Effective Field Theory

Prerequisites: Canonical Quantum Field Theory, Renormalization Group. Full path (24 pages)

Best explanations

Intuitive How Quantum Field Theory Becomes “Effective” | Sean Carroll Blog
Concrete Effective Field Theory by Howard Georgi

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

Natural phenomena, when observed with poorer resolution, become simpler. This is simple fact. When we describe the Moon trajectory orbiting around Earth, we can safely approximate both bodies with material points and neglect all the complexity of geography, crystalline structure, chemical composition, and terrestrial biology. This is technically known as the “spherical-cow approximation”, which is a basic tool of effective field theory. The observation that the universe is roughly uniform and isotropic at large scales supports the point of view that simplicity dominates the far IR.

https://arxiv.org/pdf/1710.07663.pdf

Three major ways in which we use EFT:

Bottom-up physics, or “phenomenology” […] Here one does not know the physics that UV completes one’s EFT, but the Wilson coefficients contain dimensionful factors which point to the scale by which the EFT must be completed.

Simplify calculations. […] We are wise enough to know that we cannot just brazenly compare a theory at the TeV scale with constraints at different scales; at least in principle one has to run the theory down to those scales via the renormalization group. Generally going form the TeV scale down to MW is no problem since all of the effective couplings (even SU(3)c) are perturbative and the scales are only one order of magnitude apart. However, running down to ΛQCD is a different story completely! This is the domain of Georgi’s brown muck and adventurers are required to slog through with multi-loop calculations to maintain any control over anything. This is hard work! […] Thus all we need are some very technically capable groups to actually do the multi-loop RG calculations from MW to ΛQCD for an effective theory composed of gluons and quarks whose masses are below the energy threshold as we run the RG scale down. (There are ‘threshold corrections’ as one flows past each quark mass.) At the hadronic scale we can sensibly place constraints on the Wilson coefficients and use this to compare to the parameter space back at MW (and ideally back at the TeV scale). Thus one finds several flavor physics updates where theorists and experimentalists convert experimental bounds to constraints on Wilson coefficients at, say MW . Then any TeV scale model builder can take their theory and write down the MW -scale Wilson coefficients in terms of their TeV scale parameters. One can then interpret the bounds on the Wilson coefficients from low energies (which enterprising flavor physicists have done for us) as bounds on combinations of the ‘more fundamental’ parameters of the UV theory.

http://www.physics.uci.edu/~tanedo/files/notes/FlavorNotes.pdf

There is a very naive and almost dismissive belief that effective theories are somehow lesser than ‘fundamental’ theories. This belief is somehow planted deep into our consciousness as physics students. […] We major physics rather than chemistry because the latter is just an application of the former, therefore the former must be more pure. We learn that physics is about Taylor expansions and approximations, which can—at best—reproduce ‘full’ calculations. All of these things leave the impression that something which is ‘effective’ is just a poor man’s version of the correct tool. It is important at an ideological level to address this bias and to explain the value of effective field theory in particle physics. After all, one of the great paradigm shifts in our field came from Ken Wilson, who taught us that all models of nature should be understood as effective theories.

http://www.physics.uci.edu/~tanedo/files/notes/FlavorNotes.pdf

Intuitive

To compute the orbit of the Moon we do not need to know that the Moon is made of rock, that rock is made of atoms or that atoms contain quarks. The details of what happens at short distances barely affect what happens at long distances. Where they do, their influence shows up only through a few numbers, such as the masses of Earth and Moon and perhaps a small correction from their shapes.

An effective field theory turns this observation into a method. We choose the energy (or length) scale we care about and keep only the particles and fields that can actually be produced or excited at that scale. Everything heavier or shorter-ranged is left out. Its influence is not lost: it gets absorbed into the values of the couplings of the remaining fields and into a list of extra interaction terms. These extra terms are organized by how much they matter, and each one is suppressed by a power of the ratio (our energy) / (energy of the omitted physics). At low energies only the first few terms are needed.

Two things follow. First, we can make precise predictions without knowing the correct theory at very high energies, which is the normal situation in physics. Fermi's theory of beta decay worked for decades before the W boson was found. Second, when we measure the small correction terms, their size tells us roughly at which energy the omitted physics must appear. From this point of view every theory we know, including the Standard Model and general relativity, is an effective description valid below some energy. - How Quantum Field Theory Becomes “Effective” | Sean Carroll Blog

Concrete

An effective field theory is one where we have factorized information about possibly-but-notnecessarily unknown UV physics from low-energy ‘active’ degrees of freedom. The UV information lives in $C_i$ , the Wilson coefficient, which is simply a number—a coupling which generally depends on the scale at which it is probed. The IR information lives in Oi , the effective operator. These are composed of all of the ‘low energy’ excitations that are physically accessible at the scales where the EFT is valid. […] [T]he map sending UV information into $C_i$ is not invertible. [..] The sum over Wilson coefficient should run over all allowed operators, including nonrenormalizable operators. […] [W]e have chosen an energy scale so that physics above that scale goes into the Ci while physics below that scale goes into the Oi. […] By construction, the EFT is only useful (but we’ll see it’s very useful) for answering questions about physics below Λ. […] This scale also serves to balance the dimension of the effective operators Oi so that very non-renormalizable (large dimension) operators are suppressed by large powers of 1/Λ. [..] The UV physics that we factorize into the Wilson coefficients need not be known.

http://www.physics.uci.edu/~tanedo/files/notes/FlavorNotes.pdf

Recommended Resources:

Effective Field Theory by Howard Georgi
How to use the Standard Model effective field theory by Brian Henning et. al.
No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, chapter "Effective Field Models and the Origin of Simplicity"

Abstract

We work in natural units $\hbar=c=1$ and Euclidean signature. Split the fields of a theory into modes with momenta below and above a cutoff $\Lambda$ , $\phi=\phi_<+\phi_>$ . The Wilsonian effective action at scale $\Lambda$ is defined by integrating out the high modes, $e^{-S_\Lambda[\phi_<]}=\int\mathcal{D}\phi_>\,e^{-S[\phi_<+\phi_>]} .$ By construction it reproduces all correlation functions of $\phi_<$ . $S_\Lambda$ is in general nonlocal, but for external momenta $p\ll\Lambda$ (and when no massless modes are integrated out) it can be expanded in local operators, $S_\Lambda=\int d^dx\,\sum_i g_i(\Lambda)\,\mathcal{O}_i(x),$ where the sum runs over all local operators built from the light fields that are compatible with the symmetries. Lowering $\Lambda$ defines a flow on the space of couplings $\{g_i\}$ , the renormalization group flow.

Power counting. Write $g_i=c_i\,\Lambda^{d-\Delta_i}$ with $\Delta_i$ the scaling dimension of $\mathcal{O}_i$ and $c_i$ dimensionless. Near a fixed point (for weak coupling, near the free theory, where $\Delta_i$ is the engineering dimension) the contribution of $\mathcal{O}_i$ to an observable at momentum $p$ scales as $(p/\Lambda)^{\Delta_i-d}$ . Operators with $\Delta_i<d$ are relevant, $\Delta_i=d$ marginal, $\Delta_i>d$ irrelevant. Up to a given accuracy $(p/\Lambda)^n$ only finitely many operators contribute. This is why a nonrenormalizable effective theory is still predictive: at each order in $p/\Lambda$ it has finitely many parameters.

Matching. In the top-down approach a heavy field of mass $M$ is integrated out at tree level and in loops, and the Wilson coefficients $c_i(\mu)$ are fixed by requiring that the full and effective theories give the same low-energy amplitudes at a matching scale $\mu\sim M$ . Below $M$ , the coefficients are run with the renormalization group equations of the effective theory. A standard example is the four-fermion interaction obtained from W exchange, $\frac{g^2}{8M_W^2}=\frac{G_F}{\sqrt2}.$ In mass-independent schemes such as $\overline{\text{MS}}$ heavy particles do not decouple automatically, and integrating them out explicitly is how decoupling is implemented. In a mass-dependent scheme the Appelquist and Carazzone decoupling theorem states that, under suitable conditions, the effects of heavy particles on low-energy amplitudes are either absorbed into renormalized couplings or suppressed by powers of $1/M$ .

Redundancies. The effective Lagrangian is defined only up to operators that vanish by the leading-order equations of motion, since these can be removed by field redefinitions without changing the S-matrix, and up to total derivatives. The set of physically distinct operators at a given dimension (an operator basis) is therefore a quotient of the space of all symmetric local operators. For the Standard Model effective field theory at dimension six this gives the Warsaw basis.

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Leads to: Seesaw Models.