- #1

tryingtolearn1

- 58

- 5

- Homework Statement
- Fermi Coupling Constant (muon)

- Relevant Equations
- ##\Rightarrow \frac{G_F}{(\hbar c)^3}=\sqrt{\frac{\hbar}{\tau_\mu}\cdot\frac{192\pi^3}{(m_\mu c^2)^5}}##

I am trying to determine the Fermi Coupling Constant which is measured to be ##1.1663787 *10^{-5}\text{Ge}V^{-2}##. The formula for Fermi is ##\frac{G_F}{(\hbar c)^3}=\sqrt{\frac{\hbar}{\tau_\mu}\cdot\frac{192\pi^3}{(m_\mu c^2)^5}},## where ##m_\mu## is the mass of a muon which is ##\approx 0.105##GeV, ##c## is the speed of light ##\approx 3*10^8##m/s, ##\hbar## is reduced plank constant which is ##\hbar\approx 6.582\cdot 10^{-25}\,\rm{GeV}\cdot\rm{s}## and ##\tau_\mu## is the mean decay time of a muon which is ##\approx 2.2*10^{-6} s##. Now plugging all these values into the formula gives $$\frac{G_F}{(\hbar c)^3}=\sqrt{\frac{\hbar}{\tau_\mu}\cdot\frac{192\pi^3}{(m_\mu c^2)^5}}$$

$$=\sqrt{\frac{6.582*10^{-25}*192*\pi^3}{2.2*10^{-6}*(0.105*(3*10^8)^2)^5}}\approx 4.83157 × 10^{-48}$$ which is totally off from the expected value of ##1.1663787 *10^{-5}\text{Ge}V^{-2}##. Not sure what I am doing wrong?

$$=\sqrt{\frac{6.582*10^{-25}*192*\pi^3}{2.2*10^{-6}*(0.105*(3*10^8)^2)^5}}\approx 4.83157 × 10^{-48}$$ which is totally off from the expected value of ##1.1663787 *10^{-5}\text{Ge}V^{-2}##. Not sure what I am doing wrong?