C parity

Template:Short description 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 $𝒞$ that transforms a particle into its antiparticle,

$𝒞|\psi ⟩=|\overline{\psi }⟩.$

Both states must be normalizable, so that

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

which implies that $𝒞$ is unitary,

$𝒞{𝒞}^{†}=\mathrm{𝟏}.$

By acting on the particle twice with the $𝒞$ operator,

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

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

$𝒞={𝒞}^{†},$

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

Eigenvalues

For the eigenstates of charge conjugation,

$𝒞|\psi ⟩={\eta }_C|\psi ⟩$.

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

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

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

Eigenstates

The above implies that for eigenstates, $𝒞|\psi ⟩=|\bar{\psi }⟩=\pm |\psi ⟩.$ Since antiparticles and particles have charges of opposite sign, only states with all quantum charges equal to zero, such as the photon and particle–antiparticle bound states like [[pi meson|Template:Math]], [[eta meson|Template:Math]], or positronium, are eigenstates of $𝒞.$

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 mesons there is an additional component due to the orbital angular momentum. For example, in a bound state of two pions, Template:Math with an orbital angular momentum Template:Math, exchanging Template:Math and Template:Math 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 Template:Math, where Template:Mvar is the angular momentum quantum number associated with Template:Math.

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

With a two-fermion system, two extra factors appear: One factor comes from the spin part of the wave function, and the second by considering the intrinsic parities of both the particles. Note that a fermion and an antifermion always have opposite intrinsic parity. Hence,

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

Bound states can be described with the spectroscopic notation Template:Math (see term symbol), where Template:Mvar is the total spin quantum number (not to be confused with the S orbital), Template:Mvar is the total angular momentum quantum number, and Template:Mvar the total orbital momentum quantum number (with quantum number Template:Math etc. replaced by orbital letters S, P, D, etc.).

Example

positronium is a bound state electron-positron similar to a hydrogen atom. The names parapositronium and orthopositronium are given to the states ¹S₀ and ³S₁.

• With Template:Math, the spins are anti-parallel, and with Template:Nobr they are parallel. This gives a multiplicity Template:Nobr of 1 (anti-parallel) or 3 (parallel)
• The total orbital angular momentum quantum number is Template:Nobr (spectroscopic S orbital)
• Total angular momentum quantum number is Template:Nobr
• C parity Template:Nobr depending on Template:Mvar and Template:Mvar. Since charge parity is preserved, annihilation of these states in photons Template:Nobr must be:

Orbital:	1S0 → Template:Mvar	3S1 → Template:Mvar
Template:Math :	+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, Template:Nobr We can infer that the pion therefore has ${\eta }_C=(-1{)}^2=1,$ but each additional Template:Mvar introduces a factor of Template:Math to the overall C-parity of the pion. The decay to Template:Math would violate C parity conservation. A search for this decay was conducted using pions created in the reaction ${\pi }^{-}+p\to {\pi }^0+n.$

• $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$: Decay of the eta meson.

• $p\overline{p}$ annihilations

See also

• G-parity