Abstract
In this paper, we study a new application of the positive Grassmannian to Wilson loop diagrams (or MHV diagrams) for scattering amplitudes in N= 4 Super Yang–Mill theory (N = 4 SYM). There has been much interest in studying this theory via the positive Grassmannians using BCFW recursion. This is the first attempt to study MHV diagrams for planar Wilson loop calculations (or planar amplitudes) in terms of positive Grassmannians. We codify Wilson loop diagrams completely in terms of matroids. This allows us to apply the combinatorial tools in matroid theory used to identify positroids (non-negative Grassmannians) to Wilson loop diagrams. In doing so, we find that certain non-planar Wilson loop diagrams define positive Grassmannians. While non-planar diagrams do not have physical meaning, this finding suggests that they may have value as an algebraic tool, and deserve further investigation.
Highlights
During the last decade, the computation of scattering amplitudes in N = 4 SYM has evolved away from old-school Feynman diagrams to the use of twistors and recursive methods that are much more efficient computationally
In this paper we are concerned with the representation of this rational function as a sum of contributions coming from MHV diagrams
While it is well known that all M H V Wilson loop amplitudes are trivial, there is no such result for the large class of diagrams that contain M H V subdiagrams
Summary
This section is an introduction to the combinatorics of Wilson loop diagrams for mathematicians. We show that Wilson loop diagrams define a subspace of the kernel of the matrix (Z∗μ)T , for a given twistor configuration, Z∗. We have shown that for an overdefined Wilson loop, the matrix M(W (Z∗))|VP∗ does not have full rank in a generic twistor configuration. Given a generic twistor configuration, the matrix M(W (Z∗)) has full rank only if the Wilson loop diagram is well defined. Given a Wilson loop W = (P, [n]), and a Z∗, a generic twistor configuration, the matrix M(W (Z∗)) ∈ G(|P|, [n]), if and only if W is well defined. If two exact Wilson loop diagrams define the same subspace of ker Z∗μ are equivalent, and. Theorem 3.39 shows that any well defined Wilson loop diagram with non-crossing propagators is admissible. At least in the exact case, an exact Wilson loop diagram with crossing propagators is admissible if and only if it is equivalent to an exact diagram with non-crossing propagators
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