Some time ago I wrote about doing QCD without the top quark. Now we go one step further and consider weak interactions.

## Effective weak interactions

Unlike before, we can’t just throw away the bottom quark when considering heavy quark decay, because it is lighter than the $W^\pm$ boson! Instead, we have to take a more subtle approach, and start by rewriting our Standard Model Lagrangian as the five-flavour QCD sector, the QED sector, and some effective rephrasing of the weak interactions.

Schematically, we’ll consider the sort of processes were a “down-type” object ($d,s,b,e^-,\mu^-,\tau^-$) emits a $W^\pm$ boson and is transmuted to an “up-type” object ($u,c,t,\nu_e,\nu_\mu,\nu_\tau$):

### Tree-level

Tree-level matching allows us to discard the heavy boson and consider the following 4-point interaction instead:

Writing $\frac{G_F}{\sqrt{2}}\equiv\frac{g_W^2}{8M_W^2}$, we read off:

As expected, the leading term is of dimension 6 and hence, non-renormalizable. Since the effective coupling then becomes $g_W^2\cdot (E/M_W)^2$, this explains why the weak interaction is, well, weak at low energies. Writing out the leading terms of the effective Lagrangian, we recover Fermi’s theory:

and since these higher dimensional operators are at most of order $(m_b/m_W)^2\sim 0.25$%, they can be (and usually are) safely ignored.

### Loops

To go beyond tree-level, we add all gauge invariant operators up to dimension 6 with undetermined couplings, i.e. $\mathcal{L}_\text{eff}\supset\sum_iC_i(\mu)\mathcal{O}_i$. The Wilson coefficients $C_i$ are determined by matching suitable amplitudes in perturbation theory. Looking at the one-loop diagram:

And from the following diagrams

we get expressions of the type (to be written with various combinations of Dirac matrices and colour contractions)

The loop integral contains a term of the usual form $1/(k^2-M_W^2)$, which we cannot simply expand out in $M_W$ as it becomes non-analytic from $k\sim M_W$. Instead, we write:

where the first term on the RHS is essentially $C_i(\mu)$ (independent of the low-energy scale $p$) and the second term looks like the 4-point vertex $\mathcal{O}_i$ (note that this loop is UV-divergent, contrary to the SM equivalent, but independent of $M_W$). In the literature, we find:

To eliminate large logarithms, we want to evaluate at $\mu\sim p$. Introducing the anomalous dimension $\gamma$, the RG flow of the Wilson coefficient is governed by:

Solving this equation will then allow us to evolve $C_i(\mu_\text{high})$ (from the equation above) to $C_i(\mu_\text{low})$ (for our integral). Computing the anomalous dimension to $k$ loops permits the summation of all $\sum_n\alpha_s^{k-1}\left(\alpha_s\ln(M_W^2/\mu_{\text{low}})\right)^n$.

A numerical example for more insight: if we set $C_1=1$ and $C_2=0$ for large $\mu$ at tree-level, and compute the 1-loop anomalous dimension, we get $C_1(m_b)\simeq 1.12$ and $C_2(m_b)\simeq -0.27$.

Having derived this effective theory, we can now go down to energies below $m_b$: Heavy Quark Effective Theory (HQET) and Soft Collinear Effective Theory (SCET).