\rangle }.
\label{ff}\end{gathered}$$ The approximation $f=1$ ignores the dispersion in the expectation value of the inverse of the FP operator, $$\Bigg\langle \left[ \frac{g}{1 - \lambda} \right]^2 \Bigg\rangle 
 \to  \Bigg\langle \frac{g}{1 - \lambda} \Bigg\rangle^2. \label{disp}$$ This approximation has been extensively used, [*e.g.*]{} in Refs. [@zw1; @zwan]. The three Dyson equations were analyzed in Ref. [@as1] where it was found that the solution of $\omega$ can be well approximated by the simple function $\omega(p) = \theta( \mu - p) \mu + \theta( p -\mu)
p$. The renormalization scale $\mu$, being the only parameter in the theory, can constrained by the long range part of the Coulomb kernel $\langle V_C \rangle \propto fd^2$. We will discuss this more in the subsection below. The low momentum, $p<\mu$ dependence of $d(p)$ and of the Coulomb potential $V_C(p) = f(p)d(p)^2$ is well approximated by a power-law, $$d(p) = d(\mu) \left( \frac{\mu}{p}\right)^\alpha, f(p) = f(\mu) 
\left( \frac{\mu}{p}\right)^\beta \label{df}$$ with $\alpha \sim 0.5$ and $\beta \sim 1$. The exponents are bounded by $ 2\alpha + \beta \le 2$ and the upper limit corresponds to the linearly rising confining potential. At large momentum, $p >> \mu$, as expected from asymptotic freedom, both $d$ and $f$ are proportional to $1/\log^\gamma(p)$, 