Naturalness and GUT scale Yukawa coupling ratios in the constrained MSSM

We analyze the fine-tuning in the constrained minimal supersymmetric standard model in the light of the present and expected ATLAS and CMS searches for supersymmetry. Even with $10{\mathrm{fb}}^{-1}$ of data and no discovery of supersymmetry valid regions might remain with fine-tuning less than 20. Moreover we investigate the fine-tuning price of grand unification scale Yukawa coupling relations. Considering a $2\sigma$ constraint for $(g-2{)}_{\mu }$ and fine-tuning less than 30 yields an allowed range of $y_{\tau }/y_b=[1.31,1.70]$, which points towards the alternative grand unification prediction $y_{\tau }/y_b=3/2$. Relaxing the $(g-2{)}_{\mu }$ constraint to $5\sigma$ extends the possible region to [1.02, 1.70], allowing for approximate $b-\tau$ Yukawa coupling unification.

Figure 1

Lowest fine-tuning in the $m_0-M_{1/2}$ plane consistent with current bounds on $b\to s\gamma$, $B_s\to {\mu }^{+}{\mu }^{-}$, collider bounds and $(g-2{)}_{\mu }$ at $5(2)\sigma$ in the upper (lower) plot. $a_0=A_0/M_{1/2}$ has been varied from $-20$ to 20 and $\tan \beta$ from 10 to 60 (for details, see main text). The green and red lines correspond to current CMS [1] and ATLAS [2] bounds. The dashed dotted line represents the extrapolated CMS exclusion line for $5{\mathrm{fb}}^{-1}$ ($10{\mathrm{fb}}^{-1}$) of collected data.

Figure 2

Lowest fine-tuning in the $A_0/M_{1/2}-M_{1/2}$ plane consistent with current bounds on $b\to s\gamma$, $B_s\to {\mu }^{+}{\mu }^{-}$, collider bounds and $(g-2{)}_{\mu }$ at $5(2)\sigma$ in the upper (lower) plot. $m_0$ has been varied from 0 to 4.5 TeV and $\tan \beta$ from 10 to 60 (for details, see main text).

Figure 3

Lowest fine-tuning in the GUT scale $y_{\tau }/y_b-y_t/y_b$ plane consistent with current bounds on $b\to s\gamma$, $B_s\to {\mu }^{+}{\mu }^{-}$, collider bounds and $(g-2{)}_{\mu }$ at $5(2)\sigma$ in the upper (lower) plot for $\tan \beta =10$, 15, 20, 30, 40, 50 from right to left. For $\tan \beta =60$ there is no region allowed in our scan. The $1\sigma$ errors on the quark masses [27] are taken into account by scaling the data points correspondingly.