Chiral Symmetry - Example: u and d Quarks in QCD

Example: u and d Quarks in QCD

Consider quantum chromodynamics (QCD) with two massless quarks u and d. The Lagrangian is

In terms of left-handed and right-handed spinors it becomes

(Hereby i is the imaginary unit and the well-known Dirac operator.)

Defining

it can be written as

The Lagrangian is unchanged under a rotation of by any 2 x 2 unitary matrix L, and by any 2 x 2 unitary matrix R. This symmetry of the Lagrangian is called flavor symmetry or chiral symmetry, and denoted as . It can be decomposed into

The vector symmetry acts as

and corresponds to baryon number conservation.

The axial symmetry acts as

and it does not correspond to a conserved quantity because it is violated due to quantum anomaly.

The remaining chiral symmetry turns out to be spontaneously broken by quark condensate into the vector subgroup, known as isospin. The Goldstone bosons corresponding to the three broken generators are the pions. In real world, because of the differing masses of the quarks, is only an approximate symmetry to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons.