How is the effective (or renormalized) coupling constant at a given momentum scale scale measured?(adsbygoogle = window.adsbygoogle || []).push({});

If one wants a definition which makes it easy to measure I would think it would be natural to use the LSZ formula which connects the measurable Feynman amplitude with the amputated greens function. For example in phi-four theory

$$i \mathcal{M} = Z^{(n/2)} \Gamma^{(n)}_{\text{on shell}}$$

where ##\mathcal{M}## is the feynman amplitude and ##\Gamma^{(n)}## is the vertex function and is identical to the amputated greens function. Since ##M## should be easy to measure (and in some sense indicates the strength of the interaction) I would guess natural definition of the effective coupling would be that for example ##g = Z^{(2)} \Gamma^{(4)}(p_i = \mu)## at some momentum scale ##\mu##. This is exactly the definition of the effective coupling used in conventional renormalization of phi-four theory.

This would indeed correspond to ##i\mathcal{M}## for a scalar particle scattering if ##\Gamma## was evaluated on shell in the definition of ##g##. However for an arbitrary scale ##\mu## this does not longer correspond to the LSZ formula. And I no longer see how the effective coupling is easily measured, nor how it is connected to the 'strength' of the interaction.

A similar thing also happens in QED where

$$ie \equiv Z_2 \sqrt{Z_3}ie_0\Gamma^{\mu}|_{\text{on shell}} = \frac{Z_2 \sqrt{Z_3}}{Z_1} ie_0 = i \sqrt{Z_3}e_0$$

and ##Z_3## is related to the vacuum polarization through ##Z_3^{-1} = 1 - \Pi(q^2 = 0)##. Now at an arbitrary scale I've seen the definition of ##\alpha = e^2/(4\pi)##

$$\alpha(\mu) = \alpha_0 (1 + \Pi(q^2 = - \mu^2))$$

where the photon with momentum ##q## is no longer on shell. Now I do not longer see how this is easily measured or how it connects to the strength of the interaction.

So my question is how is the effective coupling measured in a theory (for example QED) at an arbitrary momentum scale and how does the definition of the coupling then correspond to something we can call the 'strength' of the interaction?

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# Measuring effective (renormalized) coupling constants

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