Search for $CP$ violation in $D^0 \rightarrow K_S^0K_S^0$ decays at LHCb

$CP$ is the discrete symmetry given by the combined application of the spatial parity ($P$) and the charge conjugation operation ($C$). $CP$ is known to be a broken symmetry, at this could potentially explain the asymmetry between matter and anti-matter present in the Universe. The first experimental evidence for $CP$-violating effects ($CPV$ ) was found in weak transitions of the $strange$ quark in year 1964. $CPV$ in decays of $up$-type quarks has remained elusive until very recently. Amongst them, only the $charm$ quark offers concrete possibilities of observation of $CPV$ effects. But even in $charm$ hadrons $CPV$ is difficult to observe, due to the smallness of effects expected ($O\sim 10^{-3}-10^{-4}$), that are not even precisely calculable. Only in March of this year $CPV$ has finally been observed in the charm sector by the LHCb experiment at LHC measuring the integrated decay asymmetry difference of $D^0 \rightarrow K^+K^-$ and $D^0 \rightarrow \pi^+\pi^-$ decays as $(-15.4 \pm 2.9) \times 10^{-4}$. The large observed value gives hope that significant effects may be found also in other channels, and that precise enough measurements can be made to investigate this new field in detail. One of the most promising candidates for further $\mathcal{A}_{CP}$ measurements is the $D^0 \rightarrow K^0_S K^0_S$ decay, as it is possible for $\mathcal{A}_{CP}$ to take particularly large values ($\simeq 1\%$). However, this is balanced by the greater experimental difficulty of this mode, due to the pair of long-lived neutral particles in the final state. This represents a particular challenge at LHCb because it is more tuned to shorter-lived decays of heavy flavored hadrons. However, the LHCb can count on a huge production of charm particles and excellent detector performance, so it is worth investigating its potential also in this interesting mode. The aim of my thesis is to look for ways to enhance the sensitivity of LHCb to this decay mode. I concentrated on the sizable sample collected by LHCb since 2017. First, the online selection of this sample is different from the past, in that it has increased efficiency for those decays occurring outside the VELO acceptance ("Downstream" decays). While such samples have been used in past measurements, they have different features in this new samples that require different analysis methods; in addition, they represent a larger proportion of the sample, increasing their relative importance in the result. Moreover, it is in principle possible to further enhance their collection efficiency in future runs, which underlines the importance of determining the limits of their actual usability. For this purpose I have introduced a novel analysis methodology tuned to the specifics of this particular decay. While a standard LHCb analysis procedure is to apply cuts to reject charm decays coming from the decay of a bottom hadron ("secondaries"), keeping only those promptly produced in the $pp$ collision, I have studied the possibility of keeping also the secondaries in the analysis to increase the available yields. The motivation is that those selection cuts are much more inefficient in the $D^0 \rightarrow K^0_S K^0_S$ channel than in any other charm mode, due to the lower resolution, causing a much larger loss of signal. Loosening the selection however brings together new difficulties that are absent in all other charm analyses in LHCb. The most important issue is the additional asymmetry that affects secondary decays. I have demonstrated the possibility of subtracting this confounding asymmetry by means of a special control sample of $D^0 \rightarrow K^+K^-$. This required however to redesign completely the selection procedures of both samples to ensure the same proportion of secondaries, and evaluate and correct for the residual differences while keeping systematic uncertainties under control. I conclude my work with an evaluation of the $CPV$ resolution achievable with this new analysis approach, and a look at the prospects from future data taking runs.