Perturbative renormalisation of the $Φ^4_{4-\varepsilon}$ model via generalized Wick maps
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| Formato: | Preprint |
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2025
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| author | Berglund, Nils Klose, Tom Tapia, Nikolas |
| author_facet | Berglund, Nils Klose, Tom Tapia, Nikolas |
| contents | We consider the perturbative renormalisation of the $Φ^4_d$ model from Euclidean Quantum Field Theory for any, possibly non-integer dimension $d<4$. The so-called BPHZ renormalisation, named after Bogoliubov, Parasiuk, Hepp and Zimmermann, is usually encoded into extraction-contraction operations on Feynman diagrams, which have a complicated combinatorics. We show that the same procedure can be encoded in the much simpler algebra of polynomials in two unknowns $X$ and $Y$, which represent the fourth and second Wick power of the field. In this setting, renormalisation takes the form of a \lq\lq Wick map\rq\rq\ which maps monomials into Bell polynomials. The construction makes use of recent results by Bruned and Hou on multiindices, which are algebraic objects of intermediate complexity between Feynman diagrams and polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03820 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perturbative renormalisation of the $Φ^4_{4-\varepsilon}$ model via generalized Wick maps Berglund, Nils Klose, Tom Tapia, Nikolas Probability Mathematical Physics Combinatorics 81T15, 81T18, 16T05 We consider the perturbative renormalisation of the $Φ^4_d$ model from Euclidean Quantum Field Theory for any, possibly non-integer dimension $d<4$. The so-called BPHZ renormalisation, named after Bogoliubov, Parasiuk, Hepp and Zimmermann, is usually encoded into extraction-contraction operations on Feynman diagrams, which have a complicated combinatorics. We show that the same procedure can be encoded in the much simpler algebra of polynomials in two unknowns $X$ and $Y$, which represent the fourth and second Wick power of the field. In this setting, renormalisation takes the form of a \lq\lq Wick map\rq\rq\ which maps monomials into Bell polynomials. The construction makes use of recent results by Bruned and Hou on multiindices, which are algebraic objects of intermediate complexity between Feynman diagrams and polynomials. |
| title | Perturbative renormalisation of the $Φ^4_{4-\varepsilon}$ model via generalized Wick maps |
| topic | Probability Mathematical Physics Combinatorics 81T15, 81T18, 16T05 |
| url | https://arxiv.org/abs/2507.03820 |