Arbitrary Loop Order Research Articles

Explicit Expressions for Timelike and Spacelike Observables of Quantum Chromodynamics in Analytic Perturbation Theory

We study the possibility of expressing the invariant QCD coupling function (i.e., the effective coupling constant) in an explicit analytic form in two- and three-loop approximations as well as in the case of the Pade-transformed β-function. Both the timelike and spacelike domains are investigated. Technical aspects of the Shirkov–Solovtsov analytic perturbation theory are considered. Explicit expressions for the two- and three-loop effective coupling functions in the timelike domain are obtained. In the last case, we apply a new method of expanding functions represented in an arbitrary loop order of perturbation theory in powers of the two-loop function. The comparison with numerical data in the infrared region shows that the obtained explicit expressions for the three-loop functions agree fully with the exact numerical results.

The renormalization group approach to the confining string

The renormalization group approach towards the string representation of non-abelian gauge theories translates, in terms of the string sigma model beta function equations, the renormalization group evolution of the gauge coupling constant and Zamolodchikov`s c-function. Tachyon stability, glueball mass gap, renormalization group evolution of the c-function and the area law for the Wilson loop are studied for a critical bosonic string vacuum corresponding to a non-abelian gauge theory in four-dimensional space-time. We prove that the same intrinsic geometry for the string vacuum is universal in some sense, reproducing the Yang–Mills beta function to arbitrary loop order in perturbation theory.

Recursion rules for scattering amplitudes in non-Abelian gauge theories

We present a functional derivation of recursion rules for scattering amplitudes in a non-Abelian gauge theory in a form valid to arbitrary loop order. The tree-level and one-loop recursion rules are explicitly displayed.

The equivalence theorem for the heavy-Higgs Standard Model and the gauged nonlinear σ-model

The equivalence theorem states that the leading part of the amplitude for a process with external longitudinally polarized vector bosons is given by the amplitude in which the longitudinal vector bosons are replaced by the corresponding pseudo-Goldstone bosons. The validity of this theorem within the standard model with a heavy Higgs boson and within the gauged nonlinear sigma-model (in which the Higgs boson is absent) is shown. Furthermore it is examined to what extent also internal lines other than scalar lines can be neglected. A simple power-counting method is developed which determines the leading diagrams for a given process at an arbitrary loop order. This method is also applied to effective Lagrangians with additional nonstandard interaction terms of higher dimension (chiral Lagragians).

Evaluation of loop diagrams using quantum mechanics

In using the proper time formalism, Schwinger demonstrated that one-loop processes in quantum field theory can be expressed in terms of a matrix element whose form is encountered in quantum mechanics, and which can be evaluated using the Heisenberg formalism. We demonstrate how instead this matrix element can be computed using standard results in the path-integral formalism. The technique of operator regularization allows one to extend this approach to arbitrary loop order. No loop-momentum integrals are ever encountered. This technique is illustrated by computing the two-point function in [Formula: see text] theory to one-loop order.

Weyl invariance of the Green-Schwarz heterotic sigma model

Weyl invariance at one loop is proven for the heterotic non-linear sigma model coupled to curved superspace with non-trivial fermionic and bosonic background fields. A general formula is given relating the Weyl anomaly to the sigma model β-functions at arbitrary sigma-model loop order. The Weyl anomaly is also considered in the presence of Yang-Mills backgrounds.

Superinvolutions on super-Riemann surfaces and the supergeometry of type-I superstrings

The supergeometry appropriate to the perturbation theory of type-I superstrings is described, to arbitrary loop order, by means of a superinvolution on a closed super-Riemann surface. It is known that the superfields of relevant local U(1) weights restricted to be even under the superinvolution reproduce the required type-I superstring boundary conditions. Such fields can be unambiguously integrated on the world sheet. More interestingly, the ambiguity in the supermoduli integral of the string measure is shown to be removable by vacuum shifts. The constraints imposed by the superinvolution on the super-period matrix (for both even and odd spin structures) are derived and it is also shown that the gauge-fixing superdeterminant factors into two equal components.

Diagrammatic operator formalism for string amplitudes

A diagrammatic operator method based on Fock space formulations of 3-string vertices is developed for computing the integration measure for open or closed string S-matrix to arbitrary loop order. The proposed vertex does not triangulate the moduli space, thus a single diagram incorporates a number of string field theory diagrams. However, the formalism does not naturally lead to a single cover of moduli space, and the restriction to the fundamental domain must be put in by hand, as in the Polyakov functional integral approach.

The background field method and the non-linear σ-model

We prove the renormalizability in the background field method of the two-dimensional bosonic non-linear σ-model with an arbitrary riemannian manifold as target space. Particular attention is paid to the question of non-linear renormalization of the quantum fields and its effect on subdivergences. We give an algorithm that allows one to compute to arbitrary loop order.

Absence of divergences in type II and heterotic string multi-loop amplitudes

A detailed analysis is given of the two main types of degeneration of Riemann surface of arbitrary genus by domain variational theory. Explicit estimates for first and third Abelian functions are given. These estimates are used to analyse the possible divergences of type II or heterotic superstring multiloop amplitudes for the scattering of massless particles. They are all shown to be finite at arbitrary loop order.

Supergraph analysis of the ultraviolet finiteness of gauge supersymmetry

The detailed proof of the ultraviolet finiteness of the S-matrix of gauge supersymmetry for internal symmetry index N ⩾ 2 is presented (where 4 N is the number of Fermi coordinates in superspace). The theorem is established to arbitrary loop order in the linearized harmonic gauge when the spontaneous symmetry breaking of gauge supersymmetry preserves global supersymmetry. The asymptotic properties in the deep euclidean region of the tree-approximation propagators are calculated. These enter importantly in the derivation of the theorem.

Ultraviolet Finiteness of All Quantum Loops in Gauge Supersymmetry

The arbitrary $n$-point Green's functions of spontaneously broken gauge supersymmetry are shown to be ultraviolet finite to arbitrary loop order for $Ng~2$ (where $4N$ is the number of Fermi coordinates) when the spontaneous breaking preserves global supersymmetry.