Editing Chiral model

{{Short description|Model of mesons in the massless quark limit}}
[[File:Integrable chiral model soliton scattering.gif|alt=Soliton scattering process for two solitons in the integrable chiral model. The plot shows the energy density of the system, and the maxima represent the solitons. They approach along one axis, collide to form a single lump, then scatter at 90 degrees.|thumb|480x480px|Soliton scattering process for two solitons in the integrable chiral model. The plot shows the energy density of the system, and the maxima represent the solitons.<ref>{{cite journal |last1=Ward |first1=R.S |title=Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system |journal=Physics Letters A |date=November 1995 |volume=208 |issue=3 |pages=203–208 |doi=10.1016/0375-9601(95)00782-X|arxiv=solv-int/9510004 |s2cid=123153627 }}</ref><ref name="dunajski">{{cite book |last1=Dunajski |first1=Maciej |title=Solitons, instantons, and twistors |date=2010 |publisher=Oxford University Press |location=Oxford |isbn=9780198570639 |page=159}}</ref>]]

In [[nuclear physics]], the '''chiral model''', introduced by [[Feza Gürsey]] in 1960, is a [[Phenomenology (particle physics)|phenomenological]] model describing [[effective field theory|effective]] interactions of [[meson]]s in the [[chiral symmetry|chiral limit]] (where the masses of the [[quark]]s 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]] <math>G</math> as its [[target manifold]]. When the model was originally introduced, this Lie group was the [[special unitary group|SU(''N'')]], where ''N'' is the number of quark [[Flavour (particle physics)|flavor]]s. 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 <math>G_L \times G_R</math>, the left and right copies, respectively; where the left copy acts as the [[Group action (mathematics)|left action]] upon the target space, and the right copy acts as the [[Group action (mathematics)|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 [[chiral symmetry breaking|spontaneously broken]] so that the corresponding scalar fields are the requisite [[Goldstone boson|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 [[Ludwig Faddeev|Faddeev]] and [[Nicolai Reshetikhin|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 soliton]]s called [[skyrmion]]s.

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

== Mathematical formulation ==
On a [[manifold]] (considered as the [[spacetime]])  {{mvar|M}} and a choice of [[compact group|compact]] [[Lie group]]  {{mvar|G}}, the field content is a function <math>U: M \rightarrow G</math>. This defines a related field <math>j_\mu = U^{-1}\partial_\mu U</math>, a <math>\mathfrak{g}</math>-valued [[vector field]] (really, covector field) which is the [[Maurer–Cartan form]]. The '''principal chiral model''' is defined by the [[Lagrangian density]] 
<math display = block>\mathcal{L} = \frac{\kappa}{2}\mathrm{tr}(\partial_\mu U^{-1} \partial^\mu U) = -\frac{\kappa}{2}\mathrm{tr}(j_\mu j^\mu),</math>
where <math>\kappa</math> is a dimensionless coupling. In [[differential geometry|differential-geometric]] language, the field <math>U</math> is a [[section (fiber bundle)|section]] of a [[principal bundle]] <math>\pi: P \rightarrow M</math> with [[fiber bundle|fibre]]s isomorphic to the [[principal homogeneous space]] for {{mvar|M}} (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,
:<math>\begin{cases} q_\mathsf{L} \mapsto q_\mathsf{L}'= L\ q_\mathsf{L} = \exp{\left(- i {\boldsymbol{\theta}}_\mathsf{L} \cdot \tfrac{1}{2}\boldsymbol{\tau} \right)} q_\mathsf{L} \\ q_\mathsf{R} \mapsto q_\mathsf{R}'= R\ q_\mathsf{R} = \exp{\left(- i \boldsymbol{\theta}_\mathsf{R} \cdot \tfrac{1}{2}\boldsymbol{\tau} \right)} q_\mathsf{R} \end{cases}</math>
where '''{{mvar|τ}}''' denote the Pauli matrices in the flavor space and {{nobr|'''{{mvar|θ}}'''<sub>L</sub> ,}} '''{{mvar|θ}}'''<sub>R&nbsp;</sub> are the corresponding rotation angles.

The corresponding symmetry group <math>\ \text{SU}(2)_\mathsf{L} \times \text{SU}(2)_\mathsf{R}\ </math> is the chiral group, controlled by the six conserved currents 
:<math>L_\mu^i = \bar q_\mathsf{L} \gamma_\mu \tfrac{\tau^i}{2} q_\mathsf{L} , \qquad R_\mu^i = \bar q_\mathsf{R} \gamma_\mu \tfrac{\tau^i}{2} q_\mathsf{R}\ ,</math>
which can equally well be expressed in terms of the vector and axial-vector currents 
:<math> V_\mu^i = L_\mu^i + R_\mu^i, \qquad A_\mu^i = R_\mu^i - L_\mu^i ~.</math>

The corresponding conserved charges generate the algebra of the chiral group,
:<math> \left[ Q_{I}^i,  Q_{I}^j \right] = i \epsilon^{ijk} Q_{I}^k \qquad \qquad \left[ Q_\mathsf{L}^i,  Q_\mathsf{R}^j \right] = 0,</math>
with {{nobr|{{math|''I'' {{=}}}} L, R ,}} or, equivalently, 
:<math> \left[ Q_{V}^i,  Q_{V}^j \right] = i \epsilon^{ijk} Q_V^k, \qquad \left[ Q_{A}^i,  Q_{A}^j \right] = i \epsilon^{ijk} Q_{V}^k, \qquad \left[ Q_{V}^i,  Q_{A}^j \right] = i \epsilon^{ijk} Q_A^k.</math>

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 <math>\ \text{SU}(2)_\mathsf{L} \times \text{SU}(2)_\mathsf{R}\ </math> group is [[spontaneous symmetry breaking|spontaneously broken]] down to <math>\text{SU}(2)_V\ ,</math> by the [[QCD vacuum]]. That is, it is realized ''nonlinearly'', in the [[Goldstone boson|Nambu–Goldstone mode]]: The {{math|''Q<sub>V</sub>''}} annihilate the vacuum, but the ''Q<sub>A</sub>'' do not! This is visualized nicely through a geometrical argument based on the fact that the Lie algebra of <math>\text{SU}(2)_\mathsf{L} \times\text{SU}(2)_\mathsf{R}\ </math> is isomorphic to that of SO(4). The unbroken subgroup, realized in the linear Wigner–Weyl mode, is <math>\ \text{SO}(3) \subset \text{SO}(4)\ </math> which is locally isomorphic to SU(2) (V: isospin). 
 
To construct a [[Nonlinear realization|non-linear realization]] of SO(4), the representation describing four-dimensional rotations of a vector 
:<math> \begin{pmatrix} {\boldsymbol{ \pi}} \\ \sigma \end{pmatrix} \equiv \begin{pmatrix} \pi_1 \\ \pi_2 \\ \pi_3 \\ \sigma \end{pmatrix},</math> 
for an infinitesimal rotation parametrized by six angles 
:<math>\left \{ \theta_i^{V,A} \right \}, \qquad i =1, 2, 3,</math>
is given by
:<math> \begin{pmatrix} {\boldsymbol{ \pi}} \\ \sigma \end{pmatrix} \stackrel{SO(4)}{\longrightarrow} \begin{pmatrix} {\boldsymbol{ \pi}'} \\ \sigma' \end{pmatrix} = \left[ \mathbf{1}_4+ \sum_{i=1}^3 \theta_i^V\ V_i + \sum_{i=1}^3 \theta_i^A\ A_i \right] \begin{pmatrix} {\boldsymbol{ \pi}} \\ \sigma \end{pmatrix}</math>
where 
:<math> \sum_{i=1}^3 \theta_i^V\ V_i =\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}.</math>

The four real quantities {{math|('''''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 {{math|('''''π''''', ''σ'')}} are independent with respect to four-dimensional rotations. These three independent components 
correspond to coordinates on a hypersphere S<sup>3</sup>, where {{math|'''''π'''''}} and {{math|''σ''}} are subjected to the constraint 
:<math>{\boldsymbol{ \pi}}^2 + \sigma^2 = F^2\ ,</math>
with  {{mvar|F}}  a [[pion decay constant|pion decay constant]] with [[dimensional analysis|dimension]] = ''mass''.

Utilizing this to eliminate {{mvar|σ}} yields the following transformation properties of '''{{mvar|π}}''' under SO(4),
:<math>\begin{cases} \theta^V : \boldsymbol{\pi} \mapsto \boldsymbol{\pi}'= \boldsymbol{\pi} + \boldsymbol{\theta}^V \times \boldsymbol{\pi} \\ \theta^A: \boldsymbol{\pi} \mapsto \boldsymbol{\pi}'= \boldsymbol{ \pi } + \boldsymbol{\theta}^A \sqrt{ F^2 - \boldsymbol{ \pi}^2} \end{cases} \qquad \boldsymbol{\theta}^{V,A} \equiv \left \{ \theta^{V,A}_i \right \}, \qquad i =1, 2, 3. </math>

The nonlinear terms (shifting '''{{mvar|π}}''') on the right-hand side of the second equation underlie the nonlinear realization of SO(4). The chiral group <math>\ \text{SU}(2)_\mathsf{L} \times \text{SU}(2)_\mathsf{R} \simeq \text{SO}(4)\ </math> is realized nonlinearly on the triplet of pions – which, however, still transform linearly under isospin <math>\ \text{SU}(2)_V \simeq \text{SO}(3)\ </math> rotations parametrized through the angles <math>\ \left\{ \boldsymbol{\theta}_V \right\} ~.</math> By contrast, the <math>\ \left\{ \boldsymbol{\theta}_A \right\}\ </math> represent the nonlinear "shifts" (spontaneous breaking).

Through the [[Lorentz group#Relation to the Möbius group|spinor map]], these four-dimensional rotations of  {{math|('''''π''''', ''σ'')}} can also be conveniently written using 2×2 matrix notation by introducing the unitary matrix 
:<math> U = \frac{1}{F} \left( \sigma \mathbf{1}_2 + i \boldsymbol{ \pi} \cdot \boldsymbol{ \tau} \right)\ ,</math> 
and requiring the transformation properties of {{mvar|U}} under chiral rotations to be
:<math> U \longrightarrow U' = L U R^\dagger\ ,</math> 
where <math>~ \theta_\mathsf{L} = \theta_V - \theta_A\ , \quad \theta_\mathsf{R} = \theta_V+ \theta_A ~.</math>

The transition to the nonlinear realization follows,
:<math> U = \frac{1}{F} \left( \sqrt{F^2 - \boldsymbol{ \pi}^2\ }\ \mathbf{1}_2 + i \boldsymbol{ \pi} \cdot \boldsymbol{ \tau} \right)\ , \qquad \mathcal{L}_\pi^{(2)} = \tfrac{1}{4}F^2\ \langle\ \partial_\mu U\ \partial^\mu U^\dagger\ \rangle_\mathsf{tr}\ ,</math> 
where <math>\ \langle \ldots \rangle_\mathsf{tr}\ </math> denotes the [[Trace (linear algebra)|trace]] in the flavor space. This is a [[non-linear sigma model]].

Terms involving <math>\textstyle\ \partial_\mu \partial^\mu\ U\ </math> or <math>\textstyle\ \partial_\mu \partial^\mu\ U^\dagger\ </math> are not independent and can be brought to this form through partial integration. 
The constant  {{small|{{sfrac|1|4}}}}{{mvar|F}}<sup>2</sup>  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,
:<math>\ \mathcal{L}_\pi^{(2)} = \frac{1}{2} \partial_\mu \boldsymbol{\pi} \cdot \partial^\mu \boldsymbol{\pi} + \frac{1}{2} \left( \frac{ \partial_\mu \boldsymbol{\pi} \cdot \boldsymbol{\pi} }{ F } \right)^2 + \mathcal{O} ( \pi^6 ) ~.</math>

===Alternate Parametrization===
{{see also|Chiral symmetry breaking|Nonlinear realization}}
An alternative, equivalent (Gürsey, 1960), parameterization 
:<math> \boldsymbol{\pi}\mapsto \boldsymbol{\pi}~ \frac{\sin (|\pi/F|)}{|\pi/F|},</math>
yields a simpler expression for ''U'',
:<math>U=\mathbf{1} \cos |\pi/F| + i \widehat{\pi}\cdot \boldsymbol{\tau} \sin |\pi/F| =e^{i~\boldsymbol{\tau}\cdot \boldsymbol{\pi}/F}. </math>

Note the reparameterized {{math|'''''π'''''}} transform under 
:<math>L U R^\dagger=\exp(i\boldsymbol{\theta}_A\cdot \boldsymbol{\tau}/2 -i\boldsymbol{\theta}_V\cdot \boldsymbol{\tau}/2 ) \exp(i\boldsymbol{\pi}\cdot \boldsymbol{\tau}/F ) \exp(i\boldsymbol{\theta}_A\cdot \boldsymbol{\tau}/2 +i\boldsymbol{\theta}_V\cdot \boldsymbol{\tau}/2 )</math> 
so, then, manifestly identically to the above under isorotations, {{mvar|V}}; and similarly to the above, as 
:<math>\boldsymbol{\pi} \longrightarrow \boldsymbol{\pi} +\boldsymbol{\theta}_A F+ \cdots =\boldsymbol{\pi} +\boldsymbol{\theta}_A F ( |\pi/F| \cot |\pi/F| )</math> 
under the broken symmetries, {{mvar|A}}, the shifts. This simpler expression generalizes readily (Cronin, 1967) to {{mvar|N}} light quarks, so <math>\textstyle \text{SU}(N)_L \times \text{SU}(N)_R/\text{SU}(N)_V.</math>

==Integrability==
===Integrable chiral model===
Introduced by [[Richard S. Ward]],<ref>{{cite journal |last1=Ward |first1=R. S. |title=Soliton solutions in an integrable chiral model in 2+1 dimensions |journal=Journal of Mathematical Physics |date=February 1988 |volume=29 |issue=2 |pages=386–389 |doi=10.1063/1.528078|doi-access=free }}</ref> the '''integrable chiral model''' or '''Ward model''' is described in terms of a matrix-valued field <math>J: \mathbb{R}^3 \rightarrow U(n)</math> and is given by the partial differential equation
<math display = block>\partial_t(J^{-1}J_t)- \partial_x(J^{-1}J_x) - \partial_y(J^{-1}J_y) - [J^{-1}J_t, J^{-1}J_y] = 0.</math>
It has a Lagrangian formulation with the expected [[kinetic term]] together with a term which resembles a [[Wess–Zumino–Witten model|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 {{harvs|txt|last=Dunajski|year=2010}}.

Many exact solutions are known.<ref>{{cite journal |last1=Ioannidou |first1=T. |last2=Zakrzewski |first2=W. J. |title=Solutions of the modified chiral model in (2+1) dimensions |journal=Journal of Mathematical Physics |date=May 1998 |volume=39 |issue=5 |pages=2693–2701 |doi=10.1063/1.532414|arxiv=hep-th/9802122 |s2cid=119529600 }}</ref><ref>{{cite journal |last1=Ioannidou |first1=T. |title=Soliton solutions and nontrivial scattering in an integrable chiral model in (2+1) dimensions |journal=Journal of Mathematical Physics |date=July 1996 |volume=37 |issue=7 |pages=3422–3441 |doi=10.1063/1.531573|arxiv=hep-th/9604126 |s2cid=15300406 }}</ref><ref>{{cite journal |last1=Dai |first1=B. |last2=Terng |first2=C.-L. |title=Bäcklund transformations, Ward solitons, and unitons |journal=Journal of Differential Geometry |date=1 January 2007 |volume=75 |issue=1 |doi=10.4310/jdg/1175266254|s2cid=53477757 |url=http://projecteuclid.org/euclid.jdg/1175266254 |arxiv=math/0405363 }}</ref>

===Two-dimensional principal chiral model===
Here the underlying manifold <math>M</math> is taken to be a [[Riemann surface]], in particular the cylinder <math>\mathbb{C}^*</math> or plane <math>\mathbb{C}</math>, conventionally given ''real'' coordinates <math>\tau, \sigma</math>, where on the cylinder <math>\sigma \sim \sigma + 2\pi</math> is a periodic coordinate. For application to [[string theory]], this cylinder is the [[world sheet]] swept out by the closed string.<ref name="driezen">{{cite arXiv 
|last1=Driezen |first1=Sibylle 
|title=Modave Lectures on Classical Integrability in $2d$ Field Theories |date=2021
|class=hep-th 
|eprint=2112.14628}}</ref>

==== Global symmetries ====
The global symmetries act as internal symmetries on the group-valued field <math>g(x)</math> as <math>\rho_L(g') g(x) = g'g(x)</math> and <math>\rho_R(g) g(x) = g(x)g'</math>. The corresponding conserved currents from [[Noether's theorem]] are
<math display = block>L_\alpha = g^{-1}\partial_\alpha g, \qquad R_\alpha = \partial_\alpha g g^{-1}.</math>
The [[equations of motion]] turn out to be equivalent to conservation of these currents,
<math display = block>\partial_\alpha L^\alpha = \partial_\alpha R^\alpha = 0, ~ \text{ or, in coordinate-free form, } ~d * L = d * R = 0.</math>
The currents additionally satisfy the flatness condition,
<math display = block>dL + \frac{1}{2}[L,L] = 0 ~~~\text{ or, in coordinates, } ~~~\partial_\alpha L_\beta - \partial_\beta L_\alpha + [L_\alpha, L_\beta] = 0,</math>
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 [[Wess–Zumino–Witten_model|WZWN term]].

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

==See also==
* [[Sigma model]]
* [[Chirality (physics)]]

==References==
{{reflist}}
*{{Cite journal | last1 = Gürsey | first1 = F. | title = On the symmetries of strong and weak interactions | doi = 10.1007/BF02860276 | journal = Il Nuovo Cimento | volume = 16 | issue = 2 | pages = 230–240 | year = 1960 | bibcode = 1960NCim...16..230G | s2cid = 122270607 }}
* {{cite journal | last=Gürsey | first=Feza | title=On the structure and parity of weak interaction currents | journal=Annals of Physics | publisher=Elsevier BV | volume=12 | issue=1 | year=1961 | issn=0003-4916 | doi=10.1016/0003-4916(61)90147-6 | pages=91–117| bibcode=1961AnPhy..12...91G }}
*{{Cite journal | last1 = Coleman | first1 = S. | last2 = Wess | first2 = J. | last3 = Zumino | first3 = B. | doi = 10.1103/PhysRev.177.2239 | title = Structure of Phenomenological Lagrangians. I | journal = Physical Review | volume = 177 | issue = 5 | pages = 2239 | year = 1969 |bibcode = 1969PhRv..177.2239C }}; {{Cite journal | last1 = Callan | first1 = C. | last2 = Coleman | first2 = S. | last3 = Wess | first3 = J. | last4 = Zumino | first4 = B. | title = Structure of Phenomenological Lagrangians. II | doi = 10.1103/PhysRev.177.2247 | journal = Physical Review | volume = 177 | issue = 5 | pages = 2247 | year = 1969 |bibcode = 1969PhRv..177.2247C }}
*Georgi, H. (1984, 2009). ''Weak Interactions and Modern Particle Theory'' (Dover Books on Physics) {{ISBN|0486469042}} [http://www.people.fas.harvard.edu/~hgeorgi/weak.pdf online] .
*{{cite journal|doi=10.1063/1.533204|title=Chiral limit of the two-dimensional fermionic determinant in a general magnetic field|year=2000|last1=Fry|first1=M. P.|journal=Journal of Mathematical Physics|volume=41|issue=4|pages=1691–1710|arxiv = hep-th/9911131 |bibcode = 2000JMP....41.1691F |s2cid=14302881}}
*{{Citation | last2=Lévy | first1=M. | last1=Gell-Mann | first2=M. | title=The axial vector current in beta decay | publisher=Italian Physical Society | doi=10.1007/BF02859738 | year=1960 | journal=Il Nuovo Cimento | issn=1827-6121 | volume=16 | issue=4 | pages=705–726| bibcode=1960NCim...16..705G | s2cid=122945049 }}
*{{cite journal | last=Cronin | first=Jeremiah A. | title=Phenomenological Model of Strong and Weak Interactions in ChiralU(3)⊗U(3) | journal=Physical Review | publisher=American Physical Society (APS) | volume=161 | issue=5 | date=1967-09-25 | issn=0031-899X | doi=10.1103/physrev.161.1483 | pages=1483–1494| bibcode=1967PhRv..161.1483C }}
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