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From Wikipedia Coupling Constants, using QED as an example. I realise that the one-loop beta function in quantum chromodynamics is negative.

If a beta function is positive, the corresponding coupling increases with increasing energy. An example is quantum electrodynamics (QED), where one finds by using perturbation theory that the beta function is positive. In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127. Moreover, the perturbative beta function tells us that the coupling continues to increase, and QED becomes strongly coupled at high energy. In fact the coupling apparently becomes infinite at some finite energy.

My questions are based on pure curiosity (and a total lack of experimental experience, so my apologies if this combination displays naivety on my part).

Have we tested coupling constants at the lowest temperature/energy to confirm a reduction at the far end of the energy scale from the LHC?

It may be that low temperature experiments have to take any changes in values as a matter of routine, to correspond to theoretical predictions, so "yes, of course!!!" is an perfectly acceptable answer.

If a reduction has been observed in the value of any arbitrary constant, at these extremely low temperatures, can we compare this to conditions if the "heat death of the universe" scenario is true and predict what effects will occur as the temperature drops?

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The fine structure constant $\alpha\approx\frac1{137}$ appears in the Coulomb force between fundamental charges: $$ \alpha\hbar c = e^2/4\pi\epsilon_0, \quad\text{so}\quad |E_\text{Coulomb}| = \frac{e^2}{4\pi\epsilon_0} \frac1r = \frac{\alpha\hbar c}{r} $$ Quantum electrodynamics is pretty well tested down into the radio frequencies, with techniques like magnetic resonance, and radio frequencies correspond to micro-eV photons. This is zero temperature as compared to the LHC.

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    $\begingroup$ Compared to the LHC, yes, but it's extremely hot compared to the rest mass limit of the electromagnetic field, so we actually don't know how photons really behave when they "feel really low". :-) In my opinion that's an extremely important question, since the future of the universe may depend heavily on the effective photon-photon interaction at energies below $10^{-18} eV$. $\endgroup$ – CuriousOne Jul 1 '16 at 20:26
  • $\begingroup$ By "rest mass limit" are you referring to the possibility of a finite photon mass? That's really cold. However a quick and dirty calculation suggests that the number of CMB photons with energies in the regime you discuss is nonzero over parsec-sized volumes. Intriguing. $\endgroup$ – rob Jul 2 '16 at 0:01
  • $\begingroup$ The particle data book lists that as their best accepted limit for the rest mass, so I went with it. Wow... you think the thermal photon density is already non-zero at that energy (and they should overlap, right?)? Hmmm... I am amazed! That could actually be useful as a theoretical limit argument, couldn't it? $\endgroup$ – CuriousOne Jul 2 '16 at 0:25
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    $\begingroup$ Here is the argument. The CMB has locally about 400 photons/cm$^3$, or $10^{58}/\rm parsec^3$. The distribution of energies is like $\epsilon(\nu) d\nu \propto (h\nu)^3(e^{h\nu/kT}-1)^{-1} d\nu$, where most of the photons have $h\nu\approx kT \approx 250\rm\,\mu eV$. For the ratio $\epsilon_\text{cold}/\epsilon_\text{CMB}$ I find only $(h\nu_\text{cold}/kT)^2$. You can get the ratio down to $(h\nu_\text{cold}/kT)^3$ if you naively integrate over all the relevant energies. But $(10^{-18}/10^{-4})^3 = 10^{-42}$ leaves a surprisingly large number of local photons below your cutoff energy. $\endgroup$ – rob Jul 2 '16 at 3:16
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    $\begingroup$ I was also expecting the density of photons that cold to be zero. The wavelength is about 10 AU. Later I'll figure out whether there are any bonkers-cold photons that overlap. $\endgroup$ – rob Jul 2 '16 at 3:26

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