Getting natural units right in particle physics

[QuantumSkippy]

Hi guys. I have been a mathematical physicist and have recently been taking great interest in theoretical particle physics. With some effort I can do calculations in the International System of Units (SI) with complete success if I focus very carefully.

With the particle physics natural units which have h/2π = c = 1 , I have recently found myself rather embarrassed.

I want to use g = e/sinθ_W and g' = e/cosθ_W in calculations and when I came to convert e from coulombs to natural units, I really did not know where to start. These natural units seem to be really easy for you guys.

Read somewhere that converting from these natural units is pretty easy; the prescription seems to be to know what is the particular physical quantity that you are to calculate, such as cross-section, and you simply write out what should be its dimensions in the usual MLT format and then say 'what's missing, given what's in this formula that will give it the right dimensions?' and you simply multiply by the appropriate power of h/2π and c (powers may be negative or fractional), and it is not possible to go wrong with this prescription. This seems fair enough. First of all, have I got this right?

Secondly, would be really grateful if someone could please show me how to calculate g = e/sinθ and g' = e/cosθ in these natural units as well as the formula for the mass of the W which is M_W = (e²/4)√2G_Fsin²θ_W also in these natural units.

Thanks for your help. I think your reply will really help anyone else just starting out on natural units.

 

[Bill_K]

The fine structure constant e²/ħc ≈ 1/137 is dimensionless. Therefore in units where ħ = c = 1, e ≈ sqrt(1/137).

 

[QuantumSkippy]

Thanks for the help. There is still a problem:- This contradicts the formula given in Kane, 'Modern Elementary Particle Physics' page 112 which gives the formula v = 2M_W/g' ≈ 246 where v is the vacuum expectation value of the Higgs. With M_W = 80.4 this gives g' ≈ 0.65, while the value e ≈ √(1/137) gives g' = e/cosθ_W ≈ 0.097 which is very much smaller, where cosθ_W = 0.88.

This matter is easily solved if someone knows where the published values of g and g' are. In giving tables of such numbers on the internet, they seem to only want to publish the Weinberg angle for the reasons of sheer elegance.

This is just the sort of problem which I have been encountering. Further assistance here would be greatly appreciated! Surely others would have had this problem.

 

[fzero]

There's a couple of things giving you problems here. The fine-structure constant in natural units with the Lorenz-Heaviside units is

$$\alpha = \frac{e^2}{4\pi}.$$

One could omit the $4\pi$ and have Gaussian units, but this is not the usual convention in HEP. Also, $\alpha$ depends on the mass scale, so while

$$\alpha(m_e)\sim \frac{1}{137},$$

at the electroweak transition,

$$\alpha(M_Z) \sim \frac{1}{128}$$

is more accurate.

Second, your $g'$ should be

$$g' =\frac{e}{\sin\theta_W}.$$ With these corrections, you should be able to find Kane's result.

 

[QuantumSkippy]

Thanks very much for this. I will do the calculations.

But this also raises an important natural philosophical question:

As we are dealing only with units and the very well known values of c and h, the conversion should be possible without reference to the fine structure constant at all, but only to the values of c and h, which are used to go from SI units (presumably) to the natural units. How to get the value of e then, without reference to the fine structure constant?

Would be very grateful for your help here.

 

[fzero]

Thanks very much for this. I will do the calculations.

But this also raises an important natural philosophical question:

As we are dealing only with units and the very well known values of c and h, the conversion should be possible without reference to the fine structure constant at all, but only to the values of c and h, which are used to go from SI units (presumably) to the natural units. How to get the value of e then, without reference to the fine structure constant?

Would be very grateful for your help here.

Oh, it's simple, the fine structure constant is really

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

so in addition to choosing $\hbar=c=1$, we have to decide what units to choose for $\epsilon_0$. In Lorenz-Heaviside units, $\epsilon_0=1$, whereas in Gaussian units, $\epsilon_0=1/(4\pi)$.

Now if you wanted to not refer to the fine structure constant for some reason, you would have to translate the units of $\epsilon_0$ to figure out how to translate Coulombs into natural units.

 

[QuantumSkippy]

Thanks for that.

Finally, what would be the form of g and g' expressed in the usual SI units (Lorentz-Heaviside), and what would be their dimensions in MLT format?

Really appreciate your help.

 

[fzero]

Thanks for that.

Finally, what would be the form of g and g' expressed in the usual SI units (Lorentz-Heaviside), and what would be their dimensions in MLT format?

Really appreciate your help.

There shouldn't be any new factors introduced in SI units. The units are set to be those of $e$, so they're Coulombs.