Haggling over the fine-tuning price

TL;DR

This paper quantifies how LEP-era Higgs mass bounds tighten the fine-tuning required in the MSSM by using the full one-loop effective potential across all ${\tan\beta}$. It compares a minimal supergravity scenario with universal soft terms to alternative assumptions, including ${b-\tau}$ Yukawa unification, linear correlations among high-scale parameters, non-universal Higgs masses, and string-inspired inputs, assessing their impact on the tuning measure ${\Delta_0}$. The main findings are that ${\Delta_0}$ is most favorable at intermediate ${\tan\beta}$ in the minimal model, but can be substantially reduced for large ${\tan\beta}$ with non-universal Higgs masses (down to ${\Delta_0} \sim 10$ in some cases) at the cost of tighter constraints (e.g., ${b\to s\gamma}$). String-inspired parameterizations currently do not meaningfully alleviate tuning, while correlation among high-scale parameters can lower tuning substantially, indicating naturalness remains a useful heuristic for model-building rather than a hard bound on sparticle spectra.

Abstract

We amplify previous discussions of the fine-tuning price to be paid by supersymmetric models in the light of LEP data, especially the lower bound on the Higgs boson mass, studying in particular its power of discrimination between different parameter regions and different theoretical assumptions. The analysis is performed using the full one-loop effective potential. The whole range of $\tanβ$ is discussed, including large values. In the minimal supergravity model with universal gaugino and scalar masses, a small fine-tuning price is possible only for intermediate values of $\tanβ$. However, the fine-tuning price in this region is significantly higher if we require $b-τ$ Yukawa-coupling unification. On the other hand, price reductions are obtained if some theoretical relation between MSSM parameters is assumed, in particular between $μ_0$, $M_{1/2}$ and $A_0$. Significant price reductions are obtained for large $\tanβ$ if non-universal soft Higgs mass parameters are allowed. Nevertheless, in all these cases, the requirement of small fine tuning remains an important constraint on the superpartner spectrum. We also study input relations between MSSM parameters suggested in some interpretations of string theory: the price may depend significantly on these inputs, potentially providing guidance for building string models. However, in the available models the fine-tuning price may not be reduced significantly.

Figures (16)

• Figure 1: The price of fine tuning for $\tan\beta=1.65$, as a function of various variables in the minimal supergravity model. An upper limit of 1.2 TeV on the heavier stop mass is imposed in the scanning. All experimental constraints described in the text are included. In all plots, except for $\Delta_0$ versus $M_h$, the bound $M_h>90$ GeV is included. The mass of the lighter physical chargino and of the heavier physical stop are denoted by $m_{C1}$ and $M_{\tilde{t}_2}$, respectively.
• Figure 2: As in Fig. 1, but for $\tan\beta=2.5$.
• Figure 3: As in Fig. 1, but for $\tan\beta=10$.
• Figure 4: Fine-tuning measures as functions of $\tan\beta$. In panels (a),(c) and (e), lower limits on the Higgs boson mass of 90 GeV (solid), 100 GeV (long-dashed), 105 GeV (dashed) 110 GeV (dotted) and 115 GeV (dot-dashed) have been assumed. In panels (b), (d) and (f), $M_h$ has been fixed to 95 GeV (solid), 100 GeV (long-dashed), 105 GeV (dashed) 110 GeV (dotted) and 115 GeV (dot-dashed). Panels (a) and (b) correspond to independent $M_{1/2}$, $A_0$ and $\mu$ parameters. In panels (c), (d) and (e), (f), linear dependences $M_{1/2}=c_{M\mu}\mu_0$ and $A_0=c_{A\mu}\mu_0$, respectively, have been assumed.
• Figure 5: a) The running mass $m_b(M_Z)$ obtained from strict $b-\tau$ Yukawa coupling unification at $M_{GUT}=2\times10^{16}$ GeV for different values of $\alpha_s(M_Z)$, before inclusion of one-loop supersymmetric corrections. b) The minimal departure from $Y_b=Y_\tau$ at $M_{GUT}$ measured by the ratio $Y_b/Y_\tau -1$, which is necessary for obtaining the correct $b$ mass in the minimal supergravity model with one-loop supersymmetric corrections included.