Is four-dimensional RG flow a gradient flow of the a-function
In plain words
In four dimensions a certain count of degrees of freedom is proven to be lower at long distances than at short ones. A stronger claim, that the couplings always move steadily downhill along a height function given by this count, is checked only in perturbation theory.
Precise statement
Jack and Osborn (1990) showed perturbatively that the beta functions $\beta^{j}$ of a 4D renormalizable QFT satisfy $d_{i} a\sim = \chi_{ij}\beta^{j} + (d_{i} w_{j} - d_{j} w_{i})\beta^{j}$ for a function $a\sim(g)$ equal to the a-anomaly at fixed points. Determine whether, beyond perturbation theory, $\chi_{ij}$ is positive definite so that $a\sim$ decreases monotonically along every flow (strong a-theorem). The weak version $a_{\mathrm{UV}} > a_{\mathrm{IR}}$ between fixed points is proven (Komargodski and Schwimmer 2011).
What would settle it
A nonperturbative proof that $\chi_{ij}$ is positive definite in unitary 4D QFT, or a unitary flow along which $a\sim$ increases.
Status in the literature
$\chi_{ij}$ is positive at leading orders and $a\sim$ has been constructed to high loop order for general gauge theories (Jack and Poole 2015); the nonperturbative statement is open.