# Nambu-Jona-Lasinio model

Post-publication activity

Curator: Giovanni Jona-Lasinio

The Nambu-Jona-Lasinio model is an effective chiral field theory realizing the spontaneous symmetry breaking mechanism.

## History

Spontaneous breakdown of symmetry (SSB) is a concept that is applicable only to systems with infinitely many degrees of freedom. Although it pervaded the physics of condensed matter for a very long time, magnetism is a prominent example, its formalization and the recognition of its importance has been an achievement of the second half of the 20th century. Strangely enough the name was adopted only after its introduction in particle physics: it is due to Baker and Glashow (1962).

What is SSB? In condensed matter physics it means that the lowest energy state of a system can have a lower symmetry than the forces acting among its constituents and on the system as a whole. As an example consider a long elastic bar on top of which we apply a compression force directed along its axis. Clearly there is rotational symmetry around the bar which is maintained as long as the force is not too strong: there is simply a shortening according to the Hooke's law. However when the force reaches a critical value the bar bends and we have an infinite number of equivalent lowest energy states which differ by a rotation.

Heisenberg (1959, 1960) was probably the first to consider SSB as a possibly relevant concept in particle physics but his proposal was not the physically right one. The theory of superconductivity of Bardeen, Cooper and Schrieffer (1957) provided the key paradigm for the introduction of SSB in relativistic quantum field theory and particle physics on the basis of an analogy proposed by Nambu (1960a).

To appreciate the innovative character of this concept in particle physics one should consider the strict dogmas which constituted the foundation of relativistic quantum field theory at the time. One of the dogmas stated that all the symmetries of the theory, implemented by unitary operators, must leave the lowest energy state, the vacuum, invariant. This property does not hold in presence of SSB and degenerate vacua. These vacua cannot be connected by local operations and are orthogonal to each other giving rise to different Hilbert spaces. If we live in one of them SSB will be manifested by its consequences, in particular the particle spectrum.

The BCS theory of superconductivity, immediately after its appearance, was reformulated and developed by several authors including Bogolyubov, Valatin, Anderson, Ricayzen and Nambu. The following facts were emphasized

1. The ground state proposed by BCS is not invariant under gauge transformations.
2. The elementary fermionic excitations (quasi-particles) are not eigenstates of the charge as they appear as a superposition of an electron and a hole.
3. In order to restore charge conservation these excitations must be the source of bosonic excitations described by a long range (zero mass) field. In this way the original gauge invariance of the theory is restored.

The peculiarity of the paper of Nambu (1960b), was that he used a language akin to quantum field theory, that is the Green's functions formalism, and the role of gauge invariance was discussed in terms of vertex functions and the associated Ward identities. The search for analogies in particle physics became quite natural. In particular, following the suggestion of Nambu (1960a), the study of chiral symmetry breaking was developed in detail in two papers by Nambu and Jona-Lasinio (1961a,1961b) which had a considerable influence on the evolution of elementary particle theories.

## The analogy with superconductivity

Let us illustrate the elements of the analogy.

Electrons near the Fermi surface are described by the following equation \begin{align} E \psi_{p,+} &= \epsilon_p \psi_{p,+} + \phi \psi_{-p,-}^{\dagger} \\ E \psi_{-p,-}^{\dagger} &= -\epsilon_p \psi_{-p,-}^{\dagger} + \phi \psi_{p,+} , \end{align} with eigenvalues $E = \pm \sqrt{\epsilon_p^2 + \phi^2}.$

Here, $$\psi_{p,+}$$ and $$\psi_{-p,-}^{\dagger}$$ are the wavefunctions for an electron and a hole of momentum $$p$$ and spin $$+\ ;$$ $$\phi$$ is the gap.

In the Weyl representation, the Dirac equation reads \begin{align} E \psi_{1} &= \mathbf{\sigma} \cdot \mathbf{p} \psi_{1} + m \psi_{2} \\ E \psi_{2} &= -\mathbf{\sigma} \cdot \mathbf{p} \psi_{2} + m \psi_{1}, \end{align} with eigenvalues $E = \pm \sqrt{p^2 + m^2}.$ Here, $$\psi_{1}$$ and $$\psi_{2}$$ are the eigenstates of the chirality operator $$\gamma_5\ .$$ Particles with mass are superpositions of states of opposite chirality. The similarity is obvious.

The bosonic excitations necessary to restore gauge invariance in a superconductor appear in the approximate expressions for the charge density and the current in a BCS superconductor (Nambu Y, 1960b), \begin{align} \rho(x,t) &\simeq \rho_0 + \frac{1}{\alpha^2} \partial_t f \\ \mathbf{j}(x,t) &\simeq \mathbf{j}_0 - \mathbf{\nabla} f , \end{align} where $$\rho_0 = e \Psi^{\dagger} \sigma_3 Z \Psi$$ and $$\mathbf{j}_0 = e \Psi^{\dagger} (\mathbf{p}/m) Y \Psi$$ are the contributions of the quasi-particles, $$Y\ ,$$ $$Z\ ,$$ $$\alpha$$ are constants and $$f$$ satisfies the wave equation $\left( \nabla^2 - \frac{1}{\alpha^2} {\partial_t}^2 \right) f \simeq -2 e \Psi^{\dagger} \sigma_2 \phi \Psi.$ Here, $$\Psi^{\dagger} = (\psi^{\dagger}_1, \psi_2)\ .$$

In the elementary particle context the axial current $$\bar{\psi} \gamma_5 \gamma_{\mu} \psi$$ is the analog of the electromagnetic current in BCS theory. In the hypothesis of exact conservation, the matrix elements of the axial current between nucleon states of four-momentum $$p$$ and $$p'$$ have the form $\Gamma_{\mu}^A (p', p) = \left( i \gamma_5 \gamma_{\mu} - 2m \gamma_5 q_{\mu} / q^2 \right) F(q^2), \qquad q = p' - p .$ Exact conservation is compatible with a finite nucleon mass $$m$$ provided there exists a massless pseudoscalar particle.

Assuming exact conservation of the chiral current, a picture of chiral SSB may consist in a vacuum of a massless Dirac field viewed as a sea of occupied negative energy states, and an attractive force between particles and antiparticles having the effect of producing a finite mass, the counterpart of the gap. The pseudoscalar massless particle, which may be interpreted as a forerunner of the pion, corresponds to the bosonic field associated to the fermionic quasi-particles in a superconductor.

To implement this picture the construction of a relativistic field theoretic model was required. At that time Heisenberg and his collaborators had developed a comprehensive theory of elementary particles based on a non linear spinor interaction: the physical principle was that spin ½ fermions could provide the building blocks of all known elementary particles. Heisenberg was however very ambitious and wanted at the same time to solve in a consistent way the dynamical problem of a non renormalizable theory. This made their approach very complicated and not transparent.

## The Nambu-Jona-Lasinio model

A Heisenberg type Lagrangian was adopted without pretending to solve the non-renormalizability problem and introducing a relativistic cut-off to cure the divergences. This model is known in the literature with the acronym NJL (Nambu-Jona-Lasinio model). The energy scale of interest was of the order of the nucleon mass and one hoped that higher energy effects would not change substantially the picture.

The Lagrangian of the NJL model is $\mathcal {L} = -\bar{\psi} \gamma_{\mu} \partial_{\mu} \psi + g \left[ \left( \bar{\psi} \psi \right)^2 - \left( \bar{\psi} \gamma_5 \psi \right)^2 \right] .$ It is invariant under ordinary and chiral gauge transformations \begin{align} \psi &\to e^{i \alpha} \psi, \qquad &\bar{\psi} &\to \bar{\psi} e^{-i \alpha} \\ \psi &\to e^{i \alpha \gamma_5} \psi, \qquad &\bar{\psi} &\to \bar{\psi} e^{i \alpha \gamma_5} . \end{align}

To investigate the content of the model a simple mean field approximation for the mass was adopted \begin{align} m &= -2g \left[ <\bar{\psi} \psi> - \gamma_5 <\bar{\psi} \gamma_5 \psi> \right] \\ &= 2g \left[ tr S^{(m)}(0) - tr \gamma_5 S^{(m)}(0) \right], \end{align} where $$S^{(m)}$$ is the propagator of the Dirac field of mass $$m\ ,$$ or more explicitly, $\frac{2 \pi^2}{g \Lambda^2} = 1 - \frac{m^2}{\Lambda^2} \ln \left(1 + \frac{\Lambda^2}{m^2} \right),$ where $$\Lambda$$ is the invariant cut-off. This equation is very similar to the gap equation in BCS theory. If $$\frac{2\pi^2}{g\Lambda^2} < 1$$ there exists a solution $$m>0\ .$$

From this relationship a rich spectrum of bound states follows, $\begin{array}{cccc} \hline \hline \text{nucleon} & \text{mass} \mu & \text{spin-parity} & \text{spectroscopic} \\ \text{number} & & & \text{notation} \\ \hline 0 & 0 & 0^- & ^1S_0 \\ 0 & 2m & 0^+ & ^3P_0 \\ 0 & \mu^2 > \frac{8}{3} m^2 & 1^- & ^3P_1 \\ \pm 2 & \mu^2 > 2 m^2 & 0^+ & ^1S_0 \\ \hline \hline \end{array}$

The bosonic field in the superconductor and the pseudoscalar particle in the NJL model are special cases of a general proposition formulated by Goldstone (1961).

Whenever the original Lagrangian has a continuous symmetry group, which does not leave the ground state invariant, massless bosons appear in the spectrum of the theory.

Other examples are:

physical system broken symmetry massless bosons
ferromagnets rotational invariance spin waves
crystals translational and rotational invariance phonons

These massless bosons are now known in the literature as Nambu-Goldstone bosons. In nature, however, the axial current is only approximately conserved. The model made contact with the real world under the hypothesis that the small violation of axial current conservation gives a mass to the massless boson, which is then identified with the $$\pi$$ meson. Actually the NJL model, reinterpreted in terms of quarks, has been the starting point of a successful effective theory of low energy QCD, see e.g. the review by Hatsuda and Kunihiro (1994).

After the NJL model, SSB became a key concept in elementary particle physics, in particular electroweak unification (Weinberg S, 1967) requires SSB.

## References

• Baker, M and Glashow, S L (1962). Spontaneous Breakdown of Elementary Particle Symmetries. Physical Review 128(5): 2462-2471.
• Heisenberg, W; Dürr, H P; Mitter, H; Schlieder, S and Yamazaki, K (1959). Zur Theorie der Elementarteilchen. Zeitschrift für Naturforschung 14a: 441-485.
• Heisenberg, W (1960). Recent research on the nonlinear spinor theory of elementary particles. Proc. 1960 Annual Intern. Conf. on High Energy Physics, Rochester : 851-858.
• Bardeen, J; Cooper, L N and Schrieffer, J R (1957). Microscopic Theory of Superconductivity. Physical Review 106(1): 162-164.
• Nambu, Y (1960a). Axial Vector Current Conservation in Weak Interactions. Physical Review Letters 4: 380-382.
• Nambu, Y (1960b). Quasi-Particles and Gauge Invariance in the Theory of Superconductivity. Physical Review 117: 648-663.
• Nambu, Y and Jona-Lasinio, G (1961a). Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I. Physical Review 122(1): 345-358.
• Nambu, Y and Jona-Lasinio, G (1961b). Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II. Physical Review 124(1): 246-254.
• Goldstone, J (1961). Field theories with Superconductor solutions. Nuovo Cimento 19: 154-164.
• Hatsuda, T and Kunihiro, T (1994). QCD phenomenology based on a chiral effective Lagrangian. Physics Reports 247(5-6): 221-367.
• Weinberg, S (1967). A Model of Leptons. Physical Review Letters 19(21): 1264-1266.