Chiral model

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In nuclear physics, the chiral model, introduced by Feza Gürsey in 1960, is a phenomenological model describing effective interactions of mesons in the chiral limit (where the masses of the quarks go to zero), but without necessarily mentioning quarks at all. It is a nonlinear sigma model with the principal homogeneous space of the Lie group SU(N) as its target manifold, where N is the number of quark flavors. The Riemannian metric of the target manifold is given by a positive constant multiplied by the Killing form acting upon the Maurer-Cartan form of SU(N).

The internal global symmetry of this model is SU(N)L×SU(N)R, the left and right copies, respectively; where the left copy acts as the left action upon the target space, and the right copy acts as the right action. 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 spontaneously broken so that the corresponding scalar fields are the requisite Nambu−Goldstone bosons.

This model admits topological solitons called Skyrmions.

Departures from exact chiral symmetry are dealt with in chiral perturbation theory.

An outline of the original, 2-flavor model[edit]

The chiral model of Gürsey (1960; also see Gell-Mann and Lévy) is now appreciatedd to be an effective theory of QCD with two light quarks, u, and d. The QCD Lagrangian is approximately invariant under independent global flavor rotations of the left- and right-handed quark fields,

where τ denote the Pauli matrices in the flavor space and θL,R are the corresponding rotation angles.

The corresponding symmetry group is the chiral group, controlled by the six conserved currents

which can equally well be expressed in terms of the vector and axial-vector currents

The corresponding conserved charges generate the algebra of the chiral group,

with I=L,R, or, equivalently,

(Application of these commutation relations to hadronic reactions dominated current algebra calculations in the early seventies of the last century.)

At the level of hadrons, pseudoscalar mesons, the ambit of the chiral model, the chiral group is spontaneously broken down to , by the QCD vacuum. That is, it is realized nonlinearly, in the Nambu-Goldstone mode: The QV annihilate the vacuum, but the QA do not! This is visualized nicely through a geometrical argument based on the fact that the Lie algebra of is isomorphic to that of SO(4). The unbroken subgroup, realized in the linear Wigner-Weyl mode, is which is locally isomorphic to SU(2) (V: isospin).

To construct a non-linear realization of SO(4), consider the representation describing four-dimensional rotations of a vector . For an infinitesimal rotation parametrized by six angles , with i =1, 2, 3,


The four real quantities (π, σ) define the smallest nontrivial chiral multiplet and represent the field content of the linear sigma model. To switch from the above linear realization of SO(4) to the nonlinear one, we observe that, in fact, only three of the four components of (π, σ) are independent with respect to four-dimensional rotations. These three independent components correspond to coordinates on a hypersphere , where π and σ are subjected to the constraint

with F a (pion decay) constant of dimension mass.

Utilizing this to eliminate σ yields the following transformation properties of π under SO(4),

where θV,A with i = 1,2,3.

The nonlinear terms (shifting π) on the right-hand side of the second equation underlie the nonlinear realization of SO(4). The chiral group is realized nonlinearly on the triplet of pions which, however, still transform linearly under isospin rotations parametrized through the angles .

Through the spinor map, these four-dimensional rotations of (π, σ) can also be conveniently written using 2×2 matrix notation by introducing the unitary matrix

and requiring the transformation properties of U under chiral rotations to be

where .

The transition to the nonlinear realization follows,

where denotes the trace in the flavor space. This is a non-linear sigma model.

Terms involving or are not independent and can be brought to this form through partial integration. The constant F²/4 is chosen in such a way that the Lagrangian matches the usual free term for massless scalar fields when written in terms of the pions,

  • An alternative, equivalent (Gürsey, 1960), parameterization
yields a simpler expression for U,
Note the reparameterized πs transform under
so, then, manifestly identically to the above under isorotations, V; and similarly to the above, as
under the broken symmetries, A.

This simpler expression generalizes readily (Cronin, 1967) to N light quarks, so .