Re: How can we find the mass of weak gauge bosons(W&Z) by knowing their range?

Dear Sanam,

One of the rules of Quantum Mechanics, called the Heisenberg
Uncertainty Principle, states that it is impossible to know precisely
the energy of a state of a system and simultaneously the lifetime of
the state.  Mathematically, the Principle looks like

               dE * dt >= hbar/2

where dE is the uncertainty in the energy, dt is the uncertainty in the
time, and hbar is Planck's constant divided by (2 pi).

The maximum range for a virtual gauge boson (force carrying particle),
or any exchanged particle, can be estimated as the speed of light
multiplied by the uncertainty in the time

            Range = c * dt

Using the Uncertainty Principle above, and the minimum energy
uncertainty m*c^2 for a particle of mass m, the maximum range is
on the order of

            Range = hbar / (2 m * c)
   
Now you can solve for the mass of the exchanged particle in terms of
the range.

For a more rigorous derivation of the relationship between
exchanged particle mass and range of the force, look in any
Quantum Mechanics text for "Yukawa potential".  Japanese physicist
Hideki Yukawa estimated the mass of the pion as 100 MeV/c^2
given a typical nuclear size of about one femtometer.

You should also be aware that there are much more accurate ways
of determining the W and Z masses than working backwards from
their estimated ranges.

--Dr. Randall J. Scalise    http://www.phys.psu.edu/~scalise/