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Dirk Kreimer

Publications and source records attributed to Dirk Kreimer.

At least 37 records · Page 2Linked to original sources
arXiv2012

Renormalization and Mellin transforms

We study renormalization in a kinetic scheme using the Hopf algebraic framework, first summarizing and recovering known results in this setting. Then we give a direct combinatorial description of renormalized amplitudes in terms of Mellin transform coefficients, featuring the universal property of rooted trees H_R. In particular, a special class of automorphisms of H_R emerges from the action of changing Mellin transforms on the Hochschild cohomology of perturbation series. Furthermore, we show how the Hopf algebra of polynomials carries a refined renormalization group property, implying its coarser form on the level of correlation functions. Application to scalar quantum field theory reveals the scaling behaviour of individual Feynman graphs.

hep-th↗
arXiv2012

Properties of the corolla polynomial of a 3-regular graph

We investigate combinatorial properties of a graph polynomial indexed by half-edges of a graph which was introduced recently to understand the connection between Feynman rules for scalar field theory and Feynman rules for gauge theory. We investigate the new graph polynomial as a stand-alone object.

math.CO↗
arXiv2012

Field diffeomorphisms and the algebraic structure of perturbative expansions

We consider field diffeomorphisms in the context of real scalar field theories. Starting from free field theories we apply non-linear field diffeomorphisms to the fields and study the perturbative expansion for the transformed theories. We find that tree level amplitudes for the transformed fields must satisfy BCFW type recursion relations for the S-matrix to remain trivial. For the massless field theory these relations continue to hold in loop computations. In the massive field theory the situation is more subtle. A necessary condition for the Feynman rules to respect the maximal ideal and co-ideal defined by the core Hopf algebra of the transformed theory is that upon renormalization all massive tadpole integrals (defined as all integrals independent of the kinematics of external momenta) are mapped to zero.

hep-th↗
arXiv2011

Angles, scales and parametric renormalization

We decompose renormalized Feynman rules according to the scale and angle dependence of amplitudes. We use parametric representations such that the resulting amplitudes can be studied in algebraic geometry.

hep-th↗
arXiv2010

Renormalization and resolution of singularities

Since the seminal work of Epstein and Glaser it is well established that perturbative renormalization of ultraviolet divergences in position space amounts to extension of distributions onto diagonals. For a general Feynman graph the relevant diagonals form a nontrivial arrangement of linear subspaces. One may therefore ask if renormalization becomes simpler if one resolves this arrangement to a normal crossing divisor. In this paper we study the extension problem of distributions onto the wonderful models of de Concini and Procesi, which generalize the Fulton-MacPherson compactification of configuration spaces. We show that a canonical extension onto the smooth model coincides with the usual Epstein-Glaser renormalization. To this end we use an analytic regularization for position space. The 't Hooft identities relating the pole coefficients may be recovered from the stratification, and Zimmermann's forest formula is encoded in the geometry of the compactification. Consequently one subtraction along each irreducible component of the divisor suffices to get a finite result using local counterterms. As a corollary, we identify the Hopf algebra of at most logarithmic Feynman graphs in position space, and discuss the case of higher degree of divergence.

hep-th↗
arXiv2009

New algebraic aspects of perturbative and non-perturbative Quantum Field Theory

In this expository article we review recent advances in our understanding of the combinatorial and algebraic structure of perturbation theory in terms of Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf algebras of Feynman graphs, perturbative renormalization is rephrased algebraically. The Hochschild cohomology of these Hopf algebras leads the way to Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent progress in solving simple Dyson-Schwinger equations in the high energy sector using the algebraic machinery. Finally there is a short account on a relation to algebraic geometry and number theory: understanding Feynman integrals as periods of mixed (Tate) motives.

hep-th↗
arXiv2009

The QCD beta-function from global solutions to Dyson-Schwinger equations

We study quantum chromodynamics from the viewpoint of untruncated Dyson-Schwinger equations turned to an ordinary differential equation for the gluon anomalous dimension. This nonlinear equation is parameterized by a function P(x) which is unknown beyond perturbation theory. Still, very mild assumptions on P(x) lead to stringent restrictions for possible solutions to Dyson-Schwinger equations. We establish that the theory must have asymptotic freedom beyond perturbation theory and also investigate the low energy regime and the possibility for a mass gap in the asymptotically free theory.

hep-th↗
arXiv2009

Three Etudes in QFT

We review QFT from the algebraic structure of its perturbative expansion and point out three new results.

hep-th↗
arXiv2009

Algebra for quantum fields

We give an account of the current state of the approch to quantum field theory via Hopf algebras and Hochschild cohomology. We emphasize the versatility and mathematical foundation of this algebraic structure, and collect algebraic structures here in one place which are either scattered over the literature, or only implicit in previous writings. In particular we point out mathematical structures which can be helpful to farther develop our mathematical understanding of quantum fields.

hep-th↗
arXiv2009

Combinatorial Dyson-Schwinger equations in noncommutative field theory

We give here the Hopf algebra structure describing the noncommutative renormalization of a recently introduced translation-invariant model on Moyal space. We define Hochschild one-cocyles $B_+^γ$ which allows us to write down the combinatorial Dyson-Schwinger equations for noncommutative quantum field theory. One- and two-loops examples are explicitly worked out.

hep-th↗
arXiv2008

Not so non-renormalizable gravity

We review recent progress with the understanding of quantum fields, including ideas how gravity might turn out to be a renormalizable theory after all.

hep-th↗
arXiv2008

The QED beta-function from global solutions to Dyson-Schwinger equations

We discuss the structure of beta functions as determined by the recursive nature of Dyson--Schwinger equations turned into an analysis of ordinary differential equations, with particular emphasis given to quantum electrodynamics. In particular we determine when a separatrix for solutions to such ODEs exists and clarify the existence of Landau poles beyond perturbation theory. Both are determined in terms of explicit conditions on the asymptotics for the growth of skeleton graphs.

hep-th↗