Renormalization conditions and resurgence

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ArticleAIHPD 2023/169

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, which in turn 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 in certain special cases it is possible to transform the integral equation into an explicit 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 has so far only been possible in kinematic renormalization.