SUSY breaking mechanisms

As discussed above, if SUSY is broken spontaneously, unacceptably light sparticles appear in the theory, which are experimentally ruled out. In the MSSM, an explicit, soft breaking is used instead (see [33], section 9.1). “Explicit” here refers to the addition of new terms to the Lagrangian density, and “soft” means that the new terms leave the cancellation of quadratic divergences to all orders of perturbation theory intact. Lsoft includes explicit mass terms for all sparticles (sfermions, gauginos, and higgsinos), an independent set of Yukawa matrices (A_u, A_d, A_e) similar to y_u, y_d and y_e introduced in equation (2.31),

Figure 2.2: The additional particles in the MSSM affect the gauge coupling evolution.

Using a χ²-fit for a single unification point, the scale at which SUSY particles become relevant and the slopes change was determined to be around1 TeV, resulting in a GUT scale of about„10¹⁶ GeV (from [34]).

but affecting sfermions only, as well as a contribution to the Higgs mass term,bH₁¨H₂. The MSSM introduces 105 new physical parameters [55].

A theory with this number of free parameters has limited predictive power, and it is clear that the soft parameters’ values eventually need to be derived from a common underlying principle. The problem of additional light particles appearing in the theory can be avoided if the spontaneous symmetry breakdown takes places in a “hidden sector”

of particles without SM gauge interactions. The effects of the spontaneous breakdown are then transmitted to the visible sector by messenger particles that are significantly more massive than the MSSM fields.

One can also choose gravity to play the messenger role, in which case the result-ing theory is called minimal supergravity (mSUGRA). One introduces universal SUSY breaking parameters: a common tree-level mass and a common Yukawa coupling for the sfermions (m₀ and A₀, respectively), a common gaugino mass M_1{2 and two constraints for the Higgs fields: the ratio of the their vev’s tanβ “

H₂⁰ {

H₁⁰

, and the sign of the mixed term coefficient, sgnpµq. These masses and couplings are defined at some high mass scale and are evolved down to the weak scale using renormalisation group equations (see [33], chapter 11). Other important approaches of this kind are anomaly mediated SUSY breaking (AMSB) and gauge mediated SUSY breaking (GMSB), dis-cussed in chapters 12 and 13 of the same reference.

Many of the MSSM parameters are already severely restricted by experiments, for example several off-diagonal elements in the slepton and squark mass matrices that can result in CP violation and flavour changing neutral currents (FCNC) beyond experimental

limits. We now illustrate this using the flavour changing transition between neutral kaons K⁰ and antikaonsK¯⁰, that in the SM occurs through loop contributions from the weak interaction (figure 2.3).

Figure 2.3: SM diagrams contributing to neutral kaon mixing and the K_L-K_S mass difference.

The CP eigenstates are K₁ “ pK₀`K¯₀q{?

2 and K₂ “ pK₀´K¯₀q{?

2. These two are distinguished by their decays: K₁ predominantly decays into two pions, while K₂ exclusively decays into three pions. However, K₁ and K₂ are not mass eigenstates, but mix slightly into KS9K1 `K2 and KL9K2`K1 (|| « 2.2ˆ10^´3), which are distinguished by their lifetime. Experimentally one finds that K_S has a short lifetime of cτ “ 2.7 cm and K_L has a longer lifetime of cτ «15 m. The diagrams in figure 2.3 contribute to the mass difference∆mK“mpKLq´mpKSq, measured to be about3.5µeV (tiny compared to the kaon mass of 497.6 MeV; all results in this paragraph are taken from [2]).

The MSSM allows several new FCNC loop diagrams contributing to the neutral kaon mixing (figure 2.4), and hence also to∆m_K. The contribution from squark-gluino loops has the form (see section 9.1.1 in [32])

∆m_K,MSSM“

where U and U˜ are unitary matrices diagonalising the quark and squark mass-squared matrices, f is some function of the squark mass eigenvalues and i and j label the two flavours of external squarks (s and d in this case, so i‰ j). There are several ways in which ∆mK,MSSM can vanish or remain small:

• mass degeneracy: if all squarks have identical mass,f will be a constant, and (2.34) will be proportional toδ_ij,

• alignment: if the mass matrices of quarks and squarks can be diagonalised by the same unitary transformation, then UU˜^: “1, again making the off-diagonal term contributing to neutral kaon mixing vanish,

• decoupling: if the sparticles in the loop are heavier than40TeV, their contributions are sufficiently suppressed.

All of these assumptions lead to a simplified phenomenology and have to be clearly stated for the interpretation of an experimental result. Excluded parameter regions may become significantly smaller when the assumptions are relaxed [56].

sL

d¯L

˜

g,χ˜⁰ g,˜ χ˜⁰

dL

¯ sL

˜ sL d˜L

d˜L ˜sL

sL

d¯L

˜

sL d˜L

dL

¯ sL

˜ g,χ˜⁰

˜

g,χ˜⁰ s˜L

d˜L

Figure 2.4: Some of the MSSM diagrams contributing to neutral kaon mixing and the K_L-K_S mass difference. Another type of diagram has charginos and up-type squarks running in the loop.

It is possible to remain agnostic about the details of the SUSY breaking mechanism, and use other inputs to reduce the parameter space. Natural SUSY models put particular emphasis on naturalness requirements and the constraints derived from those [57, 58]. As an example, the following MSSM relation holds at lowest order in perturbation theory:

´m²_Z{2“ |µ|²`m²_H2. (2.35) It directly relates a known SM mass to the µ parameter affecting Higgs and higgsino masses and to the soft mass term of the up-type Higgs doublet, m²_H2, which is ă0 as required for EWSB. For this relation to be fulfilled, the terms on the right-hand side need to be balanced against each other. If the superpartners are too heavy, this balance requires fine-tuning of the SUSY breaking parameters; it becomes more delicate and theoretically less well motivated. This directly affects higgsino masses (controlled by µ) as well as the masses of sparticles with a strong Higgs coupling, most notably top and bottom squarks, but also gluinos through second-order loops.

Naturalness constraints allow to build models with a light Higgs boson mass com-patible to the experimentally observed value and in accordance with the experimental sparticle mass limits, without relying too much on particular values for the remaining model parameters. Natural SUSY predicts the top squarks, the left-chiral bottom squark, the higgsinos, and the gluinos to be the lightest sparticles (figure 2.5).

The phenomenological MSSM (pMSSM) [59, 60] assumes CP-conservation and min-imal flavour violation [61] and uses a variety of existing theoretical and experimental constraints to reduce the number of independent, real model parameters to 19 (or 22, ref. [59]): 10 sfermion masses (assuming the first and second generation to be mass

de-H˜

˜tL

˜b_L

˜tR

˜

g