# running of coupling constant as a function of distance?

There are many papers about the running of coupling strength as a function of momentum/energy scale, but are there any experimental papers about coupling strength as function of distance? Also, are there any good books about this topic?

oops ,really sorry i forgot to specify "experimental"paper

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what you're inquiring about is the beta function as a function of distance. This is what the position-space RG attempts to answer refs. These references should be of some help, I hope. – user346 Mar 16 '11 at 0:53
really thanks for reply,i'm looking for experimental paper which i forgot to mention – user1702 Mar 16 '11 at 7:50

I'm not sure if this is what you're asking about, but the energy scale has a one-to-one correspondence with the distance scale. For example, a particle of energy $E$ and mass $m$ has a wavelength

$$\lambda = \frac{h}{p} = \frac{hc}{\sqrt{E^2 - m^2c^4}}$$

This wavelength also represents the minimum scale of features that the particle is able to resolve, when it is used as a probe. So if you are analyzing a process that occurs on some characteristic distance scale $d$, you can find the energy associated with that scale by setting $\lambda = d$ and inverting the above equation,

$$E = \sqrt{m^2c^4 + \biggl(\frac{hc}{\lambda}\biggr)^2}$$

This tells you the approximate (minimum) energy of particles you will need to probe the process in question, and you can use that energy to evaluate the running coupling.

At high energies, or short distances (specifically, shorter than the Compton wavelength of the probe, $x \ll \frac{h}{mc}$), you can even neglect the mass and just use

$$E = \frac{hc}{\lambda}$$

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So basically write the beta function as a function of scale $\beta(E):=\beta(E[\lambda])$, right? I suppose you can do that and it is also a lot simpler than resorting to complicated RG arguments. But then again is such a substitution allowed where the RG is concerned? In any case, +1 for simplicity. – user346 Mar 16 '11 at 3:01
Well, I'm not familiar enough with the underlying renormalization procedures to say whether they support this method or not. But we do this sort of thing in QCD, so I'm pretty sure it's valid in at least some situations. – David Z Mar 16 '11 at 3:12
Really? Sweet. Another place where practical knowledge comes to the rescue! – user346 Mar 16 '11 at 7:17