Multiloop integrals in dimensional regularization made simple

Abstract

Scattering amplitudes at loop level can be expressed in terms of Feynman integrals. The latter satisfy partial differential equations in the kinematical variables. We argue that a good choice of basis for (multi-)loop integrals can lead to significant simplifications of the differential equations, and propose criteria for finding an optimal basis. This builds on experience obtained in supersymmetric field theories that can be applied successfully to generic quantum field theory integrals. It involves studying leading singularities and explicit integral representations. When the differential equations are cast into canonical form, their solution becomes elementary. The class of functions involved is easily identified, and the solution can be written down to any desired order in epsilon within dimensional regularization. Results obtained in this way are particularly simple and compact. In this letter, we outline the general ideas of the method and apply them to a two-loop example.

4 pages, 1 figure; v2: discussion of boundary conditions improved, minor changes, to be published in PRL

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• Renormalization of twist-two operators in covariant gauge to three loops in QCD
• The next-to-next-to-leading order soft function for top quark pair production
• Feynman Integrals from Positivity Constraints
• A Calabi-Yau-to-Curve Correspondence for Feynman Integrals
• Two-loop splitting in double parton distributions: the colour non-singlet case
• Four-loop QCD MOM beta functions from the three-loop vertices at the symmetric point
• Analytic decay width of the Higgs boson to massive bottom quarks at next-to-next-to-leading order in QCD
• Conservative Scattering of Reissner-Nordström Black Holes at Third Post-Minkowskian Order
• Natural Boundaries for Scattering Amplitudes
• A $QQ\to QQ$ planar doublebox in canonical form
• Two dialects for KZB equations: generating one-loop open-string integrals
• Bootstrapping spinning two body problem in dynamical Chern-Simons gravity using worldline QFT
• Planar master integrals for the two-loop light-fermion electroweak corrections to Higgs plus jet production
• Alphabet of one-loop Feynman integrals
• Intersection Numbers from Companion Tensor Algebra
• Multi-loop Integrand Reduction with Computational Algebraic Geometry
• Analytic decay width of the Higgs boson to massive bottom quarks at order ${\alpha }_s^3$
• Two-Loop master integrals for heavy-to-light form factors of two different massive fermions
• Total cross sections of $e\gamma \to eX\bar{X}$ processes with $X=\mu ,\gamma ,e$ via multiloop methods
• Position-space cuts for Wilson line correlators
• Massive two-loop four-point Feynman integrals at high energies with AsyInt
• Anomalies without an action
• Non-planar elliptic vertex
• Non-factorizable virtual corrections to Higgs boson production in weak boson fusion beyond the eikonal approximation
• Quark mass effects in two-loop Higgs amplitudes
• Maximal transcendental weight contribution of scattering amplitudes
• Systematic approximation of multi-scale Feynman integrals
• On series and integral representations of some NRQCD master integrals
• Twisted Riemann bilinear relations and Feynman integrals
• One-loop squared amplitudes for hadronic $tW$ production at next-to-next-to-leading order in QCD
• Electron-positron annihilation to photons at $O({\alpha }^3)$ revisited
• Analytic two-loop amplitudes for $tW$ production: leading color and light fermion-loop contributions
• Highly accurate simulations of asymmetric black-hole scattering and cross validation of effective-one-body models
• Integration-by-parts identities in FDR
• The recursive structure of Baikov representations II: the top-down reduction with intersection theory
• Single-inclusive hadron production in electron-positron annihilation at next-to-next-to-next-to-leading order in QCD
• Fragmentation function of $g\to Q\bar{Q}{(}^3{S}_1^{[8]})$ in soft gluon factorization and threshold resummation
• A Note on Kinematic Flow and Differential Equations for Two-Site One-Loop Graph in FRW Spacetime
• Three-loop helicity amplitudes for photon+jet production
• Matching the heavy-quark fields in QCD and HQET at four loops
• Two-loop Vertices with Vacuum Polarization Insertion
• Master integrals for $\mathcal{O}(\alpha {\alpha }_s)$ corrections to $H\to Z{Z}^{\ast }$
• Master integrals for bipartite cuts of three-loop photon self energy
• Three-loop planar master integrals for heavy-to-light form factors
• Double-Real-Virtual and Double-Virtual-Real Corrections to the Three-Loop Thrust Soft Function
• One-loop corrections to the double-real emission contribution to the zero-jettiness soft function at N3LO in QCD
• Fuchsia and Master Integrals for Energy-Energy Correlations at NLO in QCD
• A toolbox to solve coupled systems of differential and difference equations
• Learning Feynman integrals from differential equations with neural networks
• Two-loop triangle integrals with 4 scales for the $HZV$ vertex
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• Deciphering the Maximal Transcendentality Principle via Bootstrap
• The analytic leading color contribution to the Higgs-gluon form factor in QCD at NNLO
• Analytic two-loop master integrals for $tW$ production at hadron colliders. Part II
• Expansion of EYM Amplitudes in Gauge Invariant Vector Space
• All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering
• Differential equations for tree--level cosmological correlators with massive states
• A first computation of three-loop master integrals for the production of two off-shell vector bosons with different masses
• Double virtual QCD corrections to $t\bar{t}+$jet production at the LHC
• Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case
• Evaluating the last missing ingredient for the three-loop quark static potential by differential equations
• On the finite basis topologies for multi-loop high-multiplicity Feynman integrals
• A new reduction strategy for special negative sectors of planar two-loop integrals without Laporta algorithm
• Logarithmic Singularities and Maximally Supersymmetric Amplitudes
• The Curious Case of Leading Transcendentality: Three Point Form Factors
• Open-string integrals with multiple unintegrated punctures at genus one
• Analytic double-soft integrated subtraction terms for two massive emitters in a back-to-back kinematics
• $Z_c$ and $Z_{cs}$ systems with operator mixing at NLO in QCD sum rules
• Three-loop vertex integrals at symmetric point
• Region analysis of $H\to \gamma \gamma$ via a bottom quark loop
• Illustrations of Integrand-Basis Building at Two Loops
• Specific three-loop contributions to $b\to s\gamma$ associated with the current-current operators
• A high-precision result for a full-color three-loop three-point form factor in $\mathcal{N}=4$ SYM
• A Feynman Integral and its Recurrences and Associators
• A walk on sunset boulevard
• Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations
• Canonical differential equations and intersection matrices
• Master integrals for mixed QCD-QED corrections to charged-current Drell-Yan production of a massive charged lepton
• An Elliptic One-Loop Amplitude in Anti-de-Sitter Space
• Classical gravitational scattering from a gauge-invariant double copy
• Electroweak double-box integrals for Moller scattering
• Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study
• Summary of the Topical Workshop on Top Quark Differential Distributions 2014
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Three-loop effective potential of general scalar theory via differential equations
• Three-loop master integrals for the Higgs boson self-energy with internal top-quarks and W-bosons
• Four-dimensional differential equations for the leading divergences of dimensionally-regulated loop integrals
• Fermionic corrections to quark and gluon form factors in four-loop QCD
• NNLO QED contribution to the $\mu e\to \mu e$ elastic scattering
• Half-BPS half-BPS twist two at four loops in N=4 SYM
• Multiloop integrals made simple: applications to QCD processes
• Type II superstring amplitude at one-loop and transcendentality
• Analytic NNLO QCD corrections to top quark pair production in electron-positron collisions
• Master Integrals for Electroweak corrections to $gg\to \gamma \gamma$ -- Light quark contributions
• Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes
• One-loop pentagon integral with one offshell leg in $d$ dimensions from differential equations in $\epsilon$-form
• Reclassifying Feynman integrals as special functions
• Gluon fragmentation into ${}^3{P}_J^{[1,8]}$ quark pair and test of NRQCD factorization at two-loop level
• Calculation of master integrals in terms of elliptic multiple polylogarithms
• Two-Loop QCD-EW Master Integrals for Z Plus Jet Production at Large Transverse Momentum
• CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators
• Color-octet contributions for $J/\psi$ inclusive production at B factories in soft gluon factorization
• Cuts and coproducts of massive triangle diagrams
• The nonplanar cusp and collinear anomalous dimension at four loops in $\mathcal{N}=4$ SYM theory
• Rationalizing roots: an algorithmic approach
• On the two-loop amplitude for $gg\to ZZ$ production with full top-mass dependence
• Analytic Result of Higgs Boson Decay to Gluons with Full Quark Mass Dependence
• Three-Loop QCD corrections to the production of a Higgs boson and a Jet
• MultivariateResidues - a Mathematica package for computing multivariate residues
• Master Integrals for double real radiation emission in heavy-to-light quark decay
• A new method for finding more symmetry relations of Feynman integrals
• Two-loop master integrals for a planar and a non-planar topology relevant for single top production
• Monte Carlo evaluation of divergent one-loop integrals without contour deformation
• Integrated Unitarity for Scattering Amplitudes
• 3-Loop Corrections to the Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering
• HQET vertex diagram: $\epsilon$ expansion
• Multi-loop calculations: numerical methods and applications
• The HiggsTools Handbook: Concepts and observables for deciphering the Nature of the Higgs Sector
• QCD and High Energy Interactions: Moriond 2016 Theory Summary
• Two-loop infrared singularities in the production of a Higgs boson associated with a top-quark pair
• pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning
• Three-loop helicity amplitudes for four-quark scattering in massless QCD
• Next-to-next-to-leading order threshold soft function for $tW$ production
• Top-Yukawa contributions to $pp\to b\bar{b}H$: two-loop leading-colour amplitudes
• Massive Two-Loop Heavy Particle Diagrams
• Two-photon exchange in leptophilic dark matter scenarios
• Charge asymmetry in electron/positron energy loss in nuclear Bremsstrahlung
• Analytic Four-Point Lightlike Form Factors and OPE of Null-Wrapped Polygons
• Charge asymmetry in the spectra of bremsstrahlung and pair production
• One-Loop QCD Corrections to $\bar{u}d\to t\bar{t}W$ at $\mathcal{O}({\epsilon }^2)$
• Analytic auxiliary mass flow to compute master integrals in singular kinematics
• From elliptic curves to Feynman integrals
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• Two-loop non-planar hexa-box integrals with one massive leg
• Dihadron Angular Correlations in the $e^{+}{e}^{-}$ Collision
• $\text{history}$: A tool for fully-differential cross sections at next-to-next-to-leading order
• Universal Decomposition of Phase-Space Integrands
• Next-to-leading power two-loop soft functions for the Drell-Yan process at threshold
• Feynman integrals at large loop order and the $\log$-$\Gamma$ distribution
• Differential equations for Feynman integrals beyond multiple polylogarithms
• One-loop Feynman integrals for $2\to 3$ scattering involving many scales including internal masses
• Baryons and baryoniums in the perspective of QCD sum rules
• Higgs-boson production in top-quark fragmentation
• The spectrum of Feynman-integral geometries at two loops
• Three loop QCD corrections to heavy quark form factors
• Two-loop tensor integral coefficients in OpenLoops
• Analytic calculation of Energy-Energy Correlation in $e^{+}{e}^{-}$ annihilation at NLO
• Multi-parton contributions to $\bar{B}\to X_s\gamma$ at NLO
• Electroweak corrections to Higgs+jet production in gluon fusion
• The canonical form has got baubles
• Analytic results for two-loop planar master integrals for Bhabha scattering
• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics
• Co-homology of Differential Forms and Feynman diagrams
• Tropical Feynman integration in the physical region
• The diagrammatic coaction beyond one loop
• Towards NNLO corrections to muon-electron scattering
• The integrand form of infrared singularities of two-loop QCD scattering amplitudes
• Integral representation for three-loop banana graph
• Higher-order corrections to the splitting functions from differential equations in QCD
• Mixed QCD-EW corrections to $W$-pair production at electron-positron colliders
• QCD Beyond Leading Power
• One-loop amplitudes for $t\bar{t}j$ and $t\bar{t}\gamma$ productions at the LHC through $\mathcal{O}({\epsilon }^2)$
• Three-loop four-particle QCD amplitudes
• QCD Amplitudes: new perspectives on Feynman integral calculus
• The Stratification of Rigidity
• NNLO real corrections to $gg\to HH$ in the large-$m_t$ limit
• On the Four-Loop Higgs--Gluon Form Factor in Nonlocal Quantum Field Theory
• Non-leptonic B-decays at two loops in QCD Factorization
• Master integrals for two-loop $C$-odd contribution to $e^{+}{e}^{-}\to {\ell }^{+}{\ell }^{-}$ process
• Higgs boson decay to massive bottom quarks at order ${\alpha }_s^4$ induced by top-quark Yukawa couplings
• Anomalous Ward identities for on-shell amplitudes at the conformal fixed point
• A Feynman integral depending on two elliptic curves
• Soft Integrals and Soft Anomalous Dimensions at N${}^3$LO and Beyond
• Recent new methods and applications of the differential equation approach to master integrals
• CoLoRFulNNLO for hadron collisions: integrating the iterated single unresolved subtraction terms
• Recent progress for five-particle two-loop scattering amplitudes with an off-shell leg
• Two-loop scalar functions with $I$ internal lines, $I\le 5$