One-Loop Integrals from Spherical Projections of Planes and Quadrics

Understand

We initiate a systematic study of one-loop integrals by investigating the connection between their singularity structures and geometric configurations in the projective space associated to their Feynman parametrization.

• We analyze these integrals by two recursive methods, which leads to two independent algebraic algorithms that determine the symbols of any one-loop integrals in arbitrary spacetime dimensions.
• The discontinuities of Feynman diagrams are shown to arise from taking certain "spherical contour" residues in Feynman parameter space, which is geometrically interpreted as a projection of the quadric surface (associated to the Symanzik polynomial at one loop) through faces of the integration region (which is a simplex).
• This geometry also leads to a manifestly Lorentz-invariant understanding for perturbative unitarity at one loop.