Editing Chirality (physics) (section)

=== Example: u and d quarks in QCD ===
Consider [[quantum chromodynamics]] (QCD) with two ''massless'' [[quarks]] {{math|u}} and {{math|d}} (massive fermions do not exhibit chiral symmetry). The Lagrangian reads
: <math>\mathcal{L} = \overline{u}\,i\displaystyle{\not}D \,u + \overline{d}\,i\displaystyle{\not}D\, d + \mathcal{L}_\mathrm{gluons}~.</math>

In terms of left-handed and right-handed spinors, it reads
: <math>\mathcal{L} = \overline{u}_{\rm L}\,i\displaystyle{\not}D \,u_{\rm L} + \overline{u}_{\rm R}\,i\displaystyle{\not}D \,u_{\rm R} + \overline{d}_{\rm L}\,i\displaystyle{\not}D \,d_{\rm L}  + \overline{d}_{\rm R}\,i\displaystyle{\not}D \,d_{\rm R} + \mathcal{L}_\mathrm{gluons} ~.</math>
(Here, {{math|''i''}} is the imaginary unit and <math>\displaystyle{\not}D</math> the [[Dirac operator]].)

Defining
: <math>q = \begin{bmatrix} u \\ d \end{bmatrix} ,</math>
it can be written as
: <math>\mathcal{L} = \overline{q}_{\rm L}\,i\displaystyle{\not}D \,q_{\rm L} + \overline{q}_{\rm R}\,i\displaystyle{\not}D\, q_{\rm R} + \mathcal{L}_\mathrm{gluons} ~.</math>

The Lagrangian is unchanged under a rotation of ''q''<sub>L</sub> by any 2×2 unitary matrix {{mvar|L}}, and ''q''<sub>R</sub> by any 2×2 unitary matrix {{mvar|R}}.

This symmetry of the Lagrangian is called ''flavor chiral symmetry'', and denoted as {{math|U(2)<sub>L</sub> × U(2)<sub>R</sub>}}.   It decomposes into
: <math>\mathrm{SU}(2)_\text{L} \times \mathrm{SU}(2)_\text{R} \times \mathrm{U}(1)_V \times \mathrm{U}(1)_A ~.</math>

The singlet vector symmetry, {{math|U(1)<sub>''V''</sub>}},  acts as 
: <math>
q_\text{L} \rightarrow e^{i\theta(x)} q_\text{L} \qquad
q_\text{R} \rightarrow e^{i\theta(x)} q_\text{R} ~,
</math>
and thus invariant under {{math|U(1)}} gauge symmetry. This corresponds to [[baryon number]] conservation.

The singlet axial group  {{math|U(1)<sub>''A''</sub>}}  transforms as the following global transformation
: <math>
q_\text{L} \rightarrow e^{i\theta} q_\text{L} \qquad
q_\text{R} \rightarrow e^{-i\theta} q_\text{R} ~.
</math>
However, it does not correspond to a conserved quantity, because the associated axial current is not conserved. It is explicitly violated by a [[anomaly (physics)|quantum anomaly]].

The remaining chiral symmetry {{math|SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} turns out to be [[spontaneous symmetry breaking|spontaneously broken]] by a [[quark condensate]] 
<math>\textstyle \langle \bar{q}^a_\text{R} q^b_\text{L} \rangle = v \delta^{ab}</math> formed through nonperturbative action of QCD gluons, into the diagonal vector subgroup {{math|SU(2)<sub>''V''</sub>}} known as [[isospin]]. The [[Goldstone bosons]] corresponding to the three broken generators are the three [[pions]]. 
As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this [[chiral symmetry breaking]] induces the bulk of hadron masses, such as those for the [[nucleon]]s &mdash; in effect, the bulk of the mass of all visible matter.

In the real world, because of the nonvanishing and differing masses of the quarks, {{math|SU(2)<sub>L</sub> × SU(2)<sub>R</sub>}} is only an approximate symmetry<ref>{{Cite journal | last1 = Gell-Mann | first1 = M. | last2 = Renner | first2 = B. | doi = 10.1103/PhysRev.175.2195 | title = Behavior of Current Divergences under SU<sub>3</sub>×SU<sub>3</sub> | journal = Physical Review | volume = 175 | issue = 5 | pages = 2195 | year = 1968 |bibcode = 1968PhRv..175.2195G | url = https://authors.library.caltech.edu/3634/1/GELpr68.pdf }}</ref> to begin with, and therefore the pions are not massless, but have small masses: they are [[Chiral symmetry breaking|pseudo-Goldstone boson]]s.<ref>{{Cite book |last1=Peskin |first1=Michael |last2=Schroeder |first2=Daniel | title = An Introduction to Quantum Field Theory | publisher = Westview Press | year = 1995 | pages = 670 | isbn = 0-201-50397-2}}</ref>