Analytic Computing Methods for Precision Calculations in Quantum Field Theory

Abstract

An overview is presented on the current status of main mathematical computation methods for the multi-loop corrections to single scale observables in quantum field theory and the associated mathematical number and function spaces and algebras. At present massless single scale quantities can be calculated analytically in QCD to 4-loop order and single mass and double mass quantities to 3-loop order, while zero scale quantities have been calculated to 5-loop order. The precision requirements of the planned measurements, particularly at the FCC-ee, form important challenges to theory, and will need important extensions of the presently known methods.

29 pages LATEX

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• Collider Physics at the Precision Frontier
• A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable
• The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach
• Gravitational scattering at the seventh order in $G$: nonlocal contribution at the sixth post-Newtonian accuracy
• The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
• Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
• The Heavy Fermion Contributions to the Massive Three Loop Form Factors
• The three-loop single mass polarized pure singlet operator matrix element
• Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series
• The Polarized Transition Matrix Element $A_{gq}(N)$ of the Variable Flavor Number Scheme at $O({\alpha }_s^3)$
• The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$
• Subleading Logarithmic QED Initial State Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/{Z^0}^{\ast }$ to $O({\alpha }^6{L}^5)$
• The Polarized Two-Loop Massive Pure Singlet Wilson Coefficient for Deep-Inelastic Scattering
• The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg,Q}$ and $\Delta A_{gg,Q}$
• The Two-mass Contribution to the Three-Loop Polarized Operator Matrix Element $A_{gg,Q}^{(3)}$
• The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$
• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
• The unpolarized two-loop massive pure singlet Wilson coefficients for deep-inelastic scattering
• Iterated integrals over letters induced by quadratic forms
• $O({\alpha }_s^2$) Polarized Heavy Flavor Corrections}to Deep-Inelastic Scattering at $Q^2\gg m^2$
• The inverse Mellin transform via analytic continuation
• q-derivatives of multivariable q-hypergeometric function with respect to their parameters
• High Precision QED Initial State Corrections for $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z^{\ast }$ Annihilation
• Next-to-Leading Order QCD Corrections to Heavy-Flavour Production in Neutral Current DIS
• NNLO mixed EW-QCD corrections to single vector boson production