This is an ongoing research project with the goal to examine to which extent the solutions of Dyson-Schwinger equations are dependent on particular features of the equation. For this project, I mostly consider DSEs with a single kernel. Physically, this amounts to inserting one type of Feynman diagram into itself in all possible ways.

Varying insertion exponents

When recursively inserting a diagram into itself, this can be done in many different ways. Here, we are looking at inserting a propagator correction into an internal edge of the diagram. The propagator Green function is a 1PI Green function, that means, the actual quantum correction of an edge should be an infinite geometric series of the 1PI correction. Formally, one inserts instead of .

Since the insertion of itself is a “base case” (the DSE is then linear in ), it is natural to encode the exponent in the form , where is some constant. The physically relevant case of inserting a geometric series is then . We immediately note that is a trivial case, in this case, nothing is being inserted at all and the solution of the DSE is simply the underlying kernel graph, without any recursive insertion. Setting would correspond to inserting “two geometric series”. This is what happens when corrections are being inserted into more than one edge, but note that for the actual computation, it of course makes a difference whether the correction is inserted into distinct edges or two times into the same edge. For the beginning, we are looking at insertions into only one edge, then is the unique physically sensible answer and everything else amounts to summing Feynman graphs with non-standard combinatorial factors.

The outcomes of inserting arbitrary exponents are explained in my thesis. Qualitatively, one finds that small variations of have relatively little effect on the overall features of the solution. In particular, unless or , the formal power series solution is factorially divergent. At the point , which corresponds to the linear Dyson-Schwinger equation, the growth parameters of the perturbative solutions diverge, indicating a qualitative change in the shape of the resummed solution function. This is indeed what one sees from plotting numerical solutions for different . The plot below shows the anomalous dimension. Numerical solutions are shown as grey lines. The green line is the trivial DSE , the red line is the (convergent) solution of the linear DSE . The blue line is the exact resummed solution for given by Broadhurst and Kreimer.

Non-kinematic schemes as shifted kinematic schemes

Dyson-Schwinger equations can be formulated in many different ways. In one of them, they are integral equations for Green functions. Perturbatively, the Green functions are formal power series in the coupling and one (or more) kinematic parameters. In this setting, kinematic renormalization conditions are explicit physical boundary conditions for these power series: Introducing e.g. a logarithmic energy scale L, one demands that the Green function is unity at the kinematic renormalization point L=0. This greatly simplifies the analytic treatment, and it allows to transform the integral equation into a (pseudo-)differential equation for the anomalous dimension, which can then be solved in terms of power series to extremely high loop order (several hundreds of loops).

The most popular non-kinematic renormalization scheme is probably the Minimal Subtraction scheme, which is defined “algorithmically”. Its defining feature is that it produces the easiest possible counterterms. This makes it easy to use when renormalizing complicated graphs individually, but it is unsuitable for an analytic treatment of Dyson-Schwinger equations because it does not enforce any particular analytic condition. Instead, to solve a Dyson-Schwinger equation in MS, one needs to compute the counterterms order by order to “see what one gets”.

The central outcome of the article AIHPD 2023/169 and my thesis is that in fact all renormalization schemes, including MS, can be interpreted as kinematic schemes, but with a renormalization point different from L=0. It is possible to compute this point iteratively. Moreover, for the special case of (a certain class of) linear Dyson-Schwinger equations, it is possible to derive the shifted renormalization point that belongs to the MS scheme analytically from the Mellin transform of the kernel graph. With that, it is possible to solve certain linear DSEs exactly to all orders in MS, something that had so far only been possible in kinematic renormalization.