Although the main topic of this book is the study of QCD, it is in practice impossible to ignore the electroweak interactions. From an experimental point of view they are inextricably linked with QCD. The success of the present day e+e− colliders in the study of QCD is due to the large event rates which occur at the Z boson resonance. In addition, many of the event signatures at hadronic colliders involve either the production of vector bosons or leptons coming from the weak decays of heavy quarks. This chapter describes the Standard Electroweak Model [1], which, together with the discovery of QCD, was the triumph of the 1960s and 1970s. The model provides a unified description of the weak and electromagnetic interactions in terms of a renormalizable, gauge invariant field theory. We shall present the Lagrangian of the model and the Feynman rules which follow from it, together with some consequences of the standard model couplings. A more complete treatment of the electroweak sector of the Standard Model can be found, for example, in ref. [2].

Gauge boson interactions

In QED the Lagrangian density has the property of local gauge invariance. The analogous property for QCD was explained in Section 1.3.

The genesis of the theory which we now call Quantum Chromodynamics was the result of the assembling of many ideas and experimental results. In this chapter we start by reviewing the main evidence for the colour degree of freedom, which lies at the heart of the theory. We then write down the QCD Lagrangian and the Feynman rules which follow from it. We go on to discuss the exact and approximate symmetries of the theory.

Colour SU(3)

The most fundamental tenet of QCD is that hadronic matter is made of quarks. The idea of quarks arose from the need to have a physical manifestation for the SU(3) of flavour [SU(3)ƒ] observed in the spectrum of the lowest-mass mesons and baryons. The properties of the six known quarks are shown in Table 1.1. The observed baryons are interpreted as three-quark states. The quark constituents of the baryons are forced to have half-integral spin in order to account for the spins of the low-mass baryons. The quarks in the spin-3/2 baryons are then in a symmetrical state of space, spin and SU(3)ƒ degrees of freedom. However the requirements of Fermi-Dirac statistics imply the total antisymmetry of the wave function. The resolution of this dilemma was the introduction of the colour degree of freedom: a colour index a with three possible values (usually called red, green, blue for a = 1, 2, 3) is carried by each quark. The baryon wave functions are totally antisymmetric in this new index.

There are some interesting applications of thermal field theory to astro-physical processes; to be fair, all necessary computations may also be performed in the framework of standard kinetic theory, but experience with thermal field theory has recently allowed us to improve earlier calculations, resulting in large corrections in some circumstances. We shall be interested in the energy losses of stars due to the emission of weakly interacting particles. The relevant astrophysical systems are the following.

1. (i) The core of type II supernovae, which is a plasma with temperature T ∼ 30–60 MeV and density π ∼ 1015 g cm−3. The electron chemical potential is µ ∼ 350 MeV and the plasma frequency is ωp ∼ 20 MeV.

2. (ii) The core of red giants before the ‘helium flash’: in this case T ∼ 108 K (10 keV), π ≃ 106 g cm−3, corresponding to an almost degenerate electron gas with Fermi momentum PF ≃ 400 keV/c and a plasma frequency ωp ≃ 20 keV.

3. (iii) The core of young white dwarves, in which typical conditions are T ∼ 106 − 107 K (0.1–1 keV) and π ≃ 2 × 106 g cm−3, which again corresponds to an almost degenerate electron gas with Fermi momentum PF ≃ 500 keV.

We now start what will be the subject of this book: quantum field theory at non-zero temperature and/or non-zero chemical potential. Two competing formalisms have been used at zero temperature in order to study field theory: the operator formalism, which is the older one, and the path integral formalism, which represents the more modern approach. Actually, it may often be illuminating to look at a given problem from both points of view, although in some cases one of the formalisms may prove to be definitely superior to the other: for example, path integrals are much simpler when quantizing gauge theories. At finite temperature, both formalisms are useful, and it is instructive to be able to switch from one approach to the other. Since the simplest case of field theory is field theory with zero space dimension, or, in other words, quantum mechanics, we shall begin with a short description of quantum mechanics at finite temperature, or, equivalently, quantum statistical mechanics. We first deal with the path integral approach, and later on revert to the more conventional operator formalism. We shall be particularly interested in time-ordered products of position operators, which will generalize to time-ordered products of field operators in field theory.

Quantum chromodynamics

Most high-energy physicists will readily agree that quantum chromodynamics (QCD) is today the well-established theory of strong interactions; at least it has no serious competitor. Quantum chromodynamics is a non-Abelian gauge field theory whose gauge group is the colour group SU(3); it will sometimes be convenient to let the number of colours vary, and to take SU(N) as the gauge group: then the number of colours is N. There are (N2 − 1) = 8 gauge bosons, called gluons, and the matter particles are spin ½ quarks. There are six families of quarks, or six different flavours: up, down, strange, charm, beauty and top. The number of flavours will be denoted by Nf. The last three quarks are heavy and will not play any role at all in the development of this book because their mass is much larger than the characteristic energy scale of a few hundred MeV that we are interested in, while the role of the strange quark will be intermediate. The up and down quarks will often be taken as massless since their mass, of the order of a few MeV, is much smaller than our characteristic energy scale. Note that we use a system of units where ħ = c = KB = 1, where ħ, c and KB are, respectively, the Planck constant, the speed of light and the Boltzmann constant. Masses and temperatures will be often measured in MeV or GeV, lengths and times in MeV−1 or GeV−1.

This chapter deals with some infrared problems which arise in gauge theories at finite temperature. Most of these problems are still open, and represent interesting – and difficult – challenges for future investigations.

At zero temperature, the existence of singularities in field theories with massless particles has been known for a long time. It is convenient to classify these singularities into infrared and mass singularities. In order to be definite, let us take QCD as an example.

1. (i) A quark can emit a soft gluon, whose momentum k → 0: because the gluon is massless, this leads to infrared singularities, even when the quark is massive.

2. (ii) A massless quark can emit a gluon whose momentum makes a small angle θ with the initial quark momentum. The region θ → 0 leads to mass (or collinear) singularities, even if the gluon momentum k remains finite.

In both cases, the singularities arise from the fact that the final state quark + gluon is degenerate with the initial one when k → 0 in case (i) and when θ → 0 in case (ii).

Much is known about these infrared and mass singularities at T = 0; the most important information is contained in the Bloch–Nordsieck and Kinoshita–Lee–Nauenberg (KLN) theorems, which allow, in well-defined situations, control of these singularities.

In the preceding chapter we have learned that the leading behaviour in temperature of the gauge particle and fermion self-energy is proportional to T2, and that this behaviour is obtained without too much effort in the HTL approximation. In the present chapter we generalize these results to N-point functions, computed at the one-loop approximation. We shall show that some (but not all!) N-point functions also behave as T2, and that these N-point functions have rather simple expressions in the HTL approximation. This situation should be constrasted with that of the ϕ4-theory, where only the two-point function behaves as T2: once more, gauge theories are much richer than scalar theories.

We shall also discover that these N-point functions obey remarkable Ward identities, and that there are again striking similarities between QED and QCD. All our results can be expressed in a compact way by writing an effective Lagrangian. Most importantly, we shall show how to correct naïve perturbation theory, which breaks down for soft external momenta, by using a resummed (or effective) perturbative expansion. Some applications to physical processes will be given in section 7.3, and more will be given in the following chapters. We conclude with a kinetic derivation of hard thermal loops, which generalizes the results of section 6.4.

Non-relativistic field theory at finite temperature and finite density was invented in the late 1950s for the theoretical description of condensed matter and nuclear matter under standard laboratory conditions. Although it does not involve concepts beyond the Schrödinger equation and statistical mechanics, it offers a most convenient theoretical framework with which to deal with a large number of particles. It is often referred to as the ‘Many-Body Problem’ or as the ‘N-Body Problem’. Relativistic field theory at finite temperature was first studied by Fradkin (1965) and rediscovered ten years later, the main motivation at the time being a description of the phase transition which occurs in the electroweak theory, at a temperature of the order of 200 MeV. This transition is of course of great interest for the history of the early Universe.

In the early 1980s, lattice gauge theories suggested the existence of a deconfined phase of quarks and gluons, which has been called the ‘quark–gluon plasma phase’, above a temperature which is estimated nowadays to lie around 150 MeV. The possibility of observing this new state of matter in ultrarelativistic heavy ion collisions gave a further boost to the study of finite temperature field theory. Furthermore, Braaten and Pisarski (1990a,b) made a theoretical breakthrough, which enabled them to reorganize perturbation theory, whose original formulation had led to nonsensical results in some circumstances.

It is well-known that the properties of elementary particles are modified when they propagate in a medium, as they become ‘dressed’ by their interactions: they acquire, for example, an effective mass which is different from the mass as measured in the vacuum. More generally, one speaks of the propagation of collective modes, or quasi-particles; in some cases, these quasi-particles can easily be identified with ordinary particles whose properties are only slightly modified by their interactions with the medium. In other cases collective modes bear little resemblance with particles in the vacuum.

Collective modes are characterized by a dispersion law ω(q) giving their energy ω as a function of their momenta q. Their lifetime is not infinite, contrary to that of stable particles in the vacuum; thus another relevant quantity is the decay (or damping) rate γ(q) of the collective modes.

In general, collective modes appear mathematically as poles of propagators with well-defined quantum numbers in the complex plane of the energy: the real part of the pole gives the dispersion law, while the imaginary part gives the damping rate; see, however, the remarks following (6.19). In the present chapter we shall study the propagation of gauge bosons and fermions in a plasma, and we shall compute explicitly the dispersion laws to a first approximation.