FAQ: Introduction to EFT
Homework:
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
\overline{MS}, in part (b).
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?
[I get that "integrating out" heavy particles means "remove". I wonder if there is any actual integration involved? Is there any math procedure to remove heavy degrees of freedom from the Lagrangian?]
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.
3) Are Wilsonian Renormalization Group Equations (RGE) always equivalent to the Continuum Renormalization Group Equations of Gell-Mann and Low?
No, in general for power law RGE behavior which is captured by the Wilsonian approach, but can be missed if we formulate our Continuum EFT using
\overline{MS} and dimensional regularization (Dim.Reg.). But the answer is yes for the dominant logarithmic behavior. If we know what theory we are considering then it is often sufficient to treat only the logarithmic corrections, but power law corrections can be relevant for flows where a theory moves from one power counting regime into another. With a soft regulator like Dim.Reg. we can also formulate a renormalization group for power law terms, by noting that power divergences still show up as poles in other dimensions. An example we will talk about later on is the power divergence subtraction scheme. We will use this scheme to show an explicit example of a flow between regimes with two different power countings.
4) Is there any EFT which best describes the physics of RHICs?
[In the lecture, Iain mentioned that there are many different EFTs that are used to describe different behaviors in quantum chromodynamics (QCD). One of the interesting places QCD is studied is in relativistic heavy ion collisions (RHICs). -SA]
Here is a response from Xiaojun Yao, who is a former student of EFTx and, at the time of writing, is a postdoc studying EFTs at MIT:
So far, there is no single EFT that can describe every observable in heavy ion collisions. But for some observables, EFTs have been constructed and have achieved phenomenological success. For example, for light particle production spectrum at low pT, (pion, kaon and proton with pT smaller than a few GeV), hydrodynamics and simple hadronization models can describe the data.
Softer observables:
For softer observables, hydrodynamics itself is an effective theory. The modern construction of hydrodynamics is as follows
- First, write down the most general form of the stress energy tensor, that is consistent with the symmetry of the system. The building blocks include the metric and flow velocity fields.
- The construction then is organized by the order of the derivatives on the flow velocity fields. So the power counting here is the order of the derivatives. Lowest order terms have no derivatives. In this sense, the construction is similar to the two-nucleon EFT, covered in Chapter 10 of this course.
For a review along this philosophy of EFT construction, see https://arxiv.org/abs/1712.05815. When thinking in terms of the quantum fields, we are really expanding out short wavelength fields and keeping the long wavelength modes. So hydrodynamics is a EFT of long wavelength modes. For discussion along this perspective, see https://arxiv.org/abs/1805.09331 and references in the list of course projects (PFL, RSE, VHY).
5) Is it possible to apply EFT to String Theory?
Sure, the general ideas apply to string theory too. For example, in the classical limit, string theory reproduces general relativity, so you can think of string theory as a mother theory that can be used to systematically match onto a theory that is general relativity plus higher order corrections. The suggested video project "Quantum Gravity in Perturbation Theory" (QGP) explores the resulting EFT in detail, but from a bottom up perspective.
Hard Observables
For hard observables (by hard, I mean jets and heavy flavor particles), different EFTs have been used. When jets travel through the quark-gluon plasma described by hydrodynamics, their transverse momentum with respect to the jet axis will change constantly due to interactions with the plasma. This phenomenon, [known as momentum broadening -SA], has been studied by using SCET with the Glauber exchange in the forward scattering regime. For SCET in the forward scattering, you can find references in the list of course projects (SMX). For its application in the jets in the plasma, see https://arxiv.org/abs/1211.1922 and https://arxiv.org/abs/2004.11403. For quarkonium production in heavy ion collisions (at low pT), potential NRQCD has been applied. Again, you can find relevant papers of pNRQCD in the list of course projects (PNR). A recent development is to use pNRQCD to derive the Boltzmann equation for quarkonium in the plasma in a systematic expansion. One can clearly see the connection between the validity of the semiclassical transport approach and the EFT power counting. See https://arxiv.org/abs/1811.07027 for details. For other developments using pNRQCD in plasma, see https://arxiv.org/abs/1711.04515.
5) Iain mentioned that EFT is particularly useful in understanding logarithmic divergences. Why logarithmic divergences in particular rather than, say, power divergences?
The logarithmic UV divergences in the EFT are related to what would have been logarithms involving two widely separated scales in the full theory (mother theory) of a particular EFT. We will discuss this in quite a bit of detail in Chapter 4.
6) Once we have derived a tree level effective action which includes loop effects in a full theory, why do we still account for loop level effects (in the effective theory) in our calculations?
The EFT is replacing the full theory at low energies. So the leading Lagrangian in the EFT is quantized in the usual fashion, and once you have a QFT you have no choice but to consider the loops that that QFT yields. Indeed, unitarity of the EFT connects the loop diagrams to tree diagrams [(e.g. through the optical theorem) --SA], so in order to have a unitary calculational framework you must include the loops.
This question is a common one that people asked in the early days of Chiral Perturbation theory once it was understood to come from QCD. For example, it is reasonable to ask "Do loops of pions in ChPT make sense, versus loops of quarks in the full theory of QCD?" They do, and we can see that the logarithms of the quark masses that the ChPT loops yield are precisely those found in lattice QCD calculations of full QCD. All the information of the full theory is contained in the EFT for low energy observables, just organized in a different fashion.