EFT's $\hbar$ counting at loop level

Question

In the Saclay Lectures on EFT, the author Falkowski claims under eq. (2.29) on p. 22:

Note that $\hbar$ counting still works at the loop-level. To see this, one should take into account that, when $\hbar$ is retrieved in the action, the loop expansion parameter is really $\frac{\hbar}{16\pi^2}$.

$\hbar$ counting at tree level is fine: the Lagrangian $\mathcal{L}$ is $\hbar^1$, the kinetic-mass term is classical so it doesn't have any powers of $\hbar$, so the fields are all $\hbar^{1/2}$ and from that information, one can find the $\hbar$-power of all the coupling constants. But how do I find the $\hbar$-power of a loop parameter, which is basically a number? My only idea was that if $$S=S^{\text{tree}}+\hbar S^{\text{1-loop}}+\hbar^2 S^{\text{2-loop}}+\dots$$ then $S^{\text{1-loop}}$ coefficient must have dimension $\hbar^{1}$, but I don't know if the above expansion is correct (and if it is, I don't know why).

Update: this answer seems to include the desired information, so I'll try to gather something from there. I'd still appreciate an answer explicitly explaining the fact that the "loop parameter" is $\hbar^1$.

Answer 1

• Ref. [1] is considering

  1. an UV action $S_{UV}[\phi,H]$ with no explicit $\hbar$-dependence with a cubic interaction $g_3\phi^2 H$ and a quartic interaction of the form $g_4\phi^2 H^2$ and $g_4\phi^4$; and

  2. an EFT action $S_{EFT}[\phi]$ with no explicit $\hbar$-dependence and where the gauge-field $H$ has been integrated out.

• Next we perform a dimensional analysis, where we focus on implicit and explicit powers of $\hbar$.

• Reminder: Recall the standard $\hbar$/loop expansion, where the order of $\hbar$ in a diagram is the number $L$ of loops, cf. e.g. my linked Phys.SE answer. Also recall there is one $\hbar$ for each internal propagator, none for each external leg, and one $\hbar^{-1}$ for each vertex.

• Now in Ref. 1, the Planck constant $\hbar$ is summoned to play a dual role: Aside from identifying number of loops $L$ as usual, it will now also be tied to the dimension of fields & coupling constants, cf. p. 6-7 in Ref. 1.

• To simplify the analysis/avoid clutter, we set the dimension of length $$[\text{length}]~=~1$$ to be trivial. Moreover, we see that the mass parameter $m$ in the kinetic terms have dimension of inverse length, so $$[m]~=~[\text{length}^{-1}]~=~1$$

• Since $[S_{UV}]=\hbar$, we see from the kinetic terms that $$[\phi]~=~\hbar^{1/2}~=~[H],$$ so the coupling constants $$[g_3]~=~\hbar^{-1/2}$$ and $$[g_4]~=\hbar^{-1}~=~[g_3]^2.$$

• Since $[S_{EFT}]=\hbar$ then for an $n$-vertex term in $S_{EFT}$ the coupling constant $$[c_n]~=~\hbar^{1-n/2}~=~\hbar^L[g_3]^{n-2+2L},$$ where $L$ is the number of gauge loops to create the effective $n$-vertex term.

• Example: The mass-term is a 2-vertex, cf. eq. (2.29), so the coupling constants appears in the powers $g_3^2$, $g_4$ in the $L=1$ terms.

References:

1. A. Falkowski, Saclay Lectures on EFT, 2017.