Why does the range of an inter. vary inv. w/ the mass of its mediator?

[MWBratton]

Greetings PF! This is my first post...

I keep reading over and over again either "... so and so mediator is massive. Therefore, its interaction is of finite range" or the converse or "... mediator is massless... its interaction is of infinite range..."

Makes sense intuitively (inertia?) However, I am wondering if the true "proof" could be actually somewhat subtle in some way, especially since, apparently, it is impossible to have long range gluon fields even though the gluon is massless

(Follows from asserting that every physical particle is "colorless," so no long range gluon fields are generated to begin with!)

 

[The_Duck]

If you have an electric charge, you can imagine field lines radiating out from it in every direction. The fields lines never end, so if you draw a sphere of radius R around the charge, the same number of field lines pass through the sphere no matter what R is. The surface area o the sphere is $\pi R^2$, so the number of field lines per unit area, which is proportional to the field strength, falls off as $1/R^2$. That's why the electric field and other "long-range" forces fall off like $1/R^2$.

Now, it turns out that if the intermediate particle has a mass, then one thing changes: Mathematically we now have to solve the differential equation

$(\nabla^2 + m^2)\phi = 0$

(m is the mass of the intermediate particle) instead of the equation we had for $m = 0$:

$\nabla^2 \phi = 0$.

Intuitively, the difference between these equations is that in solutions to the massive equation, the field lines are no longer never-ending but tend to terminate after a characteristic distance $1/m$. (Some make it farther, and some not as far, but the typical length of a field line is now $1/m$). Again, the field strength is proportional to the density of field lines, so this picture tells us that the field strength is only appreciable within a distance of order $1/m$.

You correctly note that the situation with the strong force is more subtle than this picture suggests. The strong force has a finite range but massless intermediate particles. This is possible due to confinement. In fact there is no mathematical proof of confinement, but we can for example observe it in numerical simulations of the strong force (or in absence of a long-range strong force!).

Maybe a way to imagine what happens with the strong force is that field lines sprout new field lines after a characteristic distance of about 1 femtometer, and these new field lines in turn sprout new field lines, and so on. These endlessly branching field lines produce an extremely powerful force that pulls in an opposite color charge to cancel the color charge that is the source of the field lines. The opposite color charge gets pulled into a distance of about 1 femtometer, and if you look at the system from a larger distance than this the the fields of the two opposite color charges cancel and so there is not much field beyond a distance of 1 femtometer. But by now this is a crude analogy to a complicated quantum mechanical process.

 

[MWBratton]

Cool, thanks! Wow, fast response time