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Contents

Idea

While fundamental physics is at some level well described by quantum field theory, a typical Lagrangian used to define such a QFT can reasonably be expected to define only degrees of freedom and interactions that are relevant up to some given energy scale. In this perspective one speaks of the theory as being the effective quantum field theory of some – possibly known but possibly unspecified – more fundamental theory.

An example (historically the first to be successfully considered) is the Fermi theory of beta decay of hadrons: this contains interactions of four fermions at a time, for instance a process in which a neutron? decays into a collection consisting of a proton?, an electron and a neutrino. Later it was discovered that, more fundamentally, this is not a single reaction but is composed out of several other interactions that involve exchanges of W boson?s between these four particles. Nevertheless, Fermi’s original effective theory made very precise predictions at energy scales less than 10 MeV?. The reason is that the $W$-boson has mass several orders of magnitude higher than that (about 80 GeV?) and was thus effectively invisible at these low energies.

The low energy expansion of any unitary, relativistic, crossing symmetric? S-matrix can be described by an effective quantum field theory.

In the perspective of effective field theory notably unrenormalizable Lagrangians can still make perfect sense as effective theories and give rise to well defined predictions: they can be effective approximations to renormalizable or even degreewise finite more fundamental theories. This is sometimes called a UV completion of the given effective theory.

For instance gravity – which is notoriously non-renormalizable – makes perfect sense as an effective field theory (see for instance the introduction in (Donoghue). It is in principle possible that there is some more fundamental theory with plenty of excitations at high energies that is however degreewise finite in perturbation theory, whose effective description at low energy is given by the unrenormalizable Einstein-Hilbert action. (For instance, string theory is meant to be such a theory.)

Theory

The technique of effective field theory is based on the following fact:

This is due to (Weinberg 1979) and (Leutweyler94).

Based on this fact, one obtains an effective approximation to a given more fundamental theory (which may or may not be actually known) by

1. choosing the (sub)set of fields to be considered;

2. writing down a Lagrangian

   $$L_{\mathrm{eff}}=\sum _i{c}_i{O}_i$$L_{eff} = \sum_i c_i O_i

   that contains all the possible polynomial interaction terms $O_i$ of these fields scaled by their expected/known energy scale $[O_i]=d_i$, up to a maximal energy scale

   (this will in general contain lots of direct interaction that in the fundamental theory are really compound interactions)

   with $c_i\propto \frac{1}{{\Lambda }^{d_i-\dim X}}$;

3. finally one fixes all the coupling constants of all these interactions by

   • either deriving them from a known fundamental theory by integrating out higher energy effects in that theory;

   • or, otherwise, measuring them in the laboratory. The point being that due to the energy cutoff, this is guaranteed to be a finite number of parameters. After these have been determined, all remaining quantities given by the Lagrangian are then predictions of the effective theory.

Examples

Light-by-light scattering

(Pich, section 2.1)

Rayleigh scattering

(Pich, section 2.2)

Fermi theory of weak interactions

• Fermi theory of beta decay

(Pich, section 2.3)

Chiral perturbation theory

chiral perturbation theory? is an effective approximation of QCD in the light quark sector.

Heavy quark effective field theory

(…)

String theory and gravity coupled to gauge theory

The string scattering amplitudes for superstrings are finite (fully proven so for low loop order and with various plausibility arguments for higher loop order, see at string scattering amplitudes for more), hence define a UV-complete S-matrix. The corresponding low energy effective field theories are theories of supergravity coupled to gauge theory. (type II supergravity, heterotic supergravity).

See also at string theory FAQ – What is string theory?.

Related concepts

• effective action

References

The modern picture of effective low-energy QFT goes back to

• L. P. Kadanoff, Scaling laws for Ising models near $T_c$ , Physica 2 (1966);

• Kenneth Wilson, Renormalization group and critical phenomena , I., Physical review B 4(9) (1971).

• Joe Polchinski, Renormalization and effective Lagrangians , Nuclear Phys. B B231 (1984).

• Steven Weinberg, Physica 96 A (1979) 327

• H. Leutwyler, Ann. Phys., NY 235 (1994) 165.

A standard textbook adopting this perspective is

• Steven Weinberg The Quantum Theory of Fields (Cambridge University Press,Cambridge,1995).

whose author describes his goal as: “This is intended to be a book on quantum field theory for the era of effective field theory.” Another book which takes the effective-field-theory approach to QFT is

• Anthony Zee? Quantum Field Theory in a Nutshell (Princeton University Press, second edition, 2010).

Introductory lecture notes are for instance in

• A. Pich, Effective Field Theory (arXiv:hep-ph/9806303)

The theory of gravity based on the standard Einstein-Hilbert action may be regarded as just an effective QFT, which makes some of its notorious problems be non-problems:

• John F. Donoghue, Introduction to the Effective Field Theory Description of Gravity (arXiv:gr-qc/9512024)

Comments on this point are also in

• Jacques Distler, blog posts

  • Motivation

  • Effective field theory and gravity

• Kevin Costello, Renormalization and Effective Field Theory