Hopf-algebraic renormalization of QED in the linear covariant gauge

Kißler, Henry

doi:10.1016/j.aop.2016.05.008

Cited by 12 publications

(10 citation statements)

References 38 publications

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“…For details, see e.g. [19][20][21][22]. First, let M(X r ) = r (X r ) sr with integers s r be an arbitrary monomial in X r .…”

Section: Graph Insertion and Dyson-schwinger Equationsmentioning

confidence: 99%

Log expansions from combinatorial Dyson–Schwinger equations

Krüger

2020

Lett Math Phys

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We give a precise connection between combinatorial Dyson-Schwinger equations and log-expansions for Green's functions in quantum field theory. The latter are triangular power series in the coupling constant α and a logarithmic energy scale L -a reordering of terms as G(α, L) = 1± j≥0 α j H j (αL) is the corresponding log-expansion. In a first part of this paper, we derive the leading-log order H 0 and the next-to (j) -leading log orders H j from the Callan-Symanzik equation. In particular, H j only depends on the (j + 1)-loop β-function and anomalous dimensions. For the photon propagator Green's function in quantum electrodynamics (and in a toy model, where all Feynman graphs with vertex sub-divergences are neglected), our formulas reproduce the known expressions for the next-to-next-to-leading log approximation in the literature. In a second part of this work, we review the connection between the Callan-Symanzik equation and Dyson-Schwinger equations, i.e. fixed-point relations for the Green's functions. Combining the arguments, our work provides a derivation of the log-expansions for Green's functions from the corresponding Dyson-Schwinger equations.

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“…For details, see e.g. [19][20][21][22]. First, let M(X r ) = r (X r ) sr with integers s r be an arbitrary monomial in X r .…”

Section: Graph Insertion and Dyson-schwinger Equationsmentioning

confidence: 99%

Log expansions from combinatorial Dyson–Schwinger equations

Krüger

2020

Lett Math Phys

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“…This is an example well known in the literature, see, e.g., Ref. [422], of the sensitivity of the contraction procedure to the Lorentz structure of subdiagrams (once the contraction done, the operator reduces here too to simple multiplication). Interestingly, the transverse part is non-zero only in the reduced case (ε e > 0).…”

Section: Two-loop Fermion Self-energymentioning

confidence: 87%

“…25 A nice account on the relation between the BPHZ renormalization prescription and the Hopf-algebraic approach to renormalization, with applications to QED4, can be found in Ref. [422].…”

Section: Renormalization Methodsmentioning

confidence: 99%

Field theoretic study of electron-electron interaction effects in Dirac liquids

Teber¹

2018

Preprint

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L'objectif de cette thèse d'habilitation est de présenter des résultats récents, obtenus dans la période 2012-2017, concernant l'effet des interactions dans les liquides planaires de type Dirac, e.g., le graphène, les états de surface de certains isolants topologiques voire même les systèmes à effet Hall quantique fractionnaire à demi-remplissage (pour les fermions composite de Dirac). Ces liquides sont caracterisés par des bandes sans gap, de fortes interactions électronélectron et une invariance de Lorentz émergeant dans l'infra-rouge profond. Les problèmes abordés prennent leur source dans des expériences portant sur ces systèmes et présentent un large spectre d'intérêt aussi bien en physique de basse (physique de la matière condensée) que de haute (physique des particules) énergie. On traitera en particulier de l'influence subtile des interactions sur les proprietés de transport ainsi que de leur rôle déterminant dans la potentielle génération dynamique d'une masse. La résolution de ces problèmes nous guidera de l'examen approfondi de la structure perturbative de théories de champs de jauge au développement et l'application de méthodes non-perturbatives issues de l'électro/chromo-dynamique quantique pour aborder le régime de couplage fort.The aim of this habilitation thesis is to present recent results, obtained during the period 2012-2017, related to interaction effects in condensed matter physics systems such as planar Dirac liquids, e.g., graphene and graphene-like systems, the surface states of some topological insulators and possibly half-filled fractional quantum Hall systems (for their Dirac composite fermions). These liquids are characterized by gapless bands, strong electron-electron interactions and emergent Lorentz invariance deep in the infra-red. We address a number of important issues raised by experiments on these systems covering subjects of wide current interest in low-energy (condensed matter) as well as high-energy (particle) physics. We shall consider in particular the subtle influence of interactions on transport properties and their supposedly crucial influence on a potential dynamical mass generation. The resolution of these problems will guide us from the thorough examination of the perturbative structure of gauge field theories to the development and application of non-perturbative approaches known from quantum electro/chromo-dynamics to address strong coupling issues.

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“…Subsequent formulas along these lines can be found as Theorem 1 of [50], for QED, as Equations 46 and 47 of [51], for QCD as Equation 3.75 of [52], and generalized to super-and non-renormalizable theories as Proposition 4.2 of [53].…”

Section: 31mentioning

confidence: 99%

Algebraic Interplay between Renormalization and Monodromy

Kreimer¹,

Yeats²

2021

Preprint

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We investigate combinatorial and algebraic aspects of the interplay between renormalization and monodromies for Feynman amplitudes. We clarify how extraction of subgraphs from a Feynman graph interacts with putting edges onshell or with contracting them to obtain reduced graphs. Graph by graph this leads to a study of cointeracting bialgebras. One bialgebra comes from extraction of subgraphs and hence is needed for renormalization. The other bialgebra is an incidence bialgebra for edges put either onor offshell. It is hence related to the monodromies of the multivalued function to which a renormalized graph evaluates. Summing over infinite series of graphs, consequences for Green functions are derived using combinatorial Dyson-Schwinger equations. Contents 2.1. Graphs 2.2. Cuts 2.3. Sub-and co-graphs 3. Hopf algebras 3.1. The core Hopf algebra H core 3.2. Quotient Hopf algebras 3.3. The Hopf algebra H pC 3.4. Pre-Cutkosky Necklaces 3.5. Extension to a core coproduct for pairs (Γ, F ) 3.6. Counting spanning trees 4. Coactions 4.1. H pC → H core ⊗ H pC 4.2. Cointeracting bialgebras 4.3. Sector Decomposition 4.4. Two more coactions 5. DSEs in core and quotient Hopf algebras 5.1. Dyson-Schwinger and cut Dyson-Schwinger setup 5.2. Assembly maps vs Hochschild 1-cocycles 5.3. Core, no cuts 5.4. Core, with cuts 5.5. The coaction on Green functions 5.6. Pairs (Γ, F ) or H C 1

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Hopf-algebraic renormalization of QED in the linear covariant gauge

Cited by 12 publications

References 38 publications

Log expansions from combinatorial Dyson–Schwinger equations

Log expansions from combinatorial Dyson–Schwinger equations

Field theoretic study of electron-electron interaction effects in Dirac liquids

Algebraic Interplay between Renormalization and Monodromy

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