Divergent Parts Research Articles

NumExp: Numerical epsilon expansion of hypergeometric functions

It is demonstrated that the well-regularized hypergeometric functions can be evaluated directly and numerically. The package NumExp is presented for expanding hypergeometric functions and/or other transcendental functions in a small regularization parameter. The hypergeometric function is expressed as a Laurent series in the regularization parameter and the coefficients are evaluated numerically by using the multi-precision finite difference method. This elaborate expansion method works for a wide variety of hypergeometric functions, which are needed in the context of dimensional regularization for loop integrals. The divergent and finite parts can be extracted from the final result easily and simultaneously. In addition, there is almost no restriction on the parameters of hypergeometric functions. Program summaryProgram title:NumExpCatalogue identifier: AEPE_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPE_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1682No. of bytes in distributed program, including test data, etc.: 18431Distribution format: tar.gzProgramming language: Mathematica and/or Python.Computer: Any computer where Mathematica or Python is running.Operating system: Linux, Windows.Classification: 4.4, 5, 11.1.External routines: mpmath library (for Python)Nature of problem:Expansion of hypergeometric functions and/or other transcendental functions in a small parameter ε. These expansions are needed in the context of dimensional regularization for loop integrals.Solution method:The hypergeometric function is expressed as a Laurent series in the regularization parameter ε, where the coefficients are evaluated numerically by the multi-precision finite difference method.Restrictions:The calculation may be inefficient if the arguments of hypergeometric functions are close to the convergent boundaries.Running time:Generally it is less than a few seconds, depending on the complexity of the problem.

(Un)determined finite regularization-dependent quantum corrections: The Higgs boson decay into two photons and the two-photon scattering examples

We investigate the appearance of arbitrary, regularization-dependent parameters introduced by divergent integrals in two a priori finite but superficially divergent amplitudes: the Higgs decay into two photons and the two-photon scattering. We use a general parametrization of ultraviolet divergences which makes explicit such ambiguities. Thus we separate in a consistent way using implicit regularization the divergent, finite, and regularization-dependent parts of the amplitudes which in turn are written as surface terms. We find that, although finite, these amplitudes are ambiguous before the imposition of physical conditions, namely, momentum routing invariance in the loops of Feynman diagrams. In the examples we study, momentum routing invariance turns out to be equivalent to gauge invariance. We also discuss the results obtained by different regularizations and show how they can be reproduced within our framework allowing for a clear view on the origin of regularization ambiguities.

Steady subsonic potential flows through infinite multi-dimensional largely-open nozzles

We proved existence, regularity and uniqueness of steady subsonic potential flows in n-dimensional (n ≥ 3) infinite nozzles with largely-open convergent and divergent parts when the total mass flux is less than a certain value. Such nozzles consist of two cones with arbitrary open angles and an arbitrary smooth bounded tubular part. The existence of a weak solution is proved by applying the direct method of calculus of variation to a carefully chosen functional defined on a Hilbert space based upon Hardy inequality. Holder gradient regularity of weak solution is shown by using Moser iteration to quasilinear elliptic equations in divergence form. Also, the obtained solution is unique in the class of functions with finite kinetic energy by modulo a constant.

Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Holder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.

Kondo lattice with heavy fermions: peculiarities of spin kinetics

A model of spin relaxation of Kondo lattices is proposed to explain the angular dependence of the electron spin resonance (ESR) parameters in the heavy fermion compounds Y bIr2Si2 and Y bRh2Si2. A perturbational scaling approach reveals a collective spin motion of Yb ions with conduction electrons in the bottleneck regime. A common energy scale due to the Kondo effect regulates the temperature dependence of different kinetic coefficients to result in a mutual cancelation of all divergent parts in a collective spin mode. The angular dependence of the ESR intensity, linewidth and resonant frequency is shown to be in good agreement with experimental data on Y bIr2Si2 and Y bRh2Si2. In particular, the unexpectedly weak dependence of the ESR intensity on the orientation of the microwave magnetic field agrees with the properties of the discussed model.

Problems concerned with local erosion-corrosion of welded connections in the pipelines of a power unit at a nuclear power station

Results from a study aimed at analyzing the damageability and determining the mechanism due to which thinning occurs in the metal of welded connections in the feed water supply control assemblies used in power units of nuclear power stations equipped with RBMK-1000 reactors are presented together with the outcomes from calculation studies aimed at analyzing the effect of hydrodynamic characteristics on the kinetics characterizing the thinning of a welded joint and adjacent elements of a pipeline. Data from a calculation of the erosion-corrosion rate of welded connections in the divergent parts of assemblies controlling the supply of feed water in power units of nuclear power stations equipped with RBMK-1000 reactors are given.

A Systematization for One-Loop 4D Feynman Integrals-Different Species of Massive Fields

A systematization for the manipulations and calculations involving divergent (or not) Feynman integrals, typical of the one loop perturbative solutions of Quantum Field Theory, is proposed. A previous work on the same issue is generalized to treat theories and models having different species of massive fields. An improvement on the strategy is adopted so that no regularization needs to be used. The final results produced, however, can be converted into the ones of reasonable regularizations, especially those belonging to the dimensional regularization (in situations where the method applies). Through an adequate interpretation of the Feynman rules and a convenient representation for involved propagators, the finite and divergent parts are separated before the introduction of the integration in the loop momentum. Only the finite integrals obtained are in fact integrated. The divergent content of the amplitudes are written as a combination of standard mathematical object which are never really integrated. Only very general scale properties of such objects are used. The finite parts, on the other hand, are written in terms of basic functions conveniently introduced. The scale properties of such functions relate them to a well defined way to the basic divergent objects providing simple and transparent connection between both parts in the assintotic regime. All the arbitrariness involved in this type of calculations are preserved in the intermediary steps allowing the identification of universal properties for the divergent integrals, which are required for the maintenance of fundamental symmetries like translational invariance and scale independence in the perturbative amplitudes. Once these consistency relations are imposed no other symmetry is violated in perturbative calculations neither ambiguous terms survive at any theory or model formulated at any space-time dimension including nonrenormalizable cases. Representative examples of perturbative amplitudes involving different species of massive fermions are considered as examples. The referred amplitudes are calculated in detail within the context of the presented strategy (and systematization) and their relations among other Green functions are explicitly verified. At the end a generalization for the finite functions is presented.

Spin relaxation in Kondo lattices

A model of spin relaxation in Kondo lattices is proposed to explain the presence of an electron spin resonance (ESR) signal in the heavy fermion compounds Yb2Rh2Si2 and Yb2Ir2Si2. Coupled equations for dynamical susceptibilities of Kondo ions and conduction electrons are derived by means of the functional derivative method. The perturbational scaling approach reveals the collective spin motion of Yb-ions with conduction electrons in the bottleneck regime. A common energy scale due to the Kondo effect regulates the temperature dependence of the different kinetic coefficients and results in a mutual cancelation of all divergent parts in a collective spin mode. The angular dependence of the ESR linewidth is shown to be in a qualitative agreement with experimental data on Yb2Rh2Si2 and Yb2Ir2Si2. Linewidth contributions other than the Kondo interaction are also discussed.

On Projective Hoops: Loops in Hyperspace

We (re)derive the propagators and Feynman rules for the massless scalar and vector multiplets in $N=2$ projective superspace (``projective hyperspace''). With these, we are able to calculate both the divergent and finite parts of 2, 3, 4-point functions at 1-loop for $N=2$ super-Yang-Mills theory), explicitly in projective hyperspace itself. We find that effectively only the coupling constant needs to be renormalized, unlike in the $N=1$ case where an independent wavefunction renormalization is also required. This feature is similar to that of the background field gauge, even though we are using ordinary Fermi-Feynman gauge. The computation of 1-hoop beta-function is then straightforward and matches with the known result. We also show that it receives no 2-hoops contributions. All these calculations provide an alternative proof of the finiteness of $N=4$ super-Yang-Mills.

Erratum: One-loop corrections to the string tension of the vortex in the Abelian Higgs model [Phys. Rev. D78, 085008 (2008)

We present an exact numerical computation of the one-loop correction of the string tension for the Nielsen-Olesen vortex in the Abelian Higgs model. The computations proceeds via the computation of the Euclidean Green's function for the gauge, Higgs and Faddeev-Popov fields using mode functions, and taking the appropriate trace. Renormalization is an essential part of this computation, it is done by removing leading order contributions from the numerical results so as to make these finite, and to add the divergent parts back, after suitable regularization and renormalization. We encounter and solve some problems which are specific to gauge theories and topological solutions. The corrections to the energy are found to be sizeable, but still smaller than the classical energy as long as g**2 is smaller than unity.

MUSTANG-MR Structural Sieving Server: Applications in Protein Structural Analysis and Crystallography

BackgroundA central tenet of structural biology is that related proteins of common function share structural similarity. This has key practical consequences for the derivation and analysis of protein structures, and is exploited by the process of “molecular sieving” whereby a common core is progressively distilled from a comparison of two or more protein structures. This paper reports a novel web server for “sieving” of protein structures, based on the multiple structural alignment program MUSTANG.Methodology/Principal Findings“Sieved” models are generated from MUSTANG-generated multiple alignment and superpositions by iteratively filtering out noisy residue-residue correspondences, until the resultant correspondences in the models are optimally “superposable” under a threshold of RMSD. This residue-level sieving is also accompanied by iterative elimination of the poorly fitting structures from the input ensemble. Therefore, by varying the thresholds of RMSD and the cardinality of the ensemble, multiple sieved models are generated for a given multiple alignment and superposition from MUSTANG. To aid the identification of structurally conserved regions of functional importance in an ensemble of protein structures, Lesk-Hubbard graphs are generated, plotting the number of residue correspondences in a superposition as a function of its corresponding RMSD. The conserved “core” (or typically active site) shows a linear trend, which becomes exponential as divergent parts of the structure are included into the superposition.ConclusionsThe application addresses two fundamental problems in structural biology: First, the identification of common substructures among structurally related proteins—an important problem in characterization and prediction of function; second, generation of sieved models with demonstrated uses in protein crystallographic structure determination using the technique of Molecular Replacement.

Equilibrium charge density on a thin curved wire

We address the electrostatic problem of a thin, curved, and cylindrical conductor (a conducting filament) and show that the corresponding linear charge density slowly tends to uniformity as the inverse of the logarithm of a characteristic parameter, which is the ratio of the diameter to the smaller of the length and minimum radius of curvature of the filament. An alternative derivation of this result based on energy minimization is given. These results follow from a general asymptotic analysis of the electric field components and potential near a charged filament in the limit of vanishing diameter. It is found that the divergent parts of the radial and azimuthal electric field components are determined by the local charge density, while the axial component is determined by the local dipole density. For a straight filament our results reduce to those known for conducting needles. For curved filaments, the configuration of charges and fields is no longer azimuthally symmetric, and there is an additional length scale arising from the finite radius of curvature of the filament. The basic uniformity result survives the added complications, which include an azimuthal variation in the surface charge density of the filament. As with the variations in linear charge density along the filament, the azimuthal variations vanish with the characteristic parameter, only more rapidly. These findings allow us to derive an asymptotic formula for the capacitance of a curved filament that generalizes a result first obtained by Maxwell. The examples of a straight filament with uniform and linearly varying charge densities and a circular filament with a uniform charge distribution are treated analytically and found to agree with the general analysis. Numerical calculations illustrating the slow convergence of linear charge distribution to uniformity for an elliptical filament are presented. An interactive computer program implementing and animating the numerical calculations is available.

Two-loop QCD helicity amplitudes for ([formula omitted])-jet production in deep inelastic scattering

We derive the two-loop QCD helicity amplitudes for the processes ℓq→ℓqg (ℓq¯→ℓq¯g) and ℓg→ℓqq¯, which are the partonic reactions yielding (2+1)-jet final states in deep inelastic lepton nucleon scattering. The amplitudes are obtained by analytic continuation of the known helicity amplitudes for e+e−→qq¯g. We separate the infrared divergent and finite parts of the amplitudes using Catani's infrared factorization formula. The analytic results for the finite parts of the amplitudes are expressed in terms of one- and two-dimensional harmonic polylogarithms. To evaluate these functions numerically, we list in detail the non-trivial (and kinematic region dependent) variable transformations one needs to perform.

Recent developments in the Casimir effect

In this paper I review various developments in the past year concerning quantum vacuum energy, the Casimir effect. In particular, there has been continuing controversy surrounding the temperature correction to the Lifshitz formula for the Casimir force between real materials, be they metals or semiconductors. Consensus has emerged as to how Casimir energy accelerates in a weak gravitational field; quantum vacuum energy, including the divergent parts which renormalize the masses of the Casimir plates, accelerates indeed according to the equivalence principle. Significant development has been forthcoming in applying the multiple scattering formalism to describe the interaction between nontrivial objects. In weak coupling, closed-form expressions for the Casimir force between the bodies, which for example reveal significant discrepancies from the naive proximity force approximation, can be achieved in many cases.

Film Cooling of Accelerated Flow in a Subscale Combustion Chamber

Experimental investigations have been carried out to examine film cooling effectiveness of an accelerated hot gas in a subscale rocket combustion chamber. In support of future first-stage high-performance rocket combustion chambers, a Vulcain2-like test case has been examined with combustion pressure levels up to 12 MPa. The effectiveness of an almost tagentially injected film of hydrogen with an initial temperature of approximately 280 K has been determined. Axial distributions of temperature were measured inside the copper liner as well as on the chamber surface in the convergent and divergent parts of the nozzle segment. An existing film cooling model has been modified for application in a combined convective and filmcooled combustion chamber with an accelerated hot gas. The new model predicts film cooling effectiveness at different combustion-chamber pressures and film blowing rates at sub-, trans-, and supersonic conditions.

One-loop corrections to the string tension of the vortex in the Abelian Higgs model

We present an exact numerical computation of the one-loop correction of the string tension for the Nielsen-Olesen vortex in the Abelian Higgs model. The computation proceeds via the computation of the Euclidean Green's function for the gauge, Higgs, and Faddeev-Popov fields using mode functions, and taking the appropriate trace. Renormalization is an essential part of this computation. It is done by removing leading order contributions from the numerical results so as to make these finite, and to add the divergent parts back, after suitable regularization and renormalization. We encounter and solve some problems which are specific to gauge theories and topological solutions. The corrections to the energy are found to be sizable, but still smaller than the classical energy as long as g{sup 2} is smaller than unity.

Computational Fluid Dynamics Study for Flow of Natural Gas through High-pressure Supersonic Nozzles: Part 2. Nozzle Geometry and Vorticity

The computational fluid dynamics technique is used to study the behavior of high-pressure natural gas when it flows through nozzles with supersonic velocities. Effect of nozzle geometry is discussed by inserting a constant area channel between the convergent and divergent parts of the system. Various conduit lengths are analyzed to show how the minimum temperature could be influenced by the geometry of the nozzle. The results also show that changing channel length can affect the position of shockwave. The results for the effect of vorticity on the performance of the nozzles show that, although losses in pressure increase due to inlet swirl flow, vorticity increases very sharply in the vicinity of the shock. It could be concluded that the region just before the shock spot is the main region where fine particles can be separated because of the large vorticity strength. Shock with reasonable strength may be favored in practical applications where fine particles separation is desired.

Four-point amplitudes in [formula omitted] supergravity and Wilson loops

Prompted by recent progress in the study of N = 4 super Yang–Mills amplitudes, and evidence that similar approaches are relevant to N = 8 supergravity, we investigate possible iterative structures and applications of Wilson loop techniques in maximal supergravity. We first consider the two-loop, four-point MHV scattering amplitude in N = 8 supergravity, confirming that the infrared divergent parts exponentiate, and we give the explicit expression which represents the failure for this to occur for the finite part. We observe that each term in the expansion of the one- and two-loop amplitudes in the dimensional regularisation parameter ϵ has a uniform degree of transcendentality. We then turn to consider Wilson loops in supergravity, showing that a natural definition of the loop, involving the Christoffel connection, fails to reproduce the one-loop amplitude. An alternative expression, which involves the metric explicitly, is shown to have a close relationship with the physical amplitude. We find that in a gauge in which the cusp diagrams vanish, the remaining diagrams for this Wilson loop correctly generate the full one-loop, four-point N = 8 supergravity amplitude.

How does Casimir energy fall? III. Inertial forces on vacuum energy

We have recently demonstrated that Casimir energy due to parallel plates, including its divergent parts, falls like conventional mass in a weak gravitational field. The divergent parts were suitably interpreted as renormalizing the bare masses of the plates. Here, we corroborate our result regarding the inertial nature of Casimir energy by calculating the centripetal force on a Casimir apparatus rotating with constant angular speed. We show that the centripetal force is independent of the orientation of the Casimir apparatus in a frame whose origin is at the center of inertia of the apparatus.

Predictive formulation of the Nambu–Jona-Lasinio model

A novel strategy to handle divergences typical of perturbative calculations is implemented for the Nambu--Jona-Lasinio model and its phenomenological consequences investigated. The central idea of the method is to avoid the critical step involved in the regularization process, namely, the explicit evaluation of divergent integrals. This goal is achieved by assuming a regularization distribution in an implicit way and making use, in intermediary steps, only of very general properties of such regularization. The finite parts are separated from the divergent ones and integrated free from effects of the regularization. The divergent parts are organized in terms of standard objects, which are independent of the (arbitrary) momenta running in internal lines of loop graphs. Through the analysis of symmetry relations, a set of properties for the divergent objects are identified, which we denominate consistency relations, reducing the number of divergent objects to only a few. The calculational strategy eliminates unphysical dependencies of the arbitrary choices for the routing of internal momenta, leading to ambiguity-free, and symmetry-preserving physical amplitudes. We show that the imposition of scale properties for the basic divergent objects leads to a critical condition for the constituent quark mass such that the remaining arbitrariness is removed. The model becomes predictive in the sense that its phenomenological consequences do not depend on possible choices made in intermediary steps. Numerical results are obtained for physical quantities at the one-loop level for the pion and sigma masses and pion-quark and sigma-quark coupling constants.