Tubing solutions to Dyson-Schwinger equations

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Dyson-Schwinger equations for Feynman graphs

There are many alternative views of what a Dyson-Schwinger equation is. The fundamental idea is that an interacting quantum field theory needs to be self-consistent in the sense that an interaction process that can in principle happen must be included in all applicable situations. Formulated for (not necessarily perturbative) Green functions, this means that a Green function with higher valence must receive contributions from lower-valence ones, or conversely that low-valence Green functions must be recovered from high-valence ones upon identification of external legs.

In perturbation theory, these identities become first of all statements about certain graphs: For example, if the theory admits a 4-valent interaction, then the corresponding 4-valent Feynman graphs must be included in all places where a Feynman graph has a 4-valent subgraph. Since subgraphs are sub-integrals of Feynman integrals, the Dyson-Schwinger equation turns into an infinite sequence of integral equations. In the special case of 1-scale Feynman graphs, it is under certain further assumptions possible to recast the Dyson-Schwinger equation as an ordinary (pseudo-) differential equation, compare my other research project.

Rooted trees

In renormalizable quantum field theories, the superficial degree of divergence is given by the number and type of external edges of a Feynman integral. A Feynman graph will in general have divergent subgraphs (namely, propagator and vertex corrections as dictated by the Dyson-Schwinger equations), and these subgraphs themselves may have further divergent subgraphs. This gives the nested divergences the structure of a rooted tree. A consistent renormalization scheme requires that free parameters of the theory should be redefined in terms of measurable quantities, this gives rise to counterterms with a very specific, well-defined structure. This “redefinition of formal power series” has an algebraic structure, which turns out to induce a Hopf algebra structure on the rooted trees which encode the nested divergences. Hence, the question which subgraphs and counterterms need to be included with which signs and combinatorial factors can be entirely answered by examining the rooted trees corresponding to a Feynman graph.

A priori, the rooted trees arise from the desire to give a clean systematic description of the renormalization procedure. On the other hand, since the rooted trees describe the presence of subgraphs, which in turn is required from Dyson-Schwinger equations, it is plausible to ask whether the Dyson-Schwinger equation can be formulated directly in terms of rooted trees, instead of Feynman graphs. This can indeed be done, the operator “insert a given subgraph into a graph” becomes “add a given tree as a child to a root“. This operator, called B_+\lates, turns out to be a Hochschild-1-cocycle in the Hopf algebra of rooted trees. Using this operator, a Dyson-Schwinger equation schematically has the form

    \begin{align*} G = 1 + \alpha B_+ (G Q), \end{align*}

where G is some Green function, \alpha is a coupling constant, and Q is the “invariant charge”, which is itself a monomial in the various Green functions of the theory. In realistic theories, there are multiple coupled equations of this type and they have multiple (or even infinitely many) summands on the right hand side. The solution to a Dyson-Schwinger equation of this form is a power series of rooted trees, where \alpha is the series expansion variable. A much more detailed exposition of this formalism can be found in my thesis. As an example, the single-equation case with Q=1 is called a linear DSE (since the argument of B_+ is a linear function of G), it has a solution in terms of ladder trees

The Dyson-Schwinger equation with Q=G leads to binary trees,

Mellin transforms and tree Feynman rules

We now have two distinct solutions of Dyson-Schwinger equations: On the one hand the ordinary perturbation series in terms of Feynman graphs, on the other hand a series in rooted trees encoding the divergence structure. It would be nice to recover the former from the latter, but in general, this is impossible without giving additional information: Many distinct Feynman graphs can correspond to the same rooted tree, and the Feynman integral is a function of various energies and angles of its external particles. The situation improves if we, instead of computing the full functional form of Feynman amplitudes, restrict ourselves to just their contribution to the beta function, because this contribution is just a real number, not a function, for each Feynman graph.

The Mellin transform is an integral transform of the Feynman integral. The relevant Feynman integrals for our application are the “kernel” graphs of Dyson-Schwinger equations, these are the superficially divergent graphs without subdivergences. Their Mellin transform has a simple pole at zero, reflecting the superficial divergence (this is analogous to, but not the same as, the fact that in dimensional regularization these graphs have a simple pole in the regularization parameter \epsilon). If the coefficients c_{-1},c_0,c_1,c_2,\ldots of a power series expansion of the relevant Mellin transforms are known, the Dyson-Schwinger equation is essentially a combinatorial algorithm of how these coefficients should be combined and added to form physical quantities such as the beta function. Indeed, every rooted tree corresponds to a polynomial in the Mellin coefficients, this mapping is known as “Tree Feynman rules”. So far, the only way to compute tree Feynamn rules was by iteratively integrating and deriving certain formal power series according to the algebraic structure of renormalization encoded in the tree. Let L be the logarithmic momentum scale, then the first few renormalized tree Feynman rules read

Only the part proportional to L is relevant in these expressions, the coefficients of all higher orders L^k can be recovered trivially from the renormalization group.


Our new contribution, which is joint work with Nick Olson-Harris, Karen Yeats, Kurusch Ebrahimi-Fard, Amelia Cantwell and Lukas Nabergall, is an explicit combinatorial procedure to obtain the tree Feynman rules without any manipulation of power series. The physical intuition is that the coproduct in renormalization splits a given integral into parts which are replaced by counterterms, and parts which are unaltered. The renormalized integral is then the sum over all possible such splittings, which must be nested such that eventually all involved graphs are decomposed into subdivergence-free subgraphs. In the language of rooted trees, this means that we want to iteratively split a rooted tree into two pieces until all pieces are single vertices (because every vertex in a rooted tree represents a superficially divergent graph). There is a beautiful graphical notation for this, the “binary tubings”: Split the tree by cutting one edge, draw one “tube” each around the upper and around the lower part, and keep splitting the individual parts until every vertex resides in its own tube. The beta function is then given by first a sum over all relevant rooted trees, and then, for each of the trees, a sum over all of its tubings (where the number of tubings is not necessarily the same for two trees of the same number of vertices). It turns out that each of the tubings corresponds to exactly one monomial in the Mellin coefficients, and it is trivial to determine this monomial by counting, for each vertex, for how many tubes it is the root vertex.

A similar combinatorial expansion of renormalized amplitudes had been known before, it was constructed in terms of certain chord diagrams. The new tubings expansion reproduces all results of the chord diagram expansion, and has the benefit that it operates directly on rooted trees which have a clear physical interpretation, whereas the chord diagrams were auxiliary objects introduced purely for combinatorial purposes.


One strength of Feynman graphs is that they can be interpreted as depicting a sequence of elementary interactions contributing to an observable. At high order in perturbation theory, there are extremely many distinct Feynman graphs (see my other project), and it is increasingly troublesome to compute all these integrals. Nevertheless, even then they allow for qualitative conclusions, e.g. to find out how quickly the perturbation series grows by counting the number of graphs.

Much of the interpretation of Feynman graphs is lost upon renormalization: The original Feynman graph shows unrenormalized processes. While it is possible to denote the renormalization process graphically by introducing counterterm vertices, this makes it increasingly obscure which terms, with which signs and factors, will be present at a given order.

The beauty of the tubings expansion is that it restores the systematic graphical notation for the QFT perturbation series, but on the level of renormalized amplitudes. By considering each tubing, instead of merely each Feynman graph, we have one “graphical object” for every term that appears in the perturbation series. Of course, summing all tubings reproduces the original, more “coarse” expansion in terms of Feynman graphs, in the same way that summing all Feynman graphs reproduces the more coarse expansion in terms of powers of the coupling (i.e. many Feynman graphs contribute to the same order in the coupling, and many tubings contribute to the same Feynman graph).

An expansion in Feynman graphs is not always the most efficient way to solve a problem in quantum field theory. Similarly, constructing all the tubings might not necessarily be computationally faster (with a computer) than other methods to compute renormalized Green functions. The strength of the tubings is rather to clarify and organize the terms in the expansion. In future work, it might turn out that these tubings can themselves be grouped in different ways. As a first hint towards this, we note that tubings naturally have the structure of associahedra: