In this paper, we investigate the relation between disk mass and mass accretion rate to constrain the mechanism of angular momentum transport in protoplanetary disks. We find a correlation between dust disk mass and mass accretion rate in Chamaeleon I with a slope that is close to linear, similar to the one recently identified in Lupus. We investigate the effect of stellar mass and find that the intrinsic scatter around the best-fit M_dust_-M_ and dM_acc/dt-M_ relations is uncorrelated. We simulate synthetic observations of an ensemble of evolving disks using a Monte Carlo approach and find that disks with a constant {alpha} viscosity can fit the observed relations between dust mass, mass accretion rate, and stellar mass but overpredict the strength of the correlation between disk mass and mass accretion rate when using standard initial conditions. We find two possible solutions. In the first one, the observed scatter in M_dust and dM_acc_/dt is not primordial, but arises from additional physical processes or uncertainties in estimating the disk gas mass. Most likely grain growth and radial drift affect the observable dust mass, while variability on large timescales affects the mass accretion rates. In the second scenario, the observed scatter is primordial, but disks have not evolved substantially at the age of Lupus and Chamaeleon I owing to a low viscosity or a large initial disk radius. More accurate estimates of the disk mass and gas disk sizes in a large sample of protoplanetary disks, through either direct observations of the gas or spatially resolved multiwavelength observations of the dust with ALMA, are needed to discriminate between both scenarios or to constrain alternative angular momentum transport mechanisms such as MHD disk winds.

Cone search capability for table J/ApJ/847/31/table1 (Stellar and disk properties for Chamaeleon I and Lupus)