Physics Notes by Jakob Schwichtenberg

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quantum_field_theory:methods:effective_field_theory

Effective Field Theory

A great colloquial introduction is Effective Quantum Field Theories by H. Georgi in P. Davis (ed.) The New Physics, Cambridge University Press, Cambridge, pp. 46-57..

We must keep in mind how and why an effective Lagrangian is constructed. The word “effective” is meant to imply that such a model is not intended to describe the system in all circumstances. Rather, the model gives a simplified treatment of the system in a given range of energies and momenta.

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.

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.
• Strong coupling […]

Quotes

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.

Nathan likes to call the complicated structure of confining QCD associated with the light antiquark in a heavy meson (or light quarks in a heavy baryon) the “brown muck” of hadron physics. I’ll adopt this phrase, because it is a nice reminder of the difficulties associated with the strong QCD interactions. When you have to descend into the brown muck, you abandon all pretense of doing elegant, pristine, first-principles calculations. You have to get your hands dirty with uncontrolled approximations and models. When you are finished with the brown muck, you should wash your hands.