# Power-series expansion in coupling/Planck constant

feynman-diagramsperturbation-theoryphysical constantsquantum-field-theoryscattering

By using Feynman rules of the interacting theory, one obtains the scattering amplitude

$$\mathcal{M} = \mathcal{M}_0 + \mathcal{M}_1 + \cdots = \sum^{\infty}_{i = 0}\mathcal{M}_i\tag{1}$$

Where $$\mathcal{M}_i$$ denotes $$i$$-level loop scattering amplitude. If $$i = 0$$, then obtains tree-level contribution to the process. However, it seems that we can power-series expand in coupling constant $$g$$ or Planck constant $$\hbar$$:

$$\mathcal{M}(g) = \sum^{\infty}_{i = 0}g^i\mathcal{M}_i\tag{2}$$

$$\mathcal{M}(\hbar) = \sum^{\infty}_{i = 0}\hbar^i\mathcal{M}_i\tag{3}$$

What exactly is the difference between $$(2)$$ and $$(3)$$, if there is?

From I understand, power-series expanding around a small parameter $$\epsilon$$ makes higher-order contributions (generally) less impactful:

$${\displaystyle \mathcal{M}(\varepsilon) = \mathcal{M}_{0}+\varepsilon^{1}\mathcal{M}_{1}+\varepsilon ^{2}\mathcal{M}_{2}+\varepsilon^{3}\mathcal{M}_{3}+\cdots}\approx \mathcal{M}_0+\epsilon\mathcal{M}_1\quad (\epsilon\ll 1)$$

Diagramatically, in a theory like scalar $$\phi^3$$ or QED with a photon and an electron where you have one vertex, an expansion in the coupling constant is an expansion in the number of vertices in the diagram.
Meanwhile, an expansion in $$\hbar$$ is an expansion in terms of the number of loops in the diagram.