Supernatural MSSM

We point out that the electroweak fine-tuning problem in the supersymmetric standard models (SSMs) is mainly due to the high energy definition of the fine-tuning measure. We propose supernatural supersymmetry which has an order one high energy fine-tuning measure automatically. The key point is that all the mass parameters in the SSMs arise from a single supersymmetry breaking parameter. In this paper, we show that there is no supersymmetry electroweak fine-tuning problem explicitly in the minimal SSM (MSSM) with no-scale supergravity and Giudice-Masiero mechanism. We demonstrate that the $Z$-boson mass, the supersymmetric Higgs mixing parameter $\mu$ at the unification scale, and the sparticle spectrum can be given as functions of the universal gaugino mass $M_{1/2}$. Because the light stau is the lightest supersymmetric particle (LSP) in the no-scale MSSM, to preserve $R$ parity, we introduce a non-thermally generated axino as the LSP dark matter candidate. We estimate the lifetime of the light stau by calculating its two-body and three-body decays to the LSP axino for several values of axion decay constant $f_a$, and find that the light stau has a lifetime ${\tau }_{{\tilde{\tau }}_1}$ in $[{10}^{-4},100]\mathrm{s}$ for an $f_a$ range $[{10}^9,{10}^{12}]\mathrm{GeV}$. We show that our next to the LSP stau solutions are consistent with all the current experimental constraints, including the sparticle mass bounds, B-physics bounds, Higgs mass, cosmological bounds, and the bounds on long-lived charged particles at the LHC.

Figure 1

$\mu (\mathrm{GUT})$ (left-vertical axis) and EWFT measure ${\mathrm{\Delta }}_{\mathrm{EENZ}}$ (right-vertical axis) versus $M_{1/2}$. The thick blue line represents data generated, and the thin blue line is the line fitted to the data. Maroon points depict the values of ${\mathrm{\Delta }}_{\mathrm{EENZ}}$.

Figure 2

Parameter $c$ versus $M_Z$ plot.

Figure 3

Plots in the $\tan \beta ‐M_{1/2}$ plane.

Figure 4

Plot in the $\tan \beta -M_Z$ plane.

Figure 5

Sparticle masses as functions of $M_{1/2}$. See text for color coding.

Figure 6

The viable parameter spaces in the $M_{1/2}‐\tan \beta$ plane. The gray points have the successful REWSB and no tachyonic sfermions or pseudoscalar Higgs bosons. The blue points further satisfy sparticle mass bounds, green points form a subset of blue points and satisfy B-physics constraints, and red points are a subset of green points and satisfy Higgs mass bounds.

Figure 7

$m_{{\tilde{\tau }}_1}$ versus $\mathrm{\Omega }{h}^2$. The gray points have the successful REWSB and no tachyonic sfermions or pseudoscalar Higgs bosons. The aqua points further satisfy the sparticle mass bounds, as well as the B-physics and Higgs mass bounds. And the purple points are a subset of aqua points and satisfy $|B_0|\lesssim 1\mathrm{GeV}$. The horizontal black solid line represents $\mathrm{\Omega }{h}^2=0.11$, while the dashed black line shows $\mathrm{\Omega }{h}^2=0.137$.

Figure 8

$m_{\tilde{a}}$ versus $m_{{\tilde{\tau }}_1}$.

Figure 9

The lifetimes of staus (${\tau }_{{\tilde{\tau }}_1}$) in seconds ($s$) as functions of stau masses $m_{{\tilde{\tau }}_1}$ for $|e_Q|=1/3$ and $y=1$. Here, the light-green, orange, blue and red points represent the axion decay constants $f_a={10}^9,{10}^{10},{10}^{11}$, and ${10}^{12}\mathrm{GeV}$, respectively. The horizontal black solid line represents ${\tau }_{{\tilde{\tau }}_1}=1$ second.

Figure 10

In the left panel, the lifetimes of staus (${\tau }_{{\tilde{\tau }}_1}$) in seconds as functions of stau masses $m_{{\tilde{\tau }}_1}$ for $y=1$ and $f_a=7\times {10}^8\mathrm{GeV}$. The aqua and purple points respectively represent $|e_Q|=1/3$ and $|e_Q|=1$. The right panel, with the same color coding, shows corresponding decay lengths ($c{\tau }_{{\tilde{\tau }}_1}$) in kilometers (km) as functions of $m_{{\tilde{\tau }}_1}$.