Abstract
The clockwork mechanism would be completely spoiled by the presence of a cosmological constant in the bulk of its 5D construction, or by analogous terms in the discrete case. It is believed that supersymmetry is required to forbid the appearance of these fatal terms, thus apparently tying the fate of the clockwork to the presence of supersymmetry. In this letter we argue that a robust clockwork can be obtained also without supersymmetry, by providing a model for the clockwork/linear dilaton EFT where this protection is instead related to diffeomorphism invariance in a higher number of extra dimensions. We show that the clockwork mechanism is a quite general setup that emerges, rather minimally, from pure gravity in the presence of D−5 additional sufficiently flat extra dimensions. The linear dilaton model is obtained asymptotically in the limit of infinitely many extra dimensions. We then study the finite-D theory, which is a deformation of the linear dilaton one, but still a clockwork theory for D large enough.
Highlights
Introduction and fragility of the clockworkThe clockwork mechanism [1,2,3] is an elegant way to generate hierarchies ∼ X in a theory with no hierarchy in the fundamental parameters, by making use of a chainlike structure of interactions between a number ∼ log X of fields
Interesting is the limit in which these different fields are identified with a single field in different points of an extra dimension
Instead, we argue that one can have a robust clockwork mechanism even without supersymmetry
Summary
The clockwork mechanism [1,2,3] is an elegant way to generate hierarchies ∼ X in a theory with no hierarchy in the fundamental parameters, by making use of a chainlike structure of interactions between a number ∼ log X of fields. From the low-energy/EFT perspective, the consistent definition of the clockwork is: a theory with no exponential hierarchy in the fundamental parameters along the chain/extra dimension, that gives rise to an exponential hierarchy between the coupling of a symmetry-protected light mode and of heavier states to the same external sector, for instance the Standard Model This is invariant under field redefinitions, and independent on UV features beyond the EFT approach. In [11] it was pointed out that an additional serious tuning is implicit: a cosmological constant Λ in the 5D bulk would be in principle allowed (and generated radiatively) unless forbidden by symmetries This would modify drastically the clockwork solution 1; for instance, the size of the extra dimension RΛ corresponding to the same value of MP for fixed M5, k, would become. For flat additional D − 5 extra dimensions, pure gravity provides a dilaton field, with action of the same form as (3), providing at the same time a simple, non-stringy, UV origin for the clockwork Lagrangian
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