Abstract
We construct a theory of real scalar fields that interpolates between two different theories: a Lee-Wick theory with $N$ propagator poles, including $N-1$ Lee-Wick partners, and a nonlocal infinite-derivative theory with kinetic terms modified by an entire function of derivatives with only one propagator pole. Since the latter description arises when taking the $N\rightarrow\infty$ limit, we refer to the theory as "asymptotically nonlocal." Introducing an auxiliary-field formulation of the theory allows one to recover either the higher-derivative form (for any $N$) or the Lee-Wick form of the Lagrangian, depending on which auxiliary fields are integrated out. The effective scale that regulates quadratic divergences in the large-$N$ theory is the would-be nonlocal scale, which can be hierarchically lower than the mass of the lightest Lee-Wick resonance. We comment on the possible utility of this construction in addressing the hierarchy problem.
Highlights
When the Higgs boson was discovered at the Large Hadron Collider (LHC) in 2012 at a mass of 125 GeV, an immediate question was this: why is it so light compared to the Planck scale at 1019 GeV? The standard model, in its present form, does not provide a mechanism that protects the Higgs mass from picking up large radiative corrections from Planck-scale physics
Higher-derivative quadratic terms have received substantial attention as a means of obtaining improved ultraviolet behavior of loop amplitudes. When these terms are of finite order in the number of derivatives, as in the Lee-Wick standard model [2], additional poles in the two-point function appear that correspond to new particles; these might be expected at the TeV scale if they participate in a solution to the hierarchy problem
In theories with higher-derivative quadratic terms of infinite order, as in models where the □ operator appears as the argument of an entire function, there are no new particles, but there are other complications: the simplest formulation of such nonlocal theories in Minkowski spacetime violate unitarity [29,31,32,33,34], a problem related to the fact that there are directions in the complex p0 plane where loop integrands diverge
Summary
When the Higgs boson was discovered at the Large Hadron Collider (LHC) in 2012 at a mass of 125 GeV, an immediate question was this: why is it so light compared to the Planck scale at 1019 GeV? The standard model, in its present form, does not provide a mechanism that protects the Higgs mass from picking up large radiative corrections from Planck-scale physics. The Lee-Wick standard model [2] provides an inherent mechanism to address the hierarchy problem: supplementing the spectrum of the standard model by TeV-scale LeeWick partner particles with wrong-sign kinetic and mass terms leads to a cancellation of quadratic divergences in scalar self-energies, with the precise spectrum of Lee-Wick resonances left for experiments to determine. These models have been shown to be unitary, provided certain prescriptions are adopted in momentum space to deal with the different pole structure [3] (see [4]), are causal at the. Vand suggest that no impediments exist to generalizing this framework to Abelian gauge theories, non-Abelian gauge theories, and, in principle, the entire standard model
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