Abstract
We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces — an algebraic invariant called the intersection number — extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data. In this paper, we introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. A second paper of this series will detail intersection pairings and their use to extract integral coefficients.
Highlights
Unitarity has a long history in quantum field theory since the optical theorem and Cutkosky rules [36]
We review the relevant ideas from relative cohomology that are applicable to dual forms while subsection 2.4 explains why dual forms are localized to cuts
We review relative cohomology, which deals with integration-by-parts on manifolds with boundaries
Summary
A defining property of dimensionally-regulated integrands, originally from [63, 64], is the product rule: a d-dimensional integrand is the product of a 4-dimensional one with a (−2ε)-. We assume that all external momenta lie within the (dint = 4)-dimensional physical subspace; such integrals are referred to as being near four dimensions. The integrand depends on angles or dot products between the extra-dimensional components. The standard integration contour for a Feynman integral is the Rn subspace consisting of real Minkowski momenta (with the usual Feynman i0 prescription), times the region over which the Gram matrix i⊥· j⊥ is positive definite. As stressed already, this contour is irrelevant for the integral reduction problem that is the focus of this paper: the intersection pairing involves a (2n)-dimensional integral over all of Cn
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