Quantum scale invariance, cosmological constant and hierarchy problem (0809.3406v3)

Abstract: We construct a class of theories which are scale invariant on quantum level in all orders of perturbation theory. In a subclass of these models scale invariance is spontaneously broken, leading to the existence of a massless dilaton. The applications of these results to the problem of stability of the electroweak scale against quantum corrections, to the cosmological constant problem and to dark energy are discussed.

• The paper introduces a scale-invariant model that buffers the Higgs mass against radiative corrections while canceling the cosmological constant.
• It employs dimensional regularization and specific renormalization prescriptions to maintain exact symmetry across all perturbative orders, even with gravity.
• The approach incorporates unimodular gravity, offering a dynamic explanation for dark energy and insights into resolving fundamental SM extensions.

Scale Invariance, Quantum Cosmological Constant, and the Hierarchy Problem

The paper explores the intricate relationship between scale invariance at the quantum level and its implications on high energy physics, particularly focusing on the cosmological constant problem and the hierarchy issue associated with the Standard Model (SM). The authors, Shaposhnikov and Zenhausern, propose a class of scale-invariant theories that are maintained across all orders of perturbation theory, where scale invariance is fundamentally ingrained even at the quantum level.

Key Concepts and Theoretical Framework

The essence of this research lies in the construction of theories characterized by exact scale invariance in the quantum field. Classical field theories without dimensionful parameters inherently exhibit this scale-invariance. However, the challenge arises in preserving this invariance at the quantum level due to anomalies that typically result in the divergence of the dilatation current. The authors explore this through a model involving two scalar fields, focusing on the impact of scale invariance on the Higgs mass stability and the cosmological constant problem.

The proposed theories distinguish themselves by exhibiting spontaneous breaking of scale invariance, resulting in the presence of a massless dilaton. This feature addresses two significant fine-tuning issues: the radiative stability of the Higgs mass and the vanishing of the cosmological constant. Additionally, the incorporation of unimodular gravity within this framework suggests the potential for a dynamical explanation of dark energy, while simultaneously negating the cosmological constant.

Perturbative Approach and Scale Invariance

The paper emphasizes a perturbative construct where scale invariance reduces the theoretical landscape to one characterized by simpler dynamics. By leveraging dimensional regularization, the authors demonstrate how scale invariance remains intact across different perturbative orders. Particularly, the inclusion of functions of scalar combinations, such as $\chi$ and $h$, in renormalization prescriptions ensures scale invariance even when gravity is considered.

The potential for ground states where dilatational symmetry is spontaneously broken implies that the Higgs mass is buffered against radiative corrections while keeping the cosmological constant at zero, mitigating traditional instabilities arising in SM extensions. These properties emerge due to the exact pole structure and interactions dictated by appropriate renormalized constants and relations, such as those involving $\lambda_R$ or $\zeta_R$, where modifications through scale-preserving prescriptions assure that functional outputs remain consistent with observed low-energy phenomena.

Implications and Future Directions

From a phenomenological standpoint, the proposed unimodular models offer a viable pathway for explaining dark energy dynamics, linked to the initial conditions modulated by the background dilaton field. This removal of large radiative corrections to the Higgs mass potentially redefines the effective cutoff associated with fundamental interactions, enabling robust models with consistent electroweak parameters that extend to cosmological scales.

Future explorations should aim at probing the non-perturbative aspects of these theories, assessing renormalization and unitarity constraints, particularly around gravity’s interplay at higher derivatives. These evaluations will inform the feasibility of constructing self-consistent, all-encompassing scale-invariant theories that possess predictive power while accommodating the Standard Model's extensions.

Moreover, ongoing endeavors should entail analyzing potential lattice implementations and their bearing on scale-invariant behaviors in curved spacetime contexts. Addressing these elements enriches the theoretical structure and validates whether this perturbatively motivated framework holds the key to resolving enduring puzzles in fundamental physics, notably those tied to the nature of mass scales and cosmic acceleration.