3$\,TeV. We list in the second column of table~\ref{tab:spectra} some details of a sample parameter set from the model plane in figure~\ref{fig:200}. The Higgs mass in this case is again very slowly varying across the plane and takes the value $m_H \sim 125$\,GeV for this choice of $\tan \beta = 3$. In comparison, had we chosen $\tan \beta = 5$, the $(M_1, M_3)$ plane would look almost identical but with $m_H \sim 131$\,GeV. This and larger values of $\tan \beta$ are therefore excluded for this value of $A_0/m_0$. We have also studied smaller values of $A_0/m_0$ and found no consistent solutions of the electroweak symmetry-breaking conditions for $M_3 \lesssim 500$\,GeV for $A_0/m_0 = 1$ and no consistent solutions across the plane at somewhat lower $A_0/m_0$. This is also the case for $\tan \beta = 10$ and $A_0/m_0 = 1.5$, for which $m_H \sim 134$\,GeV. Larger values of $A_0/m_0$ also give values that tend to increase $m_H$ and, if increased too much, the stop becomes the LSP and eventually tachyonic. Next we consider some sample $(M_1, M_3)$ planes with $m_0 = 20$\,TeV and $\tan \beta = 5$, corresponding to the lower end of the $m_\chi$ plateau in figure~\ref{fig:elofig6}. Figure~\ref{fig:20.2} is for the case $A_0/m_0 = 1$, where we see in the left panel that electroweak symmetry breaking is possible up to values of $M_1 \lesssim 14$\,TeV. There is a gluino coannihilation strip close to the colored LSP boundary for $M_1 \lesssim 9$\,TeV. This is terminated by a spur extending to large $M_3$ when 9\,TeV $\lesssim M_1 \lesssim 10$\,TeV, where the lighter chargino is the LSP. There is no chargino coannihilation strip along the boundary of this region at large $M_3$, because the relic density is too high: for these values of $M_1$ and $M_3$, the Higgsino mass is too large and other coannihilations are not sufficient to bring the relic density down. At larger values, 10\,TeV $\lesssim M_1 \lesssim$ 11\,TeV, there is a Higgsino-gluino coannihilation strip, which is followed at larger $M_1$ by a focus-point strip~\cite{fp,fp.m001,fp.m002,fp.m003,fp.m004,fp.m005} hugging the electroweak symmetry breaking boundary where the neutralino is well-tempered~\cite{wt}.\footnote{A one-parameter extension of the CMSSM with lighter gluinos was considered in~\cite{hb} in the context of a mixed Higgsino-Bino neutralino.} In this case we see both the 124 and 125\,GeV Higgs mass contours and, as in the previous example, $m_H$ is compatible with experiment whenever the dark matter density falls within the allowed range. Because the relic density strip is a multi-valued function of $M_3$, the structure of the gluino-neutralino mass difference $\Delta M$ (blue curve) and the neutralino mass (red curve) shown in the right panel of figure~\ref{fig:20.2} are more complicated than in the previous cases. After growing to a local maximum $\sim 170$\,GeV when $M_3 \sim 1$\,TeV, $\Delta M$ starts to fall at larger $M_3$. We then see a change in behaviour at $M_3 \approx 1800$\,GeV along the gluino coannihilation strip. Here, the neutralino becomes Higgsino-like and, as $M_1$ is increased, the coannihilation strip tends toward lower $M_3$ with an increasing mass difference, as seen in the lower branch of the blue curve. A Higgsino LSP emerges for larger $M_1$ because it gives a positive contribution to the up Higgs soft mass from renormalization group running. As the up Higgs soft mass goes to zero so does $\mu$ and the Higgsino becomes the LSP. Once $\mu$ is small enough, the Higgsino can be a thermal relic without any assistance in setting the relic density from other particles. In this focus-point-like region, the mass difference increases beyond the range displayed. This behaviour is correlated with the value of $m_\chi$ (red curve), which increases monotonically to $\sim 4$\,TeV. When the Bino/Higgsino transition occurs at $M_3 \approx 1800$\,GeV, $m_\chi$ doubles back down to $M_3 \approx 500$\,GeV. Then, on the focus-point branch of the relic density strip, the LSP is mostly Higgsino, the value of $M_3$ grows, and the lightest neutralino mass takes the characteristic value $m_\chi \sim 1$\,TeV. We list in the third, fourth and fifth columns of table~\ref{tab:spectra} some details of sample parameter sets from the model plane in figure~\ref{fig:20.2}. In figure~\ref{fig:20.3} we choose a larger value of $A_0/m_0 = 1.5$, and we see in the left panel a gluino coannihilation strip that extends to $M_1 \sim 14$\,TeV, along which $m_H$ varies between 124 and 126\,GeV as seen by the three Higgs mass contours. The focus-point Higgsino dark matter region has disappeared, due to the large $A_0$ driving the Higgs mass to large negative values. The end-point of the gluino coannihilation strip is clearly seen in the right panel of figure~\ref{fig:20.3}, where $\Delta M \to 0$ (blue curve) at $M_3 \simeq 3300$\,GeV. Qualitatively, this case is similar to that shown in figure~\ref{fig:200}, rather than to figure~\ref{fig:20.2} with its truncated gluino coannihilation strip. In this case, the LSP mass (red curve) rises monotonically to $m_\chi \sim 7.5$\,TeV at the end-point of the strip. We list in the sixth column of table~\ref{tab:spectra} some details of a sample parameter set from the model plane in figure~\ref{fig:20.3}. \begin{figure} \centering \includegraphics[height=7cm]{M3vM18as_5_20000_15-eps-converted-to.pdf} \quad \includegraphics[height=7cm]{md5-eps-converted-to.pdf} ]]>

3$. The extended theory now has four parameters: $m_{3/2}$, $\tan \beta$, $c_{10}$, and $y_t'$. The gaugino masses in these models are generated by anomalies~\cite{anom,anom.m001,anom.m002,anom.m003,anom.m004}. Because the contributions to gaugino masses are proportional to $m_{3/2}$ times the gauge coupling beta function, the scalar masses tend to be much heavier than the gaugino masses, reminiscent of split supersymmetry~\cite{split,split.m001,split.m002,split.m003,split.m004}. With the addition of the $\mathbf{10}$ and $\mathbf{\overline{10}}$, the anomaly-mediated contributions to the gaugino masses are\footnote{The anomaly mediated gaugino masses are proportional to the gauge coupling beta functions. The addition of a $\mathbf{10}$ and $\mathbf{\overline{10}}$ alters the gauge coupling beta functions and so alters the gaugino masses. As it turns out, the additional contribution of the $\mathbf{10}$ and $\mathbf{\overline{10}}$ to the $\SU(3)$ beta function completely cancels the SM contribution at one-loop. Therefore, the total anomaly mediated contribution to the gluino mass is zero at one-loop.} \begin{eqnarray} M_{1} &=& \frac{48}{5} \frac{g_{1}^{2}}{16 \pi^{2}} m_{3/2}\,, \label{eq:M1} \\[\jot] M_{2} &=& \frac{g_{2}^{2}}{4 \pi^{2}} m_{3/2} \,, \label{eq:M2} \\[\jot] M_{3} &=& 0\ . \label{eq:M3} \end{eqnarray} In addition, the gauginos then get rather large threshold corrections from the $\mathbf{10}$ and $\mathbf{\overline{10}}$ when they are integrated out, which is in addition to the large threshold correction coming from integrating out the Higgsinos: for more details see~\cite{evo}. Since the only contribution to the mass of the gluino comes from the threshold corrections, it tends to be lighter than in typical PGM models. Hence there are regions where the gluino can coannihilate with the Bino, yielding the possibility of a relatively heavy Bino dark matter candidate. Our results for the PGM model with vector $\mathbf{10}$ and $\mathbf{\overline{10}}$ multiplets can be displayed in $(c_{10}, m_{3/2})$ planes for fixed values of the Yukawa coupling, $y_t'$ and $\tan \beta$. Two examples of these planes are shown in figures~\ref{fig:EVLSM1} and~\ref{fig:EVLSM2}. In the former, we have fixed $\tan \beta = 3$ (mainly to get an acceptable value for the Higgs mass, $m_H$, over the range of $m_{3/2} \le 600$\,TeV shown) and $y_t'^2 = 0.15$. In the left panel, we see a large red shaded region at small $c_{10}$ where the gluino is the LSP. To the right of this boundary, we see the gluino coannihilation strip.\footnote{The strip becomes less well defined at $m_{3/2} \gtrsim 350$\,TeV due to inaccuracies of the relic density calculation at such large masses.} In the lower right corner, the pink shaded region is excluded because one or more of the scalar components of the new vector matter have become tachyonic. As in previous figures, the Higgs mass contours are shown as red dot-dashed curves as labelled. Within the theoretical uncertainties, the Higgs mass agrees with experiment over the part of the plane that is shown. In the right panel we see, as before, the gluino-neutralino mass difference $\Delta M$ along the gluino coannihilation strip (blue) and the neutralino mass along the strip (red). We see that the curve for $\Delta M$ has the same characteristic shape due to strong coannihilations involving the gluino and peaks at $\simeq 170$\,GeV at $m_{3/2} \simeq 200$\,TeV when $m_\chi \simeq 3$\,TeV. The end-point of the coannihilation strip occurs at $m_{3/2} \simeq 500$\,TeV where $m_\chi \simeq 8.3$\,TeV.\footnote{The maximal value of the LSP mass that is compatible with it being a viable dark matter candidate in this case is similar to the CMSSM case, even though there are additional squarks to mediate $\chi q\to q \tilde g$. This is because the limiting reaction is gluino-gluino annihilation, which is the same in the two cases.} We list in the last two columns of table~\ref{tab:spectra} some details of sample parameter sets from the model planes in figures~\ref{fig:EVLSM1} and~\ref{fig:EVLSM2}. We recall that $A_0 = 0$ in these models, and display in parentheses (\ldots) the model values of $c_{10}$. \begin{figure} \centering \includegraphics[height=7cm]{m0vc56s_3_0_15.pdf} \quad \includegraphics[height=7cm]{mdY1f.pdf} ]]>

128$\,GeV (yellow).\footnote{We recall also that the Higgs mass is sensitive to the choice of $\tan \beta$.} The right panel shows that the gluino-neutralino mass difference is almost independent of $y_t'^2$. We do not show the neutralino mass for these cases, as it is largely independent of $y_t'$ and is determined from the gravitino mass and can be read from either of the two previous figures. \begin{figure} \centering \includegraphics[height=7cm]{m0vc56s_3_0_65.pdf} \quad \includegraphics[height=7cm]{mdy2fs-eps-converted-to.pdf} ]]>