The Cluster's Chiral Perturbation Theory
A page about chiral perturbation theory (ChPT) — the effective field theory of QCD at low energies.
Chiral perturbation theory
Chiral perturbation theory (ChPT) is the effective field theory of QCD at low energies (E << Lambda_QCD). It is based on the spontaneous breaking of chiral symmetry SU(2)_L x SU(2)_R -> SU(2)_V. The Goldstone bosons of this breaking are the pions. ChPT organizes calculations as an expansion in E / Lambda_chi, where Lambda_chi ~ 4 pi F_pi ~ 1 GeV is the chiral symmetry breaking scale. In the cluster, ChPT is the effective edit theory of the edit QCD at low edit energies, based on the spontaneous breaking of edit chiral symmetry.
The chiral Lagrangian
The leading-order chiral Lagrangian is L_2 = (F_pi^2 / 4) Tr(partial_mu U partial^mu U^dagger) + (F_pi^2 m_pi^2 / 4) Tr(U + U^dagger), where U = exp(i pi^a tau^a / F_pi) is the pion field. This Lagrangian contains all terms with two derivatives or one mass insertion. In the cluster, the edit chiral Lagrangian contains all edit terms with two edit derivatives or one edit mass insertion.
The pion mass
The pion mass is given by the Gell-Mann-Oakes-Renner relation: m_pi^2 F_pi^2 = -(m_u + m_d) <qbar q> + O(m_q^2). The pion mass is proportional to the square root of the quark masses. In the cluster, the edit pion mass is given by the edit Gell-Mann-Oakes-Renner relation. The edit pion mass is proportional to the square root of the edit quark masses.
Loop expansions
ChPT is calculable order by order in the loop expansion. The one-loop corrections give logarithmic terms ~ E^4 log(E^2 / Lambda_chi^2). The two-loop corrections give E^6 terms. Low-energy constants (LECs) parameterize the short-distance physics and are determined from experiment. In the cluster, edit ChPT is calculable order by order in the edit loop expansion. The edit LECs parameterize the edit short-distance physics.
This theory
This page is about ChPT. The chiral symmetry is SU(2)_L x SU(2)_R -> SU(2)_V. The Goldstone bosons are pions. The Lagrangian is L_2 = (F_pi^2 / 4) Tr(partial U partial U^dagger) + ... The GMOR relation gives m_pi^2 ~ m_q. The expansion is in E / Lambda_chi. The theory is real.