The difference between a neutron's and muon's decay lifetime

Both neutrons and muons decay via the weak force/a weak interaction, with the neutron having a mean lifetime of around 900 seconds, and the muon having a lifetime of around 2.2$$\mu$$s, i.e., a factor of roughly $$4 \cdot10^8$$ between their lifetimes. Their "mass difference factor" $$m_n/m_\mu$$, however, is only around 9 (approx. 900 MeV/$$c^2$$ vs 100 MeV/$$c^2$$).

I would think that their lifetimes somehow are linked to their mass (and charge?) and the mass of their decay product, i.e., a proton and electron, respectively. But somehow all these numbers don't add up to the massive difference in decay lifetime. What's the mechanism/formula describing this discrepancy?

The Wikipedia article on muons, specifically the section linked here to decay, mentions a theoretical description of the decay width via "Fermi's golden rule [which] follows Sargent's law of fifth-power dependence on $$m_\mu$$", but I do not really understand that section.

• You could think the strong interaction binding neutron let it decays slower. – Turgon May 16 at 13:55
• This answer gives you the roughly 9 orders of magnitude ratio, in terms of $(m_p-m_n)/m_\mu \sim 10^{-2}$, which Sargent's rule raises to the fifth power, so a small slop between 10 and 9 orders of magnitude is forgivable. The 5th power of available energy rule is sheer dimensional analysis, given the square of Fermi's constant in the rate, which has units of energy. – Cosmas Zachos May 16 at 16:09

It is deceptive to look at the mass of the neutron. The decay is a decay of a down quark to an up quark by virtual emission of $$W^-$$ . That is the weak decay vertex to be compared. The neutron looks something like the proton here, except with the appropriate valence quarks :

In this link the difference with the muon is discussed:

Because the recoil parameter $$δ$$ is so small it follows that the momentum transfer dependence of all form factors may be neglected.This is also the reason why the neutron lifetime is so long.... greater by a factor of $$4.0x10^8$$ than he lifetime of the muon which is the next longest lived elementary particle.

Mainly it is the available energy for the decay ($$~1.3MeV$$), not the mass of the neutron that enters, and the kinematic considerations.

Here is a similar question answered here >

It gets worse:

Here is a talk of derivation of the neutron lifetime using lattice QCD, which is the theory used to calculate strong interactions, because this is an interplay between strong existence and weak decay.

It ain't simple.

Decay lifetimes depend on two factors. 1) the strength of the interaction ( Strong, weak, or electro-magnetic) and 2) the decay energetics (states and masses of the decaying participants. Fermi's Golden Rule does provide some guidance. As your example shows the energetics can still result in a large spread in the decay life times.