Focus Points and Naturalness in Supersymmetry

TL;DR

The paper identifies a robust focus-point behavior in the RG evolution of the up-type Higgs mass $m_{H_u}^2$ within a class of SUSY models including minimal supergravity, such that $m_{H_u}^2$ becomes largely insensitive to the high-scale scalar mass $m_0$ at a weak-scale focus point $Q_F^{(H_u)}$ (near ${\cal O}(100\ \text{GeV})$) for $m_t\approx 174$ GeV. This reduces naturalness constraints on heavy scalar masses, allowing $m_0$ to reach 2–3 TeV without significant fine-tuning, while requiring gaugino/higgsino masses to stay near the weak scale. Naturalness is quantified via a sensitivity measure $c$, showing regions with heavy scalars can be as natural as lighter ones, though gaugino masses remain bounded (e.g., $M_{1/2}\lesssim 400$ GeV for $c\le 50$). The work analyzes the implications for collider searches, predicting challenging discovery prospects for scalars at the LHC and linear colliders, while predicting accessible gauginos/higgsinos and a relatively light Higgs sector in natural regions, especially at large $\tan\beta$. It also discusses the dependence on the top mass, boundary scale, and whether SM parameters like $y_t$ should be included in naturalness assessments, and suggests broader applicability to non-minimal frameworks.

Abstract

We analyze focus points in supersymmetric theories, where a parameter's renormalization group trajectories meet for a family of ultraviolet boundary conditions. We show that in a class of models including minimal supergravity, the up-type Higgs mass has a focus point at the weak scale, where its value is highly insensitive to the universal scalar mass. As a result, scalar masses as large as 2 to 3 TeV are consistent with naturalness, and {\em all} squarks, sleptons and heavy Higgs scalars may be beyond the discovery reaches of the Large Hadron Collider and proposed linear colliders. Gaugino and Higgsino masses are, however, still constrained to be near the weak scale. The focus point behavior is remarkably robust, holding for both moderate and large \tanβ, any weak scale gaugino masses and A parameters, variations in the top quark mass within experimental bounds, and for large variations in the boundary condition scale.

Figures (17)

• Figure 1: The RG evolution of $m_{H_u}^2$ for (a) $\tan\beta=10$ and (b) $\tan\beta=50$, several values of $m_0$ (shown, in GeV), $M_{1/2} = 300$ GeV, $A_0=0$, and $m_t = 174$ GeV. For both values of $\tan\beta$, $m_{H_u}^2$ exhibits an RG focus point near the weak scale, where $Q_{\rm F}^{(H_u)}\sim {\cal O}(100~{\rm GeV})$, irrespective of $m_0$.
• Figure 2: The focus point renormalization scale $Q_{\rm F}^{(H_u)}$ as a function of $\tan\beta$ for $m_0 = 500$, 1000, and 1500 GeV (from above), $M_{1/2} = 300$ GeV, $A_0=0$, and $m_t = 174$ GeV. The focus point scale is defined as the scale at which $\partial m_{H_u}^2 / \partial m_0 = 0$.
• Figure 3: Contours of constant sensitivity parameter $c_{m_0}$ in the $(m_0, M_{1/2})$ plane for (a) $\tan\beta = 10$ and (b) $\tan\beta = 50$, $A_0=0$, $\mu>0$, and $m_t = 174$ GeV. The bottom and right shaded region is excluded by the chargino mass limit of 90 GeV. The top left region is also excluded if a neutral LSP is required.
• Figure 4: As in Fig. \ref{['fig:c_m0']}, but for the sensitivity parameter $c_{M_{1/2}}$.
• Figure 5: As in Fig. \ref{['fig:c_m0']}, but for the sensitivity parameter $c_{\mu_0}$.