# Help calculating the Fermi coupling constant from the muon lifetime

• martinhiggs
In summary, the conversation is about finding the Fermi Coupling Constant using the measurement of the muon lifetime. The formula used for this calculation is G_F = √ (192π³ / τm_μ), but there were some errors in the attempt at a solution, including missing factors of hbar and c, and incorrect units for the mass of the muon. After correcting these errors, a reasonable result was obtained.
martinhiggs

## Homework Statement

I have to find the Fermi Coupling Constant using my measurement of the muon lifetime. My measurement is 2.1786x10^-6s

## Homework Equations

$$G_F=\sqrt{\frac{192.pi^{3}}{\tau.m_{\mu}}}$$

## The Attempt at a Solution

I tried plugging in the numbers that I have and I get ~495,000ish which obviously cannot be right as the value of the Fermi Coupling Constant is:

1.166 37x10-5 GeV-2

Mass of the muon as 0.105GeV

I think I have to put in factors of hbar and c somewhere, or change units or something, but it's driving me crazy and I can't figure it out!

martinhiggs said:
Mass of the muon as 0.105GeV/c²

You are missing the 1/c² in the mass of the muon.

I get an even larger result if I divide by c^2...

Indeed, a few h's and c's are missing. I looked up the value for the http://physics.nist.gov/cgi-bin/cuu/Value?gf".

$$\frac{G_F}{(\hbar\,c)^3}=1.16637\cdot 10^{-5}\,(\rm{GeV})^{-2}$$

Are you using the following formula for the muon decay?

$$\Gamma_{\mu}=\frac{\hbar}{\tau_\mu}\approx\frac{G_F^2}{192\pi^3(\hbar c)^6}\cdot(m_\mu c^2)^5$$

Apart from SI/natural unit conversion troubles, the muon mass gets a different exponent compared to your formula.

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

$$\hbar\approx 6.582\cdot 10^{-25}\,\rm{GeV}\cdot\rm{s}$$

If you now insert your value for the muon life time, you should get a reasonable result.

Last edited by a moderator:
Ah yes, got it! Thank you for your help, I've spent all morning trying to work it out! :)

## 1. What is the Fermi coupling constant?

The Fermi coupling constant, also known as GF, is a fundamental physical constant that describes the strength of the weak nuclear force. It is used to calculate the probability of certain subatomic particles decaying.

## 2. How is the Fermi coupling constant related to the muon lifetime?

The Fermi coupling constant is related to the muon lifetime through the Fermi theory of beta decay. This theory states that the probability of a muon decaying into an electron, neutrino, and antineutrino is directly proportional to the square of the Fermi coupling constant.

## 3. What is the formula for calculating the Fermi coupling constant from the muon lifetime?

The formula for calculating the Fermi coupling constant from the muon lifetime is GF = 1.16637 x 10-5 / (τμ x (1 - ΔR)), where τμ is the muon lifetime in seconds and ΔR is the radiative correction factor.

## 4. What is the significance of calculating the Fermi coupling constant from the muon lifetime?

Calculating the Fermi coupling constant from the muon lifetime allows us to better understand the weak nuclear force and its role in the decay of subatomic particles. It also helps to validate the predictions of the Fermi theory of beta decay and can lead to new insights into the fundamental laws of physics.

## 5. Are there any challenges in calculating the Fermi coupling constant from the muon lifetime?

Yes, there are several challenges in calculating the Fermi coupling constant from the muon lifetime. One challenge is accurately measuring the muon lifetime, as it is a very short timescale. Another challenge is accounting for the radiative corrections, which are small but can have a significant impact on the final calculated value of the Fermi coupling constant.

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