Terms Of Polylogarithms Research Articles

Two-loop vertices with vacuum polarization insertion

We present the analytic evaluation of the second-order corrections to the massive form factors, due to two-loop vertex diagrams with a vacuum polarization insertion, with exact dependence on the external and internal fermion masses, and on the squared momentum transfer. We consider vector, axial-vector, scalar and pseudoscalar interactions between the external fermion and the external field. After renormalization, the finite expressions of the form factors are expressed in terms of polylogarithms up to weight three.

Approximation of the McCumber-Sturge equation for the photoluminescence bandshift at low temperatures

Temperature-induced bandshift of luminescence peaks is frequently described with McCumber-Sturge relation which is difficult for fitting or practical applications due to its mathematical complexity. Here for the first time, we give its exact solution in terms of polylogarithms, and an excellent approximation that is better at lower temperatures than the approximation by Walsh and Di Bartolo (WB). The novel approximation is analytically compared with that of WB, clearly giving the preferred temperature and Debye temperature ranges for each approximation. Three additional approximations are given, each with lowered complexity, until the equivalent quadratic form of Magomedov’s approximation is reached. All approximations are compared for their complexity vs applicable temperature range.

Special values of Goss L-series attached to Drinfeld modules of rank 2

Inspired by the classical setting, Goss defined the $L$-series of Drinfeld $A$-modules corresponding to representations of the absolute Galois group of a rational function field. In this paper, for a given Drinfeld $A$-module $\phi$ of rank 2 defined over the finite field $\mathbb{F}_q$, we give explicit formulas for the values of Goss $L$-series at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$.

Analytic results for two-loop planar master integrals for Bhabha scattering

We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equations with canonical bases. We obtain results in terms of multiple polylogarithms for all the master integrals but one, for which we derive a compact result in terms of elliptic multiple polylogarithms. As a byproduct, we also provide a compact analytic result in terms of elliptic multiple polylogarithms for an integral belonging to the first family of planar Bhabha integrals, whose computation in terms of polylogarithms was addressed previously in the literature.

Polarised amplitudes and soft-virtual cross sections for b overline{b} \u2192 Z H at NNLO in QCD

Production of the Higgs boson, H in association with a massive vector boson, V, i.e., the V H process, plays an important role in the explorations of Higgs physics at the Large Hadron Collider, both for a precise study of Higgs’ Standard Model couplings and for probing New Physics. In this publication we present the two-loop corrections in mass- less quantum chromodynamics (QCD) to the amplitude of the Higgs production associated with a Z boson via the bottom quark-antiquark annihilation channel with a non-vanishing bottom-quark Yukawa coupling, which is a necessary ingredient of the full next-to-next- to-leading-order QCD corrections to the V H process in the five-flavour scheme. The computation is performed by projecting the D-dimensional scattering amplitude directly onto an appropriate set of Lorentz structures related to the linear polarisation states of the Z boson. We provide analytic expressions of the complete set of renormalised polarised amplitudes in terms of polylogarithms of maximum weight four. To give an estimation of the size of contributions from amplitudes considered in this work, we compute numerically the resulting cross sections under the soft-virtual approximation. We also take the opportunity to make a dedicated discussion regarding an interesting subtlety appearing in the conventional form factor decomposition of amplitudes involving axial currents regularised in D dimensions.

Analytic structure of thermodynamic systems with repulsive interactions

Thermodynamic properties of systems with repulsive interactions, are considered in the grand canonical ensemble. The analytic structure of the excluded-volume model in the complex plane of the system chemical potential (fugacity) is elaborated, based on the fact that the pressure function can be given in terms of the Lambert W-function. Even though the excluded volume model has no phase transitions at real values of the chemical potential, it does exhibit a branch cut singularity in the complex plane, thus limiting the convergence range of the Taylor expansion in the chemical potential. Close similarities to analytic properties of the other models with repulsive interactions, such as a cluster expansion model, the mean-field model, and the ideal Fermi gas model, are pointed out. As an example, repulsive baryonic interactions in a hadron gas, with a focus on the fugacity/virial and Taylor expansion methods used in lattice QCD, are presented. The asymptotic behavior of the Fourier expansion coefficients in these various models suggests that the singular part of net baryonic density can to leading order be universally expressed in terms of polylogarithms.

Sorption isotherm restricted by multilayer hindered adsorption and its relation to nanopore size distribution

Hindered adsorbed layers completely filling the nanopores cause significant deviations from the classical BET isotherms for multimolecular adsorption of vapor in porous solids. Since the point of transition from free to hindered adsorption moves into wider nanopores as vapor pressure increases, the surface area exposed to vapor is decreased by an area reduction factor that decreases with increasing adsorbed volume, and thus also with increasing vapor pressure (or humidity). The area reduction factor does not affect the rates of the local process of direct adsorption or condensation of individual vapor or gas molecules, but it imposes a lateral constraint on the total area and volume of the free portion of the adsorption layer that is in direct contact with vapor. A reasonable assumption for the dependence of the area reduction factor on the number of molecular layers is a selfsimilar function, i.e., a power law. This leads to a sorption isotherm expressed in terms of polylogarithms (aka Jonquière functions). The power-law exponent is a property that serves as an additional data fitting parameter, which is related to the pore size distribution. Compared to BET isotherm with the same initial slope, the proposed isotherm reduces the growth of the BET isotherm at low and intermediate humidity and the deviation increases with the exponent. The fitting of isotherm data is reduced to either a series of linear regressions or the minimization of a quadratic expression with respect to one parameter only. It is shown how to use the optimum fit to calculate the size (or width) distribution of nanopores < 6 nm. Comparisons with several published isotherms and pore size data measured on hardened cement pastes show that the present theory gives excellent fits. Finally, the semi-empirical GAB adsorption model is considered, but its additional parameters are not adopted because they weaken the physical foundation and are not constants as they need to be varied empirically with temperature and, for cements, with the degree of hydration.

Three-loop massive tadpoles and polylogarithms through weight six

We evaluate the three-loop massive vacuum bubble diagrams in terms of polylogarithms up to weight six. We also construct the basis of irrational constants being harmonic polylgarithms of arguments ekiπ/3.

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form

We present Fuchsia — an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients ∂xJ(x,ϵ)=A(x,ϵ)J(x,ϵ) finds a basis transformation T(x,ϵ), i.e., J(x,ϵ)=T(x,ϵ)J′(x,ϵ), such that the system turns into the epsilon form : ∂xJ′(x,ϵ)=ϵS(x)J′(x,ϵ), where S(x) is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator ϵ. That makes the construction of the transformation T(x,ϵ) crucial for obtaining solutions of the initial system.In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals. Program summaryProgram Title:FuchsiaProgram Files doi:http://dx.doi.org/10.17632/zj6zn9vfkh.1Licensing provisions: MITProgramming language:Python 2.7Nature of problem: Feynman master integrals may be calculated from solutions of a linear system of differential equations with rational coefficients. Such a system can be easily solved as an ϵ-series when its epsilon form is known. Hence, a tool which is able to find the epsilon form transformations can be used to evaluate Feynman master integrals.Solution method: The solution method is based on the Lee algorithm (Lee, 2015) which consists of three main steps: fuchsification, normalization, and factorization. During the fuchsification step a given system of differential equations is transformed into the Fuchsian form with the help of the Moser method (Moser, 1959). Next, during the normalization step the system is transformed to the form where eigenvalues of all residues are proportional to the dimensional regulator ϵ. Finally, the system is factorized to the epsilon form by finding an unknown transformation which satisfies a system of linear equations.Additional comments including Restrictions and Unusual features: Systems of single-variable differential equations are considered. A system needs to be reducible to Fuchsian form and eigenvalues of its residues must be of the form n+mϵ, where n is integer. Performance depends upon the input matrix, its size, number of singular points and their degrees. It takes around an hour to reduce an example 74 × 74 matrix with 20 singular points on a PC with a 1.7 GHz Intel Core i5 CPU. An additional slowdown is to be expected for matrices with complex and/or irrational singular point locations, as these are particularly difficult for symbolic algebra software to handle.

A new method for obtaining polylogarithmic Mahler measure formulas

Given a formula for the Mahler measure of a rational function expressed in terms of polylogarithms, we describe a new method that allows us to construct a rational function with 2 more variables and whose Mahler measure is still expressed in terms of polylogarithms. We use this method to exhibit three new examples of Mahler measure and higher Mahler measure formulas. One of them involves a single term with \(\zeta (5)\).

Analytical treatment of Trivich–Flinn and Shockley–Queisser photovoltaic efficiency limits using polylogarithms

ABSTRACTSolutions for the Trivich–Flinn and Shockley–Queisser formulations of the limiting efficiency of photovoltaic conversion are derived in terms of polylogarithms. Earlier Trivich–Flinn limits are shown to be zero temperature limits of the Shockley–Queisser approach. The limiting efficiency of an infinite stack of tandem cells was also investigated with an analytically based solution derived, the limit on photovoltaic conversion for a time‐symmetric system, which is compared to the time‐asymmetric limit. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Dispersion in a thermal plasma including arbitrary degeneracy and quantum recoil

The longitudinal response function for a thermal electron gas is calculated including two quantum effects exactly, degeneracy, and the quantum recoil. The Fermi-Dirac distribution is expanded in powers of a parameter that is small in the nondegenerate limit and the response function is evaluated in terms of the conventional plasma dispersion function to arbitrary order in this parameter. The infinite sum is performed in terms of polylogarithms in the long-wavelength and quasistatic limits, giving results that apply for arbitrary degeneracy. The results are applied to the dispersion relations for Langmuir waves and to screening, reproducing known results in the nondegenerate and completely degenerate limits, and generalizing them to arbitrary degeneracy.

Exponential suppression with four legs and an infinity of loops

The L-loop 4-point ladder diagram of massless phi^3 theory is finite when all 4 legs are off-shell and is given in terms of polylogarithms with orders ranging from L to 2L. We obtain the exact solution of the linear Dyson-Schwinger equation that sums these ladder diagrams and show that this sum vanishes exponentially fast at strong coupling.

Dynamics of uniform quantum gases, I: Density and current correlations

A unified approach valid for any wavenumber q , frequency ω , and temperature T is presented for uniform ideal quantum gases allowing for a comprehensive study of number density and particle-current density response functions. Exact analytical expressions are obtained for spectral functions in terms of polylogarithms. Also, particle-number and particle-current static susceptibilities are presented which, for fugacity less than unity, additionally involve Kummer functions. The q - and T -dependent transverse-current static susceptibility is used to show explicitly that current correlations are of long range in a Bose-condensed uniform ideal gas but for bosons at T > T c and for Fermi and Boltzmann gases at all temperatures these correlations are of short range. Contact repulsive interactions for systems of neutral quantum particles are considered within the random phase approximation. The expressions for particle-number and transverse-current susceptibilities are utilized to discuss the existence or nonexistence of superfluidity in the systems under consideration.

Unclosed asymptotic expansions and mock theta functions

In his last letter to Hardy, Ramanujan defined 17 functions f(q), (|q|<1), which he called mock theta functions. Each f(q) has infinitely many exponential singularities at roots of unity, and under radial approach to every such singularity, f(q) has an asymptotic approximation consisting of a finite number of terms with closed exponential factors, plus an error term O(1). We give an example of a q-series in Eulerian form having an approximation with an unclosed exponential factor. Complete asymptotic expansions as q→1 of some shifted q-factorials are given in terms of polylogarithms and Bernoulli polynomials.

TSIL: a program for the calculation of two-loop self-energy integrals

TSIL is a library of utilities for the numerical calculation of dimensionally regularized two-loop self-energy integrals. A convenient basis for these functions is given by the integrals obtained at the end of O.V. Tarasov's recurrence relation algorithm. The program computes the values of all of these basis functions, for arbitrary input masses and external momentum. When analytical expressions in terms of polylogarithms are available, they are used. Otherwise, the evaluation proceeds by a Runge–Kutta integration of the coupled first-order differential equations for the basis integrals, using the external momentum invariant as the independent variable. The starting point of the integration is provided by known analytic expressions at (or near) zero external momentum. The code is written in C, and may be linked from C/C++ or Fortran. A Fortran interface is provided. We describe the structure and usage of the program, and provide a simple example application. We also compute two new cases analytically, and compare all of our notations and conventions for the two-loop self-energy integrals to those used by several other groups. Program summary Title of program: TSIL Version number: 1.0 Catalogue identifier: ADWS Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWS Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Programming language: C Platform: Any platform supporting the GNU Compiler Collection (gcc), the Intel C compiler (icc), or a similar C compiler with support for complex mathematics No. of lines in distributed program, including test data, etc.: 42 730 No. of bytes in distributed program, including test data, etc.: 297 101 Distribution format: tar.gz Nature of physical problem: Numerical evaluation of dimensionally regulated Feynman integrals needed in two-loop self-energy calculations in relativistic quantum field theory in four dimensions. Method of solution: Analytical evaluation in terms of polylogarithms when possible, otherwise through Runge–Kutta solution of differential equations. Limitations: Loss of accuracy in some unnatural threshold cases that do not have vanishing masses. Typical running time: Less than a second.

Analytical result for dimensionally regularized massive on-shell planar double box

The dimensionally regularized master planar double box Feynman diagram with four massive and three massless lines, powers of propagators equal to one, all four legs on the mass shell, i.e., with p i 2 = m 2 , i =1,2,3,4, is analytically evaluated for general values of m 2 and the Mandelstam variables s and t . An explicit result is expressed in terms of polylogarithms, up to the third order, depending on special combinations of m 2 , s and t .

Two-loop QCD corrections to the scattering of massless distinct quarks

We present the two-loop virtual QCD corrections to the scattering of distinct massless quarks, qq→q′q′, in conventional dimensional regularisation. The structure of the infrared divergences agrees with that predicted by Catani while expressions for the finite remainder are given for each of the s-, t- and u-channels in terms of polylogarithms. The results presented here form a vital part of the next-to-next-to-leading order contribution to inclusive jet production in hadron colliders and will play a crucial role in improving the theoretical prediction for jet cross sections in hadron–hadron collisions.

Two-loop QCD corrections to massless identical quark scattering

We present the two-loop virtual QCD corrections to the scattering of massless identical quarks, q qbar to q qbar, in conventional dimensional regularisation and using the MSbar scheme. The structure of the infrared divergences agrees with that predicted by Catani while expressions for the finite remainder are given for the q qbar to qbar q and the q q to q q (qbar qbar to qbar qbar) scattering processes in terms of polylogarithms. The results presented here form a vital part of the next-to-next-to-leading order contribution to inclusive jet production in hadron colliders and will play a crucial role in improving the theoretical prediction for jet cross sections in hadron-hadron collisions.

Two-loop correction to Bhabha scattering

We present the two-loop virtual QED corrections to e^+ e^- to mu^+ mu^- and Bhabha scattering in dimensional regularization. The results are expressed in terms of polylogarithms. The form of the infrared divergences agrees with previous expectations. These results are a crucial ingredient in the complete next-to-next-to-leading order QED corrections to these processes. A future application will be to reduce theoretical uncertainties associated with luminosity measurements at e^+ e^- colliders. The calculation also tests methods that may be applied to analogous QCD processes.