Why is it that there's a precise relationship between the mass of a mediator particle and its range? Because mass shouldn't directly affect decay time, right?
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What directly affects the ""decay time"" of a virtual boson is not exactly its mass but its energy.
Recall that virtual particles are but interpretations of certain terms in the calculation of the scattering amplitude of a process, with no proper physical meaning; they are said to be "off-shell", because they don't obey momentum-energy dispersion relations. However, they can be assigned a mass, a momentum,or an energy, and in the diagrammatic description of the interaction, a vague notion of "travel time" and range.
The reasoning goes something like this, at least for massive bosons: suppose the virtual particle were a "real" particle. Then the duration of its travel (from being emitted to being absorbed) will allow an uncertainty in its energy satisfying $\Delta E \Delta t \sim\hbar/2$. This means that if you assign this particle an energy (in its rest frame ) of $mc^2$ and assume that the uncertainty is at most of this scale, then you'll have $\Delta t \sim\frac{\hbar}{2mc^2}$. Then, at most, this particle will move at a velocity of $c$, so its displacement in this process will be $\Delta x \sim \frac{\hbar}{2mc}$. Thus we take this as a vague metric of "range" of the associated interaction and put $$R= \frac{\hbar}{mc}$$
I stress the use of the various quotation marks and refer you to this answer.
Lourenco Entrudo
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