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 $A_{FB}^c$, the CP asymmetry $A_{CP}^c$ and the CP asymmetry in the forward-backward asymmetry $A_{FB}^{CP}$ – have two sources: for $D^{\pm }$ they represent a pure $\mathrm{\Delta }C=1$ & $\mathrm{\Delta }Q=0$ 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 $A_{FB}^c$, $A_{CP}^c$ and $A_{FB}^{CP}$ 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 $A_{FB}^c$, $A_{CP}^c$ and $A_{FB}^{CP}$ 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 $A_{FB}^c$ and $A_{FB}^{CP}$ which get enhanced by orders of magnitudes. Even such enhancements hardly reach absolute sizes for observable experimental effects for $A_{FB}^c$ and $A_{CP}^c$. However LHT contributions to $A_{FB}^{CP}$ 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.