Since ancient times, we have been curious to knowthe origin and the nature of the universe. Numerous ancient philosophers and scientists have tried to answer these fundamental questions. It is only at the present time of the twentieth millennium that we can provide a partial answer to these questions, as some significant progress has been accomplished in both particle physics and astrophysics, which are two areas of research in two apparently opposite scale directions (see Fig. 1.1).

On the one hand, this progress is due to our ability to explore the heart of matter, with powerful accelerators (where the accelerated particle has a velocity near to the velocity of light), which reveal their infinitely small, deepest structure (see Fig. 1.2).

As an example, we show in Figs. 1.3 and 1.4, the large electron-positron (LEP) accelerator and the reaction inside the detector after the collision of the electron and the anti-electron (positron). Notice that at LEP, the energy of the electron is in the range of 90-180 GeV which is about (5–10)×106 times the energy of our home TV screen. On the other hand, powerful telescopes (see Fig. 1.5) explore the enormous structure of the universe, and may reach the time of its origin.

We anticipated this discussion in Chapter 27 when we discussed the anatomy of the SVZ expansion. Here, we shall review the different determinations of the QCD condensates from QSSR.

Indeed, a good control of the values of the QCD condensates is necessary in the phenomenological applications of QSSR. The non-vanishing value of the light quark condensate, which we shall discuss in the next section, is intimately related to theGMORrealization of chiral symmetry, as can be inferred from the PCAC relation. SVZ [1,654] have also postulated that QCD is spontaneously broken by the gluon condensate, which they confirm from their analysis of the charmonium sum rule. The non-vanishing value of the gluon condensate and the gluon correlation length has been also checked on the lattice [402]. Since the pioneering work of SVZ, [1] a lot of effort has been devoted to this issue as can be found in the long list of published papers in this subject (for reviews see e.g. [3],[51,46], [356–363]). The condensates have been extracted from the light mesons [403–409], [325], [33,328], [341,387], and in [329] (Section 52.10), from the baryons [424–430], from the heavy quarkonia [433,434], and [313] (Section 51.3), and from the heavy-light mesons [401].

The e+e– →hadrons and τ-decays data have been always used as a laboratory for testing the perturbative and non-perturbative structure of QCD[1,3,325], [403–409], [346,338,341] and [329,161] (Sections 19.4 and 52.10).

The thermodynamic background

Let's begin our discussion of QCD in extreme environments that are rich in baryons by reviewing some basic concepts of thermodynamics. We want to remind the reader of the partition function, its relation to Euclidean field theory, and the physics and formalism of the chemical potential in simple settings. These discussions will set the stage for more-challenging applications to QCD.

The reader should appreciate that QCD is a remarkable, yet frustrating theory. On the one hand, we know the theory's “first principles” thoroughly, in the sense that we have a well-defined prescription for calculating any quantum-mechanical amplitude governed by strong interactions. The principles are simple and beautiful. However, the phenomena which result from application of these principles are very complicated. For example, asymptotic freedom indicates that a natural language, both for physical ideas and for quantitative estimates, for the short-distance behavior of QCD is that of weakly interacting quarks and gluons. For larger distances and times, this description becomes inadequate in several ways. First, the running coupling grows large and perturbative estimates are no longer reliable. Second, the assumption that physical quantities should be analytic functions of the coupling becomes untenable. Quarks and gluons are not weakly fluctuating degrees of freedom at these length scales. Other dynamical variables, perhaps instantons, color-bearing monopoles, and flux tubes, provide a more relevant description of the physics. These facts indicate that reliable calculations in QCD are beyond our existing mathematical abilities.

Color superconductivity and color–flavor locking

Lattice-gauge theory has produced unique insights and predictions for QCD at high temperature, but it has had little to say about QCD in environments rich in baryons. There are several reasons for this impasse. One of them is the fact that the fermion determinant is complex in this environment, so the standard algorithms of the subject fail. This point will be demonstrated later. Other more conventional methods, such as weak-coupling approximations, apparently apply in this environment and suggest that the theory experiences color superconductivity at asymptotically large baryon chemical potentials. There is the hope that the transition to this exotic state of matter actually occurs at a chemical potential of just a few hundred MeV. The subject of color superconductivity produces interesting conjectures for the QCD phase diagram in the T–µB plane where T is small (below a few tens of MeV) and µB is very large. In this chapter we will discuss the major points, both strengths and weaknesses, of this approach and refer the reader to the growing literature for more recent, developing events. Once lattice-gauge theory develops an algorithm that applies to this environment, these ideas will be put to stringent tests. However, for much of the rest of this chapter, we must ignore lattice-gauge theory and focus on weak-coupling methods and the BCS theory of superconductivity and its application and extensions to QCD.

Finite-temperature transitions at strong coupling

When QCD is heated sufficiently it will undergo a phase transition from a state of colorless strongly interacting particles to a quark-gluon plasma. The properties of this phase transition and properties of the high-temperature plasma are the subjects of a great deal of modern research in QCD.

We have seen that pure gauge-field lattice dynamics confines quarks at strong coupling. If we use the quenched approximation in which there are no internal quark loops, then external static quarks experience a linear confining potential and the only finite-energy states must be color singlets consisting of confined quarks and/or anti-quarks. It is not difficult to show that, if this theory is heated, it will experience a transition to a quark-gluon plasma [43]. The point is that, as the temperature is increased, thermal fluctuations will become large enough to melt the linear flux tube which had produced confinement. At high temperature the vacuum itself becomes a tangle of closed flux loops, bubbling and fluctuating. When a static quark is placed into this vacuum, the extra flux line emanating from it blends into a vacuum already rich in flux and the resulting state does not cost an energy that linearly diverges. In the hot gauge-field state, there is no barrier to generating more or less flux and complete screening of the external, static quark color can occur, no matter what representation of SU(3) it lies in.

The problems of quark confinement, chiral-symmetry breaking, and strongly interacting matter in extreme environments are converging and much is being learned, both experimentally and theoretically. We hope that this book helps researchers find common ground in this developing field.

The breadth of the field is quite remarkable. Concepts from critical behavior, effective-field theory and random-matrix models, lattice-gauge theory and numerical methods, spin models and duality, equilibrium and nonequilibrium thermodynamics, and even string theory play important roles here. It is impossible to tell which approach will lead to the next major advance in the field.

The field depends on a “new culture” in theoretical physics, one in which theorists use lattice-gauge theory as a realistic laboratory to test new ideas through numerical methods and computer simulations.The theorist is no longer limited, in many but, alas, not all cases, to unrealistic models for concrete inspiration. Lattice-gauge simulations have led to the sharpening of several approaches to QCD that have influenced the field very productively. We can look forward to the development of new algorithms that can address SU(3) QCD at nonzero baryon chemical potential and find new states of matter at high densities.

Free fermions on the lattice in one and two dimensions

Lattice fermions, their symmetries, and their continuum properties are issues central to lattice studies of dense and hot matter. It is a subtle subject because there are fundamental restrictions on the number of fermion species, their handedness, and gauge invariance. To see what the challenges are, we will illustrate lattice forms of the Dirac equation in various settings.

First consider the free Klein–Gordon equation and free Dirac equation on a spatial lattice with continuum time variable. In a Hamiltonian lattice-gauge theory, one would have a three-dimensional lattice and a continuum of time. Excitations in the system could hop from site to site given the rules of the Hamiltonian, the discrete version of the spatial derivatives in the energy, etc. Although Hamiltonian lattice-gauge theory is an important subject, our emphasis here continues to be on the Euclidean version of the theory. However, a short look at Hamiltonian methods is very elementary and enlightening. The details of the problems of lattice versions of the Dirac equation are different depending on whether time is treated as a continuum variable or a discrete one. Euclidean lattice fermions will be discussed in detail below after our introduction.

Let there be a free boson field φ(x, t) in 1 + 1 dimensions, namely one discrete spatial axis and one continuum temporal axis.

Lattice formulation, local gauge invariance, and the continuum action

In its standard formulation, lattice-gauge theory replaces the continuum spacetime of Euclidean quantum-field theory with a four-dimensional lattice of spacetime points and links. If the lattice is hypercubic, then its lattice spacing a provides an ultraviolet cutoff. Although such a spacetime cutoff is not convenient in most analytic calculations, it has several extraordinary advantages. First, it allows strong-coupling calculations both through expansions and by simulation methods. These approaches provide a framework for formulating and developing physical pictures of confinement, chiral-symmetry breaking, and other nonperturbative phenomena that are so challenging in ordinary continuum formulations of quantum-field Theory.

The various fields of the lattice version of quantum chromodynamics exist on the sites and links of the regular lattice. In the standard formulation, the fermion fields reside on the sites and the gauge fields reside on the links [10]. The fermion fields, matter fields in general, carry the color quantum number of the gauge group SU(3). It is convenient, therefore, to imagine that there is a color frame of reference at each site so that the fermion field with color index α, ψα, can be visualized. (This geometrical point of view dates back to the original Yang–Mills paper which introduced non-Abelian gauge fields into high-energy physics when they generalized the notion of isospin symmetry from the familiar global symmetry of nuclear physics to a local symmetry of high-energy field theory [11].)