Geometric and topological Higgs mechanisms: Mass generations and massive scattering amplitudes

In this talk, I present an overview on the mass-generation mechanisms via the geometric "Higgs" mechanism in the compactified higher dimensional Kaluza-Klein (KK) gauge/gravity theories and via the topological "Higgs" mechanism in the 3d dimensional Chern-Simons gauge/gravity theories. These are compared to the conventional Higgs mechanism of mass generations in the 4d standard model (SM) of particle physics. Such geometric mass-generations can be formulated at the scattering.. -matrix level through the KK Gauge and Gravitational Equivalence Theorem (GAET and GRET), whereas the topological mass-generations can be formulated at the..-matrix level through the Topological Equivalence Theorem (TET). Using the (GAET, GRET) and TET, we prove that such.. -point massive scattering amplitudes have strikingly large energy cancellations and exhibit the desirable ultraviolet behaviors as good as that of the 4d SM and that of the 4d massless Einstein gravity. We propose the extended double-copy methods to construct the massive graviton scattering amplitudes from the corresponding massive gauge boson scattering amplitudes in the higher-dimensional KK gauge/gravity theories (as the field theory limit of the KK string theory) and in the 3d ChernSimons gauge/gravity theories. This builds up the massive realizations of the deep gauge-gravity connection: (Gravity) =( Gauge Theory)(2). With these, we establish a new correspondence between the two types of large energy cancellations in the four-point longitudinal KK gauge/gravity amplitudes at tree level:E-4 -> E-0 in the KK Yang-Mills (YM) gauge theory and E-10 -> E-2 in the KK Einstein gravity. For the 3d Chern-Simons gauge/gravity theories, we uncover a series of strikingly large energy cancellations E-12 -> E(1)of the four-graviton scattering amplitude under high-energy expansion; we establish a new correspondence between the two energy cancellations in the topologically massive YM gauge theory and the topologically massive gravity theory. This presentation is based upon our recent series of works [1-6].