Chiral model

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 a Lie group

When the model was originally introduced, this Lie group was the SU(N), where N is the number of quark flavors.

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 spontaneously broken so that the corresponding scalar fields are the requisite Nambu−Goldstone bosons.

Its integrability was shown by Faddeev and Reshetikhin in 1982 through the quantum inverse scattering method.

This model admits topological solitons called skyrmions.

On a manifold (considered as the spacetime) M and a choice of compact Lie group G, the field content is a function

The principal chiral model is defined by the Lagrangian density

The chiral model of Gürsey (1960; also see Gell-Mann and Lévy) is now appreciated 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, θR are the corresponding rotation angles.

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 1970s.

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

The unbroken subgroup, realized in the linear Wigner–Weyl mode, is

To construct a non-linear realization of SO(4), the representation describing four-dimensional rotations of a vector for an infinitesimal rotation parametrized by six angles is given by where 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 S3, 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), The nonlinear terms (shifting π) on the right-hand side of the second equation underlie the nonlinear realization of SO(4).

is realized nonlinearly on the triplet of pions— which, however, still transform linearly under isospin

represent the nonlinear "shifts" (spontaneous breaking).

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 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 π transform under so, then, manifestly identically to the above under isorotations, V; and similarly to the above, as under the broken symmetries, A, the shifts.

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

Introduced by Richard S. Ward,[3] the integrable chiral model or Ward model is described in terms of a matrix-valued field

It also has a formulation which is formally identical to the Bogomolny equations but with Lorentz signature.

The corresponding conserved currents from Noether's theorem are

The equations of motion turn out to be equivalent to conservation of the currents,

The currents additionally satisfy the flatness condition,

and therefore the equations of motion can be formulated entirely in terms of the currents.

, that is, the equations of motion from the principal chiral model (PCM).