Evaluating single-scale and/or non-planar diagrams by differential equations

Abstract

We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are form-factor integrals with two external legs on the light cone. We introduce one more scale by taking one more leg off-shell, $p_2^2\ne 0$. We analytically solve the differential equations for the master integrals in a Laurent expansion in dimensional regularization with $\epsilon =(4-D)/2$. Then we show how to obtain analytic results for the corresponding one-scale integrals in an algebraic way. An essential ingredient of our method is to match solutions of the differential equations in the limit of small $p_2^2$ to our results at $p_2^2\ne 0$ and to identify various terms in these solutions according to expansion by regions. The second family consists of four-point non-planar integrals with all four legs on the light cone. We evaluate, by differential equations, all the master integrals for the so-called $K_4$ graph consisting of four external vertices which are connected with each other by six lines. We show how the boundary constants can be fixed with the help of the knowledge of the singular limits. We present results in terms of harmonic polylogarithms for the corresponding seven master integrals with six propagators in a Laurent expansion in $\epsilon$ up to weight six.

27 pages, 2 figures

References in corpus (13)

• Algorithm FIRE -- Feynman Integral REduction
• Quark and gluon form factors to three loops
• Three-Loop Superfiniteness of N=8 Supergravity
• Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA)
• The massless higher-loop two-point function
• On the Resolution of Singularities of Multiple Mellin-Barnes Integrals
• Expansion by regions: revealing potential and Glauber regions automatically
• FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions
• Geometric approach to asymptotic expansion of Feynman integrals
• New differential equations for on-shell loop integrals
• Nine-Propagator Master Integrals for Massless Three-Loop Form Factors
• Master Integrals for Fermionic Contributions to Massless Three-Loop Form Factors
• Analytic Epsilon Expansions of Master Integrals Corresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-Loop Transcendentality Weight

Cited by in corpus (73)

• FIRE5: a C++ implementation of Feynman Integral REduction
• Lectures on differential equations for Feynman integrals
• Scattering Amplitudes and Conservative Binary Dynamics at $\mathcal{O}(G^4)$
• Reducing differential equations for multiloop master integrals
• Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at $O(G^4)$
• Magnus and Dyson Series for Master Integrals
• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions
• Elliptic Feynman integrals and pure functions
• Iterative structure of finite loop integrals
• Pentagon Functions for Scattering of Five Massless Particles
• Constructing d-log integrands and computing master integrals for three-loop four-particle scattering
• The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions
• Feynman integrals and hyperlogarithms
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• On hyperlogarithms and Feynman integrals with divergences and many scales
• Four-loop photon quark form factor and cusp anomalous dimension in the large-$N_c$ limit of QCD
• Four-gluon scattering at three loops, infrared structure and Regge limit
• Solving differential equations for Feynman integrals by expansions near singular points
• Matter dependence of the four-loop cusp anomalous dimension
• Analytic result for the nonplanar hexa-box integrals
• Cluster algebras for Feynman integrals
• Deriving canonical differential equations for Feynman integrals from a single uniform weight integral
• From multiple unitarity cuts to the coproduct of Feynman integrals
• Four-Loop Nonplanar Cusp Anomalous Dimension in N=4 Supersymmetric Yang-Mills Theory
• Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons
• Three-loop master integrals for ladder-box diagrams with one massive leg
• Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering
• Four-loop quark form factor with quartic fundamental colour factor
• On the Computation of Form Factors in Massless QCD with Finite Master Integrals
• Next-to-Leading Order QCD Corrections to the Decay Width $H\to Z\gamma$
• The Sudakov form factor at four loops in maximal super Yang-Mills theory
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Simplified differential equations approach for Master Integrals
• Quark and gluon form factors in four-loop QCD
• Evaluation of the general 3-loop vacuum Feynman integral
• Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Master integrals for the four-loop Sudakov form factor
• Master integrals for the two-loop penguin contribution in non-leptonic B-decays
• Four-graviton scattering to three loops in $\mathcal{N}=8$ supergravity
• Real-Virtual-Virtual contributions to the inclusive Higgs cross section at N3LO
• MPL - a program for computations with iterated integrals on moduli spaces of curves of genus zero
• Adequate bases of phase space master integrals for $gg\to h$ at NNLO and beyond
• Differential equations on unitarity cut surfaces
• Transverse-momentum-dependent wave functions and Soft functions at one-loop in Large Momentum Effective Theory
• Analytic results for planar three-loop integrals for massive form factors
• Numerical Multi-Loop Calculations via Finite Integrals and One-Mass EW-QCD Drell-Yan Master Integrals
• Subleading Regge limit from a soft anomalous dimension
• First look at the evaluation of three-loop non-planar Feynman diagrams for Higgs plus jet production
• Master Integrals for two-loop QCD corrections to Quasi PDFs
• Master Integrals for the two-loop, non-planar QCD corrections to top-quark pair production in the quark-annihilation channel
• Two-Loop integrals for CP-even heavy quarkonium production and decays: Elliptic Sectors
• Same-hemisphere three-gluon-emission contribution to the zero-jettiness soft function at N3LO QCD
• Planar master integrals for four-loop form factors
• Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions
• Two-loop master integrals for non-leptonic heavy-to-heavy decays
• Scattering Amplitudes and N-Body Post-Minkowskian Hamiltonians in General Relativity and Beyond
• Soft Theorem to Three Loops in QCD and $\mathcal{N}=4$ Super Yang-Mills Theory
• On the calculation of soft phase space integral
• Two-Loop integrals for CP-even heavy quarkonium production and decays
• Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations
• Two-Loop master integrals for heavy-to-light form factors of two different massive fermions
• A first computation of three-loop master integrals for the production of two off-shell vector bosons with different masses
• Evaluating the last missing ingredient for the three-loop quark static potential by differential equations
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study
• All-loop-orders relation between Regge limits of $\mathcal{N}=4$ SYM and $\mathcal{N}=8$ supergravity four-point amplitudes
• Fermionic corrections to quark and gluon form factors in four-loop QCD
• Generalizations of polylogarithms for Feynman integrals