Hensel's lemma

In physics, the C parity or charge parity is a multiplicative quantum number of some particles that describes their behavior under the symmetry operation of charge conjugation.

Charge conjugation changes the sign of all quantum charges (that is, additive quantum numbers), including the electrical charge, baryon number and lepton number, and the flavor charges strangeness, charm, bottomness, topness and Isospin (I₃). In contrast, it doesn't affect the mass, linear momentum or spin of a particle.

Formalism

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

$\mathcal{𝒞}|\psi ⟩=|\overline{\psi }⟩.$

Both states must be normalizable, so that

$1=⟨\psi |\psi ⟩=⟨\overline{\psi }|\overline{\psi }⟩=⟨\psi \left|{\mathcal{𝒞}}^{†}\mathcal{𝒞}\right|\psi ⟩,$

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

$\mathcal{𝒞}{\mathcal{𝒞}}^{†}=\mathbf{1}.$

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

${\mathcal{𝒞}}^2|\psi ⟩=\mathcal{𝒞}|\overline{\psi }⟩=|\psi ⟩,$

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

$\mathcal{𝒞}={\mathcal{𝒞}}^{†},$

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

Eigenvalues

For the eigenstates of charge conjugation,

$\mathcal{𝒞}|\psi ⟩={\eta }_C|\overline{\psi }⟩$.

As with parity transformations, applying $\mathcal{𝒞}$ twice must leave the particle's state unchanged,

${\mathcal{𝒞}}^2|\psi ⟩={\eta }_C\mathcal{𝒞}|\overline{\psi }⟩={\eta }_C^2|\psi ⟩=|\psi ⟩$

allowing only eigenvalues of ${\eta }_C=\pm 1$ the so-called C-parity or charge parity of the particle.

Eigenstates

The above implies that $\mathcal{𝒞}|\psi ⟩$ and $|\psi ⟩$ have exactly the same quantum charges, so only truly neutral systems – those where all quantum charges and the magnetic moment are zero – are eigenstates of charge parity, that is, the photon and particle-antiparticle bound states like the neutral pion, η or the positronium.

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.

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

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.

$\mathcal{𝒞}|f\overline{f}⟩=(-1{)}^L(-1{)}^{S+1}(-1)|f\overline{f}⟩=(-1{)}^{L+S}|f\overline{f}⟩$

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 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 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)

Experimental tests of C-parity conservation

• ${\pi }^0\to 3\gamma$: 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 ${\pi }^{-}+p\to {\pi }^0+n$.

• $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^{-}$ Decay of the Eta meson.

• $p\overline{p}$ annihilations