Physics: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:

$\widehat{{𝒢}}\left(\begin{array}{c}{\pi }^{+}\\ {\pi }^0\\ {\pi }^{-}\end{array}\right)={\eta }_G\left(\begin{array}{c}{\pi }^{+}\\ {\pi }^0\\ {\pi }^{-}\end{array}\right)$

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

$\widehat{{𝒢}}=\widehat{{𝒞}}{e}^{(i\pi {\hat{I}}_2)}$

where $\widehat{{𝒞}}$ is the C-parity operator, and ${\hat{I}}_2$ is the operator associated with the 2nd component of the isospin "vector", which in case of isospin $I=1/2$ takes the form ${\hat{I}}_2=i{\sigma }_2/2$, where ${\sigma }_2$ 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

$\overline{Q}=\overline{B}=\overline{Y}=0$

(see Q, B, Y).

In general

${\eta }_G={\eta }_C(-1{)}^I$

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, ${\eta }_C$ always equals to $(-1{)}^{L+S}$, we have

${\eta }_G=(-1{)}^{S+L+I}$.

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. doi:10.1007/BF02744530. Bibcode: 1956NCim....3..749L. 
• Charles Goebel (1956). "Selection Rules for NN̅ Annihilation". Phys. Rev. 103 (1): 258–261. doi:10.1103/PhysRev.103.258. Bibcode: 1956PhRv..103..258G.