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# Effective field theory

In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies).

Presently, effective field theories are discussed in the context of the renormalization group (RG) where the process of integrating out short distance degrees of freedom is made systematic. Although this method is not sufficiently concrete to allow the actual construction of effective field theories, the gross understanding of their usefulness becomes clear through an RG analysis. This method also lends credence to the main technique of constructing effective field theories, through the analysis of symmetries. If there is a single mass scale M in the microscopic theory, then the effective field theory can be seen as an expansion in 1/M. This technique is useful for scattering or other processes where the maximum momentum scale k satisfies the condition k/M<<1. Since effective field theories are not valid at small length scales, they need not be renormalizable.

The most well-known example of an effective field theory is the Fermi theory of beta decay. This theory was developed during the early study of weak decays of nuclei when only the hadrons and leptons undergoing weak decay were known. The typical reactions studied were:

$n\to p+e^-+\overline\nu_e$
$\mu^-\to e^-+\overline\nu_e+\nu_\mu$

This theory posited a pointlike interaction between the four fermions involved in these reactions. The theory had great phenomenological success and was eventually understood to arise from the gauge theory of electroweak interactions, which forms a part of the standard model of particle physics. In this more fundamental theory, the interactions are mediated by a flavour-changing gauge boson, the W±. The immense success of the Fermi theory was because the W particle has mass of about 80 GeV, whereas the early experiments were all done at an energy scale of less than 10 MeV. Such a separation of scales, by over 3 orders of magnitude, has not been met in any other situation as yet.

Another famous example is the BCS theory of superconductivity. In this case the underlying theory is the theory of electrons in a metal interacting with lattice vibrations called phonons. The phonons cause attractive interactions between appropriate electrons, causing them to form Cooper pairs. The length scale of these pairs is much larger than the wavelength of phonons. As a result, it is possible to neglect the dynamics of phonons and construct a theory in which two electrons interact at a point. This is the BCS theory, which has had remarkable success in describing and predicting the results of experiments.

Presently, effective field theories are written for many situations.

• One major branch of nuclear physics is quantum hadrodynamics, where the interactions of hadrons are treated as a field theory, which one hopes to derive from quantum chromodynamics, the true underlying theory, in the future. Due to the smaller separation of length scales here, this effective theory has some classificatory power, but not the spectacular success of the Fermi theory.
• In particle physics the effective field theory of QCD called chiral perturbation theory has had better success. This theory deals with the interactions of hadrons with pions or kaons, which are the Goldstone bosons of spontaneous chiral symmetry breaking. The expansion parameter is the pion energy/momentum.
• For hadrons containing one heavy quark (such as the bottom or charm), an effective field theory which expands in powers of the quark mass, called the heavy-quark effective theory (HQET), has been found useful.
• For hadrons containing two heavy quarks, an effective field theory which expands in powers of the relative velocity of the heavy quarks, called non-relativistic QCD (NRQCD), has been found useful, especially when used in conjunctions with lattice QCD.
• For hadron reactions with light energetic (collinear) particles, the interactions with low-energetic (soft) degrees of freedom are described by the soft-collinear effective theory (SCET).
• General relativity is expected to be the low energy effective theory of a full theory of quantum gravity, such as string theory. The expansion scale is the Planck mass.
• All of condensed matter physics consists of writing effective field theories for the particular property of matter being studied.
• Effective field theories have also been used to simplify problems in General Relativity (NRGR). In particular in calculating post-Newtonian corrections to the gravity wave signature of inspiralling finite-sized objects. [1]