Properties of the Corolla Polynomial of a 3-regular Graph

Kreimer, Dirk; Yeats, Karen

doi:10.37236/2633

Cited by 14 publications

(58 citation statements)

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“…The occurrence of this polynomial and especially the interesting properties discussed in the lemmata used to prove the main theorem suggest a deeper importance of cycle subgraphs in gauge theory, which is in accordance with [19], where a combination of graph and cycle homology was used to derive the parametric integrand for general gauge theories. Since the properties of the cycle polynomial and even its appearance in the derivatives of Φ Γ (α, ξ) is independent of the specific case of QED it should be possible to generalise our results to general gauge theories by replacing the special case of D Γ with the general Corolla differential [19,20,25]. The cycle polynomial and generation of the integrand has been implemented and checked via computer algebra for all photon, fermion and vertex functions up to two loops and some three-loop photon functions.…”

Section: Resultsmentioning

confidence: 99%

New graph polynomials in parametric QED Feynman integrals

Golz

2017

Annals of Physics

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In recent years enormous progress has been made in perturbative quantum field theory by applying methods of algebraic geometry to parametric Feynman integrals for scalar theories. The transition to gauge theories is complicated not only by the fact that their parametric integrand is much larger and more involved. It is, moreover, only implicitly given as the result of certain differential operators applied to the scalar integrand exp(−, where Ψ Γ and Φ Γ are the Kirchhoff and Symanzik polynomials of the Feynman graph Γ. In the case of quantum electrodynamics we find that the full parametric integrand inherits a rich combinatorial structure from Ψ Γ and Φ Γ . In the end, it can be expressed explicitly as a sum over products of new types of graph polynomials which have a combinatoric interpretation via simple cycle subgraphs of Γ.

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Section: Resultsmentioning

confidence: 99%

New graph polynomials in parametric QED Feynman integrals

Golz

2017

Annals of Physics

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“…• The corolla polynomial is similarly distinguished amongst half-edge polynomials having recursive half-edge deletion properties [1].…”

Section: Discussionmentioning

confidence: 99%

“…Both homologies can be implemented using a new graph polynomial on half-edges, the corolla polynomial [1].…”

Section: Scalar Vs Gauge Theory Amplitudesmentioning

confidence: 99%

The corolla polynomial: a graph polynomial on half-edges

Kreimer¹

2018

Proceedings of Loops and Legs in Quantum Field Theory — PoS(LL2018)

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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.

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“…From [31] we know that all graphs contributing to such a connected amplitude can be obtained by applying the Corolla polynomial [39] to a corresponding sum of 3-regular scalar graphs. Underlying this is a bi-complex in graph and cycle homology studied in [31] which puts the approach of [28,29,30] on a firm mathematical footing.…”

Section: Hopf-algebra Derivationmentioning

confidence: 99%

Self-consistency of off-shell Slavnov-Taylor identities in QCD

2019

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Using Hopf-algebraic structures as well as diagrammatic techniques for determining the Slavnov-Taylor identities for QCD we construct the relations for the triple and quartic gluon vertices at one loop. By making the longitudinal projection on an external gluon of a Green's function we show that the gluon self-energy of that leg is consistently replaced by a ghost selfenergy. The resulting identities are then studied by evaluating all the graphs for an off-shell non-exceptional momentum configuration. In the case of the 3-point function this is for the most general momentum case and for the 4-point function we consider the fully symmetric point.

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Properties of the Corolla Polynomial of a 3-regular Graph

Cited by 14 publications

References 1 publication

New graph polynomials in parametric QED Feynman integrals

New graph polynomials in parametric QED Feynman integrals

The corolla polynomial: a graph polynomial on half-edges

Self-consistency of off-shell Slavnov-Taylor identities in QCD

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