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Inphysics, aneffective field theory is a type of approximation, oreffective theory, for an underlying physical theory, such as aquantum field theory or astatistical mechanics model. An effective field theory includes the appropriatedegrees of freedom to describe physical phenomena occurring at a chosenlength 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 inparticle physics,statistical mechanics,condensed matter physics,general relativity, andhydrodynamics. They simplify calculations, and allow treatment ofdissipation andradiation effects.^[1][2]

Presently, effective field theories are discussed in the context of therenormalization group (RG) where the process ofintegrating 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 ofsymmetries. If there is a single energy scale${\displaystyle M}$ in themicroscopic theory, then the effective field theory can be seen as an expansion in${\displaystyle 1/M}$. The construction of an effective field theory accurate to some power of${\displaystyle 1/M}$ requires a new set of free parameters at each order of the expansion in${\displaystyle 1/M}$.

This technique is useful forscattering or other processes where the maximum momentum scale${\displaystyle \mathbf{k}}$ satisfies the condition${\displaystyle |\mathbf{k}|/M\ll 1}$. Since effective field theories are not valid at small length scales, they need not berenormalizable. Indeed, the ever expanding number of parameters at each order in${\displaystyle 1/M}$ required for an effective field theory means that they are generally not renormalizable in the same sense asquantum electrodynamics which requires only the renormalization of two parameters (the fine structure constant and the electron mass).

Steven Weinberg's "folk theorem" stipulates how to build an effective field theory that is well behaved. The "theorem" states that the most generalLagrangian that is consistent with the symmetries of the low energy theory can be rendered into an effective field theory at low energies that respects the symmetries and respectsunitarity, analyticity, andcluster decomposition.^[3][4]

The best-known example of an effective field theory is theFermi theory of beta decay. This theory was developed during the early study of weak decays ofnuclei when only thehadrons andleptons undergoing weak decay were known. The typicalreactions studied were:

This theory posited a pointlike interaction between the fourfermions involved in these reactions. The theory had greatphenomenological success and was eventually understood to arise from thegauge theory ofelectroweak interactions, which forms a part of theStandard Model of particle physics. In this more fundamental theory, the interactions are mediated by aflavour-changinggauge 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.

Another famous example is theBCS theory ofsuperconductivity. Here, the underlying theory is the theory ofelectrons in ametal interacting with lattice vibrations calledphonons. The phonons cause attractive interactions between some electrons, causing them to formCooper pairs. 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.

General relativity (GR) itself is expected to be the low energy effective field theory of a full theory ofquantum gravity, such asstring theory orloop quantum gravity. The expansion scale is thePlanck mass.Effective field theories have also been used to simplify problems in general relativity, in particular in calculating thegravitational wave signature of inspiralling finite-sized objects.^[5] The most common EFT in GR is non-relativistic general relativity (NRGR),^[6][7][8] which is similar to thepost-Newtonian expansion.^[9] Another common GR EFT is the extreme mass ratio (EMR), which in the context of the inspiralling problem is calledextreme mass ratio inspiral.

Presently, effective field theories are written for many situations.

• One major branch ofnuclear physics isquantum hadrodynamics, where the interactions ofhadrons are treated as a field theory, which should be derivable from the underlying theory ofquantum chromodynamics (QCD). Quantum hadrodynamics is the theory of thenuclear force, similarly to quantum chromodynamics being the theory of thestrong interaction and quantum electrodynamics being the theory of theelectromagnetic 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.
• Inparticle physics the effective field theory of QCD calledchiral perturbation theory has had better success.^[10] This theory deals with the interactions ofhadrons withpions orkaons, which are theGoldstone bosons ofspontaneous chiral symmetry breaking. The expansion parameter is thepion energy/momentum.
• Forhadrons containing one heavyquark (such as thebottom orcharm), an effective field theory which expands in powers of the quark mass, called theheavy quark effective theory (HQET), has been found useful.
• Forhadrons containing two heavy quarks, an effective field theory which expands in powers of therelative velocity of the heavy quarks, called non-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).
• Much ofcondensed matter physics consists of writing effective field theories for the particular property of matter being studied.
• Dissipationlesshydrodynamics can also be treated using effective field theories.^[11]

• Form factor (quantum field theory)
• Renormalization group
• Quantum field theory
• Quantum triviality
• Ginzburg–Landau theory

• 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

Wikiquote has quotations related toEffective field theory.

• Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (1998). "Effective Field Theory".arXiv:hep-ph/9806303.
• Hartmann, Stephan (2001)."Effective Field Theories, Reductionism and Scientific Explanation"(PDF).Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics.32 (2):267–304.Bibcode:2001SHPMP..32..267H.doi:10.1016/S1355-2198(01)00005-3.
• Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (1997)."Aspects of Heavy Quark Theory".Annual Review of Nuclear and Particle Science.47:591–661.arXiv:hep-ph/9703290.Bibcode:1997ARNPS..47..591B.doi:10.1146/annurev.nucl.47.1.591.S2CID 13843227.
• Effective field theory (Interactions, Symmetry Breaking and Effective Fields - from Quarks to Nuclei. an Internet Lecture by Jacek Dobaczewski)

• Quantum field theory
• Statistical mechanics
• Renormalization group
• Chemical physics
• Nuclear physics
• Condensed matter physics

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