Behavior Of non-Abelian Gauge Theories Research Articles

Strings from large charged fields

It has been known for a long time that the low energy dynamics of open-strings is described by non-abelian gauge theories in the same way that the low energy effective description of closed strings is governed by Einstein's gravity. In this short note, we review and comment on some examples where conversely, an effective behavior of non-abelian gauge theories in some particular limits are described by some type of extended objects like strings or branes. We constrain the discussion to a few examples sharing some similarities.

Counting Gribov copies

We show that an explicit counting of Gribov copies can shed light on the infrared behavior of non-Abelian gauge theories. A power-law growth of the number of copies suppresses gluon propagation while the distribution of copies along a gauge orbit implies an enhanced density of small eigenvalues of the Fadeev-Popov operator. Both of these phenomena are related to confinement. The discreteness in the number of copies and the associated nonlocality also has implications for vacuum energy.

Vacuum Nodes in QCD atθ=π: Exact Results

We show that the vacuum functional of 3+1 dimensional non-abelian gauge theories vanishes for some classical field configurations $\psi_0(A)=0$ when the coefficient of the CP violating $\theta$--term becomes $\theta=\pi$ (mod. $2\pi$). Some of these classical configurations are explicitly identified and include sphalerons. The results shed new light into the non-perturbative behavior of non-abelian gauge theories and suggest a relevant role for these classical configurations in the confinement mechanism at $\theta=0$.

HAMILTONIAN DYNAMICS OF YANG-MILLS FIELDS ON A LATTICE

We review recent results from studies of the dynamics of classical Yang-Mills fields on a lattice. We discuss the numerical techniques employed in solving the classical lattice Yang-Mills equations in real time, and present results exhibiting the universal chaotic behavior of non-abelian gauge theories. The complete spectrum of Lyapunov exponents is determined for the gauge group SU(2). We survey results obtained for the SU(3) gauge theory and other nonlinear field theories. We also discuss the relevance of these results to the problem of thermalization in gauge theories.

Near-mass-shell infrared behavior of non-Abelian gauge theories and the renormalization group

We show the existence of renormalization-group equations that completely determine the infrared structure of near-mass-shell quantum-chromodynamics Green's functions. The formal solutions of these equations are governed completely by the behavior of the long-distance invariant charge $g({k}^{2})$. In the quark-gluon sector, the existence of these equations depends on choosing the particular near-mass-shell renormalization scheme where gluon and fermion momenta $q$ and $p$ are respectively subtracted at the "equal-rate-on-shell" point ${q}^{2} = \ensuremath{-}{\ensuremath{\lambda}}^{2}$, $\ensuremath{\gamma}\ifmmode\cdot\else\textperiodcentered\fi{}p\ensuremath{-}m = \ensuremath{-}\ensuremath{\lambda}$, where $m$ is the physical quark mass. The function that describes the evolution of $g({k}^{2})$, ${\ensuremath{\beta}}_{\mathrm{IR}}(g({k}^{2})) = {\mathrm{lim}}_{{k}^{2}\ensuremath{\rightarrow}0}{k}^{2}(\frac{\ensuremath{\partial}}{\ensuremath{\partial}{k}^{2}})g({k}^{2})$, is determined from the infrared structure of the gluon-quark sector alone, and it is shown to equal, explicitly at the one-loop level, ${\ensuremath{\beta}}_{\mathrm{YM}}(g({k}^{2}))$, the corresponding function of a pure Yang-Mills theory. We have computed explicitly, up to the two-loop level, the near-mass-shell behavior of the color-singlet form factor. Reorganizing all contributions as suggested by our renormalization-group results, we find that, in the Landau gauge, they sum up to the simple form $\mathrm{exp}[{B}_{1}(p;{p}^{\ensuremath{'}};{g}^{2}({k}^{2}))]$, where ${B}_{1}(p;{p}^{\ensuremath{'}};{g}^{2}({k}^{2}))$ is the one-loop contribution calculated by replacing the bare vertex ($\ensuremath{-}i{g}_{\ensuremath{\lambda}}$) by the invariant charge [$\ensuremath{-}ig(k)$]; where $k$ is the internal soft-gluon momentum. This result is valid to all orders in a near-mass-shell leading-log approximation. The connection of our work with the issue of infrared-driven quark confinement is briefly discussed.

Infrared behavior of non-Abelian gauge theories. II

We continue and extend our earlier work on infrared singularities of non-Abelian gauge theories (exemplified by quantum chromodynamics, or QCD), emphasizing the nature of these singularities in space-time rather than in momentum space. The leading logarithms of non-Abelian gauge theories sum to an eikonal (semiclassical) form for processes involving massive particles. This form is like the corresponding Abelian (QED) form, except that the coupling constant $g$ is replaced by the invariant coupling $\overline{g}(t)$ which depends on the momentum transfer $t$ to the gluon. It is not known whether the nonleading logarithms exactly cancel order by order, but there are strong cancellations which are governed by Ward identities. Besides four-dimensional QCD, two analogous theories which show confinement are studied: Two-dimensional QCD is developed in the light-cone gauge without recourse to the large-$N$ approximation of 't Hooft and is shown to have infrared singularities like those of a string theory with longitudinal modes. Three-dimensional QED is briefly discussed, and it is argued that confinement is revealed in the anomalous infrared behavior of the fermion propagator which is an entire function (of momentum) with no pole. Some speculations to this effect are made for four-dimensional QCD.

Infrared behavior of non-Abelian gauge theories

Infrared singularities in non-Abelian gauge (Yang-Mills) theories are studied in the leading-logarithm approximation. In addition to the usual infrared limit in which the infrared cutoff $\ensuremath{\mu}$ approaches zero for fixed on-shell momenta, we consider high-energy wide-angle scattering and form factors at large momentum transfer $t$ in which infrared-induced powers of ${g}^{2}{\mathrm{ln}}^{2}t$ ($g=\mathrm{coupling}\mathrm{constant}$) typical of vector field theories appear in perturbation expansions. Our techniques include asymptotic estimates of Feynman integrals (to sixth order) and a nonperturbative approach based on a conjectured formula for soft-meson emission. Remarkably, the logarithms sum up into exponential factors much as in QED. Unlike QED, cross sections for production of nonsinglet particles, even including an indefinite number of soft gauge quanta, vanish as $\ensuremath{\mu}\ensuremath{\rightarrow}0$, a phenomenon we interpret as evidence of particle confinement. We discuss exclusive hadronic processes in the context of the non-Abelian quark-gluon theory based on color SU(3). We find justification for the scaling laws at large momentum transfers ("quark counting rules") in that the infrared logarithms cancel in the scaling pieces of hadronic amplitudes while providing a fast damping of the "pinch" contributions associated with Landshoff graphs.

High-energy behavior of non-Abelian gauge theories

This paper is a detailed account of a study in perturbation theory of the high-energy behavior of non-Abelian gauge theories. We calculate the fermion-fermion scattering amplitude up to sixth order in the coupling constant in the high-energy limit $s\ensuremath{\rightarrow}\ensuremath{\infty}$ with fixed $t$, in the approximation of keeping only the leading logarithmic terms. Our results indicate that the high-energy behavior of non-Abelian gauge theories are complicated, and quite different from the known behaviors of other field theories studied so far.

Comment on High-Energy Behavior of Non-Abelian Gauge Theories

The high-energy behavior of the sixth-order fermion-fermion scattering amplitude in the Yang-Mills theory is recalculated, and found to be qualitatively different from that given previously.

Asymptotic Behavior of Non-Abelian Gauge Theories to Two-Loop Order

We have calculated the value of the Callan-Symanzik $\ensuremath{\beta}$ function to order ${g}^{5}$ for non-Abelian gauge theories with fermions. We discuss internal consistency of the calculation, and consider the approach to the aymptotic energy range in such theories.

High-Energy Behavior of Non-Abelian Gauge Theories

This paper is a detailed account of a study in perturbation theory of the high-energy behavior of non-Abelian gauge theories. We calculate the fermion-fermion scattering amplitude up to sixth order in the coupling constant in the high-energy limit $s\ensuremath{\rightarrow}\ensuremath{\infty}$ with fixed $t$, in the approximation of keeping only the leading logarithmic terms. Our results indicate that the high-energy behavior of non-Abelian gauge theories are complicated, and quite different from the known behaviors of other field theories studied so far.

Ultraviolet Behavior of Non-Abelian Gauge Theories

It is shown that a wide class of non-Abelian gauge theories have, up to calculable logarithmic corrections, free-field-theory asymptotic behavior. It is suggested that Bjorken scaling may be obtained from strong-interaction dynamics based on non-Abelian gauge symmetry.