Built on

N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen
1909
Earlier work this paper cites.
Interscience, New York, 1957
F. G. Tricomi, Integral equations · 1957
Earlier work this paper cites.
Pergamon Press, Oxford, 1966
F. D. Gakhov, Boundary value problems · 1966
Earlier work this paper cites.
K. S. Kölbig, J. A. Mignaco and E. Remiddi, On Nielsen’s generalized polylogarithms and their numerical calculation , BIT Numerical Mathematics 10
1970
Earlier work this paper cites.
K. S. Kölbig, Nielsen’s generalized polylogarithms , SIAM Journal on Mathematical Analysis 17
1986
Earlier work this paper cites.
J. M. Gracia-Bondía and J. C. Várilly, Algebras of distributions suitable for phase space quantum mechanics. I. , J. Math. Phys. 29
1988
Earlier work this paper cites.
M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Commun. Math. Phys. 147
1992
Earlier work this paper cites.
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function , Adv. Comput. Math. 5
1996
Earlier work this paper cites.
J. M. Borwein, D. M. Bradley, D. J. Broadhurst and P. Lisoněk, Special values of multiple polylogarithms , Trans. Amer. Math. Soc. 353
2001
Earlier work this paper cites.

Similar

H. Grosse and R. Wulkenhaar, Renormalisation of ϕ 4 \phi^{4} -theory on noncommutative ℝ 2 \mathbb{R}^{2} in the matrix base , JHEP 2003
2003
Cited alongside, same era.
M. Disertori, R. Gurau, J. Magnen and V. Rivasseau, Vanishing of beta function of non commutative ϕ 4 4 \phi^{4}_{4} theory to all orders , Phys. Lett. B649
2007
Cited alongside, same era.
F. C. S. Brown, Multiple zeta values and periods of moduli spaces 𝔐 ¯ 0 , n \overline{\mathfrak{M}}_{0,n} , Ann. Sci. Éc. Norm. Supér. (4) 42
2009
Cited alongside, same era.
J. Ablinger, J. Blümlein and C. Schneider, Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms , J. Math. Phys. 54
Original
2013
Cited alongside, same era.
J. M. Henn and V. A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I , JHEP 1311
Original

Then

Maplesoft, a division of Waterloo Maple Inc., “Maple 2015.”
2015
Later among the works it cites.
E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals , Computer Physics Communications 188
Original
2015
Later among the works it cites.
C. Bogner, MPL-a program for computations with iterated integrals on moduli spaces of curves of genus zero , Comput. Phys. Commun. 203
Original
2016
Later among the works it cites.
H. Grosse and R. Wulkenhaar, On the fixed point equation of a solvable 4D QFT model , Vietnam J. Math. 44
Original
2016
Later among the works it cites.
H. Grosse, A. Sako and R. Wulkenhaar, Exact solution of matricial Φ 2 3 \Phi^{3}_{2} quantum field theory , Nucl. Phys. B925
Original

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…

2018

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

Panzer, Erik, Wulkenhaar, Raimar

Understand

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-B\"urmann resummation.

Built on

N. Nielsen, Der Eulersche Dilogarithmus und seine Verallgemeinerungen
1909
Earlier work this paper cites.
Interscience, New York, 1957
F. G. Tricomi, Integral equations · 1957
Earlier work this paper cites.
Pergamon Press, Oxford, 1966
F. D. Gakhov, Boundary value problems · 1966
Earlier work this paper cites.
K. S. Kölbig, J. A. Mignaco and E. Remiddi, On Nielsen’s generalized polylogarithms and their numerical calculation , BIT Numerical Mathematics 10
1970
Earlier work this paper cites.
K. S. Kölbig, Nielsen’s generalized polylogarithms , SIAM Journal on Mathematical Analysis 17
1986
Earlier work this paper cites.
J. M. Gracia-Bondía and J. C. Várilly, Algebras of distributions suitable for phase space quantum mechanics. I. , J. Math. Phys. 29
1988
Earlier work this paper cites.
M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function , Commun. Math. Phys. 147
1992
Earlier work this paper cites.
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function , Adv. Comput. Math. 5
1996
Earlier work this paper cites.
J. M. Borwein, D. M. Bradley, D. J. Broadhurst and P. Lisoněk, Special values of multiple polylogarithms , Trans. Amer. Math. Soc. 353
2001
Earlier work this paper cites.

Similar

H. Grosse and R. Wulkenhaar, Renormalisation of ϕ 4 \phi^{4} -theory on noncommutative ℝ 2 \mathbb{R}^{2} in the matrix base , JHEP 2003
2003
Cited alongside, same era.
M. Disertori, R. Gurau, J. Magnen and V. Rivasseau, Vanishing of beta function of non commutative ϕ 4 4 \phi^{4}_{4} theory to all orders , Phys. Lett. B649
2007
Cited alongside, same era.
F. C. S. Brown, Multiple zeta values and periods of moduli spaces 𝔐 ¯ 0 , n \overline{\mathfrak{M}}_{0,n} , Ann. Sci. Éc. Norm. Supér. (4) 42
2009
Cited alongside, same era.
J. Ablinger, J. Blümlein and C. Schneider, Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms , J. Math. Phys. 54
Original
2013
Cited alongside, same era.
J. M. Henn and V. A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I , JHEP 1311
Original

Then

Maplesoft, a division of Waterloo Maple Inc., “Maple 2015.”
2015
Later among the works it cites.
E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals , Computer Physics Communications 188
Original
2015
Later among the works it cites.
C. Bogner, MPL-a program for computations with iterated integrals on moduli spaces of curves of genus zero , Comput. Phys. Commun. 203
Original
2016
Later among the works it cites.
H. Grosse and R. Wulkenhaar, On the fixed point equation of a solvable 4D QFT model , Vietnam J. Math. 44
Original
2016
Later among the works it cites.
H. Grosse, A. Sako and R. Wulkenhaar, Exact solution of matricial Φ 2 3 \Phi^{3}_{2} quantum field theory , Nucl. Phys. B925
Original

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

2013

Cited alongside, same era.

H. Grosse and R. Wulkenhaar, Self-dual noncommutative ϕ 4 \phi^{4} -theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory , Commun. Math. Phys. 329

Original

2014

Cited alongside, same era.

PhD thesis, Humboldt-Universität zu Berlin, 2014

Original

E. Panzer, Feynman integrals and hyperlogarithms · 2014

Cited alongside, same era.

arXiv:1310.0160 [cs.SC]

Original

C. Schneider, Modern Summation Methods for Loop Integrals in Quantum Field Theory: The Packages Sigma, EvaluateMultiSums and SumProduction , in Journal of Physics Conference Series , vol. 523 of Journal of Physics Conference Series , p. 012037, June, 2014 · 2014

Cited alongside, same era.

Springer Japan, 2014

I. Todorov, Polylogarithms and multizeta values in massless Feynman amplitudes , in Lie Theory and Its Applications in Physics (V. Dobrev, ed.), vol. 111 of Springer Proceedings in Mathematics & Statistics , pp. 155–176 · 2014

Cited alongside, same era.

J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel and C. Schneider, The three-loop splitting functions P q g ( 2 ) P_{qg}^{(2)} and P g g ( 2 , N F ) P_{gg}^{(2,N_{F})} , Nuclear Physics B 922

Original

Cited in the paper.

Bürmann, H., Essai de calcul fonctionnaire aux constantes ad-libitum

Cited in the paper.

J. L. Lagrange, Nouvelle méthode pour résoudre des équations littérales par le moyen de séries

Cited in the paper.

2017

Later among the works it cites.

S. Laporta, High-precision calculation of the 4-loop contribution to the electron g − 2 g-2 in QED , Physics Letters B 772

Original