Ivo Vrkoč

On the ambiguity of field correlators represented by asymptotic perturbation expansions

Starting from the divergence pattern of perturbation expansions in quantum field theory and the (assumed) asymptotic character of the series, we address the problem of ambiguity of a function determined by the perturbation expansion. We consider functions represented by an integral of the Laplace–Borel type along a general contour in the Borel complex plane. Proving a modified form of Watsons lemma, we obtain a large class of functions having the same asymptotic perturbation expansion. Some remarks on perturbative QCD are made, using the particular case of the Adler function.

Asymptotic stability of tri-trophic food chains sharing a common resource

One of the key results of the food web theory states that the interior equilibrium of a tri-trophic food chain described by the Lotka-Volterra type dynamics is globally asymptotically stable whenever it exists. This article extends this result to food webs consisting of several food chains sharing a common resource. A Lyapunov function for such food webs is constructed and asymptotic stability of the interior equilibrium is proved. Numerical simulations show that as the number of food chains increases, the real part of the leading eigenvalue, while still negative, approaches zero. Thus the resilience of such food webs decreases with the number of food chains in the web.

A new expression for the high‐energy upper bound on the scattering amplitude

The two Froissart–Martin high‐energy upper bounds for forward and nonforward scattering are combined into one formula under the additional assumption that the scattering amplitude is polynomially bounded in energy for all scattering angles inside the Lehmann–Martin ellipse. The method used presents a modification of that of Kinoshita, Loeffel, and Martin. The analogous bound for the scattering of particles with spin is obtained as well. Using the same method, a bound for the case of complex scattering angles is also derived and ways leading to its improvement by using the solution of the Dirichlet problem are suggested.

On the ambiguity of functions represented by divergent power series

Assuming the asymptotic character of divergent perturbation series, we address the problem of ambiguity of a function determined by an asymptotic power expansion. We consider functions represented by an integral of the Laplace-Borel type, with a curvilinear integration contour. This paper is a continuation of results recently obtained by us in a previous work. Our new result contained in Lemma 3 of the present paper represents a further extension of the class of contours of integration (and, by this, of the class of functions possessing a given asymptotic expansion), allowing the curves to intersect themselves or return back, closer to the origin. Estimates on the remainders are obtained for different types of contours. Methods of reducing the ambiguity by additional inputs are discussed using the particular case of the Adler function in QCD.

The operator-product expansion away from euclidean region

Abstract The role of the operator-product expansion in QCD calculations is discussed. Approximating the two-point correlation function by several terms and assuming an upper bound on the truncation error along the euclidean ray, we consider two model situations to examine how the bound develops with increasing deflection from the euclidean ray towards the cut. We obtain explicit bounds on the truncation error and show that they worsen with the increasing deflection. This result does not support the conventional believe that the remainder is constant for all angles in the complex energy plane. Further refinements of the formalism are discussed.

OPERATOR-PRODUCT EXPANSION AND ANALYTICITY

We discuss the current use of the operator-product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum expectation value of the operator product by several terms and assuming a bound on the remainder along the Euclidean region, we observe how the bound varies with increasing deflection from the Euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane down to the Minkowski region is not justified. Making specific assumptions on the properties of the expanded function, we obtain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions considered. The results obtained are discussed in connection with calculations of the coupling constant αs from the τ decay.

Asymptotic Power Series of Field Correlators

We address the problem of ambiguity of a function determined by an asymptotic perturbation expansion. Using a modified form of the Watson lemma recently proved elsewhere, we discuss a large class of functions determined by the same asymptotic power expansion and represented by various forms of integrals of the Laplace-Borel type along a general contour in the Borel complex plane. Some remarks on possible applications in QCD are made.

Stochastic Processes for Bounded Noise

AbstractThe scalar differential inclusionx˙ ∈ f(x) + g(x) u, u ∈ [-1,1], x(0) = x0 (0.1) is considered as a model of the dynamical system x˙ = f(x) perturbed by the bounded noise g(x)u, u ∈ [-1,1], and the problem of constructing a nontrivial probability measure on the set {\cal S} of solutions to (0.1) is studied. In particular, it is shown that: (i) every Markov process whose probability measure is supported on {\cal S} is degenerate, in a sense to be specified (see Theorem 3.1); (ii) given a flow of probability measures μt on the reachable sets Rt of (0.1), satisfying a certain compatibility condition, a Markov process Xt is constructed such that its marginals are exactly μt and (0.1) is satisfied “from one side” (see Theorem 4.1); its finite-dimensional distributions are computed and the regularity of its sample paths is investigated (see Section 5.2); (iii) given a process of a type previously considered, another process Yt is constructed through its finite-dimensional distributions, and its distribution is shown to be supported exactly on {\cal S}. Finally, a model example is considered (see Section 7).

On the intrerplay between perturbative and nonperturbative QCD

We discuss the current use of the operator-product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variables and assuming a bound on the remainder along the euclidean ray, we observe how the bound develops with increasing deflection from the euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane is not justified. Explicit bounds on the remainder can be obtained under additional conditions, which are still far from the realistic physical situation. Possible generalizations of the scheme are considered.

Improvement of the Froissart-Martin bound for complex scattering angles

Extension of the Froissart–Martin bound for complex scattering angles is improved using the solution of the Dirichlet boundary value problem for doubly connected domains. The Froissart–Martin bound for physical scattering angles is used as input value on one of the two boundaries. The obtained bound is valid in an ellipse smaller than the Lehmann–Martin one. Possibilities for further improvements and applications are discussed.