# C parity

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/C_parity
> Markdown URL: https://mediated.wiki/source/C_parity.md
> Source: https://en.wikipedia.org/wiki/C_parity
> Source revision: 1276358434
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{short description|Unitary operation that transforms a particle in its antiparticle}}
In [physics](/source/physics), the '''C parity''' or '''charge parity''' is a [multiplicative quantum number](/source/multiplicative_quantum_number) of some particles that describes their behavior under the symmetry operation of [charge conjugation](/source/charge_conjugation).

Charge conjugation changes the sign of all quantum charges (that is, additive [quantum number](/source/quantum_number)s), including the [electrical charge](/source/electrical_charge), [baryon number](/source/baryon_number) and [lepton number](/source/lepton_number), and the flavor charges [strangeness](/source/strangeness), [charm](/source/charm_(quantum_number)), [bottomness](/source/bottomness), [topness](/source/topness) and [Isospin](/source/Isospin) (''I''<sub>3</sub>). In contrast, it doesn't affect the [mass](/source/mass), [linear momentum](/source/linear_momentum) or [spin](/source/Spin_(physics)) of a particle.

==Formalism==
Consider an operation <math>\mathcal{C}</math> that transforms a particle into its [antiparticle](/source/antiparticle),
:<math>\mathcal C \, |\psi\rangle = | \bar{\psi} \rangle.</math>
Both states must be normalizable, so that
:<math> 1 = \langle \psi | \psi \rangle = \langle \bar{\psi} | \bar{\psi} \rangle = \langle \psi |\mathcal{C}^\dagger \mathcal C| \psi \rangle,</math>
which implies that <math>\mathcal C</math> is unitary, 
:<math>\mathcal C \mathcal{C}^\dagger =\mathbf{1}.</math>
By acting on the particle twice with the <math>\mathcal{C}</math> operator,
:<math> \mathcal{C}^2 |\psi\rangle = \mathcal{C} |\bar{\psi}\rangle = |\psi \rangle,</math>
we see that <math>\mathcal{C}^2=\mathbf{1}</math> and <math>\mathcal{C}=\mathcal{C}^{-1}</math>. Putting this all together, we see that 
:<math>\mathcal{C}=\mathcal{C}^{\dagger},</math>
meaning that the charge conjugation operator is [Hermitian](/source/self-adjoint_operator) and therefore a physically observable quantity.

===Eigenvalues===
For the eigenstates of charge conjugation,
:<math>\mathcal C \, |\psi\rangle = \eta_C \, | {\psi} \rangle</math>.

As with [parity transformations](/source/parity_(physics)), applying <math>\mathcal{C}</math> twice must leave the particle's state unchanged,
:<math>\mathcal{C}^2|\psi\rangle = \eta_C \mathcal{C} |{\psi} \rangle = \eta_{C}^{2} |\psi\rangle = | \psi \rangle</math>
allowing only eigenvalues of <math>\eta_C = \pm 1</math> the so-called ''C-parity'' or ''charge parity'' of the particle.

===Eigenstates===
The above implies that for [eigenstate](/source/eigenstate)s, <math>\ \operatorname{\mathcal C} |\psi\rangle = | \overline{\psi} \rangle = \pm | \psi \rangle ~.</math> Since antiparticles and particles have charges of opposite sign, only states with all quantum charges equal to zero, such as the [photon](/source/photon) and particle–antiparticle bound states like [{{math|&pi;{{sup|0}}}}](/source/pi_meson), [{{math|η{{sup|0}}}}](/source/eta_meson), or [positronium](/source/positronium), are eigenstates of <math>\mathcal C ~.</math>

==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 [meson](/source/meson)s there is an additional component due to the orbital angular momentum. For example, in a bound state of two [pions](/source/pions), {{math|π<sup>+</sup> π<sup>−</sup>}} with an orbital [angular momentum](/source/angular_momentum) {{math|'''L'''}}, exchanging {{math|π<sup>+</sup>}} and {{math|π<sup>−</sup>}} inverts the relative position vector, which is identical to a [parity](/source/parity_(physics)) operation. Under this operation, the angular part of the spatial wave function contributes a phase factor of {{math| (−1)<sup>''L''</sup>}}, where {{mvar|L}} is the [angular momentum quantum number](/source/angular_momentum_quantum_number) associated with {{math|'''L'''}}.
:<math>\mathcal C \, | \pi^+ \, \pi^- \rangle = (-1)^L \, | \pi^+ \, \pi^- \rangle</math>.
With a two-[fermion](/source/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,
:<math>\mathcal C \, | f \, \bar f \rangle = (-1)^L (-1)^{S+1} (-1) \, | f \, \bar f \rangle = (-1)^{L + S} \, | f \, \bar f \rangle ~.</math>

Bound states can be described with the [spectroscopic notation](/source/spectroscopic_notation) {{math|<sup>2''S''+1</sup>L<sub>''J''</sub>}} (see [term symbol](/source/term_symbol)), where {{mvar|S}} is the total spin quantum number (not to be confused with the S orbital), {{mvar|J}} is the [total angular momentum quantum number](/source/total_angular_momentum_quantum_number), and {{mvar|L}} the total [orbital momentum quantum number](/source/azimuthal_quantum_number) (with quantum number {{math|''L'' {{=}} 0, 1, 2,}} etc. replaced by [orbital letters](/source/spectroscopic_notation) S, P, D, etc.).

;Example: ''[positronium](/source/positronium)'' is a bound state [electron](/source/electron)-[positron](/source/positron) similar to a [hydrogen](/source/hydrogen) [atom](/source/atom). The names ''parapositronium'' and ''orthopositronium'' are given to the states <sup>1</sup>S<sub>0</sub> and <sup>3</sup>S<sub>1</sub>. 
* With {{math|''S'' {{=}} 0}}, the spins are anti-parallel, and with {{nobr| {{math|''S'' {{=}} 1}} }} they are parallel. This gives a multiplicity {{nobr|( {{math|2 ''S'' + 1}} )}} of 1 (anti-parallel) or 3 (parallel)
* The total [orbital angular momentum quantum number](/source/Azimuthal_quantum_number) is {{nobr| {{math|''L'' {{=}} 0 }} }} ([spectroscopic](/source/spectroscopy) S orbital)
* [Total angular momentum quantum number](/source/Total_angular_momentum_quantum_number) is {{nobr| {{math|''J'' {{=}} 0 }} or  {{math|1}} }}
* C parity {{nobr| {{math|''η''<sub>{{small|''C''}}</sub> {{=}} (−1)<sup>''L'' + ''S''</sup> {{=}} +1 }} or {{math|−1}} ,}} depending on {{mvar|L}} and {{mvar|S}}. Since charge parity is preserved, annihilation of these states in [photon](/source/photon)s {{nobr|( {{math|''η''<sub>{{small|''C''}}</sub>(''γ'') {{=}} &minus;1 }} ) }} must be:
:{|
|-
| Orbital:  
| <sup>1</sup>S<sub>0</sub> → {{mvar| γ + γ }}
| <sup>3</sup>S<sub>1</sub> → {{mvar| γ + γ + γ }}
|-
| {{math|''η<sub>{{small|C}}</sub>''}} :  
|style="padding-right:2em;"| +1 = (−1) × (−1) 
| −1 = (−1) × (−1) × (−1)
|}

==Experimental tests of C-parity conservation==
* <math> \pi^0 \rightarrow 3 \gamma</math>: The neutral pion, <math>\pi^0</math>, is observed to decay to two photons, {{nobr| {{mvar|γ+γ}} .}} We can infer that the pion therefore has <math>\ \eta_C=(-1)^2 = 1\ ,</math> but each additional {{mvar|γ}} introduces a factor of {{math|&minus;1}} to the overall C-parity of the pion. The decay to {{math|3''γ''}} would violate C parity conservation. A search for this decay was conducted<ref>{{cite journal |last1=MacDonough |first1=J. |display-authors=etal |year=1988 |title=New searches for the ''C''-noninvariant decay {{math|π<sup>0</sup>→3''γ''}} and the rare decay {{math|π<sup>0</sup>→4γ}} |journal=[Physical Review D](/source/Physical_Review_D) |volume=38 |issue=7 |pages=2121–2128 |bibcode = 1988PhRvD..38.2121M |doi = 10.1103/PhysRevD.38.2121 |pmid=9959363}}</ref> using pions created in the reaction <math>\ \pi^{-} + p \rightarrow \pi^0 + n ~.</math>

*<math>\eta \rightarrow \pi^{+} \pi^{-} \pi^{0}</math>:<ref>{{cite journal|last1=Gormley |first1=M. |display-authors=etal |year=1968 |title=Experimental test of {{mvar|C}} invariance in {{math|η → π<sup>+</sup>π<sup>−</sup>π<sup>0</sup>}} |journal=[Physical Review Letters](/source/Physical_Review_Letters) |volume=21 |issue=6 |page=402 |bibcode = 1968PhRvL..21..402G |doi = 10.1103/PhysRevLett.21.402 }}</ref> Decay of the [eta meson](/source/eta_meson).

* <math>p \bar{p}</math> annihilations<ref>{{cite journal |last1=Baltay |first1=C. |display-authors=etal |year=1965 |title=Mössbauer effect in K<sup>40</sup> using an accelerator |journal=[Physical Review Letters](/source/Physical_Review_Letters) |volume=14 |issue=15 |page=591 |bibcode = 1965PhRvL..14..591R |doi = 10.1103/PhysRevLett.14.591 }}</ref>

==See also==
*[G-parity](/source/G-parity)

==References==
<references />

<!-- footer templates -->
{{C, P and T}}

<!-- categories -->
Category:Quantum mechanics
Category:Quantum field theory

---
Adapted from the Wikipedia article [C parity](https://en.wikipedia.org/wiki/C_parity) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/C_parity?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.