Charge Conjugation

Charge conjugation is the operation that replaces a particle with its antiparticle, reversing the sign of all additive internal quantum numbers associated with charge.

In physical terms, charge conjugation is a particularly simple transformation: a given particle is exchanged for its corresponding antiparticle. For instance:

an electron $ e^- $ becomes a positron $ e^+ $
a proton $ p $ becomes an antiproton $ \bar p $
a positive pion $ \pi^+ $ becomes a negative pion $ \pi^- $

This transformation is conventionally denoted by the letter C.

Note. The term "charge" can be misleading. Charge conjugation does not apply only to electrically charged particles, but to all particles. It acts on all additive internal quantum numbers by reversing their sign, such as electric charge, baryon number, and lepton number, while leaving spin, mass, energy, and momentum unchanged. For this reason, charge conjugation can also be applied to neutral particles. For example, the charge conjugate of a neutron $ n $ is an antineutron $ \bar n $.

What is it used for?

Charge conjugation C is a symmetry that is conserved in electromagnetic and strong interactions.

This conservation law allows one to determine which reactions and decay processes are permitted in nature and which are forbidden.

As will be discussed later, however, this symmetry on its own applies only to a restricted class of particles and is violated in the weak interaction. To overcome these limitations, one introduces G-parity.

Properties of charge conjugation
G-parity

Properties of charge conjugation

Charge conjugation is a multiplicative quantum number, unlike additive quantum numbers such as electric charge or baryon number. Its values do not add but multiply when composite systems are considered.

As in the case of parity $ P $, charge conjugation $ C $ is a quantum number that is conserved in strong and electromagnetic interactions.

Moreover, applying the charge conjugation operator twice returns the system to its original state.

$$ C \cdot C = C^2 = I $$

Here $ I $ denotes the identity operator, corresponding to the particle in its initial state prior to the application of C.

If the application of C produces a different particle, then C cannot be regarded as a quantum number, because one is no longer "measuring" an intrinsic property of the particle but transforming it into a different physical entity.

For this reason, only particles that coincide with their own antiparticle can be eigenstates of C and therefore possess a well-defined value of charge conjugation.

In such cases, the operation does not change the particle species, but may introduce an overall phase factor of ±1, which is identified with the value of the quantum number.

For example, the photon, the neutral pion, and certain neutral mesons have a well-defined C-parity, whereas an electron or a charged pion does not. The charge conjugation of the photon $ \gamma $ yields the same photon $ \gamma $. The same holds for the neutral pion $ \pi^0 $, whose charge conjugate is again $ \pi^0 $, that is, the same particle.

The charge conjugation quantum number of the photon is -1, because the electromagnetic field changes sign under charge conjugation even though the particle itself remains unchanged.

By contrast, the charge conjugation quantum number of the neutral pion $ \pi^0 $ is +1.

Particle	Symbol	Antiparticle	C eigenstate	Value of C
Photon	γ	γ	Yes	-1
Neutral pion	π⁰	π⁰	Yes	+1
Eta	η	η	Yes	+1
Eta prime	η′	η′	Yes	+1
Neutral rho	ρ⁰	ρ⁰	Yes	-1
Omega	ω	ω	Yes	-1
Phi	φ	φ	Yes	-1
J/ψ	ψ	ψ	Yes	-1
Positive pion	π⁺	π⁻	No	not defined
Electron	e⁻	e⁺	No	not defined
Muon	μ⁻	μ⁺	No	not defined
Proton	p	p̄	No	not defined
Neutron	n	n̄	No	not defined
Neutrino	ν	ν̄	No	not defined

When a particle has $ C = +1 $, it is invariant under charge conjugation. When it has $ C = -1 $, the particle itself remains unchanged, but its quantum state acquires an overall minus sign. This information is essential because it determines which reactions and decays are allowed or forbidden, since in electromagnetic and strong decays the value of $ C $ must be conserved.

Systems composed of a quark and an antiquark, such as neutral mesons, form an eigenstate of charge conjugation C with eigenvalue

$$ C = (-1)^{l+s} $$

where $ l $ is the orbital angular momentum and $ s $ is the total spin.

For example, pseudoscalar mesons with $ l = 0 $ and $ s = 0 $ have

\[ C = (-1)^{0+0} = +1 \]

Vector mesons with $ l = 0 $ and $ s = 1 $ have

\[ C = (-1)^{0+1} = -1 \]

Note. It is important to keep in mind that charge conjugation is well defined only for neutral mesons that coincide with their own antiparticle, and not for all mesons. This explains why $ C $ is not a universal quantum number for every meson, but only for a specific and well-defined class of them.

A practical example

A neutral pion undergoes electromagnetic decay into two photons.

$$ \pi^0 \to \gamma + \gamma $$

Because charge conjugation C is conserved in electromagnetic interactions, the value of the quantum number C must be identical before and after the decay:

$$ \underbrace{ \pi^0 }_{C=+1} \to \underbrace{ \gamma + \gamma }_{C=+1} $$

Since each photon carries charge conjugation $ C=-1 $,

$$ \underbrace{ \pi^0 }_{C=+1} \to \underbrace{ \gamma + \gamma }_{C=(-1)\cdot(-1)=+1} $$

The decay is therefore allowed, as it respects charge conjugation symmetry.

By contrast, a decay into three photons would have $ C=(-1)^3=-1 $, violating charge conjugation conservation, and is therefore forbidden.

What are the limitations of charge conjugation?

The central issue is identifying when the operation C yields a physically meaningful result.

A fundamental limitation immediately emerges. Most particles found in nature are not eigenstates of C.

Furthermore, the weak interaction violates C, and even within strong interactions charge conjugation, considered in isolation, is not always a good symmetry. As a result, C by itself has limited practical utility, since it applies only in a restricted set of situations.

To overcome this limitation, the concept of G-parity was introduced.

G-parity

G-parity is a symmetry obtained by combining charge conjugation $ C $ with a rotation by 180° about the second isospin axis $ I_2 $: $$ G = C R_2 $$ It is conserved only in strong interactions.

The effect of this rotation is to reverse the third component of isospin, mapping $ I_3 $ into $ -I_3 $.

If the combined action of the isospin rotation and charge conjugation $ C $, evaluated using the neutral member of the multiplet, reproduces the original particle, then the particle is said to be an eigenstate of G.

This construction makes it possible to assign a quantum number conserved in strong interactions even to particles that are not eigenstates of C.

In this way, G-parity becomes a genuinely useful symmetry, because unlike charge conjugation alone, it is not restricted to self-conjugate particles.

It should be stressed that G-parity is defined and employed only for non-strange mesons, that is, mesons that do not contain strange, charm, beauty, or top quarks and that participate in strong interactions.

It does not apply to baryons, leptons, or mesons carrying strangeness or heavier flavors.

Despite its restricted scope, G-parity is an extremely powerful tool for analyzing strong decays of non-strange mesons.

Example. Let us determine the G-parity of the charged pion $ \pi^+ $. The pion $ \pi^+ $ is not an eigenstate of charge conjugation $ C $, because its antiparticle is distinct, namely $ \pi^- $. For this reason, applying $ C $ directly to $ \pi^+ $ is not meaningful. This is precisely the difficulty that G-parity resolves. We first perform a rotation by 180° about the second isospin axis $ I_2 $, which reverses the sign of $ I_3 $, taking $ \pi^+ $ with $ I_3=+1 $ into $ \pi^- $ with $ I_3=-1 $: $$ \pi^+ \longrightarrow -\pi^- $$ We then apply charge conjugation $ C $, using the neutral pion $ \pi^0 $, since it is the only member of the pion multiplet with a well-defined charge conjugation: $$ C(\pi^0)=+1 $$ The action of $ C $ exchanges the charged pions, mapping $ \pi^- $ back into $ \pi^+ $: $$ \pi^- \longrightarrow \pi^+ $$ Combining the two steps gives: $$ \pi^+ \xrightarrow{R_2} -\pi^- \xrightarrow{C} -\pi^+ $$ The final result is the same initial particle, $ \pi^+ $, multiplied by -1. This shows that the charged pion $ \pi^+ $ is an eigenstate of G-parity with eigenvalue -1: \[ G(\pi^+)= -1 \] Since G-parity is a symmetry of the strong interaction, all pions $(\pi^+,\pi^0,\pi^-)$ share the same value: \[ G=-1 \]

In general, G-parity is computed using the formula

\[ G = (-1)^I C \]

where $ I $ is the isospin and $ C $ is the charge conjugation of the neutral member of the multiplet.

This expression takes a particularly simple form for states composed exclusively of pions, because each pion has isospin $ I=1 $ and G-parity is a multiplicative quantum number. Consequently, the factor $ (-1) $ multiplies itself once for each pion, leading to $ G=(-1)^n $, where $ n $ is the number of pions:

A state with an odd number $ n $ of pions has G = -1
A state with an even number $ n $ of pions has G = +1

This rule requires no dynamical calculation and is remarkably powerful.

Example 1
The $ \rho $ meson has isospin $ I=1 $ and charge conjugation $ C=-1 $. Its G-parity is therefore \[ G=(-1)^I \cdot C = (-1)^1 \cdot (-1)=+1 \] This implies that the $ \rho $ meson can decay into a two-pion final state, which has $ G=+1 $, but cannot decay into a three-pion final state, which would have $ G=-1 $. An allowed decay is thus \[ \underbrace{\rho}_{G=+1} \rightarrow \underbrace{\pi^+ + \pi^-}_{G=(-1)\cdot(-1)=+1} \] A decay into three pions is forbidden by the selection rules imposed by G-parity in strong interactions. By contrast, mesons such as ω or φ, which have G = -1, naturally decay into three pions, while decays into two pions are forbidden by the strong interaction. No Lagrangians or Feynman diagrams are required: the symmetry alone determines the outcome.

Example 2
The $ \omega $ meson has isospin $ I=0 $ and charge conjugation $ C=-1 $, and therefore G-parity $ G=-1 $: \[ G(\omega)=(-1)^I \cdot C = (-1)^0 \cdot (-1)=-1 \] In this case, a decay into three pions is allowed because it has $ G=-1 $: \[ \underbrace{\omega}_{G=-1} \longrightarrow \underbrace{\pi^+ + \pi^- + \pi^0}_{G=(-1)\cdot(-1)\cdot(-1)=-1} \] This decay is therefore permitted. By contrast, a decay of the $ \omega $ meson into two pions, which would have $ G=+1 $, is forbidden by G-parity conservation in strong interactions.

Below is a list of the G-parity values of representative states.

Particle	Symbol	Isospin I	C (neutral member)	G-parity	Notes
Pions	π⁺, π⁰, π^-	1	+1	-1
Rho	ρ	1	-1	+1
Omega	ω	0	-1	-1
Eta	η	0	+1	+1
Two-pion state	ππ	-	-	+1	G = (-1)²
Three-pion state	πππ	-	-	-1	G = (-1)³

The essential point is this: G-parity is not a theoretical complication, but an elegant extension of charge conjugation to realistic physical systems, made possible by exploiting the internal symmetry structure of particles.

It provides a clear and practical answer to a concrete question: how many pions can emerge from a strong decay using symmetry arguments alone, without appealing to microscopic dynamical details.

And so on.