# Calculating Fermi Coupling Constant (muon)

• tryingtolearn1
In summary: Actually I figured out my issue. I was include an extra factor of ##c^5##. Inside the square root it should have been ##.105^5## instead of ##(.105*(3*10^8)^2)^5##.
tryingtolearn1
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?

tryingtolearn1 said:
##m_\mu## is the mass of a muon which is ##\approx 0.105##GeV
Consider why the mass of the muon is expressed in units of energy.

TSny said:
Consider why the mass of the muon is expressed in units of energy.
Ops I meant the mass of the muon is ##\approx 0.105GeV/c^2## but even with that unit correction I still get a value that is way off.

tryingtolearn1 said:
Ops I meant the mass of the muon is ##\approx 0.105GeV/c^2## but even with that unit correction I still get a value that is way off.
Did you change anything in your first attempt shown below?
tryingtolearn1 said:
$$\frac{G_F}{(\hbar c)^3}=\sqrt{\frac{6.582*10^{-25}*192*\pi^3}{2.2*10^{-6}*(0.105*(3*10^8)^2)^5}}\approx 4.83157 × 10^{-48}$$
If so, can you show your new calculation? It would help if you show all units for all the numbers in your calculation. See if the units cancel to give the desired units for the answer.

tryingtolearn1 said:
the mass of the muon is ##\approx 0.105GeV/c^2##
Using this value for m, what is the value of mc2 in units of GeV?

TSny said:
Did you change anything in your first attempt shown below?

If so, can you show your new calculation? It would help if you show all units for all the numbers in your calculation. See if the units cancel to give the desired units for the answer.

My new calculation using only dimensional analysis is $$\sqrt{\frac{GeV\cdot s}{s}\cdot\frac{1}{[(\frac{GeV\cdot s}{c^2}) \cdot c^2]^5}}=GeV^{-2}\cdot s^{-5/2}$$ which has an extra ##s^{-5/2}##.

TSny said:
Using this value for m, what is the value of mc2 in units of GeV?

Actually I figured out my issue. I was include an extra factor of ##c^5##. Inside the square root it should have been ##.105^5## instead of ##(.105*(3*10^8)^2)^5##. Ty

## What is the Fermi coupling constant?

The Fermi coupling constant, also known as the Fermi constant or GF, is a fundamental constant in particle physics that describes the strength of the weak nuclear force. It is used to calculate the rate of decay for certain subatomic particles, such as muons.

## How is the Fermi coupling constant calculated?

The Fermi coupling constant is calculated by measuring the rate of decay for a known particle, such as a muon, and using the relationship GF = (1.1663787 ± 0.0000006) × 10-5 GeV-2 to determine its value. This relationship was first derived by Enrico Fermi in the 1930s.

## Why is the Fermi coupling constant important?

The Fermi coupling constant is important because it is a fundamental constant that describes the strength of the weak nuclear force, which is one of the four fundamental forces in the universe. It is also used in various calculations and theories in particle physics.

## What is the role of the muon in calculating the Fermi coupling constant?

The muon, a subatomic particle similar to an electron but with a larger mass, is used in experiments to measure the rate of decay and determine the value of the Fermi coupling constant. This is because muons are unstable and decay at a known rate, making them useful for these types of calculations.

## How has the value of the Fermi coupling constant changed over time?

The value of the Fermi coupling constant has been refined and updated over time as new experimental techniques and technologies have become available. The current value of GF was determined in the 1980s and has remained relatively constant since then.

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