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Unless I'm mistaken, physical masses that one goes out an measures in experiments corresponding to the location of poles in the propagator and such pole masses are independent of the energy scale of the experiment doing the measuring.

However, when one does a QFT calculation, the location of the pole mass is determined in terms of a running mass parameter (and other running couplings) which changes with energy scale. The scale dependence of the running mass parameter can have physical effects, but everything runs in such a way that the pole mass stays invariant with energy scale (again, unless I've misunderstood something).

My question, then, is there any analogue for the pole mass for the other running parameters? That is, I can naturally associate the running mass to an invariant quantity: the pole mass. Can I associate, for instance, a Yukawa coupling to some other measurement which doesn't change with energy?

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  • $\begingroup$ The RGEs in general are found from invariance of Green's functions (Callan–Symanzik equation), so maybe that's the answer $\endgroup$
    – innisfree
    Commented Apr 2, 2015 at 20:53
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    $\begingroup$ Re-reading the Q, there is possible confusion. The normalization scale at which a parameter is defined, $\mu$, is a priori unrelated to the energy scale, $E$, of e.g. the colliding particles the experiment. $\mu=E$ might be sensible, however, because it minimizes higher-order corrections. $\endgroup$
    – innisfree
    Commented Apr 2, 2015 at 20:59
  • $\begingroup$ See here: physics.stackexchange.com/questions/435266/renormalized-mass/… $\endgroup$
    – MadMax
    Commented Jan 12, 2021 at 21:16

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The difference is that "running" parameters are parameters we, humans, use to describe the real world, while things like the pole mass you mention are real-world things independent from any human (or non-human) description.

So, if you can let a particle fly around in a detector to see how fast it goes, you can measure its mass. Particles that don't "fly around" (such as quarks and gluons, which are confined inside hadrons) won't have such a clear-cut physical mass, but only a running mass.

When it comes to couplings, the situation is similar. The electromagnetic coupling goes to a constant at zero renormalization scale, so you could say that's the "physical" coupling, as that's the one governing static situations like the hydrogen atom and we can measure it (kind of) directly. In QCD, again, the situation is not as simple, and there is no consensus on how a "physical" (non-running) coupling could be defined.

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  • $\begingroup$ I'm a bit late to the party, but wouldn't the running charge be the "physical" coupling? It gives corrections to the Coulomb potential that are measurable (like the Lamb shift). $\endgroup$
    – TheQuantumMan
    Commented Jan 12, 2021 at 18:07
  • $\begingroup$ The Lamb shift can be measured, that’s true, but the relationship between the Lamb shift and the running coupling depends on the number of loops you include, the gauge fixing and the renormalization scheme. $\endgroup$
    – David Vercauteren
    Commented Mar 29, 2023 at 8:28
  • $\begingroup$ But say that, in principle, we could calculate the exact mathematical result giving us the Lamb shift. Which would then be the charge/coupling assosiated with it? Wouldn't it be a running coupling? $\endgroup$
    – TheQuantumMan
    Commented Mar 29, 2023 at 12:33

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