Multi-loop Feynman integrals and conformal quantum mechanics

Abstract

New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless ${\varphi }^3$ theory. Using our algebraic approach we show that the problem of evaluation of special classes of Feynman diagrams reduces to the calculation of the Green functions for specific quantum mechanical problems. In particular, the integrals for ladder massless diagrams in the ${\varphi }^3$ scalar field theory are given by the Green function for the conformal quantum mechanics.

15 pages, Latex, corrected some typos and list of Refs

References in corpus (7)

• Nested Sums, Expansion of Transcendental Functions and Multi-Scale Multi-Loop Integrals
• Superconformal Symmetry, Correlation Functions and the Operator Product Expansion
• New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams
• Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables
• Exceptional non-renormalization properties and OPE analysis of chiral four-point functions in N=4 SYM_4
• Analytical Result for Dimensionally Regularized Massless Master Non-planar Double Box with One Leg off Shell
• The Leading Power Regge Asymptotic Behaviour of Dimensionally Regularized Massless On-Shell Planar Triple Box

Cited by in corpus (46)

• Magic identities for conformal four-point integrals
• Massive Feynman diagrams and inverse binomial sums
• Hexagonalization of Correlation Functions
• Quantum Gravity from Conformal Field Theory
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Gluing Ladders into Fishnets
• Multipoint Conformal Blocks in the Comb Channel
• Conformal Partial Wave Expansions for N=4 Chiral Four Point Functions
• New differential equations for on-shell loop integrals
• Conformal algebra: R-matrix and star-triangle relation
• The Double Pentaladder Integral to All Orders
• Exactly solvable magnet of conformal spins in four dimensions
• Multi-loop techniques for massless Feynman diagram calculations
• New Dual Conformally Invariant Off-Shell Integrals
• Generalised ladders and single-valued polylogarithms
• Quantum Field Theory, Feynman-, Wheeler Propagators, Dimensional Regularization in Configuration Space and Convolution of Lorentz Invariant Tempered Distributions
• Systematics of the cusp anomalous dimension
• Exponential suppression with four legs and an infinity of loops
• Exactly solvable single-trace four point correlators in $\chi$CFT${}_4$
• Amplitudes at Weak Coupling as Polytopes in AdS_5
• Hexagons and Correlators in the Fishnet Theory
• Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C) magnet
• Yangian Bootstrap for Massive Feynman Integrals
• Factorization of R-matrix and Baxter Q-operators for generic sl(N) spin chains
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• Recursive computation of Feynman periods
• Operator approach to analytical evaluation of Feynman diagrams
• Dimensionally regularized Tsallis' Statistical Mechanics and two-body Newton's gravitation
• Cosmic Wheels: From integrability to the Galois coaction
• Laurent series expansion of sunrise-type diagrams using configuration space techniques
• Expansion in Feynman Graphs as Simplicial String Theory
• Transformations of triangle ladder diagrams
• Long Range, Large Charge, Large $N$
• From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
• Ladder and zig-zag Feynman diagrams, operator formalism and conformal triangles
• Dimensionally regularized Boltzmann-Gibbs Statistical Mechanics and two-body Newton's gravitation
• Newton's gravitation-force's classical average proof of a Verlinde's conjecture
• Gluon self-interaction in the position space in Landau gauge
• On conformal invariant integrals involving spin one-half and spin-one particles
• Conformal Triangles and Zig-Zag Diagrams
• Two-fold Mellin-Barnes transforms of Usyukina-Davydychev functions
• Instability in scalar channel of fermion-antifermion scattering amplitude in massless QED_3 and anomalous dimensions of composite operators
• External leg amputation in conformal invariant three-point function
• Three-point Green function of massless QED in position space to lowest order
• The thermal representation of conformal ladder integrals
• BC-type open $SL(2,\mathbb{C})$ spin chain