Subsections

• 2.1 Problems with the SM
• 2.2 Supersymmetry
• 2.3 The sparticles
• 2.4 R parity

2. Supersymmetry

2.1 Problems with the SM

One major problem with the Standard Model is something called the Higgs divergence problem. Figure 2 shows a Feynman diagram involving a Higgs boson (if you aren't familiar with Feynman diagrams and how to use them for calculations, now would be a good time to review appendix A). As was mentioned earlier, the Higgs boson gives masses to particles; this diagram is one of many that contributes to the Higgs boson's own mass. There are infinitely many such diagrams, involving more than one such fermion loop, and if one tries to calculate the Higgs mass correction, one gets a value which diverges to infinity. This is not very good.

Figure 2: A Higgs boson dissociating into a virtual fermion-antifermion pair.

Supersymmetry can solve this problem.

2.2 Supersymmetry

Supersymmetry postulates that for every Standard Model particle there is a corresponding supersymmetric particle (or ``sparticle'') which has a spin that is different by 1/2 unit. For example, the spin-1/2 electron will have a spin-0 supersymmetric partner called a ``selectron'', and the spin-1 photon has a spin-1/2 partner named a ``photino''³.

The existence of particles with exactly the same properties as the Standard Model particles, except for different spins, helps solve the divergence problem mentioned in the last section. For every diagram like Figure 2 there is a diagram that looks like Figure 3; these diagrams have the same vertices and coupling constants, and hence the same magnitude for the amplitude. But since the particle spins are different, the amplitude has the opposite sign. So when calculating the cross section using the prescription in appendix A, the amplitudes cancel yielding a finite interaction probability.

Figure: A Higgs boson dissociating into a virtual sfermion-antisfermion pair; this diagram cancels the one in figure 2.

We know that supersymmetry cannot be an exact symmetry--if it were, there would be particles with exactly the properties of the electrons (including mass) as the electron, except for the spin. But a charged spin-0 particle with exactly the same mass of the electron would certainly have been seen already. If supersymmetry is broken, then the sparticles may have much greater masses than ordinary particles (though they must have masses less than about 1 TeV in order for the cancellation of Figure 3 to work).

2.3 The sparticles

The minimal supersymmetric standard model (MSSM) is the Standard Model with the fewest changes such that supersymmetry can be incorporated. (There is a slight change in the Higgs sector: instead of a single Higgs boson: there must be five: $h^0$, $H^0$, $H^+$, $H^-$ and $A^0$.) We then postulate a superpartner for every Standard Model particle with the same coupling strengths. Unfortunately, this leads to a lack of predictive power--just as the masses in the Standard Model all the particle masses are arbitrary and must be measured, the same is true in the MSSM. In our studies, we work with more constrained models (for example, mSUGRA, a minimal-supergravity inspired model).

Table 3 shows the supersymmetric particles compared to the Standard Model particles.

Table 3: The Standard Model and supersymmetric particles.

Standard Model	Supersymmetry
$\gamma$, $Z^0$, $h^0$, $H^0$	$\chionez$, $\chitwoz$, $\chithreez$, $\chifourz$
$W^+$, $H^+$	$\chionep$, $\chitwop$
$e^-$, $\nu_e$, $\mu^-$, $\nu_\mu$, $\nu_\tau$	$\erm$, $\elm$, $\sneue$, $\murm$, $\mulm$, $\sneum$, $\sneut$
$\tau^-$	$\tauone$, $\tautwo$
$u$, $d$, $s$, $c$	$\upr$, $\upl$, $\downr$, $\downl$, $\stranger$, $\strangel$, $\charmr$, $\charml$
$b$	$\bottomone$, $\bottomtwo$
$t$	$\topone$, $\toptwo$

We can make a number of observations:

• There are four neutralinos $\SSp{\chi}_i^0$; each is a mixture of the photino ($\photino$), zino ($\zino$), and the two neutral Higgsinos ($\higgsinoz$ and $\Higgsinoz$).
• There are two charginos $\SSp{\chi}_i^+$; each is a mixture of the wino ($\winop$) and charged Higgsino ($\Higgsinop$).
• In the Standard Model, we don't bother distinguishing between left- and right-handed fermions, since they have the same mass and can be easily converted into each other by flipping the spins. Their supersymmetric partners are spin-0, so the partner of the left-handed fermion can be completely unrelated to the partner of the right-handed fermion, so the sfermions appear as two different states with different masses.
• Exception: in the Standard Model, there are only left-handed neutrinos (although this isn't quite true if neutrinos have mass), so there is only one sneutrino from each generation.
• Another exception: because the tau lepton and the bottom and top quarks are massive, their left- and right-handed partners can mix to form states known as $\tauone$, etc. (These states would be characterized by different branching ratios and anomalously low masses.)

2.4 R parity

In our studies we assume that R-parity is conserved (if that statement doesn't make sense, take a look at appendix B for more about conservation laws). The practical impact of this statement is that every supersymmetric interaction must involve two supersymmetric particles. In other words, all supersymmetric particles are produced in pairs, and if a SUSY particle decays, it must decay into another SUSY particle (plus any number of normal particles). This means that the lightest supersymmetric particle (LSP) cannot decay, since there aren't any supersymmetric particles that it can decay into. In our studies, we usually assume that the LSP is the lightest neutralino, $\chionez$.

Note, however, that there are a number of popular theories of supersymmetry called ``Gauge-mediated supersymmetry breaking'' which not only predict that R-parity is not conserved, but that the LSP will live a significant amount of time (and therefore travel macroscopic distances inside the detector) before decaying into Standard Model particles. One challenge is designing a detector that can detect slow-moving long-lived particles that don't decay until well after the time of the electron-positron collision.