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It is well known that couplings change depending on the scale $Q$ at they are measured. This effect is experimentally well documented:

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From a theoretical point of view, the running $\alpha_S(\mu)$ can be calculated using the renormalized group equations. However, it is regularly argued that the parameter $\mu$ appears here is a "fake parameter", has no meaning and can always be chosen at will. In particular, it is argued that we can use the fact that "physical observables must be independent of the fake parameter $\mu$ to figure out how the lagrangian parameters $m$ and $g$ must change with $\mu$."

The renormalized group equations that we find this way are then regularly used to calculate the running of parameters like $\alpha_S$:

enter image description here

How does this fit together? In particular, what's the exact relationship between the measured energy dependence of parameters and the running of parameters with the fake parameter $\mu$?

Or formulated differently, if the renormalization group equations encode the dependence on the "fake parameter" $\mu$, which equations describe the dependence on the physical energy scale $Q$?

(I'm aware that there are lots of related questions, but none of them seemed to answer this question unambiguously.)

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  • $\begingroup$ I've asked a few questions about this, which I guess you saw -- is there something specifically wrong with the answers to them? They satisfied me back when I had the same question. $\endgroup$ – knzhou Jul 10 at 14:18
  • $\begingroup$ Yes, I think I've read all of them. My problem is why it's regularly emphasized that $\mu$ is an arbitrary parameter and, on the other hand, why then people are interested in how parameters run with $\mu$. The reason seems to be that parameters are indeed scale dependent as can be measured in experiments. But the physical scale $Q$ at which a parameter is probed is not equal to $\mu$. So I'm trying to pin down the exact relationship between the two. $\endgroup$ – jak Jul 10 at 14:22
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/178754/50583 $\endgroup$ – ACuriousMind Jul 10 at 17:20
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if the renormalization group equations encode the dependence on the "fake parameter" $\mu$, which equations describe the dependence on the physical energy scale Q?

Let's look at a (oversimplified) case of running (where Q is the measurable probing energy (momentum) in a realistic scattering process), $$ g(Q) = g_{\mu} + ln(\frac{Q}{\mu}), $$ which is the solution to the differential equation $$ \frac{dg}{d(lnQ)} = 1, $$ with the initial condition $$ g|_{Q=\mu} = g_{\mu}. $$

Equivalently, the above logarithmic $g(Q)$ can be viewed as the solution to a different differential equation (the garden variety renormalization group equation) $$ \frac{dg}{d(ln\mu)} = -1, $$ with the initial condition $$ g|_{\mu=Q} = g_{\mu}. $$ The first differential equation (of $\frac{dg}{d(lnQ)}$) interprets the running as "g running with the probing/scattering energy (momentum) Q" and the second differential equation (of $\frac{dg}{d(ln\mu)}$, beta function in the usual parlance) interprets the running as "g running with renormalization scale $\mu$". Most text books take the second view, while I prefer the first. Nonetheless, they depict the same underlying physics.

The parameter $\mu$ is "fake" in the sense that, you can fix the solution of the differential equation of $\frac{dg}{d(lnQ)}$ with the initial condition set at $\mu'$, rather than at $\mu$, $$ g(Q) = g_{\mu'} + ln(\frac{Q}{\mu'}), $$ where $$ g|_{Q=\mu'} = g_{\mu'} = g_{\mu} + ln(\frac{\mu'}{\mu}). $$

In other words, the "fake parameter" $\mu$ is merely an arbitrary anchor point to frame the initial condition. Nothing more, nothing less!

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