Griffiths Introduction to Elementary Particles · Problem 10.7

Weak Interactions — David J. Griffiths

Physics — QFT · neutron-decay-total-rate

Problem

Integrating the electron energy distribution (10.62) over

E

, obtain the total decay rate for the neutron (10.63):

\Gamma = \frac{1}{4\pi^3 \hbar}\left(\frac{g_w}{2M_W c^2}\right)^4 (m_e c^2)^5 \times \left[\frac{1}{15}(2a^4 - 9a^2 - 8)\sqrt{a^2-1} + a\ln(a + \sqrt{a^2-1})\right]

where

a \equiv (m_n - m_p)/m_e