Chiral model

 ( Quantum Field Theory ) 

In nuclear physics, the chiral model, introduced by Feza Gürsey in 1960, is a phenomenological model describing effective interactions of in the chiral symmetry (where the masses of the 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 $G$ 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 $G\_L\; \backslash times\; G\_R$, 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. 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 Goldstone boson.

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 Ludwig 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 .

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

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Mathematical formulation On a manifold (considered as the spacetime) and a choice of compact group Lie group , the field content is a function $U:\; M\; \backslash rightarrow\; G$. This defines a related field $j\_\backslash mu\; =\; U^\{-1\}\backslash partial\_\backslash mu\; U$, a $\backslash mathfrak\{g\}$-valued vector field (really, covector field) which is the Maurer–Cartan form. The principal chiral model is defined by the Lagrangian density $$\backslash mathcal\{L\}\; =\; \backslash frac\{\backslash kappa\}\{2\}\backslash mathrm\{tr\}(\backslash partial\_\backslash mu\; U^\{-1\}\; \backslash partial^\backslash mu\; U)\; =\; -\backslash frac\{\backslash kappa\}\{2\}\backslash mathrm\{tr\}(j\_\backslash mu\; j^\backslash mu),$$ where $\backslash kappa$ is a dimensionless coupling. In differential-geometric language, the field $U$ is a section of a principal bundle $\backslash pi:\; P\; \backslash rightarrow\; M$ with fiber bundle isomorphic to the principal homogeneous space for (hence why this defines the principal chiral model).

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Phenomenology

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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,

$\backslash begin\{cases\}\; q\_\backslash mathsf\{L\}\; \backslash mapsto\; q\_\backslash mathsf\{L\}\text{'}=\; L\backslash \; q\_\backslash mathsf\{L\}\; =\; \backslash exp\{\backslash left(-\; i\; \{\backslash boldsymbol\{\backslash theta\}\}\_\backslash mathsf\{L\}\; \backslash cdot\; \backslash tfrac\{1\}\{2\}\backslash boldsymbol\{\backslash tau\}\; \backslash right)\}\; q\_\backslash mathsf\{L\}\; \backslash \backslash \; q\_\backslash mathsf\{R\}\; \backslash mapsto\; q\_\backslash mathsf\{R\}\text{'}=\; R\backslash \; q\_\backslash mathsf\{R\}\; =\; \backslash exp\{\backslash left(-\; i\; \backslash boldsymbol\{\backslash theta\}\_\backslash mathsf\{R\}\; \backslash cdot\; \backslash tfrac\{1\}\{2\}\backslash boldsymbol\{\backslash tau\}\; \backslash right)\}\; q\_\backslash mathsf\{R\}\; \backslash end\{cases\}$

where ' denote the Pauli matrices in the flavor space and '_R  are the corresponding rotation angles.

The corresponding symmetry group $\backslash \; \backslash text\{SU\}(2)\_\backslash mathsf\{L\}\; \backslash times\; \backslash text\{SU\}(2)\_\backslash mathsf\{R\}\backslash$ is the chiral group, controlled by the six conserved currents

$L\_\backslash mu^i\; =\; \backslash bar\; q\_\backslash mathsf\{L\}\; \backslash gamma\_\backslash mu\; \backslash tfrac\{\backslash tau^i\}\{2\}\; q\_\backslash mathsf\{L\}\; ,\; \backslash qquad\; R\_\backslash mu^i\; =\; \backslash bar\; q\_\backslash mathsf\{R\}\; \backslash gamma\_\backslash mu\; \backslash tfrac\{\backslash tau^i\}\{2\}\; q\_\backslash mathsf\{R\}\backslash \; ,$

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

$V\_\backslash mu^i\; =\; L\_\backslash mu^i\; +\; R\_\backslash mu^i,\; \backslash qquad\; A\_\backslash mu^i\; =\; R\_\backslash mu^i\; -\; L\_\backslash mu^i\; ~.$

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

$\backslash left\; =\; i\; \backslash epsilon^\{ijk\}\; Q\_\{I\}^k\; \backslash qquad\; \backslash qquad\; \backslash left\; =\; 0,$

with or, equivalently,

$\backslash left\; =\; i\; \backslash epsilon^\{ijk\}\; Q\_V^k,\; \backslash qquad\; \backslash left\; =\; i\; \backslash epsilon^\{ijk\}\; Q\_\{V\}^k,\; \backslash qquad\; \backslash left\; =\; i\; \backslash epsilon^\{ijk\}\; Q\_A^k.$

Application of these commutation relations to hadronic reactions dominated current algebra calculations in the early 1970s.

At the level of hadrons, pseudoscalar mesons, the ambit of the chiral model, the chiral $\backslash \; \backslash text\{SU\}(2)\_\backslash mathsf\{L\}\; \backslash times\; \backslash text\{SU\}(2)\_\backslash mathsf\{R\}\backslash$ group is spontaneously broken down to $\backslash text\{SU\}(2)\_V\backslash \; ,$ by the QCD vacuum. That is, it is realized nonlinearly, in the Goldstone boson: The annihilate the vacuum, but the Q_A do not! This is visualized nicely through a geometrical argument based on the fact that the Lie algebra of $\backslash text\{SU\}(2)\_\backslash mathsf\{L\}\; \backslash times\backslash text\{SU\}(2)\_\backslash mathsf\{R\}\backslash$ is isomorphic to that of SO(4). The unbroken subgroup, realized in the linear Wigner–Weyl mode, is $\backslash \; \backslash text\{SO\}(3)\; \backslash subset\; \backslash text\{SO\}(4)\backslash$ 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

$\backslash begin\{pmatrix\}\; \{\backslash boldsymbol\{\; \backslash pi\}\}\; \backslash \backslash \; \backslash sigma\; \backslash end\{pmatrix\}\; \backslash equiv\; \backslash begin\{pmatrix\}\; \backslash pi\_1\; \backslash \backslash \; \backslash pi\_2\; \backslash \backslash \; \backslash pi\_3\; \backslash \backslash \; \backslash sigma\; \backslash end\{pmatrix\},$

for an infinitesimal rotation parametrized by six angles

$\backslash left\; \backslash \{\; \backslash theta\_i^\{V,A\}\; \backslash right\; \backslash \},\; \backslash qquad\; i\; =1,\; 2,\; 3,$

is given by

$\backslash begin\{pmatrix\}\; \{\backslash boldsymbol\{\; \backslash pi\}\}\; \backslash \backslash \; \backslash sigma\; \backslash end\{pmatrix\}\; \backslash stackrel\{SO(4)\}\{\backslash longrightarrow\}\; \backslash begin\{pmatrix\}\; \{\backslash boldsymbol\{\; \backslash pi\}\text{'}\}\; \backslash \backslash \; \backslash sigma\text{'}\; \backslash end\{pmatrix\}\; =\; \backslash left\; \backslash begin\{pmatrix\}\; \{\backslash boldsymbol\{\; \backslash pi\}\}\; \backslash \backslash \; \backslash sigma\; \backslash end\{pmatrix\}$

where

$\backslash sum\_\{i=1\}^3\; \backslash theta\_i^V\backslash \; V\_i\; =\backslash begin\{pmatrix\}$

0 & -\theta^V_3 & \theta^V_2 & 0 \\ \theta^V_3 & 0 & -\theta_1^V & 0 \\ -\theta^V_2 & \theta_1^V & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \qquad \qquad \sum_{i=1}^3 \theta_i^A\ A_i = \begin{pmatrix} 0 & 0 & 0 & \theta^A_1 \\ 0 & 0 & 0 & \theta^A_2 \\ 0 & 0 & 0 & \theta^A_3 \\ -\theta^A_1 & -\theta_2^A & -\theta_3^A & 0 \end{pmatrix}.

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

$\{\backslash boldsymbol\{\; \backslash pi\}\}^2\; +\; \backslash sigma^2\; =\; F^2\backslash \; ,$

with a pion decay constant with dimension = mass.

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

$\backslash begin\{cases\}\; \backslash theta^V\; :\; \backslash boldsymbol\{\backslash pi\}\; \backslash mapsto\; \backslash boldsymbol\{\backslash pi\}\text{'}=\; \backslash boldsymbol\{\backslash pi\}\; +\; \backslash boldsymbol\{\backslash theta\}^V\; \backslash times\; \backslash boldsymbol\{\backslash pi\}\; \backslash \backslash \; \backslash theta^A:\; \backslash boldsymbol\{\backslash pi\}\; \backslash mapsto\; \backslash boldsymbol\{\backslash pi\}\text{'}=\; \backslash boldsymbol\{\; \backslash pi\; \}\; +\; \backslash boldsymbol\{\backslash theta\}^A\; \backslash sqrt\{\; F^2\; -\; \backslash boldsymbol\{\; \backslash pi\}^2\}\; \backslash end\{cases\}\; \backslash qquad\; \backslash boldsymbol\{\backslash theta\}^\{V,A\}\; \backslash equiv\; \backslash left\; \backslash \{\; \backslash theta^\{V,A\}\_i\; \backslash right\; \backslash \},\; \backslash qquad\; i\; =1,\; 2,\; 3.$

The nonlinear terms (shifting ) on the right-hand side of the second equation underlie the nonlinear realization of SO(4). The chiral group $\backslash \; \backslash text\{SU\}(2)\_\backslash mathsf\{L\}\; \backslash times\; \backslash text\{SU\}(2)\_\backslash mathsf\{R\}\; \backslash simeq\; \backslash text\{SO\}(4)\backslash$ is realized nonlinearly on the triplet of pions – which, however, still transform linearly under isospin $\backslash \; \backslash text\{SU\}(2)\_V\; \backslash simeq\; \backslash text\{SO\}(3)\backslash$ rotations parametrized through the angles $\backslash \; \backslash left\backslash \{\; \backslash boldsymbol\{\backslash theta\}\_V\; \backslash right\backslash \}\; ~.$ By contrast, the $\backslash \; \backslash left\backslash \{\; \backslash boldsymbol\{\backslash theta\}\_A\; \backslash right\backslash \}\backslash$ 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

$U\; =\; \backslash frac\{1\}\{F\}\; \backslash left(\; \backslash sigma\; \backslash mathbf\{1\}\_2\; +\; i\; \backslash boldsymbol\{\; \backslash pi\}\; \backslash cdot\; \backslash boldsymbol\{\; \backslash tau\}\; \backslash right)\backslash \; ,$

and requiring the transformation properties of under chiral rotations to be

$U\; \backslash longrightarrow\; U\text{'}\; =\; L\; U\; R^\backslash dagger\backslash \; ,$

where $~\; \backslash theta\_\backslash mathsf\{L\}\; =\; \backslash theta\_V\; -\; \backslash theta\_A\backslash \; ,\; \backslash quad\; \backslash theta\_\backslash mathsf\{R\}\; =\; \backslash theta\_V+\; \backslash theta\_A\; ~.$

The transition to the nonlinear realization follows,

$U\; =\; \backslash frac\{1\}\{F\}\; \backslash left(\; \backslash sqrt\{F^2\; -\; \backslash boldsymbol\{\; \backslash pi\}^2\backslash \; \}\backslash \; \backslash mathbf\{1\}\_2\; +\; i\; \backslash boldsymbol\{\; \backslash pi\}\; \backslash cdot\; \backslash boldsymbol\{\; \backslash tau\}\; \backslash right)\backslash \; ,\; \backslash qquad\; \backslash mathcal\{L\}\_\backslash pi^\{(2)\}\; =\; \backslash tfrac\{1\}\{4\}F^2\backslash \; \backslash langle\backslash \; \backslash partial\_\backslash mu\; U\backslash \; \backslash partial^\backslash mu\; U^\backslash dagger\backslash \; \backslash rangle\_\backslash mathsf\{tr\}\backslash \; ,$

where $\backslash \; \backslash langle\; \backslash ldots\; \backslash rangle\_\backslash mathsf\{tr\}\backslash$ denotes the trace in the flavor space. This is a non-linear sigma model.

Terms involving $\backslash textstyle\backslash \; \backslash partial\_\backslash mu\; \backslash partial^\backslash mu\backslash \; U\backslash$ or $\backslash textstyle\backslash \; \backslash partial\_\backslash mu\; \backslash partial^\backslash mu\backslash \; U^\backslash dagger\backslash$ are not independent and can be brought to this form through partial integration. The constant ² 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,

$\backslash \; \backslash mathcal\{L\}\_\backslash pi^\{(2)\}\; =\; \backslash frac\{1\}\{2\}\; \backslash partial\_\backslash mu\; \backslash boldsymbol\{\backslash pi\}\; \backslash cdot\; \backslash partial^\backslash mu\; \backslash boldsymbol\{\backslash pi\}\; +\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash frac\{\; \backslash partial\_\backslash mu\; \backslash boldsymbol\{\backslash pi\}\; \backslash cdot\; \backslash boldsymbol\{\backslash pi\}\; \}\{\; F\; \}\; \backslash right)^2\; +\; \backslash mathcal\{O\}\; (\; \backslash pi^6\; )\; ~.$

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Alternate Parametrization An alternative, equivalent (Gürsey, 1960), parameterization

$\backslash boldsymbol\{\backslash pi\}\backslash mapsto\; \backslash boldsymbol\{\backslash pi\}~\; \backslash frac\{\backslash sin\; (|\backslash pi/F|)\}$,

yields a simpler expression for U,

$U=\backslash mathbf\{1\}\; \backslash cos\; |\backslash pi/F|\; +\; i\; \backslash widehat\{\backslash pi\}\backslash cdot\; \backslash boldsymbol\{\backslash tau\}\; \backslash sin\; |\backslash pi/F|\; =e^\{i~\backslash boldsymbol\{\backslash tau\}\backslash cdot\; \backslash boldsymbol\{\backslash pi\}/F\}.$

Note the reparameterized transform under

$L\; U\; R^\backslash dagger=\backslash exp(i\backslash boldsymbol\{\backslash theta\}\_A\backslash cdot\; \backslash boldsymbol\{\backslash tau\}/2\; -i\backslash boldsymbol\{\backslash theta\}\_V\backslash cdot\; \backslash boldsymbol\{\backslash tau\}/2\; )\; \backslash exp(i\backslash boldsymbol\{\backslash pi\}\backslash cdot\; \backslash boldsymbol\{\backslash tau\}/F\; )\; \backslash exp(i\backslash boldsymbol\{\backslash theta\}\_A\backslash cdot\; \backslash boldsymbol\{\backslash tau\}/2\; +i\backslash boldsymbol\{\backslash theta\}\_V\backslash cdot\; \backslash boldsymbol\{\backslash tau\}/2\; )$

so, then, manifestly identically to the above under isorotations, ; and similarly to the above, as

$\backslash boldsymbol\{\backslash pi\}\; \backslash longrightarrow\; \backslash boldsymbol\{\backslash pi\}\; +\backslash boldsymbol\{\backslash theta\}\_A\; F+\; \backslash cdots\; =\backslash boldsymbol\{\backslash pi\}\; +\backslash boldsymbol\{\backslash theta\}\_A\; F\; (\; |\backslash pi/F|\; \backslash cot\; |\backslash pi/F|\; )$

under the broken symmetries, , the shifts. This simpler expression generalizes readily (Cronin, 1967) to light quarks, so $\backslash textstyle\; \backslash text\{SU\}(N)\_L\; \backslash times\; \backslash text\{SU\}(N)\_R/\backslash text\{SU\}(N)\_V.$

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Integrability

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Integrable chiral model Introduced by Richard S. Ward, the integrable chiral model or Ward model is described in terms of a matrix-valued field $J:\; \backslash mathbb\{R\}^3\; \backslash rightarrow\; U(n)$ and is given by the partial differential equation $$\backslash partial\_t(J^\{-1\}J\_t)-\; \backslash partial\_x(J^\{-1\}J\_x)\; -\; \backslash partial\_y(J^\{-1\}J\_y)\; -\; J^\{-1\}J\_t,\; =\; 0.$$ 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.

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Two-dimensional principal chiral model Here the underlying manifold $M$ is taken to be a Riemann surface, in particular the cylinder $\backslash mathbb\{C\}^*$ or plane $\backslash mathbb\{C\}$, conventionally given real coordinates $\backslash tau,\; \backslash sigma$, where on the cylinder $\backslash sigma\; \backslash sim\; \backslash sigma\; +\; 2\backslash pi$ is a periodic coordinate. For application to string theory, this cylinder is the world sheet swept out by the closed string.

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Global symmetries The global symmetries act as internal symmetries on the group-valued field $g(x)$ as $\backslash rho\_L(g\text{'})\; g(x)\; =\; g\text{'}g(x)$ and $\backslash rho\_R(g)\; g(x)\; =\; g(x)g\text{'}$. The corresponding conserved currents from Noether's theorem are $$L\_\backslash alpha\; =\; g^\{-1\}\backslash partial\_\backslash alpha\; g,\; \backslash qquad\; R\_\backslash alpha\; =\; \backslash partial\_\backslash alpha\; g\; g^\{-1\}.$$ The equations of motion turn out to be equivalent to conservation of these currents, $$\backslash partial\_\backslash alpha\; L^\backslash alpha\; =\; \backslash partial\_\backslash alpha\; R^\backslash alpha\; =\; 0,\; ~\; \backslash text\{\; or,\; in\; coordinate-free\; form,\; \}\; ~d\; *\; L\; =\; d\; *\; R\; =\; 0.$$ The currents additionally satisfy the flatness condition, $$dL\; +\; \backslash frac\{1\}\{2\}L,L\; =\; 0\; ~~~\backslash text\{\; or,\; in\; coordinates,\; \}\; ~~~\backslash partial\_\backslash alpha\; L\_\backslash beta\; -\; \backslash partial\_\backslash beta\; L\_\backslash alpha\; +\; L\_\backslash alpha,\; =\; 0,$$ and therefore the equations of motion can be formulated entirely in terms of the currents.

Upon quantization, the axial combination of these currents develop chiral anomalies, summarized in the above-mentioned topological WZWN term.

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Lax formulation Consider the worldsheet in light-cone coordinates $x^\backslash pm\; =\; t\; \backslash pm\; x$. The components of the appropriate Lax matrix are $$L\_\backslash pm(x^+,\; x^-;\; \backslash lambda)\; =\; \backslash frac\{j\_\{\backslash pm\}\}\{1\; \backslash mp\; \backslash lambda\}.$$ The requirement that the zero-curvature condition on $L\_\backslash pm$ for all $\backslash lambda$ is equivalent to the conservation of current and flatness of the current $j\; =\; (j\_+,\; j\_-)$, that is, the equations of motion from the principal chiral model (PCM).

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See also

• Sigma model
• Chirality (physics)

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

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