Measurement of $q_T$-weighted transverse-spin-dependent azimuthal asymmetries at COMPASS
COMPASS is a fixed-target experiment in operation at the CERN North Area (SPS, M2 beam-line) since 2002. An important part of the broad physics programme of the experiment is dedicated to the exploration of the transverse spin-structure of the nucleon studying target transverse spin dependent azimuthal asymmetries (TSAs) arising in the Semi-Inclusive DIS (SIDIS) and Drell-Yan (DY) cross-sections. In addition to those measurements, COMPASS has recently studied also the TSAs weighted by powers of the hadron transverse momentum (in SIDIS) and virtual photon transverse momentum, $q_T$ (in DY). In the transverse momentum dependent (TMD) QCD approach, the conventional DY TSAs are interpreted as convolutions of the beam pion and of the transversely polarized target proton TMD parton distribution functions (PDFs), while the $q_T$-weighted TSAs can be interpreted as simple products of transverse moments of the TMD PDFs. 
In 2015 and 2018 COMPASS performed two years of Drell-Yan data taking with a 190 GeV/$c$ $\pi^-$ beam impinging on a transversely polarized NH$_3$ target. The analysis of the $q_T$-weighted TSAs performed on these two data sets is presented in this paper. The results for DY Sivers $q_T$ weighted TSA are compared with the expectations based on the studies of the weighted Sivers asymmetry measured in the SIDIS process. Combining the information from SIDIS and DY measurements, the pion Boer-Mulders TMD PDF is also studied.