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Nonperturbative Phenomena in QCD at Finite Temperature

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Nonperturbative Phenomena in QCD at Finite Temperature Physics of Atomic Nuclei, Vol. 68, No. 5, 2005, pp. 723–747. Translated from Yadernaya Fizika, Vol. 68, No. 5, 2005, pp. 755–779. Original Russian Text Copyright c 2005 by Agasian. Nonperturbative Phenomena in QCD at Finite Temperature N. O. Agasian* Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117259 Russia Received July 26, 2004 Abstract—The main objective of the present study is to analyze various nonperturbative phenomena in QCDboth at low, T<Tc, and at high, T>Tc, temperatures. New methods are developed that make it possible, on one hand, to describe data obtained by numerically simulating QCDon a lattice and, on the other hand, to study new physical phenomena in QCDat finite temperature. c 2005 Pleiades Publish- ing, Inc. 1. INTRODUCTION which was predicted theoretically in recent years (for an overview, see [3–5]), can be realized. The physics of strong interactions, QCD, has been flourishing over the past three decades. For many Quantum chromodynamics is a quantum the- years, investigations into the behavior of strongly in- ory of Yang–Mills gauge fields interacting with quark fermion fields. The asymptotic-freedom phe- teracting matter under various external effects have nomenon [6, 7] and a topologically nontrivial struc- been of great topical interest. In the real world, these ture of the vacuum of non-Abelian gauge theories [8– are primarily temperature and the baryon density. In- 12] were discovered in the 1970s. A further devel- terest in the behavior of matter under extreme con- opment of the theory revealed that it is precisely the ditions (high temperatures commensurate with the complicated nonperturbative structure of the vacuum characteristic QCDscale, T ∼ 200 MeV, and high −3 (nonperturbative fluctuations of vacuum fields) that baryon densities, n>n0 0.17 fm ,wheren0 is the is responsible for the phenomena of confinement and normal nuclear density) is motivated by the fact that spontaneous chiral-symmetry breaking and, thereby, an increase in energy was achieved in experiments for the formation of the physical hadron spectrum. aimed at studying heavy-ion collisions. In view of Thus, the development of new theoretical approaches this, it is expected that densities and temperatures at was required for describing nonperturbative phenom- which a phase transition to a new state of strongly ena in quantum field theories. interacting matter, quark–gluon plasma [1, 2], is pos- Investigations of the vacuum state at finite tem- 1) sible are reached in such experiments. perature and finite values of the chemical potential Extreme conditions existed at the initial stage of and external fields lead to new interesting phenom- expansion of the Universe. Within the time interval ena, including various phase transitions. Accordingly, − − there arises the need for developing a theoretical for- t ∼ 10 6−10 5 s after the Big Bang, the Universe malism for exploring the behavior of a quantum-field passed the stage of a strong phase transition in system and its vacuum state under external effects. which the system transformed into a hadronic phase dominated by the essentially nonperturbative phe- Quantum field theory at finite temperature and a nomena of confinement and spontaneous chiral- finite value of the chemical potential has been studied symmetry breaking. Extremely high baryon densities predominantly along three lines: (i) Perturbative calculations have been performed, (n ∼ 10n0) also exist in central regions of neu- tron stars, where the color-superconductivity phase, and various resummation schemes for perturbation- theory series have been constructed. Advances have *e-mail: [email protected] predominantly been made in developing the hard- 1)It should be emphasized from the outset that there is thermal-loop (HTL) and hard-dense-loop (HDL) ap- presently no consensus on the theoretical interpretation of proximations (for an overview, see [13, 14]). the experiments in question. The main reason is that it is not (ii) Various effective models that describe one quite clear whether the initial stage of a heavy-ion collision can be justifiably described in terms of a steady-state ther- nonperturbative phenomenon in actual QCDor an- modynamically equilibrium system in the phase of quark– other have been constructed. For example, various gluon plasma. approaches based on employing sigma models to 1063-7788/05/6805-0723$26.00 c 2005 Pleiades Publishing, Inc. 724 AGASIAN describe the chiral order parameter in QCDhave metries and constraints that these symmetries im- been developed; investigations within effective chiral pose on physical characteristics of the system are theory at T =0 have been performed; and various of particular importance in QCD, which is a the- generalizations of the Nambu–Jona-Lasinio model ory that involves confinement and where composite to the case of finite temperature, a finite value of the states (hadrons) appear as observables. Low-energy chemical potential, and finite external fields have been theorems, or Ward identities (scale and chiral ones), introduced. Other effective models have also been play a crucial role in the understanding of nonpertur- developed (for an overview, see [3, 5]). bative vacuum properties of QCD. Stricly speaking, (iii) The complex nonperturbative structure of the low-energy theorems were formulated almost simul- vacuum of non-Abelian gauge theories—in partic- taneously with the beginning of the application of ular, QCDat finite T and n—has been studied by quantum-field methods in particle physics (see, for means of numerical simulations with the aid of com- example, Low theorems [26]). In QCD, they were puters. Interesting and important results have been obtained in the early 1980s [27]. Low-energy theo- obtained along this line, and it is expected that an rems in QCD, which follow from general symmetry increase in the power of computers and the creation of properties and which are independent of the details new computational schemes would make it possible of the confinement mechanism, make it possible to to obtain deeper insights into the nonperturbative obtain information that sometimes cannot be deduced dynamics of QCD(for an overview, see [15 –17]). in any other way; they can also be used as “physi- It should be noted that the high-temperature cally reasonable” constraints in constructing effective phase (T>Tc), which involves restored chiral sym- theories and various models of the QCDvacuum. In metry and deconfined quarks and gluons, is described the present study, we develop a method that makes in a conventional way as a quark–gluon plasma it possible to generalize low-energy QCDtheorems by analogy with an electromagnetic plasma. In this to the case of finite temperature. The nonperturbative approach, the approximation of noninteracting quark QCDvacuum and condensates at T =0are studied and gluon gases is used for a zero-order approxima- by using this method. tion; further, the interaction is taken into account in As was indicated above, nonperturbative fluc- the form of a series in powers of the QCDrunning tuations of gluon fields play a crucial role in the coupling constant g(T ). In view of the asymptotic- vacuum of non-Abelian gauge theories and determine freedom phenomenon in QCD, the coupling constant many features of QCD. The instanton-liquid model, must decrease with temperature. In order to improve which was proposed in [28, 29], is an elaborate theory the convergence of a relevant series—for example, in that explains quite successfully some phenomena in calculating pressure, which is among the main ther- QCD. In this pattern, well-separated and not very modynamic quantities that characterize the system— strongly interacting instantons and anti-instantons use is made of the method for summing hard ther- (from here, the name “liquid” comes) are the main mal loops, which effectively reduces to introducing nonperturbative fields. The instanton density is ap- −4 the so-called magnetic gluon mass. The presence proximately N/V4 =1 fm . This model makes it of a nonperturbative magnetic mass, ∝g2(T )T ,in possible to solve a number of problems in QCD—in the theory is necessary for removing well-known particular, the phenomenon of spontaneous chiral- infrared singularities [18]. At the same time, the symmetry breaking naturally arises there and the QCDinteraction remains strong, g(5Tc) ∼ 1,evenat η -meson mass is explained within it. Neverthe- temperatures T ∼ 5Tc, so that the use of perturbation less, the model in question possesses a number of theory, at least as a standard means for taking into serious drawbacks—namely, it is not known how account small corrections in the interaction to the the instanton–anti-instanton ensemble is stabilized zero-order approximation, is not quite legitimate. (problem of an infrared inflation of instantons) and it Moreover, the magnetic-confinement phenomenon is impossible to explain the confinement phenomenon in non-Abelian gauge theories, which occurs at all within the instanton-liquid model. However, the temperatures, inevitably requires the presence of fluc- vacuum involves, in addition to semiclassical instan- tuating nonperturbative chromomagnetic fields [19– tons, other nonperturbative fields, which enables one, 22] (see also [23–25]). Thus, QCDthermodynamics among other things, to solve the infrared problem of above the critical temperature must be described in instantons. terms of a phase that involves a strong chromomag- The nonperturbative QCDvacuum can be param- netic condensate, against whose background there eterized by a set of nonlocal gauge-invariant vacuum arise excitations of hot quarks and gluons. expectation values of gluon-field strengths [30–32]. Relations that are obtained as corollaries of the It appears that, in this way, one can describe well a symmetry properties of the theory play an impor- large number of hadron-physics phenomena (see the tant role in quantum field theory. Searches for sym- review article of Di Giacomo et al. [33]). In order to PHYSICS OF ATOMIC NUCLEI Vol. 68 No. 5 2005 NONPERTURBATIVE PHENOMENA IN QCD725 explain qualitatively relevant effects, it is sufficient, in temperature is given.
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