C parity

Consider an operation ${\displaystyle {\mathcal {C}}}$ that transforms a particle into its antiparticle,

${\displaystyle {\mathcal {C}}\,|\psi \rangle =|{\bar {\psi }}\rangle .}$

Both states must be normalizable, so that

${\displaystyle 1=\langle \psi |\psi \rangle =\langle {\bar {\psi }}|{\bar {\psi }}\rangle =\langle \psi |{\mathcal {C}}^{\dagger }{\mathcal {C}}|\psi \rangle ,}$

which implies that ${\displaystyle {\mathcal {C}}}$ is unitary,

${\displaystyle {\mathcal {C}}{\mathcal {C}}^{\dagger }=\mathbf {1} .}$

By acting on the particle twice with the ${\displaystyle {\mathcal {C}}}$ operator,

${\displaystyle {\mathcal {C}}^{2}|\psi \rangle ={\mathcal {C}}|{\bar {\psi }}\rangle =|\psi \rangle ,}$

we see that ${\displaystyle {\mathcal {C}}^{2}=\mathbf {1} }$ and ${\displaystyle {\mathcal {C}}={\mathcal {C}}^{-1}}$. Putting this all together, we see that

${\displaystyle {\mathcal {C}}={\mathcal {C}}^{\dagger },}$

meaning that the charge conjugation operator is Hermitian and therefore a physically observable quantity.

For a system of free particles, the C parity is the product of C parities for each particle.

In a pair of bound mesons there is an additional component due to the orbital angular momentum. For example, in a bound state of two pions, π⁺ π⁻ with an orbital angular momentum L, exchanging π⁺ and π⁻ inverts the relative position vector, which is identical to a parity operation. Under this operation, the angular part of the spatial wave function contributes a phase factor of (−1)^L, where L is the angular momentum quantum number associated with L.

${\displaystyle {\mathcal {C}}\,|\pi ^{+}\,\pi ^{-}\rangle =(-1)^{L}\,|\pi ^{+}\,\pi ^{-}\rangle }$.

With a two-fermion system, two extra factors appear: one comes from the spin part of the wave function, and the second from the exchange of a fermion by its antifermion.

${\displaystyle {\mathcal {C}}\,|f\,{\bar {f}}\rangle =(-1)^{L}(-1)^{S+1}(-1)\,|f\,{\bar {f}}\rangle =(-1)^{L+S}\,|f\,{\bar {f}}\rangle }$

Bound states can be described with the spectroscopic notation ^2S+1L_J (see term symbol), where S is the total spin quantum number, L the total orbital momentum quantum number and J the total angular momentum quantum number. Example: the positronium is a bound state electron-positron similar to a hydrogen atom. The parapositronium and orthopositronium correspond to the states ¹S₀ and ³S₁.

• With S = 0 spins are anti-parallel, and with S = 1 they are parallel. This gives a multiplicity (2S+1) of 1 or 3, respectively
• The total orbital angular momentum quantum number is L = 0 (S, in spectroscopic notation)
• Total angular momentum quantum number is J = 0, 1
• C parity η_C = (−1)^{L + S} = +1, −1, respectively. Since charge parity is preserved, annihilation of these states in photons (η_C(γ) = −1) must be:

	1S0	→	γ + γ		3S1	→	γ + γ + γ
ηC:	+1	=	(−1) × (−1)		−1	=	(−1) × (−1) × (−1)

• ${\displaystyle \pi ^{0}\rightarrow 3\gamma }$: The neutral pion, ${\displaystyle \pi ^{0}}$, is observed to decay to two photons,γ+γ. We can infer that the pion therefore has ${\displaystyle \eta _{C}=(-1)^{2}=1}$, but each additional γ introduces a factor of -1 to the overall C parity of the pion. The decay to 3γ would violate C parity conservation. A search for this decay was conducted[1] using pions created in the reaction ${\displaystyle \pi ^{-}+p\rightarrow \pi ^{0}+n}$.
• ${\displaystyle \eta \rightarrow \pi ^{+}\pi ^{-}\pi ^{0}}$:[2] Decay of the Eta meson.
• ${\displaystyle p{\bar {p}}}$ annihilations[3]