Softened Gravity and the Extension of the Standard Model up to Infinite Energy

Attempts to solve naturalness by having the weak scale as the only breaking of classical scale invariance have to deal with two severe difficulties: gravity and the absence of Landau poles. We show that solutions to the first problem require premature modifications of gravity at scales no larger than $10^{11}$ GeV, while the second problem calls for many new particles at the weak scale. To build models that fulfil these properties, we classify 4-dimensional Quantum Field Theories that satisfy Total Asymptotic Freedom (TAF): the theory holds up to infinite energy, where all coupling constants flow to zero. We develop a technique to identify such theories and determine their low-energy predictions. Since the Standard Model turns out to be asymptotically free only under the unphysical conditions $g_1 = 0$, $M_t = 186$ GeV, $M_\tau = 0$, $M_h = 163$ GeV, we explore some of its weak-scale extensions that satisfy the requirements for TAF.