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The SIDIS transverse spin asymmetries weighted with powers of $P_T$, thehadron transverse momentum in the $\gamma N$ reference system,have beenintroduced already twenty years ago and are considered quite interesting. Whilethe amplitudes of the modulations in the azimuthal distribution of the hadronsare the ratios of convolutions over transverse momenta of thetransverse--momentum dependent parton distributions and of the correspondingfragmentation functions, and can be evaluated analytically only makingassumptions on the transverse--momentum dependence of these functions, theweighted asymmetries allow to solve the convolution integrals overtransverse--momenta without those assumptions. Using the high statistics datacollected in 2010 on transversely polarized proton target COMPASS has evaluatedin x-bins the $P_T$ weighted Sivers asymmetry which is proportional to theproduct of the first transverse moment of the Sivers function and of thefragmentation function. The results are compared to the standard unweightedSivers asymmetry.

COMPASS measurement of the $P_T$ weighted Sivers asymmetry