Fundamental interactions

As mentioned before, three of the four known interactions between particles are described by the SM. They are listed in Table 1.2. Each one of them has an associated symmetry group. A fundamental property of the SM is the gauge invariance, dened as the invariance of the theory under local transformations. The SM theory is invariant under transformations of the type SU (3)_C × SU (2)_I × U (1)_Y. U(1)Y is the symmetry group of the electromagnetic interaction, SU (2)I of the weak interaction and SU(3)C of the strong interaction. C, I and Y correspond to the conserved quantum numbers for each symmetry: color charge, weak isospin and hypercharge, respectively.

1.2.2.1 The electromagnetic interaction

The electromagnetic interaction occurs between electrically charged particles. It is responsible for binding the electrons to the atomic nuclei to form atoms and then molecules. The electro-magnetic force carrier is the photon γ. The photon is a massless, electrically neutral and then not self-interacting gauge boson. The electromagnetic force is described by a relativistic quan-tum eld theory called Quanquan-tum Electrodynamics (QED). QED is based on a local symmetry (i.e. separately valid at each space-time point), called U(1). As in any quantum eld theory, the kinematics and the dynamics of the theory can be deduced from a lagrangian. The QED lagrangian describes the coupling between a charged fermion eld ψ to the boson eld A^µ:

L_QED = ¯ψ(iγ^µD_µ− m)ψ − 1

4F_µνF^µν, (1.1)

where γ^µ are the Dirac matrices. The covariant derivative Dµ and the eld strength Fµν are given by:

Dµ = ∂µ− ieA_µ (1.2)

F_µν = ∂_µA_ν− ∂_νA_µ, (1.3)

such as the LQED is invariant under local U(1) gauge symmetry (ψ → e^ieξ(x)ψ). The gauge invariance of the QED theory implies that the electrical charge is conserved locally. Note that the addition of a mass term for the gauge boson, of type m²A_µA^µ, will lead to a violation of the gauge invariance. The QED's gauge boson needs to be massless and it can be directly associated with the photon. e correspond to the elemental electrical charge and is given by:

e =p4πα_QED, (1.4)

where αQED is the electromagnetic coupling constant. It is a fundamental parameter of the theory and determines the strength of the electromagnetic interaction. In QED, observables are usually expressed as a function of αQED. When using perturbation theory to calculate those observables, divergences appear in calculations involving Feynman diagrams with loops

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including virtual particles. To avoid these divergences a method called renormalization is used.

The renormalization process consists in redening measurable observables at a given energy scale (called the normalization scale µ0) to include the virtual particle corrections, absorbing in this way the innities. Imposing the independence of the physical observable from µ0 reveals that αQED depends on the energy scale at which one observes the process Q². αQED(Q²) increases when the energy increases, going from 1/137 at Q² = 0 to 1/127 at energies corresponding to the mass of the Z boson.

1.2.2.2 The strong interaction

This interaction is responsible for holding the quarks together in hadrons and binding protons and neutrons together to form the atomic nucleus. It is described by a quantum eld theory called Quantum ChromoDynamics (QCD) [14]. QCD is represented by the non-Abelian symmetry group SU(3). In this representation the gluon is the gauge eld, i.e. the QCD equivalent of the QED photons. Just as the electric charge in QED, QCD introduces its own charge, known as color. Color charge comes in three varieties called red, green and blue. Antiquarks have corresponding anticolor. Quarks and antiquarks are combined in such a way that always form colorless hadrons. Leptons have no color charge. The gluon is not a charge-neutral force carrier (as its QED counterpart), it can be thought of as carrying both color charge and anticolor charge.

There are eight possible dierent combinations of (anti)color for gluons, which form an octet in color SU(3) [15]. Due to their non-Abelian nature, the gluon gauge elds exhibit self-couplings that allow for self-interactions.

The QCD lagrangian density is given by:

L_QCD=X

q

ψ¯q,j(iγ^µ(Dµ)jk− m_qδjk)ψq,k−1

4G^a_µνG^{a µν}, (1.5) where ψq,j is the quark eld for avour q and carry a color j. The covariant derivative Dµ and the gluon eld strength tensor G^a_µν are dened as:

D_µ = ∂_µ+ ig_st^aA^a_µ (1.6)

G^a_µν = ∂_µA^a_ν− ∂_νA^a_µ− g_Sf^abcA^b_µA^c_ν, (1.7) where A^aν are the gluon elds with index a, a = 1, ..., 8. t^a are the matrices generators of the SU (3) group, called Gell-Mann matrices. They satisfy [t^a, t^b] = if^abct^c, where f^abc are the group structure constants. Finally, gS is usually expressed as gS = √

4πα_S, where αS is the strong coupling constant. αS has been found to have a dependence inversely proportional to the energy (after applying a renormalization process similar to the one described in the previous section). Therefore, quarks and gluons behave as quasi free particles at high energies (short distances), while at low energies (large distances) quarks are conned into hadrons. These interesting behaviors are known as asymptotic freedom and connement, respectively. They determine the development of pp collisions and will be further discussed in Chapter3.

1.2.2.3 The weak interaction

The weak interaction is best known for being responsible for the beta decays. The weak in-teraction aects all fermions, including neutrinos. It has several massive mediators, unlike the electromagnetic and strong forces, called Z⁰ and W^±bosons. It is their heaviness that accounts for the very short range of the weak interaction. The Z⁰ and W^± bosons mediate the neutral and charged weak currents, respectively. As the lifetime of a particle is proportional to the

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inverse square of the coupling constant of the force which causes the decay, the lifetime of par-ticles relying on the weak force for their decay processes is large. The weak interaction is the only interaction able to change the avor of a quark or a lepton. In addition, it also breaks parity-symmetry since W^± bosons couple only to left-handed particles, i.e. particles with spin and momentum of opposite direction, and right-handed antiparticles.

The electroweak theory The weak and the electromagnetic interactions have been success-fully described as dierent manifestations of the same fundamental interaction by Glashow, Weinberg and Salam in the 60s. The gauge theory that describes both interactions is called unied electroweak theory and is based on the SU(2)I × U (1)_Y symmetry group. The local gauge invariance requirement leads to the existence of four bosons: Wµⁱ (i = 1, 2, 3) from SU(2) and Bµfor U(1). The elds of the electroweak bosons (Z^µ, (W^±)^µand the photon eld A^µ) are mixtures of these gauge boson elds. The lagrangian of the electroweak interaction, ruling the interaction between the gauge elds and fermions, is given by:

L_EW= ¯ψ_L(iγ^µ(D_µ))ψ_L+ ¯ψ_R(iγ^µ(D_µ))ψ_R−1

4W_µνⁱ W^{i µν}−1

4B_µνB^µν, (1.8) where L, R refers to the left- and right-handed fermions. The gauge elds, Wµνⁱ and Bµν, and the covariant derivative are given by:

D_µ = ∂_µ+1

2gτ_L,Rⁱ W_µⁱ −1

2ig⁰Y_L,RB_µ (1.9)

W_µνⁱ = ∂_µW_νⁱ− ∂_νW_µⁱ + gε_ijkW_µ^jW_ν^k (1.10)

B_µν = ∂_µB_ν− ∂_νB_µ, (1.11)

where g and g⁰ are the coupling constants associated to SU(2) and U(1), respectively. They are related to αQED by αQED = g sin θW = g⁰cos θW. θW is known as the weak mixing angle.

The generators associated with the SU(2) symmetry group are the Pauli matrices, τi, and the generator associated to U(1) is the hypercharge, Y = Q − I3, being Q the electric charge and I3 the third component of the weak isospin. The theory as described so far predicts massless SU (3)gauge elds, contradicting the experimental observations. The photon and the gluons are massless as a consequence of the exact conservation of the corresponding symmetry generators:

the electric charge and the eight color charges. The fact that the weak bosons are massive indicates that the corresponding symmetries are broken. In 1964, Higgs, Brout and Englert proposed that the breaking of the electroweak gauge symmetry is induced by the Brout-Englert-Higgs mechanism, which predicts the existence of a spin 0 particle, known as the Brout-Englert-Higgs boson, not yet experimentally observed. The Brout-Englert-Higgs mechanism consists in introducing an additional complex scalar doublet, Φ =φ⁺

φ⁰



, where:

φ⁺≡ (φ₁+ iφ₂)/√

2 (1.12)

φ⁰≡ (φ₃+ iφ4)/√

2. (1.13)

The Higgs lagrangian is given by:

L_Higgs = (DµΦ)^†(D^µΦ) − V (Φ^†Φ) (1.14) V (Φ^†Φ) = µ²Φ^†Φ + λ(Φ^†Φ)², (1.15)

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where Dµ is given by Eq. 1.9. For µ² < 0 and λ > 0, V has its minimum at Φ^†Φ = −^µ_2λ². Spontaneous symmetry breaking is induced if the minimum of V is obtained for non-vanishing Φ values. Expanding Φ around a particular minimum, say:

φ₁ = φ₂= φ₄= 0 (1.16)

φ²₃= −µ²

λ ≡ υ², (1.17)

we obtain:

Φ = r1

2

 0

υ + H



. (1.18)

Of the four scalar elds only the Higgs eld H remains. The other three scalar elds become the longitudinal modes of the W^± and Z which acquire a mass. The value of gauge bosons masses can be found by their coupling to the Higgs eld. By replacing Φ by Eq.1.18into LHiggs:

m_W = 1

2υg (1.19)

m_Z = 1 2υp

g²+ g⁰² (1.20)

mγ = 0. (1.21)

Similar to the generation of the gauge boson masses, the fermion masses can be introduced.

The coupling between the Higgs eld and massless quark and lepton elds are described by the Yukawa interactions. Fermions acquire a mass proportional to υ. But the Higgs boson has not been experimentally observed yet. A huge experimental eort is underway at LHC to reveal the Higgs sector. Its mass, mH =√

2λυ, is not predicted by the SM since λ is a free parameter. The searches at LEP, Tevatron, ATLAS and CMS have set experimental limits on its mass. Apart from the experimental constraints, there are theoretical bounds on the value of the Higgs mass.

Both, theoretical and experimental constraints in the Higgs boson mass will be briey discussed in Section 1.2.3.

The last (important for the studies presented here) missing point corresponds to the discussion of the fermion avor changes. Weak charged currents are the only interaction in the SM that changes the avor of the fermions: for example, by emission of a W boson an up-quark is turned into a down-quark, or a νe neutrino is turned into an e⁻. The mass eigenstates of fermions are not identical to the weak eigenstates. The transformation between them is described by a 3 × 3 unitary matrix: the Cabibbo-Kobayashi-Maskawa (CKM) matrix describes the mixing of the quark eigenstates, while the Maki-Nakagawa-Sakata (MNS) matrix describes the mixing for leptons. The CKM matrix is given by (as it will be referred to in the future) [13]:

V_CKM=





V_ud V_us V_ub V_cd V_cs V_cb V_td Vts V_tb



=





0.97428 0.2253 0.00347 0.2252 0.97345 0.0410 0.00862 0.0403 0.999152



. (1.22)

The probability for a quark of avor i to be transformed to a quark of avor j, emitting a W boson is proportional to |Vij|². The CKM and MNS matrix elements are free parameters of the SM and need to be determined experimentally.

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