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3.6 Tests of SUSY Breaking Mechanisms

We have seen above that the precision measurements of the masses and the mixings in the sfermion and the chargino/neutralino sector will certainly allow us to quantitatively test various supersymmetry relations, thereby firmly establishing the existence of supersymmetry as a dynamical symmetry of particle interactions. More importantly, the precision we expect for these measurements at the JLC is so high that we will be able to infer the SUSY breaking scale and the values of the SUSY breaking parameters at this high scale. Since these SUSY breaking parameters are presumably determined by the physics at the high scale, this will open up a first realistic way to systematically study the physics at this high scale experimentally.

Some examples of such studies are already mentioned above, including tests of universality among scalar masses and gaugino mass unification expected in the SUGRA models. These tests can be carried out step by step in the course of the SUSY discovery and study scenarios. We can hope that, eventually, we will be able to determine all the masses and the mixings of the sparticle spectrum, thereby fixing all of the Lagrangian parameters such as $M_1,M_2,\mu,\tan \beta$ at the electroweak (EW) scale. In this section, we review what and to what extent we can learn then from these precision measurements of the masses and the mixings.

One can use these accurately determined Lagrangian parameters at the EW scale to fit their values at the high scale. It has been shown [48] in this approach, that the projected accurate measurements of the various sparticle masses through an energy scan for each sparticle's threshold, using ten energy points with 10 ${\rm fb^{-1}}$ each, allows one to determine $M_1, M_2,m_0, \mu$, and $\tan{\beta}$ at the high scale to an accuracy better than $1 \%$12. The accuracy is much worse, however, for higher values of $\tan{\beta}$. The expected accuracy for the trilinear term is rather poor as shown in Tables 3.5 and 3.6 taken from Ref. [48], which is due to the fact that most of the physical observables are rather insensitive to the parameters Ak.

Table 3.5: Reconstruction of SUGRA parameters assuming universal masses.
$\mbox{ }$ True value Error
m0 100 0.09
m1/2 200 0.10
A0 0 6.3
tan$\beta $ 3 0.02
$\mbox{ }$ $\mbox{ }$ $\mbox{ }$
Table 3.6: Reconstruction of SUGRA parameters with nonuniversal gaugino masses.
$\mbox{ }$ True value Error
m0 100 0.09
M1 200 0.20
M2 200 0.20
A0 0 10.3
tan$\beta $ 3 0.04

In this approach model selection will be made on the basis of goodness of the fit.

There is, however, a totally different approach to test the SUSY breaking mechanisms, the bottom up approach[50], where one starts with these Lagrangian parameters extracted at the EW scale and extrapolates them to the high scale using the renormalization group evolution (RGE). As explained in Section 3.2, different SUSY breaking mechanisms predict different relations among these parameters at the high scale. In the bottom up approach, one can test these relations directly by reconstructing them from their low energy values using the RGE. In Ref.[50], experimental values of the various sparticle masses are generated in a given scenario (mSUGRA, GMSB etc.), by starting from the universal parameters at the high scale appropriate for the model under consideration and evolving them down to the EW scale. These masses are then endowed with experimental errors expected to be reached eventually in the combined analyses of data from LHC and a TeV scale linear collider with an integrated luminosity of $1 {\rm ab}^{-1}$. These measured values are then evolved back to the high scale. The left two plots in Fig. 3.43

Figure 3.43: Bottom up approach of the determination of the sparticle mass parameters for mSUGRA and GMSB[50]. Values of the model parameters as given there.
Eps files of left figure and left figure

show results of such an exercise for the gaugino and sfermion masses, respectively, for the mSUGRA case, while the one on the right for sfermion masses in GMSB. The width of the bands indicates $95\%$ C.L. limits.

We can see that with the projected accuracies of measurements, the unification of the gaugino masses will be indeed demonstrated very clearly in the mSUGRA case. The extrapolation errors for the evolution of the slepton masses are also small, since only the EW gauge couplings contribute to it, and allows one to test the unification of the slepton masses at the high scale. This is in contrast to the squark and higgs masses, for which the extrapolation errors are rather large. In the case of the Higgs mass parameters this insensitivity to the common scalar mass m0 is due to an accidental cancellation between different contributions in the loop corrections to these masses which in turn control the RG evolution. In the case of the squarks the extrapolation errors are caused by the stronger dependence of the radiative corrections on the common gaugino mass, due to the strong interactions of the squarks. As a result of these the small error at the EW scale expands rapidly, when extrapolated to the unification scale. It should also be noted that the trilinear A coupling for the top shows a pseudo fixed point behavior, which also makes the EW scale value insensitive to m0. If the universal gaugino mass m1/2 is larger than the m0 then this pseudo fixed point behavior increases the errors in the determination of the third generation squark mass at the EW scale. This picture shows us clearly the extent to which the unification at high scale can be tested. If we compare this with the results of Tables 3.5 and 3.6, we see that with the bottom up approach we have a much clearer representation of the situation. It should be stressed that the $95\%$ C.L. bands on the squark and the higgs mass parameters become much wider if one only assumes the accuracies expected to be reached at the LHC[51]. The linear collider's precision measurements are indispensable to get a clearer picture of the SUSY breaking mechanism at the high scale.

The right-most plot in Fig. 3.43 shows the results of a similar exercise but for the GMSB model, where with the assumed values of the model parameters, one would need to have a 1.5 TeV linear collider to access the full sparticle spectrum. In this case the doublet slepton mass and the Higgs mass parameter are expected to unify at the messenger scale, which the data show quite clearly. It is remarkable that the extrapolation of the reconstructed slepton and squark masses to the high scale is stable enough to reveal the entirely different unification patterns expected in this case as opposed to the mSUGRA case. The bottom up approach of testing the structure of the SUSY breaking sector definitely requires the high accuracy that can be achieved only with the linear collider[51].

All of these discussions assume that most of the sparticle spectrum are accessible jointly by the LHC and a TeV scale linear collider. If we are unlucky and the squarks are super-heavy13, then perhaps the only clue to their existence can be obtained through the analogue of precision measurements of the oblique correction to the SM parameters at the Z pole. These super-oblique corrections [53] modify the tree-level supersymmetry relations between various couplings as already mentioned in Section 3.3.2. These modifications arise if there is a large mass splitting between the sleptons and the squarks. The expected radiative corrections imply

$\displaystyle {{\delta g_Y} \over g_Y}$ $\textstyle \simeq$ $\displaystyle {{11 g_Y^2} \over {48 \pi^2}}
ln \left({m_{\tilde{q}}} \over {m_{\tilde l}}\right),$ (3.18)

which amounts to about $0.7 \%$, if the mass splitting is a factor of 10. We have shown in the previous sections that it might well be possible to make a measurement with such accuracy at the JLC.

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