View source for Effective field theory

{{short description|Type of approximation to an underlying physical theory}}
{{More footnotes|date=May 2013}}
{{Quantum field theory|cTopic=Some models}}
{{merge to|Effective theory|discuss=Talk:Effective theory#Merge proposal|date=November 2024}}

In [[physics]], an '''effective field theory''' is a type of approximation, or [[effective theory]], for an underlying physical theory, such as a [[quantum field theory]] or a [[statistical mechanics]] model. An effective field theory includes the appropriate [[degrees of freedom (physics and chemistry)|degrees of freedom]] to describe physical phenomena occurring at a chosen [[length scale]] or energy scale, while ignoring substructure and degrees of freedom at shorter distances (or, equivalently, at higher energies). Intuitively, one averages over the behavior of the underlying theory at shorter length scales to derive what is hoped to be a simplified model at longer length scales. Effective field theories typically work best when there is a large separation between length scale of interest and the length scale of the underlying dynamics. Effective field theories have found use in [[particle physics]], [[statistical mechanics]], [[condensed matter physics]], [[general relativity]], and [[hydrodynamics]]. They simplify calculations, and allow treatment of [[Dissipative system|dissipation]] and [[radiation]] effects.<ref>{{Cite journal|doi=10.1103/PhysRevLett.110.174301|pmid=23679733|title=Classical Mechanics of Nonconservative Systems|journal=Physical Review Letters|volume=110|issue=17|pages=174301|year=2013|last1=Galley|first1=Chad R.|arxiv=1210.2745|bibcode=2013PhRvL.110q4301G|s2cid=14591873|doi-access=free}}</ref><ref>{{Cite journal |arxiv = 1402.2610|last1 = Birnholtz|first1 = Ofek|title = Radiation reaction at the level of the action|journal = International Journal of Modern Physics A|volume = 29|issue = 24|pages = 1450132–1450190|last2 = Hadar|first2 = Shahar|last3 = Kol|first3 = Barak|year = 2014|doi = 10.1142/S0217751X14501322|bibcode = 2014IJMPA..2950132B|s2cid = 118541484}}</ref>

==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 [[symmetry|symmetries]]. If there is a single energy scale <math>M</math> in the ''microscopic'' theory, then the effective field theory can be seen as an expansion in <math>1/M</math>. The construction of an effective field theory accurate to some power of <math>1/M</math> requires a new set of free parameters at each order of the expansion in <math>1/M</math>. This technique is useful for [[scattering]] or other processes where the maximum momentum scale <math>\mathbf k</math> satisfies the condition <math>|\mathbf{k}|/M\ll 1</math>. Since effective field theories are not valid at small length scales, they need not be [[Renormalization#Renormalizability|renormalizable]]. Indeed, the ever expanding number of parameters at each order in <math>1/M</math> required for an effective field theory means that they are generally not renormalizable in the same sense as [[quantum electrodynamics]] which requires only the renormalization of two parameters (the fine structure constant and the electron mass).

==Examples==
===Fermi theory of beta decay===
The best-known example of an effective field theory is the [[Fermi's interaction|Fermi theory of beta decay]]. This theory was developed during the early study of weak decays of [[Atomic nucleus|nuclei]] when only the [[hadron]]s and [[lepton]]s undergoing weak decay were known. The typical [[elementary particle reaction|reactions]] studied were:

::<math>
\begin{align}
n & \to p+e^-+\overline\nu_e \\
\mu^- & \to e^-+\overline\nu_e+\nu_\mu.
\end{align}
</math>

This theory posited a pointlike interaction between the four [[fermion]]s involved in these reactions. The theory had great [[phenomenology (particle physics)|phenomenological]] success and was eventually understood to arise from the [[gauge theory]] of [[electroweak interaction]]s, which forms a part of the [[standard model]] of particle physics. In this more fundamental theory, the interactions are mediated by a [[flavour (particle physics)|flavour]]-changing [[gauge boson]], the W<sup>±</sup>. 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.

===BCS theory of superconductivity===
Another famous example is the [[BCS theory]] of [[superconductivity]]. Here the underlying theory is the theory of [[electron]]s in a [[metal]] interacting with lattice vibrations called [[phonon]]s. The phonons cause attractive interactions between some electrons, causing them to form [[Cooper 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 on superconductivity.

===Gravitational field theories ===
[[General relativity]] (GR) itself is expected to be the low energy effective field theory of a full theory of [[quantum gravity]], such as [[string theory]] or [[loop quantum gravity]]. The expansion scale is the [[Planck mass]].
Effective field theories have also been used to simplify problems in general relativity, in particular in calculating the [[gravitational wave]] signature of inspiralling finite-sized objects.<ref>{{Cite journal |arxiv = hep-th/0409156|last1 = Goldberger|first1 = Walter|title = An Effective Field Theory of Gravity for Extended Objects|journal = Physical Review D|volume = 73|issue = 10|last2 = Rothstein|first2 = Ira|year = 2004| page=104029 |doi = 10.1103/PhysRevD.73.104029|s2cid = 54188791}}</ref> The most common EFT in GR is non-relativistic general relativity (NRGR),<ref>{{Cite web |last1=Porto |first1=Rafael A. |last2=Rothstein |first2=Ira |last3=Goldberger |first3=Walter |title=EFT meets GR |url=https://online.kitp.ucsb.edu/online/numrel-m08/buonanno/pdf1/Porto_NumRelData_KITP.pdf |access-date=3 November 2023 |website=online.kitp.ucsb.edu}}</ref><ref>{{Cite journal |arxiv = 0712.4116|last1 = Kol|first1 = Barak|title = Non-Relativistic Gravitation: From Newton to Einstein and Back|journal =  Classical and Quantum Gravity|volume = 25|issue = 14|pages = 145011|last2 = Smolkin|first2 = Lee|year = 2008|doi = 10.1088/0264-9381/25/14/145011|bibcode = 2008CQGra..25n5011K|s2cid = 119216835}}</ref><ref>{{Cite journal |arxiv = gr-qc/0511061|last1 = Porto|first1 = Rafael A|title = Post-Newtonian corrections to the motion of spinning bodies in NRGR|journal =  Physical Review D|volume = 73|issue = 104031|pages = 104031|year = 2006|doi = 10.1103/PhysRevD.73.104031|s2cid = 119377563}}</ref> which is similar to the [[post-Newtonian expansion]].<ref>{{Cite journal |doi = 10.1103/PhysRevD.88.104037|title = Theory of post-Newtonian radiation and reaction|journal = Physical Review D|volume = 88|issue = 10|pages = 104037|year = 2013|last1 = Birnholtz|first1 = Ofek|last2 = Hadar|first2 = Shahar|last3 = Kol|first3 = Barak|arxiv = 1305.6930|bibcode = 2013PhRvD..88j4037B|s2cid = 119170985}}</ref> Another common GR EFT is the extreme mass ratio (EMR), which in the context of the inspiralling problem is called [[extreme mass ratio inspiral]].

===Other examples===
Presently, effective field theories are written for many situations.
*One major branch of [[nuclear physics]] is [[quantum hadrodynamics]], where the interactions of [[hadron]]s are treated as a field theory, which should be derivable from the underlying theory of [[quantum chromodynamics]] (QCD). Quantum hadrodynamics is the theory of the [[nuclear force]], similarly to quantum chromodynamics being the theory of the [[strong interaction]] and quantum electrodynamics being the theory of the [[electromagnetic force]]. 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.<ref>{{Cite journal |arxiv = hep-ph/9311274|last1 = Leutwyler|first1 = H|title = On the Foundations of Chiral Perturbation Theory|journal =  Annals of Physics|volume = 235|pages = 165–203|year = 1994|issue = 1|doi = 10.1006/aphy.1994.1094|bibcode = 1994AnPhy.235..165L|s2cid = 16739698}}</ref> This theory deals with the interactions of [[hadron]]s with [[pion]]s or [[kaon]]s, which are the [[Goldstone boson]]s of [[spontaneous chiral symmetry breaking]]. The expansion parameter is the [[pion]] energy/momentum.
*For [[hadron]]s containing one heavy [[quark]] (such as the [[bottom quark|bottom]] or [[Charm quark|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 [[hadron]]s 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).
*Much of [[condensed matter physics]] consists of writing effective field theories for the particular property of matter being studied.
*Dissipationless [[hydrodynamics]] can also be treated using effective field theories.<ref>{{Cite journal |arxiv = 1211.6461|last1 = Endlich|first1 = Solomon|title = Dissipation in the effective field theory for hydrodynamics: First order effects|journal =  Physical Review D|volume = 88|issue = 10|pages = 105001|last2 = Nicolis|first2 = Alberto|last3 = Porto|first3 = Rafael|last4 = Wang|first4 = Junpu|year = 2013|doi = 10.1103/PhysRevD.88.105001|bibcode = 2013PhRvD..88j5001E|s2cid = 118441607}}</ref>

==See also==
*[[Form factor (quantum field theory)]]
*[[Renormalization group]]
*[[Quantum field theory]]
*[[Quantum triviality]]
*[[Ginzburg–Landau theory]]

==References==
{{Reflist}}

==Books==
* A.A. Petrov and A. Blechman, ‘’Effective Field Theories,’’ Singapore: World Scientific (2016). {{ISBN|978-981-4434-92-8}}
* C.P. Burgess, ‘’Introduction to Effective Field Theory,‘’ Cambridge University Press (2020). {{ISBN|978-052-1195-47-8}}

==External links==
{{wikiquote}}
*{{cite arXiv |eprint=hep-ph/9806303|last1=Birnholtz|first1=Ofek|title=Effective Field Theory|last2=Hadar|first2=Shahar|last3=Kol|first3=Barak|year=1998}}
*{{cite journal |doi=10.1016/S1355-2198(01)00005-3 |url=http://philsci-archive.pitt.edu/93/1/Hartmann.pdf|title=Effective Field Theories, Reductionism and Scientific Explanation|journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics|volume=32|issue=2|pages=267–304|year=2001|last1=Hartmann|first1=Stephan|bibcode=2001SHPMP..32..267H}}
*{{Cite journal |arxiv=hep-ph/9703290|last1=Birnholtz|first1=Ofek|title=Aspects of Heavy Quark Theory|journal= [[Annual Review of Nuclear and Particle Science]]|volume=47|pages=591–661|last2=Hadar|first2=Shahar|last3=Kol|first3=Barak|year=1997|doi=10.1146/annurev.nucl.47.1.591|bibcode=1997ARNPS..47..591B| doi-access=free |s2cid=13843227}}
*[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)

{{Industrial and applied mathematics}}

[[Category:Quantum field theory]]
[[Category:Statistical mechanics]]
[[Category:Renormalization group]]
[[Category:Chemical physics]]
[[Category:Nuclear physics]]
[[Category:Condensed matter physics]]