Search for BSM phenomena is one of the fundamental goals of the LHC experiments. Many BSM models foresee long-lived particles which decay far from the production vertex and a big effort has been done both by the ATLAS and CMS collaborations to detect these long-lived particles. A specific, widely studied, example for such searches is made through the measurement of displaced vertices produced by a narrow resonance decaying to a muon pair. This paper will first analyze what has been done so far and will try to evaluate if there are improvements which may allow enhancing the discovery potential at LHC for such particles. The outcome of this work is that while long-lived particles decaying more than 1 cm from the primary vertex have been carefully studied by ATLAS and CMS, the region within a few millimeters from the primary vertex may have been partially neglected. A new approach for searching such resonances, fully based on data, will then be proposed which allows minimizing systematic uncertainties while keeping at a maximum the discovery potentiality for long-lived neutral resonances which decay a few millimeters from the primary vertex.