Vector Space of Feynman Integrals and Multivariate Intersection Numbers

Abstract

Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for constructing multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the derivation of contiguity relations for special functions admitting multi-fold integral representations, and to the decomposition of a few Feynman integrals at one-and two-loops, as first steps towards potential applications to generic multi-loop integrals.

From infinity to four dimensions: higher residue pairings and Feynman integrals

“…where C × := C−{0}, in agreement with [31,78]. Physically it counts the number of linearlyindependent Feynman integrals that involve the set of propagators {D a } P a=1 over Q(K, ε, δ a ), where K in the set of kinematic variables appearing in Q, L, c. 4 It is the most convenient to compute |χ(M )| by invoking Morse-theory arguments, which for sufficiently generic W imply that it is equal to the number of critical points Crit(W ) determined by the condition dW = 0.…”

“…It would be fascinating to understand a similar geometric condition that leads to an ε-form differential equations, or decide whether such a basis could even exist. Although we used a representation in terms of Symanzik polynomials, as in (3.12), there is no substantial difficulty in repeating our analysis in other ways, e.g., using the original loop-momentum variables [72] or Baikov representation [30,31,68], where the answer to this question might prove easier.…”

“…As remarked before, many quantities of physical interest can be written as integrals on M of the general form: [5] for an exposition), other than the fact that it leads to the above pairing [58]. For physical applications see, e.g., [5,30,31,[59][60][61][62][63][64][65][66][67][68][69][70][71][72] and references therein.…”

“…Thus the problem amounts to finding the matrix Ω. It was recently realised that fibers of the vector bundle can be described by the cohomology of (Ω • M , d+τ dW ∧) and hence the entries of Ω can be computed by the same intersection numbers (1.3) described above [30,31].…”

Correlation functions on the lattice and twisted cocycles

“…Quite recently it has been become clear, that Feynman integral reduction can be formulated in the language of twisted cocycles [7][8][9][10][11]. This formalism comes with an inner product, given by the intersection number of twisted cocycles.…”