Lambert-W solves the noncommutative $Φ^4$-model

The closed Dyson-Schwinger equation for the 2-point function of the noncommutative $λϕ^4_2$-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 $λ$ inside a domain which contains $(-1/\log 4,\infty)$. Our methods include the Hilbert transform, perturbation series and Lagrange-Bürmann resummation.

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