Pion-Pion Lagrangian

The remaining fields $\vec{z}$ can be identified with the $\pi$ mesons forming the pseudoscalar isovector multiplet $\vec{\pi}=(\pi_+,\pi_0,\pi_-$),

$$\vec{\pi} = F_\pi\vec{z} ,$$ (37)

where $F_\pi$=$2\sigma_0$ is a normalization constant. The pion-pion Lagrangian density then equals

$${\cal L}_{\pi\pi} = -2\sigma_0^2\frac{\partial_\mu\vec{z}\c... ..._\pi^2)^2} = -\frac{F_\pi^2}{2}\vec{D}_\mu\circ \vec{D}^\mu ,$$ (38)

where we have defined the O(4) covariant derivative

$$\vec{D}_\mu = \frac{\partial_\mu\vec{z}}{1+\vec{z}^{\,2}} .$$ (39)

First of all we notice that Lagrangian density (38) contains only one isovector multiplet of mesons - the parity-inversed chiral partners have disappeared. This is good. The mechanism of the chiral symmetry breaking explains this experimental fact very well. In reality, the chiral partners still exist, but they have been hidden in the $\sigma$ field and pushed up to high excitation energies. They can only be revealed by exciting an (unknown) internal structure of the meson.

Second, Lagrangian density (38) contains no mass term (term proportional to $\vec{z}^{\,2}$), so the pions we have obtained are massless. This is no accident, but a demonstration of a very general fact that for dynamically broken symmetry there must exist a massless boson. This fact is called the Goldstone theorem Gol61, and the particle is called the Goldstone boson. It sounds very sophisticated, but in fact it is a very simple observation. Even in classical mechanics, if a particle is put into the ``Mexican hat'' potential and treated within the small-vibration approximation, one immediately obtains a zero-frequency mode that corresponds to uniform motion around the hat. The Goldstone boson is just that.

Third, we have derived the particular dependence of the pion-pion Lagrangian (38) on the derivatives of the pion field. Every such derivative must be combined with the particular denominator to form the covariant derivative $\vec{D}_\mu$ (39). This guarantees the proper transformation properties of the pion field with respect to the chiral group. When we later proceed with constructing other Lagrangian densities of composite particles, we shall use such a dependence on the pion fields.

Experimental masses of pions are not equal to zero, so the obtained pion-pion Lagrangian density is too simplistic. However, we can now recall that the quark mass terms do break the chiral symmetry explicitly (see Sec. 2.4). This corresponds to a slight tilt of the ``Mexican hat'' to one side. (To which side, is perfectly well defined by the O(4) structure of the quark mass terms in Eq. (31) - but we shall not discuss that.) Such a tilt creates a small curvature of the potential along the valley within the hat's brim, and this curvature gives the pion-mass term $-{\textstyle{\frac{1}{2}}}{}m_\pi^2\vec{\pi}^{\,2}$ in the pion-pion Lagrangian density. So the non-zero quark masses result in a non-zero pion mass. By the way, the difference in masses of neutral and charged pions results from a coupling to virtual photons - its origin is therefore in the QED, and not in the QCD.

It is amazing how much can be deduced from considerations based on the idea of the dynamical symmetry breaking. Considering the complication of the problem, that is unavoidable on the quark-gluon level, we have reached important results at a very low cost. This happens again and again in almost every branch of physics of the micro-world. Dynamical breaking of the local gauge symmetry gives masses to the electroweak bosons $Z^0$ and $W^\pm$, and leaves the photon massless. Dynamical breaking of the rotational symmetry in nuclei creates the collective moment of inertia and rotational bands. Dynamical breaking of the particle-number symmetry gives superconducting condensates in nuclei and in crystals. Dynamical breaking of the parity symmetry in nuclei and molecules gives collective partner bands of opposite parities. Dynamical breaking of the chiral symmetry (in a different sense, pertaining to the time-reversal symmetry) has been suggested to explain pairs of nuclear rotational bands having the same parity. The story just does not end. Dynamical symmetry breaking rules the world.