Chiral symmetry
From Wikipedia, the free encyclopedia
In quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently. The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as vector symmetry, and a component that actually treats them differently, known as axial symmetry.
[edit] 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
Defining
it can be written as
The Lagrangian is unchanged under a rotation of qL by any 2 x 2 unitary matrix L, and qR 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 masses of the quarks and electromagnetism, is only an approximate symmetry to begin with, therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons.
