Lambert-W solves the noncommutative ${\Phi }^4$-model

Abstract

The closed Dyson-Schwinger equation for the 2-point function of the noncommutative $\lambda {\varphi }_2^4$-model is rearranged into the boundary value problem for a sectionally holomorphic function in two variables. We prove an exact formula for a solution in terms of Lambert's $W$-function. This solution is holomorphic in $\lambda$ inside a domain which contains $(-1/\log 4,\infty )$. Our methods include the Hilbert transform, perturbation series and Lagrange-Bürmann resummation.

arXiv commentLaTeX, 28 pages, 2 figures. v2: main conjecture in v1 is now a theorem

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