Finding Range of Weak Interaction from Mass of Z Boson

[div4200]

Homework Statement

One of the mediators of the weak interactions is the Z boson, which has a mass of 91 GeV/c
2.

Use this information to find an approximate value for the range of the weak interaction.

Homework Equations

This is the part that I am having trouble with. I don't know where to look for information about this, and it doesn't seem to be in my book. All I ask is that someone point me to a resource where I can read about that topic. I would appreciate it greatly. Thanks in advance!

The Attempt at a Solution

(see above)

 

[fzero]

One way to approach the problem is just by dimensional analysis. What length scale corresponds to the mass? A better way, which gives essentially the same answer is to determine the corresponding Yukawa potential http://en.wikipedia.org/wiki/Yukawa_potential for Z-boson exchange.

 

[div4200]

Thanks. But I'm still somewhat confused. How can mass and length be related using dimensional analysis?

I also read the article about Yukawa potential, but I'm not sure how to use it. I would think that you would set V = 0 , but that yields the solution r = ∞, which cannot be right.

Thanks.

 

[div4200]

I just thought of something. What if I do,

E = mc² = hc/$$\lambda$$

And so:

$$\lambda$$ = h/(mc)

Plugging in the following values:

h = 4.13566733 * 10^-15 eV*s
c = 2.99792458 * 10⁸ m/s
m = 9.1 * 10⁹ eV/c²

I get

$$\lambda$$ $$\approx$$ 1.4 * 10^-16 m

Is that correct? Thanks a lot.

 

[fzero]

Thanks. But I'm still somewhat confused. How can mass and length be related using dimensional analysis?

Any mass is related to an energy by multiplying by $$c^2$$. Any energy is inversely related to a length by a factor of $$\hbar c$$. For instance, the relation between energy and wavelength of a photon is

$$\lambda = \frac{\hbar c}{E}.$$

I also read the article about Yukawa potential, but I'm not sure how to use it. I would think that you would set V = 0 , but that yields the solution r = ∞, which cannot be right.

Thanks.

You would be better off asking a question like: over what distance does the potential decrease by half? Be sure to avoid choosing $$r=0$$ or $$r=\infty$$ as reference points.

 

[div4200]

Ah I think I see now. So my method was correct, then?

And isn't your h-bar actually supposed to be h since E = hv?