Chiral Perturbation Theory
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Chiral perturbation theory (ChPT) is an
effective field theory 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 ...
constructed with a Lagrangian consistent with the (approximate)
chiral symmetry A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particl ...
of
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...
(QCD), as well as the other symmetries of parity and charge conjugation.Heinrich Leutwyler (2012), Chiral perturbation theory
Scholarpedia, 7(10):8708.
ChPT is a theory which allows one to study the low-energy dynamics of QCD on the basis of this underlying chiral symmetry.


Goals

In the theory of the strong interaction of the
standard model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
, we describe the interactions between quarks and gluons. Due to the running of the strong coupling constant, we can apply perturbation theory in the coupling constant only at high energies. But in the low-energy regime of QCD, the degrees of freedom are no longer
quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
and
gluons A gluon ( ) is a type of massless elementary particle that mediates the strong interaction between quarks, acting as the exchange particle for the interaction. Gluons are massless vector bosons, thereby having a spin of 1. Through the s ...
, but rather
hadrons In particle physics, a hadron is a composite subatomic particle made of two or more quarks held together by the strong nuclear force. Pronounced , the name is derived . They are analogous to molecules, which are held together by the electric ...
. This is a result of confinement. If one could "solve" the QCD partition function (such that the degrees of freedom in the Lagrangian are replaced by hadrons), then one could extract information about low-energy physics. To date this has not been accomplished. Because QCD becomes non-perturbative at low energy, it is impossible to use perturbative methods to extract information from the partition function of QCD.
Lattice QCD Lattice QCD is a well-established non- perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the ...
is an alternative method that has proved successful in extracting non-perturbative information.


Method

Using different degrees of freedom, we have to assure that observables calculated in the EFT are related to those of the underlying theory. This is achieved by using the most general Lagrangian that is consistent with the symmetries of the underlying theory, as this yields the ‘‘most general possible
S-matrix In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
consistent with analyticity, perturbative unitarity,
cluster decomposition In physics, the cluster decomposition property states that experiments carried out far from each other cannot influence each other. Usually applied to quantum field theory, it requires that vacuum expectation values of operators localized in bound ...
and the assumed symmetry. In general there is an infinite number of terms which meet this requirement. Therefore in order to make any physical predictions, one assigns to the theory a power-ordering scheme which organizes terms by some pre-determined degree of importance. The ordering allows one to keep some terms and omit all other, higher-order corrections which can safely be temporarily ignored. There are several power counting schemes in ChPT. The most widely used one is the p-expansion where p stands for momentum. However, there also exist the \epsilon, \delta, and \epsilon^ expansions. All of these expansions are valid in finite volume, (though the p expansion is the only one valid in infinite volume.) Particular choices of finite volumes require one to use different reorganizations of the chiral theory in order to correctly understand the physics. These different reorganizations correspond to the different power counting schemes. In addition to the ordering scheme, most terms in the approximate Lagrangian will be multiplied by ''
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between tw ...
s'' which represent the relative strengths of the force represented by each term. Values of these constants – also called low-energy constants or Ls – are usually not known. The constants can be determined by fitting to experimental data or be derived from underlying theory.


The model Lagrangian

The Lagrangian of the p-expansion is constructed by writing down all interactions which are not excluded by symmetry, and then ordering them based on the number of momentum and mass powers. The order is chosen so that (\partial \pi)^2 + m_^2 \pi^2 is considered in the first-order approximation, where \pi is the
pion In particle physics, a pion (, ) or pi meson, denoted with the Greek alphabet, Greek letter pi (letter), pi (), is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the ...
field and m_ the pion mass, which breaks the underlying chiral symmetry explicitly (PCAC). Terms like m_^4 \pi^2 + (\partial \pi)^6 are part of other, higher order corrections. It is also customary to compress the Lagrangian by replacing the single pion fields in each term with an infinite series of all possible combinations of pion fields. One of the most common choices is : U = \exp\left\ where F is called the
pion decay constant In particle physics, the pion decay constant is the square root of the coefficient in front of the kinetic term for the pion in the low-energy effective action. It is dimensionally an energy scale and it determines the strength of the chiral sym ...
which is 93 MeV. In general, different choices of the normalization for F exist, so that one must choose the value that is consistent with the charged pion decay rate.


Renormalization

The effective theory in general is non-renormalizable, however given a particular power counting scheme in ChPT, the effective theory is
renormalizable Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
at a given order in the chiral expansion. For example, if one wishes to compute an
observable In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
to \mathcal(p^4), then one must compute the contact terms that come from the \mathcal(p^4) Lagrangian (this is different for an SU(2) vs. SU(3) theory) at tree-level and the one-loop contributions from the \mathcal(p^2) Lagrangian.) One can easily see that a one-loop contribution from the \mathcal(p^2) Lagrangian counts as \mathcal(p^4) by noting that the integration measure counts as p^4, the
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. I ...
counts as p^, while the derivative contributions count as p^2. Therefore, since the calculation is valid to \mathcal(p^4), one removes the divergences in the calculation with the renormalization of the low-energy constants (LECs) from the \mathcal(p^4) Lagrangian. So if one wishes to remove all the divergences in the computation of a given observable to \mathcal(p^n), one uses the coupling constants in the expression for the \mathcal(p^n) Lagrangian to remove those divergences.


Successful application


Mesons and nucleons

The theory allows the description of interactions between pions, and between pions and
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number. Until the 1960s, nucleons were thought to be ele ...
s (or other matter fields). SU(3) ChPT can also describe interactions of
kaon In particle physics, a kaon, also called a K meson and denoted , is any of a group of four mesons distinguished by a quantum number called strangeness. In the quark model they are understood to be bound states of a strange quark (or antiquark ...
s and eta mesons, while similar theories can be used to describe the vector mesons. Since chiral perturbation theory assumes
chiral symmetry A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particl ...
, and therefore massless quarks, it cannot be used to model interactions of the heavier
quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
. For an SU(2) theory the leading order chiral Lagrangian is given by : \mathcal_=\frac(\partial_U \partial^U^)+\frac(m_q U+m_q^U^) where F = 93 MeV and m_q is the quark mass matrix. In the p-expansion of ChPT, the small expansion parameters are : \frac, \frac. where \Lambda_ is the chiral symmetry breaking scale, of order 1 GeV (sometimes estimated as \Lambda_ = 4\pi F). In this expansion, m_q counts as \mathcal(p^2) because m_^2=\lambda m_q F to leading order in the chiral expansion.


Hadron-hadron interactions

In some cases, chiral perturbation theory has been successful in describing the interactions between
hadrons In particle physics, a hadron is a composite subatomic particle made of two or more quarks held together by the strong nuclear force. Pronounced , the name is derived . They are analogous to molecules, which are held together by the electric ...
in the
non-perturbative In mathematics and physics, a non-perturbative function (mathematics), function or process is one that cannot be described by perturbation theory. An example is the function : f(x) = e^, which does not equal its own Taylor series in any neighbo ...
regime of the
strong interaction In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interaction, fundamental interactions. It confines Quark, quarks into proton, protons, n ...
. For instance, it can be applied to few-nucleon systems, and at next-to-next-to-leading order in the perturbative expansion, it can account for three-nucleon forces in a natural way.


References

{{Reflist


External links

* Howard Georgi, Weak Interactions and Modern Particle Theory, Benjamin Cummings, 1984
revised version 2008
* H Leutwyler
On the foundations of chiral perturbation theory
''Annals of Physics'', 235, 1994, p 165-203. * Stefan Scherer
Introduction to Chiral Perturbation Theory
Adv. Nucl. Phys. 27 (2003) 277. * Gerhard Ecker
Chiral perturbation theory
Prog. Part. Nucl. Phys. 35 (1995), pp. 1–80. Quantum chromodynamics