Soft supersymmetry breaking

As mentioned earlier, if supersymmetry exists at all, it must be broken. The theoretical expecta-tion is a spontaneous symmetry breaking mechanism, which works in a manner analogous to the electroweak symmetry breaking in the ordinary SM: the Lagrangian of the underlying model is invariant under supersymmetry, but the vacuum state is not.

Many models of spontaneous symmetry breaking have been studied. In fact, the question of how supersymmetry is exactly broken is one of the most intriguing topics within theoretical supersym-metry. In the MSSM, however, our ignorance of the exact mechanism is simply parameterised by introducing extra terms that break supersymmetry explicitly in the effective MSSM Lagrangian.

7.1.3 Soft supersymmetry breaking 89 It has been shown that a softly broken supersymmetric theory (symmetry-breaking couplings of positive mass dimension) is free of quadratic divergences in quantum corrections to scalar masses, to all orders in perturbation theory [119].

The soft symmetry breaking terms in the MSSM read L^MSSMsoft = −1

2

³

M3egeg+M2fWWf+M1BeBe+ c.c.´

−³ e

ua_uQHe _u−eda_dQHe _d−eea_eLHe _d+ c.c.´

−Qe^†m²_QQe−Le^†m²_LLe−eum²_uue^†−edm²_ded^†−eem²_eee^†

−m²_HuH_u^∗H_u−m²_HdH_d^∗H_d−(bH_uH_d+ c.c.),

where,M₃,M₂, andM₁ are the gluino, wino, and bino mass terms. The second line contains the (scalar)³couplings, each ofa_u,a_d,a_eis a complex3×3matrix in family space, with dimensions of [mass]. The third line consists of squark and slepton mass terms. Again, each shownm²_Xis a 3×3matrix in family space. Finally, the last line displays supersymmetry-breaking contributions to the Higgs potential.

Unlike the supersymmetry preserving Lagrangian (LSUSY), the soft-symmetry-breaking Lagrangian terms (L^MSSMsoft ) introduce a large number of unknown parameters. A careful count reveals105 in-dependent parameters (masses, mixing angles, and phases) in the MSSM which cannot be removed or associated to measured SM parameters [120].

Many of these105MSSM parameters imply flavour mixing or CP-violating processes, which are severely restricted by experiment:

• Slepton mixing, implying the individual lepton numbers are not conserved, is strongly lim-ited by experimental bounds, for instance from the process µ → eγ (Br(µ → eγ) <

1.2·10⁻¹¹[121]). This sets tight limits on the off-diagonal entries ofm²_eandm²_L.

• The squark (flavour) mixing has strong experimental constraints from flavour changing neu-tral current (FCNC) measurements, such as the neuneu-tral kaon system. More constraints come from the neutral D system and the process b → sγ (Br(b → sγ) = (3.55 ±0.26)· 10⁻⁴ [122]), and other beauty or strange quark decays to lighter quarks. All of these pro-cesses would be allowed (enhanced) by flavour mixing soft-symmetry-breaking terms.

• Strict constraints on CP-violating phases follow from limits on the electric dipole moments of the neutron and electron [123].

• Further constraints can arise through virtual sparticles, as for instance in the anomalous magnetic moment of the muon (aµ= (11659208±6)·10⁻¹⁰[124]).

All of these CP-violating and flavour-changing effects of the MSSM can be avoided by assuming a more universal symmetry breaking. The resulting constrained MSSM has far fewer parameters.

In order to understand such simplification patterns inLSUSY, it is necessary to consider models in which supersymmetry is spontaneously broken.

The common approach in supersymmetry breaking models is to assume that the MSSM soft terms arise indirectly or radiatively. The origin of the symmetry breaking is some “hidden sector” of particles that have no direct couplings to the chiral supermultiplets in the “visible sector” of the MSSM. The two sectors share some interaction which mediates the supersymmetry breaking from the hidden to the visible sector and thus results in the MSSM soft terms.

There are mainly two competing models that aim at describing this mediating interaction. The first model proposes gravitational interactions, which are associated to new physics that enters near the Planck scale. The energy scale of the hidden sector should be of the order of10¹¹GeV in such gravity-mediated scenarios. This model is further described below.

The second model assumes that the flavour-blind mediating interactions are the SM electroweak and QCD gauge interactions. In thisgauge-mediated supersymmetry breaking(GMSB) scenario, the MSSM soft terms come from loop diagrams involving some messenger particles. These mes-sengers are new chiral supermultiplets that couple to supersymmetry-breaking terms and also have the SM gauge interactions. The scale of supersymmetry breaking in the GMSB is only of the order of∼10⁴GeV.

Gravity mediated breaking

This is the supersymmetry breaking model that has been employed for the studies of this thesis.

In the minimal form, calledminimal supergravity (mSUGRA), the soft terms in L^MSSMsoft are all determined by just four parametersm_1/2,m₀,A₀, andB₀:

M3 =M2 =M1 =m_1/2,

m²_Q=m²_u =m²_d=m²_L=m²_e =m²₀1, m²_Hu =m²_Hd =m²₀, a_u=A₀y_u, a_d=A₀y_d, a_e=A₀y_e,

b=B₀µ,

at a renormalisation scale Q ≈ M_P. This framework avoids the soft terms that imply flavour-changing and CP-violating processes. It is further highly predictive, the entire MSSM particle spectrum can be calculated from these four soft parameters plus one MSSM parameterµ. Theµ termµH_uH_dis part of the MSSM superpotentialW.

By demanding that the soft terms generate a scalar potential that gives the correct electroweak symmetry breaking,µandB₀can be traded forsgn(µ)andtanβ. Thetanβparameter is defined as the ratio of the Higgs vacuum expectation values,

tanβ=hH_u⁰i/hH_d⁰i.

It is easy to imagine that the essential physics of supersymmetry breaking is not captured by these five parameters of mSUGRA (or by the minimal GMSB with six parameters), but rather described by the general MSSM (which includes all theoretically viable couplings). However, the high number of parameters in the general soft-symmetry-breaking terms of the MSSM also introduces a tremendous arbitrariness. This can result in many processes, e.g. strong FCNC, that are already ruled out by experiment. Furthermore, in phenomenological studies it is highly impractical if not