Scalar Integrals Research Articles

Oscillatory integrals and weighted gradient flows

We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on (x_{1} \partial_{x_1}f(x) ,\dots, x_{n} \partial_{x_n}f(x)) , where f(x) is the phase function. We prove a theorem which provides estimates that are uniform with respect to linear perturbations of the phase and investigate some consequences. When the phase function is quasi-homogeneous the theorem gives estimates for the associated surface measure Fourier transforms that are generally not too far off from being sharp. In addition, the theorem provides a new proof, up to endpoints, that the well-known oscillatory integral estimates of Varchenko when the Newton polyhedron of the phase function is nondegenerate extend to corresponding bounds for surface measure Fourier transforms when the index is less than 1/2 . A sharp version of this was originally proven by the author [J. Funct. Anal. 262 (2012), no. 5, 2314–2348].

Monte Carlo evaluation of divergent one-loop integrals without contour deformation

Reference (Pittau and Webber in Eur Phys J C 82(1):55, https://doi.org/10.1140/epjc/s10052-022-10008-6, 2022) introduces a method for computing numerically four-dimensional multi-loop integrals without performing an explicit analytic contour deformation around threshold singularities. In this paper, we extend such a technique to massless scalar one-loop integrals regularized in the framework of dimensional regularization. A two-loop example is also discussed.

One-loop integrals from volumes of orthoschemes

Recently in arXiv:2012.05599 Rudenko presented a formula for the volume of hyperbolic orthoschemes in terms of alternating polylogarithms. We use this result to provide an explicit analytic result for the one-loop scalar n-gon Feynman integral in n dimensions, for even n, with massless or massive internal and external edges. Furthermore, we evaluate the general six-dimensional hexagon integral in terms of classical polylogarithms.

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We compute the one-loop corrections to gg → \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\overline{t }H$$\\end{document} up to order \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}\\left({\\epsilon }^{2}\\right)$$\\end{document} in the dimensional-regularization parameter. We apply the projector method to compute polarized amplitudes, which generalize massless helicity amplitudes to the massive case. We employ a semi-numerical strategy to evaluate the scattering amplitudes. We express the form factors through scalar integrals analytically, and obtain separately integration by parts reduction identities in compact form. We integrate numerically the corresponding master integrals with an enhanced implementation of the Auxiliary Mass Flow algorithm. Using a numerical fit method, we concatenate the analytic and the numeric results to obtain fast and reliable evaluation of the scattering amplitude. This approach improves numerical stability and evaluation time. Our results are implemented in the Mathematica package TTH.

Tensor reduction of loop integrals

The computational cost associated with reducing tensor integrals to scalar integrals using the Passarino-Veltman method is dominated by the diagonalisation of large systems of equations. These systems of equations are sized according to the number of independent tensor elements that can be constructed using the metric and external momenta. In this article, we present a closed-form solution of this diagonalisation problem in arbitrary tensor integrals. We employ a basis of tensors whose building blocks are the external momentum vectors and a metric tensor transverse to the space of external momenta. The scalar integral coefficients of the basis tensors are obtained by mapping the basis elements to the elements of an orthogonal dual basis. This mapping is succinctly expressed through a formula that resembles the ordering of operators in Wick’s theorem.Finally, we provide examples demonstrating the application of our tensor reduction formula to Feynman diagrams in QCD 2 → 2 scattering processes, specifically up to three loops.

Factorization of covariant Feynman graphs for the effective action

We prove a neat factorization property of Feynman graphs in covariant perturbation theory. The contribution of the graph to the effective action is written as an integral over Schwinger parameters whose integrand is a product of a massless scalar momentum integral that only depends on the basic graph topology, and a background-field dependent piece that contains all the information of spin, gauge representations, masses etc. We give a closed expression for the momentum integral in terms of four graph polynomials whose properties we derive in some detail. Our results can also be useful for standard (non-covariant) perturbation theory.

One-loop fermion-photon vertex in arbitrary gauge and dimensions: A novel approach

We compute a one-loop electron-photon vertex with fully off-shell external momenta in an arbitrary covariant gauge and space-time dimension. There exist numerous efforts in literature where a one-loop off-shell vertex is calculated by employing the standard first-order Feynman rules in different covariant gauges and space-time dimensions of interest. The tensor structure which decomposes this three-point vertex into the components transverse and longitudinal to the photon momentum gets intertwined in this first-order formalism. The Ward-Takahashi identity is explicitly invoked to untangle the two pieces and the results are expressed in a preferred basis of 12 spin amplitudes. We propose a novel approach based upon an efficient combination of the first- and second-order formalisms of quantum electrodynamics to compute this one-loop vertex. Among some conspicuous advantages is the fact that this less known second-order formalism separates the spin and scalar degrees of freedom of an electron interacting electromagnetically. More noticeably, the longitudinal and transverse contributions naturally disentangle from the onset in our approach. Moreover, this decomposition leads to identities between one-loop scalar Feynman integrals with higher powers in the propagators and shifted space-time dimensions that can be used to prove the Ward-Takahashi identity at one-loop order without the need to evaluate any Feynman integral. Additionally, this natural decomposition allows us to establish the gauge independence of the Pauli form factor through explicit cancellations of scalar Feynman integrals that depend on the gauge parameter. These cancellations naturally lead to a compact expression for the Pauli form factor in arbitrary dimensions. Wherever necessary and insightful, we make comparisons with earlier works.

Hypergeometric structures in Feynman integrals

For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

Multi-variable integration with a neural network

In this article we present a method for automatic integration of parametric integrals over the unit hypercube using a neural network. The method fits a neural network to the primitive of the integrand using a loss function designed to minimize the difference between multiple derivatives of the network and the function to be integrated. We apply this method to two example integrals resulting from the sector decomposition of a one-loop and two-loop scalar integrals. Our method can achieve per-mil and percent accuracy for these integrals over a range of invariant values. Once the neural network is fitted, the evaluation of the integral is between 40 and 125 times faster than the usual numerical integration method for our examples, and we expect the speed gain to increase with the complexity of the integrand.

Feynman integrals of Grassmannians

We embed Feynman integrals in the subvarieties of Grassmannians through homogenization of the integrands in projective space, then obtain Gel'fand-Kapranov-Zelevinsky systems satisfied by those scalar integrals. The Feynman integral can be written as linear combinations of the hypergeometric functions of a fundamental solution system in neighborhoods of regular singularities of the Gel'fand-Kapranov-Zelevinsky system, whose linear combination coefficients are determined by the integral on an ordinary point or some regular singularities. Taking some Feynman diagrams as examples, we elucidate in detail how to obtain the fundamental solution systems of Feynman integrals in neighborhoods of regular singularities. Furthermore we also present the parametric representations of Feynman integrals of the two-loop self-energy diagrams that are convenient to embed in the subvarieties of Grassmannians.

HepLib: A C++ library for high energy physics (version 1.1)

HepLib is a C++ Library for computations in High Energy Physics, it builds on top of other well-established libraries or programs, including GiNaC, Fermat, Form, Fire, etc., its first version has been released in Comput. Phys. Commun. 265 (2021) 107982. Here we provide a minor upgraded version with parts of obligated aspects removed and many new features added, the main feature in this new version is to use Flint for the simplification on multi-variate polynomial, so one can replace the process-based parallelism by the thread-based parallelism for the related polynomial evaluations to achieve a higher performance. New version program summaryProgram title:HepLib 1.1CPC Library link to program files:https://doi.org/10.17632/jspjxy3z66.2Developer's repository link:https://github.com/HepLib/srcLicensing provisions: : GPLv3Programming language:C++Journal reference of previous version: Comput. Phys. Commun. 265 (2021) 107982Does the new version supersede the previous version?: YesExternal libraries and programs: : GiNaC [2], Qgraf [3], Flint [4], Qhull [5], Form [6], Fermat [7], Fire [8]Nature of problem: There are lots of independent packages or programs in high energy physics, which are developed in different programming languages, here we provide an integrated interface in C++ language, to generate Feynman diagrams/amplitudes, perform algebraic simplifications on Dirac/Color matrixes and reduce scalar integrals to master integrals. The evaluation of master integrals is the core part for the high-order calculations, one has to resolve ultraviolet/infrared divergence within dimensional regularization and the singularities inside the integration domain while performing multiple dimensional numerical integrations.Solution method:HepLib uses Qgraf to generate the Feynman diagrams/aptitudes, Form to perform the related computations on Dirac/Color matrix, Fire for the IBP reduction. A C++ implementation of sector decomposition method is used to extract both ultraviolet and infrared singularity of the integral, the contour deformation is adopted to avoid the singularity inside the integration domain, a parallelized numerical integration is chosen to achieve high efficiency or performance.Reasons for the new version: Fix bugs from GiNaC and implement new features.Summary of revisions: First, we have improved the installation procedure, now one can install the whole package with a single script install.sh, and furthermore one can install all libraries into a distinct directory, so the libraries will not affect the others which are installed on the user's computer. We have upgraded the GiNaC to the latest version with a few bugs fixed in the official site and removed the dependence on the libraries MinUits [9] and Cuba [10], which are used in performing sector decomposition in the previous version. To get high performance on the evaluation of multi-variate polynomials, we introduce Flint in the new version, and one can convert the expression from GiNaC to Flint, and vice versa. Finally, we also made adaptations to the codes and examples to match the new version.

General one-loop reduction in generalized Feynman parametrization form* *Supported by the National Natural Science Foundationof China (11935013)

The search for an effective reduction method is one of the main topics in higher loop computation. Recently, an alternative reduction method was proposed by Chen in [1, 2]. In this paper, we test the power of Chen's new method using one-loop scalar integrals with propagators of higher power. More explicitly, with the improved version of the method, we can cancel the dimension shift and terms with unwanted power shifting. Thus, the obtained integrating-by-parts relations are significantly simpler and can be solved easily. Using this method, we present explicit examples of a bubble, triangle, box, and pentagon with one doubled propagator. With these results, we complete our previous computations in [3] with the missing tadpole coefficients and show the potential of Chen's method for efficient reduction in higher loop integrals.

Functional reduction of one-loop Feynman integrals with arbitrary masses

A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail.The method is based on a repeated application of the functional relations proposed by the author. Explicit formulae are given for reducing one-loop scalar integrals to a simpler ones, the arguments of which are the ratios of polynomials in the masses and kinematic invariants. We show that a general scalar n-point integral, depending on n(n + 1)/2 generic masses and kinematic variables, can be expressed as a linear combination of integrals depending only on n variables. The latter integrals are given explicitly in terms of hypergeometric functions of (n − 1) dimensionless variables. Analytic expressions for the 2-, 3- and 4-point integrals, that depend on the minimal number of variables, were also obtained by solving the dimensional recurrence relations. The resulting expressions for these integrals are given in terms of Gauss’ hypergeometric function 2F1, the Appell function F1 and the hypergeometric Lauricella — Saran function FS. A modification of the functional reduction procedure for some special values of kinematic variables is considered.

Diameter estimate for closed manifolds with positive scalar curvature

For a simply connected closed Riemannian manifold with positive scalar curvature, we prove an upper diameter bound in terms of its scalar curvature integral, the Yamabe constant and the dimension of the manifold. When a manifold has a conformal immersion into a sphere, the dependency on the Yamabe constant is not necessary. The power of scalar curvature integral in these diameter estimates is sharp and it occurs at round spheres with canonical metric.

Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: Initial value problem

The light trajectory in the gravitational field of one body at rest with monopole and quadrupole structure is determined in the second post-Newtonian (2PN) approximation. The terms in the geodesic equation for light rays are separated into time-independent tensorial coefficients and four kind of time-dependent scalar functions. Accordingly, the first and second integration of geodesic equation can be reduced in each case to only four kind of scalar master integrals. These integrals can be solved in closed form by recurrence relations. The 2PN terms of monopole and quadrupole contribute less than $1$ nano-arcsecond to the total light deflection. There are, however, enhanced terms in the 2PN light deflection, both in case of monopole and quadrupole. These enhanced 2PN terms are caused by the use of an impact vector which is indispensable for modeling of real astrometric measurements. In case of grazing light rays at Jupiter and Saturn, the enhanced 2PN terms, caused by the quadrupole structure of the body, amount up to 0.95 micro-arcseconds and 0.29 micro-arcseconds, respectively. Thus, the 2PN quadrupole terms are relevant for high-precision astrometry on the sub-micro-arcsecond scale of accuracy.

Reduction of the planar double-box diagram for single-top production via auxiliary mass flow

The single-top production is an important process at the LHC to test the Standard Model (SM) and search for the new physics beyond the SM. Although the complete next-to-next-to-leading order (NNLO) QCD correction to the single-top production is crucial, this calculation is still challenging at present. In order to efficiently reduce the NNLO single-top amplitude, we improve the auxiliary mass flow (AMF) method by introducing the $\epsilon$ truncation. For demonstration we choose one typical planar double-box diagram for the $tW$ production. It is shown that one coefficient of the form factors on its amplitude can be systematically reduced into the linear combination of 198 scalar integrals.

Superintegrable systems with spin and second-order tensor and pseudo-tensor integrals of motion

We investigate a quantum non-relativistic system describing the interaction of two particles with spin and spin 0, respectively. Assuming that the Hamiltonian is rotationally invariant and parity conserving we identify all such systems which allow additional tensor and pseudo-tensor integrals of motion that are second-order matrix polynomials in the momenta. Previously we found all the scalar, pseudo-scalar, vector and axial vector integrals of motion. No non-obvious tensor integrals exist. However, nontrivial pseudo-tensor integrals do exist. Together with our earlier results we give a complete list of such superintegrable Hamiltonian systems allowing second-order integrals of motion.

New ideas for handling of loop and angular integrals in D-dimensions in QCD

We discuss new ideas for consideration of loop diagrams and angular integrals in D-dimensions in QCD. In case of loop diagrams, we propose the covariant formalism of expansion of tensorial loop integrals into the orthogonal basis of linear combinations of external momenta. It gives a very simple representation for the final results and is more convenient for calculations on computer algebra systems. In case of angular integrals we demonstrate how to simplify the integration of differential cross sections over polar angles. Also we derive the recursion relations, which allow to reduce all occurring angular integrals to a short set of basic scalar integrals. All order ε-expansion is given for all angular integrals with up to two denominators based on the expansion of the basic integrals and using recursion relations. A geometric picture for partial fractioning is developed which provides a new rotational invariant algorithm to reduce the number of denominators.

HepLib: A C++ library for high energy physics

HepLib is a C++ Library for computations in High Energy Physics, it works on top of GiNaC, a well-established C++ library used to perform symbolic computations. HepLib combines serval well-known packages to get high efficiency, including qgraf to generate Feynman aptitudes, FORM to perform Dirac/Color matrix related computations, and FIRE or KIRA for integration-by-parts (IBP) reduction. Another core feature of HepLib lies in the numerical evaluation of master integrals using sector decomposition, which is a general method widely used in high-order numerical computation and has been implemented in many public packages in many different languages, and we present another implementation in the language of C++ with many new features. We use GiNaC to handle the symbolic operations, and export the corresponding integrand into an optimized C++ code, that will be compiled internally and linked dynamically, a customizable numerical integrator is selected to perform the numerical integration, while the integrand can be evaluated in different float precisions, including the arbitrary precision supported by MPFR. Program summaryProgram Title:HepLibCPC Library link to program files:https://doi.org/10.17632/jspjxy3z66.1Developer's repository link:https://heplib.github.io/Licensing provisions:GPLv3Programming language:C++External libraries:GiNaC, qgraf, MPFR, Fermat, QHull, MinUit2, CUBAExternal programs:FORM, FIRE, KIRANature of problem: There are many independent programs or packages in high energy physics, which are written or developed in different programming languages, here we are trying to provide an integrated interface in the C++ language, for generating Feynman diagrams/amplitudes, performing algebraic simplifications on Dirac/Color matrix objects, and reducing scalar integrals to master integrals. The evaluation of master integrals is in the core part during the high-order calculations, one needs to resolve ultraviolet/infrared divergence within dimensional regularization and the singularities inside the integration domain while performing multiple dimensional numerical integrations.Solution method:HepLib uses qgraf to generate the Feynman diagrams/aptitudes, FORM to perform Dirac/Color matrix related computations, FIRE or KIRA for the IBP reduction. A C++ implementation of sector decomposition method is used to extract both ultraviolet and infrared singularity of the integral, the contour deformation is adopted to avoid the singularity inside the integration domain, a parallelized numerical integration is chosen to achieve high efficiency or performance.Restrictions: Depending on the complexity of the problem itself.

C1,α-regularity for variational problems in the Heisenberg group

We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"older continuity of horizontal gradient of minima.