Chiral symmetry

"Chiral" comes from the Greek for "Hand". The most straightforward example of Chiral symmetry is the mirror symmetry shown by your left and right hand. Chirality is a concept of importance in several areas of science.

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.[citation needed]
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

\( \mathcal{L} = \overline{u}_L\,i\displaystyle{\not}D \,u_L + \overline{u}_R\,i\displaystyle{\not}D \,u_R + \overline{d}_L\,i\displaystyle{\not}D \,d_L + \overline{d}_R\,i\displaystyle{\not}D \,d_R + \mathcal{L}_\mathrm{gluons} \)

(Hereby i is the imaginary unit and \displaystyle{\not}D the well-known Dirac operator.)

Defining

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

it can be written as

\( \mathcal{L} = \overline{q}_L\,i\displaystyle{\not}D \,q_L + \overline{q}_R\,i\displaystyle{\not}D\, q_R + \mathcal{L}_\mathrm{gluons} \)

The Lagrangian is unchanged under a rotation of q_L by any 2 x 2 unitary matrix L, and q_R 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 differing masses of the quarks, \( SU(2)_L \times SU(2)_R \) is only an approximate symmetry to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons.[1]
References

^ Peskin, Michael; Schroeder, Daniel (1995). An Introduction to Quantum Field Theory. Westview Press. pp. 670. ISBN 0201503972.

Chiral symmetry, Quarks, baryons and , Atsushi Hosaka, Hiroshi Toki

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