CP Symmetry

CP symmetry (Charge-Parity Symmetry) is the transformation that combines spatial inversion with the exchange of a particle and its antiparticle, and in many physical processes it leaves the outcome of interactions unchanged.

We are accustomed to thinking of left and right as equivalent, in the sense that viewing nature in a mirror should not modify the laws that govern it.

This expectation feels natural, yet it is not a physical necessity, because in nature symmetries can be broken.

For instance, parity symmetry P is violated in weak interactions. If a weak process is viewed in a mirror, the resulting process is not physically equivalent to the original one. In this respect, nature distinguishes between left and right.

CP symmetry partly mitigates the breaking of left right symmetry, even though it is not an exact symmetry of nature.

How does it work?

CP symmetry is the combined action of parity (P) and charge conjugation (C).

I observe a process in a mirror by applying the parity operation (P).
I replace particles with their corresponding antiparticles through charge conjugation (C).

If, after these two transformations, the resulting process coincides with the original one, the process is said to satisfy CP symmetry.

If this condition is not met, the process is said to violate CP symmetry.

Thus, CP symmetry may still hold even in situations where nature does not respect left right symmetry.

Note. This does not imply that the world becomes fully symmetric. Rather, it shows that a more general symmetry exists which, in many processes, remains valid. In any case, the physics does not change: the observed phenomena remain the same.

A concrete example

I examine CP symmetry in pion decay.

\[ \pi^+ \rightarrow \mu^+ + \nu_\mu \]

For clarity, I analyze separately the effect of parity P and then of charge conjugation C, even though CP is a single combined transformation and the final result does not depend on the order in which the two operations are applied.

Note. In weak decays, the neutrino $ \nu_\mu $ is always left handed, while the antineutrino $ \bar\nu_\mu $ is always right handed. This is a well established experimental fact and is essential for understanding the example.

1] Parity (P)

Parity P inverts spatial coordinates: $ \vec r \to -\vec r $. As a consequence, the momentum also reverses direction:

$$ \vec p \to -\vec p $$

The spin $ \vec s $, by contrast, does not change sign because it is an axial vector.

Since helicity depends on the scalar product $ \vec s \cdot \vec p $, it changes sign under the action of P:

\[ \vec s \cdot \vec p \to \vec s \cdot (-\vec p)= -(\vec s \cdot \vec p) \]

As a result, a right handed particle becomes left handed, and vice versa.

When parity is applied to pion decay, the particle content remains unchanged:

\[ \pi^+ \rightarrow \mu^+ + \nu_\mu \]

However, the fermions now have inverted helicity. In particular, the neutrino that was initially left handed becomes right handed.

Because right handed neutrinos do not participate in the weak interaction, the resulting process is not physically realized.

This demonstrates that parity symmetry P is violated.

2] Charge conjugation (C)

To analyze the combined CP transformation, I now apply charge conjugation C to the process obtained after applying P.

\[ \pi^+ \rightarrow \mu^+ + \nu_\mu \]

The C operation exchanges particles with their antiparticles:

$ \pi^+ \leftrightarrow \pi^- $
$ \mu^+ \leftrightarrow \mu^- $
$ \nu_\mu \leftrightarrow \bar\nu_\mu $

After applying C, the resulting process is:

\[ \pi^- \rightarrow \mu^- + \bar\nu_\mu \]

Charge conjugation does not reverse helicity.

As a consequence, the antineutrino in the final state is right handed, as required by the weak interaction. The resulting process is therefore physically allowed and satisfies CP symmetry.

Indeed, it corresponds to the actual decay of the negative pion, which is experimentally observed.

This example shows that, although parity symmetry P and charge conjugation symmetry C are individually violated in weak interactions, the combined CP operation can transform one physically allowed process into another physically allowed process.

Note. The P operation alone reverses helicity, producing neutrinos with forbidden chirality, and therefore it is not a valid symmetry. The C operation alone exchanges particles with antiparticles but does not alter helicity, so it also fails on its own. The combined CP operation, by contrast, exchanges particles with antiparticles and simultaneously reverses helicity, restoring in many cases a process that belongs to the class of phenomena actually observed.

In this sense, CP symmetry can be satisfied even when the individual C and P symmetries are violated, without altering the physics of the observed phenomena.

And so on.