Rare and Challenging Charm Decays at LHCb

This thesis presents two studies of charged charm meson decays using proton-proton collision data, corresponding to an integrated luminosity of 3 fb$^{-1}$, recorded by the LHCb experiment. A search for $D^+_{(s)} \to \pi^+ \mu^+ \mu^-$ decays is performed. Rare decays of charged charm mesons have not previously been observed unlike their counterpart $B$ meson and kaon decays. Such decays are $c \to u$ quark transitions and take place via loop diagrams within the Standard Model. Measurements of their branching fractions are an indirect test of New Physics, whose virtual contributions may enhance such quantities. No signals are observed and the 90% (95%) $CL$ limits on the branching fractions, the most stringent to date, are found to be, \begin{eqnarray*} \mathcal{B}(D^+ \to \pi^+ \mu^+ \mu^-) < 2.9 \, (3.3) \times 10^{-8},\\ \mathcal{B}(D^+_{s} \to \pi^+ \mu^+ \mu^-) < 1.76 \, (1.91) \times 10^{-7}. \end{eqnarray*} The charge asymmetry measurement of $D^+ \to \pi^+ \pi^0$ using $\pi^0 \to e^+ e^- \gamma$ decays is also described. Charge asymmetry in the Standard Model is expected to be small for charm meson decays. Any measured deviation from zero would be evidence of New Physics. This decay has not previously been observed at a hadron collider and the measurement acts as a `proof of principle' study allowing a future analysis to examine the decay mode. No evidence of CP violation is found.