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