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The tensorial equations for non trivial fully interacting fixed points at lowest order in the \${\textbackslash}varepsilon\$ expansion in \$4-{\textbackslash}varepsilon\$ and \$3-{\textbackslash}varepsilon\$ dimensions are analysed for \$N\$-component fields and corresponding multi-index couplings \${\textbackslash}lambda\$ which are symmetric tensors with four or six indices. Both analytic and numerical methods are used. For \$N=5,6,7\$ in the four-index case large numbers of irrational fixed points are found numerically where \${\textbar}{\textbar}{\textbackslash}lambda {\textbar}{\textbar}{\textasciicircum}2\$ is close to the bound found by Rychkov and Stergiou in arXiv:1810.10541. No solutions, other than those already known, are found which saturate the bound. These examples in general do not have unique quadratic invariants in the fields. For \$N {\textbackslash}geqslant 6\$ the stability matrix in the full space of couplings always has negative eigenvalues. In the six index case the numerical search generates a very large number of solutions for \$N=5\$.

@article{osborn_heavy_2020, title = {Heavy {Handed} {Quest} for {Fixed} {Points} in {Multiple} {Coupling} {Scalar} {Theories} in the \${\textbackslash}varepsilon\$ {Expansion}}, url = {http://arxiv.org/abs/2010.15915}, abstract = {The tensorial equations for non trivial fully interacting fixed points at lowest order in the \${\textbackslash}varepsilon\$ expansion in \$4-{\textbackslash}varepsilon\$ and \$3-{\textbackslash}varepsilon\$ dimensions are analysed for \$N\$-component fields and corresponding multi-index couplings \${\textbackslash}lambda\$ which are symmetric tensors with four or six indices. Both analytic and numerical methods are used. For \$N=5,6,7\$ in the four-index case large numbers of irrational fixed points are found numerically where \${\textbar}{\textbar}{\textbackslash}lambda {\textbar}{\textbar}{\textasciicircum}2\$ is close to the bound found by Rychkov and Stergiou in arXiv:1810.10541. No solutions, other than those already known, are found which saturate the bound. These examples in general do not have unique quadratic invariants in the fields. For \$N {\textbackslash}geqslant 6\$ the stability matrix in the full space of couplings always has negative eigenvalues. In the six index case the numerical search generates a very large number of solutions for \$N=5\$.}, urldate = {2020-11-09}, journal = {arXiv:2010.15915 [cond-mat, physics:hep-th]}, author = {Osborn, Hugh and Stergiou, Andreas}, month = oct, year = {2020}, note = {arXiv: 2010.15915}, keywords = {condensed matter, high energy physics, mentions sympy}, }

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