Magnification bias is a gravitational lensing effect that is normally overlooked because it is considered sub-optimal in comparison with the lensing shear .
Thanks to the demonstrated optimal characteristics of the sub-millimetre galaxies ( SMGs ) for lensing analysis , in this work we were able to measure the magnification bias produced by a sample of QSOs acting as lenses , 0.2 < z < 1.0 , on the SMGs observed by Herschel at 1.2 < z < 4.0 .
Two different methodologies were successfully applied : the traditional cross-correlation function approach and the Davis-Peebles estimator through stacking technique .
The second one was found to be more robust for analysing the strong lensing regime ( < 20 - 30 arcsec in our case ) and provides the possibility to take into account the positional errors of the sources in our samples .

From the halo modelling of the cross-correlation function , the halo mass where the QSOs acting as lenses are located was estimated to be greater than \log _ { 10 } { ( M _ { min } / M _ { \odot } ) } > 13.6 _ { -0.4 } ^ { +0.9 } , also confirmed by the mass density profile analysis ( M _ { 200 c } \sim 10 ^ { 14 } M _ { \odot } ) .
These mass values indicate that we are observing the lensing effect of a cluster size halo signposted by the QSOs , as in previous studies of the magnification bias .

Moreover , we were able to estimate the lensing convergence , \kappa ( \theta ) , for our magnification bias measurements down to a few kpcs .
The derived mass density profile is in good agreement with a Navarro-Frank-White ( NFW ) profile .
We also attempt an estimation of the halo mass and the concentration parameters , obtaining M _ { NFW } = 1.0 ^ { +0.4 } _ { -0.2 } \times 10 ^ { 14 } M _ { \odot } and C = 3.5 _ { -0.3 } ^ { +0.5 } .
This concentration value is rather low and it would indicate that the cluster halos around these QSOs are unrelaxed .
However , higher concentration values still provides a compatible fit to the data .