G parity

In particle physics, G parity is a multiplicative quantum number that results from the generalization of C parity (charge conjugation) to multiplets of particles. It was introduced by Louis Michel in 1953 as isotopic parity, and later introduced as G parity by T.D. Lee and C.N. Yang in 1956.

Description 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:

G : ( π + π 0 π − ) = η G ( π + π 0 π − ) {\displaystyle {\mathcal {G}}:{\begin{pmatrix}\pi ^{+}\\\pi ^{0}\\\pi ^{-}\end{pmatrix}}=\eta _{G}{\begin{pmatrix}\pi ^{+}\\\pi ^{0}\\\pi ^{-}\end{pmatrix}}}

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

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

where C ^ {\displaystyle {\hat {\mathcal {C}}}} is the C parity operator, and I ^ 2 {\displaystyle {\hat {I}}_{2}} is the operator associated with the 2nd component of the isospin "vector", which in case of isospin I = 1 / 2 {\displaystyle I=1/2} takes the form I ^ 2 = σ 2 / 2 {\displaystyle {\hat {I}}_{2}=\sigma _{2}/2} , where σ 2 {\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

Q ¯ = B ¯ = Y ¯ = 0 , {\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

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

where ηC is a C parity eigenvalue, and I is the isospin. Since no matter whether the system is fermion–antifermion or boson–antiboson, η C {\displaystyle \eta _{\mathrm {C} }} always equals to ( − 1 ) L + S {\displaystyle (-1)^{L+S}} , we have

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

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