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Why is the Compton wavelength $\lambda_c=\frac{\hbar}{mc}$ used as a sensible measure for the range of an interaction, where $m$ is the mass of the corresponding mediator?

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For massive force carriers, one finds (in natural units) an exponential dependence $\mathrm{e}^{-mr}$ that prevents long-range forces with massive mediators. Restoring SI units, one sees that the compton wavelength is the length at which the damping is exactly $\mathrm{e}^{-1}$.

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    $\begingroup$ @Roopam: For a derivation of ACuriousMind's answer, see D. Tong's quantum field theory notes. $\endgroup$ – JamalS Oct 4 '14 at 18:17

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