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Hypercubic Effects in semileptonic $D \to π$ decays on the lattice

We present a lattice determination of the vector and scalar form factors of the semileptonic $D \to π\ell ν$ decays, which are relevant for the extraction of the CKM matrix element $\lvert V_{cd} \rvert$ from experimental data. Our analysis is based on the gauge configurations produced by the European Twisted Mass Collaboration with $N_f = 2 + 1 + 1$ flavors of dynamical quarks. We simulated at three different values of the lattice spacing and with pion masses as small as 210 MeV. Quark momenta are injected on the lattice using non-periodic boundary conditions. The matrix elements of both vector and scalar currents are determined for a plenty of kinematical conditions in which parent and child mesons are either moving or at rest. Lorentz symmetry breaking due to hypercubic effects is clearly observed in the data and included in the decomposition of the current matrix elements in terms of additional form factors. Our preliminary estimate for the vector form factor at zero 4-momentum transfer is $f_+^{D \to π}(0) = 0.631\,(40)$, which can be compared with the latest FLAG average $f_+^{D \to π}(0) = 0.666\,(29)$ available only at $N_f = 2 + 1$.