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Lambert-W Solves the Noncommutative \(\varPhi ^4\)-Model

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Abstract

The closed Dyson–Schwinger equation for the 2-point function of the noncommutative \(\lambda \phi ^4_2\)-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 \(\lambda \) 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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Notes

  1. A behaviour \(\sum _n \sim V\) is assumed so that the sums in (8) are kept.

  2. For Hölder-continuous G(a, b), this is in fact just the ordinary integral. But since we will pull the numerators apart later, we write principal values already here.

  3. RW would like to thank Alexander Hock for pointing out this reference.

  4. Our case is isomorphic to the moduli space \(\mathfrak {M}_{0,5}\) of genus zero curves with 5 marked points.

  5. HyperInt can also find (23) by integrating directly the perturbative expansion of (1). Furthermore, HyperInt computes \(H=\int _0^{\infty } \frac{\mathrm {d}^{}p}{p-a} \tau _b(p)\) as an integral over real p, adding an imaginary part \({\mathrm {i}}\epsilon \delta _a\) to a. The Hilbert transform of \(\tau _b(a)\) is thus the real part of H (i.e. drop the \(\delta _a\)-term).

  6. At first we were unaware of (17c) and calculated the much harder \(\int _0^\infty \mathrm {d}^{}p \;e^{-{\mathscr {H}}_p[\tau _a]}\sin \tau _a(p)\).

  7. It is well-known that the expansion of Lambert-W at infinity is related to Stirling numbers, see [7]. However, we did not find the precise form we obtain here in the literature.

  8. These formulae are reproduced as (4c) and (4b) in the beginning.

  9. Cochleoid refers to ‘snail-shaped’. Its reciprocal \(\frac{2b}{\pi } \frac{\theta }{\sin \theta } e^{{\mathrm {i}}\theta }\) is the quadratrix of Hippias used in classical antiquity to trisect an angle or to square a circle.

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Acknowledgements

We are grateful to Spencer Bloch and Dirk Kreimer for invitation to the Les Houches summer school “Structures in local quantum field theories” where the decisive results of this paper were obtained. Discussions during this school in particular with Johannes Blümlein, David Broadhurst and Gerald Dunne contributed valuable ideas. RW would like to thank Alexander Hock for pointing out reference [9] and Harald Grosse for the long-term collaboration which preceded this work.

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Authors and Affiliations

  1. All Souls College, University of Oxford, OX1 4AL, Oxford, UK

    Erik Panzer

  2. Mathematisches Institut der WWU, Einsteinstr. 62, 48149, Münster, Germany

    Raimar Wulkenhaar

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  1. Erik Panzer
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Correspondence to Raimar Wulkenhaar.

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Panzer, E., Wulkenhaar, R. Lambert-W Solves the Noncommutative \(\varPhi ^4\)-Model. Commun. Math. Phys. 374, 1935–1961 (2020). https://doi.org/10.1007/s00220-019-03592-4

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