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  as its target manifold. When the model was originally introduced, this Lie group was the SU(N) , 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 , 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. 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.

The model was later studied in the two-dimensional case as an integrable system, in particular an integrable field theory. Its integrability was shown by Faddeev and Reshetikhin in 1982 through the quantum inverse scattering method. The two-dimensional principal chiral model exhibits signatures of integrability such as a Lax pair/zero-curvature formulation, an infinite number of symmetries, and an underlying quantum group symmetry (in this case, Yangian symmetry).

This model admits topological solitons called skyrmions.

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

Mathematical formulation 
On a manifold (considered as the spacetime)   and a choice of compact Lie group  , the field content is a function . This defines a related field , a -valued vector field (really, covector field) which is the Maurer–Cartan form. The principal chiral model is defined by the Lagrangian density 

where  is a dimensionless coupling. In differential-geometric language, the field  is a section of a principal bundle  with fibres isomorphic to the principal homogeneous space for  (hence why this defines the principal chiral model).

Phenomenology

An outline of the original, 2-flavor model
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. 

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), 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  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  are independent with respect to four-dimensional rotations. These three independent components 
correspond to coordinates on a hypersphere S3, where  and  are subjected to the constraint 

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

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

The nonlinear terms (shifting  ) 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   By contrast, the   represent the nonlinear "shifts" (spontaneous breaking).

Through the spinor map, these four-dimensional rotations of   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,

Alternate Parametrization

An alternative, equivalent (Gürsey, 1960), parameterization 

yields a simpler expression for U,

Note the reparameterized  transform under 
 
so, then, manifestly identically to the above under isorotations, ; and similarly to the above, as 
 
under the broken symmetries, , the shifts. This simpler expression generalizes readily (Cronin, 1967) to  light quarks, so

Integrability

Integrable chiral model
Introduced by Richard S. Ward, the integrable chiral model or Ward model is described in terms of a matrix-valued field  and is given by the partial differential equation

It has a Lagrangian formulation with the expected kinetic term together with a term which resembles a Wess–Zumino–Witten term. It also has a formulation which is formally identical to the Bogomolny equations but with Lorentz signature. The relation between these formulations can be found in .

Many exact solutions are known.

Two-dimensional principal chiral model
Here the underlying manifold  is taken to be a Riemann surface, in particular the cylinder  or plane , conventionally given real coordinates , where on the cylinder  is a periodic coordinate. For application to string theory, this cylinder is the world sheet swept out by the closed string.

Global symmetries 
The global symmetries act as internal symmetries on the group-valued field  as  and . 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.

Lax formulation 
Consider the worldsheet in light-cone coordinates . The components of the appropriate Lax matrix are

The requirement that the zero-curvature condition on  for all  is equivalent to the conservation of current and flatness of the current , that is, the equations of motion from the principal chiral model (PCM).

See also
 Sigma model
 Chirality (physics)

References

 
; 
Georgi, H. (1984, 2009). Weak Interactions and Modern Particle Theory (Dover Books on Physics)  online .

Quantum field theory
Quantum chromodynamics
Nuclear physics