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 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 dissipation and radiation effects.
== 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. The construction
of an effective field theory accurate to some
power of 1/M requires a new set of free parameters
at each order of the 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. Indeed,
the ever expanding number of parameters at
each order in 1/M 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.
== Examples of effective field theories ==
=== Fermi theory of beta decay ===
The best-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
→
p
+
e
−
+
ν
¯
e
μ
−
→
e
−
+
ν
¯
e
+
ν
μ
.
{\displaystyle {\begin{aligned}n&\to p+e^{-}+{\overline
{\nu }}_{e}\\\mu ^{-}&\to e^{-}+{\overline
{\nu }}_{e}+\nu _{\mu }.\end{aligned}}}
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.
=== BCS theory of superconductivity ===
Another famous example is the BCS theory of
superconductivity. Here the underlying theory
is of electrons in a metal interacting with
lattice vibrations called phonons. The phonons
cause attractive interactions between some
electrons, causing them to form Cooper 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.
=== Effective Field Theories in Gravity ===
General relativity 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. The most common EFT in GR is "Non-Relativistic
General Relativity" (NRGR), which is similar
to the post-Newtonian expansion. Another common
GR EFT is the Extreme Mass Ratio (EMR), which
in the context of the inspiralling problem
is called EMRI.
=== Other examples ===
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 should
be derivable from the underlying theory of
quantum chromodynamics. 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. 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).
Much of condensed matter physics consists
of writing effective field theories for the
particular property of matter being studied.
Hydrodynamics can also be treated using Effective
Field Theories
== See also ==
Form factor (quantum field theory)
Renormalization group
Quantum field theory
Quantum triviality
Ginzburg–Landau theory
