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Magnification bias is a gravitational lensing effect that is normallyoverlooked because it is considered sub-optimal in comparison with the lensingshear. Thanks to the demonstrated optimal characteristics of the sub-millimetregalaxies (SMGs) for lensing analysis, in this work we were able to measure themagnification 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 methodologieswere successfully applied: the traditional cross-correlation function approachand the Davis-Peebles estimator through stacking technique. The second one wasfound to be more robust for analysing the strong lensing regime ($<20-30$arcsec in our case) and provides the possibility to take into account thepositional errors of the sources in our samples. From the halo modelling of thecross-correlation function, the halo mass where the QSOs acting as lenses arelocated 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($M_{200c}\sim 10^{14} M_\odot$). These mass values indicate that we areobserving the lensing effect of a cluster size halo signposted by the QSOs, asin previous studies of the magnification bias. Moreover, we were able toestimate the lensing convergence, $\kappa(\theta)$, for our magnification biasmeasurements down to a few kpcs. The derived mass density profile is in goodagreement with a Navarro-Frank-White (NFW) profile. We also attempt anestimation of the halo mass and the concentration parameters, obtaining$M_{NFW}=1.0^{+0.4}_{-0.2}\times10^{14} M_\odot$ and $C=3.5_{-0.3}^{+0.5}$.This concentration value is rather low and it would indicate that the clusterhalos around these QSOs are unrelaxed. However, higher concentration valuesstill provides a compatible fit to the data.

QSOs sigposting cluster size halos as gravitational lenses: halo mass, projected mass density profile and concentration at $z\sim0.7$