I'm considering renormalization scheme for Dirac-delta potentials in 3D.

Renormalization group equation I derived is: $\frac{dg}{d\Lambda}=A \cdot \frac{g^2}{\Lambda^{2/3}}$, where A is some numerical constant which is not important, $\Lambda$ is energy scale and g is coupling constant. $\beta(g) = \Lambda \frac{dg}{d\Lambda} =A \frac{g^2}{\Lambda^{1/3}}$. I know that this is the correct equation because it reproduces know results for coupling constant. So only fixed point is $g=0$ which correspond to no interaction.

My question is: Can $\beta$ function depend explicitly on scale? From examples I know e.g. Standard Model beta function doesn't depend on scale. Do you know of any theories with scale dependence in $\beta$ function ? What does it mean physically?


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