1) HW 1A: When I entered the renormalization scale mu in f(x,q), I got a warning "mu not permitted in answer". Is there any thing wrong with mu? Isn't there always a factor of mu^(2 epsilon) for each loop in d-reg? Omitting it makes the ln(m^2 - ...) look weird, as the log of a dimensionful quantity.
This is a common question, partly because some textbooks are not careful about explaining this.
Indeed, we are used to seeing a factor of mu^(2 epsilon) in expressions involving gauge couplings. This additional factor comes from switching from the bare coupling to the dimensionless renormalized coupling:
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$$\alpha^{\rm(bare)} = Z_\alpha \mu^{2 \varepsilon} \alpha(\mu)$$
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It does look weird to have a dimensionful logarithm! However, this is not a problem. It is related to the weirdness of having a 4-d dimensional coupling, which we rectify when switching to the coupling used in
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The additional factor of mu^(2 epsilon) is not associated to the loop measure itself. For many one-loop calculations in gauge theory you can get away with associating it with the loop measure, which is why people gloss over this point. In other theories, and in gauge theory beyond one-loop, you have to be careful about it, and implement it in the correct manner as described above.
Some textbook references with correct descriptions that you can look at are: eq.(1.42) of Heavy Quark Physics by Manohar and Wise, eq.(23.23) of the Quantum Field Theory book by Matthew Schwartz.
EFT Concepts:
1) What does "integrating out" mean?
The terminology of "integrating out" a particle or high energy modes of a field comes from Ken Wilson and corresponds to explicitly integrating out field configurations at the path integral formulation. In the course we will often "integrate out" a particle by solving the equations of motion for its field at the Lagrangian level. However, one can analogously build an effective action by performing the integral for the field at the path integral level (typically using Gaussian integration over the field configurations).
As we will see during the course, that is not the only way of building an effective field theory. In particular we will build effective theories where we don't know the fields and the Lagrangian/Action of the full theory ("bottom-up" approach) and therefore won't be able to explicitly "integrate out" anything.
For this reason, the idea of "integrating out" a field is closely related to ''top-down" EFTs.
2) In the course, Iain mentions EFTs which violate the naive assumption that all spacetime directions have the same power counting (or "get resccaled equally"). What is an example of such an EFT?
If you take for example HQET or SCET, both of which we discuss later in the course, different spacetime components scale differently (i.e. they in general have different power counting). In these situations, the Lorentz symmetry of QCD manifests itself in a completely different form in the EFT (via reparameterization invariance which relates terms at different orders in the power counting, see the lecture HQET Power Corrections from Symmetry and Reparameterization Invariance). Nevertheless HQET and SCET are renormalizable order by order in their power counting parameter.