Sigma model

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Author: Dr. Sergei V. Ketov, Tokyo Metropolitan University

A (non-linear) sigma model is a scalar field theory whose (multi-component) scalar field defines a map from a `space-time' to a Riemann (target) manifold.

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Basic facts

Consider a set of D real scalar fields $\phi^a(x^{\mu})$ mapping a d-dimensional (flat) space $\Sigma$ into a D-dimensional target space M, with the action

(1)

$S[\phi]=\frac{1}{2}\int_{\Sigma}{\rm d}^dx\,g_{ab}(\phi)\partial^{\mu}\phi^a\partial_{\mu}\phi^b~,$

where $\partial^{\mu}=\eta^{\mu\nu}\partial_{\nu},\quad \partial_{\nu}=\partial/\partial x^{\nu},\quad \eta^{\mu\nu}$ is normally a Minkowski metric (though often $\eta^{\mu\nu}$ is taken to be Euclidean). It is called a Non-Linear Sigma-Model (NLSM) with the metric $g_{ab}(\phi).$ It is usually assumed that $g_{ab}(\phi)$ is a positive-definite field-dependent matrix, in order to ensure the absence of negative norm states. The standard mechanisms for a construction of compact NLSM (without ghosts) are: (i) imposing constraints on the NLSM scalars so that they take their values in a compact (usually symmetric) space, and (ii) gauging internal symmetries of the NLSM.

The geometrical significance of NLSM lies in the invariance of the action (1) under the (infinitesimal) field reparametrizations

(2)

$\phi^a \to \phi^a + \xi^a(\phi)$

that can be interpreted as diffeomorphisms of the target space M, provided that the metric $g_{ab}(\phi)$ transforms as a second-rank tensor. The latter in the field theory context is a transformation of the fields and the coupling constants (defined as the coefficients in Taylor expansion of the metric components with respect to the fields). Hence, generically, there is no Noether current for the NLSM reparametrizational invariance. The two NLSM are physically equivalent when they are related by a field redefinition alone. It is not difficult to check that it happens only if $\xi$ is a Killing vector in M. In this case the NLSM action (1) has an isometry (or an internal symmetry) leading to a conserved Noether current and the corresponding conserved charge. All the isometries of the target space M form a group G, representing the global symmetry of M. The most important case is given by the target space that is a Lie group G itself. For example, the O(n) NLSM is defined by the action

(3)

$S[\vec{n}]=\frac{1}{2\lambda^2}\int {\rm d}^dx \,\partial^{\mu}\vec{n}\cdot \partial_{\mu}\vec{n}~,$

where the real scalar fields $\vec{n}(x^{\mu})$ are restricted by a condition

(4)

$\vec{n}\cdot \vec{n}=const.$

After solving the condition in terms of some independent field variables (angles) $\{\phi\},$ and substituting the result $\vec{n}=\vec{n}(\phi)$ back into the action, one gets a NLSM action (1). The O(3) NLSM model can be interpreted as the continuum limit of an isotropic ferromagnet in statistical mechanics. The two-dimensional O(3) model is known to be integrable as a classical field theory.

To illustrate the gauging procedure, let's consider the simplest example of the CP(n) model with an action

(5)

$S[\vec{n}]=\frac{1}{2\lambda^2}\int {\rm d}^dx\,(D^{\mu}\vec{n})^{\dagger}\cdot D_{\mu}\vec{n}~,$

in terms of the (n+1)-component complex scalar field $\vec{n}$ subject to the constraint

(6)

$\vec{n}{}^{\dagger} \cdot \vec{n}=const~,$

and the covariant derivatives

(7)

$D_{\mu}\vec{n}= (\partial_{\mu} +iA_{\mu})\vec{n}~.$

The action (5) is invariant under Abelian gauge transformations,

(8)

$\vec{n} \to e^{i\Lambda(x)}\vec{n}~,\quad A_{\mu} \to A_{\mu}-\partial_{\mu}\Lambda ~,$

with the gauge parameter $\Lambda(x)$ . There are no kinetic terms for the gauge field $A_{\mu}$ in the action (5), so that the gauge field can be integrated out from the action by the use of its algebraic equation of motion. Substituting the solution $A_{\mu}=i\vec{n}^{\dagger} \cdot\partial_{\mu}\vec{n}$ back into the action (5) gives rise to a CP(n) NLSM.

A scalar potential term $V(\phi)$ may be added into the action (1). Given an isometry symmetry G of the NLSM terms, the scalar potential should be invariant under the same symmetry G. Reparametrizational invariance does not apply in the presence of a scalar potential. Nevertheless, the scalar field theory with free kinetic terms, $g_{ab}(\phi)=\delta_{ab}~,$ and a G-invariant scalar potential is called a linear Sigma-model.

A d=2 NLSM is special since its fields $\phi$ , its metric $g_{ab}$ and, hence, all of its coupling constants are dimensionless. In quantum theory it implies that the 2d NLSM (1) is renormalizable by index of divergence, i.e. its ultra-violett counterterms are of the same (mass) dimension two as the NLSM Lagrangian itself. The NLSM action (1) has the most general (parity conserving) d=2 kinetic terms, so that all the ultra-violett counterterms can be absorbed into the NLSM metric as its `quantum' deformations (geometrical renormalization). It is known as the on-shell renormalizability. Being non-linear, it should be distinguished from the usual (multiplicative) renormalizability that is only the case for a symmetric NLSM target space.

Another special d=2 feature is the existence of an extension of the NLSM action (1) by the so-called Wess-Zumino-Novikov-Witten (WZNW) term,

(9)

$S_{\rm WZNW}[\phi]=\frac{1}{2}\int {\rm d}^2x\,h_{ab}(\phi)\epsilon^{\mu\nu}\partial_{\mu}\phi^a\partial_{\nu}\phi^b~,$

where $h(\phi)$ is a two-form in M, and $\epsilon^{\mu\nu}$ is a Levi-Civita antisymmetric symbol in d=2. The most important example is given by the standard WZNW model having a Lie group G as the target space, with the torsion $T=dh$ to be constructed out of the Lie group structure constants. A quantized WZNW model at its fixed point is conformally invariant, being equivalent to a free quantum fermionic theory. This phenomenon is called a non-Abelian bosonization. There are infinitely many conserved currents in the quantized WZNW model at the fixed point. Those currents form an affine (or Kac-Moody) algebra, while the classical conformal algebra of the WZNW model is also extended to a quantum Virasoro algebra. Hence, the quantized NLSM on a group manifold G at its fixed point is the example of a conformal field theory with a chiral current algebra.

There exist yet another interesting connection between complex geometry and extended supersymmetry in d=2 NLSM. Though any action (1) can be supersymmetrized with respect to a simple (1,1) supersymmetry in d=2, requiring more supersymmetries implies geometrical restrictions on the target space M. For instance, in the case of (2,2) supersymmetry, the NLSM target space must be Kaehler ( $b=0$ ) or bi-hermitean ( $b\neq 0$ ), whereas in the case of the maximal (4,4) supersymmetry, the NLSM target space must be hyper-Kaehler ( $b=0$ ) or bi-hyper-complex ( $b\neq 0$ ).

In string theory the d=2 space $\Sigma$ represents a string world-sheet, the NLSM metric is identified with a truly space-time metric representing the gravitational background where the string propagates, and the antisymmetric field $h_{ab}$ is called a B-field.

Though a NLSM in d=4 is not renormalizable, it often arises in elementary particles physics and nuclear physics as the effective field theory of Goldstone bosons associated with spontaneous breaking of an internal symmetry. The most famous example is given by the chiral symmetry breaking in QCD with $N_f$ flavors, where the phenomenological low-energy Lagrangian of pions (Goldstone pseudo-scalar mesons made out of quark-antiquark pairs) $U_{ff'}\sim < \bar{q}^L_fq^R_{f'}>$ can be constructed as a NLSM,

(10)

$L[U] = -\frac{F^2_{\pi}}{16} tr (A_{\mu}A^{\mu}) -\frac{F^2_{\pi}}{2} tr [ \hat{M}^2(U^{\dagger}+U)]$

in terms of the Cartan field $A_{\mu}=U^{\dagger}\partial_{\mu}U$ , the $SU(N_f)$ -valued pion scalar field $U$ , the pion mass squared matrix $\hat{M}^2$ , and the pion decay constant $F_{\pi}$ whose experimental value is 186 MeV. The effective action of pions is the stating point of chiral perturbation theory. Baryons can be indentified with solitons (classical exact solutions of finite energy) when adding to the NLSM Lagrangian (10) a Skyrme term

(11)

$L_{Skyrme}[U]= \frac{1}{32e^2} tr ([A_{\mu},A_{\nu}]^2)$

with a dimensionless coupling constant $e$ whose experimental value is 5,45. From the NLSM point of view, the Skyrme term (11) represents the 4th-order derivative contribution to the NLSM (10) that is of the 2nd-order in the derivatives of its scalar field $U$ . From the phenomenological viewpoint, the Skyrme term (11) is needed for stability of solitons (skyrmions), which is ensured by their conserved topological winding number identified with the baryon number. For instance, numerical computations of the skyrmion mass with baryonic number $B=1$ yield 850 MeV that is pretty close to the nucleon mass.

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References

Zakrzewski W J (1989), Low Dimensional Sigma Models, IOP Publishers, Bristol; ISBN 978-0852742310
Olshanetsky M A, Perelomov A M (1981), Classical integrable systems related to Lie algebras, Physics Reports 71, 313-400
Ketov S V (1995), Conformal Field Theory, World Scientific, Singapore; ISBN 981-02-1608-4

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Invited by:	Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France
Action editor:	Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France

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