Double logarithmic asymptotic behavior of vertex functions in quantum chromodynamics. II. Eighth order of perturbation theory

1980 ◽  
Vol 45 (2) ◽  
pp. 957-962 ◽  
Author(s):  
V. V. Belokurov ◽  
N. I. Usyukina
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Quantum Chromodynamics ◽  
Eighth Order ◽  
Logarithmic Asymptotic Behavior

scholarly journals Spectral deformation in a problem of singular perturbation theory

2019 ◽  
Vol 484 (4) ◽  
pp. 397-400
Author(s):  
S. A. Stepin ◽  
V. V. Fufaev
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Integral Method ◽  
Deformation Parameter ◽  
Liouville Problem ◽  
Singular Perturbation Theory ◽  
Different Types ◽  
Spectral Deformation ◽  
Sturm Liouville ◽  
Spectral Complex

Quasi-classical asymptotic behavior of the spectrum of a non-self-adjoint Sturm–Liouville problem is studied in the case of a one-parameter family of potentials being third-degree polynomials. For this problem, the phase-integral method is used to derive quantization conditions characterizing the asymptotic distribution of the eigenvalues and their concentration near edges of the limit spectral complex. Topologically different types of limit configurations are described, and critical values of the deformation parameter corresponding to type changes are specified.

scholarly journals Asymptotics of the spectrum of even-order differential operators with discontinuos weight functions

2020 ◽  
Vol 22 (1) ◽  
pp. 48-70
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Boundary Value ◽  
Weight Functions ◽  
Order Differential Operator ◽  
Eighth Order ◽  
Entire Family ◽  
Indicator Diagram

The boundary-value problem for an eighth-order differential operator whose potential is a piecewise continuous function on the segment of the operator definition is studied. The weight function is piecewise constant. At the discontinuity points of the operator coefficients, the conditions of "conjugation" must be satislied which follow from physical considerations. The boundary conditions of the studied boundary value problem are separated and depend on several parameters. Thus, we simultaneously study the spectral properties of entire family of differential operators with discontinuous coefficients. The asymptotic behavior of the solutions of differential equations defining the operator is obtained for large values of the spectral parameter. Using these asymptotic expansions, the conditions of "conjugation" are investigated; as a result, the boundary conditions are studied. The equation on eigenvalues of the investigated boundary value problem is obtained. It is shown that the eigenvalues are the roots of some entire function. The indicator diagram of the eigenvalue equation is investigated. The asymptotic behavior of the eigenvalues in various sectors of the indicator diagram is found.

scholarly journals The Leutwyler–Smilga relation on the lattice

2019 ◽  
Vol 35 (01) ◽  
pp. 1950346 ◽  
Author(s):  
Gernot Münster ◽  
Raimar Wulkenhaar
Keyword(s):  
Perturbation Theory ◽  
Quantum Chromodynamics ◽  
Lattice Qcd ◽  
Chiral Perturbation Theory ◽  
Effective Action ◽  
Lattice Spacing ◽  
Chiral Perturbation ◽  
Quark Masses

According to the Leutwyler–Smilga relation, in Quantum Chromodynamics (QCD), the topological susceptibility vanishes linearly with the quark masses. Calculations of the topological susceptibility in the context of lattice QCD, extrapolated to zero quark masses, show a remnant nonzero value as a lattice artefact. Employing the Atiyah–Singer theorem in the framework of Symanzik’s effective action and chiral perturbation theory, we show the validity of the Leutwyler–Smilga relation in lattice QCD with lattice artefacts of order a2 in the lattice spacing a.

VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

1995 ◽  
Vol 05 (05) ◽  
pp. 565-585 ◽  
Author(s):  
MIGUEL LOBO ◽  
EUGENIA PÉREZ
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Small Parameter ◽  
Local Problem ◽  
Micro Structure ◽  
Spectral Perturbation Theory ◽  
Concentrated Masses ◽  
The Masses ◽  
Spectral Perturbation ◽  
Small Eigenvalues

We consider the asymptotic behavior of the vibrations of a membrane occupying a domain Ω ⊂ ℝ2. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε−m) with m>0. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. Depending on the value of the parameter m(m>2, m=2 or m<2) we describe the asymptotic behavior of the eigenvalues. Small eigenvalues, of order O(εm−2) for m>2, are approached via those of a local problem obtained from the micro-structure of the problem, while the eigenvalues of order O(1) are approached through those of a homogenized problem, which depend on the relation between ε and η. Techniques of boundary homogenization and spectral perturbation theory are used to study this problem.

Operator expansion in quantum chromodynamics beyond perturbation theory

1980 ◽  
Vol 174 (2-3) ◽  
pp. 378-396 ◽  
Author(s):  
V.A. Novikov ◽  
M.A. Shifman ◽  
A.I. Vainshtein ◽  
V.I. Zakharov
Keyword(s):  
Perturbation Theory ◽  
Quantum Chromodynamics ◽  
Operator Expansion

scholarly journals Hamiltonian approach to QCD in Coulomb gauge: Perturbative treatment of the quark sector

2015 ◽  
Vol 30 (17) ◽  
pp. 1550100 ◽  
Author(s):  
Davide R. Campagnari ◽  
Hugo Reinhardt
Keyword(s):  
Perturbation Theory ◽  
Quantum Chromodynamics ◽  
Quark Propagator ◽  
Functional Integral ◽  
Coulomb Gauge ◽  
Hamiltonian Approach ◽  
Equal Time ◽  
Quark Sector ◽  
Schrödinger Perturbation ◽  
Perturbative Treatment

We study the static gluon and quark propagator of the Hamiltonian approach to quantum chromodynamics in Coulomb gauge in one-loop Rayleigh–Schrödinger perturbation theory. We show that the results agree with the equal-time limit of the four-dimensional propagators evaluated in the functional integral (Lagrangian) approach.

The principle of minimum sensitivity and moments of nonsinglet structure functions

1983 ◽  
Vol 61 (1) ◽  
pp. 99-101 ◽  
Author(s):  
Gerry McKeon
Keyword(s):  
Perturbation Theory ◽  
Quantum Chromodynamics ◽  
Finite Order ◽  
Parton Model ◽  
Structure Functions ◽  
Renormalization Procedure ◽  
Dependent Part ◽  
Model Predictions ◽  
Minimum Sensitivity

The corrections implied by quantum chromodynamics to parton model predictions are not unique to finite order in perturbation theory on account of the possibility of choosing different renormalization schemes. Stevenson has provided a criterion for selecting the "best" renormalization procedure; the so-called "principle of minimum sensitivity" (PMS). This criterion is applied here to the Q2-dependent part of the moments of the nonsinglet structure functions in lepton–hadron scattering.

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