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We consider multi-edge or banana graphs $b_n$ on $n$ internal edges $e_i$ with different masses $m_i$. We focus on the cut banana graphs $\Im(\Phi_R(b_n))$ from which the full result $\Phi_R(b_n)$ can be derived through dispersion. We give a recursive definition of $\Im(\Phi_R(b_n))$ through iterated integrals. We discuss the structure of this iterated integral in detail. A discussion of accompanying differential equations, of monodromy and of a basis of master integrals is included., Comment: 43 pages, 7 figures, minor corrections

We consider a coaction which exists for any bridge-free graph. It is based on the cubical chain complex associated to any such graph by considering two boundary operations: shrinking edges or removing them. Only if the number of spanning trees of a graph G equals its number of internal edges we find that the graphical coaction ΔG constructed here agrees with the coaction ΔInc proposed by Britto and collaborators. The graphs for which this is the case are one-loop graphs or their duals, multi-edge banana graphs. They provide the only examples discussed by Britto and collaborators so far. We call such graphs simple graphs. The Dunce’s cap graph is the first non-simple graph. The number of its spanning trees (five) exceeds the number of its edges (four). We compare the two coactions which indeed do not agree and discuss this result. We also point out that for kinematic renormalization schemes the coaction ΔG simplifies.

This letter examines diagrammatic cancellations for Quantum Electrodynamics (QED) in the general linear gauge. These cancellations combine Feynman graphs of various topologies and provide a method to reconstruct the gauge dependence of the electron propagator from the result of a particular gauge by means of a linear Dyson-Schwinger equation. We use this method in combination with dimensional regularization to demonstrate how the 3-loop ${\epsilon}$-expansion in the Feynman gauge determines the ${\epsilon}$-expansions for all gauge parameter dependent terms to 4 loops.

Using Hopf-algebraic structures as well as diagrammatic techniques for determining the Slavnov-Taylor identities for QCD we construct the relations for the triple and quartic gluon vertices at one loop. By making the longitudinal projection on an external gluon of a Green’s function we show that the gluon self-energy of that leg is consistently replaced by a ghost self-energy. The resulting identities are then studied by evaluating all the graphs for an off-shell nonexceptional momentum configuration. In the case of the 3-point function this is for the most general momentum case and for the 4-point function we consider the fully symmetric point.

We outline ideas to connect the analytic structure of Feynman amplitudes to the structure of Karen Vogtmann’s and Marc Culler’s Outer Space. We focus on the role of cubical chain complexes in this context, and also investigate the bordification problem in the example of the 3-edge banana graph.

The study of Feynman rules is much facilitated by the two Symanzik polynomials, homogeneous polynomials based on edge variables for a given Feynman graph. We review here the role of a recently discovered third graph polynomial based on half-edges which facilitates the transition from scalar to gauge theory amplitudes: the corolla polynomial. We review in particular the use of graph homology in the construction of this polynomial., Comment: 7 pages, 4 figures, Loops and Legs in Quantum Field Theory (LL2018) St. Goar, Germany

Dyson–Schwinger equations determine the Green functions G r ( α , L ) in quantum field theory. Their solutions are triangular series in a coupling constant α and an external scale parameter L for a chosen amplitude r , with the order in L bounded by the order in the coupling. Perturbation theory calculates the first few orders in α . On the other hand, Dyson–Schwinger equations determine next-to { j } -leading log expansions, G r ( α , L ) = 1 + ∑ j = 0 ∞ ∑ M p j M α j M ( u ) . ∑ M sums for any finite j a finite number of functions M in u . Here, u is the one-loop approximation to G r , for example, for the (inverse) propagator in massless Yukawa theory, u = α L / 2 . The leading logs come then from the trivial representation, e.g. M ( u ) = 1 − 1 − 2 u in massless Yukawa theory. The respective period at j = 0 is p 0 M ( u ) = 1 . All non-leading logs are organized by corresponding suppressions in powers α j . We describe an algebraic method to derive all next-to { j } -leading log terms from the knowledge of the first ( j + 1 ) terms in perturbation theory and their filtrations. This implies the calculation of the functions M ( u ) and periods p j M . In the first part of our paper, we investigate the structure of Dyson–Schwinger equations and develop a method to filter their solutions. Applying renormalized Feynman rules maps each filtered term to a certain power of α and L in the log-expansion. Based on this, the second part derives the next-to { j } -leading log expansions. Our method is general. Here, we exemplify it using the examples of the propagator in Yukawa theory and the photon self-energy in quantum electrodynamics. In particular, we give explicit formulas for the leading log, next-to-leading log and next-to-next-to-leading log orders in terms of at most three-loop Feynman integrals. The reader may apply our method to any (set of) Dyson–Schwinger equation(s) appearing in renormalizable quantum field theories.

We present the lattice structure of Feynman diagram renormalization in physical QFTs from the viewpoint of Dyson-Schwinger-Equations and the core Hopf algebra of Feynman diagrams. The lattice structure encapsules the nestedness of diagrams. This structure can be used to give explicit expressions for the counterterms in zero-dimensional QFTs using the lattice-Moebius function. Different applications for the tadpole-free quotient, in which all appearing elements correspond to semimodular lattices, are discussed.

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 $B_+^\gamma$ 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., Comment: 29 pages, 15 figures

We overview recent results on the mathematical foundations of Cutkosky rules. We emphasize that the two operations of shrinking an internal edge or putting internal lines on the mass-shell are natural operation on the cubical chain complex studied in the context of geometric group theory. This together with Cutkosky's theorem regarded as a theorem which informs us about variations connected to the monodromy of Feynman amplitudes allows for a systematic approach to normal and anomalous thresholds, dispersion relations and the optical theorem. In this report we follow [1] closely., Comment: 8p, talk given at Loops and Legs in Quantum Field Theory, Leipzig, April 24-29 2016, Germany

We study field diffeomorphisms $\Phi(x)= F(\rho(x))=a_0\rho(x)+a_1\rho^2(x)+\ldots=\sum_{j+0}^\infty a_j \rho^{j+1}$, for free and interacting quantum fields $\Phi$. We find that the theory is invariant under such diffeomorphisms if and only if kinematic renormalization schemes are used., Comment: Removed some extraneous files that were appearing at the end. No content changes from the previous version. 32 pages

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., Comment: 8 pages, 2 figures

We show how to interpret the scalar Feynman integrals which appear when reducing tensor integrals as scalar Feynman integrals coming from certain nice matroids., 12 pages, corrections suggested by referees

We study co-ideals in the core Hopf algebra underlying a quantum field theory., 11 pages

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., Comment: 22 pages

We relate renormalization in perturbative quantum field theory to the theory of limiting mixed Hodge structures using parametric representations of Feynman graphs., 48 pages, more expository details added, typos corrected

We discuss the structure of Dyson--Schwinger equations in quantum gravity and conclude in particular that all relevant skeletons are of first order in the loop number. There is an accompanying sub Hopf algebra on gravity amplitudes equivalent to identities between n-graviton scattering amplitudes which generalize the Slavnov Taylor identities. These identities map the infinite number of charges and finite numbers of skeletons in gravity to an infinite number of skeletons and a finite number of charges needing renormalization. Our analysis suggests that gravity, regarded as a probability conserving but perturbatively non-renormalizable theory, is renormalizable after all, thanks to the structure of its Dyson--Schwinger equations., Comment: 9p, several eps figures

We solve the linear Dyson Schwinger equation for a massless vertex in Yukawa theory, iterating the first two primitive graphs.

We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson--Schwinger equations into Euler products are discussed., 41 pages, epsf figures, talk given at the "Dennisfest", "Graphs and Patterns in Mathematics and Theoretical Physics, Stony Brook, June 2001, final version, to appear in the Proceedings

We discuss similarities and differences between Green Functions in Quantum Field Theory and polylogarithms. Both can be obtained as solutions of fixpoint equations which originate from an underlying Hopf algebra structure. Typically, the equation is linear for the polylog, and non-linear for Green Functions. We argue though that the crucial difference lies not in the non-linearity of the latter, but in the appearance of non-trivial representation theory related to transcendental extensions of the number field which governs the linear solution. An example is studied to illuminate this point., 5 pages contributed to the proceedings "Loops and Legs 2004", April 2004, Zinnowitz, Germany

We prove that insertion-elimination Lie algebra of Feynman graphs, in the ladder case, has a natural interpretation in terms of a certain algebra of infinite dimensional matrices. We study some aspects of its representation theory and we discuss some relations with the representation of the Heisenberg algebra, Comment: LaTex, 17 pages, typos corrected, to appear in LMP

We discuss factorization of the Dyson--Schwinger equations using the Lie- and Hopf algebra of graphs. The structure of those equations allows to introduce a commutative associative product on 1PI graphs. In scalar field theories, this product vanishes if and only if one of the factors vanishes. Gauge theories are more subtle: integrality relates to gauge symmetries., Comment: 5pages, Talk given at "RadCor 2002 - Loops and Legs 2002", Kloster Banz, Germany, Sep 8-13, 2002

Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the role they can play in future developments., Comment: 26p,4figs., Invited Contribution to Annals of Physics, minor typos corrected

The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine., Comment: 21 pages, eps figures

We discuss structural similarities between Knizhnik--Zamolodchikov equations (in fact, their simplest version needed to introduce the Drinfeld associator) and Dyson--Schwinger equations. We emphasize that the latter allow for a filtration by co-radical degree using quasi-shuffle products and the lower central series filtration of the Lie algebra of Feynman graphs. This clarifies how they are a generalization of the KZ equations. This is a starting point for a algebraic organization of the next-to...-to leading log expansion which has been worked out in collaboration with Olaf Krueger and which will be given elsewhere [1,2]., 7 pages, Proceedings "Loops and Legs in Quantum Field Theory - LL 2014" (PoS)

We study the infrared (large separation) behavior of a massless minimally coupled scalar quantum field theory with a quartic self interaction in de Sitter spacetime. We show that the perturbation series in the interaction strength is singular and secular, i.e. it does not lead to a uniform approximation of the solution in the infrared region. Only a nonperturbative resummation can capture the correct infrared behavior. We seek to justify this picture using the Dyson-Schwinger equations in the ladder-rainbow approximation. We are able to write down an ordinary differential equation obeyed by the two-point function and perform its asymptotic analysis. Indeed, while the perturbative series-truncated at any finite order-is growing in the infrared, the full nonperturbative sum can be decaying., 4 figures

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar $\phi^3$ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension $d=4$, we obtain an integrodifferential Dyson-Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at $d=6$, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator-coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams., Comment: 20p, 2 epsf fig

We review the algebraic structures imposed on the renormalization procedure in terms of Hopf and Lie algebras of Feynman graphs, and exhibit the connection to diffeomorphisms of physical observables.

It is easy to sum chain-free self-energy rainbows, to obtain contributions to anomalous dimensions. It is also easy to resum rainbow-free self-energy chains. Taming the combinatoric explosion of all possible nestings and chainings of a primitive self-energy divergence is a much more demanding problem. We solve it in terms of the coproduct $\Delta$, antipode S, and grading operator Y of the Hopf algebra of undecorated rooted trees. The vital operator is $S\star Y$, with a star product effected by $\Delta$. We perform 30-loop Pad\'e-Borel resummation of 463 020 146 037 416 130 934 BPHZ subtractions in Yukawa theory, at spacetime dimension d=4, and in a trivalent scalar theory, at d=6, encountering residues of $S\star Y$ that involve primes with up to 60 digits. Even with a very large Yukawa coupling, g=30, the precision of resummation is remarkable; a 31-loop calculation suggests that it is of order $10^{-8}$., Comment: 10 pages, LaTeX, 1 eps figure added, comments expanded

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.

The recently discovered formalism underlying renormalization theory, the Hopf algebra of rooted trees, allows to generalize Chen’s lemma. In its generalized form it describes the change of a scale in Green functions, and hence relates to the operator product expansion. Hand in hand with this generalization goes the generalization of the ordinary factorial n! to the tree factorial t. Various identities on tree-factorials are derived which clarify the relation between Connes-Moscovici weights and Quantum Field Theory.

We investigate to what extent renormalization can be understood as an algebraic manipulation on concatenated one-loop integrals. We find that the resulting algebra indicates a useful connection to knot theory as well as number theory and report on recent results in support of this connection.

Counterterms that are not reducible to ζn are generated by 3F2 hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots (4, 3) = 819 and (5, 3) = 10124, are found in anomalous dimensions at O(1/N 3) in the large-N limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Pade resummations of e-expansions, which are compared with analytical results in 3 dimensions. The O(1/N 3) results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in 1/N one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots 10139 and 10152 are not generated by such self-energy insertions.

The package XLOOPS presented in this workshop relies on the application of parallel space techniques. We introduce these techniques covering the following topics: • - The generation of integral representations for massive two-loop diagrams. • - The treatment of tensor structures. • - The handling of the γ-algebra in this scheme. • - The separation of UV and IR divergences in analytic expressions. We present two-loop examples taken from Standard Model calculations.

We calculate the two-loop vertex function for the crossed topology, and for arbitrary masses and external momenta. We derive a double integral representation, suitable for a numerical evaluation by a Gaussian quadrature. Real and imaginary parts of the diagram can be calculated separately., Comment: 16 pages, latex, uses epsfig, amstex, amssymb. Replacement has typo corrected, references updated

We exhibit in the simple example of the Dunce's cap Feynman graph the structure of parametric renormalization on the level of integrands, and exhibit also the decomposition $\Phi^R=\Phi_{\mathrm{fin}}^{-1}(\Theta_0)\star \Phi^R_{\mathrm{\makebox{1-s}}}(S/S_0)\star\Phi_{\mathrm{fin}}(\Theta)$ into angles and scales of renormalized Feynman rules., Comment: 6p, 1 figure. Proceedings of the 11th DESY workshop "Loops and Legs in Quantum Field Theory", April 2012, Werningerode, Germany

We study renormalization in a kinetic scheme (realized by subtraction at fixed external parameters as implemented in the BPHZ and MOM schemes) 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.

We consider massless Quantum Electrodynamics in momentum scheme and carry forward an approach based on Dyson-Schwinger equations to approximate both the $\beta$-function and the renormalized photon self-energy [Y11]. Starting from the Callan-Symanzik equation, we derive a renormalization group (RG) recursion identity which implies a non-linear ODE for the anomalous dimension and extract a sufficient but not necessary criterion for the existence of a Landau pole. This criterion implies a necessary condition for QED to have no such pole. Solving the differential equation exactly for a toy model case, we integrate the corresponding RG equation for the running coupling and find that even though the $\beta$-function entails a Landau pole it exhibits a flat contribution capable of decreasing its growth, in other cases possibly to the extent that such a pole is avoided altogether. Finally, by applying the recursion identity, we compute the photon propagator and investigate the effect of flat contributions on both spacelike and timelike photons., Comment: 16 pages, 7 figures

We evaluate all the primitive divergences contributing to the 7--loop $\beta$\/--function of $\phi^4$ theory, i.e.\ all 59 diagrams that are free of subdivergences and hence give scheme--independent contributions. Guided by the association of diagrams with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams, associated with the knots $10_{124}$, $10_{139}$, and $10_{152}$, are evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered and the transcendental number associated with it is found. Thus we achieve an analytical result for the 6--loop contributions, and a numerical result at 7 loops that is accurate to one part in $10^{11}$. The series of `zig--zag' counterterms, $\{6\zeta_3,\,20\zeta_5,\, \frac{441}{8}\zeta_7,\,168\zeta_9,\,\ldots\}$, previously known for $n=3,4,5,6$ loops, is evaluated to 10 loops, corresponding to 17 crossings, revealing that the $n$\/--loop zig--zag term is $4C_{n-1} \sum_{p>0}\frac{(-1)^{p n - n}}{p^{2n-3}}$, where $C_n=\frac{1}{n+1}{2n \choose n}$ are the Catalan numbers, familiar in knot theory. The investigations reported here entailed intensive use of REDUCE, to generate ${\rm O}(10^4)$ lines of code for multiple precision FORTRAN computations, enabled by Bailey's MPFUN routines, running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines., Comment: 6 pages plain LaTex

It is shown that the two-loop four-point functions are similar in structure to the three-point two-loop functions for all mass cases and topologies. The result is derived by using a rotation to a (+,-,-,+) signature without spoiling analyticity properties., Comment: 7 pages, UTAS-PHYS-94-23, plain LATEX

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., 58 pages, many figures

A review of the Hodge and Hopf-algebraic approach to QFT., 6p, talk given at Loops and Legs 2010

We present a new calculation of the well-known one-loop two-point scalar and tensor functions. We also present a systematic reduction to a certain class of functions which minimizes the effort for calculating tensor integrals drastically. We avoid standard techniques such as Feynman parametrization and Wick rotation.

We present a calculation of the two-loop three-point scalar functions for the two overlapping topologies. These are the master functions for the ladder and the crossed ladder graphs. We also present a method for the extraction of possible (on-shell) mass singularities.