In this text, I want to discuss a particular supersymmetric search that hasn't been published yet. According to some super-secret information, this search for some cool signatures of the sbottom squark will be published in 2-3 months.

However, I want to use this opportunity to describe the list particles of the Minimal Supersymmetric Standard Model (MSSM).

The MSSM is a supersymmetric counterpart of the Standard Model (SM). In both cases, they're models with the "minimum number of fields" that are compatible with certain symmetry constraints and with some features of particle physics we have already experimentally established.

Some people are imagining that the supersymmetric models are "less natural" but this is a deep misunderstanding. The MSSM is as minimal as the SM is in its own "category". The categories differ by having a different list of required symmetries. In this sense, they're qualitatively different so we should give them comparable prior probabilities to be true.

However, the supersymmetric models are actually a priori more likely because they're more constrained than the non-supersymmetric ones, not less so, and they may naturally explain the hierarchy problem, dark matter, and gauge coupling unification. String theory sheds some independent light on the claim that supersymmetric field theories are actually more natural and fundamental, and not less so, than their non-supersymmetric counterparts.

**Spectrum of the Standard Model**

I want to begin with the list of particles in the Standard Model, the model with the minimum number of fields that is compatible with the qualitative and some quantitative observations we have made in this Universe. This model contains particles of spin\[

j \in \left\{0, \frac 12, 1\right\}

\] Let us begin with the bosonic fields of spin \(j=1\). They are inevitably gauge fields corresponding to some local or gauge, Yang-Mills symmetries: such a symmetry is needed to get rid of the time-like component that would create ghosts and allow the probabilities to go negative. The gauge symmetry of the SM is \(SU(3)_c\times SU(2)_w\times U(1)_Y\). The first factor is the 8-dimensional group \(SU(3)\) mixing the three colors of quarks; the second factor is the 3-dimensional group \(SU(2)\) that rotates the two (or three) components of the electroweak doublet; the third is the 1-dimensional electroweak hypercharge symmetry. The latter two factors are broken, by the God mechanism, to \(U(1)_\text{em}\), the electromagnetic \(U(1)\) group, which is generated by the electric charge\[

Q = \frac{Y}{2} + T_3

\] where the conventional factor \(1/2\) in the first term isn't quite guaranteed to agree with other conventions. Great. So when we want to define the spectrum of a gauge theory such as the SM, we begin with the gauge group. For each of the \(8+3+1=12\) generators of the Lie group, there is a spin-1 field \(A_\mu\). As an off-shell field, it has \(4\) polarizations. But \(1+1=2\) (a famous identity) of them are eliminated by the gauge symmetry we require from all the physical states. So we are left with \(D-2=2\) transverse polarizations, say \(x\) and \(y\), for each of these twelve Lorentz vector fields. Well, this is true for the \(U(1)_\text{em}\) electromagnetic field and the \(SU(3)_c\) colorful gluon field (8 bicolor components).

The other gauge bosons (three vectors of them) are massive so instead of two polarizations, each of them has to have three components, in order to transform as a spatial vector in the rest frame of the gauge boson. But in some sense, the extra polarization should be viewed as coming from another field, a component of the God doublet. See Who ate the Higgs. I will discuss the lunch serving the God particle momentarily.

So the list of spin-one particles of the SM is\[

\gamma,\quad g,\quad W^+,\quad W^-,\quad Z^0

\] The first two entries, the photon and the gluon, are massless fields with two physical components (although gluon can't be isolated due to confinement and its color, much like quarks can't). The first two gauge bosons have two polarizations in space; each of the last three gauge bosons has three polarizations in space but the third one comes from degrees of freedom that will be discussed later.

The photon and the \(Z\)-boson are linear superpositions of the gauge bosons corresponding to the hypercharge \(U(1)_Y\) symmetry in the SM group and the third component of the \(SU(2)_w\) gauge group in the SM group. One has to diagonalize the mass operator in this 2-dimensional space because the mass operator has no reason to be diagonal in a predetermined basis (because this off-diagonal element violates no unbroken symmetries as it links objects with the same electric charge and spin).

Fine. Now let's discuss the spin-1/2 fields. By the spin-statistics theorem due to Pauli, they have to be fermionic. The most famous fermion is the electron, the lightest charged particle. It is described by a Dirac spinor field. The electron has two heavier cousins, the muon and the tauon.

Each of the three charged leptons also has its neutrino but it's just a Majorana spinor which has the same number of degrees of freedom as a Weyl fermion – one-half of a Dirac spinor. Finally, there are three generations of up-type quarks \(u,c,t\) and three down-type quarks \(d,s,b\). Each of them is described by a Dirac spinor. In fact, the number of fields they carry is three times higher because the field has an additional color index \(i=1,2,3\).

Each Dirac fermion may be spinning up or spinning down so it has two independent components but the antiparticle, generated by the Hermitian-conjugated fields, gives independent degrees of freedom. That's why we may say that each type of a Dirac fermion has 4 different "polarizations" and each type of a Weyl or Majorana fermion carries 2 different "polarizations".

Well, a Dirac spinor actually has 4 complex components i.e. 8 real ones. But for fermions, the canonical momenta aren't made out of time derivatives but from the non-differentiated fields themselves. That's why I will divide the "totally overall" number by two and still consider the electron's Dirac field to have "4 fermionic components".

At any rate, the list of the fermions is\[

(e,\mu,\tau), \quad (\nu_e,\nu_\mu,\nu_\tau), \quad (u,c,t), \quad (d,s,b)

\] The first two triplets are leptons (the first group are charged leptons; the second group are neutrinos); the last two triplets are quarks. The neutrinos are described by Majorana spinors; all other groups are described by Dirac spinors which are kind of pairs of two Weyl spinors (a different way of looking at Majorana spinors). The quarks have an extra three-fold color degeneracy.

All these fields have Hermitian conjugate fields that produce antiparticles. That's a special feature of the fermions; for the spin-1 and spin-0 bosons, there's a natural convention in which the fields are already Hermitian which means that the particles they create or annihilate are already identical with their antiparticles. For fermions, however, they're not identical. Well, if you interpret the neutrinos as Majorana fermions, you may say that they have 4 components but they're identical to their antiparticles, too. However, charged leptons and quarks are surely different from their antiparticles.

It may be helpful to describe the number of independent 2-component spinor fields that are needed to describe one generation of fermions in the SM. The electron needs two of them, two inequivalent Weyl spinors that add up to a Dirac spinor. Its neutrino only has one 2-component spinor, as I have mentioned; the other part of the Majorana (restricted Dirac) 4-component spinor is just the Hermitian conjugate of the first one.

Each quark species has two 2-component spinors again. But their number is actually 3 times higher due to the color, so the total number of 2-component spinors per generation is\[

2+1 + 3\times 2+ 3 \times 2 = 15

\] It's fifteen. Each of them carries a 2-valued Weyl spinor index. For all these 30 component fields, there is a Hermitian conjugate. The number 15 becomes interpreted as \({\bf 5}\oplus{\bf 10}\) in \(SU(5)\) grand unified theories. One may add the missing component of the right-handed neutrino to get sixteen, \({\bf 16}\), which may transform as an irreducible complex spinor representation of \(SO(10)\), a larger grand unified group which may be "obviously" obtained from string theory's \(E_8\) and \(SO(16)\) by taking very easy-to-see subgroups.

Finally, we have spin-0 bosons. In the SM, there is only one doublet of these fields, transforming as \({\bf 2}\) under the electroweak \(SU(2)_w\). Because it's not a real representation, the Hermitian conjugate fields are independent, so we really have four real components or four particles equivalent to their antiparticles. Because three generators of the \(SU(2)_w\times U(1)_Y\) are broken, three of the four God doublet components are eaten by the gauge bosons, giving the extra third polarization to the \(W^+,W^-,Z^0\) bosons, as discussed previously. So these three components have already been mentioned or "exploited" in our list of gauge bosons and we only have one new spin-0 particle, the God particle\[

h

\] denoted according to Peter Higgs, an important co-discoverer. This particle is identical to its antiparticle, much like in the case of the photon. If you count the total number of component fields (including expansions of all hidden indices), we have \[

2\times 1+ 2\times 8+ 3\times 3 + 1 = 2 \times 12 + 4 = 28

\] Hermitian bosonic physical fields and\[

3\times \left( 2\times 15 \right) = 90

\] fermionic Hermitian fields (plus their canonical momenta, so if we really counted all the Hermitian conjugates, we would get 180); the overall factor of \(3\) comes from three generations and the factor \(2\) takes the particle/antiparticle difference into account because the spin up-down degeneracy of e.g. electron has already been counted in the number 15. In the text above, I have mentioned that the photon and the \(Z\)-boson are being mixed with each other. I should emphasize that the same mixing or "rotation" of states to get the right eigenstates also applies to three generations of quarks as well as leptons.

**Going to MSSM**

The fields of the Standard Model probably sound familiar to pretty much everyone. How do we get the fields for the MSSM? Well, roughly speaking, we just double the fields content. For 180 fermionic spin-1/2 fields (plus their 90 canonical momenta), we add 90 real bosonic physical spin-0 fields (plus their velocities); and for those 28 bosonic component spin-0 and spin-1 physical fields, we add 28 spin-1/2 fermionic component Hermitian fields (plus their momenta).

Well, not quite. For the MSSM to work, we actually have to double the God sector; see Five faces of the God particle. So instead of one God doublet with 4 real components, we have two God doublets with 8 real components. Again, 3 of them are eaten by the three gauge bosons so we are left with 5 physical spin-0 God particles in the MSSM. Two of them are charged, being antiparticles of one another; three of them are neutral, coinciding with their antiparticles. Among the three neutral ones, one of them is CP-odd and the other two are CP-even.

Note that we have added 4 real spin-0 God bosonic components. That brings us to\[

28+4 = 32

\] Hermitian components for the bosons. Add the 90 Hermitian component superpartners of the SM fermions to see that we have 122 Hermitian bosonic component fields in the MSSM and the same number of the fermionic ones (the latter are equivalent to 61 two-component Weyl spinors and their Hermitian conjugates).

**Let's organize the MSSM particle content a little bit**

The smart linguists' way of writing down the particle content of the MSSM boils down to inventing new word for each superpartner of an SM particle. If the SM particle were fermion, its bosonic partner is called "s-something" and we have

sleptons: (selectron, smuon, stau), (electron sneutrino, muon sneutrino, tau sneutrino)I've added an "e" into "sestrange" because that's what the Czech language would do in such difficult linguistic situations. ;-) On the other hand, if the original SM particle is a boson, its new MSSM fermionic partner carries the name of the type "-ino". A smart linguist would call them as follows:

squarks: (sup, scharm, stop), (sdown, sestrange, sbottom)

photino, gluino, zino, wino (all those particles were gauginos), godinoThe godino is also technically known as Jesus-Bambino \(\tilde h\), especially by the most serious physicists who insist on precise terminology. ;-) A linguist could think that they have the same number of components as the original SM particles.

However, particle physics isn't just linguistics so the linguist's treatment wasn't quite accurate. One of the simple effects in the particle spectrum that go beyond linguistics is the mixing. Any two particles with the same spin and electric charge (aside from approximately conserved quantum numbers such as the baryon and lepton number) may mix with each other. There may be an off-diagonal element of the Hamiltonian that makes them oscillate into each other. We have to rediagonalize the mass matrix to get the actual mass eigenstates.

How does it affect the list of our allowed particles?

First, look at the sfermions i.e. the spin-0 partners of the leptons and quarks. One thing I haven't discussed in detail is that the superpartner of the Dirac spinor – of a pair of Weyl spinors – actually contains two complex bosons and not just one. Even if this complex scalar field is charged and the particle is different from its antiparticle, we still have two of them.

For example, there isn't just one complex scalar field called the selectron. There are two complex scalar fields. You may say that each of them arises from one of the Weyl spinors in the Dirac spinor. However, the Dirac spinor had 4 real fermionic components because the Lorentz symmetry required it; all of them have to have the same mass. However, the scalar fields fit into smaller multiplets and the 4 real components are reducible to 2 pieces. Each of them can have different masses.

It means that there are actually two electrons, the left-handed and right-handed selectron:\[

\tilde e_L, \tilde e_R

\] The same thing holds for smuon, stau, and the six squark flavors, but not for sneutrinos. However, because the left-handed and right-handed selectron have the same spin and electric charge, they may mix with each other, so better symbols for the mass eigenstates are\[

\tilde e_1, \tilde e_2

\] which are linear superpositions of the L and R selectrons. The tildes mean we are dealing with superpartners. The subscripts always label the particles or fields with similar spins and charges starting from the lightest mass eigenvalues. The same notation applies to smuons, staus, and all the squarks. In many cases, one may say that the lighter eigenstate is "almost exactly" the right-handed or "almost exactly" the left-handed one, and so on.

We must also appreciate that not only the left-handed and right-handed sleptons are mixing with each other, and similar for quarks, but the three generations of sleptons are mixing with each other, and the same thing holds for quarks. It may look tough but it doesn't really affect the linguists' words too much. We still have sleptons, squarks, selectron, stop, sbottom, sneutrinos, and so on.

**Gauginos and godinos**

A much more nontrivial mixing occurs to the fermionic superpartners of the bosonic SM particles, i.e. to gauginos and godinos. The only sparticle that isn't affected here is the gluino which has 8 bicolorful components that have nothing to mix with because they're the only elementary fermions in MSSM with the same color charges.

However, there is still the photino, zino, winos, and godinos. The photino has 2 real physical polarizations and so does the zino, one wino, and the other wino (I am counting the divine-origin longitudinal components to be a part of the God sector). The God sector contains 8 real components. In total, there are 16 real components of non-gluino fermionic superpartner fields (plus their canonical momenta) which may be described as 8 Weyl 2-component spinors or 8 Majorana spinors but the best description is a compromise, 4 Majorana spinors plus 4 Weyl spinors (the latter are organized into 2 Dirac spinors).

You may look at the charges. You will see that exactly 1/2 of them is electrically neutral, namely zinos, photinos, and one-half of the godino doublet; the remaining 1/2 may be divided to 1/4 plus 1/4 with charges equal to \(Q=\pm 1\). So the fermionic superpartners add up to four 2-component neutral Majorana fields that are complex conjugate to themselves if represented as 4-component Dirac spinors, and two charged Dirac fields.

If you check how many different mass eigenvalues are included in this set of particles, you will see that we have four Majorana neutralinos (coming from photino, zino, and neutral godino components) and two Dirac neutralinos:\[

\chi_1^0,\chi^0_2, \chi^0_3, \chi^0_4; \qquad \chi^\pm_1, \chi^\pm_2

\] The lightest neutralino \(\chi_1\) has been believed to be the LSP, the lightest superpartner that naturally wants to be the explanation of almost all the dark matter. I think it's more likely after 2011 that the LSP is the gravitino, the 2-real-polarization spin-3/2 superpartner of the graviton.

So instead of photinos, zinos, winos, neutral godinos, and charged godinos, we talk about four Majorana neutralinos and two Dirac charginos. I forgot to say that there's one more terminology that is sometimes helpful: instead of photinos and zinos, people usually undo the mixing of the gauge bosons (which is unhelpful for the superpartners anyway, they mix differently) and talk about binos and neutral winos, the superpartners of the components of the gauge bosons (which are not mass eigenstates) associated with the \(Y\) and \(T_3\) gauge generator, respectively. In many cases, phenomenologists say that the LSP is "nearly a bino" or "nearly a wino" (meaning the neutral wino).

LSP can't be electrically charged because it would interact with the electromagnetic field and therefore emit light, so it couldn't be dark matter and we would lose the explanation for dark matter. Sneutrinos may in principle be dark matter but it has some disadvantages.

**Decays of produced superpartners**

The R-parity may be violated and I tend to believe it probably is – which still allows the gravitino to be stable enough because the gravitational force needed for its destruction is weak and slow. However, when the particles are produced at the LHC, the R-parity is approximately OK during the production. So the new, R-parity-odd particles are pair produced.

Except for the LSP, they quickly decay; the LSP is among the final products. The LSP is dark and nearly invisible so it looks like missing energy at the LHC (just like the well-known neutrinos do). The number of events with a lot of missing energy has been shown to be small so if SUSY is relevant for the LHC scale, Nature chooses a method to avoid too many events with a lot of missing energy.

Although many SUSY phenomenologists expected the LHC to discover SUSY in the missing energy events, they have been proved wrong. Nature uses a different path. There are still many viable classes of SUSY models that agree with the limits set by the LHC – natural SUSY, stealth SUSY (with some new sector), heavy scalars, maximum mixing in the top sector, and of course most of the R-parity-violating model building which usually avoids missing energy signatures completely because none of the SM superpartners is stable, not even metastable. As long as the stop squark mass is close enough to the electroweak scale, SUSY still can successfully explain the unbearable lightness of the God particle's being. And so far, the stop squark may be almost as light as the top quark.

Still, the experimenters are working hard to exclude – or discover – the remaining possibilities.

One search that will be published in 2-3 months and that I am curious about involves some decays of the sbottom squark. The sbottom squark may be comparably light as the stop squark; SUSY sfermions have the unusual property that you expect the third generation to be actually lighter than the first one – there's no reason why the ordering should agree and there are reasons why it is probably reverted.

In the search, the sbottom squark may decay to the top quark and a chargino, or the bottom quark and the second-lightest neutralino. Note that the decay products don't include the lightest neutralino itself.

This is a funny scenario because these decays are comparably likely to the decays involving the lightest neutralino directly but they have an advantage: the chargino or second-lightest neutralino keeps on decaying to the lightest neutralino and the God particle or the Z-boson or the W-boson (or the virtual versions of these particles, usually denoted by an asterisk), respectively. So without much suppression, one may produce a God particle at the same moment.

Note that Murray Gell-Mann's version of the LHC stop squark rumor says that "one and maybe two" superpartners have been discovered. If the number "two" were the more accurate one, it's likely that the discovered particles would be stop and sbottom squarks, probably the lighter versions of them (or two stops? or stop and gluino? or stop and stau?). The experimenters are carefully chasing traces of a 200-500 GeV sbottom squark in their data. The lightest neutralino may be between 60 GeV and 450 GeV but at most equal to the sbottom mass minus 50 GeV or so. The scenarios in which the lightest neutralino mass is 60 GeV or 1/2 of the next neutralino (or lightest chargino) mass are being simulated in detail. The experimenters who ordered the Monte Carlo centers to simulate those seemingly "exotic" or "obscure" decays of the sbottom squarks think that the same computer analysis will be used by a larger number of "clients" which surely sounds interesting. ;-)

My estimate for the time scale after which we have a reasonable chance to hear some truly new potentially "positive" results about new physics, after those months with lots of "negative" results – still based on the 2011 LHC run – is about 2-3 months. I can't tell you my detailed reasoning here... But don't forget that the searches at the LHC are very far from over. In fact, even the evaluation of the searches through the already accumulated 2011 dataset is far from over.

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