Chiral and gauge symmetry conspire to keep the radiative corrections to the fermions and gauge boson masses at a manageable logarithmic level. The only sector of the Standard Model theory whose mass corrections diverge quadratically is the Higgs vector boson. Al-though fermions and gauge bosons masses’ divergence is only logarithmic these very cor-rections impinge on the Higgs potential and in turn on the cut-off Λ. There is no intrinsic symmetry in the theory that can help cure this divergence. From the Feynmann rules we know that any perturbative contribution to observables swap signs when the role of fermions and bosons are interchanged. This simple property has lead to the idea of supersymmetry (SUSY) ([47],[48],[49]). This symmetry envisages the existence of a whole set of yet unseen particles mirroring the existing known particles. The spin of those new particles has to differ from the standard particles by one half to allow for the mentioned sign swap and thus for the desired divergence cancellation in the perturbative expansion. In the symmetry parlance it ought to exist a set of symmetry generators Q that relate fermions and bosons:
Q|F >= |B >; Q|B >= |F > (2.2)
By assuming such a symmetry we unavoidably achieve the desired cancellation of the diverg-ing mass terms. The generators Q carry spin 1/2 and can be proven to obey, under certain restrictions, the following (anti)commutation algebraic relations
{Q, Q†} = Pµ (2.3)
{Q, Q} = {Q†, Q†} = 0 (2.4)
µ µ †
where Pµ is the space-time translation operator. From those equations we can imply that the single particles are arranged in multiplets containing both fermions and bosons. The operator −P2 commutes with the operators Q and Q† hence the particles in the same multiplet have the same masses. The same two operators commute with the generators of the gauge transformations therefore the partners in the multiplet must have the same charge, isospin and colour and on the same line of reasoning it can be shown that each multiplet must contain an exact number of fermions and bosons for the commutation rules 2.3, 2.4 and 2.5 to hold true. It is remarkable that once we introduce this symmetry and the group of superparteners with it the expression 2.1 becomes
m2h ∼ m20+ g(m2f˜− m2f)ln(Λ/mf˜) (2.6)
it is clear that the introduction of the supersymmetry has reduced the divergence of the renormalized Higgs mass. From equation 2.6 we see that the symmetry does not have to be exact (mf˜= mf), what is required is that the dimensionless coupling constant is the same for the Standard Model particles and the superpartners and that the physical masses of the superpartners are not too far from the ones of the corresponding SM particle. In such an extension each of the known fundamental particle must be in a chiral or gauge multiplet and be coupled with a superpartner with a spin differing of 1/2 unit. The widely accepted nomenclature for this set of superpartners is to add an s (for ”scalar”) in front of the existing Standard Model particle (e.g. squark, selectron, smuon, stau) and in short notation a tilde is added to the particle name to denote the sparticle name (˜e, ˜µ, ˜u). Sleptons, although they include the symbol L or R for their helicity, do not actually carry any real handedness being bosons but this nomenclature is just kept to remind the SM lepton partner helicity. A
summarised in table 2.7
Names spin 0 spin 1/2 spin 1 SU(3)C, SU(2)L, U(1)Y squarks, quarks (˜uL, ˜dL) (uL,dL) (3, 2, 1/6)
(×3 families) u˜∗R u†R (¯3, 1, −2/3)
d˜∗R d†R (¯3, 1, 1/3)
sleptons, leptons (˜ν, ˜eL) (ν, eL) (1, 2, −1/2)
(×3 families) e˜∗R e†R (1, 1, 1)
Higgs, higgsinos (Hu+, Hu0) ( ˜Hu+, ˜Hu0) (1, 2, +1/2) (Hd0, Hd−) ( ˜Hd0, ˜Hd−) (1, 2, −1/2)
gluino, gluon ˜g g (8, 1, 0)
winos, W bosons W˜± W˜0 W± W0 (1, 3, 0)
bino, B boson B˜0 B0 (1, 1, 0)
(2.7)
In this table e = (e, µ, τ ) are the three leptons, u = (u, s, b) and d = (d, c, t) are the quarks. The fifth column indicates the transformation properties of the supermultiplet under the action of the Standard Model gauge symmetries. The superpartners of the unbroken Standard Model bosons are the the gluino, the Wino and the Bino while the Higgsinos are the counterparts of the vector bosons fields. This exact symmetry must be broken otherwise we would have observed the superpartners long time ago. After symmetry breaking the new mass eigenstates of the theory are expressed as a linear combination of the unbroken model eigenstates. The charged Higgsinos and the Winos combine to give two charginos ( ˜χ±1,2) while the neutral ones together with the Bino mix into four neutralinos ( ˜χ01,2,3,4). In the
Mχ =
The extension to the Standard Model with the minimal particle content is called MSSM.
In this model every particle has a superpartner differing only in the spin properties as already mentioned. It is interesting to highlight that there are already two know particles whose spin differs 1/2 units and with identical gauge quantum numbers: the Higgs boson and the neutrino. It must be said that those two particles cannot be part of the same supermultiplet because this would lead to lepton number violation and this is indeed ruled out on phenomenological grounds. In principle SUSY theories admit the presence of barion and lepton number violating interactions in the Lagrangian. Such an occurrence would lead to an unobserved high proton decay rate and other processes like µ → eγ or µ → eee would have already been detected. This freedom is completely different from the Standard Model where gauge invariance guarantees the absence of any barion/lepton number violating interaction. If we define the global discrete symmetry
RP = (−1)3B+L+2s (2.8)
where B(L) is the barion(lepton) number while s is the particle spin. There are two broad classes of supersymmetric theories namely the ones that violate (RPV ) and the ones that conserve (RPC) R parity. The most important phenomenological difference between those
and the decay chains always end up into the lightest supersymmetric particle (LSP). In this theories the LSP is actually stable and if electrically neutral and only weakly interacting it will escape detection at the experimental facilities accounting for a large fraction of the initial energy at collision to escape undetected (e.g. large missing energy) providing a very effective way of discriminating supersymmetric events. There is strong experimental evidence that if the LSP does exist it must be weakly interacting and neutral because otherwise it could be possible to find relic LSP bound into atoms and this is ruled out with a very good precision. Within MSSM the LSP can be only the lightest neutralino or the sneutrino and the neutralino is a top candidate for dark matter. In SUSY theories that include gravitation the gravitino can be the LSP but although of cosmological relevance such an occurrence would prove meaningless in an experimental framework because the LSP would interact only gravitationally. In RPV theories the superparticles can decay completely into Standard Model particles and the analysis is thus much more complex.