Soft masses in supersymmetric SO(10) GUTs with low intermediate scales

The specific shape of the squark, slepton and gaugino mass spectra, if measured with sufficient accuracy, can provide invaluable information not only about the dynamics underpinning their origin at some very high scale such as the unification scale $M_G$, but also about the intermediate scale physics encountered throughout their renormalization group equations evolution down to the energy scale accessible for the LHC. In this work, we study general features of the TeV scale soft supersymmetry breaking parameters stemming from a generic mSugra configuration within certain classes of supersymmetry $SO(10)$ GUTs with different intermediate symmetries below $M_G$. We show that particular combinations of soft masses show characteristic deviations from the mSugra limit in different models and thus, potentially, allow to distinguish between these, even if the new intermediate scales are outside the energy range probed at accelerators. We also compare our results to those obtained for the three minimal seesaw models with mSugra boundary conditions and discuss the main differences between those and our $SO(10)$ based models.

Figure 1

Gauge-coupling unification in Model I in two limits corresponding to different positions of the sliding $SU(2{)}_R\times U(1{)}_{B-L}$ breaking scale $V_R$. In solid lines, we depict the RGE behavior of the gauge couplings for $V_R$ in the vicinity of the electroweak scale $V_R\sim {10}^4\mathrm{GeV}$ while the dashed lines correspond to $V_R\sim {10}^{14}\mathrm{GeV}$. The position of the intersection region shifts slightly up with rising $V_R$ but the corresponding scale remains intact.

Figure 2

Gauge-coupling unification in Model II in two limits corresponding to different positions of the sliding $SU(2{)}_R\times U(1{)}_{B-L}$ breaking scale $V_R$. In solid lines, we depict the RGE behavior of the gauge couplings for $V_R$ in the vicinity of the electroweak scale $V_R\sim {10}^4\mathrm{GeV}$ while the dashed lines correspond to $V_R\sim {10}^{14}\mathrm{GeV}$. The position of the intersection region shifts slightly up with rising $V_R$ but the corresponding scale remains intact.

Figure 3

Running in the Model III variant of the low-LR scale SUSY SO(10). Here the SO(10) gauge symmetry is broken first into a Pati-Salam intermediate stage stretching from the unification point down to the relevant energy scale $V_{\mathrm{PS}}$ (in the middle) and, subsequently, to the $L\mathrm{\text{−}}R$ symmetry stage. The value of $V_{\mathrm{PS}}$ is correlated to the position of the $L\mathrm{\text{−}}R$ breaking scale $V_R$ which can again slide from as low as few TeV up to roughly ${10}^{14}\mathrm{GeV}$, c.f. FIG. 8.

Figure 4

Running in Model IV with a low $B-L$ scale. Note the effects of the $U(1)$ mixing in the running and matching; the lowest curve corresponds to the off-diagonals of the $(G{G}^T/4\pi {)}^{-1}$ matrix.

Figure 5

The MSSM-like nonunification triangle in Models I and III with $v_R={10}^{14}\mathrm{GeV}$ for two different values of the unknown soft-SUSY breaking scale ($m_{\mathrm{SUSY}}=1\mathrm{TeV}$ for the upper one and $m_{\mathrm{SUSY}}=500\mathrm{GeV}$ for the lower). The upper sides of the triangles corresponds to ${\alpha }_L^{-1}$ while the lower-left sides depict the effective ${\alpha }_Y^{-1}$ defined as $\frac{3}{5}{\alpha }_R^{-1}+\frac{2}{5}{\alpha }_{B-L}^{-1}$. The light blue area surrounding the ${\alpha }_S^{-1}$ line represents the $1\sigma$ uncertainty in ${\alpha }_s(M_Z)$ as given in 57. Both triangles move down for lower values of $v_R$, see Figs. 1, 2.

Figure 6

The $v_R$-dependence of the leading-log invariants in Model I, c.f. Sec. 1b1a. The bands represent the error due to the nonexact gauge-coupling unification depicted in Fig. 5. For practical reasons, the numerical values of the invariants QE and DL have been scaled down by a factor of 10.

Figure 7

The same as in Fig. 6 but for Model II of Sec. 1b1b. The QU and LE behavior differs from that in Fig. 6 mainly in the low-$v_R$ regime.

Figure 8

The correlation of the intermediate symmetry-breaking scales in Model III (allowed region coloured). The contours correspond to the quality of the fit of ${\alpha }_S(M_Z)$ for each choice of the Pati-Salam breaking scale $v_{\mathrm{PS}}$ an the LR breaking scale $v_R$, within $1\sigma$ (white area within the colored band), $2\sigma$ (yellow) and $3\sigma$ (orange) of the range quoted in 57.

Figure 9

Intermediate-scale-dependence of the RGE invariants Model III, see Sec. 1b2. For each of the four invariants, the solid curve corresponds to ${\alpha }_S(M_Z)$ fixed at its central value and the dashed and dotted lines refer to the $-1\sigma$ and $+1\sigma$ trajectories, respectively; c.f. Fig. 8.

Figure 10

The parameter space of Model IV of Sec. 1c. The contours correspond to the different quality of the ${\alpha }_S(M_Z)$ fit for different choice of the $L\mathrm{\text{−}}R$ breaking scale $v_R$, namely, to $1\sigma$ (white), $2\sigma$ (yellow) and $3\sigma$ (orange) values for the range quoted in 57.

Figure 11

The $v_{BL}$-scale dependence of the RGE invariants in Model IV. For practical purposes the figure has been split into two panels. For each of the four invariants, the solid curves correspond to ${\alpha }_S(M_Z)$ fixed at its central value while the dashed and dotted lines refer to the $-1\sigma$ and $+1\sigma$ trajectories corresponding to roughly $v_R\sim 2\times {10}^{15}\mathrm{GeV}$ and $v_R\sim 4\times {10}^{15}\mathrm{GeV}$, respectively; c.f. Fig. 10.

Figure 12

The MSSM squark and slepton spectra mSugra and Models I, II and III calculated for the SPS3 benchmark point, i.e. for $m_0=90\mathrm{GeV}$ and $M_{1/2}=400\mathrm{GeV}$. In all cases, $v_R={10}^3\mathrm{GeV}$ and $v_{\mathrm{PS}}\sim {10}^7\mathrm{GeV}$ in Model III. From bottom to top the horizontal lines correspond to: $m_{{\tilde{e}}^c}$ (light blue), $m_{\tilde{l}}$ (blue), $m_{{\tilde{u}}^c}$ (orange), $m_{{\tilde{d}}^c}$ (light orange) and $m_{\tilde{q}}$ (purple). We do not show the results for Model IV in this figure, since they are very similar to the mSugra case.