Abstract
We consider general renormalizable scalar field theory and derive six-loop beta functions for all parameters in d = 4 dimensions within the overline{mathrm{MS}} -scheme. We do not explicitly compute relevant loop integrals but utilize O(n)-symmetric model counter-terms available in the literature. We consider dimensionless couplings and parameters with a mass scale, ranging from the trilinear self-coupling to the vacuum energy. We use obtained results to extend renormalization-group equations for several vector, matrix, and tensor models to the six-loop order. Also, we apply our general expressions to derive new contributions to beta functions and anomalous dimensions in the scalar sector of the Two-Higgs-Doublet Model.
Highlights
Corresponding unknown coefficients by matching them to specific models
The anomalous dimension5 γφφ of the corresponding operator is related to the so-called crossover exponent and can be found in our approach as γφφ = −βgφφ + 2γφ with βgφφ being the beta function of gφφ and γφ corresponding to the anomalous dimension of the field computed via eq (1.3)
Our new result is related to the six-loop contribution to the beta function of the vacuum energy βΛ for gφφ = 0
Summary
We decided to use an approach similar to the one in ref. [22], based on the direct computation of the necessary counter-terms from individual diagrams. [22] considered all the required six-loop graphs in the context of the O(n)-symmetric model and made the corresponding counter terms available in a computer-readable form One can adopt the latter for more complicated theories by changing model-dependent prefactors. We provide a table containing a minimal set of unique tensor structures formed by different contractions between λabcd indices and the corresponding coefficients.. We provide a table containing a minimal set of unique tensor structures formed by different contractions between λabcd indices and the corresponding coefficients.3 Given these tables, we derive the beta functions for dimensionful parameters entering (1.1) employing the so-called dummy field method [6, 31, 32]. The tensor structures, including the corresponding graphs and coefficients for all the considered RG functions, can be found in the form of supplementary Mathematica files
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