HET Journal club: Jacob Bourjaily

Speaker:  Jacob Bourjaily

Title: Constructing Cluster Coordinates for Loop Integration

Abstract: In recent years, our understanding of loop integrands for scattering amplitudes has far surpassed (with few exceptions) our understanding of the functional form of amplitudes after loop integration. Much of this disparity is due to the (rapidly improving, but still) woefully inadequate technology that exists for loop integration. I briefly review the current "state of the art" for loop integration, and then outline a (surprisingly, but often much) better strategy for loop integration—applicable to a wide variety of loop integrals/observables. The two key insights of this "new" approach are: first, to insist that all symmetries (such as (dual) conformal invariance) and simplicities (such as finiteness) be preserved; and secondly, to make use of "good" kinematic variables (namely, cluster coordinates): those that "linearize" all boundary configurations—the locations of branch cuts, discontinuities, (landau) singularities, etc. It turns out that these two ideas alone suffice to define a new (and rapidly improving) "state of the art" for many loop integrals of interest.
After broadly outlining this quixotic state of affairs, I outline how the well-developed understanding of the "positive Grassmannian" can be used to construct "optimal" cluster coordinates individually tailored for a particular integral of interest.