Last week I attended an interesting course about Supersymmetry theory (SUSY) at CP3. The course was meant to be an introduction to SUSY, but in fact it treated this theory in deep detail from a mathematical point of view. It developed all the (endless!) calculations from which the supersymmetric lagrangian can be inferred.

I’m not going to write this post in such a detail, of course. Instead, I would be happy to give you a general idea about this theory so discussed today by scientists all over the world.

The so called Standard Model (SM) is a theory concerning the classification of all the subatomic particles, explaining how they interact with each other through the weak, electromagnetic and strong interactions. Despite the great phenomenological success, the Standard Model has several theoretical limits and it can be considered as part of a more general and fundamental theory. Among all the theories beyond the Standard Model, SUSY is one of the most accredited. According to SUSY, the SM particles have supersymmetric partners, called “sparticles”, characterized by masses at the scale of TeV.

Let’s now see in detail why SUSY can be considered a valid extension of the Standard Model.

The Standard Model has been proven experimentally several times: the discovery of the neutral currents (1973) and of the W and Z bosons (1982-1983) are just two examples of its validity. The Large Electron-Positron collider (LEP) at CERN confirmed the quantitative predictions of the SM with an extraordinary precision, at higher perturbative orders with respect to the tree-level.

Despite the impressive phenomenological success, the SM has some “theoretical problems”:

- it has 18 free parameters, maybe a too high number for a complete theory of fundamental interactions;
- the gravitational interaction is not treated within it; a quite reasonable choice, you may think, since the intensity of its coupling is 10
^{-38}… but a theory of fundamental interactions has to take it into account; - it does not answer the fundamental questions: “Why three families of fermions?”, “Why do some interactions have the typical structure left/right?”;
- according to the SM, all the neutrinos have exactly zero mass, but the neutrino oscillations, experimentally seen, imply that their mass is different from zero;
- in the strong interaction sector, the SM provides for a term that breaks CP symmetry, but experimentally no such violation has been found;
- the Hierarchy problem.

*A clarifying note about the Hierarchy problem…*

*According to the SM, the corrected mass of the Higgs Boson taking into account scalar loops is m ^{2}_{H }(corr) ∼ m^{2}_{H}+cΛ^{2}, where Λ is the cut-off parameter, that provides the scale of energy to which the SM is not valid anymore. We can reasonably suppose that this scale is comparable to that of the great unification (about 10^{16} GeV): but this assertion implies that the corrected value of the Higgs mass is on the order of the cut-off parameter. However, we know experimentally that the mass of the Higgs boson does not exceed hundreds of GeV. The only way to solve this problem seems to admit the existence of a theory that can be valid to a scale comparable to energies of TeV, to make sure that the corrections to the Higgs mass remain small enough to fit the experimental value.*

Supersymmetry is a theory that puts in relation fermionic and bosonic states, according to the following rules:

Q|B〉 = |F〉

Q^{†}|F〉 = |B〉

In other words, the Q-operator transforms fermions into bosons and viceversa.

Within this theory, the fermionic states and the corresponding bosonic ones have the same mass. You can see the superpartners of all the fundamental particles in fig. 1.

But you can easily deduce that these superpartners can’t have the same mass as the SM particles… otherwise we would have already detected them! So, we have to break this symmetry by introducing a *soft-breaking* term inside the SUSY lagrangian.

The spontaneous electroweak symmetry breaking implies that states with same charge, spin and color can mix; also in the SUSY scenario (given the fact that the higgsinos and the gauginos are the supersymmetric partners of the Higgs boson and of the gauge bosons respectively), the neutral higgsino and gaugino states mix, and this fact leads to the definition of four mass states, the *neutralinos*: χ̃_{1}^{0}, χ̃_{2}^{0}, χ̃_{3}^{0}, χ̃_{4}^{0}.

In the same way, the mixing of the charged higgsino and gaugino states leads to the definitions of four *charginos: χ̃ _{1}^{±}, χ̃_{2}^{±}*.

One of the main consequences of SUSY is that the coupling constants (that vary with energy) will have a different behavior with respect to the SM scenario. As you can see from fig. 2, in a SUSY framework, the strong, weak and electromagnetic coupling constant will intersect in the same region at 10^{17} GeV; the SM does not provide such a unification. Could it be a proof of the Great Unification? Maybe!

Moreover, SUSY seems to give a satisfactory solution to the problem of Dark Matter (the matter that makes up 27% of our universe): it could originate from the Lightest Supersymmetric Particle (LSP), the neutralino χ̃_{1}^{0}, supposed that it is stable.

Finally, this theory seems to solve the Hierarchy problem since the SUSY particles’ masses are not supposed to exceed the TeV scale.

Of course, this post is meant to give just a general idea about the Supersymmetry theory. If you’re interested and you want to know more, you can download and read the article at the following link:

http://arxiv.org/abs/hep-ph/9709356

À plus!

12 March 2016 at 16:44

Hi there, nice summary. But for serious amateur mathematical physicists, can I ask: is there anyway to get a copy of notes for the course you cited that gave “all the (endless!) calculations from which the supersymmetric lagrangian can be inferred”.

Every few years I think it’s time I really understood more about supersymmetry from a more technical viewpoint, but all the books I’ve collected are dense and poorly written, and so a usually demure because after a few hours of intense effort I find I have other things in life more important. So that’s why I’m asking. When I find a really great book, well written, and highly non-trivial in the mathematics, then I can afford to make the time to read it through. to the guts

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16 March 2016 at 11:34

Hi,

Sorry for the little delay in my reply. However I’m pleased you appreciate this sort of mini-introduction and I understand you perfectly when you say that it’s a big effort to deal with this kind of calculations. The theory behind SUSY is not trivial at all and I believe it requires dedicated (and also quite long) courses. The article I pointed you to is quite specific in the calculations, but if you want to know more you can contact Christopher Smith (Grenoble): he’s the guy who gave the lectures! If you want to try with a real SUSY Theory book, you can have a look at “Weak scale Supersymmetry” by Baer and Tata. I found it pretty nice. Hope this can help!

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