In Perkins (2000, link to Google Books) the following statement is made about Sargent rule:

Discovered empirically in nuclear $\beta$-decay, but applicable to any three-body weak decay, ... (pg 405 4th ed)

To me this statement seem dubious; the author himself demonstrates it for $\beta$ decay only in the relativistic limit (pg 200). My question is therefore:

For a given weak, 3-body, decay: \[A\rightarrow B+C+D\] Under what conditions does Sargent rule hold?

For those who don't know - Sargent rule states that the Decay rate is proportional to the 5th power of the disintegration energy ($Q$-value)

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    $\begingroup$ Sargent's rule works best for a one-scale model, i.e. when the available energy quantifying the phase space of available states in Fermi's Golden rule is larger than product masses. For a weak decays, $\Gamma \propto G_F^2$. If the only dominant scale around is Q, dimensional analysis dictates $\Gamma \propto G_F^2 ~ Q^5$. $\endgroup$ May 16, 2019 at 22:07
  • $\begingroup$ PS the 5th power of the density of states in Fermi's Golden Rule is discussed here. $\endgroup$ May 16, 2019 at 22:18

1 Answer 1


Sargent's rule is derived in the four fermi interaction model of weak decays.

A more general derivation (within the fermi model) than the relativistic approximation can be seen here, 4th page .


The four fermi interaction is the low energy limit of the weak decay described through gauge boson exchanges, with only three track vertices allowed.


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