Dimensional Regularization Parameter Research Articles

MultiHypExp: A Mathematica package for expanding multivariate hypergeometric functions in terms of multiple polylogarithms

We present the Mathematica package MultiHypExp that allows for the expansion of multivariate hypergeometric functions (MHFs), especially those likely to appear as solutions of multi-loop, multi-scale Feynman integrals, in the dimensional regularization parameter. The series expansion of MHFs can be carried out around integer values of parameters to express the series coefficients in terms of multiple polylogarithms. The package uses a modified version of the algorithm prescribed in [1]. In the present work, we relate a given MHF to a Taylor series expandable MHF by a differential operator. The Taylor expansion of the latter MHF is found by first finding the associated partial differential equations (PDEs) from its series representation. We then bring the PDEs to the Pfaffian system and further to the canonical form, and solve them order by order in the expansion parameter using appropriate boundary conditions. The Taylor expansion so obtained and the differential operators are used to find the series expansion of the given MHF. We provide examples to demonstrate the algorithm and to describe the usage of the package, which can be found in this url. Program summaryProgram Title:MultiHypExp.wl version 1.0CPC Library link to program files:https://doi.org/10.17632/zxwkx38y7b.1Developer's repository link:https://github.com/souvik5151/MultiHypExpLicensing provisions: GNU General Public License 3Programming language: Wolfram Mathematica version 11.3 or higherExternal routines/libraries: HolonomicFunctions, CANONICA, HYPERDIRE, PolyLogTools, HPLNature of problem: Obtaining the series expansion of multivariate hypergeometric functions around integer values of Pochhammer parameters in terms of multiple polylogarithmsSolution method: Mathematica implementation of a modified version of the algorithm prescribed in Nucl. Phys. B 989 (2023) 116145.

Numerical scattering amplitudes with pySecDec

We present a major update of the program pySecDec, a toolbox for the evaluation of dimensionally regulated parameter integrals. The new version enables the evaluation of multi-loop integrals as well as amplitudes in a highly distributed and flexible way, optionally on GPUs. The program has been optimised and runs up to an order of magnitude faster than the previous release. A new integration procedure that utilises construction-free median Quasi-Monte Carlo rules is implemented. The median lattice rules can outperform our previous component-by-component rules by a factor of 5 and remove the limitation on the maximum number of sampling points. The expansion by regions procedures have been extended to support Feynman integrals with numerators, and functions for automatically determining when and how analytic regulators should be introduced are now available. The new features and performance are illustrated with several examples. Program summaryProgram Title: pySecDecCPC Library link to program files:https://doi.org/10.17632/dnrkf5jxzh.4Developer's repository link:https://github.com/gudrunhe/secdec, https://secdec.readthedocs.io (Online documentation)Licensing provisions: GNU Public License v3Programming language: Python, Form, C++, CudaExternal routines/libraries:GSL [1], NumPy [2], SymPy [3], Nauty [4], Cuba [5], Form [6], GiNaC and CLN [7], Normaliz [8], GMP [9].Journal reference of previous version: Comput. Phys. Commun. 273 (2022) 108267 [1].Does the new version supersede the previous version?: Yes.Nature of problem: Scattering amplitudes at higher orders in perturbation theory are typically represented as a linear combination of coefficients — containing the kinematic invariants and the space-time dimension — multiplied with loop integrals which contain singularities and whose analytic representation might be unknown.Solution method: Extraction of singularities in the dimensional regularization parameter as well as in analytic regulators for potential spurious singularities is done using sector decomposition. The combined evaluation of the integrals with their coefficients is performed in an efficient way.Restrictions: Depending on the complexity of the problem, limited by memory and CPU/GPU time.

Bananas of equal mass: any loop, any order in the dimensional regularisation parameter

We describe a systematic approach to cast the differential equation for the l-loop equal mass banana integral into an ε-factorised form. With the known boundary value at a specific point we obtain systematically the term of order j in the expansion in the dimensional regularisation parameter ε for any loop l. The approach is based on properties of Calabi-Yau operators, and in particular on self-duality.

Taming Calabi-Yau Feynman Integrals: The Four-Loop Equal-Mass Banana Integral.

Certain Feynman integrals are associated to Calabi-Yau geometries. We demonstrate how these integrals can be computed with the method of differential equations. The four-loop equal-mass banana integral is the simplest Feynman integral whose geometry is a nontrivial Calabi-Yau manifold. We show that its differential equation can be cast into an ϵ-factorized form. This allows us to obtain the solution to any desired order in the dimensional regularization parameter ϵ. The method generalizes to other Calabi-Yau Feynman integrals. Our calculation also shows that the four-loop banana integral is only minimally more complicated than the corresponding Feynman integrals at two or three loops.

The massless single off-shell scalar box integral — branch cut structure and all-order epsilon expansion

We investigate the single off-shell scalar box integral with massless internal lines in dimensional regularization. A special emphasis is given to higher orders in the dimensional regularization parameter epsilon, its branch cut structure, and kinematic limits. Common representations of the box integral introduce superficial branch cuts, which we eliminate to all orders in the epsilon expansion. In the literature so far a satisfactory solution for this issue only exists up to finite order in the epsilon expansion. However, for certain calculations at NNLO in perturbation theory, higher orders in epsilon of this integral are required. In this paper, we present results to all orders in epsilon in terms of single-valued polylogarithms and explicitly determine the real and imaginary part of the box integral in all kinematic regions.

The three-loop equal-mass banana integral in ε-factorised form with meromorphic modular forms

We show that the differential equation for the three-loop equal-mass banana integral can be cast into an ε-factorised form with entries constructed from (meromorphic) modular forms and one special function, which can be given as an iterated integral of meromorphic modular forms. The ε-factorised form of the differential equation allows for a systematic solution to any order in the dimensional regularisation parameter ε. The alphabet of the iterated integrals contains six letters.

Complete collection of one-loop triple-collinear splitting operators for dimensionally-regulated QCD

We provide results for the one-loop triple-collinear color/spin-space splitting operators for the five possible processes, q → {qq}^{prime }{overline{q}}^{prime } , q → qqoverline{q} , q → qgg, g → gqoverline{q} and g → ggg. The expressions are exact in dimensionally-regulated massless QCD up to a single integral, which we expand to second order in the dimensional-regularisation parameter. We also evaluate the related splitting functions. Our results are both sufficient and indispensable for the construction of subtraction and integrated-subtraction terms for triple-collinear singularities of one-loop double-real-emission cross-section contributions as part of a next-to-next-to-next-to leading order subtraction scheme.

Expansion by regions with pySecDec

We discuss the technique of expansion by regions from a geometric perspective, and its implementation within pySecDec, a toolbox for the evaluation of dimensionally regulated parameter integrals. The program offers an automated way to perform asymptotic expansions and provides a new mechanism for efficiently evaluating amplitudes, as well as individual integrals. The usage of the new features available within pySecDec is illustrated with several examples. Program summaryProgram Title: pySecDecCPC Library link to program files:https://doi.org/10.17632/dnrkf5jxzh.3Developer's repository link:https://github.com/gudrunhe/secdec, https://secdec.readthedocs.io (online documentation)Licensing provisions: GNU Public License v3Programming language: Python, Form, C++, CudaExternal routines/libraries:GSL [1], NumPy [2], SymPy [3], Nauty [4], Cuba [5], Form [6], GiNaC [7]; optionally Normaliz [8].Journal reference of previous version:Comput. Phys. Commun. 240 (2019) 120–137.Does the new version supersede the previous version?: YesNature of problem: Expansion of Feynman integrals in different kinematic limits, regularisation of ultraviolet, infrared and spurious singularities, numerical integration in the presence of integrable singularities (e.g. kinematic thresholds).Solution method: After specification of the desired kinematic limit by the user (definition of a smallness parameter), the regions contributing to the integral are determined and expansion in the smallness parameter up to the desired order is performed. Extraction of singularities in the dimensional regularization parameter as well as in analytic regulators for potential spurious singularities is done using sector decomposition. This leads to a Laurent series in the regularization parameters, where the coefficients are finite integrals over the unit hypercube. Integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way. The integrals expanded in the different regions are summed and evaluated numerically.

Coaction and double-copy properties of configuration-space integrals at genus zero

We investigate configuration-space integrals over punctured Riemann spheres from the viewpoint of the motivic Galois coaction and double-copy structures generalizing the Kawai-Lewellen-Tye (KLT) relations in string theory. For this purpose, explicit bases of twisted cycles and cocycles are worked out whose orthonormality simplifies the coaction. We present methods to efficiently perform and organize the expansions of configuration-space integrals in the inverse string tension α′ or the dimensional-regularization parameter ϵ of Feynman integrals. Generating-function techniques open up a new perspective on the coaction of multiple polylogarithms in any number of variables and analytic continuations in the unintegrated punctures. We present a compact recursion for a generalized KLT kernel and discuss its origin from intersection numbers of Stasheff polytopes and its implications for correlation functions of two-dimensional conformal field theories. We find a non-trivial example of correlation functions in ( mathfrak{p} , 2) minimal models, which can be normalized to become uniformly transcendental in the mathfrak{p} → ∞ limit.

NNLO zero-jettiness beam and soft functions to higher orders in the dimensional-regularization parameter epsilon

We present the calculation of the next-to-next-to-leading order (NNLO) zero-jettiness beam and soft functions, up to the second order in the expansion in the dimensional regularization parameter epsilon . These higher order terms are needed for the computation of the next-to-next-to-next-to-leading order (hbox {N}^3hbox {LO}) zero-jettiness soft and beam functions. As a byproduct, we confirm the {mathcal {O}}{(epsilon ^0)} results for NNLO beam and soft functions available in the literature by Gaunt et al. (JHEP 4:113, 2014), Gaunt et al. (JHEP 8:20, 2014), Boughezal et al. (Phys Rev D 96:34001, 2017), Monni et al. (JHEP 8:10, 2011) and Kelley et al. (Phys Rev D 84:45022, 2011).

Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach

We calculate the motion of binary mass systems in gravity up to the fourth post–Newtonian order. We use momentum expansions within an effective field theory approach based on Feynman amplitudes in harmonic coordinates by applying dimensional regularization. We construct the canonical transformations to ADM coordinates and to effective one body theory (EOB) to compare with other approaches. We show that intermediate poles in the dimensional regularization parameter ε vanish in the observables and the classical theory is not renormalized. The results are illustrated for a series of observables for which we agree with the literature.

Cusp and Collinear Anomalous Dimensions in Four-Loop QCD from Form Factors.

We calculate the complete quark and gluon cusp anomalous dimensions in four-loop massless QCD analytically from first principles. In addition, we determine the complete matter dependence of the quark and gluon collinear anomalous dimensions. Our approach is to Laurent expand four-loop quark and gluon form factors in the parameter of dimensional regularization. We employ finite field and syzygy techniques to reduce the relevant Feynman integrals to a basis of finite integrals, and subsequently evaluate the basis integrals directly from their standard parametric representations.

The unequal mass sunrise integral expressed through iterated integrals on [formula omitted

We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter ε. In order to do so, we transform the system of differential equations for the master integrals to an ε-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space M1,3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M‾1,3. On the hypersurface τ=const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.

Soft corrections to inclusive deep-inelastic scattering at four loops and beyond

We study the threshold corrections for inclusive deep-inelastic scattering (DIS) and their all-order resummation. Using recent results for the QCD form factor, related anomalous dimensions and Mellin moments of DIS structure functions at four loops we derive the complete soft and collinear contributions to the DIS Wilson coefficients at four loops. For a general SU(nc) gauge group the results are exact in the large-nc approximation and for QCD with nc = 3 we present precise approximations. We extend the threshold resummation exponent GN in Mellin-N space to the fifth logarithmic (N4LL) order collecting the terms {alpha}_{mathrm{s}}^3 (αs ln N)n to all orders in the strong coupling constant αs. We study the numerical effect of the N4LL corrections using both the fully exponentiated form and the expansion of the coefficient function in towers of logarithms. As a byproduct, we derive a numerical result for the complete pole structure of the QCD form factor in the parameter of dimensional regularization ε at four loops.

All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order.

We evaluate analytically all previously unknown nonplanar master integrals for massless five-particle scattering at two loops, using the differential equations method. A canonical form of the differential equations is obtained by identifying integrals with constant leading singularities, in D space-time dimensions. These integrals evaluate to Q-linear combinations of multiple polylogarithms of uniform weight at each order in the expansion in the dimensional regularization parameter and are in agreement with previous conjectures for nonplanar pentagon functions. Our results provide the complete set of two-loop Feynman integrals for any massless 2→3 scattering process, thereby opening up a new level of precision collider phenomenology.

Planar master integrals for four-loop form factors

We present the complete set of planar master integrals relevant to the calculation of three-point functions in four-loop massless Quantum Chromodynamics. Employing direct parametric integrations for a basis of finite integrals, we give analytic results for the Laurent expansion of conventional integrals in the parameter of dimensional regularization through to terms of weight eight.

Analytic Result for a Two-Loop Five-Particle Amplitude.

We compute the symbol of the full-color two-loop five-particle amplitude in N=4 super Yang-Mills theory, including all nonplanar subleading-color terms. The amplitude is written in terms of permutations of Parke-Taylor tree-level amplitudes and pure functions to all orders in the dimensional regularization parameter, in agreement with previous conjectures. The answer has the correct collinear limits and infrared factorization properties, allowing us to define a finite remainder function. We study the multi-Regge limit of the nonplanar terms, analyze its subleading power corrections, and analytically present the leading logarithmic terms.

A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec

The purely numerical evaluation of multi-loop integrals and amplitudes can be a viable alternative to analytic approaches, in particular in the presence of several mass scales, provided sufficient accuracy can be achieved in an acceptable amount of time. For many multi-loop integrals, the fraction of time required to perform the numerical integration is significant and it is therefore beneficial to have efficient and well-implemented numerical integration methods. With this goal in mind, we present a new stand-alone integrator based on the use of (quasi-Monte Carlo) rank-1 shifted lattice rules. For integrals with high variance we also implement a variance reduction algorithm based on fitting a smooth function to the inverse cumulative distribution function of the integrand dimension-by-dimension. Additionally, the new integrator is interfaced to pySecDec to allow the straightforward evaluation of multi-loop integrals and dimensionally regulated parameter integrals. In order to make use of recent advances in parallel computing hardware, our integrator can be used both on CPUs and CUDA compatible GPUs where available. Program summaryProgram Title: pySecDec, qmcProgram Files doi:http://dx.doi.org/10.17632/dnrkf5jxzh.2Licensing provisions: GNU General Public License v3Programming language: python, FORM, C++, CUDAExternal routines/libraries: catch [1], gsl [2], numpy [3], sympy [4], Nauty [5], Cuba [6], FORM [7], Normaliz [8]. The program can also be used in a mode which does not require Normaliz.Journal reference of previous version: Comput. Phys. Commun. 222 (2018) 313–326.Does 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 quantum field theory. 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 ϵ (and optionally other regulators), 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. The parameter integrals forming the coefficients of the Laurent series in the regulator(s) are provided in the form of libraries which can be linked to the calculation of (multi-) loop amplitudes.Restrictions: Depending on the complexity of the problem, limited by memory and CPU/GPU time.

Planar Double Box Integral for Top Pair Production with a Closed Top Loop to all orders in the Dimensional Regularization Parameter.

We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularization parameter ϵ. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in ϵ, where the ϵ^{0} part is strictly lower triangular. This system is easily solved order by order in the dimensional regularization parameter ϵ. This is an example of an elliptic multiscale integral involving several elliptic subtopologies. Our methods are applicable to similar problems.

Analytic results for the planar double box integral relevant to top-pair production with a closed top loop

In this article we give the details on the analytic calculation of the master integrals for the planar double box integral relevant to top-pair production with a closed top loop. We show that these integrals can be computed systematically to all order in the dimensional regularisation parameter ε. This is done by transforming the system of differential equations into a form linear in ε, where the ε0-part is a strictly lower triangular matrix. Explicit results in terms of iterated integrals are presented for the terms relevant to NNLO calculations.