Chiral perturbation theory

The Lagrangian of the ${\displaystyle p}$-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 ${\displaystyle (\partial \pi )^{2}+m_{\pi }^{2}\pi ^{2}}$ is considered in the first-order approximation, where ${\displaystyle \pi }$ is the pion field and ${\displaystyle m_{\pi }}$ the pion mass, which breaks the underlying chiral symmetry explicitly (PCAC).[4][5] Terms like ${\displaystyle m_{\pi }^{4}\pi ^{2}+(\partial \pi )^{6}}$ 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

${\displaystyle U=\exp \left\{{\frac {i}{F}}{\begin{pmatrix}\pi ^{0}&{\sqrt {2}}\pi ^{+}\\{\sqrt {2}}\pi ^{-}&-\pi ^{0}\end{pmatrix}}\right\}}$

where ${\displaystyle F}$ is called the pion decay constant which is 93 MeV.

In general, different choices of the normalization for ${\displaystyle F}$ exist, so that one must choose the value that is consistent with the charged pion decay rate.

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 ${\displaystyle {\mathcal {O}}(p^{4})}$, then one must compute the contact terms that come from the ${\displaystyle {\mathcal {O}}(p^{4})}$ Lagrangian (this is different for an SU(2) vs. SU(3) theory) at tree-level and the one-loop contributions from the ${\displaystyle {\mathcal {O}}(p^{2})}$ Lagrangian.)

One can easily see that a one-loop contribution from the ${\displaystyle {\mathcal {O}}(p^{2})}$ Lagrangian counts as ${\displaystyle {\mathcal {O}}(p^{4})}$ by noting that the integration measure counts as ${\displaystyle p^{4}}$, the propagator counts as ${\displaystyle p^{-2}}$, while the derivative contributions count as ${\displaystyle p^{2}}$. Therefore, since the calculation is valid to ${\displaystyle {\mathcal {O}}(p^{4})}$, one removes the divergences in the calculation with the renormalization of the low-energy constants (LECs) from the ${\displaystyle {\mathcal {O}}(p^{4})}$ Lagrangian. So if one wishes to remove all the divergences in the computation of a given observable to ${\displaystyle {\mathcal {O}}(p^{n})}$, one uses the coupling constants in the expression for the ${\displaystyle {\mathcal {O}}(p^{n})}$ Lagrangian to remove those divergences.

The theory allows the description of interactions between pions, and between pions and nucleons (or other matter fields). SU(3) ChPT can also describe interactions of kaons 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[1]

${\displaystyle {\mathcal {L}}_{2}={\frac {F^{2}}{4}}{\rm {tr}}(\partial _{\mu }U\partial ^{\mu }U^{\dagger })+{\frac {\lambda F^{3}}{4}}{\rm {tr}}(m_{q}U+m_{q}^{\dagger }U^{\dagger })}$

where ${\displaystyle F=93}$ MeV and ${\displaystyle m_{q}}$ is the quark mass matrix. In the ${\displaystyle p}$-expansion of ChPT, the small expansion parameters are

${\displaystyle {\frac {p}{\Lambda _{\chi }}},{\frac {m_{\pi }}{\Lambda _{\chi }}}.}$

where ${\displaystyle \Lambda _{\chi }}$ is the chiral symmetry breaking scale, of order 1 GeV (sometimes estimated as ${\displaystyle \Lambda _{\chi }=4\pi F}$). In this expansion, ${\displaystyle m_{q}}$ counts as ${\displaystyle {\mathcal {O}}(p^{2})}$ because ${\displaystyle m_{\pi }^{2}=\lambda m_{q}F}$ to leading order in the chiral expansion.[6]

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 perturbative expansion, it can account for three-nucleon forces in a natural way.[7]