Dodgson polynomial identities

Dodgson polynomials appear in Schwinger parametric Feynman integrals and are closely related to the well known Kirchhoff (or first Symanzik) polynomial. In this article a new combinatorial interpretation and a generalisation of Dodgson polynomials are provided. This leads to two new identities that relate large sums of products of Dodgson polynomials to a much simpler expression involving powers of the Kirchhoff polynomial. These identities can be applied to the parametric integrand for quantum electrodynamics, simplifying it significantly. This is worked out here in detail on the example of superficially renormalised photon propagator Feynman graphs, but works much more generally.

• Publication year

  2018

• Venue

  Communications in Number Theory and Physics

• Publication date

  2018-10-15

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.4310/cntp.2019.v13.n4.a1arXiv 1810.06220
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Feynman integral relations from parametric annihilators

  2017cited by this paper
• New graph polynomials in parametric QED Feynman integrals

  2017influential reference
• Contraction of Dirac matrices via chord diagrams

  2017cited by this paper
• Renormalization group functions of $\phi^4$ theory in the MS-scheme to six loops

  2016cited by this paper
• Cutkosky Rules and Outer Space

  2015cited by this paper
• Feynman integrals and hyperlogarithms

  2015cited by this paper
• Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals

  2014cited by this paper
• Quantization of gauge fields, graph polynomials and graph homology

  2012cited by this paper
• Properties of c_2 invariants of Feynman graphs

  2012cited by this paper
• Angles, Scales and Parametric Renormalization

  2011cited by this paper
• Enumerative Combinatorics: Volume 1

  2011cited by this paper
• A K3 in $\phi^{4}$

  2010cited by this paper
• On the periods of some Feynman integrals

  2009cited by this paper
• Spanning Forest Polynomials and the Transcendental Weight of Feynman Graphs

  2009cited by this paper
• The Massless Higher-Loop Two-Point Function

  2008cited by this paper
• Condensation of Determinants

  2007cited by this paper
• Communications in Mathematical Physics On Motives Associated to Graph Polynomials

  2006cited by this paper
• Graph-polynomials

  2004cited by this paper
• Special issue on the Tutte polynomial

  2004cited by this paper
• Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture

  1999cited by this paper
• Unknotting the polarized vacuum of quenched QED

  1995cited by this paper
• The analytic four-loop corrections to the QED β-function in the MS scheme and to the QED ψ-function. Total reevaluation

  1991cited by this paper
• Matrix Tree Theorems

  1978cited by this paper
• Bogolubov–Parasiuk theorem in the α‐parametric representation

  1976cited by this paper
• Feynman-Dyson rules in parametric space

  1974cited by this paper
• Renormalization of Feynman amplitudes and parametric integral representation

  1974cited by this paper
• The Art of Computer Programming—Volume 1: Fundamental Algorithms (Donald E. Knuth)

  1969cited by this paper
• The Art of Computer Programming

  1968cited by this paper
• Sixth-Order Contribution to Z3 in Finite Quantum Electrodynamics

  1966cited by this paper
• Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)

  1965cited by this paper
• Mass singularities of Feynman amplitudes

  1962cited by this paper
• Parametric Integral Formulas and Analytic Properties in Perturbation Theory

  1961cited by this paper
• DISPERSION RELATIONS AND VERTEX PROPERTIES IN PERTURBATION THEORY

  1958cited by this paper
• General Integral Formula of Perturbation Term in the Quantized Field Theory

  1957cited by this paper
• Parametric representations of general Green’s functions

  1957cited by this paper
• A Contribution to the Theory of Chromatic Polynomials

  1954cited by this paper
• The Coloring of Graphs.

  1931cited by this paper
• Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird

  year unknowncited by this paper
• A Determinant Formula for the Number of Ways of Coloring a Map

  year unknowncited by this paper

• Combinatorial proof of a non-renormalization theorem

  2024influential citation
• Generalised Graph Laplacians and Canonical Feynman Integrals with Kinematics

  2022cites this paper
• Numerators in Parametric Representations of Feynman Diagrams

  2022cites this paper
• Into the Woods: A graph theoretic proof of the Dodgson/Muir identity

  2021cites this paper
• Invariant Differential Forms on Complexes of Graphs and Feynman Integrals

  2021cites this paper
• Gauge Symmetries and Renormalization

  2019cites this paper
• The Axiomatic Destiny of the Theorems of Pappus and Desargues

  2019cites this paper
• Lewis Carroll and the Red Hot Potato: A graph theoretic approach to a linear algebraic identity

  2018cites this paper