A brief discussion on Effective Feild Theories
1
Effective Field Theories
This is a general discussion on effective field theories (EFT’s), which aims to complement course material. Many of the
technical details are discussed in the Standard model (SM) course.
Let’s start the discussion by saying that it is important to understand that every QFT actually comes with an energy
scale. When someone gives you a QFT, they should also give you an energy scale with it. This energy scale tells us the
range of validity of the QFT. We do not expect our theory to be accurate at energies above this energy scale. Whenever
we apply our theory above this energy scale, it is often the case that we find infinities. These infinities signify the start of
some new physics that our theory does not contain information on1 .
What happens if the energy scale that we use in an experiment is much smaller than the energy scale of the QFT? Then
the QFT is going to contain a lot more information than we actually need. The calculations with this extra information
will be a lot more difficult. The information about the high energy (UV) modes, that the QFT holds, relative to our
experimental energy scale complicates our theoretical analysis.
By the time-energy uncertainity principal, since UV modes have much higher energy than our experimental scale,
they can only exist for short times at our experimental energy scale. When our experiment is carried out at a lower energy
scale, the UV modes of the QFT are less important for understanding the dynamics of our experiment at this lower energy
scale.
Let’s clarify some ideas by using an example. Take the weak interactions as described in Dr. Wingates SM notes.
Let’s make an EFT out of these weak interactions. The EFT constructed from the weak interactions is called Fermi
Theory.
2
Fermi Theory
In the ElectroWeak section of the SM course, we see that the weak interactions are mediated by the very heavy W ± , Z 0
bosons, where mW ± ,Z 0 ∼ 80 Gev.
Say we have an experiment running at a much lower energy scale. Say the transfer momentum at each vertex
connecting to a W ± , Z 0 is q 2 mW ± ,Z 0 . The W ± , Z 0 are UV modes in this case. This is the case in certain leptonic
transitions like e+ e− → µ+ µ− .
While the experiment is running, a strange scenerio plays out in front of you! The funding body for the experiment
have told your supervisor that they need a result one day before the experiment finishes. A big scandal is coming out and
they want to look good before hand. Your supervisor tells you to give them any theoretical prediction as soon as possible.
The scene is set. You need a theoretical prediction soon. You realise the difference in energy scales and proceed by
constructing an EFT in order to make the calculation simpler. Let’s see how you would do this to leading order for the
W ± , Z 0 bosons. First consider the Z 0 boson. The arguments carry over identically for the W ± bosons. We have seen in
the SM notes that the Z 0 propagator is:
∼
1
qµ qν
z
D µν (q) = 2
−gµν + 2
q − m2z
mµ
gµν
2
2
≈ 2 as q mz
mz
This result is only to leading order by expanding 1/(1 −
∼
z
course. Taking the fourier transform of D
µν (q)
q2
m2z ).
yields
z
Dµν
(x) ≈
1 This
A more detailed analysis of EFT’s is covered in the SM
gµν (4)
δ (x)
m2z
is discussed more in the renormalisation notes on my webpage: http://www.damtp.cam.ac.uk/user/ch558
1
GF
W ±, Z 0
Going to the Fermi EFT
Figure 1: Example of what happens to W ± , Z 0 to leading order in Fermi theory.
We see that, in this leading order EFT, by the effect of the delta function, the interactions mediated by the W ± , Z 0 now
become point interactions, where the information about the W ± , Z 0 is contained in the new coupling GF . This is shown
diagramically in Figure (1). This is in agreement with the arguments in Section 1: the UV modes are less important for
understanding the dynamics at our experimental energy scale.
The W ± , Z 0 have been replaced by local ’point’ interactions. The calculations become much easier and you find the
theoretical results in less time, much to your supervisors and funding bodies delight!
To probe smaller distances, we need higher energies. So by going to lower energies, we go to larger distances, or
’zoom’ out of a diagram. This is the understanding of what happens in an EFT. In an EFT, we go to lower energies and
’zoom’ out. So as we go to the Fermi EFT from the full weak theory, we zoom out of the Feynman diagram on the LHS
of Figure (1) until we see only a point interaction (the RHS of Figure (1)). The interaction mediated by the W ± , Z 0 now
appears as a point interaction.
In Dr. Wingate’s SM lecture notes, we found the leading order Fermi Lagrangian in the section on Weak decays.
3
Outline of constructing an EFT
Once we have a clear hierarchy of energy scales in a QFT, there is a formal precedure that enables you to find an EFT. It
is covered in the end part of Dr. Wingate’s SM lecture notes. For the Fermi theory discussed above, the hierarchy of scales
is the experimental centre of mass energy q 2 and the mass of the particles mediating the force, mW ± , where q 2 << mW ± .
As a brief outline on how to do build an EFT, once you have a clear hierarchy in energy scales,
1. Find the UV degrees of freedom and your light degrees of freedom in the QFT.
2. Write down all operators for the effective Lagrangian that are allowed by the symmetries of your theory, even if the
operators are non renormalisable. This gives you an infinte series (called a tower by Georgi) of terms in the effective
Lagrangian.
3. The effective Lagrangian has to be dimension four. Count the dimensions of the couplings in the effective Lagrangian.
Pull out the dimension of the coupling as a UV scale, i.e, let bi be a coulping of dimension n. Then let bi = ci /Λn
where Λ is the UV energy scale and n is the dimension of the coupling. Λ is assumed large so terms with larger n
will become more and more negliable. This is called power counting.
4. Choose an appropiate order that you wish to calculate results to (just like in pertubation theory) and throw away
terms of higher order in 1/Λ.
5. You have unconstrained parameters in this theory, the dimensionless couplings ci . We calculate these by matching
them to the orginal theory in some way (usually pertubation theory) or by equating them to some experimentally
derived value. This is called matching (strangly!).
In this way, we have a systemic approach to study our original theory, and a complete understanding of the error from
throwing away terms from the infinite series in the effective Lagrangian.
4
Some points worth noting
• It is ok to allow non-renormalisable operators in our effective theory, as non-renormalisable theories are those where
the UV modes do not decouple from the physical observables.2 But since we are working in an EFT, the UV modes
2 Look
at the notes on my webpage for a discussion on renormalisation: http://www.damtp.cam.ac.uk/user/ch558
2
become negliable. This is why they are called irrelevent operators in Dr. Wingate’s SM notes on EFT’s. As
we go to lower and lower energies, they become more and more negliable. This means that the EFT of a nonrenormalisable theory is a valid EFT. Since the standard model has to be renormalised, it is an EFT of some other
“theory of everything”. Once we get infinities in an EFT at some energy scale, this signifies the start of some new
physics which our EFT does not contain information on.
• Since we have constructed the EFT to be a low energy theory (without all the information on the UV modes), the
EFT will only match the dynamics of the original QFT at low energies. At high energy, the EFT will breakdown
(as it doesn’t contain all the information on the UV modes) and will not match the dynamics of the original QFT.
• Because we have neglected some information from our theory, calculations in an EFT will be much simplier than
calculations in the full theory. For example, try to calculate the leptonic decay rate for a process mediated by a
W ± , Z 0 in the full theory (with massive leptons). This becomes tedious, fast. An example would be question (4) on
the SM example sheet three.
• EFT’s are useful when we cannot make any obvious progression in a problem. Such is often the case in QCD. In
certain cases, we can parametrise our ignorance of the strong regime but in others it is not obvious what a progressive
step looks like. In such cases, it is often possible to construct an EFT, simplifing the problem in some way. We
can then try to understand the dynamics of the EFT. When it is possible to do this, it often offers insights into
the original theory. An example of such a case is called heavy quark effective theory (HQEFT). In this EFT, we
consider some heavy particle bound together with some other light particle. Think about the hydrogen atom with
the proton being the heavy particle and the electron being the other. Mesons consiting of a heavy quark and a light
anti quark are also an example of such a system. The heavy quark mass gives an energy scale for the UV modes in
the theory. A different EFT is for a meson with a heavy quark and a heavy anti-quark bound together. In such a
case, the quarks are travelling non-relativistically. We then construct an EFT using the velocity as the expansion
parameter (which is small). This EFT is called Non-Relativistic QCD (NRQCD).
Ciaran Hughes
3