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 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 () emits a boson and is transmuted to an “up-type” object ():

### Tree-level

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

Writing , we read off:

As expected, the leading term is of dimension 6 and hence, *non-renormalizable*. Since the effective coupling then becomes , 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 %, 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. . The *Wilson coefficients* 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 , which we cannot simply expand out in as it becomes non-analytic from . Instead, we write:

where the first term on the RHS is essentially (independent of the low-energy scale ) and the second term looks like the 4-point vertex (note that this loop is UV-divergent, contrary to the SM equivalent, but independent of ). In the literature, we find:

To eliminate large logarithms, we want to evaluate at . Introducing the *anomalous dimension* , the RG flow of the Wilson coefficient is governed by:

Solving this equation will then allow us to evolve (from the equation above) to (for our integral). Computing the anomalous dimension to loops permits the summation of *all* .

A numerical example for more insight: if we set and for large at tree-level, and compute the 1-loop anomalous dimension, we get and .

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