Gradient properties of $φ^3$ in $d=6-\varepsilon$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912514275540992 |
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| author | Benfatto, Lorenzo Zanusso, Omar |
| author_facet | Benfatto, Lorenzo Zanusso, Omar |
| contents | The renormalization group flow of the multiscalar interacting $φ^3$ theory in $d=6$ dimensions is known to have a gradient structure, in which suitable generalizations of the beta functions $B^{I}$ emerge as the gradient of a scalar function $A$, $\partial_I A = T_{IJ} B^J $, with a nontrivial tensor $T_{IJ}$ in the space of couplings. This has been shown directly to three loops in schemes such as $\overline{\rm MS}$ and can be argued in general by identifying $A$ with the coefficient of the topological term of the trace-anomaly in $d=6$ up to a normalization. In this paper we show that the same renormalization group has a gradient structure in $d=6-\varepsilon$. The requirement of a gradient structure is translated to linear constraints that the coefficients of the $\overline{\rm MS}$ beta functions must obey, one of which is new and pertinent only to the extension to $d \neq 6$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_20761 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient properties of $φ^3$ in $d=6-\varepsilon$ Benfatto, Lorenzo Zanusso, Omar High Energy Physics - Theory The renormalization group flow of the multiscalar interacting $φ^3$ theory in $d=6$ dimensions is known to have a gradient structure, in which suitable generalizations of the beta functions $B^{I}$ emerge as the gradient of a scalar function $A$, $\partial_I A = T_{IJ} B^J $, with a nontrivial tensor $T_{IJ}$ in the space of couplings. This has been shown directly to three loops in schemes such as $\overline{\rm MS}$ and can be argued in general by identifying $A$ with the coefficient of the topological term of the trace-anomaly in $d=6$ up to a normalization. In this paper we show that the same renormalization group has a gradient structure in $d=6-\varepsilon$. The requirement of a gradient structure is translated to linear constraints that the coefficients of the $\overline{\rm MS}$ beta functions must obey, one of which is new and pertinent only to the extension to $d \neq 6$. |
| title | Gradient properties of $φ^3$ in $d=6-\varepsilon$ |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2507.20761 |