Lambert-W solves the noncommutative $Φ^4$-model
We show that the closed Dyson-Schwinger equation for the 2-point function of the noncommutative λϕ42-model can be rearranged into the boundary value problem Ψ(a+,b+)Ψ(a−,b−)=Ψ(a+,b−)Ψ(a−,b+) for a sectionally holomorphic function Ψ. This expresses the 2-point function as Hilbert transform of an angl...
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| Format: | Working paper |
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2018
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| _version_ | 1826286307645587456 |
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| author | Panzer, E Wulkenhaar, R |
| author_facet | Panzer, E Wulkenhaar, R |
| author_sort | Panzer, E |
| collection | OXFORD |
| description | We show that the closed Dyson-Schwinger equation for the 2-point function of the noncommutative λϕ42-model can be rearranged into the boundary value problem Ψ(a+,b+)Ψ(a−,b−)=Ψ(a+,b−)Ψ(a−,b+) for a sectionally holomorphic function Ψ. This expresses the 2-point function as Hilbert transform of an angle function which itself satisfies a highly non-linear integral equation. A solution of that equation as formal power series in λ shows a surprisingly simple structure. The solution to 10th order is matched by Stirling numbers of the first kind. Its extrapolation to all orders is resummed with the Lagrange-B\"urmann formula to Lambert-W. This leads to an explicit exact formula of the 2-point function, real-analytic at any coupling constant λ>−1/(2log2), in terms of Lambert-W. |
| first_indexed | 2024-03-07T01:41:49Z |
| format | Working paper |
| id | oxford-uuid:972224ac-f3b2-42a9-a4bf-8fe4deedacd6 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:41:49Z |
| publishDate | 2018 |
| record_format | dspace |
| spelling | oxford-uuid:972224ac-f3b2-42a9-a4bf-8fe4deedacd62022-03-26T23:57:22ZLambert-W solves the noncommutative $Φ^4$-modelWorking paperhttp://purl.org/coar/resource_type/c_8042uuid:972224ac-f3b2-42a9-a4bf-8fe4deedacd6Symplectic Elements at Oxford2018Panzer, EWulkenhaar, RWe show that the closed Dyson-Schwinger equation for the 2-point function of the noncommutative λϕ42-model can be rearranged into the boundary value problem Ψ(a+,b+)Ψ(a−,b−)=Ψ(a+,b−)Ψ(a−,b+) for a sectionally holomorphic function Ψ. This expresses the 2-point function as Hilbert transform of an angle function which itself satisfies a highly non-linear integral equation. A solution of that equation as formal power series in λ shows a surprisingly simple structure. The solution to 10th order is matched by Stirling numbers of the first kind. Its extrapolation to all orders is resummed with the Lagrange-B\"urmann formula to Lambert-W. This leads to an explicit exact formula of the 2-point function, real-analytic at any coupling constant λ>−1/(2log2), in terms of Lambert-W. |
| spellingShingle | Panzer, E Wulkenhaar, R Lambert-W solves the noncommutative $Φ^4$-model |
| title | Lambert-W solves the noncommutative $Φ^4$-model |
| title_full | Lambert-W solves the noncommutative $Φ^4$-model |
| title_fullStr | Lambert-W solves the noncommutative $Φ^4$-model |
| title_full_unstemmed | Lambert-W solves the noncommutative $Φ^4$-model |
| title_short | Lambert-W solves the noncommutative $Φ^4$-model |
| title_sort | lambert w solves the noncommutative φ 4 model |
| work_keys_str_mv | AT panzere lambertwsolvesthenoncommutativeph4model AT wulkenhaarr lambertwsolvesthenoncommutativeph4model |