en>SmackBot: Standard headings/general fixes

2009-05-19T17:38:04Z

Standard headings/general fixes

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'''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 (physics)|parity]] and charge conjugation. 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 the low-energy regime of QCD, the degrees of freedom are no longer [[quarks]] and [[gluons]], but rather [[hadrons]]. This is a result of [[color confinement|confinement]]. If one could "solve" the QCD [[partition function (quantum field theory)|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. 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]] (LECs), are associated with terms in the Lagrangian that can be determined by fitting to experimental data or be derived from underlining theory.

There are several power counting schemes in ChPT. The most widely used one is the <math>p</math>-expansion. However, there also exist the <math>\epsilon</math>, <math>\delta,</math> and <math>\epsilon^{\prime}</math> expansions. All of these expansions are valid in finite volume, (though the <math>p</math> 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.

The Lagrangian of the <math> p </math> 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 <math>(\partial \pi)^2 + m_{\pi}^2 \pi^2</math> is considered in the first approximation, and terms like <math>m_{\pi}^4 \pi^2 + (\partial \pi)^6</math> 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

<math>
U = \exp\left\{\frac{i}{F} \begin{pmatrix} \pi^0 & \sqrt{2}\pi^+ \\ \sqrt{2}\pi^- & - \pi^0 \end{pmatrix}\right\}
</math>
where <math>F = 93</math> MeV. In general different choices of the normalization for <math>F</math> exist and one must choose the value that is consistent with the charged pion decay rate.

The theory allows the description of interactions between [[pion]]s, and between pions and [[nucleon]]s (or other matter fields). SU(3) ChPT can also describe interactions of [[kaon]]s and eta mesons, while similar theories can be used to describe the vector mesons. Since chiral perturbation theory assumes [[chiral symmetry]], and therefore massless quarks, it cannot be used to model interactions of the heavier [[quarks]].

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

<math>
\mathcal{L}_{2}=\frac{F^2}{4}{\rm tr}(\partial_{\mu}U \partial^{\mu}U^{\dagger})+\frac{\lambda F^3}{4}{\rm tr}(m_q U+m_q^{\dagger}U^{\dagger})
</math>

where <math>F = 93</math> MeV and <math>m_q</math> is the quark mass matrix. In the <math>p</math>-expansion of ChPT, the small expansion parameters are

<math>
\frac{p}{\Lambda_{\chi}}, \frac{m_{\pi}}{\Lambda_{\chi}}.
</math>

where <math>\Lambda_{\chi}</math> is the chiral symmetry breaking scale, of order 1 GeV (sometimes estimated as
<math>\Lambda_{\chi} = 4\pi F</math>).
In this expansion, <math>m_q</math> counts as <math>\mathcal{O}(p^2)</math> because <math>m_{\pi}^2=\lambda m_q F</math> to leading order in the chiral expansion ({{cite doi|10.1103/PhysRev.175.2195|noedit}}).

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. For example, if one wishes to compute an [[observable]] to <math>\mathcal{O}(p^4)</math>, then one must compute the [[contact terms]] that come from the <math>\mathcal{O}(p^4)</math> Lagrangian (this is different for an SU(2) vs. SU(3) theory) at [[tree-level]] and the [[one-loop]] contributions from the <math>\mathcal{O}(p^2)</math> Lagrangian.) One can easily see that a one-loop contribution from the <math>\mathcal{O}(p^2)</math> Lagrangian counts as <math>\mathcal{O}(p^4)</math> by noting that the integration measure counts as <math>p^4</math>, the [[propagator]] counts as <math>p^{-2}</math>, while the derivative contributions count as <math>p^2</math>. Therefore, since the calculation is valid to <math>\mathcal{O}(p^4)</math>, one removes the divergences in the calculation with the renormalization of the [[low-energy constants]] (LECs) from the <math>\mathcal{O}(p^4)</math> Lagrangian. Therefore, if one wishes to remove all the divergences in the computation of a given observable to <math>\mathcal{O}(p^n)</math>, one uses the [[coupling constants]] in the expression for the <math>\mathcal{O}(p^n)</math> 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]]. For instance, it can be applied to few-nucleon systems, and at next-to-next-to-leading order in the [[perturbation theory|perturbative expansion]], it can account for [[three-nucleon force]]s in a natural way.

== External links ==
* Howard Georgi, Weak Interactions and Modern Particle Theory, Benjamin Cummings, 1984; [http://www.people.fas.harvard.edu/~hgeorgi/weak.pdf revised version 2008]
* [[Heinrich Leutwyler]], [http://arxiv.org/abs/hep-ph/9311274 On the foundations of chiral perturbation theory], Annals of Physics, v 235, 1994, p 165-203.
* [http://www.scholarpedia.org/article/Chiral_perturbation_theory Heinrich Leutwyler (2012), Chiral perturbation theory], Scholarpedia, 7(10):8708. doi 10.4249/scholarpedia.8708

[[Category:Particle physics]]
[[Category:Quantum chromodynamics]]

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en>SmackBot: Standard headings/general fixes