Abstract
Abstract We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space. Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop n-gon integrals in n dimensions. These are known to be given by volumes of hyperbolic simplices. We explicitly compute the five-dimensional pentagon integral in full generality using Schläfli’s formula. Then, as a first step to understanding higher loops, we use spline technology to construct explicitly the 6d hexagon and 8d octagon integrals in two-dimensional kinematics. The fully massive hexagon and octagon integrals are then related to the double box and triple box integrals respectively. We comment on the classes of functions needed to express these integrals in general kinematics, involving elliptic functions and beyond.
Highlights
Theory has been completely constructed [23,24,25,26], new mathematical techniques are necessary to efficiently describe the integrated objects
We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space
When the number of kinematic variables is sufficiently small, it has even been possible with sufficient effort to obtain analytic formulas for certain amplitudes [40,41,42,43,44], but our analytic knowledge of more complicated integrals appearing in multi-loop SYM scattering amplitudes is still quite limited
Summary
The multi-dimensional Mellin transform formalism was introduced in the work of Mack [45, 46] and quickly applied to both AdS/CFT [48,49,50,51,52,53] and flat space calculations [54, 70]. The Mellin transform can be applied to any conformally invariant function of several points xi, with given conformal weights ∆i. Where xij ≡ (xi − xj) and the δij parameters satisfy the constraints δij = 0, i=j δii = −∆i. The function M (δij) is usually called the Mellin transform of φ∆1(x1) · · · φ∆n(xn). It is important to note that the constraints can formally be solved by introducing a set of Mellin momenta ki, satisfying momentum conservation, i ki = 0, such that δij = ki · kj, ki2 = −∆i. This parameterization provides some intuition for the δij parameters.
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