How Do I Express Mass and e in Natural Units?

[QuantumSkippy]

Hi Everyone,

I have to calculate the coefficients of the massive Higgs Lagrangian in natural units, ħ = c = 1.

Do I assume firstly that, in these units the masses of the Higgs, W and Z are the usual,
125.09±0.21, 80.4, 91.2 respectively, and secondly that e has to have the value 2√(απ), where α is the Fine Structure Constant? This seems to make sense.

Please tell me if these values are correct and if not, what the values are and how they come about.

Your assistance will be greatly appreciated!

Have fun with Physics!

 

[king vitamin]

Setting $\hbar = c = 1$ doesn't get rid of all of your units - masses still have dimension, so you need to specify the units when you cite a mass.

Similarly, you should be clear about electromagnetic units. There are units where $e$ is dimensionless, but unless you specify that unit system, I can't tell you the relation between electric charge and the fine structure constant.

 

[QuantumSkippy]

Setting $\hbar = c = 1$ doesn't get rid of all of your units - masses still have dimension, so you need to specify the units when you cite a mass.

Similarly, you should be clear about electromagnetic units. There are units where $e$ is dimensionless, but unless you specify that unit system, I can't tell you the relation between electric charge and the fine structure constant.

Thanks for replying.

The unit system being used would be the one most commonly employed by particle physicists. I would be happy to know the values in the most commonly used system.

Cheers

 

[king vitamin]

In QFT textbooks one often uses energy for all units, and sets $\epsilon_0 = 1$. If temperature is ever used, also set $k_B = 1$.

So you should be sure to put the units on the numbers you gave to the particle masses you mentioned (they are all in $GeV$). The fine structure constant in SI units is

$$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c}$$

so clearly

$$e = \sqrt{4 \pi \alpha}$$

as you said.

 

[QuantumSkippy]

In QFT textbooks one often uses energy for all units, and sets $\epsilon_0 = 1$. If temperature is ever used, also set $k_B = 1$.

So you should be sure to put the units on the numbers you gave to the particle masses you mentioned (they are all in $GeV$). The fine structure constant in SI units is

$$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c}$$

so clearly

$$e = \sqrt{4 \pi \alpha}$$

as you said.

Thanks very much for your help. I have to give a talk and I wanted to make sure I got it right. Really need to do more calculations more often, so that I will be comfortable with the various systems of units. Thanks again! Cheers.