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The moment $\langle x\rangle_{u-d}$ of the nucleon from $N_f=2$ lattice QCD down to nearly physical quark masses

We present an update of our analysis [1] which includes additional ensembles at different quark masses, lattice spacings and volumes, all with high statistics. We use $N_f=2$ mass-degenerate quark flavours, employing the non-perturbatively improved clover action. The lattice matrix elements are converted to the $\overline{\rm MS}$ scheme via renormalization factors determined non-perturbatively in the RI$^\prime$-MOM scheme. We have systematically investigated excited state contributions, in particular, at the smallest, near physical, pion mass. While our results~(with much increased precision) are consistent with Ref.~[1], comparing with previous determinations we find that excited state contributions can be significant if the quark smearing is not suitably optimized, in agreement with other recent studies. The difference with respect to the value for $\langle x\rangle_{u-d}$ extracted from experimental data is reduced but not resolved. Using lattice sizes in the range $L m_π\sim 3.4-6.7$, no significant finite volume effects were observed. Performing a controlled continuum limit that may remove the discrepancy will require simulations at lattice spacings $a< 0.06$ fm.