C parity

Inphysics,theC parityorcharge parityis amultiplicative quantum numberof some particles that describes their behavior under the symmetry operation ofcharge conjugation.

Charge conjugation changes the sign of all quantum charges (that is, additivequantum numbers), including theelectrical charge,baryon numberandlepton number,and the flavor chargesstrangeness,charm,bottomness,topnessandIsospin(I₃). In contrast, it doesn't affect themass,linear momentumorspinof a particle.

Formalism

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

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

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 isHermitianand therefore a physically observable quantity.

Eigenvalues

For the eigenstates of charge conjugation,

${\mathcal {C}}\,|\psi \rangle =\eta _{C}\,|{\psi }\rangle$.

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

${\mathcal {C}}^{2}|\psi \rangle =\eta _{C}{\mathcal {C}}|{\psi }\rangle =\eta _{C}^{2}|\psi \rangle =|\psi \rangle$

allowing only eigenvalues of$\eta _{C}=\pm 1$the so-calledC-parityorcharge parityof the particle.

Eigenstates

The above implies that for eigenstates,${\mathcal {C}}|\psi \rangle =|{\overline {\psi }}\rangle =\pm |\psi \rangle$.Since antiparticles and particles have charges of opposite sign, only states with all quantum charges equal to zero, such as thephotonand particle–antiparticle bound states like the neutral pion, η or positronium, are eigenstates of${\mathcal {C}}$.

Multiparticle systems

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

In a pair of boundmesonsthere is an additional component due to the orbital angular momentum. For example, in a bound state of twopions,π⁺π⁻with an orbitalangular momentumL,exchanging π⁺and π⁻inverts the relative position vector, which is identical to aparityoperation. Under this operation, the angular part of the spatial wave function contributes a phase factor of (−1)^L,whereLis theangular momentum quantum numberassociated withL.

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

With a two-fermionsystem, two extra factors appear: one 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,

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

Bound states can be described with thespectroscopic notation^2S+1L_J(seeterm symbol), whereSis the total spin quantum number,Lthe totalorbital momentum quantum numberandJthetotal angular momentum quantum number. Example: thepositroniumis a bound stateelectron-positronsimilar to ahydrogenatom.Theparapositroniumandorthopositroniumcorrespond to the states¹S₀and³S₁.

• WithS= 0 spins are anti-parallel, and withS= 1 they are parallel. This gives a multiplicity (2S+1) of 1 or 3, respectively
• The totalorbital angular momentum quantum numberisL= 0 (S, in spectroscopic notation)
• Total angular momentum quantum numberisJ= 0, 1
• C parity η_C= (−1)^L+S= +1, −1, respectively. Since charge parity is preserved, annihilation of these states inphotons(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

• $\pi ^{0}\rightarrow 3\gamma$: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 conductedusing pions created in the reaction$\pi ^{-}+p\rightarrow \pi ^{0}+n$.
• $\eta \rightarrow \pi ^{+}\pi ^{-}\pi ^{0}$:Decay of theEta meson.
• $p{\bar {p}}$annihilations

See also

• G-parity