# When is a zero of a beta function a stable infrared fixed point?

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Suppose I have a beta function for a quantum field theory in $d-\epsilon$ dimensions. I calculate the beta function in MS-bar scheme to be:

$$\beta(g) = {\partial{g} \over \partial{\ln \mu}}=-A\epsilon g + B g^2$$

The fixed point is at $g_* = \epsilon A/B$.  Is my beta function definition correct? Is the fixed point is an IR fixed point if ${\partial \beta(g) \over \partial g}= -A\epsilon+2 B g_* = \epsilon A$ is positive? In exactly $d$ dimensions, how does the sign of $B$ determine whether the theory is asymptotically free or not?

How does this generalize to a theory with say two coupling constants?

$$\beta(g) = {\partial{g_i} \over \partial{\ln \mu}}=-A_i\epsilon g_i + B_{ij} g_i g_j$$ asked Jul 4, 2018
retagged Jul 5, 2018

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