PhD thesis - CP3-Origins

Beyond Standard Model PhysicsMarch 18, 2013Tuomas HapolaInstitut for Fysik, Kemi og FarmaciCP 3 - OriginsParticle Physics & Origin of MassAcademic Advisor: Prof. Francesco SanninoA dissertation submitted to the University of Southern Denmarkfor the degree of Doctor of Philosophy

vResuméEn æra sluttede i den 4. juli sidste år, da to eksperimenter på LargeHadron Collider annoncerede opdagelsen af en resonans i overensstemmelsemed egenskaber af Standardmodellens Higgs boson. Med målingenaf Higgsbosonens masse er alle Standarmodellens nitten frie parametrefastlagt. Standardmodellen er med høj præcision i overensstemmelsemed et stort antal af observabler ved forskellige energi-skalaer.I denne afhandling undersøger vi udvidelser af standardmodellen,hvor standard Higgs sektor erstattes af en stærkt vekselvirkende sektor.Vi begynder med en kort introduktion til Standardmodellen og gennemgåårsagerne til, at der må være noget ud over Standardmodellen. Efterudviklingen af værktøjer til at studere stærkt vekselvirkende teorier,fortsætter vi med Minimal Walking Technicolor modellen og undersøger,hvordan det muligvis manifesterer sig ved Large Hadron Collider.Det sidste kapitel i afhandlingen er dedikeret til en forlængelse, somkun tilføjer flere partikler i forlængelse af Standardmodellen. En af denye partikler, der er stabil og elektrisk neutral, forklarer eksistensen afen slags stof i Universet, som ikke skinner lys.

viiPrefaceThe research for this thesis was done at the Centre for Cosmology and Particle PhysicsPhenomenology (CP 3 -Origins), University of Southern Denmark. I would like to thankmy advisor Francesco Sannino for the opportunity to do this PhD in Odense, and for allthe guidance during these years. I would also like to thank all the people in CP 3 for theencouraging working atmosphere.Finally, I would like express my deepest gratitude to Krista for her love, support andpatience.Tuomas Hapola

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ixList of PublicationsThe following publications are included in the thesis:I Pseudo Goldstone Bosons Phenomenology in Minimal Walking TechnicolorT. Hapola, F. Mescia, M. Nardecchia and F. SanninoEur. Phys. J. C72, 2063 (2012) [arXiv:1202.3024 [hep-ph]]II W’ and Z’ limits for Minimal Walking TechnicolorJ. R. Andersen, T. Hapola and F. SanninoPhys. Rev. D85, 055017 (2011) [arXiv:1105.1433 [hep-ph]]III Composite Higgs to two Photons and GluonsT. Hapola and F. SanninoMod. Phys. Lett. A26, 2313 (2011) [arXiv:1102.2920 [hep-ph]]IV Perturbative Extension of the Standard Model with 125 GeV Higgs andMagnetic Dark MatterK. Dissauer, M. T. Frandsen, T. Hapola and F. SanninoAccepted for publication in Phys. Rev. D [arXiv:1211.5144 [hep-ph]]

Contents1. Introduction 32. Introduction to Elementary Particle Physics 52.1. Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2. Goldstone’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3. Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4. Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.1. Custodial symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5. Going Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.1. Empirical proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.2. Theoretical Arguments . . . . . . . . . . . . . . . . . . . . . . . . 172.5.3. Something potentially dangerous . . . . . . . . . . . . . . . . . . 182.5.4. How to go Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . 203. QCD and Chiral Symmetry Breaking 213.1. Effective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2. Banks-Casher Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3. Vafa-Witten Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4. Gauging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5. Vacuum Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6. Weinberg Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334. Technicolor 354.1. Historical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2. Extended technicolor and Walking . . . . . . . . . . . . . . . . . . . . . . 37xi

Contents 15. Minimal Walking Technicolor 475.1. Goldstone Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1.1. Effective Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2. Vector Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2.1. Non-linear Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 555.2.2. Linear Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.3. Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3. Technicolor Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.1. Hybrid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656. Dark Matter Model Building 676.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70A. Witten Anomaly 71B. Dyson-Schwinger Equation 75Bibliography 83List of Figures 89List of Tables 91

Chapter 1.IntroductionThe Large Hadron Collider (LHC) is the biggest scientific instrument ever created. Itaccelerates and collides protons along the 27 kilometers long tunnel excavated beneaththe Franco-Swiss border. In addition to the main accelerator, several other smallermachines are needed to inject the protons into the LHC. Protons, which result fromhydrogen atoms by removing electrons, are first accelerated in the linear collider. Fromthe linear collider, protons are injected into the proton-booster synchrotron, followedby the proton synchrotron and the super proton synchrotron. From the super protonsynchrotron, protons with 450 GeV energy are finally injected into the LHC. Protons arethen accelerated to obtain the wanted collision energy.Proton beams consist of 2808 bunches with 1.15 · 10 11 protons in each bunch. At thecollision point, the transverse size of both beams is squeezed to a size of tens of microns.Together with bunch crossing interval of the order of tens of nano seconds this shouldyield an instantaneous luminosity up to 10 34 cm −2 s −1 . High instantaneous luminosityand frequent collision rate are desired to be able to study rare processes. The downsideis that there will be tens of simultaneous events per bunch crossing i.e. events are pilingup.When two protons collide they break apart into quarks and gluons. High energycollision of these quarks and gluons carrying a small fraction of the total collision energyis what we are actually interested in, and what we can calculate using the StandardModel (SM). This hard process is accompanied by low momentum transfer processes andthe experimental challenge is to pin down what happens in the hard interaction.The main physics goal of the LHC is to complete the dynamical picture of the SM. Afirst step towards this goal was taken when the two biggest experiments, ATLAS and3

4 IntroductionCMS, announced the discovery of a resonance consistent with the properties of the SMHiggs boson [1, 2]. This particle was the final missing piece of the SM and after its massis measured, all the nineteen free parameters of the SM are fixed. The next big step is tofind out what lies beyond the SM. Despite of its success it cannot accommodate all theobservations and there are a number of theoretical reasons why it cannot be the ultimatetheory of nature.This thesis consists of four original research papers [I-IV] and an introductory andsummary parts presented below. In Chapter 2, the SM of the particle physics and itsproblems are discussed. The chapter begins with a discussion about the symmetriesand why the symmetries are the primary reason for the predictive power of the SM. InChapter 3, we introduce chiral symmetry breaking using the quantum chromodynamics(QCD) as an example. Chapter 4 gives an introduction to Technicolor and Chapter 5summarizes the results of papers [I-III]. The last Chapter 6 introduces the Dark Matterproblem and summarizes the results of paper [IV].

Chapter 2.Introduction to Elementary ParticlePhysics2.1. SymmetriesSymmetry means invariance under a set of transformations. For, example a square is asymmetric geometric object which looks exactly the same if we rotate it by an angle of90 degrees, or by n ∈ Z times this angle. The set of all symmetry transformations forma symmetry group of the object. Of course, not all transformations are symmetric. Arotation is a continuous transformation, but we could have used a discrete transformationas an example as well.Similarly, the laws of nature can be symmetric. This means the form of the equationsdescribing the law is maintained under a change of variables and/or space-time coordinates.Hence, it is natural to categorize the symmetries accordingly to geometric symmetriesand internal symmetries. The geometric symmetries act on space-time coordinates andthe internal symmetries do not act on them.We can relate the continuos symmetries of the equation of motions to conservedquantities. The precise mathematical formulation of this is given by the Noether’stheorem [3].Theorem. (Noether) If the equations of motion are invariant under a continuous transformationwith n parameters, there exist n conserved quantities.This applies to both geometric and internal symmetries.5

6 Introduction to Elementary Particle PhysicsIn classical mechanics the basic space time dependent transformations are translationin time, translations in space and rotations in space. If the equations of motion are invariantunder these transformations, the corresponding conserved quanties, or conservationlaws, areTime independence → Energy conservationTranslation independence → Momentum conservationRotational independence → Angular momentum conservationIn order to discuss the relativistic particles, it is convenient to introduce Minkowskispace M which is a real 4-dimensional vector space parametrized by coordinates x µ = (t, ⃗r)and equipped with the metric(ds) 2 = (dx) 2 = dt 2 − (d⃗r) 2 . (2.1)The Lorentz transformations x → x ′ = Λx and the space-time translations x → x ′ = x + a(a ∈ M) both form a group of transformations leaving the Minkowski metric invariant. Asemi-direct product 1 of these groups form the group of isometries of Minkowski space-time,called the Poincaré group.Describing an elementary particle should not depend on its position in space-timeor if the observer is in uniform motion relative to it. Thus, all the elementary particlescan be classified according to representations of the Poincaré group. Moreover, themathematical description of a particle and its interactions should be invariant under thePoincaré group.Let us write down a Lagrangian which is invariant under the Poincaré transformationsL = ¯ψ(iγ µ ∂ µ ψ) = ¯ψ(i/∂ψ). (2.2)It describes a Dirac fermion ψ(x) and is in addition invariant under a global U(1) phasetransformationψ(x) → e ieα ψ(x), (2.3)where e and α are space-time independent constants. If we take the α to be a space-timedependent function, equation (2.3) is no longer invariant under the U(1) transformation.1 The group of translations is Abelian.

Introduction to Elementary Particle Physics 7In order to restore the invariance, we replace the partial derivative with the covariantderivativeD µ = ∂ µ + ieA µ , (2.4)where the gauge field A µ transforms asA µ → A µ − ∂ µ α(x). (2.5)Local phase transformations are called gauge transformations and the invariance underthese transformations is known as gauge invariance. The procedure is called gauging andit fixes the form of interactions. The local, and also global, U(1) symmetry is a continuoussymmetry and the corresponding conserved quantity, in this case, is the electric charge.This can be generalized to non-abelian compact Lie groups. It is important to note therole of internal and space time symmetries.The internal and space-time symmetries are described in terms of Lie groups. There isno non trivial way to combine these symmetries according to Coleman-Mandula theorem[4]. If we consider a more general algebraic structure than a Lie group, namely a gradedLie group, this can be omitted. The Haag-Lopuszánski-Sohnius theorem [5] states thatthe most general graded Lie group of symmetries of a local field theory is the N-extendedsuper Poincaré group. This allows non trivial mixing between internal and space-timesymmetries leading to symmetry relating bosons and fermions.Up to this point, we have considered only exact symmetries. Later, it will be importantto differentiate what actually is symmetric, Lagrangian or the solutions, and at whatscale the symmetry manifests itself and, if broken, how it is broken.Explicit breaking can occur via non-invariant terms in the Lagrangian. This does notmean that the symmetry cannot be used to draw conclusions if the breaking is small.Some of the Lagrangian’s classical symmetries can be spoiled by the quantum effects.This is called an anomalous breaking and the term driving the breaking is called ananomaly. It is important that in the end all the anomalies are cancelled. Otherwisethe renormalizability of the theory is destroyed. If the Lagrangian is more symmetricthan the quantum states, the symmetry is said to be spontaneously broken. Especiallyinteresting is if the vacuum is not invariant under the same symmetries as the Lagrangian.

8 Introduction to Elementary Particle Physics2.2. Goldstone’s TheoremConsider an action which is invariant under the infinitesimal transformationφ i → φ ′ i = φ i + θ a (δφ) α i , (2.6)where , for generality, φ i can be either fermonic or bosonic. Following from the Noether’stheorem, we have a conserved charge∫Q a (t) = d 3 xJ0 a (x), (2.7)where the corresponding current isJµ(x) a = −δLδ(∂ µ φ i ) (δφ)a i . (2.8)Let us take a small side step at this point for further convenience. The conjugatefield is defined asπ i (x) =δLδ∂ 0 φ i. (2.9)Using the canonical commutation relations, or the anti-commutation relations in thefermion case,[π i (⃗x, t), φ j (⃗y, t)] = −iδ 3 (⃗x − ⃗y)δ ij , (2.10)one can show that the conserved charges generate the algebra[Q a (t), Q b (t)] = if abc Q c (t). (2.11)From this relation, it follows that also the unintegrated currents satisfy[J a 0 (⃗x, t), J b 0(⃗x, t)] = if abc J c 0(⃗x, t)δ 3 (⃗x − ⃗y). (2.12)This is the basis of current algebra which was extensively employed during the earlydays of modern particle physics.

Introduction to Elementary Particle Physics 9Let Q be a symmetry charge, and the corresponding conserved current J µ , so thatQ|0〉 ≠ 0 (2.13)The vacuum state |0〉 is not annihilated by Q and therefore does not represent the truevacuum of the theory. Current conservation implies that∫0 = d 3 x [∂ µ J µ (x), φ(0)]∫= ∂ 0 d 3 x [ J 0 (x), φ(0) ] ∫+Sd ⃗ S[⃗J(x), φ(0)],(2.14)for the field φ(0). We take the surface S to be large enough so that the last term vanishes.This givesThend[Q(t), φ(0)] = 0. (2.15)dt〈0| [Q(t), φ(0)] |0〉 = C ≠ 0, (2.16)which, after inserting a complete set of intermediate states and integrating over ⃗x gives∑(2π) 3 δ 3 (⃗p n ) [ 〈0|J 0 (0)|n〉〈n|φ(0)|0〉e −iEnt − 〈0|φ(0)|n〉〈n|J 0 (0)|0〉e iEnt] = C ≠ 0.n(2.17)Now, if E n ≠ 0 the opposite frequency parts cannot cancel and the expression cannot beconstant. In order to satisfy the previous equation the intermediate states have to bemassless. The existence of these states is proven by the fact that C ≠ 0. Thus for everybroken generator there must be massless state satisfying〈n|φ(0)|0〉 ≠ 0, 〈0|J 0 (0)|n〉 ≠ 0. (2.18)This result is called the (Nambu-)Goldstone’s theorem [6].

10 Introduction to Elementary Particle Physics2.3. SuperconductivityThe original motivation to formulate the Higgs mechanism came from condensed matterphysics and superconductivity. This section serves as an introduction to spontaneoussymmetry breaking and will also be used later on to motivate the idea of dynamicalelectroweak symmetry breaking.Superconductivity is the phenomenon of vanishing electrical resistivity at low temperaturesand expulsion of magnetic fields from the interior of the sample (the Meissnereffect). This phenomenon can be described near the critical temperature, T c , withthe Ginzburg-Landau model which is a generalization of the Landau model for phasetransitions.Let us begin with a LagrangianL = − 1 4 F 2 µν + |(∂ µ + iqA µ )φ| 2 + m 2 |φ| 2 + λ|φ| 4 , (2.19)which couples a complex scalar field φ to Maxwell’s theory [7]. This Lagrangian isinvariant under local U(1) phase rotations. In the static limit, (2.19) reduces to theGinzburg-Landau LagrangianL = 1 2 (∇ × A)2 + |(∇ − iqA)φ| 2 + m 2 |φ| 2 + λ|φ| 4 , (2.20)where m 2 = a(T − T c ) near the critical temperature. Above the critical temperaturem 2 > 0 and the minimum of the potential is at |φ| = 0. When the temperature is belowthe critical one, the minimum is at |φ| 2 = − m2 . The Vacuum is not anymore invariant2λunder U(1) rotations, which in the static limit areφ(x) → e iα(x) φ(x),A ⃗ → A ⃗i + ∇α(x). (2.21)qThe conserved current, associated with U(1) rotations, is⃗j = −i (φ ∗ ∇φ − φ∇φ ∗ ) − 2q|φ| 2 ⃗ A. (2.22)When T < T c and the field φ varies slowly over the medium:⃗j = − qm22λ ⃗ A ≡ −k 2 ⃗ A. (2.23)

Introduction to Elementary Particle Physics 11This is called the London equation. The Ohm’s law⃗E = R⃗j (2.24)defines the resistance R. From the London equation and current conservation one candeduce that the resistance R must be zero. Thus, we have superconductivity. TheMeissner effect can be derived taking the curl of Ampere’s equation∇ × (∇ × ⃗ B) = ∇(∇ · ∇) − ∇ 2 ⃗ B = ∇ × ⃗j, (2.25)⇒ ∇ 2 ⃗ B = k2 ⃗ B, (2.26)where we have used Gauss law. In one dimension B x ∼ e −kx , indicating that the magneticfields are expelled from a superconductor with penetration depth characterized by 1/k.By writing equation (2.26) in a covariant form✷A µ = −k 2 A µ , (2.27)we can see that the photon has acquired a mass k.This behavior can be explained in terms of a microscopic theory known as the BCStheory of superconductivity [8]. The main assumption of the BCS theory is that thereexists an attractive force between the electrons inside the medium. Thus, the electronswill pair and form a bound state called the Cooper pair. At low temperatures thesefall into the same quantum state forming a Bose-Einstein condensate. In this case, thecomplex φ above would be a many particle wave function. The coefficients m 2 and λ canbe calculated from the BCS theory which effectively coincides with the Ginzburg-Landaumodel near the phase transition. The attractive interaction arises when the electronphonon interactions overcome the repulsive Coulomb interaction. The nuclei in a crystaloscillate about their equilibrium positions and the quanta of these vibrations are thephonons.2.4. Standard ModelThe Standard Model (SM) of particle physics is a gauge field theory based on thegauge symmetry group SU(3) × SU(2) L × U(1) Y [9–11]. The field content of the SM issummarized in Table 2.1 The following notation is used:

12 Introduction to Elementary Particle PhysicsGauge FieldsSymbol Associate charge Group Coupling RepresentationB Weak hypercharge U(1) Y g ′ (1, 1, 0)W Weak isospin SU(2) L g (1, 3, 0)G color SU(3) g s (8, 1, 0)FermionsSymbol Name Baryon number Lepton number Representation1Q L Left-handed quark30 (3, 2, 1)3(u c ) i 1L Right-handed up quark30 (¯3, 1, − 4)3(d c ) i 1L Right-handed down quark30 (¯3, 1, 2)3L L Left-handed lepton 0 1 (1, 2, −1)(e c ) i L Right-handed lepton 0 1 (1, 1, 2)ScalarsSymbol Name RepresentationH Higgs boson (1, 2, 1)Table 2.1.: The field content of the SM.Q i L =⎛⎞⎠d i⎝ uiLL i L =⎛⎞⎠e i⎝ νi eL, (2.28)where u(d) is called an up(down) type quark, e a lepton, ν e a neutrino and i is thegeneration index. The SM contains three generations with a naming convention:u ∈ {u, c, t},d ∈ {d, s, b},e ∈ {e, µ, τ},ν e ∈ {ν e , ν µ , ν τ }.For further convenience the Lagrangian of the SM is useful to divide into four parts:L SM = L Gauge + L Kinetic + L Higgs + L Yukawa . (2.29)

Introduction to Elementary Particle Physics 13The first term contains kinetic terms for the gauge bosonsL Gauge = − 1 4 B µνB µν − 1 4 W i µνW iµν − 1 4 W a µνW aµν , (2.30)whereB µν = ∂ µ B ν − ∂ ν B µ , (2.31)W i µν = ∂ µ W i ν − ∂ ν W i µ + gɛ ijk W j µW k ν , (2.32)G a µν = ∂ µ G a ν − ∂ ν G a µ + g s f abc G b µG c ν. (2.33)The structure constants are defined as [τ i , τ j ] = iɛ ijk τ k and [λ a , λ b ] = if abc λ c , where τ iand λ a are the generators of SU(2) and SU(3) groups respectively. The second termcontains kinetic terms for the fermionsL = ¯Q L i /DQ L + ū R i /Du R + ¯d R i /Dd R + ¯L L i /DL L + ē R i /De R , (2.34)where the covariant derivative isD µ = ∂ µ − ig ′ Y B µ − ig τ i2 W i µ − ig sλ a2 Ga µ. (2.35)Here Y denotes the hypercharge of a respective particle. Of course the last two terms in thecovariant derivative can be absent if the particle is not charged under the correspondingforce.At this stage all the fermions are massless. A Majorana mass term is not possiblebecause all the fermions carry hypercharge. A Dirac mass term is not allowed becausethe left-handed and right-handed fermions are not in complex conjugated representations.Thus the fermion sector possesses five accidental global U(3) symmetries for the rightand left-handed quarks and leptons.The Yukawa Lagrangian violates these symmetriesL Y = −Y iju¯Q i LɛHu j R − Y ijdwhere Y u , Y d and Y e are 3 × 3 matrices andH =¯Q i LHd j R − Y e ij ¯L i LHe j R+ h.c., (2.36)⎛⎞⎠ . (2.37)h 0⎝ h+

14 Introduction to Elementary Particle Physicsis the Higgs doublet. A subset of [U(3)] 5 symmetries is left intact corresponding tothe baryon and lepton numbers. In the end, only a linear combination of these twosymmetries is an accidental global symmetry of the SM, separately these symmetries areanomalously broken. When the Higgs field acquires a vacuum expectation value (vev),⎛⎞〈H〉 = ⎝ 0 ⎠ , (2.38)vthe Yukawa interactions form mass terms for the fermions. The neutrinos are stillmassless because the minimal SM does not contain right-handed neutrinos. Formationof a Majorana mass term after the symmetry breaking is forbidden by the accidentalbaryon minus lepton number symmetry.The last part of the Lagrangian is the part containing only the Higgs and theelectroweak gauge bosonsL H = (D µ H) † D µ H − V (H),V (H) = −µ 2 H † H + λ ( H † H ) 2,(2.39)where the covariant derivative isD µ = ∂ µ + i g 2 σ · W µ + i g′2 B µ. (2.40)The mass terms for the gauge bosons follow from the kinetic term after the Higgs fieldacquires the vev.2.4.1. Custodial symmetryThe Higgs Lagrangian is SU(2) L × U(1) Y invariant by construction, but has also anaccidental symmetry. To illustrate better this accidental symmetry we can write theLagrangian in another form. Instead of a doublet, let us form a bi-doublet:Φ =⎛⎞⎝ h0∗ h +⎠ . (2.41)−h − h 0

Introduction to Elementary Particle Physics 15Now we can write the Higgs Lagrangian asL H = Tr (D µ Φ) † D µ Φ + µ 2 TrΦ † Φ − λ ( TrΦ † Φ ) 2, (2.42)where the covariant derivative isThe electroweak symmetry acts as followsD µ φ = ∂ µ Φ + i g 2 σ · W µΦ − i g′2 B µΦσ 3 . (2.43)SU(2) L :Φ → LΦ,U(1) Y : Φ → Φe − i 2 σ3θ .(2.44)We can make the global symmetry manifest by taking the hypercharge interactions tovanish, g ′ → 0. In this limit the Higgs Lagrangian has a global SU(2) R symmetrySU(2) R : Φ → ΦR † . (2.45)Therefore the Higgs Lagrangian has SU(2) L × SU(2) R symmetry which breaks down toSU(2) L+R when the Higgs field aquires a vev⎛ ⎞〈Φ〉 = √ 1 ⎝ v 0 ⎠ . (2.46)2 0 vThis breaking pattern yields three Goldstone bosons which are eaten by the Higgsmechanism providing masses to the weak gauge bosons.M 2 W = 1 4 g2 v 2M 2 Z = 1 4 (g2 + g ′2 )v 2 .(2.47)Thus, at tree levelρ =M 2 WM 2 Z cos2 θ W= 1, (2.48)where the θ W is the Weinberg angle. In the limit g ′ → 0 the W + , W − and Z bosonsform a triplet under the SU(2) L+R explaining why the masses are degenerate in this

16 Introduction to Elementary Particle Physicslimit. Notably the radiative corrections must be proportional to g ′2 protecting the treelevel value ρ = 1. For this reason this symmetry is called the custodial symmetry.2.4.2. CommentsThere are a few notes which should be kept in mind for later reference. The theory isnot invariant under axial gauge transformations and one has to check that the gaugeanomalies are cancelled. The gauge anomaly cancellation is shown in many quantumfield theory text books (see, for example, [7]). Because of the SU(2) L gauge group, theSM suffers also from the Witten topological anomaly, reviewed in Appendix A. Thisanomaly cancels because there is an even number of left-handed fermion doublets in thetheory.There is no reason for the Yukawa couplings in equation (2.36) to be diagonal.These matrices can be brought to diagonal form using bi-unitary transformations. Theelectromagnetic and neutral weak currents will remain intact under the diagonalizationand there is no flavor changing neutral currents. This is easily seen as all generationsare replicas of each other and unitary diagonalization matrices always give rise to theunit matrix. Also, the charged current for the leptons is flavor diagonal provided thatthe neutrinos are mass degenerate. This is true in the SM. Only the system of chargedweak currents involving quarks is effected by the mixing. The non-existence of the flavorchanging neutral currents in the SM is called the GIM mechanism [12]. Nowadays itlooks trivial, but back in the day it was proposed only three quarks were thought to existand it predicted the charm quark.The gauge structure SU(2) L × U(1) Y and the electroweak sector was tested thoroughlyat the Large Electron-Positron collider (LEP). Colliding electron- positron pairs at theZ-pole enabled the precise measurements of several observables [13]. These precisionobservables agree well with the theoretical predictions. By doing a global fit one canexamine how consistent the theory is or determine the validity of the model by overconstraining the system. This is now possible because we know the Higgs mass [14].

Introduction to Elementary Particle Physics 172.5. Going BeyondBefore discussing different strategies and ways to go beyond the standard model, it isnecessary to discuss why there should be something more than the SM. I will first listsome examples of observations which are not explained within the SM and then give afew theoretical arguments.2.5.1. Empirical proofThe first solid result which requires something beyond the SM is that neutrinos mustbe massive because they are observed to oscillate. The most minimal extension of theSM would be to add right-handed neutrinos when one would be able to write downMajorana mass terms for neutrinos or/and Yukawa interactions. It is interesting topoint out that the most stringent limits on the neutrino mass scale comes from thecosmological measurements [15] and not from the dedicated neutrino experiments. Theneutrino experiments, which study neutrino oscillations, can access only to the massdifferences [16].The next observational evidence does not come from the particle physics experimentsbut from the cosmology (see for example [17]). Only roughly five percents of theenergy content of the universe is made up of ordinary matter. Dark energy accountsapproximately 72 percents and dark matter (DM) circa 23 percents. We do not knowmuch about dark energy. The same applies for DM but at least we know that it behavesgravitationally like ordinary matter but does not shine light, which is where the namecomes from.2.5.2. Theoretical ArgumentsAt the Planck scale around 10 19 GeV, gravity becomes a strong force. Due to thenon-renormalizabilty of gravity, the SM can be at most an effective theory up to thePlanck scale. Once we have accepted the existence of a cut-off, we run into problems. Theradiative corrections to the Higgs mass diverge quadratically with this high energy cut-offdriving theory to be strongly coupled because λ H ∼ m 2 H /v2 . The quadratic correctionsand the huge hierarchy between the electroweak scale v ≈ 246 GeV and the Planckscale implies that there must be new physics at lower scales to meet the observed Higgs

18 Introduction to Elementary Particle Physicsmass. Otherwise, one must unnaturally fine-tune the Higgs mass order by order sothat the physical mass is the observed one. To this respect the experimental success ofthe SM is unreasonable.One could also ask that, why there is only one scalar field and three families offermions. The hierarchy in the fermion masses is also an interesting question. Withoutprior knowledge one would expect the masses to be in the order of the electroweak scale.Only the top quark mass fits this expectation.2.5.3. Something potentially dangerousLet us examine two potential problems, namely triviality and the vacuum stability. TheHiggs self-coupling runs as a function of the renormalization scale µ as [18]where t = ln µ. At one-loop the beta function readsdλdt = β λ, (2.49)β λ = 3 [λ 2 + 1 4π 2 2 λy2 t − 1 4 y4 t − 1 8 λ(3g2 + g ′2 ) + 1 ]64 (3g4 + 2g 2 g ′2 + g ′4 ) , (2.50)where y t is the top quark Yukawa coupling. Studying two different regimes λ >> g, g ′ , y tand λ > g, g ′ , y t . At this limit it is straightforward tosolve equation (2.49)λ(µ) =λ(v)1 − 3λ(v)4π 2ln(µ 2v 2 ). (2.51)The self coupling will hit a Landau pole at the scale µ = ve 2π/3λ(v) . This allows us todetermine λ max (v) by taking λ(µ) = ∞. This yieldsλ max (v) =4π(2) (2.52)µ3 ln2v 2

Introduction to Elementary Particle Physics 19The Higgs mass is related to the self-coupling through the relation m 2 h = 2λv2 . Pluggingin the maximum value for the self coupling gives:m h < √ 8π2 v(2). (2.53)µ3 ln2v 2Using the Planck mass as a cut-off for the theory, the numerical value of the upper limitis m h < 160 GeV.The lower limit is set by examine the region λ 0 we obtain a condition( ) µ2λ(v) + β λ ln > 0. (2.55)v 2Again, using the relation between the self-coupling and the Higgs mass we get( ) µm 2 h > −2v 2 2β λ ln . (2.56)v 2If we now plug in the numbers, we find that the lower bound is actually larger than theupper bound. The one loop result thus does not make any sense. A more careful analysisusing a 2-loop renormalization group improved effective potential suggests that we arecurrently just below the lower bound for stability [19].Notice that the uncertainties are still sizable; mainly coming from the Higgs massand from the top quark mass uncertainties. Even if the Higgs potential is not absolutelystable, it is enough if the decay time to another vacuum is longer than the age of theuniverse.

20 Introduction to Elementary Particle Physics2.5.4. How to go BeyondAs we saw in the last section we have some solid observational reasons to extend the SM.The theoretical problems are more difficult to use as a guiding principle. There is notheorem which states that nature should respect conceptual nicety. I would like to getrid of these problems. However, one must consider that after the discovery of the Higgs,and if it is confirmed to be SM like, one could argue that maybe these are not so severeafter all. Even if there is a more fundamental model solving these theoretical problems,we do not know the scale of the new physics where this new model manifests itself. If thescale is high enough, particle physics experiments cannot directly explore it. This is, ofcourse, a problem because, in the end, physics should be discipline explaining observedphysical phenomena.If we want to extend the SM, which problem we should tackle first. Or should we tryto kill all the problem at once. Traditionally, the hierarchy problem has been the drivingforce for BSM physics. The cure to other unsatisfactory features are then adapted toone’s favorite model to solve the hierarchy problem, if possible. The point here is thatthese, usually tremendously complicated, models give raise to a bunch of new problemswhich the SM does not suffer.From the experimental point of view, it might be too restrictive to only explore”complete” models which try to explain everything. A more meaningful approach maybe signature based studies [20]. The LHC collaboration have already published resultsbased on the simplified models. From the theorist point of view this can look like goingback in time, but without any hint of new physics the situation is difficult.I should stress that there is still much to do within the SM. Our understanding, forexample, of QCD is still mostly limited to the perturbative regime.

Chapter 3.QCD and Chiral SymmetryBreakingThe QCD Lagrangian for N f quark flavors readswhere the covariant derivative isL QCD = − 1 4 Ga µνG aµν +N f∑i=1¯ψ i (i /D − m i )ψ i , (3.1)D µ = ∂ µ − ig s G µ . (3.2)The basic parameters of QCD are the dimensionless bare coupling g s and the bare quarkmasses m i . In order to keep physics invariant under the renormalization, changing therenormalization point must be offset by changes in the renormalized physical parametersas a function of the energy. The amount by which the bare coupling must be shifted isgiven by the beta functionThe leading contribution to the beta function readsg 3 sβ(g s ) = −β 0(4π 2 ) + O(g5 s) = − g3 s(4π 2 )µ dg sdµ = β(g s). (3.3)(11 − 2 )3 N f + O(gs) 5 (3.4)The number of flavors in QCD is six. This means that the beta function is negative andthe theory is asymptotically free.21

22 QCD and Chiral Symmetry BreakingIntegrating equation (3.3) and defining a scale Λ QCD at which g s diverges, we haveg s (µ) 2 =(4π) 2β 0 ln ( µ 2 /Λ 2 QCD). (3.5)The scale Λ QCD is renormalization group invariant and the coupling g s depends onit. Hence, the theory is actually characterized by this scale and not by the dimensionlesscoupling constant in the Lagrangian. This phenomenon is called dimensionaltransmutation.The renormalization point dependence of the quark mass is given by the γ functionAt one loop level the γ function is given asµ dm idµ = −γ(g s)m i . (3.6)γ(g s ) = g2 s2π 2 + O(g4 s). (3.7)Let us forget the masses for a moment. The Lagrangian without mass term is invariantunder independent left-handed and right-handed (chiral) U(N f ) rotationsψ L → U L ψ L , ψ R → U R ψ R , U L(R) ∈ U(N f ), (3.8)where ψ L(R) = 1 2 (1 ± γ5 )ψ. The axial current is not conserved at quantum level and thetrue global symmetry of massless QCD isG = SU(N f ) L × SU(N f ) R × U(1) V . (3.9)A generally accepted picture is that the quarks condensate and pick up a non-zerovacuum expectation value〈 ¯ψi ψ i〉 ≠ 0. (3.10)This breaks the global chiral symmetry down to maximal diagonal subgroupSU(N f ) V × U(1) V . (3.11)

QCD and Chiral Symmetry Breaking 23According to the Goldstone theorem N f − 1 massless Goldstone bosons should appear.Next, we will discuss how to write down an effective theory describing the Goldstonebosons.3.1. Effective LagrangianIf a theory with a global symmetry G posseses a vacuum state which respects only asubgroup H of G, the action of G on this state generates a manifold of degenerate vacua.Any given state in this manifold is unchanged by a group of transformations isomorphic toH. Hence the set of transformations of degenerate vacua is isomorphic to the coset spaceG/H. Transformations in G/H correspond to directions of variations of the Lagrangianin which the effective action is level at its minimum. Quantizing the excitations alongthese direction produces zero mass particles one for each orthogonal direction in G/H.Next we will figure out how the Goldstone bosons fields transform following reference[21].The Goldstone theorem states that after the breaking there are dim(G)−dim(H)Goldstone bosons. The group G acts on the Goldstone bosons through some representationπ g → φ(g, π), g ∈ G. (3.12)Because φ is a representation, we have the composition lawφ(g 1 , φ(g 2 , π)) = φ(g 1 g 2 , π). (3.13)Let us consider the image of the origin, φ(g, 0). The set of elements h which map theorigin onto itself forms a subgroup H ∈ G. Furthermore φ(gh, 0) = φ(g, 0) according tothe composition law for any g ∈ G, h ∈ H. The function φ(g, 0) thus maps G/H on thespace of Goldstone bosons fields. As the dimension of the coset space is equal to thenumber of the Goldstone bosons, these can be identified with the coordinates of G/H.Thus the Goldstone bosons are said to live on the coset space.Every element of G can be uniquely decomposed as g = ξh where ξ is an element ofone of the equivalent classes {gh, h ∈ H}. Using the composition propertyφ(h, ξ ′ ) = φ(h, φ(g ′ , ξ)) = φ(hg ′ , ξ) (3.14)

24 QCD and Chiral Symmetry Breakingwe can find out the transformation law for ξgξ = ξ ′ h. (3.15)i.e. the standard acton of G on the G/H space. Thus the only freedom left is the choiceof representatives in the coset space.In the case of G = SU(2) L × SU(2) R and H = SU(2) V quotient space is SU(2). Thepion fields are the three coordinates needed to parametrize the manifold, or group in thiscase. The usual choice of coordinates is [22]U(x) = e i π F , π(x) = π a (x)τ a . (3.16)The pion decay constant, F , is defined via relation〈0|j µ5a |π b (p) 〉 = −ip µ F δ ab e ipx , (3.17)where j µ5a is the relevant current to considerer the pion as a Goldstone boson. Thetransformation law of U(x) follows from the relation gξ(x) = ξ(x) ′ hU ′ (x) = V R U(x)V † L , (3.18)where V R(L) is the SU(2) R(L) transformation. The leading term in the derivativeexpansion isL = F 24 Tr ( ∂ µ U∂ µ U †) . (3.19)If we include masses for the quarks, one can write down, in leading order, a term thatbreaks symmetryL sb = 1 2 F 2 ( BTr [ MU †] + B ∗ Tr [ M † U ]) , (3.20)where B is a normalization constant. Expanding the symmetry breaking term we canread of the pion massm π = (m u + m d )B + ... (3.21)

QCD and Chiral Symmetry Breaking 25The first term in the expansion is a constant which is related to the vacuum expectationvalue of the fermion condensate:〈0|ūu|0〉 = 〈0| ¯dd|0〉 = −F 2 B + ... (3.22)The two flavors are denoted with u and d. Combining the two equations above we endup with the Gell-Mann-Oakes-Renner relation [23]F 2 m 2 π = (m u + m d )|〈0|ūu|0〉| + ... (3.23)This allows one to study chiral symmetry breaking on the lattice by examining howthe mass of the pion scales as a function of quark masses. If the condensate has a nonzero value, the pion mass squared should approach zero linearly when the quark massesapproach to zero.3.2. Banks-Casher RelationThere is a way to compute the value of the condensate directly [24]. First step isto evaluate the fermion propagator as an eigenmode expansion [25]. Let us start byexpanding ψ in terms of eigenfunctions of the Dirac operator i /Dψ = ∑ nb n u n , (3.24)where the b n are Grassman variables andi /Du n (x) = λ n u n (x),∫d 4 xū m u n = δ mn . (3.25)We can write the QCD partition function in a specific background gauge field configurationas∫∫Z = D[ ¯ψ]D[ψ]D[A µ ]e i ∫ d 4 xL QCD= Π n db ∗ ndb n D[A µ ]e (iλn−m)b∗ nb n∫(3.26)= − D[A µ ]Π n (iλ n − m).

26 QCD and Chiral Symmetry BreakingThe quark propagator is then∫〈 ∣ ∣ 〉0 ∣ψ(x) ¯ψ(y) 1 0 =Z∑Π n db ∗ ndb n b i u i (x) ∑ b ∗ jū j (y)e (iλn−m)b∗ nb nij= 1 ∑u n (x)u ∗Zn(y)Π n≠m (m − iλ n ) (3.27)nu n (x)u ∗ n(y)m − iλ n= ∑ nThe non-zero eigenvalues come in complex-conjugate pairs. The quark condensateis evaluated by taking x = y, integrating over volume and averaging over all gaugeconfigurations〈0|¯qq|0〉 = 1 V∫d 4 x 〈 ¯ψ(x)ψ(x)〉= −2mV∑λ>01λ 2 n + m 2 . (3.28)Call the mean number of eigenvalues in an interval dλ per unit volume, ρ(λ)dλ. Thespectral density ρ(λ) can be introduced into above equation by taking the infinite volumelimit, at which 1/V ∑ n → ∫ dλρ(λ), yielding〈0|¯qq|0〉 = −2m∫ ∞0dλρ(λ)λ 2 + m . (3.29)2By taking the zero mass limit we arrive to the Banks-Casher relation〈0|¯qq|0〉 = −πρ(0). (3.30)Although this is, in principle, a straightforward way to study the chiral symmetrybreaking using lattice techniques, the practice is not so straightforward. See for example[26] and references therein.3.3. Vafa-Witten TheoremA natural question to ask at this point: Is there something special with the chiralsymmetries compared to vector symmetries? Vafa and Witten [27] have proved thatthis is indeed the case; vector symmetries cannot be spontaneously broken. Followingreference [25], let us go through the argument.

QCD and Chiral Symmetry Breaking 27Considering the two flavor QCD with equal quark masses m u = m d = m ≠ 0 thevector symmetry in question is the SU(2) V . If this symmetry were broken there wouldbe Goldstone bosons associated with the scalar currents.Let us consider Euclidean correlator〈〉∣C Γ = 0 ∣J ūd (x)J ¯du (y) ∣ 0 , (3.31)where J ūd = ūΓd are quark currents withΓ = 1, γ 5 , iγ µ , γ µ γ 5 , iσ µν . (3.32)Using the results derived in the section 3.2, we can present the correlators asC Γ (x, y) = 1 Z [Π n(m − iλ)] 2 Tr {ΓG(x, y)ΓG(y, x)} , (3.33)where G(x, y) is the Euclidean Green function of the u- and d-quarks in a given gaugefield backgroundG(x, y) = ∑ ku k (x)u † k (y)m − iλ k. (3.34)Note that we have explicitly employed the fact that the masses are degenerate. Inaddition, we have assumed that the common mass is real i.e. the θ angle of the QCDvacuum is zero.Using the symmetry u k → γ 5 u k , λ k → − λ k we can show thatγ 5 G(x, y)γ 5 = ∑ k[γ 5 u k (x)][γ 5 u k (y)] †m − iλ k= ∑ k[u k (x)u † k (y) ∑=m + iλ kk]u k (y)u † k (x)†m − iλ k(3.35)= G † (y, x).The Green function can be expanded over the full basisG(x, y) = s(x, y) + γ 5 p(x, y) + iγ µ v µ (x, y) + γ µ γ 5 a µ a µ (x, y) + 1 2 iσ µνt µν , (3.36)

28 QCD and Chiral Symmetry BreakingyieldingTr { γ 5 G(x, y)γ 5 G(y, x) } = Tr { |G(x, y)| 2}= 4 ( |s| 2 + |p| 2 + |v µ | 2 + |a µ | 2 + |t µν | 2) .(3.37)On the other hand, using equation (3.35), we haveTr {G(x, y)G(y, x)} = Tr { G(x, y)γ 5 G † (x, y)γ 5}= 4 ( |s| 2 + |p| 2 − |v µ | 2 − |a µ | 2 + |t µν | 2) .(3.38)This then implies|C γ 5(x, y)| ≥ |C Γ (x, y)| (3.39)in any given background gauge field configuration.By inserting a complete set of states and explicitly displaying the time evolutionoperator〈〉∣0 ∣J ūd (x)J ¯du (y, t) ∣ 0 = ∑ n〈 ∣0 ∣J ūd (x, 0) ∣ 〉 〈 〉∣ n n ∣e −Ent J ¯du (y, 0) ∣ 0(3.40)we see that the asymptotic behavior is dominated by the lightest stateC Γ ∝ e −m Γt(3.41)The equation (3.39) implies that a pseudo scalar state is lighter than any other state,and, in particular, lighter than any scalar statem P S ≤ m S . (3.42)If the vector symmetry were broken there would be massless scalar Goldstone bosons inthe spectrum. The mass inequality above shows that there must also be massless pseudoscalar states in the spectrum. We have assumed that m u = m d ≠ 0 meaning that thetheory has no exact axial symmetry and thus no reason for massless pseudo scalar statesto exist. This completes the argument that vector symmetries cannot be spontaneouslybroken.

QCD and Chiral Symmetry Breaking 293.4. GaugingIf we gauge a SU(2) L × U(1) Y subgroup of SU(2) L × SU(2) R , the formation of the quarkcondensate spontaneously breaks the SU(2) L × U(1) Y down to U(1) Q . Thus we canachieve spontaneous symmetry breaking and massive gauge bosons within the StandardModel without the standard Higgs sector. This is called dynamical electroweak symmetrybreaking.Gauging the subgroup means that we have to replace the ordinary derivative inequation (3.19) with the following covariant derivative:D µ U = ∂ µ Φ + i g 2 σ · W µU − i g′2 B µUσ 3 . (3.43)Expanding the matrix U, gives the following mass terms for the gauge bosonsm W = gF π2 , m Z = √ g ′2 + g 2 F π2 , m A = 0. (3.44)The value of the decay constant in our normalization is F π ≈ 93MeV . Hence QCDcannot provide the observed masses for the SM gauge bosons.We have to remember here that the symmetry is already broken explicitly by gauging.In the next section, we will study how the spontaneous and explicit breaking worktogether.3.5. Vacuum AlignmentHere we will follow original references [28, 29]. Let us consider a theory with fermions inthe representation r of SU(N), with global symmetry G breaking to its subgroup H. Thegenerators of G are denoted as G a , the generators of H as T i , and the generators of G/Has X z (orthogonal generators of G). The generators are normalized as TrG a G b = δ ab . Toeach generator of G corresponds a symmetry current J a µ (x) = ¯ψ L (x)γ µ G a ψ L (x) . BecauseG/H is a symmetric space, we can define a parity operator P so thatP 2 = 1, P T i P = +T i , P X z P = −X z . (3.45)

30 QCD and Chiral Symmetry BreakingThe global symmetry G of the theory is spontaneously broken by the vacuum state|0〉 to a subgroup H. The generators T i satisfy T i |0〉 = 0. The set of degenerate vacuummay be written as {exp (iα z X z ) |0〉} = {|α〉}. According to the Goldstone theorem, eachcurrent involving an X z can crete a single massless boson from the vacuum.〈π y (p)|J µ z (0)|0〉 = −ip µ f π δ yz (3.46)The generators X z correspond to a single irreducible representation of H, thus the decayconstant is the same for all the Goldstone bosons.Depending on if the representation r is real or complex we can have different globalsymmetries and symmetry breaking patterns.SU(2N)⊃ SU(N) × SU(N) × U(1)↓↓O(2N) or Sp(2N) ⊃ SU(N) × U(1)(3.47)A small perturbation, ∆H, breaking the symmetry may lift the degeneracy of thevacuum∆E(α) = 〈α|∆H|α〉 = 〈0|e −iαzXz (∆H)e iαzXz |0〉. (3.48)To identify |0〉 with the true vacuum ∆E(α) should have a minimum at α y = 0. Hence,we have two requirements for the vacuum energy∂∆E(α)|∂α α=0= i 〈0 |[X y , ∆H]| 0〉 = 0,y(3.49)∂ 2∆E(α)|∂α y ∂α α=0= − 〈0 |[X y , [X z , ∆H]]| 0〉 ≥ 0.z(3.50)The second derivative is proportional to the Goldstone boson mass matrix. Thenormalization is derived in [30, 31]m 2 yz = − 1 〈0 |[Xfπ2 y , [X z , ∆H]]| 0〉 . (3.51)

QCD and Chiral Symmetry Breaking 31If we consider two flavor QCD, in which the perturbation is given by the quark masses,we find:m 2 π ∼ m u + m df 2 π(3.52)We are interested here in a situation where the perturbation originates from theelectroweak gauge boson exchange. The weak interactions determine their own symmetrybreaking pattern by their own choice of vacuum orientation. Let us gauge a subgroupG w of G. The spontaneous breaking determines another subgroup H of G. The relativealignment is physically important. There can be also another subgroup S which is themaximal set of elements of G which commutes with G w .IIIISIVHIIG WGFigure 3.1.: Alignment of subgroups.nday, January 13, 13The G w × S symmetry remains the exact symmetry of the Lagrangian and thereforethe Goldstone bosons remain massless in regions I and II of Fig. Figure 3.1. TheGoldstone bosons in region III acquire mass, because they do not correspond to exactsymmetries. The gauge bosons in region I are coupled to broken symmetries and willreceive mass through the Higgs mechanism. The Goldstone bosons in II will remainin the spectrum as physical states. The gauge bosons in IV will remain massless. Theoverlap of G w and H is dynamically determined. It depends on which of the initiallydegenerate vacua is preferred by the perturbations.The G w couplings to fermions can be written as:L = A A µ ¯ψγ µ G A ψ = A A µ J µ A , (3.53)

32 QCD and Chiral Symmetry Breakingwhere the G A with capital indices are defined to absorb all the numerical factors. TheG A can be decomposed as:G A = T I(A) + X Z(A) (3.54)and the corresponding currents as:J µ A = J µ I(A) + J µ Z(A) . (3.55)Here we have used the parity we defined earlier, it forbids the mixed terms.The leading order perturbation is due to one-gauge bosons exchange,∆H = − 1 ∫d 4 x∆ µν (x)T [J µA (x)J νA ] , (3.56)2where ∆ µν is the gauge boson propagator.From the parity and Schur’s lemma it follows that the only H invariant term in theproducts X x X y and T i T j are δ xy and δ ij respectively. This allows us to write〈0|T J µi J νj |0〉 = 〈0|J T J T |0〉 δ ij = 〈0|J T J T |0〉 Tr (T i T j ) , (3.57)〈0|T J µz J νy |0〉 = 〈0|J X J X |0〉 δ zy = 〈0|J X J X |0〉 Tr (X z X y ) . (3.58)Thus〈0|T J µA J νA |0〉 = 〈 0|T J µI(A) J νI(A) |0 〉 + 〈 0|T J µZ(A) J νZ(A) |0 〉= 〈0|J T J T |0〉 Tr ( T I(A) T I(A))+ 〈0|JX J X |0〉 Tr ( X Z(A) X Z(A))= 〈0|J T J T |0〉 Tr ( )T I(A) G (A) + 〈0|JX J X |0〉 Tr ( ) (3.59)X Z(A) G (A)= 〈0|J T J T |0〉 Tr ( )G (A) G (A) + 〈0|JX J X − J T J T |0〉 Tr ( )X Z(A) G (A) .The first term in the last line is α independent. The α dependence is factored out inthe second term and the vacuum energy density reads:{ ∫ 1〈0|∆H|0〉 = ∆E(0) = E 0 +2}d 4 x∆ µν 〈0 |J µT J νT − J µX J νX | 0〉 Tr ( ) 2X Z(A) .(3.60)

QCD and Chiral Symmetry Breaking 33The preferred vacuum is the one which minimizes Tr ( ) 2.X Z(A) The second condition inequation (3.49) can now be written as:∂ 2m 2 xy = 1 Tr (X Z (A)) 2 × M 2 ≥ 0, (3.61)2 ∂α x ∂α ywhereM 2 = 1 ∫fπ2d 4 x∆ µν 〈0|J µT J νT − J µX J νX |0〉 . (3.62)The α dependence is, as stated earlier, only in the trace. In next section we will see howone can estimate the factor M 2 .3.6. Weinberg Sum RulesProperties of the underlying theory can be linked to the effective theory via Weinbergsum rules (WSRs) [32]. To derive the sum rules let us consider the time ordered productof two currentsiΠ abµν(q) ≡∫d 4 xe −iqx [ 〈 0|J a µ,V (x)J b ν,V (0)|0 〉 − 〈 0|J a µ,A(x)J b ν,A(0)|0 〉 ] (3.63)where a, b = 1, . . . , N 2 − 1 and the current read asJ a µ,V = ¯QT a γ µ Q, J a µ,A = ¯QT a γ µ γ 5 Q. (3.64)In the chiral limitΠ abµν(q) = (q µ q ν − g µν q 2 )δ ab Π(q 2 ), (3.65)where the function Π(q 2 ) obeys the unsubtracted dispersion relationΠ(Q 2 ) = 1 π∫ ∞0ds ImΠ(s)s + Q 2 . (3.66)

34 QCD and Chiral Symmetry BreakingWe are looking now a situation where SU(N f ) L × SU(N f ) R symmetry breaks downto SU(N f ) V . For the next step let us study the high energy behavior of∫G abµν(q) ≡d 4 xe −iqx [ 〈 0|J a µ,L(x)J b ν,R(0)|0 〉 ], (3.67)which is equal to 1/4 times the V V − AA product and transforms as the adjointrepresentation (N 2 f −1, N 2 f −1) under SU(N f) L × SU(N f ) R . We can employ the operatorproduct expansion to this task because the expansion coefficient functions respect theglobal symmetry. The asymptotic behavior is dictated by the lowest-dimension operatorin expansion of Jµ,L a (x)J ν,R b (0) which has non-zero vacuum expectation value. It mustbe a singlet under the stability group and transform as the adjoint under the globalsymmetry. Lowest dimensional operator to satisfy these constraints is a four-fermionoperator of dimension [mass] 6 . Thus G abµν(q) ∼ q −4 and Π(Q 2 ) ∼ Q −6 .Therefore, by expanding the right hand side of equation (3.66), we find the first andthe second WSR:1π∫ ∞0dsImΠ(s) = 0,1π∫ ∞0ds sImΠ(s) = 0. (3.68)Assuming that it is reasonable saturate the integral with the lowest lying narrowresonances, the vector and axial vector mesons as well as the Goldstone boson, we canwriteImΠ(s) = πF 2 V δ(s − M 2 V ) − πF 2 Aδ(s − M 2 A) − πF 2 π δ(s). (3.69)Substituting this to WSRs, we arrive to following relationsF 2 V − F 2 A = F 2 π , F 2 V M 2 V − F 2 AM 2 A. (3.70)The precision parameter S [33] is related to the V V − AA vacuum polarization andcan be expressed as∫ ∞[dsF2S = 40 s Im¯Π(s) = 4π VMV2− F ]A2 , (3.71)MA2where Im¯Π is the same as ImΠ without the Goldstone boson contribution. This iscommonly referred as the zeroth WSR.

Chapter 4.TechnicolorAn attractive feature of dynamical electroweak symmetry breaking is that it does notsuffer from the naturalness, hierarchy and triviality problems like the elementary scalarHiggs in the SM. In order to understand that large separation of scales arises naturallyin asymptotically free, strongly coupled theories, let us solve equation (3.3) for a largescale Λ in terms of the bare couplingΛ QCD = Λe −8πg (0)s b 0 . (4.1)If we take the scale to be the Planck scale, a coupling of order g s(0) ∼ 0.4 will generatea strong coupling scale Λ QCD ∼ 300 MeV. Therefore, the scale of symmetry breakinggenerated by the dimensional transmutation is naturally exponentially smaller than thecut-off of the theory.In technicolor theories new massless fermions, so called techniquarks, are includedinto the SM without the Higgs sector [34, 35]. They feel a new QCD-like strong force,corresponding to a gauge group SU(N TC ), which causes the formation of techniquarkcondensate breaking the global chiral symmetry G down to H ∈ G. In order to achievecorrect breaking of the electroweak sector, SU(2) L × U(1) Y group has to be embeddedproperly into G, as discussed in Section 3.5.There is an intriguing similarity between superconductivity and technicolor. TheSM with the minimal Higgs sector corresponds to the Ginzburg-Landau theory andtechnicolor would correspond to the BCS theory.35

36 Technicolor4.1. Historical SetupLet us consider a model with N f techniquarks in the fundamental representation ofSU(N T C ). The left-handed techniquarks are arranged into SU(2) L doublets while theright handed ones are singlets⎛ ⎞Q L = ⎝ U ⎠ , U R , D R . (4.2)DLThe hypercharge assignment is taken to be anomaly free. In analogue with QCD, thistheory has a scale Λ T C at which the technicolor gauge coupling ˜g diverges. Also thetechni-pion decay constant F T C is defined like the pion decay constant. Choosing thetechni-pion decay constant appropriately, we reproduce correct masses for the gaugebosons.Approximating QCD with the Nambu-Jona-Lasinio (NJL) model and using theDyson-Schwinger equations, one can show that QCD obey the following scaling rules [36]f π ∼ √ N c Λ QCD , 〈¯qq〉 ∼ N c Λ QCD . (4.3)Additionally, the technicolor model we are considering should follow these scaling rules,allowing us to writeF T C ∼√NT C3( )ΛT Cf π ,Λ QCDv = √ N D F T C ∼√ND N T C3( )ΛT Cf π , (4.4)Λ QCDRearranging the latter expression gives an estimate for the technicolor scaleΛ T C ∼ Λ QCDv √ 3f π√ND N T C. (4.5)Let us summarize: Using a new strong sector to replace the SM Higgs sector, we cancorrectly break the electroweak symmetry and give correct masses for the electroweakgauge bosons. At the same time we get rid of the hierarchy, fine-tuning and naturalnessproblems. A downside is that this mechanism alone cannot generate masses for the SMfermions.

Technicolor 374.2. Extended technicolor and WalkingEffective Couplings and Mass TermsLet us extend the symmetry of the theory so that above the scale Λ ET C > Λ T C thesymmetry group of the theory is G ET C [37, 38]. Here ETC refers to extended technicolorand TC to technicolor. Accommodating technifermions and fermions into a sameirreducible representation of G ET C , there are ETC gauge bosons connecting the SMfermions to technifermions. When the symmetry breaks,G ET C → G T C × G SM , (4.6)gauge boson of the G ET C become massive. At low energies, we can write down effectivefour-fermion interactions:α ab( ¯Q L T a Q R ¯ψR T b ψ L )Λ 2 ET C+ β ab( ¯QT a Q ¯QT b Q)Λ 2 ET C+ γ ab( ¯ψ L T a ψ R ¯ψR T b ψ L )Λ 2 ET CThe first term is responsible for giving masses for the SM fermions+ . . . , (4.7)m f ∼ g2 ET C〈M ¯QQ〉ET 2 ET C . (4.8)CIn the preceding equation g ET C is the ETC gauge coupling constant, M ET C ETC gaugeboson mass and 〈 ¯QQ〉 ET C the techniquark condensate evaluated at the ETC-scale. Themass hierarchy between families can be achieved breaking G ET C in several stepsG ET C → G n → . . . → G 1 → G T C × G SM . (4.9)During the every step some of the gauge bosons become massive and produce a massterm for desired fermions. This scenario is called tumbling.The second term in (4.7) can induce mass for the pseudo-Goldstone bosons[36]. Usingequation (3.51), mass term for the technician readsg2 ET Cm 2 π T C∼ 〈(Fπ 2 M ¯QQ) 2 〉ET 2 ET C . (4.10)C

38 TechnicolorThe two scales, Λ T C and Λ T C , can be connected using the renormalization groupequations [39]:(∫〈 ¯QQ〉ΛET C)ET C = exp d(ln µ)γ(α(µ)) 〈 ¯QQ〉 T C , (4.11)Λ T Cwhere γ is the anomalous dimension of the operator ¯QQ. We are talking about QCD-likeasymptotically free gauge theory with running coupling constant. Hence, γ ≪ 1 at largeenergies (see equation (3.7)) and 〈 ¯QQ〉 ET C ∼ 〈 ¯QQ〉 T C . Using the scaling relations, wecan write the mass terms as:m f ∼ g2 ET C F 3 πM 2 ET C, m πT C∼ g ET CF 2 πM ET C. (4.12)Flavor Changing Neutral CurrentFinally, consider the γ ab -term in (4.7). This operator mediates flavor changing neutralcurrents which must be suppressed. For example, operator£ |∆S|=2 = 4g2 ET C V ds2¯dγ µ PMET 2 L s ¯dγ µ P L s + h.c., (4.13)Ceffects on the mass difference, ∆M K , between the mixed eigenstates of the neutral kaons[40, 41]. In the above equation V ds is the mixing factor of the quarks and P L is theprojection operator. The kaon decay constant is defined as 〈 0| ¯dγ µ γ 5 s| ¯K(q) 〉 = i √ 2f K q µtaking numerical value f K ≈ 110MeV. The mass difference is given as∆M K = 2Re 〈 K|£ |∆S|=2 | ¯K 〉= 4g2 ET C Re(V ds 2 ) 〈K|MET 2 C M ¯dγ µ P L s ¯dγ µ P L s| ¯K 〉 (4.14)K≈ g2 ET C Re(V ds 2 ) fKM 2 K2M 2 ET CThe experimental value for this is ∆M K ≈ 3.5 × 10 −15 GeV [42]. Assuming the mixingfactor is the same oder of magnitude as the corresponding Cabibbo angle V ds ≈ 0.1, wecan obtain numerical estimate for the technipion mass using (4.12):m πT C 10GeV. (4.15)

Technicolor 39A particle with this small mass should be seen in the experiments, thus the mass hasto be bigger. We can increase the mass by making Λ ET C to be smaller. On the otherhand large suppression of the flavor changing neutral currents requires just the opposite.Again, we need something more.Walking DynamicsIntuitively the problem can be solved if there is a great difference between condensateat scales Λ T C and Λ ET C , since m f ∝ 〈 ¯QQ〉 ET C . Because of this, we can achieve largeenough masses and suppressed flavor changing neutral currents at the same time. Thisidea was first introduced in reference [43] without providing an explicit model.The coupling constant of a given Yang-Mills theory can behave also differently than inQCD, see Figure 4.1. If the β-function has a non-trivial fixed point α ∗ , the scale evolutionof the theory stops when it flows to this point β(α ∗ ) = 0. If we formulate the theory suchthat it almost reaches the fixed point, its scale evolution slows down near fixed pointbut does not stop. Technicolor model with β(µ) ≪ 1 between Λ T C ≤ µ ≤ Λ ET C is calledwalking technicolor model.Βg 2walkingQCDlikeFigure 4.1.: Different possible beta functions for SU(N) gauge theory.It is also important that α ∗ α c where the subscript refers to the critical value atwhich the condensate forms. If the fixed point is achieved first, the scale evolution stopsand the condensate can not form. On the other hand, formation of the condensate can

40 Technicoloraffect the behavior of the β-function, and drive the evolution away from the vicinity ofthe fixed point too soon.Dyson-Schwinger AnalysisDyson-Schwinger equation for the fermion self energy is derived in Appendix B∫/p − m 0 + C 2 (r)d 4 k(2π) 4 α(p, k)γµ D µν (p − k)S(K)Γ ν (p − k; k, p)= iS −1 (p),(4.16)where S(p) is the fermion propagator, D µν (p − k) the gluon propagator and Γ ν (p − k; k, p)the three point function. C 2 (r) denotes the Casimir invariant of the fermion representation.Note also that the coupling constant and the generator of the fermion representation aretaken out from the three point function. Writing the fermion propagator in the form:iS −1 (p) = Z(p 2 )/p − Σ(p 2 ), (4.17)where Z(p 2 ) is the wave function renormalization factor and Σ(p 2 ) is the self-energy,yields:Σ(p 2 ) + [1 − Z(p 2 )]/p =∫ d 4 km 0 + C 2(2π) α(p, 4 k)γµ G µν (p − k) Z(k2 )/k + Σ(k 2 )Z(k 2 ) 2 k 2 + Σ(k 2 ) 2 Λν (p − k; k, p).(4.18)Let us approximate the gluon propagator with the free propagator in the Landau gaugeand the three point function with the tree level vertex factor. Writing the substitutionsexplicitly, we can identify equations for the self-energy and the renormalization factor∫ dΣ(p 2 4 k) =m 0 + 3C 2(2π) α(p, 1 Σ(k 2 )4 k)2 (4.19)(p − k) 2 Z(k 2 ) 2 k 2 + Σ(k 2 )∫2 dZ(p 2 4 k) =1 + C 2(2π) α(p, 1 Z(k 2 )4 k)2 p 2 (p − k) 2 Z(k 2 ) 2 k 2 + Σ(k 2 )[ ]2(4.20)k · p(p − k) 2 + 2p · (p − k)k · (p − k).(p − k) 2

Technicolor 41Performing angular integrals we find Z(p 2 ) = 1 andΣ(p 2 ) = m 0 + 3C 24π∫ Λ 20[ ]dk 2 Σ(k 2 ) k2α(p, k)k 2 + Σ(k 2 ) 2 p 2 θ(p2 − k 2 ) + θ(k 2 − p 2 ) . (4.21)We can convert this to a differential equation by differentiating with respect to p 2 ,multiplying by p 4 and differentiating again. Using a notation λ = 3C 2α4πwe get2 dΣ(x)dxwith the following boundary condition+ xd2 Σ(x)dx 2+ λΣ(x)m 2 + x= αα c4and x = p2= 0, (4.22)ddx (xΣ(x)) | x=Λ 2 = m 0 . (4.23)The solution of this differential equation is a hypergeometric function [44]Σ(x) = MF( 12 + ω 2 , 1 2 − ω 2 , 2; − x M 2 ), (4.24)where ω = √ 1 − α α c. In the asymptotic region this solution can be expressed asΣ(x)M = c ( xM 2 ) −(1−ω)/2+ d( xM 2 ) −(1+ω)/2. (4.25)For strong coupling region, α > α c , the solution becomes oscillating:Σ(x)M√ ( M = 2√ 2 ω′cdx sin 2 ln x M + 1 ( )) −c2 2i ln , (4.26)dwhere ω ′ = √ αα c− 1. In the weak coupling region, the solution readsΣ(x)M= 2√ −cd√M2( ωx sinh 2 ln x M + 1 ( −c2 2 ln d)), (4.27)Only the oscillating solution can non-trivially satisfy the boundary condition (4.23)in the chiral limit m 0 → 0 [45]. Thus we can identify the coupling α c with the criticalcoupling for the chiral symmetry breaking. Using these solutions one can calculate theanomalous dimension at the critical coupling [46]:γ(α c ) = 1. (4.28)

42 TechnicolorLet us return to consider the renormalization group analysis (4.11), when β(µ) ≪ 1during the interval Λ T C ≤ µ ≤ Λ ET C . Now we know that anomalous dimension ofthe operator ¯QQ is in this case γ = 1, leading to the following relation between thetechniquark condensates〈 ¯QQ〉 ET C ≈ Λ ET CΛ T C〈 ¯QQ〉 T C . (4.29)Note the enhancement in comparison with the QCD like case, γ ≪ 1, yielding a correctionto the fermion and technipion mass terms (4.12). Walking over two orders of magnitudeproduces an upper limit for the technipion massΛ ET CΛ T C∼ 10 2 ❀ m πT C 1TeV, (4.30)which is large enough to be out of range of the todays accelerators.The Conformal WindowThe lack of full understanding of dynamics of strongly coupled theories using the perturbationtheory, makes model building a challenging problem. We would like to constructnear conformal theories, but we cannot just simply using pen and paper tell exactlywhich models fit into this category. As we have seen, lattice calculations can be used forthis task. However, before doing some time consuming calculations, we would like toknow what is worth calculating. Let us use the results from Dyson-Schwinger analysis toestimate which theories could be interesting.The two loop β function for a generic SU(N) gauge theory with fermions in therepresentation R isβ(g) = −β 0g 3g 5(4π) − β 2 1(4π) , 42Nβ 0 = 113 C 2(G) − 4 3 T (R)(2N)β 1 = 343 C2 2(G) − 203 C 2(G)T (R) − 4C 2 (R)T (R),(4.31)

Technicolor 43Table 4.1.: Group theoretical quantities.tableaux.Representations are written in terms of Youngr T (r) C 2 (r) d(r)12N 2 −12NNG N N N 2 − 1N+22(N−1)(N+2)NN(N+1)2where C 2 (R) is the quadratic Casimir and T (R) the trace normalization factor. Thesetwo are related viaN f C 2 (R)d(R) = T (R)d(G), (4.32)where d(R) is the dimension of the representation. The group theoretical quantities forrepresentations we need are given in Table 4.1.Theory loses asymptotic freedom when the first coefficient changes sign. This definesthe upper limit for the conformal window:N I f = 114C 2 (G) d(G)T (r) d(R) . (4.33)When the second coefficient changes sign the theory develops a Banks-Zaks fixed point.Therefore, if the number of flavors is smaller thanN IIIf =17C 2 (G) d(G)C 2 (G)10C 2 (G) + 6C 2 (r) d(R)C 2 (r)(4.34). the spontaneous symmetry breaking cannot be triggered. Thus the lower boundary ofthe conformal window must be somewhere between these two values. Using the criticalcoupling, given by the DS analysis, and identifying it with the coupling at the fixed point,we acquire the DS estimate for the lower boundaryN IIf SD = 17C 2(G) + 66C 2 (r)10C 2 (G) + 30C 2 (r)C 2 (G)T (r) . (4.35). The conformal windows for the fundamental, adjoint, two-index symmetric and twoindexanti-symmetric representation are showed in figure 4.2.

44 Technicolor2015N f10502 4 6 8 10NFigure 4.2.: Phase diagram for SU(N) theory with fermions in the fundamental (upper lines)and two-index symmetric representation (lower lines). Dashed line is the lowerbound achieved via SD-analysis.Minimal models of Walking TechnicolorUsing the conformal windows in Figure 4.2 as a guide for model building, several walkingmodels with minimal flavor content have been constructed.• The Minimal Walking Technicolor (MWT) is SU(2) T C gauge theory with twoDirac flavors of technifermions transforming according to the two-index symmetricrepresentation of the gauge group.• The Next-to Minimal Walking Technicolor (MWT) is SU(3) T C gauge theory withthree Dirac flavors of technifermions transforming according to the two-index symmetricrepresentation of the gauge group.• The Ultra Minimal Walking Technicolor (MWT) is SU(2) T C gauge theory withmatter in two different representations of the gauge group. Two Dirac fermions inthe fundamental representation and two Weyl fermions in the adjoint of SU(2) T C .

Technicolor 45The global symmetry breaking patterns for these models are:SU(2) L × SU(2) R × U(1) V → SU(2) × U(1) VSU(4) → SO(4)SU(4) × SU(2) × U(1) → Sp(4) × SO(2) × Z 2NMWTMWTUMT(4.36)We have omitted here the fundamental representation which is disfavored by electroweakprecision measurements (for detailed discussion see, for example, [47, 48]).Ideal WalkingLet us take a step backwards and what has been stated. The MWT model seemsto already be inside the conformal window. This is further confirmed by the latticesimulations [49]. Should we discard the model based on this observation?The conformal windows in Fig Figure 4.2 are drawn for technicolor theories inisolation without taking into account effects from the SM interactions and from theETC interactions. The framework of ideal walking [50] consistently takes into accountthe effects from the four-fermion interactions for different representations as a functionof number of favors and number of colors. As a result of this analysis the conformalwindow is shown to shrink for all the representations. In other words, the four-fermioninteractions can drive a conformal theory to non-conformal phase.Another desired results is that the anomalous dimension of the mass increases beyondunity, which was the value from the DS analysis. This helps to yield the correct mass forthe top quark.

Chapter 5.Minimal Walking TechnicolorIn the MWT, the extended gauge group is SU(2) T C × SU(3) C × SU(2) L × U(1) Y andthe field content of the technicolor sector is constituted by the techni-fermions, Q, U cand D c , and one techni-gluon all transforming according to the adjoint representation ofSU(2) T C .The model suffers from the Witten topological anomaly [51] which is cured by addinga new fermionic weak doublet L singlet under technicolor gauge group [48]. Furthermore,the gauge anomalies cancel when introducing the SU(2) L singlets E c and N c with thehypercharge assignment below 1 :Field SU(2) T C SU(3) C SU(2) L U(1) Y⎛ ⎞TechniquarksNew LeptonsQ = ⎝ U ⎠ 3 1 2DU c 3 1 1 − y+12D c 3 1 1 − y−12L 1 1 2 − 3y 2N c 1 1 1E c 1 1 1y23y−123y+12(5.1)The parameter y can take any real value [48]. We refer to the states L, E c and N c asthe New Leptons. The condensate which correctly breaks the electroweak symmetry1 We use the two component Weyl notation throughout this section.47

48 Minimal Walking Technicoloris 〈UU c + DD c 〉. To discuss the symmetry properties of the theory it is convenient toarrange the technifermions as a column vector, transforming according to the fundamentalrepresentation of SU(4)⎛ ⎞UDˆQ =, (5.2)⎜U ⎝c ⎟⎠D cThe breaking of SU(4) to SO(4) is driven by the following condensate〈 ˆQ T E ˆQ〉 (5.3)The matrix E is a 4 × 4 matrix defined in terms of the 2-dimensional unit matrix as⎛E = ⎝ 0 ⎞1 ⎠ . (5.4)1 0The above condensate is invariant under an SO(4) symmetry.5.1. Goldstone Bosons5.1.1. Effective TheoryThe symmetry breaking pattern of the MWT model is SU(4) → SO(4). This leaves uswith nine broken generators with associated Goldstone bosons. As in Section 3.1 ,theresulting low energy effective theory can be organized in a derivative expansion withcut-off scale 4πF , where F is the Goldstone boson decay constant. We first introducethe matrix:U = exp(i√2F Πa X a )E , (5.5)

50 Minimal Walking Technicolorof the SM gauge bosons while the remaining 6 Goldstone bosons carry technibaryonnumber and will be denoted by Π UU , Π UD and Π DD . Because Goldstone bosons carrytechnibaryon number, we refer to these states also as technibaryons.The bosons can be classified according to the unbroken group U(1) V × U(1) Q in thefollowing way:Boson U(1) Vcharge U(1) Qcharge Linear CombinationW + L0 +1Π 1 −iΠ 2 √2W − L0 −1Π 1 +iΠ 2 √2Z L 0 0 Π 3Π UU +1 y − 1Π 4 +iΠ 4 +Π 6 +iΠ 72Π DD +1 y + 1Π 4 +iΠ 4 +Π 6 +iΠ 72(5.10)ΠΠ UD +1 y 8 √+iΠ 92Π † ΠUU−1 −y + 1 4 −iΠ 4 +Π 6 −iΠ 72Π † ΠUD−1 −y 8 √−iΠ 92Π † DD−1 −y − 1Π 4 −iΠ 4 +Π 6 −iΠ 72Electroweak interactions split the technipion masses according to the following pattern[52]:∆m 2 Π UU∆m 2 Π DD= m2 [walk g2g1 2 + g22 1 (1 + 2y) 2 + g2]2[g21 (4y 2 − 1) + g2]2= m2 walkg 2 1 + g 2 2(5.11)(5.12)∆m 2 Π UD= m2 walkg 2 1 + g 2 2[ ]g21 (1 − 2y) 2 + g22 . (5.13)In models with walking dynamics the value of m walk can be even of few hundreds GeV[52].Furthermore it is also possible to introduce a common mass term for the pseudo Goldstonebosons by adding the following term to the Lagrangian (5.7):− m2 etcF 2Tr [ ]U † B V UB V4(5.14)

Minimal Walking Technicolor 51with⎛B V = ⎝ 1 00 −1⎞⎠ . (5.15)This term was already added in [53] and it is expected to emerge from a more completetheory of SM fermion mass generation. It is expected to emerge from a four-techniquarkinteraction term, and preserves SU(2) L × SU(2) R × U(1) V of SU(4), which contains theSU(2) V custodial symmetry group.It is useful to know the transformation properties of the matrix U with respect tothe electroweak gauge group. Under SU(4) the matrix U transforms as a two indexsymmetric tensor, or in other words U transforms like the irreducible representation 10of SU(4):U → gUg T , g ∈ SU(4). (5.16)Knowing the embedding of the electroweak generators in the SU(4) algebra it is possibleto decompose the representation 10 according to the SU(2) L × U(1) Y group:10 → 3 y + 2 1/2 + 2 −1/2 + 1 −y+1 + 1 −y + 1 −y−1 . (5.17)The identification of these representations inside the U matrix is given by:3 y → T LU = 1 F2 1/2 → H 22 −1/2 → H 11 −y−1 → S 11 −y+1 → S 31 −y → S 2⎛⎜⎝T LH 1 √2H 2 √2⎞⎟⎠√2 S 3H√1T 2S 1 √2S 2H√2T 2S 2(5.18)By expanding the exponential in equation (5.5) to the second order in the numberpseudo Goldstone boson fields, we identify the following phenomenologically relevant

52 Minimal Walking Technicolorinteraction Lagrangian for the first LHC searches:⎛T L = i ⎝ Π UUΠ √2 UD⎞Π √2 UDΠ DD⎠ S 1 = iΠ † UUS 2 = iΠ † UDS 3 = iΠ † DD(5.19)H 1√2 F=⎛⎞⎠2 √ 2 F 2⎝ 1 − Π† UD Π UD+2 Π † UU Π UU4 F 2− Π† UD Π DD+Π † UU Π UDH 2√2 F=⎛⎝ − Π† DD Π UD+Π † UD Π UU2 √ 2 F 21 − Π† UD Π UD+2 Π † DD Π DD4 F 2 ⎞⎠ . (5.20)Depending on the specific choice of the hypercharge assignment y, one can constructdifferent Yukawa-type interactions involving the matrix U. Odd integer values arerequired for y to avoid stable composite states with fractional electric charge 2 . In general,electrically charge stable states are excluded by cosmology, see for example discussionin [54].Within this setup the allowed Yukawa terms areL 2HDM = H 2 qu c + H 1 qd c + H 1 le c + H 2 LN c + H 1 LE c (5.21)+ H1qu c c + H2qd c c + H2le c c + H1LN c c + H2LE c c + h.c.⎧y = −3 S 3e † c e c + h.c.⎪⎨ y = −1 S 1e † c e c + S 2 ll + T † Lll + h.c.L Πψψ =(5.22)y = +1 S 3 e c e c + S 2ll † + T L ll + h.c.⎪⎩ y = +3 S 1 e c e c + h.c.⎧⎨y = −1 S 1 E c E c + h.c.L ΠLL =(5.23)⎩y = +1 S 3 N c N c + h.c.⎧y = −5 S 1 E c e c + h.c.⎪⎨ y = −3 S 1Ll † + S 1 N c e c + S 2 E c e c + h.c.L ΠLψ =(5.24)y = −1 S 2Ll † + S 2 N c e c + T † LLl + h.c.⎪⎩ y = +1 H1Le c c + H 2 Le c + H 1 N c l + H2N c c l + S 3 Ll + S 3 N c e c + h.c.Hic is defined as Hi c ≡ −iσ 2 Hi ∗ , and the e c and l symbols stand respectively for SU(2) Lsinglet and doublets states of the 3 lepton families (e, µ, τ). In equations (5.21)-(5.24),2 Bound states made by one technifermion and one technigluon have electric charge − y ± 12

Minimal Walking Technicolor 53for simplicity, we omitted the appropriate SU(2) L contractions, the flavor indices andthe coupling constants in front of each term. The interactions in (5.21)-(5.24) are validfor generic ETC models. Specific models can provide further symmetries which canbe exploited to reduce the number of operators in (5.21)-(5.24). Moreover, it is worthnoticing that quarks can couple, to this order, only to the Higgs sector (via L 2HDM ), i.e.quadratic in the number of technipions.5.1.2. ResultsThe allowed values for y can be further restricted by requiring that the lightest state,among the pseudo Goldstone bosons and the new leptons, is not electrically charged andstable. According to this, we discard the cases y = {−5, 3} and the surviving values of yare {−3, −1, 1}.For every y, the neatest process to be investigated at the LHC involving the PGBsfor which we will derive relevant constraints ispp → Π † 2Π 2 → (l − i l− j )(l+ h l+ k) where i, j, h, k = e, µ, τ. (5.25)The flavor structure of the coupling λ ij Π 2 l + i l+ j depends on the ETC sector, but giventhat this sector is unknown it is possible to have different lepton pairs in the final state.In order to simplify our analysis, we assume decays only into a pair of same sign muons.The ATLAS collaboration has studied the production of a doubly charged Higgs bosonat the LHC in [55] with 1.6 fb −1 of integrated luminosity. This study was performedby examining the invariant mass distribution of the same sign muon pairs. The doublycharged particle is assumed to decay into two muons with 100% branching fraction.We have used the number of observed events and the number of expected backgroundevents reported in table 2 of [55] to calculate the 95% exclusion limit for the pseudoGoldstone boson production using the modified frequentists CL s method [56, 57]. Thevalue for the coupling between the pseudo Goldstone boson and the muons is chosenaccording to [55], to ensure that the pseudo Goldstone bosons are decaying before thedetector. This yields a lower limit of 286 GeV for the pseudo Goldstone boson mass.The exclusion plot is presented in Figure 5.2 for 1.6 fb −1 .

-54 Minimal Walking Technicolor) (fb)2µ+ 2µ ±±(pp ±±21010pp @ 7 TeV, y = -1, p > 20 GeV, || < 2.5±±+(pp ±± 2µ 2µ - )-195% upper limit, ATLAS data, L dt = 1.6 fb95% expected limit, 95% expected limit, -1L dt = 5 fb-1L dt = 15 fb0.9 m < m(µ ± µ ± ) < 1.1 m1150 200 250 300 350 400 ±± mass (GeV)Figure 5.2.: Exclusion for a doubly charged PGB with y = −1 based on the ATLAS data.5.1.3. SummaryThis sector of the Minimal Walking Technicolor model was for the first time investigated in[I]. Strongly coupled theories are in many occasions studied in terms of vector resonances.Nevertheless, it is possible that pseudo Goldstone bosons are the the first sign of strongdynamics at the LHC.The signature which is identified to be the most important can also arise from manytheories with doubly charged Higgs bosons. Usually in these models, producing a singledoubly charged Higgs is more economical. If doubly charged scalar states are observedto be produced only in pairs, this would favor our model. Of course the reverse is true aswell.5.2. Vector ResonancesDifferent walking technicolor models posses different global symmetries and symmetrybreaking patterns. The precision measurements dictates that all these different extensionof the SM have to contain at least the following chiral symmetry breaking pattern:SU(2) L × SU(2) R → SU(2) V . (5.26)

Minimal Walking Technicolor 55Thus the NMWT model serves as a template to study vector resonance phenomenologyat the LHC. Based on the symmetry breaking pattern, the low energy spectrum isdescribed in terms of lightest spin one vector and axial-vector iso-triplets V ± ,0 and A ± ,0as well as lightest iso-singlet scalar resonance H. The corresponding states in the QCDare the ρ ± ,0 , a ± ,01 and the f 0 (600).All the results in paper [II] are obtained using a linearly realized effective Lagrangian.How to write down this effective Lagrangian is described in details in [53, 58]. Out ofauthor’s own interest, let us write down a non-linear effective Lagrangian, supplementedwith a scalar singlet state (i.e. the Higgs), and compare it with the Linear realization.5.2.1. Non-linear LagrangianIn order to consistently introduce mass terms for the vector mesons the GeneralizedHidden Local Symmetry (GHLS) method is employed [59]. Formalism is based on theobservation that a nonlinear sigma model based the manifold G/H is gauge equivalentto a linear model with symmetry group G Global × G local . The vector mesons are taken tobe gauge bosons of the G local symmetry and a subgroup of G Global is gauged under theelectroweak interactions. The gauge equivalence is straightforward to verify by using theequations of motion of the gauge bosons and choosing a special gauge for the local group.In our case G Global = SU(2) L × SU(2) R and G local = SU(2) L × SU(2) R . A basicdynamical variable of the G Global × G local model is a unitary matrix field U(x) [59–61]transforming asU(x) → U(x ′ ) = ˜h(x)U(x)˜g † , (5.27)where ˜h(x) ∈ G local and ˜g ∈ G Global . It is convenient to decompose the U(x) asU(x) = ξ † L (x)ξ M(x)ξ R (x) (5.28)The variables ξ(x) transform asξ L(R) → h L(R) (x)ξ L(R) (x)g † L(R) , ξ M → h L (x)ξ M (x)h † R(x), (5.29)

56 Minimal Walking Technicolorwhere g L(R) ∈ [ ]SU(2) L(R) and hGlobal L(R) ∈ [ SU(2) L(R) . We can construct the]LocalLagrangian by introducing the covariant Maurer-Cartan 1-forms defined as:ˆα µ L,R,M (x) = Dµ ξ L,R,M (x) · ξ † L,R,M(x). (5.30)The covariant derivatives for the different ξ fields are given asD µ ξ L (x) = ∂ µ ξ L (x) − iL µ (x)ξ L (x) + iξ L (x)L µ (x) (5.31)D µ ξ R (x) = ∂ µ ξ R (x) − iL µ (x)ξ R (x) + iξ R (x)R µ (x) (5.32)D µ ξ M (x) = ∂ µ ξ M (x) − iL µ (x)ξ M (x) + iξ M (x)R µ (x), (5.33)where L µ = gW a µ T a and R µ = g ′ B µ T 3 are the ordinary electroweak gauge bosonsintroduced by gauging the SU(2) L × U(1) Y subgroup of G Global symmetry. The G Localgauge bosons are defined to include the gauge coupling ˜gL µ = ˜g V µ − A µ√2, R µ = ˜g V µ + A µ√2. (5.34)Finally the Lagrangian isL = L kin + aL V + bL A + cL M + dL π , (5.35)whereL kin = − 1 2 Tr [Wµν W µν ] − 1 4 Bµν B µν − 1 2 Tr [Vµν V µν + A µν A µν ] , (5.36)L V = F 2 Tr [ˆα ‖µ ˆα ‖ µ ] , L A = F 2 Tr [ˆα ⊥µ ˆα µ ⊥ ] , L M = F 2 Tr [ˆα Mµ ˆα µ M ] , (5.37)The perpendicular and parallel projections are defined asL π = F 2 Tr [(ˆα ⊥µ + ˆα Mµ )(ˆα µ ⊥ + ˆαµ M)] . (5.38)ˆα µ ‖,⊥ = (ξ M ˆα µ R ξ† M ± ˆαµ L)/2. (5.39)

Minimal Walking Technicolor 57And the field strengths areW µν = ∂ µ W ν − ∂ ν W µ − ig [W µ , W ν ] , (5.40)B µν = ∂ µ B ν − ∂ ν B µ , (5.41)V µν = ∂ µ V ν − ∂ ν V µ − i˜g [V µ , V ν ] , (5.42)A µν = ∂ µ A ν − ∂ ν A µ − i˜g [A µ , A ν ] . (5.43)The Lagrangian (5.35) can be extended to include states which are singlets under thestability group H. After adding a scalar state the Lagrangian readsL = L kin + k a (χ)aL V + k b (χ)bL A + k c (χ)cL M + k d (χ)dL π+ k h (χ) 1 2 ∂ µh∂ µ h − V(χ),(5.44)where χ = h/v and v is an arbitrary scale. The arbitrary functions k x (χ) must satisfyk a (0) = k b (0) = k c (0) = k a (0) = k h (0) = 1 (5.45)for a proper normalization of the Goldstone boson kinetic terms. The function k d (χ)stands in front of a mixed term and the former requirement does not apply. Nevertheless,we will choose it to satisfy the same condition. One can derive the interactions with hexpanding k x (χ) around χ = 0.5.2.2. Linear LagrangianLet us write down the linearly realized effective Lagrangian for comparison:L boson = − 1 ] [˜Wµν˜W2 Tr µν− 1 4 ˜B ˜Bµν µν − 1 2 Tr [F LµνF µνL + F RµνF µν+ m 2 Tr [ C 2 Lµ + C 2 RµR ]]+12 Tr [ D µ MD µ M †] − ˜g 2 r 2 Tr [ C Lµ MC µ R M †]− i ˜g r 3Tr [ (C Lµ MD µ M † − D µ MM †) (+ C Rµ M † D µ M − D µ M † M )]4+ ˜g2 s4 Tr [ C 2 Lµ + C 2 Rµ]Tr[MM† ] + µ22 Tr [ MM †] − λ 4 Tr [ MM †] 2(5.46)where ˜W µν and ˜B µν are the ordinary electroweak field strength tensors, F L/Rµν are thefield strength tensors associated to the vector meson fields A L/Rµ , and the C Lµ and C Rµ

58 Minimal Walking Technicolorfields areThe 2 × 2 matrix M isC Lµ ≡ A Lµ − g˜g ˜W µ , C Rµ ≡ A Rµ − g′˜g ˜B µ . (5.47)M = 1 √2[v + H + 2 i π a T a ] , a = 1, 2, 3 (5.48)where π a are the Goldstone bosons produced in the chiral symmetry breaking, v = µ/ √ λis the corresponding VEV, H is the composite Higgs, and T a = σ a /2, where σ a are thePauli matrices. The covariant derivative isD µ M = ∂ µ M − i g ˜W a µ T a M + i g ′ M ˜B µ T 3 . (5.49)When M acquires its VEV, the Lagrangian of equation (5.46) contains mass mixingterms for the spin-1 fields.5.2.3. ComparisonThe new vector resonances mass mixes with the SM gauge bosons. This induces thecouplings between the new vectors and the SM fields. The masses depend on theparameters a, b, c, d in the nonlinear model and on the parameters r 1 , r 2 , r 3 , k in thelinear model. This in addition to a parameter fixing the scale and the gauge couplings.If we are interested in signatures at the LHC not involving the Higgs, it seems that atleading order we can define a mapping between the parameters of these two realizationsof the model.In the linear Lagrangian the symmetry SU(2) L × SU(2) R fixes the Higgs couplings tobe given in terms of parameters a, b, c, d. In the non-linear realization all the couplingsinvolve independent free parameters. The couplings between the new vector states andthe Higgs are important when we calculate decay widths for the vector resonances. Thiscan, without a doubt, have an effect on the results, even if we study processes withoutthe Higgs. In Figure 5.3 we have plotted the decay width of the R 1 vector resonance inthe linear model and in the non-linear model taking k x (χ) = 0. Notice here that thediscussion applies to any theory with the same global symmetry. Connection to theunderlying gauge theory can be achieved via the WSRs, which impose relations betweenthe parameters. In rest of the chapter we will use the Linear Lagrangian.

Minimal Walking Technicolor 59R 1 2 X50.0NonlinearWidth GeV10.05.01.00.5Linear0.1500 1000 1500 2000MAFigure 5.3.: The decay width of the R 1 vector resonance.5.2.4. ResultsAs already mentioned, the new vector and axial-vector states mix with the SM gaugeeigenstates yielding the ordinary SM bosons and two triplets of heavy mesons, R ± ,01 andR ± ,02 , as mass eigenstates. The couplings of the heavy mesons to the SM particles areinduced by the mixing. Important here is how the heavy vectors couple to the fermions.In the region of parameter space where R 1 is mainly an axial-vector and R 2 mainly avector sate, the dependence of the couplings to the SM fermions as a function of ˜g isvery roughlyg R1,2 f ¯f ∼ g2˜g(5.50)where g is the electroweak gauge coupling. The full coupling constant is also a functionof M A , but this dependence is very weak.To constrain the parameter space of the model we use the CMS results [62], whichreport limits for a W ′ boson decaying to a muon and a neutrino at √ s = 7 TeV in themass range 600 - 2000 GeV for the resonance.The relevant calculations are performed using MadGraph [63], using the CTEQ6Lparton distribution functions [64], and the implementation for the NMWT model [65].reference [62] reports experimental limits for the muon channel together with a combinedanalysis with the electron channel [66]. Because of the missing energy in the final state,the invariant mass of the resonance cannot be reconstructed, and the following transverse

60 Minimal Walking Technicolor1210Consistency of the theoryg 864TevatronCMS Data 15 fb 1 , 3 Σ5 fb 1 , 5 Σ13 TeV, 100 fb 1 , 3 Σ13 TeV, 100 fb 1 , 5 ΣWalkingRunning2YW0.5 1.0 1.5 2.0 2.5M A TeVFigure 5.4.: Bounds in the (M A , ˜g) plane of the NMWT parameter space: (i) CDF directsearches of the neutral spin one resonance excludes the uniformly shaded areain the left, with M H = 200 GeV and s = 0. (ii) The 95 % confidence levelmeasurement of the electroweak precision parameters W and Y excludes thestriped area in the left corner. (iii) Imposing the modified WRS’s excludes theuniformly shaded area in the right corner. (iv) The horizontal stripe is excludedimposing reality of the axial and axial-vector decay constants. (v) The area belowthe thick uniform line is excluded by the CMS data [62]. (vi) Dashed and dottedlines are expected exclusions using different values of the integrated luminosityand center of mass energy.mass variable is utilizedM T =√2 · p T E missT· (1 − cos ∆φ µ,ν ). (5.51)In the experimental analysis, the cut on the transverse mass is adjusted in bins of themass of the sought-after resonance. In addition to the transverse mass cut, the leptonacceptance |η| < 2.1 is used. The resulting cross section is then compared with the limitsreported in the experimental analyses.Exploring the signal from the process pp → R 1,2 → lν we are able to limit the possiblevalues for the parameters M A and ˜g. The theoretical limits as well as the limits from theTevatron are described in [67]. The M A , ˜g plane of the parameter space is presented inFigure 5.4 for M H = 200GeV and s = 0.

Minimal Walking Technicolor 61The uniformly shaded region on the left is excluded by the CDF searches of theresonance in the the p¯p → e + e − process. The striped region in the lower left corner isexcluded by the measurements of the electroweak W and Y parameters [68] adaptedfor models of MWT in [69]. Avoiding imaginary decay constants for the vector andaxial-vector sets an upper bound for the ˜g, i.e. excludes the uniformly shaded in theupper part of the figure. The near conformal (walking) dynamics modifies the WRS’s,compared to a running case like QCD, as explained above [70]. Imposing these modifiedsum rules excludes the lower right corner of the parameter space. The CDF exclusionlimit is sensitive, indirectly, to the mass of the composite Higgs and the coupling s viaproperties of the new heavy spin one states. However, the edge of the excluded areavaries only very weakly as a function of s and M H . The CMS search imposes a 95 % CLexclusion bound described with the thick solid (red) line. The thick dashed and dottedlines (blue) are three and five sigma exclusion limits for 7 TeV and 5 fb −1 . The thindotted and dashed lines describe the reach of the LHC with 100 fb −1 at 13 TeV. The threeand five sigma exclusion limits are calculated using poisson distribution, following [71].Due to the effective description, we have not employed the K-factors when calculatingthe exclusion limits.Comparing the three sets of lines for the LHC, the increase in the horizontal directionfollows roughly the increase in luminosity. The small role of the center of mass energycan be understood by exploring the behavior of the cross section as a function of thecenter of mass energy and comparing it with the scaling with ˜g, obtained from equation(5.50).5.2.5. SummaryThe limits on the masses are significantly lower that for the W ′ with SM like couplings.The reason for this can be traced back to fermion couplings which arise via mixing. TheATLAS Collaboration has performed a similar analysis with this model and reportedresults in [72]. The exclusion plot derived by ATLAS is shown in Figure 5.5As evidenced in Figure 5.5, the leptonic final states are not enough to explore thewhole parameter space. Interesting possibility is to use associate Higgs production to findvector resonances. For example, in the process pp → R 1 → ZH the new vector state hasonly one coupling with the fermions which will help with large values of ˜g. An exampleinvariant mass distribution is given in Figure 5.6. Here the Higgs is taken to decay to

62 Minimal Walking TechnicolorFigure 5.5.: Excluded parameter space based on the ATLAS measurements. The plot is takenfrom [72]dσ/dM (fb/10 GeV)1-110pp @s = 8TeV+ -pp → bbZ, (Z → l l )M A=700 GeV, g ~ =5, S=0.3sm, m =125 GeVh|y | < 4.5, p > 45GeV, p > 25GeVb1,b2,b-210∆R(b,b)>0.7, 110 GeV < M(b,b) < 135 GeV|y | < 2.5, p > 20GeVllM R1 =702 GeV-310M R2 =1055 GeV-410400 500 600 700 800 900 1000 1100+ -M(bbl l ) (GeV)Figure 5.6.: Production cross section as a function of invariant mass M HZ .

Minimal Walking Technicolor 63b-quark pair and the Z boson to leptons. To effectively use this channel requires a greatdeal of integrated luminosity and efficient b-tagging.5.3. Technicolor HiggsIn technicolor the Higgs sector is replaced with a strongly interacting sector. The lightestscalar state plays a role of the Higgs bosons, provided that it is lighter compared to othersates. In QCD like models, mass of the composite Higgs is estimated to be roughly of theorder of M H ∼ 1 TeV, which is of course way too heavy when compared to the observedHiggs mass. In walking models the mass of the Higgs can be as low as few hundreds ofGeV [48, 73]. In order to explain why one scalar state if much lighter than other statesthe model, it is convenient to identify it with the Goldstone boson of the approximatescale symmetry. This scenario goes under the name techni-dilation [74].What we have stated is not quite the complete picture. The Higgs here is composedof massless techniquarks and all the mass is given by the dynamics. The dynamical massis not the physical mass; one has to consider also the effect from the radiative corrections.The SM top-induced radiative corrections can reduce the composite Higgs mass downto observed value [73]. This is like an inverse of the little hierarchy problem causingproblems in several BSM models.5.3.1. Hybrid ModelInspired by the hybrid models used to calculate the two photon decay of the sigma mesonin QCD, we calculated in [I] the effect of techniqurks on the Higgs two photon decay. Thecontribution to the process is modeled by re-coupling, in a minimal way, the compositeHiggs to the techniquarks Q via the following operator:L T C−2γ = √ 2 M Qv weak[Q Lt· HDRt + Q Lt· (i τ2 H ∗ )U Rt]+ h.c. , (5.52)where M Q is the dynamical mass of the techniquark and t = 1, . . . , d[r] is the Technicolorindex and d[r] the dimension of the representation under which the techniquarks transform.If the model suffers from the Witten anomaly, a lepton family is addedL L−2γ = √ 2 M Ev weakL L · HE R + √ 2 M Nv weakL L · (i τ 2 H ∗ )N R + h.c. , (5.53)

64 Minimal Walking Technicolorwhere M E and M N are the fermion masses. The techniquark dynamical mass is intrinsicallylinked to the technipion decay constant F π via the Pagels-Stokar formula [75, 76]M Q ≈ √ 2πFπ .d(r)The one loop Higgs decay width to two photons for any weakly interacting elementaryparticle contributing to this process can be neatly summarized as [77]:Γ(h → γγ) = α2 G F M 2 h128 √ 2π 3 ∣ ∣∣∣∣ ∑in i Q 2 i F i∣ ∣∣∣∣2, (5.54)where i runs over the spins, n i is the multiplicity of each species with electric charge Q iin units of e. The F i functions are given byF 1 =2 + 3τ + 3τ(2 − τ)f(τ), (5.55)F 1/2 = − 2τ[1 + (1 − τ)f(τ)], (5.56)F 0 =τ (1 − τf(τ)) , (5.57)where τ = 4m2 iM 2 hand⎧⎪⎨f(τ) =⎪⎩− 1 4(arcsin((log√11+ √ 1−τ1− √ 1−τ) 2τ , if τ ≥ 1) ) 2 . (5.58)− iπ , if τ < 1The lower index of the function F represent the spin of each particle contributing in theprocess.5.3.2. ResultsWe plot in Figure 5.7 the intrinsic dependence on the dimension of the Technicolor matterrepresentation d[r] according to the ratio:R = Γ SM(h → γγ) − Γ T C (h → γγ). (5.59)Γ SM (h → γγ)For any odd representation we included the Lepton contribution and therefore we couldnot simply join the points. Interestingly when d[r] is around 7 the contribution from thetechniquarks and from the SM fermions (mainly the top) cancels the W contributionleading to Γ T C ≈ 0.

Minimal Walking Technicolor 651.00.5R0.00.51.00 5 10 15 20drFigure 5.7.: Difference of the Walking TC and SM decay widths divided by the SM widthplotted with respect to number of colors. m h = 120.5.3.3. SummaryAbove calculation is a very crude estimate of a result which cannot be calculatedexactly. We have also a problem with the double counting; the techni-pions eaten by theelectroweak gauge bosons are techniquark bound sates. Moreover, for reasonable matterrepresentations, the effect is too small to be observed at the LHC.

Chapter 6.Dark Matter Model BuildingLet us forget the problems with the SM Higgs sector for the rest of the thesis. A largenumber of observational evidences suggest that there is non-luminous matter in theuniverse [78]. This dark matter cannot be explained by the particles we know, thusrequires physics beyond the SM. There are a number of experiments around the worldwhich are trying to detect dark matter directly. The situation with these experimentsis quite interesting because they do not mutually agree; for example DAMA/LIBRA[79], GoGeNT [80] and CRESST [81] claims to have seen dark matter while CDMS[82], XENON [83] and PICASSO [84] experiments exclude the parameter regions whereobservations have been made. However, there are number of caveats that need to bediscussed as one tries to interpret these results as discussed for example in [85].6.1. The modelEffect of a magnetic dipole moment of the dark matter on the direct detection experimentshas been discussed in[85, 86] using effective operator approach. In in [IV] we have provideda renormalizable model which can, at low energies, provide this effective operator. Ontop of the SM we add a vector-like heavy electron (E), a complex scalar (S) and a SMsinglet Dirac fermion (χ). The associated renormalizable Lagrangian isL SEχy = L SM + ¯χi/∂χ − m χ ¯χχ + Ei /DE − m E EE − (SEχy + h.c.)+ D µ S † D µ S − m 2 SS † S − λ HS H † HS † S − λ S (SS † ) 2 , (6.1)where D µ = ∂ µ − ie swc wZ µ + ieA µ , s w and c w represent the sine and cosine of the Weinbergangle. We assume the new couplings y, λ HS and λ S to be real and the bare mass squared67

68 Dark Matter Model Buildingof the S field to be positive so that the electroweak symmetry breaks via the SM Higgsdoublet (H). The interactions among χ, our potential dark matter candidate, and theSM fields occur via loop-induced processes involving the SĒχy operator in (6.1).These extra interactions have an effect on the vacuum stability we investigated inthe introduction. However, the effect is very small because the scalar S does not acquirea vacuum expectation value, as argued in [87]. Both of the new fermions can mix withthe SM leptons. For the sake of simplicity, we do not consider the effect of the mixing,even though it can lead to interesting consequences. For example, sizable mixing canexplain the discrepancy between the measured and calculated muon anomalous magneticmoment.At the one loop level χ develops the following magnetic-type interactionsL 5 = λ χ2 ¯χσ µνχF µν − s w2 c wλ χ ¯χσ µν χZ µν , (6.2)where F µν and Z µν are the photon and Z field strength tensors, and λ χ is related to theelectromagnetic form factor F 2 (q 2 ) as follows:λ χ = F 2(q 2 )2m χe . (6.3)The explicit derivation of the one-loop–induced form factor can be found in the firstappendix of [IV]. It is enlightening to report the analytic form for a few interesting limitsto learn about the dependence upon couplings and masses. We start by considering thestatic limit F 2 (0), useful for dark matter direct detection experiments. To reduce theparameter space we take the masses of E and S to be degenerate m E = m S = M leadingto the simplified expressionwhere z = mχ. For small z we haveM( √ ( ) )F 2 (0) = y2 2 2 + zz8π 2 z 2 − z tan−1 √ − 1 , (6.4)4 − z2F 2 (0) =( )y2z + z2+ O ( z 3) , (6.5)16π 2 3which for z = 0 gives λ χ = ey 2 /(32π 2 M). It is also interesting to consider a dark mattercandidate with mass of the order of the electroweak scale or slightly higher. For thispurpose a simple estimate for the electromagnetic form factor can be deduced by setting

Dark Matter Model Building 69√3π − 3z = 1 (i.e. m χ = M), and M around the electroweak scale, yielding F 2 (0) = y 2√ 24π 23π − 3and therefore λ χ = ey 2 . From the two limits it follows that the scale of the48π 2 Mmagnetic moment is controlled by the common mass of the heavy states M.6.2. ResultsIt was shown in [85] that there is a region of parameter space able to alleviate thetension between the experiments when using magnetic moment interactions. It wasobserved that one can, in fact, bring DAMA, CoGeNT and CRESST signals to overlapwhile being marginally consistent with CDMS, XENON and PICASSO experiments(see Fig. 3 of ref. [85]). A best fit result leads to a dark matter mass around 10 GeVand a constant magnetic moment of about 1.5 × 10 −18 e cm, which corresponds to32π 2 M/y 2 = eλ χ∼ 10 TeV. For example for M ≈ 500 GeV we find y ≈ π meaning thatthe underlying dynamics of the model can be explored at the electroweak scale.The contour plot of equal λ χ in the (M,y) plane is shown in Figure 6.1 together withthe exclusion regions obtained using the following information: The CMS constraintson charged long-lived particles, The XENON100 results [83] after having taken intoconsideration the threshold effects [85] and the Higgs to two gamma constraints.1086h→γγXENONCMS90.0 ·10 −19 e · cm40.0 ·10 −19 e · cmλ HS = √ 4πm χ =10 GeV20.0 ·10 −19 e · cm1086h→γγXENONCMS90.0 ·10 −19 e · cm40.0 ·10 −19 e · cmλ HS =4πm χ =10 GeV20.0 ·10 −19 e · cmyy415.0 ·10 −19 e · cm415.0 ·10 −19 e · cm26.0 ·10 −19 e · cm2.0 ·10 −19 e · cm26.0 ·10 −19 e · cm2.0 ·10 −19 e · cm400 600 800 1000 1200M [GeV]400 600 800 1000 1200M [GeV]Figure 6.1.: Strength of the magnetic moment λ χ in the (M, y) plane with m χ = 10 GeV andfor λ HS = √ 4π (left panel) and λ HS = 4π (right panel). The unshaded region isthe one allowed by experiments.

70 Dark Matter Model Building6.3. SummaryThe extension of the SM presented here can be tested at the LHC while providing adark matter candidate interacting with ordinary matter via magnetic operators whichcan be simultaneously investigated or observed in dark matter experiments. In mostcases the link between the experiments can be established on within a renormalizablemodel. Number of papers have been studying limits on the dark matter production atthe LHC using the same effective operators as for the direct detection. This can be, andis, dangerous due to vastly different energy scales.

Appendix A.Witten AnomalyLet us consider an Euclidean SU(2) Yang-Mills theory in the compact space S 4 . FollowingWitten, the global SU(2) anomaly is based on the observation that the fourth homotopygroup of SU(2)Π 4 (SU(2)) = Z 2(A.1)is nontrivial [51]. Homotopy groups are invariant under a homeomorphism, hence theyare topological invariants [88]. The homeomorphism classify spaces according to whethercan they be continuously deformed one to another. Thus homotopy groups offers a lessrestrictive way to classify spaces since spaces that have same homotopy groups are notnecessarily homeomorphic. In physics homotopy groups are usually used to classify mapsrather than spaces.The group manifold of the SU(2) is S 3 . Thus the gauge transformations are mapsU(x) : S 4 → S 3 . The equation (A.1) can be interpreted so that there exists differenttypes of gauge transformations which approach unity at infinity 1 . The fact that fourthhomotopy group is Z 2 means that there is a topologically non-trivial gauge transformationthat cannot be smoothly deformed to the identity but when the transformation is donetwice it can be deformed to the identity.Due to the topologically non-trivial mapping, for the every gauge field there is aconjugate fieldA U µ = U −1 A µ U − iU −1 ∂ µ U.(A.2)1 S 4 is achieved from the R 4 with the one-point compactification R 4 ⋃ {∞}. Thus the gauge transformationshave to possess a constant value at the infty.71

72 Witten AnomalyAnd they both give exactly the same contribution to the functional integral∫Z =( ∫dψd ¯ψdA µ exp −)d 4 1x[2g tr(F 2 ) + ¯ψi /Dψ] , (A.3)2where ψ denotes a single left-handed fermion doublet. The Dirac operator i /D is hermitian,thus the eigenvalues of the operator are real. Since the gamma matrices γ µ and γ 5anticommute, for every eigenvalue λ there is an eigenvalue −λi /Dψ = λψ, i /D(γ 5 ψ) = −λ(γ 5 ψ). (A.4)If the doublet ψ contains Weyl fermions, the integral over the fermion field is∫( ∫dψd ¯ψ exp −)d 4 x ¯ψi /Dψ = √ det i /D, (A.5)where the determinant is the product of eigenvalues [89]. The square root stands for theproduct of the half of the eigenvalues.We can choose that the square root of the eigenvalues is the product of positiveeigenvalues. The key point is that there exists a possibility for the positive and negativeeigenvalues to interchange their places under the non-trivial gauge transformation. Onepossible eigenvalue flow is represented in the figure A.1. If the number of flowsinterchanging positive and negative eigenvalue pairs is odd, it follows thatdet i /D(A) = − det i /D(A U ).(A.6)The gauge fields A µ and A U µ contribute equally to the functional integral (A.3), whichmeans that it vanishes in this case.In order to show that this kind of eigenvalue flow happens, Witten defines an instantonlikegauge fieldA t(τ)µ = (1 − t(τ))A µ + t(τ)A U µ , 0 ≥ t(τ) ≥ 1 (A.7)and considers a five dimensional Dirac equation/D (5) Ψ =)5∑3∑γ(∂ i i + A a i T a Ψ = 0. (A.8)i=1a=1

Witten Anomaly 73Figure A.1.: Eigenvalue flow under the non-trivial mpping.The fifth component is the path parameter τ so that t(τ) = 1 when τ → +∞ and t(τ) = 0when τ → − ∞. Since A a 5 = 0 and the gamma matrices are real and symmetric we canwrite the equation (A.8) asdΨdτ = −γτ /D 4 Ψ,(A.9)where /D 4 is a four dimensional Dirac operator for each τ. To solve this we writeΨ(x, τ) = F (τ)φ τ (x),(A.10)where φ τ (x) is the eigenfunction of the operator γ τ /D 4 . In the adiabatic limit dφτ (x)dτ= 0and the equation (A.9) simpliffies to a formdF (τ)dτ= −λ(τ)F (τ). (A.11)The solution to this equation is( ∫ τ)F (τ) = F (0) exp − dxλ(x) . (A.12)0

74 Witten AnomalyNow the logic goes as follows: According to mod 2 index theorem [90], the fivedimensionalDirac operator has an odd number of zero eigenvalues. This impose thatequation (A.9) has an odd number of solutions. On the other hand equation (A.12)is normalizable only if λ(τ) is positive for τ → + ∞ and negative for τ → − ∞. Thisimplies that there have to be odd number of eigenvalue pairs interchanging their placesconfirming the equation (A.6) to hold in this case.Vanishing path integral (A.3) causes a problem, because for each gauge invariantoperator the vacuum expectation value is〈0|W |0〉 =∫D[A, ¯ψ, ψ] W e−S∫D[A, ¯ψ, ψ] e−S= ”0”0 , (A.13)which is not well defined. Thus the SU(2) gauge theory with odd number of Weyl fermiondoublets is inconsistent.The hypothesis of adiabaticy is crucial in this derivation and criticism against thispoint is represented in the reference [91]. However, there are also two other ways to statethe SU(2) anomaly. One is based on the U(1) anomaly and the rotation in the center ofthe SU(2), which is free from perturbative anomalies [92]. In the third approach SU(2)group is embedded into SU(3) group and the global anomaly result from the non-abeliananomaly of the group SU(3) [93, 94]. These alternative derivations confirm the existenceof the SU(2) anomaly.

Appendix B.Dyson-Schwinger EquationThe Schwinger-Dyson equation is based on the simple observation the (functional) integralof a (functional) derivative is zero [39]∫D[ψ] δδψ ≡ 0.(B.1)This of course requires that the field ψ vanishes at spatial infinity. If we consider thegenerating functional of a QCD like theory∫Z [J, η, ¯η] = D [ G, ψ, ¯ψ ] e iS[G,ψ, ¯ψ,J,η,¯η] ,S [ G, ψ, ¯ψ, J, η, ¯η ] ∫= d 4 x [ L + J µ G µ + etaψ ¯ + ¯ψη ] ,(B.2)where J µ , η and ¯η are the source fields, we have∫0 =D [ G, ψ, ¯ψ ] δS [ G, ψ, ¯ψ, J, η, ¯η ]δ ¯ψ iα (x)e iS[G,ψ, ¯ψ,J,η,¯η] .This can be written as[ () ]η iα + (i/∂ − m i )δγ α + gγ µ (T a ) α δ δµZ [J, η, ¯η] = 0.δiJ aµ δi¯η iγ(B.3)(B.4)Differentiating with respect to η jβ and dividing with Z giveδ 4 (x − y)δ i jδ α γ + (i/∂ x − m i )δ α β+gγ µ (T a ) α γ1ZδδJ aµ (x)i δ 2 ZZ δ ¯ψ iγ (x)δη jβ (y)δ 2 Zδη iγ (x)δη jβ (y) = 0,(B.5)75

76 Dyson-Schwinger Equationwhich can be modified to a form(δ 4 (x − y)δjδ i γ α + (i/∂ x − m i )δβ α δ 2 ln Ziδ ¯ψ iγ (x)δη jβ (y) + δ ln Z )δ ln Zδ¯η iγ (x) δη jβ (y)(( )+gγ µ (T a ) α δ 3 ZγδJ aµ (x)δη iγ (x)δη jβ (y) + δ δ ln Z δ ln ZδJ aµ (x) δ¯η iγ (x) δη jβ (y)+ δ ln Z (δ 2 ln ZδJ aµ (x) δ ¯ψ iγ (x)δη jβ (y) + δ ln Z ))δ ln Z= 0,δ¯η iγ (x) δη jβ (y)(B.6)By setting sources to zero this simplifies toδ 4 (x − y)δjδ i γ α + (i/∂ x − m i )δβ α δ 2 ln Ziδ ¯ψ iγ (x)δη jβ (y) | J,¯η,η=0+gγ µ (T a ) α γδ 3 ZδJ aµ (x)δη iγ (x)δη jβ (y) | J,¯η,η=0 = 0,(B.7)Next we need the full quark and gluon propagatorsS(x − y) i jδβ α δ 2 ln Z= −iδ¯η iα (x)iδη jβ (y) | J,¯η,η=0 ,D µν (x − y)δ ab =δ 2 ln ZiδJ µa (x)iδJ νb (y) | J,¯η,η=0 ,(B.8)(B.9)and also the full gluon-quark-quark vertex∫= −δ 3 ln ZiδJ µa (z)iδ¯η iα iδη jβ| J,¯η,η=0d 4 x ′ d 4 y ′ d 4 z ′ D µν (z − z ′ )S(x − x ′ ) i kgΓ ν (z ′ , x ′ , y ′ ) k l (T a ) β αS(y ′ − y) l j.(B.10)With the aid of identity∫d 4 zS(x − z) α γ δ i kS −1 (z − y) γ β δk j = δ α β δ i jδ 4 (x − y),(B.11)and we get the Dyson-Schwinger equation for the quark propagator+g 2 ∫iS −1 (x − y) i jδ α β = (i/∂ − m i )δ i jδ α γ δ 4 (x − y)d 4 x 1 d 4 x 2 γ µ (T a ) α γ S(x − x 1 ) i kΓ ν (x 2 , x 1 , y) k j (T a ) γ β D µν(x − x 2 ).(B.12)

Dyson-Schwinger Equation 77Similarly for the gluon propagator{ (Dµν −1 (x − y)δ ab = −i ∂ 2 g µν − 1 − 1 ) }∂ µ ∂ ν δ ab δ 4 (x − y)ξ∫+ig 2 d 4 x 1 d 4 x 2 Tr [ γ µ T a S(x − x 1 )Γ ν (y, x 1 , x 2 )T b S(x 2 − x) ](B.13)

Dyson-Schwinger Equation 79

ColophonThis thesis was made in L A TEX 2ε using the “hepthesis” class [95].81

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List of Figures3.1. Alignment of subgroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1. Different possible beta functions for SU(N) gauge theory. . . . . . . . . . 394.2. Phase diagram for SU(N) theory with fermions in the fundamental (upperlines) and two-index symmetric representation (lower lines). Dashed lineis the lower bound achieved via SD-analysis. . . . . . . . . . . . . . . . . 445.1. Spontaneous and explicit breaking of the SU(4) symmetry. . . . . . . . . 495.2. Exclusion for a doubly charged PGB with y = −1 based on the ATLASdata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3. The decay width of the R 1 vector resonance. . . . . . . . . . . . . . . . . 595.4. Bounds in the (M A , ˜g) plane of the NMWT parameter space: (i) CDFdirect searches of the neutral spin one resonance excludes the uniformlyshaded area in the left, with M H = 200 GeV and s = 0. (ii) The 95% confidence level measurement of the electroweak precision parametersW and Y excludes the striped area in the left corner. (iii) Imposing themodified WRS’s excludes the uniformly shaded area in the right corner.(iv) The horizontal stripe is excluded imposing reality of the axial andaxial-vector decay constants. (v) The area below the thick uniform line isexcluded by the CMS data[62]. (vi) Dashed and dotted lines are expectedexclusions using different values of the integrated luminosity and center ofmass energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.5. Excluded parameter space based on the ATLAS measurements. The plotis taken from[72] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.6. Production cross section as a function of invariant mass M HZ . . . . . . . 6289

90 LIST OF FIGURES5.7. Difference of the Walking TC and SM decay widths divided by the SMwidth plotted with respect to number of colors. m h = 120. . . . . . . . . 656.1. Strength of the magnetic moment λ χ in the (M, y) plane with m χ = 10GeV and for λ HS = √ 4π (left panel) and λ HS = 4π (right panel). Theunshaded region is the one allowed by experiments. . . . . . . . . . . . . 69A.1. Eigenvalue flow under the non-trivial mpping. . . . . . . . . . . . . . . . 73

List of Tables2.1. The field content of the SM. . . . . . . . . . . . . . . . . . . . . . . . . . 124.1. Group theoretical quantities. Representations are written in terms ofYoung tableaux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4391

92 LIST OF TABLESPaper Ihttp://arxiv.org/abs/arXiv:1202.3024Paper IIhttp://arxiv.org/abs/arXiv:1105.1433Paper IIIhttp://arxiv.org/abs/arXiv:1102.2920Paper IVhttp://arxiv.org/abs/arXiv:1211.5144