Operator approach to analytical evaluation of Feynman diagrams

Abstract

Abstract. The operator approach to analytical evaluation of multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of massless Feynman integrals, such as the integration by parts method and the method of "uniqueness" (which is based on the star-triangle relation), can be drastically simplified by using this operator approach. To demonstrate the advantages of the operator method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless φ 3 theory (analytical results for these diagrams are expressed in terms of multiple polylogarithms). It is shown how operator formalism can be applied to calculation of certain massive Feynman diagrams and investigation of Lipatov integrable chain model.

Conformal algebra: R-matrix and star-triangle relation

“…It remains to compare this equation with the defining equation (5.26) for the intertwining operator S which suggests that the change (5.28). Thus we have that S 2 is the operator of multiplication by the function and are both equivalent to the operator identity [49,50]:…”

Basso-Dixon correlators in two-dimensional fishnet CFT

“…This graph is presented on Fig.1. It has four external fixed coordinates and, similarly to the conformal 4-point functions, has a non-trivial dependence on two cross-ratios u, v. This Basso-Dixon (BD) formula takes the form of an N × N determinant of explicitly known "ladder" integrals [2,3]. It is one of very few examples of explicit results for Feynman graphs with arbitrary many loops.…”

“…Here and in the following we adopt the notation [z − w] α ≡ (z − w) α (z * − w * )ᾱ for propagators, see App.A for details 3. Or alternatively, due to the obvious L ↔ N symmetry of the integral, in terms of the (L − 1) × (L − 1) determinant of the same matrix elements, which will depend only on L + N combination.…”

Conformal quantum mechanics & the integrable spinning Fishnet

“…Without loss of generality we can restrict ourselves to the representations of the principal series, that means ∆ = 2 + iν with ν a real number. 1 A scale-invariant combination of the position operator x µ and momentum operator pµ = −i∂ µ satisfies the remarkable duality [2,6] p2u…”

“…This work is based on the papers [2][3][4][5]; our motivation is to clarify their various interrelations and to deduce how their results apply to the most general setup. First of all we managed to reformulate the expression given in [3] for the intertwiners S 1 , S 2 , S 3 of unitary irreducible representations of SO (1,5), and thus the infinite-dimensional R-operator, in the form of symmetric traceless tensors of coordinates and momenta [2,6]. Ultimately, these achievements are based on the new star-triangle relation derived in [4].…”