Tropicalized massive scalar QFT is exactly solvable via a non-linear recursion for effective action coefficients that computes graph moduli space volumes, enabling a polynomial-time sampling algorithm for high-order perturbative contributions.
Hopf algebraic Renormalization of Kreimer's toy model
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abstract
This masters thesis reviews the algebraic formulation of renormalization using Hopf algebras as pioneered by Dirk Kreimer and applies it to a toy model of quantum field theory given through iterated insertions of a single primitive divergence into itself. Using this example in a subtraction scheme, we exhibit the renormalized Feynman rules to yield Hopf algebra morphisms into the Hopf algebra of polynomials and as a consequence study the emergence of the renormalization group in connection with combinatorial Dyson-Schwinger equations. In particular we relate the perturbative expansion of the anomalous dimension to the coefficients of the Mellin transform of the integral kernel specifying the primitve divergence. A theorem on the Hopf algebra of rooted trees relates different Mellin transforms by automorphisms of this Hopf algebra.
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Graphical functions, defined as massless three-point position-space integrals, serve as a powerful tool for evaluating multi-loop Feynman integrals, with extensions to conformal field theory and recent algorithmic computability.
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Tropicalized quantum field theory and global tropical sampling
Tropicalized massive scalar QFT is exactly solvable via a non-linear recursion for effective action coefficients that computes graph moduli space volumes, enabling a polynomial-time sampling algorithm for high-order perturbative contributions.
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Graphical Functions by Examples
Graphical functions, defined as massless three-point position-space integrals, serve as a powerful tool for evaluating multi-loop Feynman integrals, with extensions to conformal field theory and recent algorithmic computability.