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Elementary Particles and the World of Planck Scale


Supersymmetry differs from all other symmetries in that it relates two classes of elementary particles which are so fundamentally different -- the fermions and the bosons. According to supersymmetry, every "ordinary" particle has a companion particle differing in spin by half a unit, but has the same mass. However, since no superpartners have been detected so far (as indicated by ATLAS's data), it means that supersymmetry must be broken by some unknown mechanism resulting in differences in mass. Many models are introduced to explain this fact. The SUGRA in Figure 15-12b is one of them. Anyway, all supersymmetric formulism is invariant when the particle is replaced by its superpartner as shown in Figure 15-12a, where equivalent processes appear in supersymmetry theory for photon absorption (by electron) in Standard Model.
Superpartner Note that the replacement is not so arbitrary, the processes have to obey the conservation of R-parity. Supersymmetry so simplifies the mathematics of quantum field theory and String Theory that it allows theoriests to obtain solutions that would otherwise be far beyond their calculating ability. The general idea is for the unification of all forces of nature including quantum gravity.

Figure 15-12a Superpartner
[view large image]

Since the graviton has spin 2, while the other gauge bosons have spin 1 and 0, supersymmetry is used to mix them. Starting with the graviton state of spin 2 and acting by supersymmetry generators we get the following chain of states : spin 2 spin 3/2 spin 1 spin 1/2 spin 0.

The 6th column in Figure 15-12b shows the names and symbols for all the superpartners -- "s" in front of the fermion superpartner, "ion" behind the boson superpartner, and a "~" on the top of a symbol to designate the superpartner. The mass difference produced by broken symmetry is shown in the insert in term of size (even though they are all point particles in theory).

As mentioned in the section on GUT, the X particles have mass of the order 1015 Gev, while electroweak bosons have mass of the order 102 Gev. It is expected that new physics such as GUT opened up at some high energy scale would have an effect on lower energy quantities such as the mass of the electroweak bosons and the Higgs particle. But the huge difference in mass requires fine tune of the contributions from virtual X particles (some of which are "+" large numbers, others are "-" large numbers) to better than one part in 1012. Such difficulty goes under the name
Supersymmetry State of the Universe of "hierarchy problem" (Figure 15-12d). Supersymmetry, however, leads to delicate cancellations in the computation of these masses in an entirely natural way. Hence, the enormous difference between the electroweak scale and GUT scale is an uncontrived feature of supersymmetric models, i.e., supersymmetry provides a natural explanation. This cancelling mechanism becomes more important as it is discovered that the known mass of the Higgs and top quark implies an universe in metastable state (Figure 15-12c). It requires an explanation to show why the universe is still here and we are still alive. That's why the failure (up till 2014) of the LHC to find evidence of supersymmetry almost propels a crisis in physics.

Figure 15-12b Supersymmetry
[view large image]

Figure 15-12c State of the Universe [view large image]

Hierarchy Problem

Another motivation for supersymmetry is its intimate connection with gravity. The supersymmetry considered so far is global. However, if the SUSY generator is local - meaning that it depends on spacetime, and we impose invariance to a theory under such transformation, then the formulation forces the introduction of a gauge field that turns out to have the properties of a graviton. In fact this new type of theory is just Einstein's general relativity within the framework of quantum fields and is thus called supergravity. The problem with supergravity is the divergence. Although it is not as divergent as ordinary gravity, it is still not finite. The infinities cannot be canceled out at the three loops level. Thus such attempt to merge general relativity with quantum mechanics ultimately met with failure, the more promising application is associated with ten dimensional string theories. Supergravity is the low energy limit where the structureless point particle is a good approximation. See "Unitarity Method" for an update.

Figure 15-12d Hierarchy Problem [view large image]

Supersymmetry also addresses a host of other mysteries in modern physics such as the tremendous concentration of energy in the universe (the cosmological constant problem), the origin of cosmic inflation, matter/antimatter asymmetry, the nature of cold dark matter, and the special forms of the Higgs interactions.

Supersymmetry Breaking If supersymmetry were an exact, unbroken symmetry, the superpartners would have the same mass of the ordinary particles. However, no such particles have ever been observed, and supersymmetry, therefore, if it is a true symmetry of particle physics, must be broken. If the breaking of supersymmetry is in such a way that the explanation for the hierarchy problem is still valid, then the mass of the superpartners would be in the order of 103 Gev - just at the mass range accessible to the new generation of accelerators. All current models of supersymmetry breaking predict flavor-changing interactions. These are processes that change quarks or leptons into their other generation - processes not observed in experiments. How to break supersymmetry but prevent flavor changing is a crucial challenge if supersymmetry is to succeed in addressing the hierarchy problem. Figure 15-12e depicts a model developed by Lisa Randall. It resolves the flavor-changing problem with two branes sequestered (separated) in a fifth dimension. In the model, the Standard model particles are on one brane, and particles that break supersymmetry are sequestered on the other. Gravitons in the fifth dimension serve as the intermediary particle that carry the effect of supersymmetry breaking to the Standard model particles. Such form of interaction would generate the necessary superpartner masses (in the 250 Gev range), but do not cause quarks or leptons to change to another flavor particles.

Figure 15-12e Supersymmetry Breaking [view large image]

The idea of supersymmetry can be expressed in simple mathematics such as:
Qi|Fi> = |Bi> and Qi|Bi> = |Fi>
where |Fi> and |Bi> are fermionic and bosonic states respectively. The operator Qi is called supersymmetry (SUSY) generator (also known as supercharge), which acts to transform these states into each other. The number of SUSY generators characterizes the theory when we add one or more generators to the fields of the standard model or other theories such as the theory of string. The altered theory is then adjusted to remain invariant under the SUSY transformation - resulting in new fields and associated particles (see for example the case of adding supersymmetry to the string theory). If there is one SUSY generator, then the new theory has N = 1 supersymmetry, e.g., the minimally supersymmetric standard model or MSSM. The N = 2 supersymmetry theory has 2 SUSY generators, ... and so on.

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