Chiral perturbation theory

Chiral perturbation theory (ChPT) is an effective field theory constructed with a Lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), as well as the other symmetries of parity and charge conjugation. In Physics, an effective field theory is an approximate theory (usually a Quantum field theory) that includes appropriate degrees of freedom to describe The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system In Quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate. ChPT is a theory which allows one to study the low-energy dynamics of QCD. As 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 is one alternative method that has proved successful in extracting non-perturbative information. In Physics, lattice quantum chromodynamics (lattice QCD is a theory of Quarks and Gluons formulated on a space-time lattice.

In the low-energy regime of QCD, the degrees of freedom are no longer quarks and gluons, but rather hadrons. In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( 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. A low-energy effective theory with hadrons as the fundamental degrees of freedom is a possible solution. According to Steven Weinberg, an effective theory can be useful if one writes down all terms consistent with the symmetries of the parent theory. Steven Weinberg (born May 3, 1933) is an American Physicist, and Nobel laureate in Physics for his contributions with Abdus Salam In general there are an infinite number of terms which meet this requirement. Therefore in order to make any physical predictions, one assigns the theory a power counting scheme which organizes terms by a pre-specified degree of importance which allows one to keep some terms and reject all others as higher-order corrections which can be safely neglected. In addition, unknown coupling constants, also called low-energy constants (LEC's), are associated with terms in the Lagrangian that must be determined by fitting to experimental data.

There are several power counting schemes in ChPT. The most widely used one is the $p$ -expansion. However, there also exist the $ε$ , $δ,$ and $\epsilon^{\prime}$ 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 requires one to use different reorganizations or the chiral theory in order to correctly understand the physics. These different reorganizations correspond to the different power counting schemes.

The Lagrangian of the $p$ expansion is constructed by introducing every interaction of particles which is not excluded by symmetry, and then ordering them based on the number of momentum and mass powers (so that $(\partial \pi)^2 + m_{\pi}^2 \pi^2$ is considered in the first approximation, and terms like $m_{\pi}^4 \pi^2 + (\partial \pi)^6$ are used as higher order corrections). It is also common 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

$U = \exp(\frac{i}{f} \begin{pmatrix} \pi^0 /\sqrt{2} & \pi^+ \\ \pi^- & - \pi^0/\sqrt{2} \end{pmatrix})$ where $f = 132$ MeV. In general different choices for $f$ exist and one must specify the value they choose before beginning any computations.

The theory allows the description of interactions between pions, and between pions and nucleons (or other matter fields). In Particle physics, pion (short for pi meson) is the collective name for three Subatomic particles, and. In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton. SU(3) ChPT can also describe interactions of kaons and eta mesons, while similar theories can be used to describe the vector mesons. In Particle physics, a kaon (/ˈkeɪɒn/ also called K-meson and denoted) is any one of a group of four Mesons distinguished by the fact that they Since chiral perturbation theory assumes chiral symmetry, and therefore massless quarks, it cannot be used to model interactions of the heavier quarks. In Quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle.

For an SU(2) theory the leading order chiral Lagrangian is given by

$\mathcal{L}_{2}=\frac{f^2}{8}{\rm tr}(\partial_{\mu}U \partial^{\mu}U^{\dagger})+\frac{\lambda f^2}{4}{\rm tr}(m_q U+m_q^{\dagger}U^{\dagger})$

where $f = 132$ MeV and $m q$ is the quark mass matrix. In the $p$ -expansion of ChPT, the small expansion parameters are

$\frac{p}{\Lambda_{\chi}}, \frac{m_{\pi}}{\Lambda_{\chi}}.$

In this expansion, $m q$ counts as $\mathcal{O}(p^2)$ because $m_{\pi}^2=2\lambda m_q$ to leading order in the chiral expansion.

The effective theory in general is non-renormalizable, However given a particular power counting scheme in ChPT, the effective theory is renormalizable at a given order in the chiral expansion. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection For example, if one wishes to compute an observable to $\mathcal{O}(p^4)$ , then one must compute the contact terms that come from the $\mathcal{O}(p^4)$ Lagrangian. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical (This is different for an SU(2) vs. SU(3) theory) at tree-level and the one-loop contributions from the $\mathcal{O}(p^2)$ Lagrangian. In Physics, a one-loop Sander-Feynman diagram is a connected Feynman diagram with only one cycle ( Unicyclic) ) One can easily see that a one-loop contribution from the $\mathcal{O}(p^2)$ Lagrangian counts as $\mathcal{O}(p^4)$ by noting that the integration measure counts as $p 4$ , the propagator counts as $p - 2$ , while the derivative contributions count as $p 2$ . In Quantum mechanics and Quantum field theory, the propagator gives the Probability amplitude for a particle to travel from one place to another in a given Therefore, since the calculation is valid to $\mathcal{O}(p^4)$ , one removes the divergences in the calculation with the renormalization of the low-energy constants (LEC's) from the $\mathcal{O}(p^4)$ Lagrangian. Therefore, if one wishes to remove all the divergences in the computation of a given observable to $\mathcal{O}(p^n)$ , one uses the coupling constants in the expression for the $\mathcal{O}(p^n)$ Lagrangian to remove those divergences.

In some cases, chiral perturbation theory has been successful in describing the interactions between hadrons in the non-perturbative regime of the strong interaction. In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( In Mathematics and Physics, a non-perturbative function or process is one that cannot be accurately described by Perturbation theory. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and 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. This article describes perturbation theory as a general mathematical method A three-body force is a Force that does not exist in a system of two objects but appears in a three-body system

References and external links

On the foundations of chiral perturbation theory, H. Leutwyler (Annals of Physics, v 235, 1994, p 165-203)

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