Chiral symmetry

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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

\mathcal{L} = \overline{u}i\displaystyle{\not}D u + \overline{d}i\displaystyle{\not}D d + \mathcal{L}_\mathrm{gluons}

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

\mathcal{L} = \overline{u}_Li\displaystyle{\not}D u_L + \overline{u}_Ri\displaystyle{\not}D u_R + \overline{d}_Li\displaystyle{\not}D d_L + \overline{d}_Ri\displaystyle{\not}D d_R + \mathcal{L}_\mathrm{gluons}

Defining

q = \begin{bmatrix} u \\ d \end{bmatrix}

it can be written as

\mathcal{L} = \overline{q}_Li\displaystyle{\not}D q_L + \overline{q}_Ri\displaystyle{\not}D q_R + \mathcal{L}_\mathrm{gluons}

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 U(2)_L \times U(2)_R. It can be decomposed into

SU(2)_L \times SU(2)_R \times U(1)_V \times U(1)_A

The vector symmetry U(1)_V\, acts as

q_L \rightarrow e^{i\theta} q_L \qquad q_R \rightarrow e^{i\theta} q_R

and corresponds to baryon number conservation.

The axial symmetry U(1)_A\, acts as

q_L \rightarrow e^{i\theta} q_L \qquad q_R \rightarrow e^{-i\theta} q_R

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

The remaining chiral symmetry SU(2)_L \times SU(2)_R turns out to be spontaneously broken by quark condensate into the vector subgroup SU(2)_V\,, 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, SU(2)_L \times SU(2)_R is only an approximate symmetry to begin with, therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons.