Deriving canonical differential equations for Feynman integrals from a single uniform weight integral

243

We present the computation of a full set of planar five-point two-loop master integrals with one external mass. These integrals are an important ingredient for two-loop scattering amplitudes for two-jet-associated W-boson production at leading color in QCD. We provide a set of pure integrals together with differential equations in canonical form. We obtain analytic differential equations efficiently from numerical samples over finite fields, fitting an ansatz built from symbol letters. The symbol alphabet itself is constructed from cut differential equations and we find that it can be written in a remarkably compact form. We comment on the analytic properties of the integrals and confirm the extended Steinmann relations, which govern the double discontinuities of Feynman integrals, to all orders in ϵ. We solve the differential equations in terms of generalized power series on single-parameter contours in the space of Mandelstam invariants. This form of the solution trivializes the analytic continuation and the integrals can be evaluated in all kinematic regions with arbitrary numerical precision.

Section: Constructing Pure Master Integralsmentioning

confidence: 99%

“…Devising an effective D-dimensional algorithm to find such a basis is an active field of research (see e.g. [33,[42][43][44][45][46][47][48][49][50][51]). Here we solve this problem by constructing the basis with a heuristic approach, which we then validate by constructing the differential equation and observing the canonical form.…”

Section: Introductionmentioning

confidence: 99%

Two-loop integrals for planar five-point one-mass processes

Abreu

Ita

Moriello

et al. 2020

243

“…A UT basis can be also constructed via Baikov analysis [58], and systematically via the dlog form in a general representation and the intersection theory [59]. And recently, it was discovered that the full UT basis from only one UT integral [60].…”

Section: )mentioning

confidence: 99%

IBP reduction coefficients made simple

Boehm

Wittmann

et al. 2020

We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, we observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ∼ 100. We observe that our algorithm also works well for settings without a UT basis.

“…The master integrals for massless five-point scattering processes have been a subject of extensive studies in recent years. The method of differential equations (DE) [36][37][38][39][40] in their canonical form [41][42][43][44][45], and systematic understanding of the transcendental functions appearing in calculations of multi-scale Feynman integrals [46][47][48][49][50] proved to be indispensable to obtain analytic results for five-point massless master integrals for planar [51][52][53] and non-planar [27,30,[54][55][56] topologies. Differential equations in canonical form provide a natural framework for expressing master integrals in terms of functions of uniform transcendental (UT) weight order by order in the dimensional regulator.…”

Section: Introductionmentioning

confidence: 99%

Pentagon functions for scattering of five massless particles

Chicherin

Sotnikov

2020

103

118

We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.