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Assume we have an Lagrangian $\mathcal{L}$ of matter field $\psi$ including interactive part mediated by gauge bosons $A_\mu$, eg structured "similary" to usual QED Lagrangian like

$$ D_\mu \psi^{\dagger}D_\mu \psi -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} -V(\psi) $$

where $F_{\mu \nu}=\partial_{\nu}A_\mu-\partial_{\mu}A_\nu$ and $ V(\psi) $ does not depend on derivative of $\psi$ .

Here there is no "mass term" $c \cdot A_{\mu}A^{\mu }$, and one often says that since these gauge bosons are massless in this theory, the " force/ interaction mediated by them is "long ranged", see eg in the the free available script "Global and Local symmetries" by Ling-Fong Li, but the're of course myriad scripts on QFT using this formulation.

Question: Can this phrase be expressed more precisely (in terms of mathematical notions $f(x) \sim g(x)$ for asymptotics ) how the mass of gauge bosons is related to the resulting characteristic range of carried interaction/ force?

user267839
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  • If you employ the Born approximation in the non-relativistic limit you get a direct relation through a Fourier transform between a scattering amplitude and a potential (https://en.wikipedia.org/wiki/Born_approximation). As such studying $2\to 2$ scattering amplitudes and taking a non-relativistic limit you can compare what one gets whether $A_\mu$ is massless or massive. See this thread for details on this approach https://physics.stackexchange.com/questions/142159/deriving-the-coulomb-force-equation-from-the-idea-of-virtual-photon-exchange – Gold Jul 06 '23 at 13:10
  • @Gold: what I'm wondering about is that Born approximation is a pertubative method. So is this connection between weight of gauge bosons and characteristic range of force they carry only for theories valid which are approachable by perturbation mehods? But what about QCD? – user267839 Jul 06 '23 at 13:57

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Of course. It's a bit complicated and model (the gauge group) dependent but ideally if the force carrying particle has mass $\mu$ then the potential it induces is:

$V(r) = \frac{-g^2}{4\pi} \frac{e^{-\mu r}}{r}$

The characteristic range than is $R \approx1/\mu$ as you can see by $e^{-r/R}$

The approach to obtain it is by evaluation of the gauge boson propagator, which is then fourier transformed.

LolloBoldo
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