# What can be inferred about the mass of a particle from its decay width?

If we know the decay width of some particle, the mean life of the particle can be found via Heisenberg's uncertainty principle $$\Delta E\Delta t\sim\hbar$$

Is there a way to find the range of the particle's actual mass from this information?

And how can we tell from the decay width and mean lifetime whether the particle decays by strong, weak or electromagnetic interactions?

An unstable particle of mass $$M$$ and mean lifetime $$\tau$$ has width $$\Gamma$$ that such that:

$$\Gamma\tau \approx \hbar$$

The observed mass distribution follows that Breit Wigner resonance probability:

$$f(E) =\frac k {(E^2-M^2)^2+M^2\Gamma^2}$$

with

$$k = \frac{2\sqrt 2 M\Gamma\gamma}{\pi\sqrt{M^2+\gamma}}$$

where:

$$\gamma = \sqrt{M^2(M^2+\Gamma^2)}$$

You have to have $$M\gg\Gamma$$ if you want to actually see the resonance, otherwise, there is no "bump" over background. (It's like a very over-damped oscillator, which will never oscillates..it just relaxes slowly back to 0).

Weak interactions are slow, e.g. charged pions have around $$10^{-8}\,$$s, while the neutral pion decays electromagnetically in $$10^{-15}$$ seconds.

One doesn't usually talk about widths until the strong interaction, e.g. the $$\Delta$$ resonances, which decay in $$5\times 10^{-24}\,s$$ for a width of $$130\,{\rm MeV/c^2}$$...about one tenth the mass.

Of course there are many exceptions to these rules. The $$W^{\pm}$$ decay weakly with a width of $$2\, {\rm GeV/c^2}$$, but here you are already working at the $$W$$-mass scale, where the weak interaction is no longer weak.

Nuclear decays span many orders of magnitude, with many long lived alpha (strong decay) emitters, and shorted lived beta (weak) emitters. There are many other factors affecting decay rate besides the coupling strength. Nevertheless, most electromagnetic metastable decays are fast.

Meanwhile, glow-in-the-dark plastics are electromagnetic molecular transitions that are macroscopically long...but again there are other factors like "forbidden" transitions and spin flips coming into play.

• I guess the typical decay times for strong, weak and EM interactions are determined experimentally? Or can the laws of quantum mechanics specifically show that one type of interaction leads to decays in a certain time frame? Commented Sep 9, 2021 at 18:17

τ = ħ/Γ is the expression relating the lifetime and the width. But the width is linked to the variance of the mass M, so it hardly matters, in principle, where the center of the mass distribution is -- see WP.

You may go to the PDG and see the

• $$f_0(500)$$ with M~500MeV ~ Γ;
• $$J/\psi (3100)$$ with M~3.1GeV but Γ~93kev;
• Υ(9460) with M~ 9.4GeV but Γ~18GeV;
• and the Z boson with M~ 80GeV and Γ~2.5GeV.

You thus see that, generically, the dispositive feature of them is the availability of decay channels (normally going up with mass); and the type of the interaction involved: strong interactions enhance Γ, EM are in the middle, and Weak ones suppress it (narrow it). But there is no substitute to perusing the PDG and comparing apples with apples.