Rearranging Dyson-Schwinger equations

Yeats, Karen

doi:10.1090/s0065-9266-2010-00612-4

Cited by 25 publications

(58 citation statements)

References 13 publications

(1 reference statement)

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“…When there are multiple insertion places, we take each possibility for the Feynman integral for the primitive regularized at one of the insertion places and then sum these and divide by the number of insertion places. For more on symmetric insertion see Section 2.3.3 of [16] or [17]. Symmetric insertion gives the sum over all Feynman diagrams built by inserting into each of the insertion places used in the symmetric insertion, but we are constrained to get them all together as a linear combination.…”

Section: Dyson-schwinger Equationsmentioning

confidence: 99%

$$\hbox {Next-to}{}^k$$ Leading Log Expansions by Chord Diagrams

Courtiel

Yeats

2020

Commun. Math. Phys.

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Green functions in a quantum field theory can be expanded as bivariate series in the coupling and a scale parameter. The leading logs are given by the main diagonal of this expansion, i.e. the subseries where the coupling and the scale parameter appear to the same power; then the next-to leading logs are listed by the next diagonal of the expansion, where the power of the coupling is decremented by one, and so on.We give a general method for deriving explicit formulas and asymptotic estimates for any nextto k leading-log expansion for a large class of single scale Green functions. These Green functions are solutions to Dyson-Schwinger equations that are known by previous work to be expressible in terms of chord diagrams. We look in detail at the Green function for the fermion propagator in massless Yukawa theory as one example, and the Green function of the photon propagator in quantum electrodynamics as a second example, as well as giving general theorems. Our methods are combinatorial, but the consequences are physical, giving information on which terms dominate and on the dichotomy between gauge theories and other quantum field theories. arXiv:1906.05139v1 [math-ph]

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Section: Dyson-schwinger Equationsmentioning

confidence: 99%

$$\hbox {Next-to}{}^k$$ Leading Log Expansions by Chord Diagrams

Courtiel

Yeats

2020

Commun. Math. Phys.

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“…Returning to the set up of [19] and the connection to chord diagrams, after being converted into differential form following [25] or [26], the Dyson- Schwinger equation considered in [19] becomes…”

Section: Contextmentioning

confidence: 99%

Connected Chord Diagrams and Bridgeless Maps

Courtiel

Yeats

Zeilberger

2019

Electron. J. Combin.

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“…evaluates at = 0 to the compact form [36,5]. Reference [34] is devoted to a detailed study of this type of equations and also solves the case κ = 1 with the help of the Lambert W function.…”

Section: Relation To Mellin Transformsmentioning

confidence: 99%

Renormalization, Hopf algebras and Mellin transforms

Panzer¹

2015

Feynman Amplitudes, Periods and Motives

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Abstract. This article aims to give a short introduction into Hopf-algebraic aspects of renormalization, enjoying growing attention for more than a decade by now. As most available literature is concerned with the minimal subtraction scheme, we like to point out properties of the kinematic subtraction scheme which is also widely used in physics (under the names of MOM or BPHZ).In particular we relate renormalized Feynman rules φ R in this scheme to the universal property of the Hopf algebra H R of rooted trees, exhibiting a refined renormalization group equation which is equivalent to φ R :being a morphism of Hopf algebras to the polynomials in one indeterminate.Upon introduction of analytic regularization this results in efficient combinatorial recursions to calculate φ R in terms of the Mellin transform. We find that different Feynman rules are related by a distinguished class of Hopf algebra automorphisms of H R that arise naturally from Hochschild cohomology.Also we recall the known results for the minimal subtraction scheme and shed light on the interrelationship of both schemes.Finally we incorporate combinatorial Dyson-Schwinger equations to study the effects of renormalization on the physical meaningful correlation functions. This yields a precise formulation of the equivalence of the two different renormalization prescriptions mentioned before and allows for non-perturbative definitions of quantum field theories in special cases.

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Rearranging Dyson-Schwinger equations

Cited by 25 publications

References 13 publications

$$\hbox {Next-to}{}^k$$ Leading Log Expansions by Chord Diagrams

$$\hbox {Next-to}{}^k$$ Leading Log Expansions by Chord Diagrams

Connected Chord Diagrams and Bridgeless Maps

Renormalization, Hopf algebras and Mellin transforms

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