Variants of single-kernel Dyson-Schwinger equations

The single-kernel DSEs which express the insertion of 1-loop multiedge-
or triangle graphs into each other have been solved to all orders in the
linear case by Delbourgo and collaborators in 1996, and in the
non-linear case by Broadhurst and Kreimer in 2001. Subsequent work by
Bellon, Borinsky, Broadhurst, Dunne, Kreimer, Yeats, and others has
further improved our understanding of solution methods, the asyptotic
growth of series coefficients, and the resurgence properties of these
DSEs. The divergence of the power series of the anomalous dimension in
these systems is commonly called "renormalon".

In my talk, I will first review established methods and known results
for the 1-loop multiedge DSE. Then, I will discuss to what extent the
qualitative properties of the solution depend on the particular DSE in
question. Concretely, the 1-loop multiedge DSE can be modified naturally
along three axes: Firstly, the power series coefficients of the anomalous
dimension depend on the chosen renormalization scheme, where almost
all existing work has used kinematic renormalization conditions.
Secondly, the DSE contains a parameter $w$ which controls the power of
insertion in the recurrence and hence the "non-linearity" of the DSE.
$w=0$ is the linear DSE and $w=-2$ is the physically relevant non-linear
case that has mostly been studied so far. By analytic continuation, one
can consider other, even non-integer, exponents, which smoothly
interpolate between the linear and the physical case. Thirdly, one can
insert subgraphs into both edges of the 1-loop multiedge kernel, 
potentially with different exponents.

All these choices have an influence on the power series coefficients of
the resulting anomalous dimension, and potentially even on their
asymptotics. I will present analytic and/or numerical findings for
various combinations of these choices in the multiedge DSE model, and
discuss qualitative conclusions and open questions. The talk is based on
AIHPD 2023/169 and my thesis (Springer, 2024), as well as on current
work in progress.