This question has already been asked and answered in Most general Lagrangian in CFT in 0+1D.
However I am just partially convinced with the answer. The idea is to construct the most general Lagrangian that is scale and time invariant in 0+1D (so quantum mechanics). The proposed one is
$$L=\frac{1}{2}\dot{Q}^2-\frac{g}{2Q^2} \, ,\tag{1.11}$$
but why couldn’t we add terms that have $d$ derivatives and $n$ fields with $n = 2(d-1)$ such as $\dot{Q}^3Q$, $\dot{Q}^4Q^2$ and so on $[a]$, because these terms in principle would respect both time and scaling invariance?
In the answer given in the link above it says something that because we don’t want to modify the kinetic term we propose terms of the form $g_nO^n$ but I don’t see why the terms I have mentioned would modify the kinetic term.
$[a]$: with 3 derivatives and 4 fields, obviously I am counting the derivatives and fields separately since each carry dimensions, I could have also said 3 field derivatives and 1 field.