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 φ 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 φ 3 scalar field theory are given by the Green function for the conformal quantum mechanics.

R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain

“…This line of reasoning is due to Isaev [36]. The property of commutativity of the operators H b and G a can be expressed as the integral identity which presents a more conventional form of the star-triangle relation…”

Факторизация $R$-матрицы, $Q$-оператор и разделение переменных в случае $XXX$-спиновой цепочки с группой симметрии $SL(2,\mathbb{C})$

“…Во-первых, преобразуем представление для Q-операторов, используя операторное тождество, которое называется "соотношение звезда-треугольник" [20]:…”

On conformal invariant integrals involving spin one-half and spin-one particles

“…We use i, j, k, · · · for spacetime indices, and α, β, · · · for exponents ofq 2 andp 2 . Thus,q 2α = ( iqiqi ) α (and likewisep 2α ), the parameter α being in general a complex number [13]. The fundamental commutation relation…”

“…(13) and (14) of Ref. [13].) We use the normalization of position and momentum eigenstates followed in Ref.…”

“…The evaluation of the tree level integrals is necessary for implementing the bootstrap program in a CFT 2 Star-triangle relation involving scalar fields It will be helpful to first discuss, following Ref. [13], the usual star-triangle relation involving scalar fields within the framework of the operator algebraic method. The relation evaluates 0|T (φ 1 (x 1 )φ 2 (x 2 )φ 3 (x 3 ))|0 to the lowest order in position space with a φ 1 φ 2 φ 3 interaction.…”