The purpose of an effective field theory is to represent in a simple way the dynamical content of a theory in the low-energy limit. One uses only those light degrees of freedom that are active at low energy, and treats their interactions in a full field-theoretic framework. The effective field theory is often technically non-renormalizable, yet loop diagrams are included and renormalization of the physical parameters is readily accomplished.

Effective field theory is used in all aspects of the Standard Model and beyond, from QED to superstrings. Perhaps the best setting for learning about the topic is that of chiral symmetry. Besides being historically important in the development of effective field theory techniques, chiral symmetry is a rather subtle subject, which can be used to illustrate all aspects of the method, viz., the low-energy expansion, non-leading behavior, loops, renormalization and symmetry breaking. In addition, the results can be tested directly by experiment since the chiral effective field theory provides a framework for understanding the very low-energy limit of QCD.

In this chapter we introduce effective field theory by a study of the linear sigma model, and discuss the generalization of these techniques to other settings.

Effective lagrangians and the sigma model

The linear sigma model, introduced in Sects. I–4, I–6, provides a ‘user friendly’ introduction to effective field theory because all the relevant manipulations can be explicitly demonstrated. The Goldstone boson fields, the pions, are present at all stages of the calculation. It also introduces many concepts which are relevant for the low enegy limit of QCD.

In this chapter we study the second fundamental interaction, the strong one. It binds together the quarks by exchange of gluons. The strong charges, corresponding to the electric one, are called colours and their theoretical description is called quantum chromodynamics, QCD. There are similarities with QED, but also many fundamental differences. One of them, called confinement, is that quarks are never free. We cannot break a proton and extract its quarks with whichever energy we strike it. We shall see why.

We begin by showing how the colour charges were experimentally discovered by studying the production of hadrons, mainly pions, in experiments with electron–positron colliders. Moreover, we shall see that the underlying process e+ + e− → q + q−becomes evident when the CM energies are large enough. The quarks appear as jets of hadrons. When one of the quarks radiates a gluon, this appears as a third jet. In this way the gluon was shown to exist and its spin was determined by studying its angular distribution.

We saw in Chapter 4 how hadron spectroscopy points to their internal quark composition. However, even quarks as mathematical rather than physical objects can explain the spectroscopy. And so they were considered by many till experiments were performed at SLAC that probed the proton with high-energy, high-resolution, electron probes and showed, like the Retherford experiment on atoms, the presence of an internal structure.

Elementary particles are at the deepest level of the structure of matter. Students have already met the upper levels, namely the molecules, the atoms and the nuclei. These structures are small and their physics is properly described by non-relativistic quantum mechanics, by the Schrödinger equation. It is not relativistic because the speeds of the electrons in a molecule or in an atom and of the protons and neutrons in a nucleus are much smaller than the speed of light.

Protons and neutrons contain quarks, which have very small masses, corresponding to rest energies much smaller than their kinetic energy, and their speed is close to that of light. The structure of the nucleons, and more generally of the hadrons that we shall discuss, is described by relativistic quantum mechanics. The relevant equation, the Dirac equation, will be recalled.

The relativity theory is important in particle physics also for a different reason: the study of elementary particles requires experiments with beams accelerated at very high energies. There are two reasons for this: (a) the creation of new particles by, for example, annihilating a particle–antiparticle pair requires an initial energy large enough to be converted in the mass–energy of the new particle; (b) to study the internal structure of an object we must probe it with adequate resolving power, which increases with the energy of the probe, as we shall discuss.

In this chapter the student will learn the basic notions that will be necessary for her/his further study.

The development of the theoretical model that led to the electro-weak unification started in the 1960s. In this model, a single gauge theory, with the symmetry group SU(2) ⊗ U(1), includes the electromagnetic and weak interactions, both NC and CC. In particular, the electromagnetic and weak coupling constants are not independent but correlated by the theory. On the contrary, electro-weak theory and QCD, both being gauge theories, are unfied by the theoretical framework while their coupling constants are independent. Electro-weak theory and QCD together form the Standard Model of fundamental interactions.

In Sections 9.1–9.3 we shall introduce the electro-weak theory, as usual without any theoretical rigour. The unfication characteristics appear mainly in the neutral current processes. The transition probabilities of all these processes are predicted by the theory with a single free parameter, the electro-weak mixing angle. We shall mention the several processes in which it has been determined and discuss one of them in detail as an example.

A crucial prediction of the theory is the existence of three vector bosons, W+, W− and Z0. Even if the theory does not predict their masses, it precisely states the relations of the latter with two measured quantities, the Fermi constant and the weak mixing angle. We shall describe the UA1 experiment and the discovery of the vector bosons in Sections 9.6 and 9.7.

While keeping the same target and the same design principles as the first edition, the second edition results from a complete review (more precise definition of the parity of the spinors, further clariication of the concepts of helicity versus chirality, description of the GIM mechanism extended to loop order, gauge dependence of the colour charge, etc.) and is completely updated to take into account the progress of the ield in the past six years (on the mass differences and lifetimes of the neutral mesons, limit on proton decay lifetime, quark masses values, in particular, for the light ones, CP violation in charged mesons, etc.).

A very important element of the Standard Model, the origin of all the masses, had not been experimentally proven at the time of the irst edition, and as such it had not been included. The spontaneous symmetry breaking mechanism of Englert & Brout, Higgs and Gularnik, Hagen … Kibble is now discussed at the introductory level of the textbook. The CERN LHC collider, the ATLAS and CMS experiments and their discovery of the Higgs boson are now discussed, as well as the available measurements of its characteristics. I also include the precision tests of the Standard Model and the Higgs search at the Fermilab.

In the above mechanism, massless Goldstone bosons do not appear, but rather are absorbed in the non-physical degrees of freedom of the gauge fields.

‘Symmetry’ comes from the Greek word συμμετρος, meaning well-ordered. Symmetry is present in many fields. Several objects in Nature are geometrically symmetrical, for example a butterfly, or a flower, or a fullerene molecule.

Symmetry is used in physics in several different ways, exploiting its mathematical description. In modern physics, it is used as a powerful instrument to constrain the form of equations. The equations are assumed to be invariant under the transformation of a group, which may be discrete or a continuous Lie group. This approach is of fundamental importance, in particular, for particle physics.

This chapter begins with a simple classification of the various types of symmetry, introducing the concepts that will be used in later sections. The reader should already be familiar with Lorentz invariance, i.e. the invariance of the equation of motion under the Poincaré group, as well as the Noether theorem, at least in classical physics. The situation is not different in quantum physics.

The concept of spontaneous symmetry breaking is also introduced. It will evolve into the Higgs mechanism, which gives origin to the masses of the vector bosons that mediate the weak interactions, of the quarks and of the charged leptons. This fundamental aspect of the Standard Model has been experimentally verified by the LHC experiments in 2011–2012, as will be discussed in Chapter 9. The student will meet a simpler example of spontaneous symmetry breaking in the chiral symmetry of the strong interaction, in Section 7.11.

Only a few elementary particles are stable: the electron, the proton, the neutrinos and the photon. Many more are unstable. The particles that decay by weak interactions live long enough to travel macroscopic distances between their production and decay points. Therefore, we can detect them by observing their tracks or measuring their time of flight. Distances range from a fraction of millimetre to several metres, depending on their lifetime and energy. In this chapter, we shall study the simplest properties of these particles and discuss the corresponding experimental discoveries. In the next chapter we shall present to the reader the symmetry properties of the interactions and the corresponding selection rules, and in Chapter 4 we shall discuss the hadronic resonances, i.e. the particles that decay via strong interactions with lifetimes too short to allow them to travel over observable distances.

The development of experimental sciences is never linear, rather it follows complicated paths reaching partial truths, making errors that are later corrected by new experiments, often with completely unexpected results, and gradually reaching the correct conclusions. The study of at least a few aspects of such a development requires some effort but it is worth it to gain a deeper understanding of the resulting physical laws. This is why in this chapter we shall initially follow a historical approach.

We shall start by recalling the Yukawa assumption of a meson, the pion, as the mediator of the nuclear forces.

In the previous chapters we have learnt the properties of the elementary particles, a term that includes also the hadrons, which, as we have seen, are composed of smaller structures, the quarks and the gluons.

Now, starting with this chapter, we shall discuss the electromagnetic, strong and weak interactions of the leptons and the quarks, and finally their unification in the Standard Model. All of them are quantum field theories, for all of them the interaction Lagrangian is invariant under a local gauge symmetry, corresponding to different symmetry groups.

First, consider quantum electrodynamics, which corresponds to classic electrodynamics, namely the Maxwell equations. It was the first to be historically developed and the one with the simplest structure, corresponding to the simplest gauge group, namely U(I) We shall start by recalling the gauge invariance of the Maxwell equations and its strict connection to the conservation of the electric charge. These concepts are present in quantum field theories too, in which they become even more fundamental, because they determine the form of the interaction itself.

The fundamental experiment showing that non-relativistic quantum mechanics was insufficient to describe Nature was done by Lamb and Retherford in 1947. This masterpiece of atomic experimental physics is described in Section 5.2. We shall see that, in quantum field theory, it is not only the photons that are quanta of a field but also the charged particles, like the electrons and positrons.