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

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-B"urmann resummation.