Contour Integral Research Articles

Analytic structure of three-point functions from contour deformations

We explore the analytic structure of three-point functions using contour deformations. This method allows continuing calculations analytically from the spacelike to the timelike regime. We first elucidate the case of two-point functions with explicit explanations how to deform the integration contour and the cuts in the integrand to obtain the known cut structure of the integral. This is then applied to one-loop three-point integrals. We explicate individual conditions of the corresponding Landau analysis in terms of contour deformations. In particular, the emergence and position of singular points in the complex integration plane are relevant to determine the physical thresholds. As an exploratory demonstration of this method's numerical implementation we apply it to a coupled system of functional equations for the propagator and the three-point vertex of $\phi^3$ theory. We demonstrate that under generic circumstances the three-point vertex function displays cuts which can be determined from modified Landau conditions.

Focused polarization ellipse field singularities: interaction of spin-orbital angular momentum and the formation of optical Möbius strips

We study here the intensity distribution and formation of optical polarization Möbius strips by tightly focusing of C-point singularity beams. These beams are characterized by a central circular polarization point (C-point) surrounded by a spatially varying elliptic polarization. Under tight focusing conditions, the different polarization components of the beam interfere and exhibit clear difference between left-handed and right handed input beams. The transverse polarization distribution at the focal plane is similar to the input distribution for left-handed lemon beam, but exhibits 180° rotation for right handed lemon beam. Moreover, the longitudinal polarization component exhibits spiral phase distribution, owing to spin-orbit angular momentum conversion at the focal plane, with opposite winding directions for the left-handed and right-handed input beams. We show that the shape of the resulting Möbius strip is determined by the helicity of the C-point and by the polarization singularity index, which is the contour integral of polarization ellipse angle around the singularity. It is found that inverting the helicity leads to 180° rotation in the focal plane intensity distribution, accompanied by handedness inversion for the polarization ellipses. The number of separatrices in the input polarization distribution is equivalent to the number of twist points of the Möbius strip in the focal plane, as well as to the number of intensity zeros in the z-component of the focused field. These phenomena are observed for beams with a bright C-point, but also for dark C-point, in which the electric field is zero at the center of the beam.

Damage Creep Model of Viscoelastic Rock Based on the Distributed Order Calculus

In this paper, the distributed order calculus was used to establish a creep damage theoretical model to accurately describe the creep properties of viscoelastic materials. Firstly, the definition and basic properties in math of the distributed order calculus were given. On this basis, the mechanical elements of the distributed order damper were built to describe the viscoelastic properties. Then, the distributed order damper was introduced into the three-parameter solid model to establish the distributed order three-parameter solid model. The inverse Laplace transform derived the operator’s contour integrals and the path integrals along Hankel’s path. The integral properties were analysed. Next, the creep properties and relaxation characteristics of the distributed order three-parameter solid model were studied in detail. Finally, taking the rock materials as an example, the distributed order damage damper model was established. Its operator integrals were calculated, and the properties were discussed. Meanwhile, taking the integer-order Nishihara model as the standard, the distributed order damage creep combined model of the rock mass was constructed. The calculation examples were given to study the damage creep properties of the rock mass.

On the equivalence between SRS and PCO formulations of superstring perturbation theory

We establish the equivalence between two formulations of superstring perturbation theory, one based on integration over the supermoduli space of super Riemann surfaces (SRS), the other based on integration over the bosonic moduli space with insertions of picture changing operators (PCO) on the worldsheet and the vertical integration prescription, by showing how the latter arises from a specific construction of the supermoduli integration contour.

Borel–Laplace sum rules with τ decay data, using OPE with improved anomalous dimensions

We perform numerical analysis of double-pinched Borel–Laplace QCD sum rules for the strangeless semihadronic τ-decay data. The D = 0 contribution to the theoretical contour integral in the sum rules is evaluated by the (truncated) Fixed Order perturbation theory method (FO) and by the Principal Value (PV) of the Borel integration. We use for the full Adler function the operator product expansion (OPE) with the terms ∼〈O D 〉 of dimension D = 2n where 2 ≤ n ≤ 5 for the (V+A)-channel, and 2 ≤ n ≤ 7 for the V-channel data. In our previous works Ayala, Cvetič and Teca (2021, Eur. Phys. J. C 81 930), Ayala, Cvetič and Teca (2022, Eur. Phys. J. C 82 362), only the (V+A)-channel data was analysed. In this work, the analysis of a new set of V-channel data is performed as well. Further, a renormalon-motivated construction of the D = 0 part of the Adler function is improved in the u = 3 infrared renormalon sector, by involving the recently known information on the two principal noninteger values of the effective leading-order anomalous dimensions. Additionally, the OPE of the Adler function has now the D = 6 contribution with the principal anomalous dimension (), and terms of higher dimension (with zero anomalous dimension). Cross-checks of the obtained extracted values of α s and of the condensates were performed by reproduction of the (central) experimental values of several double-pinched momenta a (2,n). The averaged final extracted values of the () coupling are: , corresponding to .

Conserved currents for a Kerr black hole and orthogonality of quasinormal modes

Using a symmetry of the Kerr spacetime, the authors determine an original bilinear form for perturbations of the Weyl scalars. They show that upon choosing appropriate integration contours, this form can also be used on quasi-normal mode data. This manuscript provides a formalism for studying ringdown beyond the leading order, with potential application to numerical relativity and gravitational wave observation.

On the 4d superconformal index near roots of unity: bulk and localized contributions

We study the expansion near roots of unity of the superconformal index of 4d SU(N) \U0001d4a9 = 4 SYM. In such an expansion, middle-dimensional walls of non-analyticity are shown to emerge in the complex analytic extension of the integrand. These walls intersect the integration contour at infinitesimal vicinities and come from both, the vector and chiral multiplet contributions, and combinations thereof. We will call these intersections vector and chiral bits, and the complementary region bulk, and show that, in the corresponding limit, the integrals along the infinitesimal bits include, among other contributions, factorized products of either Chern-Simons and 3d topologically twisted partition functions.In particular, we find that the leading asymptotic contribution to the index, which comes from collecting all contributions coming from vector bits, reduces to an average over a set of N copies of three-dimensional SU(N) Chern-Simons partition functions in Lens spaces L(m, 1) with m > 1, in the presence of background {mathbb{Z}}_m^{N-1} flat connections. The average is taken over the background connections, which are the positions of individual vector bits along the contour. We also find there are other subleading contributions, a finite number of them at finite N, which include averages over products of Chern-Simons and/or topologically A-twisted Chern-Simons-matter partition functions in three-dimensional manifolds. This shows how in certain limits the index of 4d SU(N) \U0001d4a9 = 4 SYM organizes, via an unambiguously defined coarse graining procedure, into averages over a finite number of lower dimensional theories.

Unitarity cuts of the worldsheet

We compute the imaginary parts of genus-one string scattering amplitudes. Following Witten’s i\varepsiloniε prescription for the integration contour on the moduli space of worldsheets, we give a general algorithm for computing unitarity cuts of the annulus, Möbius strip, and torus topologies exactly in \alpha'α′. With the help of tropical analysis, we show how the intricate pattern of thresholds (normal and anomalous) opening up arises from the worldsheet computation. The result is a manifestly-convergent representation of the imaginary parts of amplitudes, which has the analytic form expected from Cutkosky rules in field theory, but bypasses the need for performing laborious sums over the intermediate states. We use this representation to study various physical aspects of string amplitudes, including their behavior in the (s,t)(s,t) plane, exponential suppression, decay widths of massive strings, total cross section, and low-energy expansions. We find that planar annulus amplitudes exhibit a version of low-spin dominance: at any finite energy, only a finite number of low partial-wave spins give an appreciable contribution to the imaginary part.

Isovector meson masses from QCD sum rules

We present a calculation of the masses of the isovector mesons (vector, scalar and pseudoscalar including the established recurrences) using a new method of finite energy QCD sum rules. The method is based on the idea of choosing a suitable integration kernel which minimizes the occurring integral over the cut in the complex energy (squared) plane. We obtain remarkably stable results in a wide range [Formula: see text], where [Formula: see text] is the radius of the integration contour. The sum rule predictions agree with the experimental values within the expected accuracy showing that QCD describes single resonances.

On the Use of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions for the Calculation of Infinite Sums and the Analysis of Zeroes of Analytical Functions

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. On many occasions, this enables to facilitate the obtaining of known results thus having important methodological meaning. Additionally, some new results, to the best of our knowledge, are also obtained in this way. For example, we established new properties of the sum of inverse zeroes of a digamma function, new formulae for the sums ∑kiρi2 for zeroes ρi of incomplete gamma and Riemann zeta functions having the order ki (These results can be straightforwardly generalized for the sums ∑kiρin with integer n > 2, and so on.)

Oscillatory states of quantum Kapitza pendulum

We study quantum mechanics problem described by the Schrödinger equation with Kapitza pendulum potential, that is the asymmetric double-well potential on the circle. For the oscillatory states spatially localize around the two stable saddle positions of the potential, we obtain the perturbative eigenvalues and corresponding piecewise wavefunctions. The spectrum is computed by extending the angle coordinate to the complex plane so that the quantization condition is formulated as contour integral along a path extending in the imaginary direction. Quantum tunneling between the wells is computed.

Certain q-Analogue of Fractional Integrals and Derivatives Involving Basic Analogue of the Several Variable Aleph-Function

Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional q-integral and q-differential formulae for the q-extended several variable Aleph-function. Using the q-analogue of the Leibniz rule for the fractional q-derivative of a product of two basic functions, we also provide a formula for the q-extended several variable Aleph-function, which is expressed in terms of an infinite series of the q-extended several variable Aleph-function. Since the three main formulas presented in this article are so general, they can be reduced to yield a number of identities involving q-extended simpler special functions. In this connection, we choose only one main formula to offer some of its particular instances involving diverse q-extended special functions, for example, the q-extended I-function, the q-extended H-function, and the q-extended Meijer’s G-function. The results presented here are hoped and believed to find some applications, in particular, in quantum mechanics.

The heat equation on time scales

We present the use of a Fourier transform on time scales to solve a dynamic heat IVP. This is done by inverting a certain exponential function via contour integral. We include some specific examples and directions for further study.

On the properties of some inversion methods of the Laplace transform

The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations, in which the unknowns are either the coefficients of the series expansion in terms of special functions, or the approximate values of the desired original at a number of points. Described method of inversion using special quadrature formulas of the highest degree of accuracy and the characteristics of the accuracy and stability of this method are indicated. Quadrature inversion formulas are constructed, which are adapted for the inversion of long-term and slow processes of linear viscoelasticity. A method of deformation of the integration contour in the Riemann-Mellin inversion formula is proposed, which leads the problem to the calculation of certain integrals and allows obtaining error estimates. A method is described for determining the possible breakpoints of the original and calculating the magnitude of the jump at these points.

WHY A NEW FOREIGN POLICY CONCEPT OF THE RUSSIAN FEDERATION IS IMPORTANT FOR PACIFIC ISLANDS?

The Pacific Islands are a diverse and unique region consisting of thousands of islands scattered across the vast ocean. Although they differ in terms of geography, culture and economic weight, they all have a common question: what is their role in the geopolitics of the 21st century and in building a new world order. For a long time, the Pacific Islands were perceived by long-established major actors in the region only in terms of geostrategic benefits. However, times are changing, and there is a growing demand for a more sovereign foreign policy among the Pacific Island Countries (PICs). It's especially important to consider their specific needs and support the pursuit of economic, industrial and technological development. The potential PICs' inclusion in new international platforms, for example, in BRICS+, is seen as relevant here. Many of the countries in the association are already known to them. Common points of foreign policy and development vision can also be traced in the new Concept of the Foreign Policy of the Russian Federation. The document not only defines the vector of the country's future interaction with the Asia-Pacific, but also emphasizes the importance of independent foreign policy and fair access to economic development for all countries. The article examines tectonic shifts in the international arena, Russia's role in the Asia-Pacific, the specifics of the Pacific Islands' vision, and their importance for a multipolar world. Special attention is paid to assessing their possible involvement in BRICS+ affairs. Among other things, the article considers the advantages of cooperation (building of a broad integration contour; ensuring free access of countries to the benefits of the global economy; and reducing vulnerability to unfriendly external influences).

A short note on a extended finite secant series

<abstract><p>In this paper, a summation formula for a general family of a finite secant sum has been extended by making use of a particularly convenient integration contour method. The main theorem derived from this approach is the finite sum involving the Hurwitz-Lerch zeta function. This theorem for particular values is used to derive the finite product of the fifth roots of the quotient product of the gamma function along with finite sums and functional equations involving trigonometric functions.</p></abstract>

Contour prescriptions in string-inspired nonlocal field theories

In quantum field theory, a consistent prescription to define and deform integration contours in the complex energy plane is needed to evaluate loop integrals and compute scattering amplitudes. In some nonlocal field theories, including string field theory, interaction vertices contain transcendental functions of momenta that can diverge along certain complex directions, thus making it impossible to use standard techniques, such as Wick rotation, to perform loop integrals. The aim of this paper is to investigate the viability of several contour prescriptions in the presence of nonlocal vertices. We consider three ``different'' prescriptions, and establish their (in)equivalence in local and nonlocal theories. In particular, we prove that all these prescriptions turn out to be equivalent in standard local theories, while this is not the case for nonlocal theories where amplitudes must be defined first in Euclidean space, and then analytically continued to Minkowski. We work at one-loop level and focus on the bubble diagram. In addition to proving general results for a large class of nonlocal theories, we show explicit calculations in a string inspired nonlocal scalar model.

On the Issue of the Strategy for the Formation of the Trans-regional Integration Contour of the EAEU

The instability and uncertainty of international relations accentuate the role of new and inclusive platforms of interaction and cooperation. Eurasian Economic Union (EEU) alongside its regional and transregional partners represent an example of such a platform based on new principles and values, an element of international relation configuration.Aim. Assess the prospects for a zone (cluster) within the planned and potential transregional EEU’s expansion beyond the CIS borders.Task. To evaluate the dynamics of the consolidation concerning, first, current free trade EEU agreements (FTAs or the EEU FTAs) with Serbia, Iran, Vietnam and Singapore, second, potential agreements with listed in the EEU official strategy, third, other developing states with relevant economic and/or demographic potential.Methods. The authors use additional consolidation (share of intra-bloc trade) as a metric. It is calculated as the difference between the share of EEU intra-trade and within the existing or potential EEU FTA partnerships.Results. Based on the assessment of additional consolidation, the research revealed that the existing EEU FTAs with Iran and Vietnam are effective, while trade integration with Serbia, Singapore and China are ineffective. The presence of additional consolidation was revealed for Turkey, Egypt, and Bangladesh.Conclusions. The implementation of the transregional integration track is an important development step for the Eurasian Economic Union. However, trade partnerships have different trade efficiency.

Contour integrals in nonlinear fracture mechanics for mixed forms of deformation

Modern knowledge in the field of fracture mechanics is the first key knowledge in solving the problems of safety and strength of the objects with crack-like damages of the various origins. Nonlinear fracture mechanics in the analysis of the stress-strain state in the crack tip region is based on the one- and two-parameter approaches. The classical one-parameter studies involve the study of singular quantities, including a contour J -integral, independent of the path of integration, a stress intensity factor (SIF), etc. The values of the SIF and J -integral are interdependent. Combined methods are very popular, based on the union of numerical, experimental and analytical calculations, which make it possible to obtain the most clear description of the parameters of fracture mechanics. Calculation of the J -integral in finite element models, by the method of reactions or stresses, is very effective, but this requires sufficiently accurate analytical representations of the contour J -integral. There are certain limiting conditions when obtaining such formulas. In the numerous scientific works, it has been proved that J is an integral in the most cases does not depend on the path of integration, but is highly dependent on the methods of describing the parameters of the stress-strain state, as well as their derivatives, on the dimension of the problem and on the degree of distance of the contour of integration from the crack tip. In this paper, we review and present the author's conclusions of the contour integrals in nonlinear fracture mechanics for three cases: the classical Hutchinson - Rosengren - Rice solution (HRR), contour integrals in the gradient theory of plasticity, and the calculation of the J -integral for a general case when the components of stresses and displacements are the functions of three Cartesian coordinates. A generalized J- integral is derived and used to characterize a nonlinear amplitude fac.

Verified eigenvalue and eigenvector computations using complex moments and the Rayleigh–Ritz procedure for generalized Hermitian eigenvalue problems

We propose a verified computation method for eigenvalues in a region and the corresponding eigenvectors of generalized Hermitian eigenvalue problems. The proposed method uses complex moments to extract the eigencomponents of interest from a random matrix and uses the Rayleigh–Ritz procedure to project a given eigenvalue problem into a reduced eigenvalue problem. The complex moment is given by contour integral and approximated using numerical quadrature. We split the error in the complex moment into the truncation error of the quadrature and rounding errors and evaluate each. This idea for error evaluation inherits our previous Hankel matrix approach, whereas the proposed method enables verification of eigenvectors and requires half the number of quadrature points for the previous approach to reduce the truncation error to the same order. Moreover, the Rayleigh–Ritz procedure approach forms a transformation matrix that enables verification of the eigenvectors. Numerical experiments show that the proposed method is faster than previous methods while maintaining verification performance and works even for nearly singular matrix pencils and in the presence of multiple and nearly multiple eigenvalues.