Values of the coupling constants

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  • #1
gerald V
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I am heavily confused about the coupling constants. I primarily refer to this source https://www.physicsmasterclasses.org/exercises/keyhole/en/projects/running_alphas.html , but other sources were not able to lift my confusion either.

First, in the figure, the scales appear as quite failed. The horizontal axis clearly is meant as logarithmic, so there should not be a zero-point. The vertical scale is even more misleading, but very likely it shall be logarithmic as well. The inverse strength of electromagnetism, which is 137, can be read off quite well. For the strength of the strong force, the article gives 0,119, so its inverse is 8,40. Trying to interprete the scale, one can argue that the figure tries to show this value as well. From other sources, I got for the inverse of the weak coupling strength 31,7, and this seems to be what the figure shows.

My first question: Are the numbers 137, 8,40 and 31,7 correct or what are the correct values and what would be a good reference to look them up?

My second question: Why does this and other sources say that the weak coupling constant is about 10000 times smaller than the strong coupling constant, whereas actually the factor only is about 4? Even, the weak force is stronger than the electromagnetic force by a factor of about 4 (the inverse square of the sine of the Weinberg angle).

My third question refers to the running. I had thought that the coupling constants run to unity, whereas the figure shows that their inverses roughly run to 100. But maybe I had misunderstood this „unification“ as „becoming unity“.

Thank you very much in advance for any answer.
 

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  • #2
Dr.AbeNikIanEdL
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I am also not sure I can make sense of the figure in your link. A somewhat more authoritative source might be the Review of Particle Physics by the Particle Data Group on gauge coupling unification, http://pdg.lbl.gov/2019/reviews/rpp2018-rev-guts.pdf#page8, which has essentially the same plot (but also a lot of stuff that might be beyond I-level). Note that the running is actually not determined for the electromagnetic and weak force (which are low energy phenomena) but for the coupling constants of the SU(2)xU(1) theory at high energies (see the second footnote on the first page of that pdf for how to calculate between ##\{e,\sin\Theta_W\}## and ##\{g1,g2\}##). What they refer to as ##\alpha_3## is indeed the strong coupling. Equations (115.4), (115.5) and (115.9) give the input values they use (and PDG is a good the reference for this). Note that they actually go the other way around and predict the third coupling constant from assuming gauge coupling unification, e.g. from the assumption that all couplings are equal at some high scale (I hope this also answers your third question, “gauge coupling unification” = “all couplings become the same coupling at some high scale”).

When one says that the weak force is 10000 times smaller, I assume what is meant is the effective coupling in a low energy regime, e.g. relevant for decays of hadrons, mesons and muons with masses between hundreds of MeV to a few GeV. The weakness of the weak force here comes from the fact that the corresponding bosons are massive, and interactions below the boson mass are suppressed with the inverse of that mass squared. The Z and W masses are ##\approx 80## and ##\approx 91.2## GeV, so close to ##100## GeV, for processes around ##1## GeV we have ##(1 \mathrm{GeV} / 100 \mathrm{GeV})^2## which gives the 10000 fold suppression.
 
  • #3
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The horizontal axis clearly is meant as logarithmic, so there should not be a zero-point. The vertical scale is even more misleading, but very likely it shall be logarithmic as well.
Both axes are logarithmic and none of them includes zero. If you would extend the x-axis more to the left then the couplings would show deviations from the nice linear plot.
My first question: Are the numbers 137, 8,40 and 31,7 correct or what are the correct values and what would be a good reference to look them up?
There are publications measuring them but they can be quite technical.
Why does this and other sources say that the weak coupling constant is about 10000 times smaller than the strong coupling constant, whereas actually the factor only is about 4?
I guess they mean the actual strength here, not the coupling constant. The weak interaction has massive bosons, that makes it very weak at lower energies.
My third question refers to the running. I had thought that the coupling constants run to unity, whereas the figure shows that their inverses roughly run to 100. But maybe I had misunderstood this „unification“ as „becoming unity“.
Unification as in "union", "together": They become the same value, which doesn't have to be 1.
 
  • #4
vanhees71
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Already the first sentence in the above quoted document makes me a bit puzzled. Particularly in the case of QCD you have to be specific about in which context the strong coupling ##\alpha_{\text{s}}=g^2/(4 \pi)## is measured. The quoted value seems to be a value measured in deep inelastic scattering at energy-transfer scales around the Z-boson mass of around 90 GeV.
 

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