A Clockwork Theory

The clockwork is a mechanism where O(1) couplings exponentiate, generating tiny couplings from a theory with no exponentially small parameters at the fundamental level. In this talk I discuss the difference between mass scales and interaction scales. Then introduce the clockwork mechanism for scalars and sketch how it could work for gravitons. The linear dilaton theory naturally provides a 5D setup in which this scenario arises for gravitons, and I will sketch its phenomenology. 1 The clockwork mechanism The clockwork mechanism rests on the crucial difference between masses and interaction scales. To appreciate the difference between the two it is useful to reinstate in the Lagrangian the appropriate powers of , while working in units with c = 1. Consider a general 4D action involving scalar (φ), fermion (ψ), and vector gauge fields (A µ), normalised such that all kinetic terms and commutation relations are canonical. In our basis, there are no explicit factors of in the classical Lagrangian in position space. The dimensionality of the quantities of interest, including gauge couplings g, Yukawa couplings y, and scalar quartic couplings λ, are [] = EL , [L] = EL −3 , [φ] = [A µ ] = E 1/2 L −1/2 , [ψ] = E 1/2 L −1 , (1) [∂] = [ ˜ m] = L −1 , [g] = [y] = E −1/2 L −1/2 , [λ] = E −1 L −1. (2) Canonical dimensions in natural units with = 1 are recovered by identifying E = L −1. Unlike the case of natural units, our dimensional analysis shows that couplings, and not only masses, are dimensionful quantities. It is useful to introduce convenient units of mass˜Mmass˜ mass˜M ≡ L −1 and coupling C ≡ E −1/2 L −1/2. Let us now add to the Lagrangian an effective operator of canonical dimension d, of the general form 1 Λ d−4 ∂ n D Φ n B ψ n F. (3) Here n D is the number of derivatives, n B the number of boson fields (Φ = φ, A µ), and n F the number of fermion fields, with n D + n B + 3 2 n F = d. The dimensionful quantity Λ that defines