Chiral model

In <a href="/facts/Nuclear_physics/fzQbt1aF">nuclear physics</a>, the chiral model, introduced by <a href="/facts/Feza_G%C3%BCrsey/Yxty2pVY">Feza Gürsey</a> in 1960, is a <a href="/facts/Phenomenology_(particle_physics)/390KC4vG">phenomenological</a> model describing <a href="/facts/Effective_field_theory/oQ2nWoko">effective</a> interactions of <a href="/facts/Meson/9R0kIc1q">mesons</a> in the <a href="/facts/Chiral_symmetry/H8Xbcvzt">chiral limit</a> (where the masses of the <a href="/facts/Quark/hwuuVkV5">quarks</a> go to zero), but without necessarily mentioning quarks at all. It is a <a href="/facts/Nonlinear_sigma_model/WqQI5CRJ">nonlinear sigma model</a> with the <a href="/facts/Principal_homogeneous_space/N27maCdL">principal homogeneous space</a> of a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
 as its <a href="/facts/Target_manifold/WqQI5CRJ">target manifold</a>. When the model was originally introduced, this Lie group was the <a href="/facts/Special_unitary_group/Qgy2orcu">SU(N)</a>, where N is the number of quark <a href="/facts/Flavour_(particle_physics)/RR7rsuTD">flavors</a>. The <a href="/facts/Riemannian_metric/5KYnmEgF">Riemannian metric</a> of the target manifold is given by a positive constant multiplied by the <a href="/facts/Killing_form/u4HxRh5o">Killing form</a> acting upon the <a href="/facts/Maurer%E2%80%93Cartan_form/0Xhjm36A">Maurer–Cartan form</a> of SU(N).
The internal <a href="/facts/Global_symmetry/gJmgsooI">global symmetry</a> of this model is 
 
 
 
 
 G
 
 L
 
 
 ×
 
 G
 
 R
 
 
 
 
 {\displaystyle G_{L}\times G_{R}}
 
, the left and right copies, respectively; where the left copy acts as the <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">left action</a> upon the target space, and the right copy acts as the <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">right action</a>. Phenomenologically, the left copy represents flavor rotations among the left-handed quarks, while the right copy describes rotations among the right-handed quarks, while these, L and R, are completely independent of each other. The axial pieces of these symmetries are <a href="/facts/Chiral_symmetry_breaking/bMsTTLKw">spontaneously broken</a> so that the corresponding scalar fields are the requisite <a href="/facts/Goldstone_boson/zGBpWRYU">Nambu−Goldstone bosons</a>.
The model was later studied in the two-dimensional case as an <a href="/facts/Integrable_system/Crgzjw6G">integrable system</a>, in particular an integrable field theory. Its integrability was shown by <a href="/facts/Ludwig_Faddeev/gR8t21SP">Faddeev</a> and <a href="/facts/Nicolai_Reshetikhin/DH7r3bWv">Reshetikhin</a> in 1982 through the <a href="/facts/Quantum_inverse_scattering_method/NGmzt8TT">quantum inverse scattering method</a>. The two-dimensional principal chiral model exhibits signatures of integrability such as a <a href="/facts/Lax_pair/dyzB7Ttb">Lax pair</a>/zero-curvature formulation, an infinite number of symmetries, and an underlying <a href="/facts/Quantum_group/huNTuaXs">quantum group</a> symmetry (in this case, <a href="/facts/Yangian/3EKUD0wQ">Yangian</a> symmetry).
This model admits <a href="/facts/Topological_soliton/Hqrvx2Lj">topological solitons</a> called <a href="/facts/Skyrmion/Dw0l6Q0m">skyrmions</a>.
Departures from exact chiral symmetry are dealt with in <a href="/facts/Chiral_perturbation_theory/7E04FlDu">chiral perturbation theory</a>.

Chiral model open-in-new

Chiral model