Kaon (K meson)

The kaon (K) is a subatomic particle belonging to the family of mesons, namely particles composed of a quark and an antiquark. It is intrinsically unstable, produced in high-energy strong interactions, and it decays on very short timescales through the weak interaction.

In contemporary particle physics, interest in the kaon arises chiefly from the fundamental phenomena it makes accessible to experimental investigation.

In particular, the study of kaons enables physicists to probe the fundamental symmetries of the laws of physics and to confront one of the most profound questions in cosmology: why the observable universe is dominated by matter rather than antimatter.

In this sense, the kaon provides a paradigmatic illustration of how, in particle physics, an extremely short-lived particle can yield remarkably stable and far-reaching insights into the deep structure of nature.

Note. Kaons are not "exotic" particles in any sensational sense. Their significance is conceptual: they played a decisive role in confirming the existence of quarks, in the operational introduction of the quantum number known as strangeness, and in the discovery of CP symmetry violation. For this reason, they continue to serve as a privileged testing ground for physics beyond the Standard Model.

Characteristics

Kaons belong to the class of hadrons, that is, particles subject to the strong interaction.

More specifically, they are mesons rather than baryons, they have spin 0, and they participate in all fundamental interactions, although their decays proceed exclusively through the weak interaction.

Their defining feature is the presence of the strange quark, which introduces a new quantum number known as strangeness.

Strangeness is a quantum number associated with the presence of strange quarks ( $ s $) or strange antiquarks ( $ \overline{s} $).

• a strange quark s carries strangeness -1
• a strange antiquark s̄ carries strangeness +1

In kaons, strangeness is not conserved in weak decays.

This seemingly technical feature proved to be of fundamental importance in the historical development of the quark model.

Types of kaons

There are four fundamental kaons, two charged and two neutral:

• $ k^+ $ (positive kaon)
• $ k^- $ (negative kaon)
• $ k^0 $ (neutral kaon)
• anti-K⁰ (neutral antikaon)

Charged kaons are distinct from their antiparticles.

Neutral kaons, by contrast, exhibit a more subtle and nontrivial behavior, which will be examined shortly.

Quark composition

Each kaon consists of a light quark (up or down) bound to a strange quark, or to the corresponding antiquark.

Kaon	Composition
\(K^{+}\)	\(u\,\bar{s}\)
\(K^{-}\)	\(s\,\bar{u}\)
\(K^{0}\)	\(d\,\bar{s}\)
\(\overline{K}^{0}\)	\(s\,\bar{d}\)

This internal structure accounts directly for both the electric charge and the value of the strangeness quantum number associated with each kaon.

Main physical properties

Charged and neutral kaons share many physical properties, although they are not identical.

With regard to mass, charged kaons have a mass of approximately \( 494 ,\text{MeV}/c^2 \), whereas neutral kaons are slightly heavier, with a mass close to \( 498 ,\text{MeV}/c^2 \). The difference is small but experimentally measurable, and it reflects their different quark compositions.

As for electric charge, charged kaons carry charge \( +1 \) or \( -1 \) in units of the elementary charge, while neutral kaons, as the name indicates, have zero electric charge.

Their mean lifetime provides another clear distinction between the two families.

• Charged kaons are relatively long-lived for subatomic particles, with a mean lifetime of the order of \( 1.2 \times 10^{-8},\text{s} \).
• Neutral kaons, by contrast, do not possess a single characteristic lifetime: they exist in two distinct physical states, one very short-lived and one longer-lived, which decay on dramatically different timescales. This peculiarity makes the neutral kaon system particularly rich and interesting from a theoretical standpoint.

Kaon decays

Kaons decay predominantly through the weak interaction, producing lighter particles. The most common decay products are pions ( $ \pi^+ , \pi^-, \pi^0 $) and, in many cases, leptons such as electrons ( $ e^{\pm} $ ), muons ( $ \mu^{\pm} $ ), and neutrinos ( $ \nu $ , $ \overline \nu $ ). 

A typical example is the decay of a charged kaon into a pion and a muon, accompanied by the emission of a muon neutrino ( $ \nu_\mu $ ):

\[ K^+ \rightarrow \pi^0 + \mu^+ + \nu_\mu \]

Equivalently, in the decay of the negative kaon, a muon antineutrino \( \bar{\nu}_{\mu} \) is emitted:

\[ K^- \rightarrow \pi^0 + \mu^- + \bar{\nu}_\mu \]

From a physical standpoint, the two processes are related by charge conjugation and constitute a standard example of a semileptonic decay mediated by the weak interaction, which allows the flavor of the quarks involved to change.

Note. The fact that kaons decay on timescales that are long when compared with processes governed by the strong interaction provides a clear indication of the nature of the underlying mechanism. The relative slowness of the decay is an unmistakable signature of the involvement of the weak interaction.

The special case of neutral kaons

Neutral kaons are produced as \( K^0 \) or as \( \overline{K}^0 \) in strong interactions, since strangeness is conserved in those processes.

The initial pure state, however, is not stable and evolves in time. This happens because the weak interaction violates strangeness conservation and allows continuous transitions between the particle and its antiparticle:

\[ K^0 \rightleftarrows \overline{K}^0 \]

In other words, the neutral kaon $ K^0 $ with strangeness +1 can transform into its antiparticle $ \bar K^0 $ with strangeness -1 through the weak interaction, and vice versa. Gell-Mann and Pais were the first to anticipate the existence of this phenomenon, which was subsequently confirmed experimentally.

This Feynman box diagram illustrates the transition from $ K^0 = (d \bar s ) $ to its antiparticle $ \bar K^0  = (\bar d s ) $.

At vertex A, the down quark ( $ d $ ) emits a virtual $ W^- $ boson and changes flavor, transforming into an up quark ( $ u $ ).

$$ d \to u + W^- $$

Note. The quark $ d $ carries electric charge $ - \frac{1}{3} $, while the quark $ u $ carries charge $ + \frac{2}{3} $. The emitted boson therefore has charge $ -1 $.  $$ \underbrace{d}_{ - \frac{1}{3} } \to \underbrace{u + W^-}_{+ \frac{2}{3} - 1 = - \frac{1}{3}} $$ The process is thus physically allowed because electric charge is conserved.

At vertex B, the strange antiquark ( $ \bar s $ ) absorbs the $ W^- $ boson and is converted into an up antiquark ( $ \bar u $ ).

$$ \bar s + W^- \to \bar u $$

At vertex C, the antiquark $ \bar u $ emits a second $ W^- $ boson and turns into an antiquark $ \bar d $.

$$ \bar u \to \bar d + W^- $$

At vertex D, the quark $ u $ absorbs the $ W^- $ boson and is transformed into a strange quark $ s $.

$$ u + W^- \to s $$

At the end of this weak-interaction process, the kaon $ K^0 = (d \bar s ) $ has been converted into its antiparticle $ \bar K^0 = ( \bar d s ) $. Both constituents have changed flavor through the exchange of two virtual bosons.

In some cases, the same mechanism can occur with a charm quark ( $ c $ ) or a top quark ( $ t $ ) in place of the up quark ( $ u $ ).

As a consequence, \( K^0 \) and \( \overline{K}^0 \) are not stationary states. They undergo particle - antiparticle oscillations and mix dynamically.

Note. In quantum physics, states that continuously transform into one another are not convenient for describing time evolution. It is preferable to work with states that retain their form in time. These are known as eigenstates.

In the neutral-kaon system, the eigenstates are not \( K^0 \) and \( \overline{K}^0 \). Instead, they are two linear combinations, one antisymmetric and one symmetric, denoted respectively by \( K_1 \) and \( K_2 \).

$$ K_1 = \frac{1}{\sqrt{2}} ( K^0 - \overline{K}^0 ) $$

$$ K_2 = \frac{1}{\sqrt{2}} ( K^0 + \overline{K}^0 ) $$

These are the physical states that actually decay. In other words, the kaon is produced as $ K^0 $ but decays as $ K_1 $ or $ K_2 $.

Note. For neutral kaons, the following relations hold under parity: $$ P \lvert K^0 \rangle = - \lvert K^0 \rangle, \qquad P \lvert \overline{K}^0 \rangle = - \lvert \overline{K}^0 \rangle $$ Under charge conjugation: $$ C \lvert K^0 \rangle = \lvert \overline{K}^0 \rangle, \qquad C \lvert \overline{K}^0 \rangle = \lvert K^0 \rangle $$ Combining charge conjugation and parity, one finds: $$ CP \lvert K^0 \rangle = - \lvert \overline{K}^0 \rangle $$ $$ CP \lvert \overline{K}^0 \rangle = - \lvert K^0 \rangle $$ Consider a generic superposition: $$ \lvert \psi \rangle = a \lvert K^0 \rangle + b \lvert \overline{K}^0 \rangle $$ We require this state to be an eigenstate of CP, that is: $$ CP \lvert \psi \rangle = \lambda \lvert \psi \rangle $$ with \( \lambda = \pm 1 \). First, consider the difference: $$ \lvert K_1 \rangle = \frac{1}{\sqrt{2}} \left( \lvert K^0 \rangle - \lvert \overline{K}^0 \rangle \right) $$ Applying CP and using $ CP \lvert K^0 \rangle = - \lvert \overline{K}^0 \rangle $ and $ CP \lvert \overline{K}^0 \rangle = - \lvert K^0 \rangle $, we obtain: $$ CP \lvert K_1 \rangle = \frac{1}{\sqrt{2}} \left( - \lvert \overline{K}^0 \rangle + \lvert K^0 \rangle \right) $$ Since quantum states are vectors, reordering the terms does not change the state. Therefore: $$ CP \lvert K_1 \rangle = + \lvert K_1 \rangle $$ Hence, $ K_1 $ is a CP eigenstate with eigenvalue $ +1 $.

Now consider the sum: $$ \lvert K_2 \rangle = \frac{1}{\sqrt{2}} \left( \lvert K^0 \rangle + \lvert \overline{K}^0 \rangle \right) $$ Applying CP gives: $$ CP \lvert K_2 \rangle = - \frac{1}{\sqrt{2}} \left( \lvert K^0 \rangle + \lvert \overline{K}^0 \rangle \right) $$ Therefore: $$ CP \lvert K_2 \rangle = - \lvert K_2 \rangle $$ showing that $ K_2 $ is a CP eigenstate with eigenvalue $ -1 $. In summary, the state $ K_1 $ has $ CP = +1 $, whereas the state $ K_2 $ has $ CP = -1 $. These results hold only in the CP-conserving limit. When CP violation is taken into account, \( K_1 \) and \( K_2 \) are no longer physical states and must be replaced by the more general states \( K_S \) and \( K_L \), which will be discussed later.

Nevertheless, a striking phenomenon occurs: the identity of a neutral-kaon beam changes as it propagates.

After a beam of neutral kaons $ K^0 $ is produced, it initially evolves into an equal mixture of $ K_1 $ and $ K_2 $. After a short time, however, only the $ K_2 $ component remains. What has happened?

This behavior arises because the two states $ K_1 $ and $ K_2 $ have nearly identical masses but very different decay rates.

• $ K_1 $ (short-lived state)
• $ K_2 $ (long-lived state)

To understand the origin of the markedly different mean lifetimes of neutral kaons, one must consider CP symmetry, where $ C $ denotes charge conjugation, corresponding to the exchange of particle and antiparticle, and $ P $ denotes parity, corresponding to spatial inversion.

The state $ K_1 $ has $ CP = +1 $, whereas the state $ K_2 $ has $ CP = -1 $.

Here, \( CP = +1 \) means that under the combined charge-parity transformation the state is invariant. By contrast, \( CP = -1 \) means that under CP the state is reproduced with an overall minus sign.

$$ CP \lvert K_1 \rangle = + \lvert K_1 \rangle $$

$$ CP \lvert K_2 \rangle = - \lvert K_2 \rangle $$

If CP symmetry is conserved, a decay can occur only into final states with the same CP eigenvalue as the initial state.

Since kaons decay into pions, and pion final states respect CP symmetry, two decay channels are possible:

• $ K_1 $ has $ CP = +1 $ and can decay into two pions, since the two-pion system has $ CP(2\pi) = +1 $. The CP eigenvalues of the initial and final states therefore coincide.
• $ K_2 $ has $ CP = -1 $ and can decay into three pions, since the three-pion system has $ CP(3\pi) = -1 $. In this case as well, CP symmetry is conserved.

The decay into two pions is much faster than the decay into three pions, because it releases a larger amount of energy.

For this reason, the state $ K_1 $ has a very short mean lifetime, whereas the state $ K_2 $ has a much longer mean lifetime.

Consequently, in a neutral-kaon beam that initially contains equal amounts of $ K_1 $ and $ K_2 $, after a brief interval the $ K_1 $ component has almost completely disappeared due to decay, and only the $ K_2 $ component is observed.

CP symmetry violation

In the decays of neutral kaons, an especially important phenomenon appears, known as CP symmetry violation.

In principle, if CP symmetry were exact, the laws of physics would describe matter and antimatter in exactly the same way.

Experiments with neutral kaons show instead that this symmetry is not perfectly respected: matter and antimatter evolve in slightly different ways. The effect is extremely small, but it can be measured with high precision.

Note. This tiny asymmetry has dramatic consequences. If the laws of physics were perfectly symmetric between matter and antimatter, the two would have almost completely annihilated each other in the early universe, leaving radiation as the dominant remnant. CP violation, by contrast, allows a small excess of matter to survive, making the existence of the observable universe possible.

From a historical perspective, the first experimental evidence for CP violation was discovered precisely in the neutral-kaon system.

In 1964, an experiment performed by Cronin and Fitch showed that some neutral kaons in the long-lived state ( $ K_L $ ) occasionally decay into two pions rather than three, apparently violating CP symmetry.

$$ K_L \rightarrow 2\pi $$

This decay would be strictly forbidden if CP symmetry were exactly conserved.

Explanation. In the ideal CP-conserving limit, the long-lived state $ K_L $ would coincide with the state $ K_2 $, which has CP eigenvalue $ CP = -1 $. The two-pion final state, however, has $ CP = +1 $. The initial and final states therefore carry different CP eigenvalues. If CP symmetry were exact, such a decay could not occur.

And yet this seemingly “forbidden” decay $ K_L \rightarrow 2\pi $ was observed experimentally, raising a fundamental question.

The resolution is that the physical states observed in nature are not the pure CP eigenstates $ K_1 $ and $ K_2 $. Instead, they are two states with very different lifetimes that are linear combinations of $ K_1 $ and $ K_2 $.

• $ K_S $ (short-lived)
• $ K_L $ (long-lived)

In particular, the long-lived state $ K_L $ contains a very small admixture of $ K_1 $. As a result, it includes a tiny component with $ CP = +1 $.

$$ \vert K_L \rangle \approx \vert K_2 \rangle + \epsilon \vert K_1 \rangle $$

More precisely, including the normalization factor, one writes:

$$ \vert K_L \rangle \approx \frac{1}{ \sqrt{1+| \epsilon |^2} } \left( \vert K_2 \rangle + \epsilon \vert K_1 \rangle \right) $$

This extremely small $ CP = +1 $ component makes it possible, although very rarely, for the decay into two pions to occur through the weak interaction:

$$ K_L \rightarrow 2 \pi $$

This observation provided the first direct experimental proof of CP symmetry violation in weak interactions.

In summary, CP violation in neutral kaons is small but nonzero, and it appears in the fact that the physical states \( K_S \) and \( K_L \) do not coincide with the ideal CP eigenstates \( K_1 \) and \( K_2 \).

Note. CP violation in neutral kaons was the first experimentally observed example of CP symmetry breaking. Despite its small magnitude, it was a milestone discovery, opening the way to the study of matter - antimatter asymmetry and to modern cosmological questions about the origin of the universe. Later experiments showed that CP violation also occurs in other systems, most notably in B mesons, where the effect can be significantly larger.

Summary of kaon properties

The main properties of kaons are summarized in the following tables for quick reference.

Quantity	K⁺	K⁻	K0	anti-K0
Name	positive kaon	negative kaon	neutral kaon	neutral antikaon
Family	meson	meson	meson	meson
Spin	0	0	0	0
Electric charge	+1 e	-1 e	0	0
Mass	≈ 493.7 MeV/c²	≈ 493.7 MeV/c²	≈ 497.6 MeV/c²	≈ 497.6 MeV/c²
Quark composition	u s̄	s ū	d s̄	s d̄
Strangeness (S)	+1	-1	+1	-1
Baryon number (B)	0	0	0	0
Strong interaction	yes	yes	yes	yes
Electromagnetic interaction	yes	yes	no	no
Weak interaction	yes	yes	yes	yes
Gravitational interaction	yes	yes	yes	yes
Mean lifetime	≈ 1.24 × 10⁻⁸ s	≈ 1.24 × 10⁻⁸ s	see KS, KL	see KS, KL
Typical decay modes	π + μ + ν	π + μ + ν	ππ, πℓν	ππ, πℓν

The physical states of neutral kaons are summarized below.

State	Name	Mean lifetime	Main decay modes
KS	short-lived kaon	≈ 9 × 10-11 s	2 pions
KL	long-lived kaon	≈ 5 × 10-8 s	3 pions, πℓν

And so on.