Abstract: A k dimensional rectifiable current is given by an oriented k dimensional rectifiable set M together with a positive integer-valued density function D. The mass of the current is then simply the integral of D over M (with respect to k dimensional Hausdorff measure). In 1960 Federer and Fleming proved the existence of a rectifiable current of least mass for a given boundary. For q in [0,1] , the q-mass of the current is the integral of D^q over M . The case q = 0 corresponds to "size" , introduced by Almgren as a way of using currents to model soap films. q-mass minimizing sequences may have unbounded mass and not converge as currents. Motivated by ideas from slicing currents, we introduce a (possibly infinite mass) generalization called a "scan". The q-mass minimizing scan obtained is concentrated on a rectifiable set. We discuss the interior, boundary, and free boundary partial regularity. Other variational problems from transport theory involve fractional power densities.