1-dimensional Integral Research Articles

Mean-field tricritical polymers

We provide an introductory account of a tricritical phase diagram, in the setting of a mean-field random walk model of a polymer density transition, and clarify the nature of the density transition in this context. We consider a continuous-time random walk model on the complete graph, in the limit as the number of vertices $N$ in the graph grows to infinity. The walk has a repulsive self-interaction, as well as a competing attractive self-interaction whose strength is controlled by a parameter $g$. A chemical potential $\nu$ controls the walk length. We determine the phase diagram in the $(g,\nu)$ plane, as a model of a density transition for a single linear polymer chain. A dilute phase (walk of bounded length) is separated from a dense phase (walk of length of order $N$) by a phase boundary curve. The phase boundary is divided into two parts, corresponding to first-order and second-order phase transitions, with the division occurring at a tricritical point. The proof uses a supersymmetric representation for the random walk model, followed by a single block-spin renormalisation group step to reduce the problem to a 1-dimensional integral, followed by application of the Laplace method for an integral with a large parameter.

All orders structure and efficient computation of linearly reducible elliptic Feynman integrals

We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator in terms of a 1-dimensional integral over a polylogarithmic integrand, which we call the inner polylogarithmic part (IPP). The solution is obtained by direct integration of the Feynman parametric representation. When the IPP depends on one elliptic curve (and no other algebraic functions), this class of Feynman integrals can be algorithmically solved in terms of elliptic multiple polylogarithms (eMPLs) by using integration by parts identities. We then elaborate on the differential equations method. Specifically, we show that the IPP can be mapped to a generalized integral topology satisfying a set of differential equations in ϵ-form. In the examples we consider the canonical differential equations can be directly solved in terms of eMPLs up to arbitrary order of the dimensional regulator. The remaining 1-dimensional integral may be performed to express such integrals completely in terms of eMPLs. We apply these methods to solve two- and three-points integrals in terms of eMPLs. We analytically continue these integrals to the physical region by using their 1-dimensional integral representation.

Local-Stochastic Volatility for Vanilla Modeling: A Tractable and Arbitrage Free Approach to Option Pricing

We present a novel and flexible technique for the construction of tractable, and fully arbitrage-free, Local-Stochastic Volatility (LSV) term distribution models. The method utilizes Lamperti’s harmonic transform to combine a pure stochastic volatility (SV) process MT with a general local volatility (LV) function σ. European options prices can conveniently be expressed in terms of a 1-dimensional integral over options on the SV component MT. As a demonstration, we apply the technique to the popular SABR model family by letting MT be the Normal SABR process, and letting the LV function σ be an extended CEV function, endowed with additional controls over the backbone and the wings for the smile. The resulting model not only rectifies the arbitrage problems of commonly used expansions for the SABR class, its additional parameters also allow for adequate handling of non-positive rates and improved calibration to quoted swaptions and CMS forward rates. Finally, for variety we also illustrate the technique using Heston dynamics and a general piece-wise linear LV function.

Modifying partition functions: A way to solve the sign problem

A possible method to solve the sign problem is developed by modifying the original theory. Considering several modifications of the partition function, the observable in the original theory is reconstructed from the identity connecting the observables in the original and modified theories. We demonstrate that our method gives the correct results even if the original theory has the severe sign problem by using a simple 1-dimensional integral.

SecDec-3.0: Numerical evaluation of multi-scale integrals beyond one loop

SecDec is a program which can be used for the factorization of dimensionally regulated poles from parametric integrals, in particular multi-loop integrals, and the subsequent numerical evaluation of the finite coefficients. Here we present version 3.0 of the program, which has major improvements compared to version 2: it is faster, contains new decomposition strategies, an improved user interface and various other new features which extend the range of applicability. Program summaryProgram title: SecDec 3.0Catalogue identifier: AEIR_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIR_v3_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.: 123828No. of bytes in distributed program, including test data, etc.: 1651026Distribution format: tar.gzProgramming language: Wolfram Mathematica, perl, Fortran/C++.Computer: From a single PC to a cluster, depending on the problem.Operating system: Unix, Linux.RAM: Depending on the complexity of the problemClassification: 4.4, 5, 11.1.Catalogue identifier of previous version: AEIR_v2_1Journal reference of previous version: Comput. Phys. Comm. 184(2013)2552Does the new version supersede the previous version?: YesNature of problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in gauge theories. Numerical integration in the presence of integrable singularities (e.g. kinematic thresholds).Solution method: Algebraic extraction of singularities within dimensional regularization using iterated sector decomposition. This leads to a Laurent series in the dimensional regularization parameter, where the coefficients are finite integrals over the unit-hypercube. Those integrals are evaluated numerically by Monte Carlo integration. The integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way.Reasons for new version:•Improved user interface.•Additional new decomposition strategies.•Usage on a cluster is much improved.•Speed-up in numerical evaluation times.•Various new features (please see below).Summary of revisions:•Implementation of two new decompositions strategies based on a geometric algorithm.•Scans over large ranges of parameters are facilitated.•Linear propagators can be treated.•Propagators with negative indices are possible.•Interface to reduction programs like Reduze, Fire, LiteRed facilitated.•Option to use numerical integrator from Mathematica.•Using CQUAD for 1-dimensional integrals to improve speed of numerical evaluations.•Option to include epsilon-dependent dummy functions.Restrictions: Depending on the complexity of the problem, limited by memory and CPU time.Running time: Between a few seconds and several hours, depending on the complexity of the problem.

The LPM effect in sequential bremsstrahlung

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We analyze the case when the coherence lengths of two consecutive splitting processes overlap, which is important for understanding corrections to standard treatments of the LPM effect in QCD. Previous authors have analyzed this problem in the case of overlapping double bremsstrahlung where at least one of the bremsstrahlung gluons is soft. Here we show how to generalize to include the case where both splittings are hard. A number of techniques must be developed, and so in this paper we simplify by (i) restricting attention to a subset of the interference effects, which we call the "crossed" diagrams, and (ii) working in the large-$N_c$ limit. We first develop some general formulas that could in principle be implemented numerically (with substantial difficulty). To make more analytic progress, we then focus on the case of a thick, homogeneous medium and make the multiple scattering approximation (also known as the $\hat q$ or harmonic approximation) appropriate at high energy. We show that the differential rate $d\Gamma/dx\,dy$ for overlapping double bremsstrahlung of gluons with momentum fractions $x$ and $y$ can then be reduced to the calculation of a 1-dimensional integral, which we perform numerically. [Though this paper is unfortunately long, our introduction is enough for getting the gist of the method.]

On Wronskian Determinant Formulas of the General Hypergeometric Functions

The general hypergeometric functions of confluent type are studied. We establish a link between the general hypergeometric functions defined by 1-dimensional integrals and those defined by multi-dimensional integrals. The key point is to form an intermediate Wronskian determinant for the 1-dimensional ones and to rewrite it into a multi-dimensional integral using the generalized Veronese map.

Analytical solution of the Yukawa closure of the Ornstein-Zernik equation IV: the general 1-component case

In previous work we have studied the solution of the Ornstein-Zernike equation with a general multiyukawa closure. Here the direct correlation function is expressed by a rapidly converging sum of M (complex) exponentials. For a simple fluid the mathematical problem of solving the Ornstein-Zernike equation is equivalent to finding the solution of a linear algebraic equation of order M. The solution for the arbitrary case is given in terms of a scaling matrix Γ. For only one component this matrix is diagonal and the general solution using the properties of M-dimensional SOM Lie group is given. In the Mean Spherical Approximation (MSA) the excess entropy is obtained and expressed as a sum of 1-dimensional integrals of algebraic functions. We remark that the general solution of the M exponents-1 component case was found in our early work (Blum, L., and Hoye, J. S.,1978, J. stat. Phys., 19, 317) in implicit form. The present explicit solution agrees completely with the early one. Other thermodynamic properties such as the energy equation of state are also obtained, explicitly for 2 and 3 exponentials. The analytical solution of the effective MSA is also obtained from the simple variational form for the Helmholtz excess free energy δA (1) where (2) where both the excess energy ΔE(Γ) and the excess entropy ΔS(Γ) are functionals of Γ, which opens interesting possibilities that are discussed elsewhere. We remark that this is a non-trivial property, which is certainly true for the MSA (Chandler, D., and Andersen, H. C., 1972, J. chem. Phys., 57, 1930). It implies cross-derivative properties for the closure equations, which have been verified in all cases.

On the adaptive computation of integrals of wavelets

The numerical solution of partial differential equations involves the computation of integrals of products of given functions and (derivatives of) trial and test functions. We study this problem using adaptively chosen wavelet bases. Firstly, we reduce this problem to the computation of 1-dimensional integrals and present an algorithm for computing these integrals. Then, we consider appropriate adaptive approximations and study the induced error. Finally, we give numerical results.