- Effective field theory
In

physics , an**effective field theory**is an approximate theory (usually aquantum field theory ) that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances (or, equivalently, at higher energies).**The renormalization group**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 berenormalizable .**Examples of effective field theories****Fermi theory of beta decay**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

hadron s andlepton s undergoing weak decay were known. The typical reactions studied were:::$n\; o\; p+e^-+overline\; u\_e$::$mu^-\; o\; e^-+overline\; u\_e+\; u\_mu.$This theory posited a pointlike interaction between the fourfermion s involved in these reactions. The theory had great phenomenological success and was eventually understood to arise from the gauge theory ofelectroweak interaction s, which forms a part of thestandard 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 80GeV , whereas the early experiments were all done at an energy scale of less than 10MeV . Such a separation of scales, by over 3 orders of magnitude, has not been met in any other situation as yet.**BCS theory of superconductivity**Another famous example is the

BCS theory ofsuperconductivity . Here the underlying theory is ofelectron s in ametal interacting with lattice vibrations calledphonon s. The phonons cause attractive interactions between some electrons, causing them to formCooper pair s. The length scale of these pairs is much larger than the wavelength of phonons, making it possible to neglect the dynamics of phonons and construct a theory in which two electrons effectively interact at a point. This theory has had remarkable success in describing and predicting the results of experiments.**Other examples**Presently, effective field theories are written for many situations.

*One major branch ofnuclear physics isquantum hadrodynamics , where the interactions ofhadron s are treated as a field theory, which one hopes to derive fromquantum 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.

*Inparticle physics the effective field theory of QCD calledchiral perturbation theory has had better success. This theory deals with the interactions ofhadron s withpion s orkaon s, which are theGoldstone boson s ofspontaneous chiral symmetry breaking . The expansion parameter is the pion energy/momentum.

*Forhadron s containing one heavyquark (such as the bottom orcharm ), an effective field theory which expands in powers of the quark mass, called theheavy-quark effective theory (HQET), has been found useful.

*Forhadron s containing two heavy quarks, an effective field theory which expands in powers of the relative velocity of the heavy quarks, callednon-relativistic QCD (NRQCD), has been found useful, especially when used in conjunctions withlattice QCD .

*Forhadron reactions with light energetic (collinear) particles, the interactions with low-energetic (soft) degrees of freedom are described by thesoft-collinear effective theory (SCET).

*General relativity is expected to be the low energy effective theory of a full theory ofquantum gravity , such asstring theory . The expansion scale is thePlanck mass .

*All ofcondensed 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. [*http://arxiv.org/pdf/hep-th/0409156*]**ee also***

Renormalization group

*Quantum field theory **References and external links*** [

*http://arxiv.org/abs/hep-ph/9806303 Effective Field Theory, A. Pich*] , Lectures at the 1997 Les Houches Summer School "Probing the Standard Model of Particle Interactions."

* [*http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VH6-42WP59W-7&_user=10&_handle=V-WA-A-W-D-MsSAYWA-UUW-U-AABVCUWVAD-AABWUYBWAD-VDWCVUEAW-D-U&_fmt=summary&_coverDate=06%2F30%2F2001&_rdoc=7&_orig=browse&_srch=%23toc%236058%232001%23999679997%23246610!&_cdi=6058&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f3f507eb552a4deccd7a93977d1dde3d Effective field theories, reduction and scientific explanation, by S. Hartmann*] , "Studies in History and Philosophy of Modern Physics"**32B**, 267-304 (2001).

* [*http://arxiv.org/abs/hep-ph/9311274 On the foundations of chiral perturbation theory, H. Leutwyler*] (Annals of Physics, v 235, 1994, p 165-203)

* [*http://arxiv.org/abs/hep-ph/9703290 Aspects of heavy quark theory, by I. Bigi, M. Shifman and N. Uraltsev*] (Annual Reviews of Nuclear and Particle Science, v 47, 1997, p 591-661)

* [*http://www.fuw.edu.pl/~dobaczew/maub-42w/node18.html Effective field theory*] (Interactions, Symmetry Breaking and Effective Fields - from Quarks to Nuclei. an Internet Lecture by Jacek Dobaczewski)

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