Differential Equations for Two-Loop Four-Point Functions

Abstract

At variance with fully inclusive quantities, which have been computed already at the two- or three-loop level, most exclusive observables are still known only at one-loop, as further progress was hampered so far by the greater computational problems encountered in the study of multi-leg amplitudes beyond one loop. We show in this paper how the use of tools already employed in inclusive calculations can be suitably extended to the computation of loop integrals appearing in the virtual corrections to exclusive observables, namely two-loop four-point functions with massless propagators and up to one off-shell leg. We find that multi-leg integrals, in addition to integration-by-parts identities, obey also identities resulting from Lorentz-invariance. The combined set of these identities can be used to reduce the large number of integrals appearing in an actual calculation to a small number of master integrals. We then write down explicitly the differential equations in the external invariants fulfilled by these master integrals, and point out that the equations can be used as an efficient method of evaluating the master integrals themselves. We outline strategies for the solution of the differential equations, and demonstrate the application of the method on several examples.

26 pages, LaTeX; some explanatory comments added; several typos corrected

References in corpus (1)

• Non-planar massless two-loop Feynman diagrams with four on-shell legs

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• Beam functions for $N$-jettiness at N${}^3$LO in perturbative QCD
• Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions
• The full angle-dependence of the four-loop cusp anomalous dimension in QED
• The light-fermion contribution to the exact Higgs-gluon form factor in QCD
• Radiative Classical Gravitational Observables at $\mathcal{O}(G^3)$ from Scattering Amplitudes
• Triple-real contribution to the quark beam function in QCD at next-to-next-to-next-to-leading order
• Two loop QCD corrections for the process Pseudo-scalar Higgs $\to 3$ partons
• The Four-Loop $\mathcal{N}=4$ SYM Sudakov Form Factor
• Two-Loop QCD Helicity Amplitudes for (2+1)-Jet Production in Deep Inelastic Scattering
• Global Poles of the Two-Loop Six-Point N=4 SYM integrand
• Precision QCD Physics at the LHC
• On the calculation of soft phase space integral
• Non-perturbative computation of lattice correlation functions by differential equations
• Towards Systematic Evaluation of de Sitter Correlators via Generalized Integration-By-Parts Relations
• NNLO QCD Corrections to the Drell-Yan Cross Section in Models of TeV-Scale Gravity
• Factorization of denominators in integration-by-parts reductions
• Cutting massless four-loop propagators
• NLO effects for ${\Omega }_{QQQ}$ Baryons in QCD Sum Rules
• Antenna subtraction at NNLO with identified hadrons
• Laurent series expansion of sunrise-type diagrams using configuration space techniques
• Two-loop master integrals for the mixed QCD-electroweak corrections for $H\to b\bar{b}$ through a $Ht\bar{t}$-coupling
• Neutral and Charged Current Semi-Inclusive Deep-Inelastic Scattering at NNLO QCD
• Recent Progress in Perturbative Quantum Field Theory
• Two-loop massive QCD and QED helicity amplitudes for light-by-light scattering
• Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations
• Two-loop Feynman integrals for leading colour $t\bar{t}W$ production at hadron colliders
• Two-Loop integrals for CP-even heavy quarkonium production and decays
• The propagator seagull: general evaluation of a two loop diagram
• Reduction of Feynman Integrals in the Parametric Representation III: Integrals with Cuts
• Two-Loop QCD Correction to massive spin-2 resonance $\to$ 3 gluons
• Mixed QCD-EW corrections for Higgs leptonic decay via $H{W}^{+}{W}^{-}$ vertex
• Two loop QCD amplitudes for di-pseudo scalar production in gluon fusion
• Resummation methods for Master Integrals
• Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar{t}+$jet production at leading colour
• Multivariate hypergeometric solutions of cosmological (dS) correlators by $\text{d}\log$-form differential equations
• Self-duality from twisted cohomology
• On the reduction of Feynman integrals to master integrals
• Three-loop banana integrals with four unequal masses
• Three loop form factors of a massive spin-2 particle with nonuniversal coupling
• Renormalization of twist-two operators in covariant gauge to three loops in QCD
• Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals
• Factorization of Radiative Leptonic Decays of $B^{-}$ and $D^{-}$ Mesons Including the Soft Photon Region
• Feynman Integrals from Positivity Constraints
• Manifestly Causal Loop-Tree Duality
• Order alpha_s^2 magnetic penguin correction for B decay to light mesons
• The next-to-next-to-leading order soft function for top quark pair production
• Cyclic Mario Worlds -- Color-Decomposition for One-Loop QCD
• Two-loop splitting in double parton distributions: the colour non-singlet case
• Laurent series expansion of a class of massive scalar one-loop integrals up to ${\cal O}(\ep^2)$ in terms of multiple polylogarithms
• Planar master integrals for the two-loop light-fermion electroweak corrections to Higgs plus jet production
• Two-Loop master integrals for heavy-to-light form factors of two different massive fermions
• Bootstrapping Witten diagrams via differential representation in Mellin space
• NNLO QCD corrections to production of a spin-2 particle with non-universal couplings in the DY process
• Recoil corrections to the energy levels of hydrogenic atoms
• The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales
• Bootstrapping spinning two body problem in dynamical Chern-Simons gravity using worldline QFT
• A $QQ\to QQ$ planar doublebox in canonical form
• On canonical differential equations for Calabi-Yau multi-scale Feynman integrals
• The real-virtual antenna functions for $S\to Q\bar{Q}X$ at NNLO QCD
• Analytic evaluation of Feynman graph integrals
• Massive two-loop four-point Feynman integrals at high energies with AsyInt
• Quark mass effects in two-loop Higgs amplitudes
• Systematic approximation of multi-scale Feynman integrals
• Position-space cuts for Wilson line correlators
• Conformal Fields from Neural Networks
• Twisted Riemann bilinear relations and Feynman integrals
• Evaluating double and triple (?) boxes
• Dimensional Regularization and Two-Loop Vacuum Polarization Operator: Master Integrals, Analytic Results and Energy Shifts
• Master integrals for $\mathcal{O}(\alpha {\alpha }_s)$ corrections to $H\to Z{Z}^{\ast }$
• Highly accurate simulations of asymmetric black-hole scattering and cross validation of effective-one-body models
• Single-inclusive hadron production in electron-positron annihilation at next-to-next-to-next-to-leading order in QCD
• Some recent results on evaluating Feynman integrals
• Integration-by-parts identities in FDR
• Two-loop Vertices with Vacuum Polarization Insertion
• Learning Feynman integrals from differential equations with neural networks
• Status of jet cross sections to NNLO
• NNLOCAL: completely local subtractions for color-singlet production in hadron collisions
• Two-Loop Diagrammatics in a Self-Dual Background
• Les Houches 2023 -- Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• The gravitational Compton amplitude from flat and curved spacetimes at second post-Minkowskian order
• The analytic leading color contribution to the Higgs-gluon form factor in QCD at NNLO
• Developments since Kira 2.0
• Differential equations for tree--level cosmological correlators with massive states
• All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering
• Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case
• Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change
• Four-loop large-n_f contributions to the non-singlet structure functions F_2 and F_L
• A first computation of three-loop master integrals for the production of two off-shell vector bosons with different masses
• Harmonic polylogarithms for massive Bhabha scattering
• Two-loop Doubly Massive Four-Point Amplitude Involving a half-BPS and Konishi Operator
• Spin-2 Form Factors at Three Loop in QCD
• Methods for the Reduction of Three-Loop QCD Form Factors
• Two-loop QED Corrections to Bhabha Scattering
• Canonical differential equations and intersection matrices
• RI-(S)MOM to $\overline{\mathrm{MS}}$ conversion for $B_K$ at two-loop order
• Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations
• Non-planar double-box, massive and massless pentabox Feynman integrals in negative dimensional approach
• On the finite basis topologies for multi-loop high-multiplicity Feynman integrals
• The Curious Case of Leading Transcendentality: Three Point Form Factors
• QCD at high energy
• Some methods to evaluate complicated Feynman integrals
• Evaluating the last missing ingredient for the three-loop quark static potential by differential equations
• $Z_c$ and $Z_{cs}$ systems with operator mixing at NLO in QCD sum rules
• Analytic double-soft integrated subtraction terms for two massive emitters in a back-to-back kinematics
• Integral Reduction by Unitarity Method for Two-loop Amplitudes: A Case Study
• Two-Loop Four-Fermion Scattering Amplitude in QED
• Jet Physics at Two-Loop Accuracy
• Two-loop QCD Correction to Massive Spin-2 Resonance $\to q\bar{q}g$
• Classical gravitational scattering from a gauge-invariant double copy
• Reduction to master integrals and transverse integration identities
• Four-dimensional differential equations for the leading divergences of dimensionally-regulated loop integrals
• Electroweak double-box integrals for Moller scattering
• NNLO QED contribution to the $\mu e\to \mu e$ elastic scattering
• Two-Loop QCD-EW Master Integrals for Z Plus Jet Production at Large Transverse Momentum
• QCD: Theoretical Developments
• Master Integrals for Electroweak corrections to $gg\to \gamma \gamma$ -- Light quark contributions
• Fermionic corrections to quark and gluon form factors in four-loop QCD
• Differential equations and high-energy expansion of two--loop diagrams in D dimensions
• Color-octet contributions for $J/\psi$ inclusive production at B factories in soft gluon factorization
• Evaluation of three-loop self-energy master integrals with four or five propagators
• Reduction of Feynman Integrals in the Parametric Representation
• Gluon fragmentation into ${}^3{P}_J^{[1,8]}$ quark pair and test of NRQCD factorization at two-loop level
• CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators
• Calculation of master integrals in terms of elliptic multiple polylogarithms
• On the two-loop amplitude for $gg\to ZZ$ production with full top-mass dependence
• Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes
• The HiggsTools Handbook: Concepts and observables for deciphering the Nature of the Higgs Sector
• On non-factorisable contributions to $t$-channel single-top production
• Kite diagram through Symmetries of Feynman Integrals
• Master Integrals for double real radiation emission in heavy-to-light quark decay
• Integrated Unitarity for Scattering Amplitudes
• Monte Carlo evaluation of divergent one-loop integrals without contour deformation
• Unitarity methods for Mellin moments of Drell-Yan cross sections
• Next-to-next-to-leading order threshold soft function for $tW$ production
• Three-Loop QCD corrections to the production of a Higgs boson and a Jet
• The real photon structure functions in massive parton model in NLO
• Next-to-leading order contributions to the pole mass of gluino in minimal gauge mediation
• Two-loop master integrals for a planar and a non-planar topology relevant for single top production
• The polarized photon structure function $g_1^{\gamma }(x,Q^2)$ in massive parton model in NLO
• Three-loop helicity amplitudes for four-quark scattering in massless QCD
• NNLO mixed EW-QCD corrections to single vector boson production
• Two-photon exchange in leptophilic dark matter scenarios
• Higgs-boson production in top-quark fragmentation
• Jet physics at hadron colliders
• Two-loop non-planar hexa-box integrals with one massive leg
• Two-loop amplitude for mixed QCD-EW corrections to $gg\to Hg$
• The massless non-adjacent double off-shell scalar box integral -- branch cut structure and all-order epsilon expansion
• QCD two-loop amplitudes for e+e- --> 3 jets: the fermionic contribution
• Analytic auxiliary mass flow to compute master integrals in singular kinematics
• Scattering amplitudes for e^+e^- --> 3 jets at next-to-next-to-leading order QCD
• Top-Yukawa contributions to $pp\to b\bar{b}H$: two-loop leading-colour amplitudes
• $\text{history}$: A tool for fully-differential cross sections at next-to-next-to-leading order
• Dihadron Angular Correlations in the $e^{+}{e}^{-}$ Collision
• Integral of the double-emission eikonal function for a massive and a massless emitter at an arbitrary angle
• Evaluating Feynman Integrals through differential equations and series expansions
• One-Loop QCD Corrections to $\bar{u}d\to t\bar{t}W$ at $\mathcal{O}({\epsilon }^2)$
• Pseudoscalar Higgs boson decay to three parton amplitudes at NNLO to higher orders in the dimensional regulator
• Two-parton contribution to the heavy-quark A_FB at NNLO QCD in e+e- collisions
• Triangle diagram, Distance Geometry and Symmetries of Feynman Integrals
• Two-loop scattering amplitude for heavy-quark pair production through light-quark annihilation in QCD
• NNLO QCD corrections to event shape variables in electron positron annihilation
• Two-loop Splitting Amplitudes
• Two-loop QCD corrections to the vector form factors for the heavy-quark photo-production
• One-loop Feynman integrals for $2\to 3$ scattering involving many scales including internal masses
• QCD Beyond Leading Power
• Electroweak corrections to Higgs+jet production in gluon fusion
• Towards NNLO corrections to muon-electron scattering
• Analytic results for two-loop planar master integrals for Bhabha scattering
• Co-homology of Differential Forms and Feynman diagrams
• Two-loop master integrals for pseudo-scalar quarkonium and leptonium production and decay
• The integrand form of infrared singularities of two-loop QCD scattering amplitudes
• The diagrammatic coaction beyond one loop
• Integral representation for three-loop banana graph
• QCD Amplitudes: new perspectives on Feynman integral calculus
• One-loop amplitudes for $t\bar{t}j$ and $t\bar{t}\gamma$ productions at the LHC through $\mathcal{O}({\epsilon }^2)$
• Soft Integrals and Soft Anomalous Dimensions at N${}^3$LO and Beyond
• Two-loop mixed QCD-EW corrections to neutral current Drell-Yan
• Two-loop corrections to $gg\to \gamma \gamma$
• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics
• Two-loop scalar functions with $I$ internal lines, $I\le 5$
• Analytic calculation of Energy-Energy Correlation in $e^{+}{e}^{-}$ annihilation at NLO
• Disentangling singularities with non-linear mappings - a new method for NNLO
• Anomalous Ward identities for on-shell amplitudes at the conformal fixed point
• Two-loop tensor integral coefficients in OpenLoops