The corolla polynomial: a graph polynomial on half-edges

Abstract

The study of Feynman rules is much facilitated by the two Symanzik polynomials, homogeneous polynomials based on edge variables for a given Feynman graph. We review here the role of a recently discovered third graph polynomial based on half-edges which facilitates the transition from scalar to gauge theory amplitudes: the corolla polynomial. We review in particular the use of graph homology in the construction of this polynomial.

Complexes of marked graphs in gauge theory

“…For a thorough discussion of the quantum field theoretical motivation and interpretation of these complexes, we refer to the original article [11] and the review [10]. A detailed discussion of the analytic approach via corolla polynomials can be found in [12], and a general reference for background material on the quantization of gauge theories is the classical work [2].…”

Off-shell Diagrammatics for Quantum Gravity

“…In the sense of graph theory, these define an operation on graphs by edge-collapsing. In combination with the ghost identity, this informs the definition of a gravitational graph and cycle homology as in the Yang-Mills case [10,11] and possibly enables the construction of a corolla polynomial for gravity as extension of [44,10,12,45].…”

Gauge Symmetries and Renormalization

“…reviewed in [36,37]. Finally, we remark that the above discussion can be also lifted to the algebra of meromorphic functions M ε := C ε −1 , ε , if a suitable regularization scheme E is chosen, 20 by setting…”

“…Definition 2.41 and Remark 6.8. The Corolla polynomial is a graph polynomial in half-edges that relates amplitudes in Quantum Yang-Mills theory to amplitudes in φ 3 4 -theory [18][19][20]. More precisely, this graph polynomial is used for the construction of a so-called Corolla differential that acts on the parametric representation of Feynman integrals [19,21,22].…”