Higgs mass naturalness and scale invariance in the UV

TL;DR

The paper investigates whether Higgs mass naturalness can be achieved if the Standard Model (SM) flows to a ultraviolet conformal field theory (CFT) with scale invariance at high energies. It shows that the Higgs mass is sensitive to nonperturbative turnover scales $M$ where SM couplings transition to UV fixed-point behavior, and that this sensitivity persists even when no heavy states are present; the corrections scale roughly as $δm_h^2 \,\sim\, M^2$ in toy models and depend on the anomalous dimension $\,\ abla\gamma_{ m UV}$ of the operator coupling to the Higgs. Crucially, the UV fixed point must be interacting and the turnover must occur near the TeV scale to avoid large Higgs-mass corrections—free UV fixed points lead to residual scale violations and cannot protect the Higgs mass. Consequently, a UV completion preserving Higgs naturalness via scale invariance implies new TeV-scale dynamics (e.g., additional gauge bosons from a non-Abelian completion of hypercharge) and a nontrivial UV fixed point, rather than a free fixed point, with gravity and gauge sectors reshaped around the TeV scale to satisfy the required RG behavior.

Abstract

It has been suggested that electroweak symmetry breaking in the Standard Model may be natural if the Standard Model merges into a conformal field theory (CFT) at short distances. In such a scenario the Higgs mass would be protected from quantum corrections by the scale invariance of the CFT. In order for the Standard Model to merge into a CFT at least one new ultraviolet (UV) scale is required at which the couplings turn over from their usual Standard Model running to the fixed point behavior. We argue that the Higgs mass is sensitive to such a turn-over scale even if there are no associated massive particles and the scale arises purely from dimensional transmutation. We demonstrate this sensitivity to the turnover scale explicitly in toy models. Thus if scale invariance is responsible for Higgs mass naturalness, then the transition to CFT dynamics must occur near the TeV scale with observable consequences at colliders. In addition, the UV fixed point theory in such a scenario must be interacting because logarithmic running near a free fixed point constitutes hard breaking of scale invariance and spoils the Higgs mass protection.

Figures (2)

• Figure 1: Top quark loop contribution to the Higgs mass.
• Figure 2: Plot of the contributions to the Higgs mass in (\ref{['eq:example3']}) as a function of the UV cutoff $y_\mathrm{UV}$ on the integral. The blue horizontal line corresponds to the full Higgs mass contribution. The green line which asymptotes to it at small $y_\mathrm{UV}$ is (\ref{['eq:example3']}) as a function of the short distance cutoff $y_\mathrm{UV}$. The two lines which grow at small $y_\mathrm{UV}$ are the original unsubtracted integral (red-dashed) and the subtraction (yellow-dotted), both evaluated with the short distance cutoff. In this example, we chose $n=4$ for the transition parameter and $\gamma_\mathrm{UV}=-1/3$ for the UV anomalous dimension.