Relation between decay probability and the energy of particle

Question

Is there any way to find the energy of a particle through its decay probability?

Answer 1

No.

If a particle can decay, it eventually will. The quantity of interest is its lifetime (or half-life, if you'd prefer; it is longer by a factor of about 1.4). The lifetime of an unstable particle is related to its "width" $\Gamma$ which is a quantity with units of energy:

$$ \tau = \frac{\hbar}{\Gamma} $$

So, if you know the lifetime of the particle, you can find its width. This does not, however, tell you the energy (or equivalently mass) of the particle.

There is a sense in which the mass and width are two components of the same quantity (which might be guessed from the fact that they have the same units):

$$ M + i \Gamma $$

This combined complex quantity shows up in quantum field theory. When computing the amplitude for a field excitation (i.e. particle) to travel a time interval $t$, it shows up in an exponential term

$$ e^{i(M + i \Gamma)t} = e^{iMt} e^{-\Gamma t} $$

So, the real component (mass) determines the "frequency" at which the field excitation oscillates (and thus the energy), while the imaginary component (decay width) dampens the excitation, giving it a finite lifetime. Measuring the lifetime of the particle only gives you access to $\Gamma$, and not the mass $M$.

Here is a list (wish I could make a table) that illustrates this point, somewhat:

• Top quark has lifetime $2.5$ to $5 \times 10^{-25}$ s, mass $173.07$ GeV
• W boson has a similar lifetime $3.2 \times 10^{-25}$ s, mass $80.4$ GeV
• Iron has a half-life of $ > 10^{22}$ years, mass about $52$ GeV
• The neutron has a lifetime of about $12$ minutes, mass about $1$ GeV
• The proton has an infinite lifetime, mass about $1$ GeV
• Tritium has a half-life of $12$ years, mass about $3$ GeV

It's evident that there is no pattern in which the mass of the particle is related directly to its lifetime.

Answer 2

It depends on which energy (total, kinetic?) and which decay probability (at rest, the effective measured one?) you are talking about, and also on other information you may be in the possession of.

There is an influence of speed (which is related to kinetic energy) on decay probability per unit of time, due to relativistic effects. So if you know the lifetime at rest, and the effective lifetime you measure, it is possible to deduce the speed the particles are travelling at. If you know the rest mass of the particle, it is possible to compute the kinetic energy given this speed (and then also the total energy if you want to).

Furthermore more massive particles decay more rapidly, all other things being equal. If you know all (or at least the most important) interactions and decay channels relevant for the particle under consideration, it is possible to compute the lifetime, and the result will be a function of the mass (i.e. the "rest energy"). For example, the top and charm quark interact and decay in the same way, and the top decays much more quickly due to its greater mass.