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Charge conjugation in particle physics
Charge parity of

Formalism
Consider an operation, $$\mathcal{C}$$, that transforms a particle into it's antiparticle
 * $$\mathcal C \, |\psi\rangle = | \bar{\psi} \rangle$$.

Both states must be normalizable, so that
 * $$ 1 = \langle \psi | \psi \rangle = \langle \bar{\psi} | \bar{\psi} \rangle = \langle \psi |\mathcal{C}^\dagger \mathcal C| \psi \rangle$$

which implies that $$\mathcal C$$ is unitary,
 * $$\mathcal C \mathcal{C}^\dagger =\mathbf{1} $$.

By acting on the particle twice with the $$\mathcal{C}$$ operator,
 * $$ \mathcal{C}^2 |\psi\rangle = \mathcal{C} |\bar{\psi}\rangle = |\psi \rangle $$,

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

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

Eigenvalues
For the eigenstates of charge conjugation,
 * $$\mathcal C \, |\psi\rangle = \eta_C \, | \psi \rangle$$.

As with parity transformation, operating twice with $$\mathcal{C}$$ is symmetric and must leave the original particle's state unchanged,
 * $$\mathcal{C}^2|\psi\rangle = \eta_C \mathcal{C} |\bar{\psi} \rangle = \eta_{C}^{2} |\psi\rangle = | \psi \rangle$$

allowing for eigenvalues of $$\eta_C = \pm 1$$, which is called the C-parity or charge parity of the particle.

Eigenstates
The above implies that $$\mathcal C|\psi\rangle$$ and $$|\psi\rangle$$ have exactly the same quantum charges, so only truly neutral systems &mdash;those where all quantum charges and magnetic moment are 0&mdash; are eigenstates of charge parity, that is, the photon and particle-antiparticle bound states: neutral pion, η, positronium... The neutron is not an eigenstate because it has a magnetic moment, and so does not have an associated C parity.

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

In a pair of bound bosons 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.
 * $$\mathcal C \, | \pi^+ \, \pi^- \rangle = (-1)^L \, | \pi^+ \, \pi^- \rangle$$.

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

Bound states can be described with the spectroscopic notation 2S+1LJ (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 an hydrogen atom. The parapositronium and ortopositronium 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 ηC = (−1)L + S = +1, −1, respectively. Since charge parity is preserved, annihilation of these states in photons (ηC(γ) = −1) must be:


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

Experimental tests of C-parity conservation

 * $$\pi^0\rightarrow 3\gamma$$: The neutral pion, $$\pi^0$$, is observed to decay to two photons,γ+γ. We can infer that the pion therefore has $$\eta_C=(-1)^2=1$$, 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 using pions created in the reaction $$\pi^{-} + p \rightarrow \pi^0 + n$$.


 * $$\eta \rightarrow \pi^{+} \pi^{-} \pi^{-}$$


 * $$p \bar{p}$$ annihilations