The formulation of the Standard Model (SM) of particle physics represents one of the great achievements of the twentieth century. A complete description of the SM is far beyond the scope of this thesis. However, a few of the most salient features are summarised.
The SM describes all fundamental matter particles (fermions) and the electromagnetic, weak, and strong interactions involving the exchange of gauge bosons, using the QFT formalism. It thereby combines the two QFT descriptions of the electroweak and strong interactions. The former description already unifies the relativistic electromagnetic theory, called quantum electrodynamics (QED), with the theory for weak interactions, as initially proposed by Glashow, Weinberg, and Salam [3,4,5]. The theory of strong interactions is called quantum chromodynamics (QCD), with an allusion to the charge of the strong force, namedcolour.
The most common approach to formulate a QFT uses the Lagrangian (L), which is a (scalar) func-tion that represents the dynamics of a given system. The equafunc-tions of mofunc-tion can then be derived using the least action principle. Feynman generalised the least action principle to the path integral formulation and the so-called Feynman diagrams [6,7]. They provide a deep physical insight into the interactions of particles: particles interact in all possible (allowed) ways; the probability for each final state is the sum over all such possibilities. A Feynman diagram represents a class of particle paths, which join and split as described by the diagram.
Both the unified electroweak theory and QCD are formulated as Yang-Mills theories [8]. This means they are gauge QFTs based on certain symmetry groups. Astonishingly, symmetry gauging gives rise to the force fields in a natural way. In the simple example of classical electrodynamics, many electromagnetic potentials describe the same electromagnetic field. This freedom of choice is calledgauge invariancebecause the physics observables are invariant in the chosen gauge (here the electromagnetic potential). The invariance is caused by a continuous symmetry, which always implies a conservation law (Noether’s theorem [9]). The idea of (local) gauge invariance as a dynamical principle to construct interacting field theories was first elaborated by Weyl [10].
In QED for instance, the requirement of local gauge invariance under theU(1)emsymmetry group leads to the introduction of a new gauge field. This field transforms just as Maxwell’s equations.
Indeed, it describes the massless spin-1 photon field. In much the same way, local invariance requirements in the electroweak theory bring in the gauge bosons of the two forces: massiveW+, W−, andZbosons which mediate the weak force, and the masslessγ boson of electromagnetism.
The underlying symmetry group of the electroweak theory isSU(2)L⊗U(1)Y, representing the weak isospin or chiral symmetry (L) and the weak hypercharge (Y) symmetry. The two neutral gauge bosonsB0,W0 ofSU(2)L⊗U(1)Y mix (described by the Weinberg angle) and thereby form theZandγ bosons. Finally, the requirement of local gauge invariance in QCD results in the eight massless spin-1gluons of the strong force. The gauge symmetry group of QCD isSU(3)C and describes three colour degrees of freedom.
The combinedSU(3)C ⊗SU(2)L⊗U(1)Ysymmetry group, together with the dynamical sym-metry gauging principle is often seen as the basis of the SM.
An important issue is related to the massive gauge bosons (W± andZ). The presence of any (fermionic or gauge) mass term violates the chiral symmetrySU(2)L. A solution to this severe
2.1. THE STANDARD MODEL 5
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mH(GeV/c2)
95% CL Limit/SM
Tevatron Run II Preliminary, L=0.9-4.2 fb-1
Expected
Figure 2.1: Global fit to electroweak precision data (left) [17], and Higgs boson mass exclusions from LEP and Tevatron (right) [16].
problem was suggested by Higgs [11,12]: the gauge invariance could bespontaneously broken with the addition of (at least) one doubletφof complex scalar fields, with Lagrangian
LHiggs = (∂µφ)†(∂µφ)−V(φ), where the potentialV(φ) =µ2(φ†φ) +λ¡
φ†φ¢2is the key to spontaneous symmetry breaking.1 This Higgs mechanism renders massiveW±,Z, and fermions possible, while retaining a massless photon. Additionally, a massive Higgs boson (h) is predicted, where the only free parameter is the Higgs mass. The Higgs boson is the only SM particle which has not yet been discovered.
Further strong arguments for a scalar Higgs boson arise from the amplitude of (longitudinal)W+ W−scattering. If only theZ andγ bosons are exchanged, then the amplitude violates unitarity.
The Higgs contributions, however, can cancel the divergence. Major searches for the Higgs boson have been conducted at the Large Electron-Positron Collider (LEP) at CERN and at the Tevatron collider at FNAC. The LEP lower limit for the Higgs boson mass ismh >114.4 GeV[15], while the Tevatron has recently excluded a Higgs boson mass between160 and170GeV [16], both at 95% confidence level. The exclusion results together with a global fit to precision electroweak measurements are shown in Fig.2.1.
In the formulation of the SM, it is also noteworthy that theSU(2)andSU(3)symmetry groups are non-abelian. As an important consequence, theW± and gluon gauge bosons carry weak and colour charge respectively. They can therefore self-couple, contributing to their self-energy and thus to the running of the coupling constants. This is one illustrative example of the deep relation between the underlying symmetries and the effective theory.
1Other physicists, Brout and Englert [13], as well as Guralniket al.[14], had reached the same conclusion indepen-dently about the same time.
Table 2.1: Fundamental fermionic particles in the SM grouped according to family. No right-handed neutrinos are included. Braces indicate weak isospin doublets. The subscriptsLandRdenote the left and right handed components respectively.
Family I II III
leptons Llep (e, νe)L (µ, νµ)L (τ, ντ)L
Rlep eR µR τR
quarks Lq (u, d)L (c, s)L (t, b)L
(×3colours) Rq uR, dR cR, sR tR, bR
QCD enjoys two special properties:
• Confinement, which describes the rapid increase of the strong force when trying to sepa-rate two coloured particles (in contrast to all other forces which diminish with increasing distance) [18].
• Asymptotic freedom, which means that at very small distances (or equivalently very high-energy reactions) coloured particles interact very weakly [19,20].
The very important consequences of QCD confinement and asymptotic freedom are: we can use perturbation theory for high-energy processes, but not in the low energy regime; coloured particles (gluons and quarks) will undergo so-called hadronisation, before we could possibly “observe”
them. In the process of hadronisation, the coloured objects fragment (group themselves) into colour singlet (neutral) objects. The resulting colour singlet objects are hadrons and mesons. A collimated “jet” of such hadrons and mesons is what we detect experimentally, if the initial parton was generated with high momentum, i.e. it originates from a hard-scattering process.
The fundamental particles of the SM are fermions. There are three generations (or families) of coloured up- and down-type quarks, and three families of charged and neutral leptons. The quarks come in three colours, as described in the quark model developed by Gell-Mann [21]. Table2.1 summarises the fermionic particles of the SM. The question why the fundamental fermionic parti-cles come in three generations cannot be answered by the SM (already the discovery of the muon led to the famous quote by Rabi: “Who ordered that?”). Theorists hope to address this question with more fundamental theories (with extended symmetries).
The quark mass eigenstates as given in Table2.1are not the weak eigenstates. The latter are mixed states where a unitary3×3matrix, calledCabibbo-Kobayashi-Maskawa (CKM)matrix [22,23], governs the transformation. The implications of this quark mixing are very important for CP-violation, see for instance Ref. [24].
For completeness, the SM Lagrangian (before electroweak symmetry breaking and without