Consider a non-Abelian gauge theory with both fermionic $\psi$ and bosonic $\phi$ matter fields in 4D, which may be called a QCD-Higgs model, $$L=-\frac{1}{4 g^2}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}i\gamma^\mu D_\mu\psi+|D_\mu\phi|^2-\frac{\lambda}{4}|\phi|^2.$$ The one-loop beta function for the gauge coupling $g$ can be found in Peskin-Schroeder book, $$\beta(g)=-\frac{g^3}{4\pi}\left(\frac{11}{3}C_2(G)-\frac{4}{3}n_f C(r_f)-\frac{1}{3}n_b C(r_b)\right).$$ Given that there are some recent progress in 5-loop beta function of QCD theory (see for example [1,2]), I am wondering what is the recent status of the beta function for the QCD-Higgs model that also incoporates the bosonic matter field. My question is what is (or where to find any reference about) the higher loop beta function for the QCD-Higgs model.

Some search on google only returns results about either QCD or the Standard Model, but I would like to know generic results about general gauge group $G$ (for example O(N) or Sp(N)) with generic representations for fermions $r_f$ and bosons $r_b$.

[1] https://arxiv.org/abs/1606.08662v2

[2] https://arxiv.org/abs/1701.01404v2


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