  Mass determinations from gravitational lensing shear and the higher order
estimator flexion are both subject to the mass sheet degeneracy. Mass sheet
degeneracy refers to a transformation that leaves the reduced shear and flexion
invariant. In general, this transformation can be approximated by the addition
of a constant surface mass density sheet. We propose a new technique to break
the mass sheet degeneracy. The method uses mass moments of the shear or flexion
fields in combination with convergence information derived from number counts
which exploit the magnification bias. The difference between the measured mass
moments provides an estimator for the magnitude of the additive constant that
is the mass-sheet. For demonstrating this, we derive relations that hold true
in general for n-th order moments and show how they can be employed effectively
to break the degeneracy. We investigate the detectability of this degeneracy
parameter from our method and find that the degeneracy parameter can be
feasibly determined from stacked galaxy-galaxy lensing data and cluster lensing
data. Furthermore, we compare the signal-to-noise ratios of convergence
information from number counts with shear and flexion for SIS and NFW models.
We find that the combination of shear and flexion performs best on galaxy and
cluster scales and the convergence information can therefore be used to break
the mass sheet degeneracy without quality loss in the mass reconstruction. In
summary, there is power in the combination of shear, flexion, convergence and
their higher order moments. With the anticipated wealth of lensing data from
upcoming and future satellite missions - EUCLID and WFIRST - this technique
will be feasible.
