A symbol bootstrap using leading singularities determines the maximal-weight symbol of the planar two-loop six-gluon amplitude in massless QCD for MHV configurations.
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Multiloop integrals in dimensional regularization made simple
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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.
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representative citing papers
Machine learning discovers a tube-seeding strategy for IBP reduction of Feynman integrals that scales linearly with numerator power, demonstrated on rank-20 2-loop 5-point integrals.
Two-loop Feynman integrals involve Riemann spheres, elliptic curves, hyperelliptic curves of genus 2 and 3, K3 surfaces, and a rationalizable Del Pezzo surface of degree 2.
At 5PM-1SF order, Calabi-Yau three-fold periods emerge in radiation-reacted observables for classical black hole scattering computed with worldline QFT and advanced IBP/DE methods.
Nonlocal-in-time conservative tail contributions to gravitational scattering are derived at 5PM and 10SF orders, expressed via polylogarithms up to weight three and agreeing with prior results through 6PN.
Four-loop non-singlet splitting functions in QCD are computed analytically for the first time, with numerical representations provided.
Derives gravitational Compton amplitude at O(G^4) and N-matrix element for scattering phase shift, verified by agreement with black-hole perturbation theory.
Mixed QCD-EW corrections to qqbar -> HH computed analytically via differential equations, matched to large-mass limit, implemented in POWHEG-BOX, showing up to +10% effect on invariant mass distribution near threshold.
An algorithm reconstructs symbolic IBP reduction coefficients via intermediate bases, demonstrated on massive box-triangle and pentagon-triangle integrals using 3289 and 13013 samplings versus over a million unknowns.
Analytic NNLO partonic coefficient functions for F2, FL, F3 in charged-current DIS with exact charm mass, expressed via Goncharov polylogarithms and Chen iterated integrals.
Proves maximal transcendentality to all orders for the vacuum-energy expansion of the double-scaled large-N PCM, with coefficients as rational polynomials in odd zeta values after a natural coupling shift.
The paper develops reduction operators from resonance in GKZ systems to contract edges in Feynman graphs for one-loop, sunrise, and banana graphs, closing differential equation systems to master integrals.
Introduces weight-shifting matrices for de Sitter diagrams, generalized with Kronecker products to arbitrary tree-level graphs, to derive massless wavefunction coefficients from conformally coupled seeds.
Determines the unique two- and three-loop symbols for the four-point NMHV form factor from an 88-letter alphabet, providing first multi-loop non-MHV data and supporting alphabet universality.
Gravitational Compton amplitude computed to third post-Minkowskian order via worldline EFT with infrared and forward divergences regulated to connect to black hole perturbation theory.
The integrated iterated single-unresolved approximate cross section in CoLoRFulNNLO for hadron collisions is a convolution of the Born cross section with an insertion operator.
Twisted Feynman integrals are introduced with graded Symanzik polynomials, classified as exponential periods, and shown to have geometry not inferable from generalized Baikov leading singularities.
Leading-colour two-loop virtual amplitudes for ttbar+jet are extracted analytically via finite-field evaluations and differential equations, then packaged in a C++ library with new numerical integration techniques.
Cluster algebras for planar conformal kinematics are identified as G(4,n) subalgebras and used to bootstrap the symbol of an 8-point three-loop wheel integral via D3 and new algebraic letters.
Kira 2.0 implements finite-field coefficient reconstruction for IBP reductions and improved user-equation handling, yielding lower memory use and faster performance on state-of-the-art problems.
First numerical evaluation of planar two-loop helicity amplitudes for W-boson plus four partons using finite-field reduction and sector decomposition on a subset of master integrals.
Feynman integrals selected for unit leading singularities in complex geometries satisfy epsilon-factorized differential equations with new transcendental functions corresponding to periods and differential forms in the Gauss-Manin connection.
First beyond-NLO tensor decomposition and higher-order analytic one-loop amplitudes for e+e- to pi+pi-gamma, paired with a fast numerical five-point integral evaluator.
A parity-split IBP system for n-propagator families in de Sitter space is identified, along with a conjecture that dlog-form differential equations extend to dS integrands with Hankel functions, verified for the one-loop bubble.
citing papers explorer
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The spectrum of Feynman-integral geometries at two loops
Two-loop Feynman integrals involve Riemann spheres, elliptic curves, hyperelliptic curves of genus 2 and 3, K3 surfaces, and a rationalizable Del Pezzo surface of degree 2.
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The four-loop non-singlet splitting functions in QCD
Four-loop non-singlet splitting functions in QCD are computed analytically for the first time, with numerical representations provided.
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Integrand Analysis, Leading Singularities and Canonical Bases beyond Polylogarithms
Feynman integrals selected for unit leading singularities in complex geometries satisfy epsilon-factorized differential equations with new transcendental functions corresponding to periods and differential forms in the Gauss-Manin connection.
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Tensor decomposition of $e^+e^-\to\pi^+\pi^-\gamma$ to higher orders in the dimensional regulator
First beyond-NLO tensor decomposition and higher-order analytic one-loop amplitudes for e+e- to pi+pi-gamma, paired with a fast numerical five-point integral evaluator.
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Planar master integrals for two-loop NLO electroweak light-fermion contributions to $g g \rightarrow Z H$
Analytic expressions for the planar master integrals in two-loop NLO EW light-fermion contributions to gg → ZH are derived via canonical differential equations and expressed using Goncharov polylogarithms or one-fold integrals over them.
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Progress on the soft anomalous dimension in QCD
A lightcone-expansion strategy using Wilson-line correlators and the Method of Regions yields the three-loop soft anomalous dimension for QCD amplitudes with one massive colored particle and arbitrary massless ones.