The bearable lightness of new physics beyond the Standard Model

The absence of new physics signals at the TeV scale has recently triggered a shift of interest towards light and weakly coupled extensions of the Standard Model (SM). We consider a light new physics scenario coupled to the SM fields in a $SM+X$ effective theory. Different types of light new physics, searched by present and future experiments at the intensity frontier, are studied and investigated, from sub-GeV abelian gauge vector bosons to right-handed neutrinos at GeV scale. In the former case, we study the dependence on the UV completion of the effective Wess-Zumino terms that appears in the IR theory when gauge bosons are coupled to a SM current whose conservation is broken at loop level. We show how to avoid the would be strong constraints of energy enhanced process due to flavor changing neutral current generated by the Wess-Zumino terms. In the latter case, we work in a minimal see-saw scenario with two right-handed neutrinos with mass at the GeV scale and highlight the prospects for testing the decay $N_2\to N_1\gamma$ induced by an effective dipole operator at future facilities targeting long-lived particles such as the SHiP experiment. In the last part of the thesis, we critically re-examine the new particle interpretation of recent experimental anomalies observed in nuclear transition from the ATOMKI collaboration. Indeed, the hypothetical particle, denoted as $X17$ and proposed by the collaboration itself, would be a light and weakly-interacting boson and we employ a multipole expansion to estimate the nucleon coupling to the light state, identifying the axial vector state as the most promising candidate. Intensity-frontier experiments like MEG II and PADME will probe the ATOMKI anomaly in the near future.