World-Line Formalism: Non-Perturbative Applications

Abstract

This review addresses the impact on various physical observables which is produced by confinement of virtual quarks and gluons at the level of the one-loop QCD diagrams. These observables include the quark condensate for various heavy flavors, the Yang-Mills running coupling with an infra-red stable fixed point, and the correlation lengths of the stochastic Yang-Mills fields. Other non-perturbative applications of the world-line formalism presented in the review are devoted to the determination of the electroweak phase-transition critical temperature, to the derivation of a semi-classical analogue of the relation between the chiral and the gluon QCD condensates, and to the calculation of the free energy of the gluon plasma in the high-temperature limit. As a complementary result, we demonstrate Casimir scaling of k-string tensions in the Gaussian ensemble of the stochastic Yang-Mills fields.Keywords: the world-line formalism; Wilson loops; analytic calculations of path integrals with the minimal-area surfaces; theoretical foundations of the stochastic vacuum model; confinement and chiral-symmetry breaking in QCD; gluonic bound states; Casimir scaling; the Yang-Mills running coupling; thermodynamics of the gluon plasma; electroweak phase transition Quark Condensate for Various Heavy FlavorsAs is well known, because of confinement in QCD, quarks and gluons do not exist as individual particles, but appear only in the form of bound states (for recent reviews, see [1,2]). The latter include mesons, baryons (as well as other possible quark bound states, such as tetra-and pentaquarks), glueballs, and the so-called hybrids consisting of a quark, an antiquark, and one or several gluons. Each bound state can be represented as an average of a certain operator over the Euclidean QCD vacuum. Since the vacuum state is gauge-invariant, only the gauge-invariant operators yield non-vanishing results once averaged over it. (This statement can be viewed as a corollary of the so-called Elitzur theorem [3,4], which states that a local gauge symmetry cannot be broken spontaneously.) For local composite operators, such asψψ or (F a µν ) 2 , gauge invariance holds automatically, and the mean values of these operators yield the QCD condensates [5,6]. Nevertheless, the quark and the gluon bound states are in general represented by the field operators which are defined at different points of the Euclidean space, so that a phase factor (also called a parallel transporter or a Schwinger string) interpolating between these points is required in order to provide gauge invariance of the full non-local operator. That is, the non-local gauge-invariant operators have, e.g., the formψ i (x)Φ ij xxwhere Φ ij xx and Φ ab xx are the phase factors transforming under the fundamental and the adjoint representations of SU(N), respectively, with i, j = 1, . . . , N and a, b = 1, . . . , N 2 − 1. (Throughout this review, we denote the number of colors by N.) Thus, the phase factors provide long-range correlations of color between individual quark...

Paramagnetic versus Diamagnetic Interaction in the SU(2) Higgs Model

“…Furthermore, by considering the electroweak phase transition as a vacuum instability resulting from the negative sign of the paramagnetic contribution to the vacuum-energy energy, it became possible-within the present approach-to correctly reproduce the known critical temperature of that phase transition [5]. In the present paper, we will apply the same techniques to the calculation of the paramagnetic and the diamagnetic contributions to the one-loop effective action of the four-dimensional Higgs model (1).…”

“…Finally, let us extrapolate the above considerations to the deconfinement phase. To this end, we notice that, at temperatures larger than the deconfinement critical temperature T c , the chromo-electric condensate (E a i ) 2 vanishes, while the chromo-magnetic condensate (H a i ) 2 does not; which leads to the so-called spatial confinement (For references, see, e.g., Section 4 of [5].). This means that only the spatial components of the surface tensor Σ µν and the current j µ remain involved in the regularized expressions (6) and 7, so that Equations (8) and (9) take the form…”

“…Yet, some similarities can be drawn between the present case and the high-temperature phase of (the bosonic sector of) the Standard Model, as the latter admits an effective description by means of a three-dimensional SU(2) Higgs model [2,3], in which all the gauge bosons have equal masses. Techniques similar to those which are used in the present paper have been utilized, in [4,5], to obtain the ratio of the paramagnetic to the diamagnetic contribution to the one-loop effective action of that three-dimensional theory. Namely, in the limit of vanishing mass of the gauge bosons, the ratio has been found [4] to be equal to −16, while a generalization of this result to arbitrary values of the gauge-boson mass has been obtained in [5]; yielding, in particular, a finite value of −24 in the opposite limit, where this mass became much larger than the inverse correlation length of the high-temperature three-dimensional Yang-Mills theory.…”

Cylinder quantum field theories at small coupling

“…The worldline formulation of quantum field theory (QFT) is an important tool for organizing perturbation theory [1], as well as for describing non-perturbative effects [2]. In this framework, the usual Feynman graphs of a QFT are reinterpreted as terms in the path integral of a relativistic quantum mechanics describing the propagation of corresponding particles.…”

Cylinder quantum field theories at small coupling

“…The worldline formulation of quantum field theory (QFT) is an important tool for organising perturbation theory [1], as well as for describing non-perturbative effects [2]. In this framework, the usual Feynman graphs of a QFT are reinterpreted as terms in the path integral of a quantum mechanics describing the propagation of corresponding particles.…”