Abstract
The one-loop beta functions for systems of Ns scalars and Nf fermions interacting via a general potential are analysed as tensorial equations in 4 − ε dimensions. Two distinct bounds on combinations of invariants constructed from the couplings are derived and, subject to an assumption, are used to prove that at one-loop order the anomalous dimensions of the elementary fields are universally restricted by γϕ ⩽ \( \frac{1}{2} \) Nsε and γψ ⩽ Nsε. For each root of the Yukawa beta function there is a number of roots of the quartic beta function, giving rise to the concept of ‘levels’ of fixed points in scalar-fermion theories. It is proven that if a stable fixed point exists within a certain level, then it is the only such fixed point at that level. Solving the beta function equations, both analytically and numerically, for low numbers of scalars and fermions, well-known and novel fixed points are found and their stability properties are examined. While a number of fixed points saturate one out of the two bounds, only one fixed point is found which saturates both of them.
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Acknowledgments
We would like to thank Hugh Osborn for enlightening discussions and comments on the manuscript. AS is funded by the Royal Society under grant URF\R1\211417.
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Pannell, W.H., Stergiou, A. Scalar-fermion fixed points in the ε expansion. J. High Energ. Phys. 2023, 128 (2023). https://doi.org/10.1007/JHEP08(2023)128
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DOI: https://doi.org/10.1007/JHEP08(2023)128