G-parity

In particle physics, G-parity is a multiplicative quantum number that results from the generalization of C-parity to multiplets of particles.

C-parity applies only to neutral systems; in the pion triplet, only π⁰ has C-parity. On the other hand, strong interaction does not see electrical charge, so it cannot distinguish amongst π⁺, π⁰ and π⁻. We can generalize the C-parity so it applies to all charge states of a given multiplet:

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

where η_G = ±1 are the eigenvalues of G-parity. The G-parity operator is defined as

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

where ${\displaystyle {\hat {\mathcal {C}}}}$ is the C-parity operator, and ${\displaystyle {\hat {I}}_{2}}$ is the operator associated with the 2nd component of the isospin "vector", which in case of isospin ${\displaystyle I=1/2}$ takes the form ${\displaystyle {\hat {I}}_{2}=i\sigma _{2}/2}$, where ${\displaystyle \sigma _{2}}$ is the second Pauli matrix. G-parity is a combination of charge conjugation and a π rad (180°) rotation around the 2nd axis of isospin space. Given that charge and isospin are preserved by strong interactions, so is G. Weak and electromagnetic interactions, though, does not conserve G-parity.

Since G-parity is applied on a whole multiplet, charge conjugation has to see the multiplet as a neutral entity. Thus, only multiplets with an average charge of 0 will be eigenstates of G, that is

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

(see Q, B, Y).

In general

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

where η_C is a C-parity eigenvalue, and I is the isospin.

Since no matter whether the system is fermion–antifermion or boson–antiboson, ${\displaystyle \eta _{C}}$ always equals to ${\displaystyle (-1)^{L+S}}$, we have

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

See also

• Quark model

References

• T. D. Lee and C. N. Yang (1956). "Charge conjugation, a new quantum number G, and selection rules concerning a nucleon-antinucleon system". Il Nuovo Cimento. 3 (4): 749–753. Bibcode:1956NCim....3..749L. doi:10.1007/BF02744530. S2CID   119539007.
• Charles Goebel (1956). "Selection Rules for NN̅ Annihilation". Phys. Rev. 103 (1): 258–261. Bibcode:1956PhRv..103..258G. doi:10.1103/PhysRev.103.258.