Abstract
Contour integrals of rational functions over ${\cal M}_{0,n}$, the moduli space of $n$-punctured spheres, have recently appeared at the core of the tree-level S-matrix of massless particles in arbitrary dimensions. The contour is determined by the critical points of a certain Morse function on ${\cal M}_{0,n}$. The integrand is a general rational function of the puncture locations with poles of arbitrary order as two punctures coincide. In this note we provide an algorithm for the analytic computation of any such integral. The algorithm uses three ingredients: an operation we call general KLT, Petersen's theorem applied to the existence of a 2-factor in any 4-regular graph and Hamiltonian decompositions of certain 4-regular graphs. The procedure is iterative and reduces the computation of a general integral to that of simple building blocks. These are integrals which compute double-color-ordered partial amplitudes in a bi-adjoint cubic scalar theory.
Highlights
The new formulas for the scattering of n particles are given as a sum over multidimensional residues [14] on M0,n
The step in the construction is a rational map from Cn−3 → Cn−3 which is a function of the entries of a symmetric n × n matrix, sab, with vanishing diagonal, i.e., saa = 0, and all rows adding up to zero
H(σ, k, ) is a rational function that depends on the theory under consideration and contains all information regarding wave functions of the particles such as polarization vectors μ a and momenta kaμ
Summary
The aim of this work is to provide an algorithm for the reduction of contour integrals on the moduli space of an n-punctured sphere of the form dμnF (σ). We discuss the basic building blocks which are special contour integrals with f (rijkl) = 1 and whose values are explicitly known [2, 4, 9]. Solving the equations Ea = 0 is a nontrivial task when n > 5 as for generic values of sab and after finding a Groebner basis one is faced with an irreducible polynomial of degree (n − 3)!. We quote the result and use these integrals as building blocks for generic ones. Each way of doing so defines a cyclic ordering of the labels {1, 2, . Let us denote the set of all graphs T consistent with the ordering α by Γ(α).
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