Particle Decay

Particle decay is the spontaneous process through which an unstable particle transforms into lighter, more stable ones.

In general, particles naturally evolve toward greater stability, unless blocked by a conservation law.

This transformation can proceed through different decay channels (strong, weak, or electromagnetic interactions), each defined by its own probability and characteristic lifetime.

The process is shaped by two key principles:

• Tendency toward lower-energy states
  An unstable particle will decay into a configuration with lower total mass. It’s a natural relaxation process, similar to a falling object under gravity.
  

  Note. In classical mechanics, a balanced object eventually drops to the ground, settling into equilibrium. In chemistry, an unstable atom rearranges itself into a more stable state. In particle physics, decay serves the same role: a particle spontaneously converts into more stable products, with a certain probability and a characteristic mean lifetime.

• Conservation laws
  Not every energetically possible decay is allowed. The transformation must respect the fundamental conservation laws - including energy and momentum, electric charge, baryon and lepton number, quark flavor (conserved in strong and electromagnetic interactions but not always in the weak), spin, parity, and others.

Decay is therefore not random: it follows precise transition probabilities and is measured through the particle’s mean lifetime.

It’s important to stress that truly stable particles are extremely rare: almost all others eventually decay.

In particle physics, decay is the norm; stability is the exception.

Example. A textbook case is the decay of a free neutron. Outside the atomic nucleus, a neutron is unstable and decays via the weak interaction into a proton, an electron, and an electron antineutrino. This is known as beta-minus decay: $$ n \to p + e^- + \bar{\nu}_e $$ Through this process, the neutron shifts into a lower-energy, more stable configuration: the proton.

Why is the proton stable? The proton remains stable because it is the lightest baryon: there is no lighter baryon it could decay into without violating baryon number. Any decay into non-baryonic particles would break baryon number conservation, which holds in all known interactions. For this reason, within the Standard Model the proton is considered stable. That said, some speculative theories predict proton decay over unimaginably long timescales, but no such decay has ever been observed.

Decay Channels

A particle can decay through different channels, depending on the fundamental interaction that governs the process.

• Strong interaction
  Responsible for the decay of hadronic resonances, which transform almost instantaneously. This is the fastest channel, with typical lifetimes of about $10^{-23}$ s. For example, the baryon $\Delta^{++}$ decays into a proton and a positive pion: $$ \Delta^{++} \to p^+ + \pi^- $$
• Weak interaction
  Enables processes that change quark flavor or involve leptons. It is the slowest channel, with lifetimes ranging from $10^{-13}$ s up to several minutes. A well-known case is the negative muon $\mu^-$, which decays into an electron, an electron antineutrino, and a muon neutrino: $$ \mu^- \to e^- + \bar{\nu}_e + \nu_{\mu} $$
• Electromagnetic interaction
  Involves photon emission and occurs when electric charge or magnetic moments play a decisive role. It is slower than strong decays, with lifetimes around $10^{-16}$ s. For instance, the neutral pion $\pi^0$ decays into two photons: $$ \pi^0 \to \gamma + \gamma $$

Identifying the decay channel

The decay channel can often be identified by examining the final products:

Photons point to an electromagnetic process. Neutrinos indicate a weak decay. If neither photons nor neutrinos appear, the process is usually strong.

In some cases, however, different interactions can produce the same initial and final states, making it less straightforward to determine which interaction is responsible.

One effective method is to study lifetimes, since each interaction operates on a characteristic timescale:

• strong interaction ≈ $10^{-23}$ s
• electromagnetic interaction ≈ $10^{-16}$ s
• weak interaction $\gtrsim 10^{-13}$ s

Strong decays are extremely short-lived. Electromagnetic decays are slower by several orders of magnitude, while weak decays are the slowest, with lifetimes ranging from fractions of a microsecond to several minutes, as in the case of the neutron.

Another criterion is statistical: each decay channel has a certain probability of occurring, expressed through branching ratios.

Branching Ratios

Branching ratios describe the relative likelihood that an unstable particle will decay through one channel rather than another.

In other words, a particle may have several possible decay modes, each with its own probability, and the total probability must add up to 100%.

Formally, the branching ratio for a given channel is defined as

$$ BR_i = \frac{\Gamma_i}{\Gamma_{\text{tot}}} $$

where $\Gamma_i$ is the partial decay width of that channel and $\Gamma_{\text{tot}}$ is the total width (the inverse of the mean lifetime).

For example, the particle $\Delta^0$ decays into a proton and a negative pion: $$ \Delta^0 \;\to\; p + \pi^- $$ In principle, this decay could proceed via either the strong or the weak interaction. In practice, the $\Delta^0$ decays almost exclusively through the strong force, with a characteristic lifetime of about $10^{-23}$ s. The weak channel, mediated by a $W^-$ boson, is theoretically possible but so suppressed that it is unobservable: the strong channel completely dominates.

Mean Lifetime of a Particle

The mean lifetime of a particle depends not only on the interaction driving the decay (strong, electromagnetic, or weak), but also on the mass difference between the initial state and the final products.

• If the mass difference is large, the decay is fast, since more energy is available and more final states are accessible.
• If the mass difference is small, the process is slowed down: the particle can only decay gradually.

The mean lifetime is therefore determined by both the interaction involved and the mass gap.

This rule generally holds, with only rare exceptions.

Example. A free neutron has a mass only slightly greater than that of a proton plus an electron. As a result, beta-minus decay has an unusually long mean lifetime - about 15 minutes - compared with the typical weak-interaction scale of $\sim 10^{-13}$ s: $$ n \;\to\; p + e^- + \bar{\nu}_e $$ By contrast, $\Delta$ resonances have a large mass gap relative to their decay products (a nucleon plus a pion) and therefore decay almost instantly via the strong interaction, with characteristic lifetimes on the order of $10^{-23}$ s: $$ \Delta^0 \;\to\; p + \pi^- $$

And so forth.