Let us assume $4$ spacetime dimensions.

QCD, the $SU(3)$ gauge theory with quarks as the matter fields, have the asymptotic freedom property as long as there are 16 quark flavors of mass below the energy scale of interest.

I wonder if there is any similar criterion for the matter fields to $SU(2)$ gauge theory so that asympotitc freedom is maintained?

I tried to find myself but not successful.

  • 2
    $\begingroup$ "I tried to find myself but not successful." What did you try? $\endgroup$
    – hft
    Commented Dec 13, 2023 at 1:47

1 Answer 1


What you are looking for is the beta function for non-abelian gauge theories. For QCD-like theories at one-loop level, it is given by $$ \beta(\alpha) = -\frac{\alpha^2}{2\pi}\left(11-\frac{N_c}{6} -\frac{2N_f}{3}\right) , $$ where $\alpha$ is the coupling constant, $N_c$ is the number of colours and $N_f$ is the number of flavours. It needs to be negative for asymptotic freedom. You can plug in the numbers to see when it works.

  • $\begingroup$ your formula is specific to vector-like theories, SU(2) is special in that it allows for chiral matter (as long as the index is even to avoid the Witten anomaly) $\endgroup$ Commented Dec 13, 2023 at 14:20
  • $\begingroup$ note also that higher order corrections to the beta function could change the behaviour; e.g. with the conformal window physics.stackexchange.com/a/714786/60080 $\endgroup$ Commented Feb 22 at 23:59

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