3.2 SUSY Breaking Scenarios

It is clear that the SUSY is a broken symmetry, if it exists at all. It should not arbitrarily be broken, however, as long as it is meant to solve the naturalness problem: only Soft Supersymmetry Braking (SSB) terms are allowed to tame the quadratic divergence of the Higgs mass correction¹. Phenomenologically viable models can thus be classified in terms of how the SSB takes place and how it is transmitted to our observable sector. In almost all of the models, SUSY is broken dynamically at a high scale and then this breaking is mediated to our low energy world. Various SSB parameters at the high scale of SUSY breaking are determined by the choice of the SSB mechanism and the mediation mechanism. Various theoretical and experimental considerations restrict the scale to a rather big range 10⁴ GeV < M_SSB < M_Pl. The low energy values of the SUSY breaking parameters are then decided by evolving them back to the weak scale via renormalization group equations. Consequently the sparticle masses and, in cases where mixing occurs, even their couplings depend on the SSB mechanism. Once these low energy values of sparticle properties are measured, therefore, we can in principle point towards the physics at high scale and hence at the SUSY breaking mechanism.

As already mentioned, in the early days of SUSY model building there existed essentially only one class of models where the SSB is transmitted via gravity to the low energy world. The past few years changed the situation drastically and now we have a set of different models that include the following:

1)

Gravity mediated models that include (minimal) SUGRA (mSUGRA), (constrained) MSSM (cMSSM), etc., where supergravity couplings of the fields in the hidden sector with the SM fields are responsible for the SSB terms. The difference between mSUGRA and cMSSM lies in the fact that the former fixes the Higgsino mixing mass parameter $\mu$by demanding the radiative breaking of the EW symmetry, while the latter leaves it as a free parameter. Both assume universality of the gaugino and sfermion masses at the high scale. These models always have extra scalar mass parameter m₀² which needs fine tuning so that the sparticle exchange does not generate FCNC effects, at an unacceptable level.

2)

Anomaly Mediated Supersymmetry Breaking (AMSB) models, for which supergravity couplings that induce mediation are absent and the SSB is caused by loop effects. The conformal anomaly, which is always present, generates the SSB terms and the sparticles acquire masses due to the breaking of scale invariance. Note that this contribution exists even in the case of mSUGRA/MSSM, but is much smaller in comparison with the tree level terms that exist in those models. This mechanism becomes a viable one for solely generating the SSB terms, when the quantum contributions to the gaugino masses due to the `superconformal anomaly' can be large [6,7], hence the name Anomaly mediation for them. The slepton masses in the simplest model of this kind are tachyonic and require some other SUSY breaking mechanism to obtain phenomenologically acceptable mass spectrum. One way to fix this problem is to introduce a scalar mass parameter m₀².

3)

Gauge Mediated Supersymmetry Breaking (GMSB) models [8], where the SSB is transmitted to the low energy world via a messenger sector through messenger fields which have gauge interactions. These models have no problems with the FCNC and do not involve any scalar mass parameter.

4)

Models where the SSB mediation is dominated by gauginos [9]. These models are based on the brane world scenarios, where the brane (our world) on which the matter particles and their superpartners live is separated in the bulk from the one that is responsible for the SUSY breaking. Consequently, the wave functions of the matter particles and their superpartners on the SUSY breaking brane are suppressed, whereas those of the gauginos are substantial, due to the fact that the gauge superfields live in the bulk. Hence the matter sector feels the effects of SUSY breaking dominantly via gauge superfields. As a result, in these scenarios, one expects $m_0 \ll m_{1/2}$, reminiscent of the `no scale' models.

All of these models clearly differ in their specific predictions for various sparticle spectra, features of some of which are summarized in Table 3.1[10], where the usual messenger scale parameter $\Lambda$ had been traded for M₂ for ease of comparison.

 

Table 3.1: Predictions of different types of SUSY breaking models for gravitino, gaugino, and scalar masses. $\alpha _{i} = {g_{i}^{2}}/{4 \pi }$ (i=1,2,3 corresponds to U(1), SU(2) and SU(3), respectively), b_i are the coefficients of the ${-g_{i}^{2}}/{(4 \pi )^{2}}$ in the expansion of the $\beta$ functions $\beta _{i}$ for the coupling g_i and a_i are the coefficients of the corresponding expansion of the anomalous dimension. The coefficients G_i are the squared gauge charges multiplied by various factors which depend on the loop contributions to the scalar masses in the different models.

Model	$m_{\tilde{G}}$	(mass)2 for gauginos	(mass)2 for scalars
mSUGRA	${{M_{SSB}^{2}} / {\sqrt{3} M_{pl}}}$ $\sim$ TeV	$({\alpha_{i}}/{\alpha_{2}})^{2}$ M22	$m_{0}^{2} + \sum_{i} G_{i} M_{i}^{2}$
cMSSM	$M_{SSB} \sim 10^{10} - 10^{11}$ GeV	$\mbox{ }$	$\mbox{ }$
GMSB	$({\sqrt{F}}/{100TeV})^{2}$ eV	$({\alpha_{i}}/{\alpha_{2}})^{2} M_2^2$	$\sum_{i} G_{i}^{'} M_{2}^{2}$
$\mbox{ }$	10 $< \sqrt{F} < 10^4$ TeV
AMSB	$\sim$ 100 TeV	$({\alpha_{i}}/{\alpha_{2}})^{2} ({b_{i}}/{b_{2}})^{2} M_2^2$	$\sum_{i} 2 a_{i} b{i} ({\alpha_{i}}/{\alpha_{2}})^{2} M_2^2$

As one can see, the expected gravitino mass varies widely in different models. The SUSY breaking scale $\sqrt{F}$ in GMSB model is restricted to the range shown in the table by cosmological considerations. Since SU(2) and U(1) gauge groups are not asymptotically free, i.e., b_i are negative, the slepton masses are tachyonic in the AMSB model, without a scalar mass parameter, as can be seen from the third column of the table. The minimal cure to this is, as mentioned before, to add an additional parameter m₀², not shown in the table, which however spoils the RG invariance. In the gravity mediated models like mSUGRA, cMSSM, and most of GMSB models, gaugino masses unify at high scale, whereas in the AMSB models the gaugino masses are given by RG invariant equations and hence are determined completely by the values of the couplings at low energies and become ultraviolet insensitive. Due to this very different scale dependence, the ratio of gaugino mass parameters at the weak scale in the two sets of models are quite different: gravity mediated models and GMSB models have M₁ : M₂ : M₃ = 1 : 2 : 7, whereas the AMSB model has M₁ : M₂ : M₃ = 2.8 : 1 : 8.3. The latter therefore, has the striking prediction that the lightest chargino $\tilde{\chi}^\pm_1$ and the lightest supersymmetric particle (LSP) $\tilde{\chi}^0_1$, are almost pure SU(2) gauginos and are almost mass-degenerate. The expected sparticle spectra in any given model can vary a lot. But still one can make certain general statements, e.g. the ratio of squark masses to slepton masses is usually larger in the GMSB models as compared to mSUGRA. In mSUGRA one expects the sleptons to be lighter than the first two generation squarks, the LSP is expected mostly to be a bino and the right-handed sleptons are lighter than the left-handed sleptons. On the other hand, in the AMSB models, the left- and right-handed sleptons are almost degenerate. The above mentioned degeneracy between $\tilde{\chi}_{1}^{\pm}$and ${\tilde{\chi}_1^0}$ is lifted by the loop effects [11]. For $\Delta M$ = $m_{\tilde{\chi}_{1}^{\pm}}$ - $m_{\tilde{\chi}_{1}^{0}}$< 1 GeV, the phenomenology of the sparticle searches in AMSB models will be strikingly different from that in mSUGRA, MSSM, etc. In the GMSB models, the LSP is gravitino and is indeed `light' for the range of the values of $\sqrt{F}$ shown in Table 3.1. The candidate for the next lightest sparticle, the NLSP, can be $\tilde{\chi}^0_1$, $\tilde{\tau}_1$, or $\tilde{e}_R$ depending on model parameters. The NLSP life time and hence the decay length of the NLSP in lab is given by $L = c\tau \beta \gamma \propto \frac{1}{(M_{NLSP})^5}$ $(\sqrt{F})^{4}$. Since the theoretically allowed values of $\sqrt{F}$ span a very wide range as shown in Table 3.1, so do those for the expected life time and this range is given by 10^-4 < c $\tau \gamma \beta$ < 10⁵ cm. Since the crucial differences in different models exist in the slepton and the chargino/neutralino sector, it is clear that the leptonic colliders which can study these sparticles with the EW interactions, with great precision, can really play a crucial role in model discrimination.

The above discussion, which illustrates the wide `range' of predictions of the SUSY models, also makes it clear that a general discussion of the sparticle phenomenology at any collider is far too complicated. This makes it even more imperative that we try to extract as much model independent information as possible from the experimental measurements. This is one aspect where the JLC can really play an extremely important role.