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).