# Relative strength of strong and weak nuclear forces

Sorry to be obsessive, but I read today a statement that “the strong force is 10[sup]5[/sup] times as strong as the weak”[sup]1[/sup]. I seemed to remember a different figure being quoted for the relative strengths and, lo and behold, found another source that says “the strong force is 10[sup]13[/sup] times stronger than the weak”[sup]2[/sup]. This value for the ratio is also stated here.

After a few more checks of my “physics for morons” bookshelf, I found another cite to support the 10[sup]5[/sup] ratio, and even one claiming a ratio of 10[sup]7[/sup] (I can give the exact refs if you want).

So, what gives? What is the currently accepted ratio of the strengths of the strong to weak nuclear forces?

[sub]1 - Greene in The Elegant Universe
2 - Ferris in Coming of Age in the Milky Way[/sub]

As I understand it, it depends on how far apart the two particles are. IIRC, the strong force is very strong for close particles but drops of very quickly as they separate, while the weak is weaker overall but acts at longer distance. Where’s Chronos when you need him?

It’s a case of apples and oranges.

Knowing the relative strengths of any two of the fundamental forces is useful, but the comparison is not straightforward. The closest-to-home comparison would be gravity and electromagnetism. You might compare the electrostatic force between two electrons with the gravitation pull they feel from each other. But what if we choose protons? They’ll feel the same electrostatic force, but the gravitational force will be 3 million times higher than it was for the electrons.

Quoted fundamental force comparisons are usually made by characterizing a force’s strenght by a coupling constant – a dimensionless number that sort of determines how likely it is for something interaction-like to happen. The actual numerical values for the strong and weak coupling constants, though, depend on the renormalization method chosen (although there is a certain method called minimal subtraction that is widely used), the mass scale chosen (analogous to the EM/gravity case), and more. The strong force coupling constant, for instance, gets tiny at very high energies. However, typical values for the coupling constants are:

• a[sub]strong[/sub] = 1
• a[sub]weak[/sub] = 10[sup]-5[/sup]

These could easily vary by factors of 10 or more, though, depending on your chosen parameters and conventions.

It is important to note, though, that physical results depend on the square of the coupling constants, so if you want to speak of the relative strength of the force, it makes more sense to compare the squares of a[sub]s[/sub] and a[sub]w[/sub]. As long as your audience knows what you’re comparing, though, it doesn’t matter which you use.

If you want a hard number to quote for the relative strengths, then, I’d chose anything between 10[sup]-8[/sup] and 10[sup]-12[/sup].

Thanks for clearing this up.