G parity

Summarize Timeline Top Qs Fact Check

Charge conjugation or C parity applies only to neutral systems. For example, in the pion triplet, only the neutral pion 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:

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

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 ${\displaystyle {\hat {I}}_{2}=\sigma _{2}/2}$, where ${\displaystyle \sigma _{2}}$ is the second Pauli matrix. G-parity is a combination of charge conjugation and a π radians (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

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

where Q is the electric charge of the multiplet, B is the baryon number and Y of the hypercharge.

In general

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

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, ${\displaystyle \eta _{\mathrm {C} }}$ always equals to ${\displaystyle (-1)^{L+S}}$, we have

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

• Quark model