Asymptotic expansion of Feynman integrals near threshold

Abstract

We present general prescriptions for the asymptotic expansion of massive multi-loop Feynman integrals near threshold. As in the case of previously known prescriptions for various limits of momenta and masses, the terms of the threshold expansion are associated with subgraphs of a given graph and are explicitly written through Taylor expansions of the corresponding integrands in certain sets of parameters. They are manifestly homogeneous in the threshold expansion parameter, so that the calculation of the given Feynman integral near the threshold reduces to the calculation of integrals of a much simpler type. The general method is illustrated by two-loop two-point and three-point diagrams. We discuss the use of the threshold expansion for problems of physical interest, such as the next-to-next-to-leading order heavy quark production cross sections close to threshold and matching calculations and power counting in non-relativistic effective theories.

24 pages, LaTeX, 4 figures included via epsf.sty

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• Improved Nonrelativistic QCD for Heavy Quark Physics
• Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension
• Generalized Recurrence Relations for Two-loop Propagator Integrals with Arbitrary Masses
• Two-loop Correction to the Leptonic Decay of Quarkonium
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• Large momentum expansion of two-loop self-energy diagrams with arbitrary masses
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• Asymptotic expansions of two-loop Feynman diagrams in the Sudakov limit
• Zero-threshold expansion of two-loop self-energy diagrams
• Threshold behavior of Feynman diagrams: the master two-loop propagator
• Small-threshold behaviour of two-loop self-energy diagrams: two-particle thresholds
• Asymptotic Expansions of Feynman Diagrams on the Mass Shell in Momenta and Masses
• Towards systematic near-threshold calculations in perturbative QFT

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• Non-relativistic and potential non-relativistic effective field theories for scalar mediators
• The QCD Static Energy using Optimal Renormalization and Asymptotic Padé-approximant Methods
• An alternative scheme for effective range corrections in pionless EFT
• Threshold factorization of the Drell-Yan quark-gluon channel and two-loop soft function at next-to-leading power
• A planar four-loop form factor and cusp anomalous dimension in QCD
• Radiation pions in two-nucleon effective field theory
• What Can Possibly Go Wrong?
• Hadronic Atoms
• Asymptotic expansions through the loop-tree duality
• On the calculation of soft phase space integral
• Finite-width effects in unstable-particle production at hadron colliders
• Infrared properties of five-point massive amplitudes in N=4 SYM on the Coulomb branch
• $\mathcal{O}(\alpha {\alpha }_s)$ corrections to the $\gamma t\bar{t}$ vertex at the top quark threshold
• Soft Theorem to Three Loops in QCD and $\mathcal{N}=4$ Super Yang-Mills Theory
• Neutrinoless double beta decay rates in the presence of light sterile neutrinos
• Identifying regions in wide-angle scattering via graph-theoretical approaches
• Resummation of Goldstone Infrared Divergences: A Proof to All Orders
• Dissecting polytopes: Landau singularities and asymptotic expansions in $2\to 2$ scattering
• Quark and gluon two-loop beam functions for leading-jet $p_T$ and slicing at NNLO
• On the contribution of the electromagnetic dipole operator ${\mathcal{O}}_7$ to the ${\bar{B}}_s\to {\mu }^{+}{\mu }^{-}$ decay amplitude
• Threshold expansion for heavy-light systems and flavor off-diagonal current-current correlators
• Effective Lagrangians in bound state calculations
• NLO fragmentation functions for a quark into a spin-singlet quarkonium: Same flavor case
• Parity-Violating Møller Scattering at NNLO: Closed Fermion Loops
• Can $B->X_c\ell \nu$ help us extract |V_{ub}| ?
• Finite Volume Corrections to the Electromagnetic Mass of Composite Particles
• Regge behavior in effective field theory
• Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations
• Recent Progress in Perturbative Quantum Field Theory
• ${\chi }_{QJ}\to {\ell }^{+}{\ell }^{-}$ within and beyond the Standard Model
• Diboson production in the SMEFT from gluon fusion
• The Sommerfeld Enhancement in the Scotogenic Model with Large Electroweak Scalar Multiplets
• A Finite $S$-Matrix
• Two-Loop integrals for CP-even heavy quarkonium production and decays
• Effective Field Theory Islands from Perturbative and Nonperturbative Four-Graviton Amplitudes
• Effective Field Theory with Dibaryon Fields: Nucleon-Nucleon amplitudes at NNLO
• Covariant calculation of a two-loop test of nonrelativistic QCD factorization
• One-loop matching of $CP$-odd four-quark operators to the gradient-flow scheme
• Asymptotic analysis of Feynman diagrams and their maximal cuts
• Three-loop banana integrals with four unequal masses
• Chiral Perturbation Theory in a Nuclear Background
• Total Born cross section of $e^{+}{e}^{-}$-pair production in relativistic ion collisions from differential equations
• Structure Function Sum rules for Systems with Large Scattering Lengths
• Resummation methods for Master Integrals
• Top quark production near threshold
• One-loop effective potential in the non-local supersymmetric gauge theory
• The NNLO soft function for N-jettiness in hadronic collisions
• On supersymmetric gauge theories with higher derivatives and nonlocal terms in the matter sector
• Renormalization group improved pQCD prediction for ${\rm Y}(1S)$ leptonic decay
• A factorization theorem connecting the light-cone distribution amplitudes of heavy-flavor mesons in QCD and HQET
• Gapped Goldstones at the cut-off scale: a non-relativistic EFT
• A factorization approach to next-to-leading-power threshold logarithms
• Details on the $\mathcal{O}(m_e{\alpha }^6)$ Positronium Hyperfine Splitting due to Single Photon Annihilation
• Two-loop QCD corrections to the $V\to q\bar{q}g$ helicity amplitudes with axial-vector couplings
• Special case of sunset: reduction and epsilon-expansion
• Lepton pair production at NNLO in QED with EW effects
• Next-to-leading order QCD corrections for the charmonium production via the channel $e^{+}{e}^{-}\to H(|c\bar{c}⟩)+\gamma$ round the $Z^0$ peak
• Factorization of low-energy gluons in exclusive processes
• Two-loop splitting in double parton distributions: the colour non-singlet case
• Subtracting IR Renormalons from Wilson Coefficients: Uniqueness and power dependences on ${\Lambda }_{\mathrm{QCD}}$
• Calculations of binding energies and masses of heavy quarkonia using renormalon cancellation
• Natural Boundaries for Scattering Amplitudes
• Radiative corrections and Monte Carlo tools for low-energy hadronic cross sections in $e^{+}{e}^{-}$ collisions
• Two-loop corrections to false vacuum decay in scalar field theory
• $Z$ boson radiative decays to a $P$-wave quarkonium at NNLO and LL accuracy
• New Covariant Feynman Rules for Effective Field Theories
• One-loop matching of the $CP$-odd three-gluon operator to the gradient flow
• Heavy meson semileptonic differential decay rate in two dimensions in the large Nc
• $Z$ boson radiative decays to a $S$-wave quarkonium at NNLO and NLL accuracy
• Subleading Effects in Soft-Gluon Emission at One-Loop in Massless QCD
• Recoil corrections to the energy levels of hydrogenic atoms
• Threshold singularities, dispersion relations and fixed-order perturbative calculations
• Theory for quarkonium: from NRQCD factorization to soft gluon factorization
• Resummation of double logarithms in loop-induced processes with effective field theory
• Massive two-loop four-point Feynman integrals at high energies with AsyInt
• Exploring slicing variables for jet processes
• Non-factorizable virtual corrections to Higgs boson production in weak boson fusion beyond the eikonal approximation
• $p$-Forms on the Celestial Sphere
• One-loop Matching of Scotogenic Model onto Standard Model Effective Field Theory up to Dimension 7
• Invariant-mass distribution of top-quark pairs and top-quark mass determination
• The leading twist light-cone distribution amplitudes for the S-wave and P-wave Bc mesons
• Complex-mass renormalization in hadronic EFT: applicability at two-loop order
• QCD radiative corrections to the soft spectator contribution in the wide angle Compton scattering
• The Higgs-gluon form factor at three loops in QCD with three mass scales
• Expansion by regions meets angular integrals
• Higher order hadronic and leptonic contributions to the muon $g-2$
• Complete One-loop Structure of the Type-(I+II) Seesaw Effective Field Theory
• Evaluating double and triple (?) boxes
• New proofs for the two Barnes lemmas and an additional lemma
• Reconciling the nonrelativistic QCD prediction and the $J/\psi \to 3\gamma$ data
• Some recent results on evaluating Feynman integrals
• On the one-loop effective potential in nonlocal supersymmetric theories
• Three-loop non-singlet matching coefficients for heavy quark currents
• XEFT, the challenging path up the hill: dim = 6 and dim = 8
• Next-to-leading order QCD corrections to the production of $B_c$ and $B_c^{\ast }$ through $W^{+}$-boson decays
• Matching the heavy-quark fields in QCD and HQET at four loops
• $\mathcal{O}({\alpha }_s^2)$ corrections to $J/\psi +{\chi }_{c0,1,2}$ production at $B$ factories
• $P$-wave quarkonium wavefunctions at the origin in the $\overline{\mathrm{MS}}$ scheme
• Multiple soft-photon emission at next-to-leading power to all orders
• Factorizing the hard and soft spectator scattering contributions for the nucleon form factor F_1 at large Q^2
• Heavy Quarkonium Physics beyond the Next-to-Next-to-Leading Order of NRQCD
• Next-to-leading-order QCD corrections to a vector bottomonium radiative decay into a charmonium
• Chiral Perturbation Theory Reflections on Effective Theories of the Standard Model
• One-loop corrections to the double-real emission contribution to the zero-jettiness soft function at N3LO in QCD
• Learning Feynman integrals from differential equations with neural networks
• Relativistic corrections to the static energy in terms of Wilson loops at weak coupling
• Post-Newtonian Dynamics of Spinning Black Hole Binaries in Einstein-Scalar-Gauss-Bonnet Gravity
• Master integrals for bipartite cuts of three-loop photon self energy
• Probing heavy meson lightcone distribution amplitudes with heavy quark spin symmetry
• Once more on the Higgs decay into two photons
• General Relativity from Intersection Theory
• Manifestly Soft Gauge Invariant Formulation of vNRQCD
• Renormalon-free definition of the gluon condensate within the large-${\beta }_0$ approximation
• On renormalons of static QCD potential at $u=1/2$ and $3/2$
• Double-Real-Virtual and Double-Virtual-Real Corrections to the Three-Loop Thrust Soft Function
• Decay constant of $B_c^{\ast }$ accurate up to $\mathcal{O}({\alpha }_s^3)$
• Charm-quark mass effects in NRQCD matching coefficients and the leptonic decay of the ${\rm Y}(1S)$ meson
• Structure-dependent QED effects in exclusive $B$-meson decays
• Soft-Collinear Supersymmetry
• The method of expansion of Feynman integrals
• Removing phase-space restrictions in factorized cross sections
• UV contribution and power dependence on ${\Lambda }_{\mathrm{QCD}}$ of Adler function
• Top quark mass definition and top quark pair production near threshold at the NLC
• Small-angle scattering and quasiclassical approximation beyond leading order
• $\mathcal{O}({\alpha }_s{v}^2)$ corrections to hadronic decay of vector quarkonia
• The Sky Remembers everything: Celestial amplitude, Shadow and OPE in quadratic EFT of gravity
• Expansion by regions with pySecDec
• Towards pair production near threshold with unstable particle effective theory
• OPITeR: A program for tensor reduction of multi-loop Feynman Integrals
• Tensor Reduction for Feynman Integrals with Lorentz and Spinor Indices
• Machine Learning Post-Minkowskian Integrals
• QCD one-loop correction to Higgs boson decay into quarkonium-pair
• Specific three-loop contributions to $b\to s\gamma$ associated with the current-current operators
• NNNLO correction to the toponium and bottomonium wave-functions at the origin
• Evaluating the last missing ingredient for the three-loop quark static potential by differential equations
• Heavy Quarkonium and Nonrelativistic Effective Field Theories
• $(Z\alpha {)}^4$ order of the polarization operator in Coulomb field at low energy
• The SCET_II and factorization
• Observation prospects of leptonic and Dalitz decays of pseudoscalar quarkonia
• NNLO electron structure functions (PDFs) from SCET
• Analytical evaluation of certain on-shell two-loop three-point diagrams
• Region analysis of $H\to \gamma \gamma$ via a bottom quark loop
• Angular distribution of thrust axis with power-suppressed contribution in e+e- annihilation
• Generalized eikonal identities for charged currents
• SM single-top production at hadron colliders
• Mass Dependence of Higgs Production at Large Transverse Momentum
• Five-loop massless propagator integrals
• A Study of Power Suppressed Contributions in $J/\psi \to p\bar{p}$ Decay
• New Physics Search in Flavour Physics
• Supersymmetric structure of electroweak Sudakov corrections
• Resummation of terms enhanced by trilinear squark-Higgs couplings in the MSSM
• Inclusive production of heavy quarkonium ${\eta }_Q$ via $Z$ boson decays within the framework of nonrelativistic QCD
• Renormalon Subtraction in OPE by Dual Space Approach: Nonlinear Sigma Model and QCD
• An Elliptic One-Loop Amplitude in Anti-de-Sitter Space
• The $q_{\mathrm{T}}$ and $\Delta {\varphi }_{t\bar{t}}$ spectra in top-antitop hadroproduction at NNLL+NNLO: the interplay of soft-collinear resummation and Coulomb singularities
• Disperon QED
• Classical gravitational scattering from a gauge-invariant double copy
• Energy and decay width of the pi-K atom
• Two-Loop QCD-EW Master Integrals for Z Plus Jet Production at Large Transverse Momentum
• Two-loop corrections to Lamb shift and hyperfine splitting in hydrogen via multi-loop methods
• Decoupling heavy particles simultaneously
• Soft-Collinear Factorization and Sudakov Resummation of Heavy Meson Decay Amplitudes with Effective Field Theories
• Dark origin of quark flavor hierarchy and mixing
• Non-relativistic high-energy physics: top production and dark matter annihilation
• Mass Dependence of Higgs Production at Large Transverse Momentum through a Bottom Quark Loop
• Photon pair production in gluon fusion: Top quark effects at NLO with threshold matching
• Threshold corrections to the gamma t anti-t vertex at O(alpha alpha(s))
• Two-loop mixed QCD-electroweak amplitudes for $Z+$jet production at the LHC: bosonic corrections
• Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes
• Three-Loop OPE Wilson Coefficients of Dimension-Four Operators for (Axial-)Vector and (Pseudo-)Scalar Current Correlators
• Color-octet contributions for $J/\psi$ inclusive production at B factories in soft gluon factorization
• Running couplings for the simultaneous decoupling of heavy quarks
• Two loop QCD corrections to $e^{+}{e}^{-}\to J/\psi +{\eta }_c$ in asymptotic expansion
• Matching the Standard Model to Heavy-Quark Effective Theory and Nonrelativistic QCD
• Radiative decays ${\chi }_{cJ}\to V\gamma$ within the QCD factorisation framework
• Sudakov Logarithms in Four-Fermion Electroweak Processes at High Energy
• Correction to the energy spectrum of ${}^1{S}_0$ heavy quarkonia due to two-gluon annihilation effect
• The HiggsTools Handbook: Concepts and observables for deciphering the Nature of the Higgs Sector
• Threshold Resummation for Dark-Matter Production at the LHC
• On multi-propagator angular integrals
• The hierarchical three-body problem at $\mathcal{O}(G^2)$
• One-loop corrections of order (Z alpha)^6m_1/m_2, (Z alpha)^7 to the muonium fine structure
• The method of direct expansions of Feynman integrals
• NLO QCD Corrections to Inclusive Charmonium and $B_c$ Meson Production in $W^{+}$ Decays
• Next-to-leading order QCD corrections to $Z\to {\eta }_Q+Q+\bar{Q}$
• Improved analysis of double $J/\psi$ production in $Z$-boson decay
• Short-range force between two Higgs bosons
• Automated One-loop Computation in Quarkonium Process within NRQCD Framework
• An improved calculation of the $D_{(s)}^{\ast }{D}_{(s)}V$ and $B_{(s)}^{\ast }{B}_{(s)}V$ couplings from light-cone sum rules
• $B_c\to {\eta }_c$ form factors at large recoil: Interplay of soft-quark and soft-gluon dynamics
• Unregulated Divergences of Feynman Integrals
• The High-Temperature Expansion of the Thermal Sunset
• A new method for finding more symmetry relations of Feynman integrals
• Analytic Result of Higgs Boson Decay to Gluons with Full Quark Mass Dependence
• Radiative distortion of kinematic edges in cascade decays
• NRQCD, Effective Field Theories and Potential Models
• Evaluating master integrals in non-factorizable corrections to $t$-channel single-top production at NNLO QCD
• QCD corrections of $e^{+}{e}^{-}\to J/\psi +c+\bar{c}$ using the principle of maximum conformality
• The soft-energy region in the radiative decay of bound states
• From Models to SMEFT and Back?
• Scattering Amplitudes for Binary Systems beyond GR
• Renormalization group analysis of electromagnetic properties of the deuteron
• Evaluating Feynman Integrals through differential equations and series expansions
• Symmetry Restoration in the SMEFT: Finite Counterterms for a Non-Anticommuting ${\gamma }_5$
• Role of Resonances in Heavy Meson Processes within Standard Model and Beyond
• Heavy Quarkonium at finite temperature and chemical potential
• Analytic auxiliary mass flow to compute master integrals in singular kinematics
• Singlet and non-singlet three-loop massive form factors
• Universal structure of radiative QED amplitudes at one loop
• Next-to-leading power two-loop soft functions for the Drell-Yan process at threshold
• One Formula To Match Them All: The Bispinor Universal One-Loop Effective Action
• Applications of QCD
• NNLO corrections to false vacuum decay rate in thin-wall approximation
• Exclusive decays ${\chi }_{cJ}\to K^{\ast }(892)K$ within the effective field theory framework
• Gauge dependence and matching procedure of a nonrelativistic QED/QCD boundstate formalism
• An Effective Field Theory Look at Deep Inelastic Scattering
• Scattering Amplitudes in Gauge Theories
• Unstable-particle effective field theory
• Nonrelativistic effective field theory of unstable top
• Effective Field Theories for the LHC
• A theoretical update on t \bar{t} production near threshold
• $\mathcal{O}({\alpha }_s)$ corrections to $e^{+}{e}^{-}\to J/\psi +{\eta }_{c2}({\chi }_{c1}^{\prime })$ at $B$ factories
• One-loop matching of the LEFT to the QCD gradient flow
• Effective string theory for vortex lines in fluids and superfluids
• Quantum dynamics of perfect fluids
• The Soft Régime and βFunction of NRQCD
• Three-loop vertex corrections
• Large mass expansion in two-loop QCD corrections of para-charmonium decay
• CoLoRFulNNLO for hadron collisions: integrating the iterated single unresolved subtraction terms
• The analytic two-loop soft function for leading-jet $p_T$
• Soft Integrals and Soft Anomalous Dimensions at N${}^3$LO and Beyond
• Mixed QCD-EW corrections to $W$-pair production at electron-positron colliders
• Subleading Effects in Soft-Gluon Emission at One-Loop in Massive QCD
• A stroll through the loop-tree duality
• Some recent progress in understanding exclusive double charmonium production at $B$ factories
• Anomalous Ward identities for on-shell amplitudes at the conformal fixed point
• Effective Angular Momentum Operators in NRQED and Matching at One-Loop Order
• A coordinate description of partonic processes
• QCD Beyond Leading Power
• NLO electroweak potentials for minimal dark matter and beyond
• Higher-Order Corrections to Higgs Boson Amplitudes with Full Quark Mass Dependence in Quantum Chromodynamics
• An Effective Field Theory of Magneto-Elasticity
• Potential Nonrelativistic QCD and Heavy Quarkonium Spectrum in Next-to-Next-to-Next-to-Leading Order
• Photon pair production in gluon fusion: Top quark effects around threshold
• QCD calculations of radiative heavy meson decays with subleading power corrections
• Impact on the decay rate of $B_s\to {\mu }^{+}{\mu }^{-}$ from the dispersive two-photon transition
• $\sigma (e^{+}{e}^{-}\to hadrons)$ and the Heavy Quark Masses
• Dark Matter Recoupling
• Higgs boson decay to massive bottom quarks at order ${\alpha }_s^4$ induced by top-quark Yukawa couplings