# EFT's $\hbar$ counting at loop level

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 $$\hbar^1$$.

• 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.