Chiral symmetry

The chiral symmetry (from the Greek χέρι hand) is a possible symmetry of the Lagrangian function in quantum field theory , which is often given - at least approximately - and then plays an important role, e.g. B. with the pions .

Here are left-handed and right-handed portion of the fermionic fields independently transformed. The chiral symmetry transformation can be divided into a component that treats left-handed and right-handed components equally ( vector symmetry ), and a component that treats them “in opposite directions” ( axial symmetry ). The latter part disappears through quark condensation in the former phase.

Example: u - and d - quarks in QCD

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

${\displaystyle {\mathcal {L}}={\overline {u}}\,i\displaystyle {\not }D\,u+{\overline {d}}\,i\displaystyle {\not }D\,d+{\mathcal {L}}_{\text{Gluonen}}\,.}$

The i means the imaginary unit and the Dirac operator in Feynman Slash notation . The u and d are the four-component Dirac spinors and the overline denotes the Dirac adjoints. ${\displaystyle \displaystyle {\not }D}$

According to quantum chromodynamics, the mesons are made up of a quark and an antiquark, e.g. B. the one and one . However, this does not fundamentally change the following derivation. ${\displaystyle \,\pi ^{+}}$ ${\displaystyle \,u}$ ${\displaystyle {\overline {d}}}$

In the representation of the left-handed and right-handed spinors , one gets first

${\displaystyle {\mathcal {L}}={\overline {u}}_{L}\,i\displaystyle {\not }D\,u_{L}+{\overline {u}}_{R}\,i\displaystyle {\not }D\,u_{R}+{\overline {d}}_{L}\,i\displaystyle {\not }D\,d_{L}+{\overline {d}}_{R}\,i\displaystyle {\not }D\,d_{R}+{\mathcal {L}}_{\text{Gluonen}}\,.}$

It is defined

${\displaystyle q={\begin{bmatrix}u\\d\end{bmatrix}}\,.}$

So it follows

${\displaystyle {\mathcal {L}}={\overline {q}}_{L}\,i\displaystyle {\not }D\,q_{L}+{\overline {q}}_{R}\,i\displaystyle {\not }D\,q_{R}+{\mathcal {L}}_{\text{Gluonen}}\,.}$

The Lagrangian remains in rotation with unitary 2 × 2 matrices L of rotation and with 2 × 2 unitary matrices R are each invariant. This symmetry of the long-range function is called flavor symmetry or chiral symmetry and is noted as. It can be broken down into the following partial symmetries ${\displaystyle q_{L}}$ ${\displaystyle q_{R}}$ ${\displaystyle U(2)_{L}\times U(2)_{R}}$

${\displaystyle SU(2)_{L}\times SU(2)_{R}\times U(1)_{V}\times U(1)_{A}\,\,.}$

The vector symmetry is ${\displaystyle U(1)_{V}\,}$

${\displaystyle q_{L}\rightarrow e^{i\theta }q_{L}\qquad q_{R}\rightarrow e^{i\theta }q_{R}}$

and corresponds to the conservation of the baryon number.

The corresponding axial operation is ${\displaystyle U(1)_{A}\,}$

${\displaystyle q_{L}\rightarrow e^{i\theta }q_{L}\qquad q_{R}\rightarrow e^{-i\theta }q_{R}\,.}$

It does not  correspond to a conserved quantity because it is broken by a quantum anomaly .

It turns out that the remaining chiral symmetry to the vector subgroup (the isospin group ) is broken spontaneously . The breaking of symmetry is expressed by a corresponding, complete quark condensate . ${\displaystyle SU(2)_{L}\times SU(2)_{R}}$ ${\displaystyle SU(2)_{V}\,}$

The Goldstone bosons that correspond to the three broken generators of the transformation are the pions . Since the masses of the quarks are not the same, this is only approximately a symmetry of the system. The pions are therefore not “real”, massless Goldstone bosons, but so-called pseudo Goldstone bosons. ${\displaystyle SU(2)_{L}\times SU(2)_{R}}$

Chiral Limes

A distinction must be made between the “chiral symmetry” and the “chiral limes” ( ) of a single Dirac equation . This limit is best realized with neutrinos or their antiparticles with their well-defined chirality: ${\displaystyle m\to 0}$

• "Left screw" with respect to spin and momentum in neutrinos: ${\displaystyle {\vec {s}}}$ ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {s}}\propto -{\vec {p}}}$
• "Right screw" regarding spin and momentum in antineutrinos: ${\displaystyle {\vec {s}}\propto +{\vec {p}}}$

as well as in solids in graphenes .

See also

• Spontaneous breaking of symmetry

Web links

• Strong force between right and left