How do fermion and scalar masses run with energy? Is the difference in their running the core of the hierarchy problem? Do fermion and scalar running masses run in the same way?  Specifically, what are the qualitative differences in the mass beta functions for, say, scalar $\lambda\phi^4$ field theory and the fermion masses in QED? 
Next, is there some obvious intuition for whether the masses should get bigger or smaller in the UV? Since particle masses become more and more irrelevant in scattering processes at higher energies I'd think that the running masses would get smaller in the UV, but this might not be the correct intuition.
Finally, can one see that scalar masses are unnatural solely from their mass beta functions?  Do they just run very quickly so that they are sensitive to their values in the UV?
 A: Référence : Steven-Weinberg-The-Quantum-Theory-of-Fields-Vol-2-Modern-Applications
p.143 (18.4.15), (18.4.16)(18.4.17), p.144 (18.4.19) (18.4.20)
Scalar $(\phi^4$ theory):
$$\mu \frac{d}{d \mu} m^2_\phi(\mu) = (-2 + \frac{g_\mu}{16 \pi^2} + O(g_\mu^2))~m_\phi^2(\mu) \quad \quad \quad(18.4.17)$$
Electron :
$$\mu \frac{d}{d \mu} m^2_e(\mu) = (-1 - \frac{e_\mu^2}{2 \pi^2} + O(e_\mu^4))~m^2_e(\mu)  \quad \quad \quad(18.4.20)$$
Weinberg comment p 144:
"As long as the coupling remain small, the $m(\mu)$ decrease in magnitude. Our previous assumption that masses may be neglected as $\mu \rightarrow +\infty$ is justified if in fact $m(\mu)$ does vanish for $\mu \rightarrow +\infty$. However it is only known to be the case in asymptotically free theories, where the couplings all do remain small for $\mu \rightarrow +\infty$; in all other cases, this assumption is just an educated guess".
[EDIT] 
Ref (Ryder, Quantum field theory) pages 314,315, 337-339
From my point of view, The difference of sign relatively to the constant coupling comes, first, from a different expression of the self-energy $\Sigma$, which gives a correction to the inverse propagator (so a correction to $m^2$ for scalars - and a correction to $m$ for fermions), secondly, for the electron, we have to take care of the renormalisation of the field $\psi$ .
For $\Phi^4$ scalars, we have, in dimensional regularisation ($\epsilon = 4 -d$) - You may replace $\frac{1}{\epsilon}$ by $\log (\frac{\Lambda}{\mu})$ for Pauli-Villars regularisation if you want :
$$\Sigma_\phi = \frac{-g m^2}{16 \pi^2 \epsilon} +  finite ~ terms$$
For the electron in QED, we have : 
$$\Sigma_e = \frac{e^2m}{2 \pi^2 \epsilon} +  finite ~ terms$$
For the electron, we have to take care of the the renormalization of the field, but this does not change the global sign (the final result is a $\frac{3}{8}$ term instead of $\frac{1}{2}$). 
