Equivalence of Recurrence Relations for Feynman Integrals with the Same Total Number of External and Loop Momenta

Abstract

We show that the problem of solving recurrence relations for L-loop (R+1)-point Feynman integrals within the method of integration by parts is equivalent to the corresponding problem for (L+R)-loop vacuum or (L+R-1)-loop propagator-type integrals. Using this property we solve recurrence relations for two-loop massless vertex diagrams, with arbitrary numerators and integer powers of propagators in the case when two legs are on the light cone, by reducing the problem to the well-known solution of the corresponding recurrence relations for massless three-loop propagator diagrams with specific boundary conditions.

Figures (6)

• Figure 1: Three types of two-loop vertex diagrams with general numerators and integer powers of propagators: (a) planar, (b) non-planar and (c) non-Abelian.
• Figure 2: Three types of three-loop propagator diagrams with general numerators and integer powers of propagators: (a) planar, (b) non-planar and (c) Mercedes.
• Figure 3: Master diagrams for 3-loop propagators.
• Figure 4: Recursively one-loop diagrams for 3-loop propagators. Integer indices of lines and numerators are arbitrary.
• Figure 5: Master diagrams for 2-loop vertices. All the lines have indices equal to one and the numerator is equal to one.