# Why is the weak force 10^-7 times the electromagnetic force?

• I
In the case of two protons in the nucleus.
I've seen charts with that info, but I don't know how... How can I calculate that ratio?
I've looked everywhere, and I can't find anything...

Thanks

Staff Emeritus
2021 Award
I've seen charts with that info

Where?

mfb
Mentor
In current theories, we cannot calculate the fundamental strength of interactions - they are purely experimental results. Once you measure that strength in one system, however (e. g. the lifetime of muons tells you how strong the weak interaction is), you can calculate the strength in other systems, based on lengthy quantum field theory calculations. To get a reasonable estimate, you can calculate something called propagator, which depends on the (measured) mass of the bosons that mediate this interaction, the distance, and take the coupling constant, everything combined gives you an idea how strong the interaction will be.

tzukishiro
phyzguy
While you're at it, why is gravitation ~10^40 times weaker than electromagnetism? We don't know.

In current theories, we cannot calculate the fundamental strength of interactions - they are purely experimental results. Once you measure that strength in one system, however (e. g. the lifetime of muons tells you how strong the weak interaction is), you can calculate the strength in other systems, based on lengthy quantum field theory calculations. To get a reasonable estimate, you can calculate something called propagator, which depends on the (measured) mass of the bosons that mediate this interaction, the distance, and take the coupling constant, everything combined gives you an idea how strong the interaction will be.
Hmm, but then how did people come into the conclusion of my question? And yes, it's from that poster and a teacher also proposed the question in a class, I've been so confused and desperate looking into it hahah

nrqed
Homework Helper
Gold Member
Hmm, but then how did people come into the conclusion of my question? And yes, it's from that poster and a teacher also proposed the question in a class, I've been so confused and desperate looking into it hahah
Do you know a bit of particle physics? For the electromagnetic force, you may simply use Coulomb's law with a typical value for the size of a nucleus.
For the weak interaction, you may use Fermi's four-fermion approximation where the coupling constant is basically Fermi's constant ##G_F/2## (at least I think there is a factor of 2 there) whose value you may look up. The units will not be the same and some work is required there. I would personally simply use ##\alpha_{em}/q^2 ## for the electromagnetic force, with ##q## the momentum corresponding to the size of a nucleus. And then the answer will basically be ##G_F## divided by ##\alpha_{em}/q^2 ##.

Do you know a bit of particle physics? For the electromagnetic force, you may simply use Coulomb's law with a typical value for the size of a nucleus.
For the weak interaction, you may use Fermi's four-fermion approximation where the coupling constant is basically Fermi's constant ##G_F/2## (at least I think there is a factor of 2 there) whose value you may look up. The units will not be the same and some work is required there. I would personally simply use ##\alpha_{em}/q^2 ## for the electromagnetic force, with ##q## the momentum corresponding to the size of a nucleus. And then the answer will basically be ##G_F## divided by ##\alpha_{em}/q^2 ##.

In a very broad way, I'm not a physics student, but I took this course because we need to have a certain amount of courses outside our own field to graduate.
We don't really go into the math and hard deep physics of it, it's very superficial.

Gf divided by alpha/q^2 would give me said ratio? That kinda lost me, sorry

mfb
Mentor
Here is my description from post #6 with numbers. Take two protons with a distance of 0.5 femtometer.

The photon is massless, therefore the electrostatic force follows an inverse square law. The force between the protons is ##F=\frac{q^2}{4 \pi \epsilon_0} \frac{1}{r^2} = 920 N##.

The Z boson also leads to a force, but the Z has a nonzero mass. This leads to a Yukawa potential. The scaling constant k there can be taken as the ratio (speed of light)/(planck constant). Multiplying this with the Z mass and 0.5 femtometer gives -36, and ##e^{-36} \approx 2.3 \cdot 10^{-16}##. The prefactor we get from this page and messing around a bit, giving 1.1*107 N. Multiply with the term calculated before and we get ##2.5\cdot 10^{-9} N##. That is 11.5 orders of magnitude below the force of the electromagnetic interaction, not 7, but my choice of a distance was completely arbitrary (all choices are arbitrary, the poster author could have chosen a different one), and I ignored other prefactors as well.

vanhees71 and nrqed