

2(H

c_{, H)} _(1.30)

where the H, Hc on the RHS are 2D complex vectors, and the H on the LHS is a 2 × 2 complex matrix2_{. The action of the custodial symmetry on the matrix H is:}

H → H0 = ULHU †

R (1.31)

where UL,R are SU (2)L,R matrices respectively. The residual subgroup after SSB is

the diagonal combination SU (2)V3:

SU (2)L× SU (2)R→ SU (2)V .

Is the custodial symmetry a full symmetry of the SM? No: otherwise the full SU (2)R

group should be gauged. Instead only the hypercharge generator Y = T3R+

B − L

2 (1.32)

is gauged, and not the full SU (2)R. This is clearly a source of explicit breaking of

the custodial symmetry. It can be shown that a source of explicit breaking of the full custodial symmetry comes from the Yukawa couplings. What are the consequences of this approximate accidental symmetry of the SM? Let’s look at the effect of radiative corrections on the mass ratio ρ:

ρ ≡ M 2 W cos2_θ wM_Z2 . (1.33)

At tree level ρ = 1 by construction. Radiative corrections to equation (1.33) can come only from the explicit breaking of the SU (2)L × SU (2)R [12], and vanish

in the exact symmetry limit. Indeed in this limit there is no difference between the elements of the gauged SU (2)L triplet. So the custodial symmetry keeps the ρ

parameter close to its tree level unity value, hence its name. We can try to extend this symmetry to act also on fermions. The SU (2)R action on the fermions is analogous

to the one of SU (2)L, but instead of acting on the SU (2)L doublets it acts on the

corresponding right-handed fields. While the uR, dR fields can be accomodated in

a SU (2)R doublet, the eR lacks a partner. This problem can be solved with the

addition of a right-handed neutrino NR. With this addition a single SM family has

an approximate symmetry Gcus:

Gcus = SU (3)c× SU (2)L× SU (2)R× U (1)X . (1.34)

2_{Sometimes it’s used a rescaled matrix Σ =}√_{2H instead of H.} 3_{Sometimes authors refer to this residual SU (2)}

V as the custodial symmetry of the SM. However

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1.3 The hierarchy problem and naturalness

So far, the SM correctly describes many particle physics phenomena, such as branching ratios, mixings, decays and so on, with a stunning accuracy. However, it is not a theory of everything: for example it does not include a description of gravitational forces. There are other physical phenomena not addressed by the SM. The vast majority of mass of the matter in the universe is made up by dark matter (DM) that apparently can’t be explained easily in terms of the known particles.

Moreover, the vast majority of energy density in the universe comes from dark energy and can’t be explained at all in the SM. In addition to cosmological and gravitational problems, there are open questions more related to particle physics. Indeed, it has been experimentally verified that neutrino do oscillate between their different flavors. This implies that at least two out of the three neutrino flavors must have a mass. In the SM as it is, the neutrino can acquire mass only through a 5D operator known as the Weinberg operator [112]:

LW =

1 ΛSM

¯

lL· Hc (lcL· Hc) . (1.35)

This operator violates the leptonic number, and in particular it contributes to rare flavor processes. From the operator in (1.35) we can get an estimate on the cutoff of the theory, or equivalently on the suppression scale ΛSM. From experimental

bounds on flavor violating processes and neutrino masses we get that it must be very high compared to the Fermi scale: ΛSM ≈ 1014GeV. As an aside, we will see

in chapter 4 that the addition of a right-handed neutrino NR, total singlet under

GSM, can be used to build a renormalizable mass term for neutrinos. Other hints of

new physics come from the muon g − 2 anomaly [51] and possibly from B meson physics (for the latter see chapter 4). All this facts hint to the possibility that the SM is not the ultimate theory of physics. It seems very likely that the SM is only an effective field theory (EFT) describing the low energy regime of a more fundamental theory. What is the fundamental cutoff of the SM, or in other words, what is the energy scale at which the SM stops working? With the addition of the right-handed neutrino, it might very well be that there is no new physics until gravitational effects become relevant, and thus that the fundamental cutoff of the SM is the Planck scale MP L = 1019GeV 4. In this scenario every problem we listed above has a solution in

this beyond-Planck scale theory, and thus it’s likely related to gravity.

This however has a bad consequence on the mass of the fundamental Higgs scalar: indeed fundamental scalars do not decouple easily from UV physics. Let ΛSM be

the SM cutoff, above which there are new physics effects, such as new particles. In general, there will be radiative corrections to the bare Higgs mass, coming both from SM particles and from the ones above ΛSM. To keep the dicussion simple, we will

employ a hard physical cut-off in the momentum to regularize 1-loop corrections, but everything can be reformulated in a gauge-invariant fashion [58]. The SM contributions to the Higgs can be estimated from the diagrams in figure 1.3. The

4_{The cutoff can also be taken as the slightly lower M}

GU T = 1015GeV, at which the SM gauge

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1.3. THE HIERARCHY PROBLEM AND NATURALNESS 13

Figure 1.1: Unnatural scenario with the SM cutoff ΛSM at the Planck scale.

Figure 1.2: Natural scenario with new physics at ΛN P ≈ 10 TeV.

Figure 1.3: Examples of processes contributing to the Higgs mass in the SM.

SM contributions δSMm2h are reported in equation (1.36) [85]:

δSMm2h = 3y2 t 8π2 − 3g_w2 32π2 − 3g_w2 64π2_cos2_θ w − 3λ 8π2 Λ2_SM . (1.36) As can be seen, the SM contributions to m2

h are quadratic in the cutoff. This could

have been expected from na¨ıve scaling arguments, since the mass operator has a classical dimension equal to 2. If the cutoff of the SM is near MP L, as in figure 1.1,

then its radiative contributions are much higher than the physical Higgs mass. In order to have a light scalar around 100 GeV, then the new physics contributions δN Pm2h must cancel the SM contributions almost exactly. To quantify the amount of

cancellation between the two contributions needed to achieve a light fundamental scalar, we define the amount of fine-tuning as the ratio of the amount of cancellation

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and the physical Higgs mass. ∆ ≈ δSMm 2 h m2 h ≈ Λ 2 SM (450 GeV)2 . (1.37) If ΛSM is at the Planck scale, then

∆ ' 1030. (1.38) Therefore in order to have light scalars in the IR theory, the parameters in the UV theory, which is a priori uncorrelated with the SM, must conspire together and achieve an almost perfect cancellation. Indeed the two uncorrelated contributions must agree at least up to the thirtieh decimal digit to ensure such a result. A theory whose parameters are fine-tuned to obtain such a miracolous cancellation is said to be unnatural [105]. Therefore in a SM-only scenario the EW scale v = 246 GeV is unnatural with respect to the Planck scale MP L ≈ 1019GeV. The lightness of the

Fermi scale compared to MP L is also called the hierarchy problem, or naturalness

problem of the Fermi scale. In particular, the naturalness assumption implies that the radiative corrections on the mass parameter m2

h should have its same order of

magnitude, setting a bound on the cutoff of the new physics ΛSM ≡ ΛN P. In other

words, in a natural theory the tuning should be of the order of unity (or not too large). Allowing for a mild tuning in the parameters at the percent level we get from equation (1.36) the bound [85]:

100 GeV . ΛN P . 10 TeV , (1.39)

as depicted in figure 1.2. It might seem that naturalness is a sort of aesthetical requirement on the theory without any real application. However it sits at the core of effective field theories, and has been used in the past to shed light on physical phenomena [59, 60]. For example, a first estimate of the mass of the charm quark was given through naturalness arguments applied to kaon oscillations [53]. Another interesting example of usage of the naturalness argument is in the explanation of the pion mass splitting. The mass splitting between charged pions π± and the neutral pion π0 _{is largely due to photon exchange. Using a hard momentum cutoff Λ for the}

Λ

γ

π

±

π

±

Figure 1.4: Process contributing to the charged pion mass. loop in figure 1.4, the contribution can be estimated as:

m2_π±− m2_π0 '

3α 4πΛ

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1.3. THE HIERARCHY PROBLEM AND NATURALNESS 15

Naturalness arguments imply that this contribution should not be much greater than the physical mass splitting:

m2_π±− m2_π0

phys ' (35.5 MeV) 2

. (1.41)

This sets a bound for Λ: Λ ≤

r 4π

3α35.5 MeV ' 850 MeV . (1.42) Indeed around this scale in the loop appears a composite particle, the ρ resonance, with mass mρ= 770 MeV. Physically what happens is that the pions at around that

scale start to reveal their composite nature, and therefore the na¨ıve loop integral is cut off by compositeness effects. Thus the naturalness criterion is a powerful argument to get hints on where new physics should appear.

After having convinced ourselves that naturalness is a useful tool, we can go back to the hierarchy problem itself. Several solutions have been proposed to address the lightness of the Fermi scale [33]. Among the many, we cite supersymmetry, stabilization from gravitational effects [97], different quantization prescriptions [63], multiverse [20] and the anthropic principle [37].

The pion example suggests another solution: afterall pions are scalars like the Higgs, and their mass splitting is protected from UV corrections by compositeness. It’s worthwhile to consider as an option the fact that the Higgs boson might be a composite particle rather than a fundamental one. In analogy to the pion analysis, we will consider as a solution of the hierarchy problem the framework of Higgs compositeness. Under this assumption, the Higgs is a composite particle, much like the pions, made by new fundamental fermionic constituents, called hyperfermions. The naturalness scale ΛN P can be identified as the compositeness scale of the theory,

and should be around the 1 ÷ 10 TeV scale to keep the Higgs mass natural, as emphasized in equation (1.39). One of the goal of this work is to study possible sectors whose dynamics is compatible with the composite Higgs scenario. The sectors are broadly characterized by the following aspects:

an asymptotically free, simple gauge group GHC, called hypercolor. This

interaction is responsible for the formation of bound states like the Higgs. a set of hyperfermions charged under GHC, exhibiting a global symmetry group

G. G should contain as a subset the IR symmetry of the SM Gcus. The SM

gauging can be seen as a weak perturbation to the strong hypercolor dynamics, and thus we can work in the limit of 0 SM gauge couplings.

Notice that in order to avoid again the naturalness problem, we do not allow any fundamental scalar in the UV sector. The full set of requirements is reported and motivated in chapter 5. Thanks to asymptotic freedom, below a certain energy threshold ΛN P, the GHC force binds together the hyperfermions. New resonances

will appear in the loops, effectively making the radiative contributions to m2_h natural. Moreover, a hyperfermionic bilinear condensate will be formed at ΛN P, breaking the

global symmetry group G to a subgroup H. The surviving IR symmetry group H must contain again Gcus since this is a symmetry of the low energy SM. As we will

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remark in chapter 3, if the Higgs is seen as a pseudo-Nambu-Goldstone boson (a hyperfermion-antihyperfermion bound state) associated to the coset G/H 5_{, then}

the Higgs mass can be made naturally lighter than the new physics scale ΛN P, which

is the typical mass of the heavy bound states. This is the same mechanism that allows pions to be lighter than baryons in QCD. Given that the hyperfermions are charged under SM interactions, the composite particles will be charged under it too. This is crucial if we want to identify some of the Goldstone bosons with the SU (2)L

doublet Higgs. An exact NGB should be massless, however via explicit symmetry breaking and radiative corrections it gains a potential, exactly like the QCD pions. In particular, in CH theories the Higgs NGB must acquire a potential that can realize electroweak symmetry breaking. These contributions are generated at the loop level via terms that explicitly break the global symmetry group G of the hyperfermionic sector. Usually these are mixing terms between hyperfermions and SM, and SM gauge interactions. Indeed the SM fermions will not be in general invariant under G transformations, but only under its subgroup Gcus (at least approximately). Such

terms destroy G invariance and are capable of generating the desired potential for the Higgs NGB. We will review in detail composite Higgs theories in chapter 3. Composite Higgs theories (CH) differ from the other options in the fact that they postulate new physics between v and MP L, built out of a strongly interacting sector

around the TeV scale. This means that usually their phenomenology is rich because of the full tower of composite resonances, of which the Higgs is simply the lightest one. An apparent problem is that we have introduced a new scale ΛN P around the TeV,

which is still much smaller than MP L. Does it suffer from the hierarchy problem?

No: the large hierarchy is generated by the running of the coupling of the strongly interacting theory, much like ΛQCD is generated via running. So the compositeness

scale is natural in our setup even if the cutoff ΛU V of the hyperfermionic sector

is again MP L. One of the disadvantages of postulating new physics at the TeV

scale is that now rare flavor processes are not suppressed by MP L as for example in

(1.35), but from a much smaller ΛN P. This is in tension with the need to protect

SM-suppressed processes such as decays that violate lepton family number. This implies that the flavor structure of the new theory beyond ΛN P must be highly

non-trivial, such as in minimal flavor violating (MFV) theories [36]. We will neglect in this work flavor-related aspects, but it’s something that must be kept in mind when discussing the natural scale of the Higgs.

1.4 Motivations

After having explained the reasons behind the idea of a composite Higgs theory, we can properly explain the goals of this work mentioned in the introduction. The starting point is the classification of the UV sectors of CH theories first done in [50]. In that work all the possible GHC and respective hyperfermionic content giving rise

to a NGB Higgs in the IR were classified. The authors made a further request on the sector, based on the partial compositeness6 _{framework, which turned out to be}

5_{see chapter 2.} 6_{See chapter 3.}

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1.4. MOTIVATIONS 17

quite restrictive on the possible UV sectors. The two purposes of our work stem from this analysis. For simplicity, we will drop the technical requirement of partial compositeness.

1.4.1 Leptocolor unification

The first purpose of this work is to extend the classification to sectors with an enlarged symmetry group, larger than the custodial group Gcus. There are many

ways to extend the symmetries of the SM into a larger group. A brief overview on the possibilities is given in chapter 4. In particular, the simplest extension of Gcus is

the Pati-Salam group [87]:

GP S= SU (4)LC× SU (2)L× SU (2)R (1.43)

where we unify the color SU (3)c and U (1)B−L in the leptocolor group. The name

comes from the fact that leptons can be seen as the fourth color of this enlarged SU (4)LC. We will gauge only the color subgroup, so the SU (4)LC is only an

approximate global symmetry of the sector. Indeed with the addition of the right-handed neutrino, the SM fermions are organized in approximate multiplets of the Pati-Salam symmetry. It’s natural to extend it to the hyperfermionic sector. In particular we focus on models in which this enlarged SU (4)LC is dynamically broken to the IR

symmetry SU (3)c× U (1)X. The peculiarity of these models is the appearance in the

spectrum of particles called leptoquarks. This term refers to scalars or vector particles that are charged, usually in the fundamental, of SU (3)c. The name comes from the

fact that they can mediate quark-lepton couplings. In other words, the request of the spontaneous breaking of the leptocolor symmetry can be reformulated in the request of having Pati-Salam scalar leptoquarks in the NGB spectrum. Leptoquarks are an old idea, however in very recent years they became popular again. Indeed they were postulated to be related to possible hints of new physics beyond the SM in third generation processes such as B meson decays [13, 14, 21, 34, 35, 40, 67, 93]. The mass range of the leptoquarks related to B anomalies is around the TeV scale, thus around the scale that would keep the Fermi scale natural. Therefore it’s tempting to find a common origin for the Higgs compositeness and the leptoquarks [79]. We stress that we will not discuss at all the phenomenology of these models, nor give an explanation to the B anomalies: the goal is to understand how a composite Higgs model can fit in leptoquarks in the most simple scenario. This is a first step toward giving an UV motivation to some of the phenomenological models based on Pati-Salam unification cited above. In chapter 5 we start by considering models that exhibit only the SU (4)LC symmetry and that dynamically break it to SU (3)c× U (1)X. We

find in particular three classes of models, based on the real, pseudoreal or complex nature of the GHC representation of the hyperfermions. Then we use this models

in conjunction with the EW cosets that contain the Higgs NGB, to get a complete classification.

1.4.2 Other dynamics?

The second purpose of this thesis is to understand if the classification given in [50] is complete. Models whose GHC content is chiral were completely neglected in that

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work. Since the SM is itself chiral, there is no reason to exclude such dynamics from the classification. In chapter 6 we discuss some chiral examples that might be of interest in composite Higgs theories. Notice that this problem is interesting even in the absence of the leptocolor symmetry request. Chiral models are not a curiosity: their NGB spectrum can be different from the one of vector-like theories, and some interesting coset that is excluded in vector-like dynamics can be realized. However the behaviour of chiral models is still unknown, so rather than attempting a full and exhaustive classification of these models we simply study the most promising ones, that can realize cosets that were not included in the classification.

In conclusion, our goal is to critically re-examine the previous work under two different points of view:

1. Can we classify, in the spirit of [50], models with a Pati-Salam enlarged symmetry in the UV, that is spontaneously broken in the IR to allow leptoquarks in the spectrum?

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Chapter 2 Spontaneous symmetry breaking

The goal of the present work is to give a description of the possible UV completions of the SM that can explain a composite Higgs, possibly having additional symmetries such as the Pati-Salam leptocolor. Before going in the details of such models, we give a review of the concept of spontaneous symmetry breaking (SSB): this theoretical framework allows the presence of naturally light scalar particles, the Goldstone bosons. Then we proceed to explain how spontaneous symmetry breaking can be achieved in strongly interacting QFTs. The prime example of dynamical SSB is the breaking of chiral symmetry in QCD. After a brief review of its behaviour we will look at deep results on the nature of SSB in similar theories. As the last section of the chapter, we review the CCWZ formalism, which is employed to describe the IR phase of a theory undergoing SSB.

2.1 Spontaneous symmetry breaking

The concept of spontaneous symmetry breaking is one of the pillars of modern physics. Its applications range from condensed matter physics to particle physics. Several different phenomena find a common explanation in its mechanism, like ferromagnetism and the very existence of scalar particles like pions. What is symmetry breaking? In a very concise way, there is a phenomenon of spontaneous symmetry breaking (SSB) when a theory is invariant under a given group G of transformation, but the ground state is invariant only under a smaller subgroup H ⊆ G. The linear sigma model [54] is a field theoretical example of SSB. It was also one of the first models able to describe the QCD pions and chiral symmetry breaking. The Lagrangian of such model is:

Lσ = 1 2∂µφ † ∂µφ − V φ†φ . (2.1)

The field φ is a scalar under Lorentz transformations and it’s a 4 dimensional real vector of an internal symmetry SO(4). Since the potential term V depends only on the SO(4) norm of the vector φ, the Lagrangian is invariant under the full SO(4) group. It’s important to keep in mind that the symmetry group for this model is continuous, so we are allowed to expand around the identity and perform small transformations. If we restrict only to renormalizable, Z2 invariant operators of

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dimension ≤ 4, the potential can be rewritten as:

V φ†φ = aφ†φ + b φ†φ2 . (2.2) Since in the potential there are no derivatives, in order to study its minimum we can restrict to analyze spatially constant field configurations. In order for the potential to be bounded from below, b > 0. If a > 0, the potential in (2.2) has a minimum in the origin, meaning φ = 0. However, if a < 0, the potential assumes the characteristic mexican-hat shape. Since we are looking at the case a < 0, b > 0, we can introduce the more usual parameters µ2_{, λ:}

V φ†φ = −1 2µ 2 φ†φ + λ 4 φ † φ2 . (2.3) In this case V develops a minimum away for the origin:

φ†φ = µ

2

λ ≡ v

2 _{> 0 .} _(2.4)

Notice that the minimum of the potential is degenerate. In particular, all field configurations with fixed length φ†φ = µ_λ2 are a minimum of the potential. However the system must choose one of the possible minima. For example it can take the value:

˜

φ = (0, 0, 0, v)t . (2.5) It’s straightforward to see that the system has lost the full SO(4) invariance since it has picked a particular direction. However it is still invariant under a subgroup H = SO(3) that acts on the first three components of φ. The elements of SO(4) not belonging to the surviving SO(3) are said to be broken. This is a na¨ıve explanation of a SO(4) → SO(3) spontaneous symmetry breaking. It’s interesting to study the theory around this minimum. In order to do so we have to Taylor-expand the field φ around the minimum v. We reparametrize the field φ as follows:

φ = π1, π2, π3, v + σt . (2.6) If we substitute (2.6) into the Lagrangian of the sigma model in (2.1), we obtain the following Lagrangian [89]: L =1 2∂µσ∂ µ σ − 1 2∂µπ k ∂µπk− µ2σ2−√λµσ3+ −√λµ πk2 σ − λ 4σ 4₋ λ 2 π k2 σ2−λ 4 π k 22 . (2.7)

In the new parametrization, we observe that there is a single massive mode, σ, and three massless modes, the πk_{. The π}k _{fields are the so-called Nambu-Goldstone}

bosons (NGBs), and their presence is a signal of SSB. In particular, each of them is associated to a generator of G not belonging to H: they are related to an infinitesimal rotation around the chosen minimum ˜φ that does not keep it fixed. The explanation of spontaneous symmetry breaking in terms of a Lagrangian invariant under G and of a degenerate ground state invariant only under H is intuitive enough for the purposes of the work. However some details and subtleties, such as the need of having infinite degrees of freedom, were omitted for the sake of clarity. A more formal and precise approach to SSB can be found in [103].

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2.2. GOLDSTONE THEOREM 21

2.2 Goldstone theorem

As seen in the linear sigma model in equation (2.7) the field φ has massless modes (also called zero-modes) associated to broken generators. The appearance of massless modes is actually a general feature of the spontaneous breaking of global symmetries. This fact is known as Goldstone theorem [61, 103].

Theorem 2.2.1 (Goldstone theorem). For every spontaneously broken continuous symmetry, the theory contains a massless scalar particle.

The idea of the proof is to exploit the symmetry of the classical potential V and expand it around the minimum ˜φ. Consider a Lagrangian invariant under a continuous group G, with field φ.

L = Lkin(φ) − V (φ) . (2.8)

The field φ lies in a representation of G, whose components are labelled by the index a. Let also ϕ ≡ φ − ˜φ. The Taylor expansion around the minimum ˜φ yields:

V (φ) = V ( ˜φ) + ∂V ∂φaϕ a₊ 1 2 ∂2_V ∂φa_∂φb( ˜φ)ϕ a_ϕb _{+ O ϕ}3 . (2.9)

By neglecting the constant V ( ˜φ) and using the fact that ˜φ is a minimum of the potential, the potential V at the quadratic order is:

V(2) = 1 2 ∂2_V ∂φa_∂φb( ˜φ)ϕ a ϕb ≡ 1 2M 2 ab ϕaϕb . (2.10) where we defined the mass matrix M2. The next step is to consider the case where the displacement ϕ from the minimum ˜φ is generated by a G transformation. Under an infinitesimal G transformation, the field φ transforms as:

φa→ φ0a = φa+ iTabφb ≡ φa+ δa(φ) , (2.11) where T is one of the generators of G. By hypothesis the potential V is invariant under the action of G. This implies the relation:

V (φ) = V (φ + δ) . (2.12) By plugging the expansion (2.9) in (2.12), and identifying δ( ˜φ) with ϕ, we get at the quadratic order the relation:

∂2_V

∂φa_∂φbδ

a_{( ˜}_φ)δb_{( ˜}_{φ) = M}2 ab_δa_{( ˜}_φ)δb_{( ˜}_{φ) = 0 .} _(2.13)

Since ˜φ is a minimum, the eigenvalues of M2 cannot be negative by definition. By differentiating the quadratic form in (2.13) we obtain a set of linear homogeneous equations:

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Thus either δ( ˜φ) = 0 or δ( ˜φ) is a massless eigenvalue of the mass matrix. By the hypothesis of SSB, the only transformations that do not leave ˜φ invariant are the ones generated by the broken generators in G/H. Thus there are at least dim(G/H) massless modes, or in other words the NGBs. For all the other cases δ( ˜φ) = 0, and nothing can be inferred on the mass matrix eigenvalue. There is a quantum analogue of the theorem [62]. In particular, one way to prove it is to substitute the classical potential V appearing in the Lagrangian with the quantum effective potential Γ. The Γ(2) _{contains the poles of the propagators, or the physical masses of the particles,}

thus in quantum systems the mass of the massless NGBs is protected from radiative corrections. We mention that it’s crucial in the argument that G is a continuous group. Indeed the NGBs are related to the amount of displacement of the minimum vector ˜φ, and therefore the argument does not apply if the broken symmetry is a discrete one, such as ZN: it was necessary in the proof to continuously displace the

minimum vector ˜φ.

2.3 Higgs mechanism

What happens if the symmetry group G is a local gauge symmetry instead of a global one? According to the arguments given in chapter 1, the massless modes can be cancelled by a particular gauge choice, the unitary gauge. In this gauge the massless modes are “eaten” by the gauge fields, which become massive. The degree of freedom of a single Goldstone boson becomes the longitudinal polarization of the gauge fields. This is how electroweak symetry breaking occurs in the SM, as seen in chapter 1. Although this picture works and gives the correct masses and couplings, there are subtleties that need a clarification. Indeed, gauge symmetries are never true symmetries of the system, but rather a redundancy of the description of the models utilized to describe a certain phenomenon. What does it mean to break a gauge symmetry, since it’s not a true symmetry? It turns out that gauge symmetries cannot be broken at all, meaning that in a fully gauge-invariant theory, non-gauge invariant operators cannot get a non-zero vacuum expectation value (vev). This result is known as Elitzur theorem [45]. How can Elitzur theorem be reconciled with the SM predictions, if these were relying on the Higgs field H getting a vev? The solution to this apparent contradiction is found in Fr¨ohlich-Morchio-Strocchi theory [52]. The idea is that, under a more thorough, non-perturbative analysis, only gauge invariant operators get a vev, such as H†H. However, when computing gauge invariant quantities such as the masses of the gauge bosons or coupling constants, the FMS approach and the na¨ıve one coincide. As a consequence, we will use the more easy, simplistic argument in later chapters. However it is intended that the more formal approach is the correct one, but for our purposes the na¨ıve approach gets the right result.

2.4 Dynamical symmetry breaking

In the previous sections we analyzed the consequences of spontaneous global and local symmetry breaking. Nevertheless, we did not give the exact details of

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2.4. DYNAMICAL SYMMETRY BREAKING 23

the mechanism behind SSB: we limited to state the existence of a potential with degenerate minimum and expanded the theory around it. In the Standard Model the spontaneous symmetry breaking of the electroweak sector SU (2)L× U (1)Y is

obtained by making the Higgs field H develop a vev (at least in the na¨ıve picture). However in a quantum system, if the quantum fluctuations become large, there might be no need of a fundamental field to achieve SSB. Since the mechanism underlying the SSB is due to the evolution of the system rather than by tuning the potential, this scenario is called dynamical symmetry breaking. One of the best known examples of this kind of symmetry breaking is QCD, but the same happens in other vector-like confining gauge theories.

2.4.1 QCD

Quantum chromodynamics (QCD) is a gauge theory with gauge group SU (3)c,

and the following matter content:

nf × ψ ∈ 3 , nf × ˜ψ ∈ ¯3 , (2.15)

where nf is the number of flavors. Both ψ, ˜ψ are left-handed (LH) Weyl fermions.

The Lagrangian is the usual Yang-Mills Lagrangian: LQCD = − 1 4F a µνFµνa+ ¯ψ /DFψ + ¯ ˜ ψ /D_F¯ψ˜ (2.16)

where an implicit diagonal sum is intended in the flavor index. Dr stands for the

covariant derivative for the representation r of the gauge group SU (3)c. In our

case the representations are the fundamental and antifundamental one (3 and ¯3 respectively). Notice that in principle the Lagrangian in equation (2.16) can contain an additional gauge invariant term, the theta term:

Lθ = θ g2 32π2F µν a_F_˜a µν ≡ θ g2 64π2µνρσF µν a_Fρσ a _. _(2.17)

However in QCD θ is experimentally very small (≤ 10−10) [23], and can be set to 0. The problem of understanding why Nature selected such a small value for θ goes under the name of strong CP problem. In this work we will only consider gauge theories with θ = 0 for the sake of simplicity. We are considering the theory in the so-called chiral limit, in which the fermions are massless. In this limit the theory has a nice symmetry group:

G = SU (nf)L× SU (nf)R, (2.18)

where SU (nf)L, (SU (nf)R) acts as a rotation on the left-handed ψ spinor

(right-handed ˜ψc). A theory like QCD in which for each Weyl LH fermion in a complex representation R of the gauge group there is a LH partner in the conjugate ¯R is called vector-like theory. The name stems from the fact that the Weyl fermions can be paired to form a Dirac vector:

Ψ = ψ ˜ ψc . (2.19)

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One of the most striking characteristics of QCD (and in general of some non-abelian Yang-Mills theories) is asymptotic freedom. Asymptotic freedom is related to the running of the coupling as a function of the typical energy µ of the scattering process one needs to compute. The behaviour of the coupling g is encoded in the β function: β (µ) = µdg dµ = −β0 g3 16π2 − β1 g5 4π2 + O g 5 . (2.20) The β function of a Yang-Mills theory is usually computed in perturbation theory, as shown in equation (2.20). In particular, consider a SU (N ) Yang-Mills theory with ni LH Weyl fermions in the representation Ri of SU (N ). The coefficients for such

a theory are known up to 5 loops [68]. In particular, the coefficient we are most interested in is the first one:

β0 = 1 3 11N − 2 X i niTRi ! , (2.21)

where TRi is the Dynkin index of Ri

1_{. The coefficient β}

0 can change sign depending

on the sum of the total Dynkin indices of the fermionic representations. Although at first it might seem like a very obscure result, it actually has a nice physical meaning. The running of the coupling g (µ) as a function of the scattering momentum µ implies that, if β is positive, then the effective charge grows at higher µ: this is related to the fact that virtual particles tend to screen the charge of a matter particle charged under SU (N ). By probing the particle with higher momenta, the scattering probe experiences a bigger charge the deeper it goes. On the contrary, if β is negative, it turns out that the screening particles are actually enhancing the bare charge. For example, in non-abelian gauge theories, the gauge fields (the gluons in QCD) interact between themselves. The interaction is such that the gluons tend to “anti-screen” their charge: they tend to create a cloud of virtual gluons around themselves that are not of “opposite” charge. Instead the effect of the fermions is to shield the charge, as can be seen by the different sign of their contribution in equation (2.21). So if there are too many fermionic fields, the SU (N ) charge is screened by the fermions, while in the opposite case the gluons self-interaction is enough to enhance the charge. If β < 0, at very high energy the charge probed is small, and it grows indefinitely when going at larger radius away from the bare SU (N ) charge. It’s important to understand that in a non-abelian gauge theory the “magnitude” of the strong charge is related to the Dynkin index of the representation, since it is the quantity appearing when computing scattering amplitudes. A theory with negative β is said to be asymptotically free. By computing the first coefficient in (2.21), it can be shown that

QCD is a theory of this kind.

The consequences of a negative β in QCD are far reaching. Indeed, it means that if the theory starts as perturbative in the UV regime, at some energy scale ΛQCD the coupling g grows too large to allow a perturbative expansion. So in the IR

the theory becomes strongly coupled. This implies that the UV degrees of freedom, the quarks, are going to form bound states. Such bound states are expected to be gauge-singlets. Indeed if the coupling is large enough, a charged bound state is

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expected to excite from the vacuum virtual gluons and quarks that cloud the naked charge. This phenomenon is also known as confinement, and it’s the reason why the QCD bound states, such as pions and baryons (protons, neutrons,...) are not charged under color SU (3)c. It’s believed that asymptotically free theories display a confining

behaviour, so we will assume that any such theory forms color (or the equivalent strongly interacting group) singlets in the IR. What happens more precisely below the confinement scale ΛQCD ≈ 200 MeV for QCD? The quarks bind together to form

the chiral condensate:

_¯

ΨΨ ≈ −Λ3/2

QCD , (2.22)

where a diagonal sum on flavor indices is left implicit, and where we employed the Dirac spinors Ψ. The bilinear condensate in equation (2.22) does not respect the full global chiral symmetry. Indeed it leaves invariant only the vectorial subgroup SU (nf)V, that simultaneously rotates with the same phase ψ and ˜ψc, so that it

rotates the full Dirac spinor Ψ. The other generators, called axial, are broken by the condensate. So in QCD the quarks form a bilinear that sponteaneously break the chiral symmetry. The NGBs of the broken axial generators are the pseudoscalar pions. The pions can loosely be thought as bound states of a quark-antiquark pair. The other bound states are the color singlet baryons, formed by 3 quarks (in the case of N = 3 colors). The baryons, unlike pions, acquire a mass of the order of ΛQCD

via condensate-interaction. So far we gave no proof for the breaking of the axial symmetry via the formation of the chiral condensate. However, as seen in section 2.4.2, 2.4.3 it can be shown that this is the true QCD behaviour in the IR. In reality, the chiral symmetry is only an approximate symmmetry of the theory. In fact, quark masses and QED interactions of quarks explicitly break the flavor symmetry. As a consequence, the pions are massive pseudo-Nambu-Goldstone bosons, although they are still lighter than the baryonic bound states. It should be remarked that the concepts of confinement and chiral symmetry breaking are distinct, although in QCD they happen at around the same scale. Indeed there are examples of theories that may confine withouth breaking any symmetry at all [16].

2.4.2 Anomaly matching

In section 2.4.1 we assumed the formation of a chiral symmetry breaking conden-sate due to the strong confining QCD force. The question of the IR behaviour of a strongly interacting force was partially addressed by ’t Hooft [105]. In particular, he gave a criterion, the ’t Hooft anomaly matching condition (THAMC), to study the spontaneous symmetry breaking of the global symmetry group G of a given confining QFT. In a confining theory with gauge group GHC (also called hypercolor

group), the UV field content is different from the content of the IR theory. Indeed below the hypercolor-confinement scale ΛHC the theory is made by composite gauge

singlets, assuming confinement occurs. The idea behind THAMC is that the two theories are simply describing two different aspects of the same theory. So there must be some quantity that is the same in the two scenarios. Let’s imagine that the global symmetry group G of the UV theory is weakly gauged with coupling constant gf. The gauge group of the theory is thus GHC × G. The original theory

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will be recovered in the limit gf → 0. In general the G group will be anomalous2.

Let Af f be the G anomaly due to the fundamental fermions in the UV theory. In

order to cancel the anomalies, we add spectator fermions, with the right G charges to cancel G anomalies, and singlets under GHC, so not to spoil the strongly interacting

dynamics. This condition can be rewritten as:

Af f+ Asf = 0 , (2.23)

where in equation (2.23) Asf is the G anomaly due to the spectator fermions. Notice

that in the UV theory the fields contributing to the G anomalies (also called ’t Hooft anomalies) are the spectator fermions and the fundamental UV fermions. If gf 1,

we expect that this perturbation to the GHC dynamics will be negligible, so that

confinement and bound states formation is not affected by the weak gauging. If there is no SSB, the composite baryons are massless: there is no condensate that can give them mass. In other words, if the chiral symmetry is unbroken in the IR theory, it prevents the construction of any massive bound states. Thus below the confinement scale ΛHC the particle spectrum of the gf 6= 0 theory is made by:

the GHC-singlet, massless bound states of the original theory.

the weakly gauged G gauge bosons. the GHC singlet spectator fermions.

Since in our setup the flavor symmetry group G is gauged, the IR theory must be free from the gauge anomalies of the G gauge bosons. The particles circulating in the triangle diagram loops, and thus the ones contributing to the anomaly, are the spectator fermions and the IR bound states. The condition on the IR theory is then: Asf + Abs = 0 , (2.24)

where Abs in equation (2.24) is the G anomaly of the composite bound states. But

the spectator fermions’ anomalies Af s are the same in the UV and IR regime. Thus

the anomaly-free conditions in equations (2.24), (2.23) imply that the G anomaly of UV fundamental fermions must be the same of the IR anomaly of the composite bound states:

Af f+ Abs = 0 . (2.25)

The condition (2.25) is not trivial to satisfy: it strongly depends on the UV field content and GHC.

There is a second condition that is used together with (2.25): the so-called persistent mass condition (PMC). The PMC states that the IR bound states must be made in such a way that if one of its constituent fermions gets a large mass in the UV theory, then also the bound state should become massive and disappear from the IR description. If we give mass to a single UV fermion, then the flavor group is explicitly broken by this mass term to a subgroup G0. The IR fermionic bound states can be decomposed in representations of G0. Since the bound states must disappear if one of the constituent becomes massive, then the IR content must allow the construction of

(33)

such massive Dirac spinors for each representation R0 of the flavor group G0 present in the IR spectrum. The PMC basically is equivalent to the condition that there must be the same number of left-handed bound states in R0 and right-handed bound states in ¯R0 _{in order to allow the decoupling. The theoretical foundation of the PMC}

is the Appelquist-Carazzone decoupling theorem [6]. However, this result works only for large bare masses of the fundamental fermions, and the m → ∞ limit can present problems of phase transitions in the initial theory, which invalidates the argument for our massless theory. However, for vector-like gauge theories like QCD, it has been proven by Vafa and Witten that a bound state gets always a mass if one of its constituent has a mass m 6= 0, independently of its value.

The anomaly matching condition is not trivial to satisfy. As an example, con-sider a SU (2N ) gauge theory with nf fermions of spin 1/2 in the fundamental F

representation of SU (2N ) and nf fermions in the conjugate representation ¯F . The

global symmetry group is G = SU (nf) × SU (nf). The axial transformation are, as

in QCD, anomalous, and the anomaly is: Af f = 2N

nf

2 . (2.26)

In order to cancel this anomaly, we add N nf hypercolor singlet transforming in the

(F, F ) representation of G. Now that the theory is anomaly free, we can weakly gauge G. Assuming no SSB occurs, the ’t Hooft anomaly of the fundamental UV fermions must be matched to the one of the IR fermionic bound states, in accordance with (2.26). But the IR, gauge-singlet bound states can’t be fermions since they are made of an even number of hypercolored fermions. Thus they can never contribute to any chiral anomaly, such as the ’t Hooft anomalies of chiral G = SU (nf) × SU (nf).

So the THAMC equation does not have any solution for this simple theory.

Usually when computing THAMC, the IR spectrum is considered to be made only by the simplest hypercolor fermionic singlets (such as baryons in QCD). It’s legitimate to ask if by adding in the IR spectrum exotic singlets (as example pentaquarks in QCD) the THAMC can be satisfied. It can be shown that if the simplest singlets cannot match the flavor anomalies, the exotics will not match either [48]. The THAMC and PMC are vastly studied in the literature, but in vector-like gauge theories, barring very few exceptions, there are practically no solutions. So at least some axial subgroup must be broken [26, 106]. There are chiral theories, such as the Gerogi-Glashow model and the Bars-Yankielowicz model that satisfy the THAMC, thus admitting massless IR composite fermions. We will briefly review these theories in section 2.4.5 and chapter 6. What can be inferred from the non-existence of solutions to the THAMC? The anomalies must be matched by some massless particle in the IR theory. In the absence of SSB, these particles had to be massless fermions. But if there is SSB, as seen in section 2.2, massless scalar particles can be found in the spectrum. So, since there can be no massless composite fermions to ensure the anomaly-free condition of the IR theory, the massless Goldstones must be the ones responsible for cancelling the anomaly via triangle diagrams. A remark: we assumed in the discussion that the only IR massless particles are either scalar or spin 1/2 fermions. This is not restrictive at all, since it can be shown that these are the only two possibilities for massless particles [114]. Therefore the lesson to be learnt is that the failure of THAMC implies the spontaneous symmetry breaking of the chiral

Leptocolor symmetric completions of Composite Higgs theories

UNIVERSIT 

A DI PISA

Dipartimento di Fisica

Corso di Laurea Magistrale in Fisica

Tesi di Laurea Magistrale

Leptocolor symmetric completions

of Composite Higgs theories

Candidato

Marco Costa

Relatore

Prof. Dario Buttazzo

Relatore interno

Prof. Alessandro Strumia

Contents

Introduction

Chapter 1

Motivations

1.1

The Standard Model

1.2

The Higgs field

1.2.1

The Higgs boson

1.2.2

Gauge bosons

1.2.3

Fermions

1.2.4

Custodial symmetry

1.3

The hierarchy problem and naturalness

Λ

γ

π

π

1.4

Motivations

1.4.1

Leptocolor unification

1.4.2

Other dynamics?

Chapter 2

Spontaneous symmetry breaking

2.1

Spontaneous symmetry breaking

2.2

Goldstone theorem

2.3

Higgs mechanism

2.4

Dynamical symmetry breaking

2.4.1

QCD

2.4.2

Anomaly matching

UNIVERSIT