Abstract
For certain quantum field theories, the Kreimer-Connes Hopf-algebraic approach to renormalization reduces the Dyson-Schwinger equations to a system of non-linear ordinary differential equations for the expansion coefficients of the renormalized Green's function. We apply resurgent asymptotic analysis to find the trans-series solutions which provide the non-perturbative completion of these formal Dyson-Schwinger expansions. We illustrate the general approach with the concrete example of four dimensional massless Yukawa theory, connecting with the exact functional solution found by Broadhurst and Kreimer. The trans-series solution is associated with the iterative form of the Dyson-Schwinger equations, and displays renormalon-like structure of integer-repeated Borel singularities. Extraction of the Stokes constant is possible due to a property we call `functional resurgence'.
Highlights
The Kreimer-Connes approach to renormalization in quantum field theory recasts the perturbative renormalization process in Hopf-algebraic terms, leading to new perspectives as well as new computational methods [1,2,3,4,5]
We illustrate the general method by considering a local quantum field theory with a Green’s function depending on a single running coupling, α, and a single kinematical variable, L = ln q2/μ2, where μ is the renormalization scale. It has been shown by Broadhurst and Kreimer [10, 11], and Kreimer and Yeats [12, 13], that the recursive Hopf-algebraic structure of the Dyson-Schwinger equations, combined with the renormalization group equations describing the anomalous scaling under re-scaling of parameters, reduces the problem to a set of non-linear ordinary differential equations (ODEs)
In this paper we study the associated ODEs using resurgent asymptotics and alien calculus, complementary approaches which yield non-perturbative trans-series solutions, whose expansions display familiar features of resurgence such as large-order/low-order relations
Summary
The Kreimer-Connes approach to renormalization in quantum field theory recasts the perturbative renormalization process in Hopf-algebraic terms, leading to new perspectives as well as new computational methods [1,2,3,4,5]. We illustrate the general method by considering a local quantum field theory with a Green’s function depending on a single running coupling, α, and a single kinematical variable, L = ln q2/μ2, where μ is the renormalization scale It has been shown by Broadhurst and Kreimer [10, 11], and Kreimer and Yeats [12, 13], that the recursive Hopf-algebraic structure of the Dyson-Schwinger equations, combined with the renormalization group equations describing the anomalous scaling under re-scaling of parameters, reduces the problem to a set of non-linear ordinary differential equations (ODEs). Renormalons have been studied recently using ideas from resurgence, in a wide variety of theories: see for example [24,25,26,27,28,29,30,31], and references therein
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