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I will discuss how a series of key contributions from Lattice Field Theory to the SM prediction of the leading order hadronic vacuum polarization term of the muon anomalous magnetic moment, g_μ -2, has led to unveil either problems in the experimental results for electron-positron annihilation into hadrons at the GeV scale, used so far in the "SM prediction" of g_μ -2, or signals of New Physics.