G parity

The G-parity is a multiplicative quantum number , which can take the values +1 and -1. It generalizes the C parity to particle multiplets .

This makes sense because the C parity is only defined for neutral systems (e.g. in the pion triplet only the π ⁰ C parity), but the strong interaction acts independently of the electrical charge (equally on π ⁰ , π ^- and π ⁺ ).

Since the G parity is applied to a whole multiplet, the charge conjugation sees the multiplet as a neutral whole. Hence only multiplets with mean charges of 0 can be eigen-states of G , i.e. H. only multiplets for which the following applies:

${\displaystyle {\bar {Q}}={\bar {B}}={\bar {Y}}=0}$

with the electric charge  Q, the baryon number  B and the hypercharge  Y.

Formulation with operators

${\displaystyle {\mathcal {G}}{\begin{pmatrix}\pi ^{+}\\\pi ^{0}\\\pi ^{-}\end{pmatrix}}=\eta _{G}{\begin{pmatrix}\pi ^{+}\\\pi ^{0}\\\pi ^{-}\end{pmatrix}}}$

Here η _{G are} the eigenvalues of G parity (for pions in particular is ). ${\displaystyle \eta _{G}(\pi )=-1}$

The operator of G parity is defined as: ${\displaystyle {\mathcal {G}}}$

${\displaystyle {\mathcal {G}}={\mathcal {C}}\,e^{(i\pi I_{2})}}$

with the operator of the C parity and the second component of the isospin . The G parity is thus a combination of charge conjugation and a 180 ° rotation around the 2-axis in isospin space. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle I_{2}}$

Formulation with eigenvalues

In general

${\displaystyle \eta _{G}=\eta _{C}\,(-1)^{I}}$

with eigenvalue η _C of C parity and the isospin I .

For fermion- antifermion systems this becomes

${\displaystyle \eta _{G}=(-1)^{S+L+I}\,}$

with the total spin S and the total angular momentum quantum number L.

and for boson- antiboson systems

${\displaystyle \eta _{G}=(-1)^{L+I}\,}$.

Invariance and conservation

The G -parity is invariant under the strong interaction, since this receives both charge conjugation and isospin. However, under the electromagnetic and the weak interaction , G parity is not invariant.

Since it is a multiplicative quantum number, the G parity for a system of n pions is:

${\displaystyle \eta _{G}(n)=\left(-1\right)^{n}}$.

This results in an interesting consequence of the conservation of G for processes in which only pions appear : under the strong interaction, the number of pions can only change by an even number.

literature

• TD Lee and CN Yang : Charge conjugation, a new quantum number G, and selection rules concerning a nucleon-antinucleon system . In: Il Nuovo Cimento . 3, No. 4, 1956, pp. 749-753. doi : 10.1007 / BF02744530 .
• Charles Goebel: Selection Rules for NN̅ Annihilation . In: Phys. Rev. . 103, No. 1, 1956, pp. 258-261. doi : 10.1103 / PhysRev.103.258 .
• Christoph Berger: Particle Physics - An Introduction . Springer, Berlin 1992, pp. 110f, ISBN 978-3-540-54218-6