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On $D\to X_u l^+ l^-$ within the Standard Model and Frameworks like the Littlest Higgs Model with T Parity

The $D\to X_u l^+ l^-$ transitions -- branching ratios, forward-backward asymmetry $A^c_{FB}$, the CP asymmetry $A^c_{CP}$ and the CP asymmetry in the forward-backward asymmetry $A^{CP}_{FB}$ -- have two sources: for $D^{\pm}$ they represent a pure $ΔC=1$ & $ΔQ=0$ current interaction whereas neutral $D$ mesons can also communicate via their anti-hadron. Standard Model (SM) contributions to BR$(D\to X_u l^+ l^-)$ come primarily from long distance dynamics, which overshadow short distance contributions by several orders of magnitude; still they fall much below the present upper experimental bounds. Even the SM contributions to $A^c_{FB}$, $A^c_{CP}$ and $A^{CP}_{FB}$ are tiny, quite unlike in beauty hadrons. The branching ratios are hardly dented by contributions from the Littlest Higgs Models with T parity (LHT) even in the short distance regime, let alone in the SM long distances dynamics. Yet the asymmetries $A^c_{FB}$, $A^c_{CP}$ and $A^{CP}_{FB}$ in these New Physics models can be enhanced over SM predictions, as they arise purely from short distance dynamics; this can occur in particular for $A^c_{FB}$ and $A^{CP}_{FB}$ which get enhanced by orders of magnitudes. Even such enhancements hardly reach absolute sizes for observable experimental effects for $A^c_{FB}$ and $A^c_{CP}$. However LHT contributions to $A^{CP}_{FB}$ could be measured in experiments like the LHCb and the SuperB Collaboration. These results lead us to draw further conclusions on FCNCs within LHT-like models through some simple scaling arguments that encapsulate the essence of flavour dynamics in and beyond the Standard Model.