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In series of papers the Boer-Mulders function for a given quark flavour hasbeen extracted: (i) from data on semi-inclusive deep inelastic scattering,using the simplifying, but theoretically inconsistent, assumption that it isproportional to the Sivers function for each quark flavour and (ii) from dataon Drell-Yan reactions. In earlier papers, using the semi-inclusive deepinelastic COMPASS deuteron data on the $\langle\cos\phi_h\rangle$ and$\langle\cos 2 \phi_h\rangle$ asymmetries, we extracted the collinear$x_{_{\!B}}$ dependence of the Boer-Mulders function for the sum of the valencequarks $Q_V=u_V + d_V$ using a small number of model dependent assumptions andfound a significant disagreement with the analysis in (i). In the presentpaper, we provide a more complete analysis of the semi-inclusive deep inelasticscattering reaction, including a discussion of higher twist andinteraction-dependent terms, and also a comparison with the Boer-Muldersfunction extracted from data on the Drell-Yan reaction. We confirm that theproportionality relation of the BM function to the Sivers function, for eachquark flavour, fails badly, but find that it holds rather well if applied tothe nonsinglet valence-quark combination, $Q_V$. We also find good agreementwith the results of the Drell-Yan analysis. Furthermore, we obtain interestinginformation on the quark transverse momentum densities in the nucleon and onthe hadron transverse momentum dependence in quark fragmentation.

New study of the Boer-Mulders function: Implications for the quark and hadron transverse momenta