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High-energy theory

Counting everything that runs

About 8 min if you have seen this material before, 25 min or so if it is new to you.

Background
Assumed background: the one-loop The rate at which a coupling changes with energy scale, β(g)=μ dg/dμ\beta(g) = \mu\, dg/d\mu, which decides a marginal coupling's fate. of a non-abelian gauge coupling, The normalisation factor T(R)T(R) of a representation, defined by tr(TaTb)=T(R)δab\mathrm{tr}(T^a T^b) = T(R)\delta^{ab}, which weights each field's contribution to a beta function., and Weyl-fermion counting. There is no idea in this problem. Every ingredient is a lookup and the only way to get it wrong is to drop something, which is what a large fraction of real calculations are like.
Conventions, since they vary between books: β(g)=−b g316π2,b=113T(G)−23∑WeylT(R)−13∑complex scalarsT(R),\beta(g) = -\frac{b\,g^3}{16\pi^2}, \qquad b = \frac{11}{3} T(G) - \frac{2}{3}\sum_{\text{Weyl}} T(R) - \frac{1}{3}\sum_{\text{complex scalars}} T(R), with T(fund)=1/2T(\text{fund}) = 1/2 and T(G)=NT(G) = N for SU(N)SU(N). A positive bb means The behaviour of a coupling that shrinks toward high energy, which is what makes the strong interaction calculable at colliders.. The model is the Standard Model with a second Higgs doublet. Charged under SU(2)LSU(2)_L: three generations, each containing one quark doublet in three colours and one lepton doublet, and two complex scalar doublets. Every right-handed fermion is an SU(2)LSU(2)_L singlet and contributes nothing.
Compute bb for SU(2)LSU(2)_L in this model.

Past 45 min on this? That usually means a missing prerequisite rather than a lack of effort — worth reading up before pushing further, starting with Asymptotic freedom (Wikipedia).

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