High-scale supersymmetry, the Higgs boson mass, and gauge unification

Suppressing naturalness concerns, we discuss the compatibility requirements of high-scale supersymmetry breaking with the Higgs boson mass constraint and gauge coupling unification. We find that to accommodate superpartner masses significantly greater than the electroweak scale, one must introduce large nondegeneracy factors. These factors are enumerated for the Minimal Supersymmetric Standard Model, and implications for the allowed forms of supersymmetry breaking are discussed. We find that superpartner masses of arbitrarily high values are allowed for suitable values of $\tan \beta$ and the nondegeneracy factors. We also compute the large, but viable, threshold corrections that would be necessary at the unification scale for exact gauge coupling unification. Whether or not high-scale supersymmetry can be realized in this context is highly sensitive to the precise value of the top quark Yukawa coupling, highlighting the importance of future improvements in the top quark mass measurement.

Figure 1

Plot showing the running of ${\lambda }_H$ in the SM (solid blue line), with $3\sigma$ contours corresponding to both the error in ${\alpha }_s(m_t)$ (inner dotted blue lines) and $y_t(m_t)$ (outer dashed blue lines). Also shown is the tree-level SUSY matching condition of Eq. (2), for values of $\tan \beta =1$, 2, 4, 50 (dark blue solid, dashed purple, dot-dashed olive, dotted green).

Figure 2

Plot showing the required threshold corrections for the tree-level SUSY ${\lambda }_H^{\text{tree}}(\tilde{m})$ to match the SM ${\lambda }_H^{\mathrm{SM}}(\tilde{m})$ at a given scale $\tilde{m}$. Again, $3\sigma$ contours are shown, with the same definition as in Fig. 1. Shown is $\mathrm{\Delta }{\lambda }_H^{\text{req}}(\tilde{m})={\lambda }_H^{\mathrm{SM}}(\tilde{m})-{\lambda }_H^{\text{tree}}(\tilde{m})$ for $\tan \beta =1$, 2, 4, 50 (blue, purple, red, orange).

Figure 3

Plot showing the required threshold corrections for the tree-level SUSY ${\lambda }_H^{\text{tree}}(\tilde{m})$ to match the SM ${\lambda }_H^{\mathrm{SM}}(\tilde{m})$ as a function of $\tan \beta$. Again, $3\sigma$ contours are shown, with the same definition as in Fig. 1. Shown is $\mathrm{\Delta }{\lambda }_H^{\text{req}}(\tilde{m})={\lambda }_H^{\mathrm{SM}}(\tilde{m})-{\lambda }_H^{\text{tree}}(\tilde{m})$ for $\tilde{m}=5\times {10}^3,{10}^6,{10}^{10},{10}^{16}\mathrm{GeV}$ (blue, purple, red, orange).

Figure 4

Plot showing the variation of different parts of the threshold corrections $\mathrm{\Delta }{\lambda }_H$ at one-loop, in units of $16{\pi }^2$, as a function of $\tan \beta$ with $\tilde{m}={10}^4\mathrm{GeV}$. Shown are the threshold corrections defined in Eqs. (c2)–(c6), divided into subcomponents, with the soft breaking masses of all superpartners $m_i$ chosen to be almost degenerate ($m_i=1.01\tilde{m}$), so that the “log” s are not all zero. In blue is the pure $m_{{\tilde{q}}_{L,3}},m_{{\tilde{t}}_{R,3}}$ part, corresponding to the first two lines of Eq. (c2). In black is the stop mixing part, corresponding to the last three lines of Eq. (c2) (black dot-dashed corresponds to stop mixing with $A_t=0$.) The first and second generation squark, and all slepton generation contribution, corresponding to Eq. (c3) is smaller, and therefore is not shown. The part of the scalar corrections which is independent of the scalar masses, corresponding to Eq. (c4), is shown in green. The $m_A$-dependent part from Eq. (c5) is shown in purple. The red line corresponds to the first three lines of Eq. (c6). The orange line shows a possible correction to the running of the gauge couplings in a split SUSY setup, due to the Higgsino mass parameter $\mu$ being considerably lighter than the rest of the spectrum. All solid lines indicate positive corrections, while dashed lines indicate negative corrections.

Figure 5

These plots show how the required corrections to ${\lambda }_H$ (solid green) compare to the upper and lower limits of the threshold corrections $\mathrm{\Delta }\lambda$ obtained from a scan, at a SUSY scale $\tilde{m}=m_{{\tilde{q}}_{L,3}}=m_{{\tilde{t}}_{R,3}}=5\times {10}^3\mathrm{GeV}$. The dashed green lines correspond to the $3\sigma$ upper and lower limits from $m_t$ uncertainty. Scanned independently are $\mu ,\{m_{{\tilde{q}}_{L,i}},m_{{\tilde{u}}_{R,i}},m_{{\tilde{d}}_{R,i}},m_{{\tilde{b}}_{R,3}},m_{{\tilde{\ell }}_{L,i}},m_{{\tilde{e}}_{R,i}},m_A\},\{M_a\}$ and $\tan \beta$. The top trilinear $A_t$ has been set to $\sqrt{6}\tilde{m}$. Each mass is allowed to vary up to $\xi =100$ relative to $\tilde{m}$, with scanned points shown here for $\xi =0$, 10, 20, 50, 75 and 100. The four plots show the results for four separate choices of $\tan \beta$.

Figure 6

These plots show how the required corrections to ${\lambda }_H$ (solid green) compare to the upper and lower limits of the threshold corrections $\mathrm{\Delta }\lambda$ obtained from a scan, at a SUSY scale $\tilde{m}=m_{{\tilde{q}}_{L,3}}=m_{{\tilde{t}}_{R,3}}={10}^6\mathrm{GeV}$. The dashed green lines correspond to the $3\sigma$ upper and lower limits from $m_t$ uncertainty. Scanned independently are $\mu ,\{m_{{\tilde{q}}_{L,i}},m_{{\tilde{u}}_{R,i}},m_{{\tilde{d}}_{R,i}},m_{{\tilde{b}}_{R,3}},m_{{\tilde{\ell }}_{L,i}},m_{{\tilde{e}}_{R,i}},m_A\},\{M_a\}$ and $\tan \beta$. The top trilinear $A_t$ has been set to zero. Each mass is allowed to vary up to $\xi =100$ relative to $\tilde{m}$, with scanned points shown here for $\xi =0$, 10, 20, 50, 75 and 100. The four plots show the results for four separate choices of $\tan \beta$.

Figure 7

These plots show how the required corrections to ${\lambda }_H$ (solid green) compare to the upper and lower limits of the threshold corrections $\mathrm{\Delta }\lambda$ obtained from a scan, at a SUSY scale $\tilde{m}=m_{{\tilde{q}}_{L,3}}=m_{{\tilde{t}}_{R,3}}={10}^{10}\mathrm{GeV}$. The dashed green lines correspond to the $3\sigma$ upper and lower limits from $m_t$ uncertainty. Scanned independently are $\mu ,\{m_{{\tilde{q}}_{L,i}},m_{{\tilde{u}}_{R,i}},m_{{\tilde{d}}_{R,i}},m_{{\tilde{b}}_{R,3}},m_{{\tilde{\ell }}_{L,i}},m_{{\tilde{e}}_{R,i}},m_A\},\{M_a\}$ and $\tan \beta$. The top trilinear $A_t$ has been set to zero. Each mass is allowed to vary up to $\xi =100$ relative to $\tilde{m}$, with scanned points shown here for $\xi =0$, 10, 20, 50, 75 and 100. The four plots show the results for four separate choices of $\tan \beta$.

Figure 8

These plots show how the required corrections to ${\lambda }_H$ (solid green) compare to the upper and lower limits of the threshold corrections $\mathrm{\Delta }\lambda$ obtained from a scan, at a SUSY scale $\tilde{m}=m_{{\tilde{q}}_{L,3}}=m_{{\tilde{t}}_{R,3}}={10}^{16}\mathrm{GeV}$. The dashed green lines correspond to the $3\sigma$ upper and lower limits from $m_t$ uncertainty. The dashed blue lines shown for $\tan \beta =1$ correspond to $m_h\pm 1\mathrm{GeV}$. Scanned independently are $\mu ,\{m_{{\tilde{q}}_{L,i}},m_{{\tilde{u}}_{R,i}},m_{{\tilde{d}}_{R,i}},m_{{\tilde{b}}_{R,3}},m_{{\tilde{\ell }}_{L,i}},m_{{\tilde{e}}_{R,i}},m_A\},\{M_a\}$ and $\tan \beta$. The top trilinear $A_t$ has been set to zero. Each mass is allowed to vary up to $\xi =100$ relative to $\tilde{m}$, with scanned points shown here for $\xi =0$, 10, 20, 50, 75 and 100. The four plots show the results for four separate choices of $\tan \beta$.

Figure 9

Shown are ${\lambda }_H^{\mathrm{SM}}$ (black), ${\lambda }_H^{\text{tree}}$ (purple dot-dashed) and $\mathrm{\Delta }{\lambda }_H^{\mathrm{tot}}$ (blue), as a function of the matching scale ${\mu }_{\text{match}}=m_{{\tilde{q}}_{L,3}}=m_{{\tilde{t}}_{R,3}}$. The four panels correspond to different characteristic SUSY scales. Also shown are the various components that add up to give $\mathrm{\Delta }{\lambda }_H^{\mathrm{tot}}$, namely the correction from $\overline{\mathrm{MS}}$ to $\overline{\mathrm{DR}}$ (orange dashed), the scalar contribution (red dotted) and the Higgsino/gaugino contribution (green solid).

Figure 10

Shown are the regions in $M_a$, $\mu$ space, for degenerate scalars $\tilde{m}$, $\tan \beta =2.5$, 3.1 and $m_{{\tilde{q}}_{L,3}}=m_{{\tilde{t}}_{R,3}}={\mu }_{\text{match}}$ where ${\lambda }_H$ is matched to within 1, 1.96 and 3 sigma (green, orange, red). The matching has been performed at the scale ${\mu }_{\text{match}}={10}^6\mathrm{GeV}$.

Figure 11

Shown are the regions in $M_a$, $\mu$ space, for degenerate scalars $\tilde{m}$, $\tan \beta =1$, 1.2 and $m_{{\tilde{q}}_{L,3}}=m_{{\tilde{t}}_{R,3}}={\mu }_{\text{match}}$ where ${\lambda }_H$ is matched to within 1, 1.96 and 3 sigma (green, orange, red). The matching has been performed at the scale ${\mu }_{\text{match}}={10}^{10}\mathrm{GeV}$.

Figure 12

Shown are the regions in $M_a$, $\mu$ space, for degenerate scalars $\tilde{m}$, $\tan \beta =1$, 0.9 and $m_{{\tilde{q}}_{L,3}}=m_{{\tilde{t}}_{R,3}}={\mu }_{\text{match}}$ where ${\lambda }_H$ is matched to within 1, 1.96 and 3 sigma (green, orange, red). The matching has been performed at the scale ${\mu }_{\text{match}}={10}^{16}\mathrm{GeV}$.

Figure 13

Plot of the threshold corrections needed for exact gauge coupling unification. The numbers along the line are the scales ${\mu }_{*}$ at which the IR couplings are evaluated for unification and at which point the needed threshold corrections are computed and then plotted in the plane. The long straight line is assuming only the SM up to the highest scale. The second line that branches downward is for the case of superpartners existing at $10^{10}\mathrm{GeV}$, which lowers the needed threshold corrections at high scales. The results for this plot are obtained by using two-loop running in the SM and MSSM, and one-loop matching.