Phenomenological and constrained supersymmetry

space, phenomenological studies are very difficult. Thus we need a guiding principle to reduce the amount of free parameters. We address this issue in the next section.

experiments. With further assumptions on the underlying SUSY breaking mechanism these models can be highly predictive with only a few free parameters (see below). Alternatively, in a bottom-up approach, the SUSY parameters can be chosen directly at the EW scale, only guided by phenomenological arguments (as discussed above) and, optionally, generic parameter relations that are typically obtained in high-scale models.

Constrained (high-scale) MSSM

As briefly mentioned in Section 2.3, one of the motivations for SUSY is the gauge coupling unification at a scale M_GUT∼2·10¹⁶GeV, suggesting that the MSSM is embedded in a grand unified theory. In that case it seems natural to have all gauginos in the same representation of a unifying simple gauge group [118,225,226], leading to equal gaugino masses at the GUT scale,

M₁=M₂ =M₃ ≡m_1/2 at M_GUT. (2.58)

In the same way, the universality conditions Eqs. (2.56) and (2.57) should be assumed at M_GUT. In minimal high-scale models, further universality of the soft-breaking masses and trilinear couplings is often assumed:

M_{`˜</sub><sup>2</sup> =M<sub>˜e</sub><sup>2</sup>=M<sub>q˜</sub><sup>2</sup> =M<sub>u˜</sub><sup>2</sup>=M<sub>d</sub><sup>2</sup><sub>˜</sub>=m<sup>2</sup><sub>H1</sub> =m<sup>2</sup><sub>H2</sub> ≡m<sub>0</sub> at M<sub>GUT</sub>, (2.59) A<sub>`} =Au =A_d≡A0 at M_GUT. (2.60) Note that, due to the RGE running, the scalar masses and trilinear interactions are in general not universal at the electroweak scale. This can lead to small contributions to FCNCs, which are however typically consistent with the experimental constraints [227].

In addition to these three universal soft-breaking parameters we need two more parameters to specify the MSSM Higgs sector, see Section 2.4.3. A convenient choice are the parameters tanβ and sgn(µ), where tanβ is the ratio of the vevs of the two Higgs doublets and sgn(µ) is the sign of the Higgs mixing parameter µ, cf. Eq. (2.52). The magnitude of µ and the soft-breaking parameterB, cf. Eq. (2.55), are fixed by the minimization of the scalar potential, see Section2.4.3. Thus, this high-scale SUSY model is completely specified by the five parameters m₀, m_1/2, A₀, tanβ, sgn(µ). (2.61) This model is known as the constrained MSSM (CMSSM). It can set in relation to the class of supergravity models [228], where supersymmetry is promoted to a local symmetry and thus provides a natural incorporation of gravity. The minimal supergravity (mSUGRA) model, which is a subset of the CMSSM parameter space, can be obtained by imposing certain simplifying assumptions on the supergravity Lagrangian [229], thus this model is also well motivated from a theoretical perspective.

The (experimentally accessible) particle spectrum at the EW scale is obtained from the evolution of the RGEs³⁰. An example of the RGE running of the soft-breaking masses from M_GUTto the EW scale is shown in Fig.2.12for a typical CMSSM parameter point, specified by m0= 80 GeV,m_1/2= 250 GeV,A0 =−500 GeV, tanβ = 10 andµ >0 atMGUT[22]. Due to the different gauge charges, the squark and slepton masses evolve quite differently, i.e. the squarks

30Explicit formulae for the one-loop RGEs can e.g. be found in Ref. [22]

2 4 6 8 10 12 14 16 18 Log10(Q/1 GeV)

0 100 200 300 400 500 600

Mass [GeV]

m₀ m_1/2 (µ²+m₀²)^1/2

squarks sleptons M₁ M₂ M₃ H_d H_u

Figure 2.12: Typical RGE running of slepton (red lines), squark (blue lines) and gaugino (black lines) soft-breaking masses fromMGUTto MZ in the CMSSM. Also shown is the running of (m²_H

1+µ²) and (m²_H

2 +µ²). The latter runs negative and thus triggers electroweak symmetry breaking. Taken from Ref. [22]. Note the slightly different notation, H_d≡H₁,H_u≡H₂.

become much heavier than the sleptons at the EW scale since they are strongly interacting.

The RGE running is furthermore influenced by the Yukawa couplings which lower the sparticle masses at the EW scale. Thus, the third generation squarks and sleptons are lighter than those of the first and second generation.

Fig.2.12 also shows the running of the quantities (m²_H

1+µ²) and (m²_H

2+µ²) which appear in the Higgs scalar potential, see Section2.4.3. The large top-Yukawa coupling leads to a large decrease of the up-type Higgs mass parameter,mH2, going from MGUT to the EW scale, such that (m²_H2+µ²) eventually runs to negative values at the EW scale, leading to EWSB. Thus, SUSY provides adynamical mechanism for EWSB [230,231].

The gaugino masses feature the same running as the corresponding gauge couplings, hence M3 increases and M1 and M2 decrease when going from MGUT to the EW scale. At every energy scale, we have the relation (up to small two-loop effects [232])

M₁ g₁² = M₂

g²₂ = M₃

g₃² = m_1/2

g²_GUT, (2.62)

with g1 = ^p5/3gY, the strong coupling constant g3 = gs and the universal gauge coupling g_GUT'0.73 at M_GUT. Expressed in terms of the weak mixing angle, we thus have

M1 = 5

3tan²θWM2, (2.63)

and an approximate relationM₃:M₂ :M₁ '7 : 2 : 1 at the EW scale.

Before the start-up of the LHC, results from global fit analyses [233,296] to low-energy and astrophysical observables suggested that the CMSSM, if realized in Nature, features a relatively light SUSY particle spectrum. In particular, these studies predicted a great coverage of the preferred CMSSM parameter space by the discovery potential of the first LHC run. However, no signal has been seen in any of the SUSY searches, excluding the previously preferred parameter

space.

The current status of the CMSSM, taking into account the exclusion limits from SUSY searches from the first LHC run as well as the measurements of the discovered Higgs boson, has been studied extensively in the literature, see e.g. Refs. [233–235,249,296]. The remaining allowed parameter space features a rather heavy sparticle spectrum, with masses of the squarks, gluino and the heavier EW gauginos being well above 1 TeV. Furthermore, the lightest CP -even Higgs boson is very SM-like in this parameter region, and the lightest neutralino and scalar tau lepton have similar masses of ∼300−600 GeV [234]. Besides these grim prospects for a potential discovery at the LHC, the model looses more and more its attractiveness when naturalness arguments are considered [249] (see Section 2.4.3 for a brief discussion of the so-called Little Hierarchy Problem).

The phenomenological MSSM (pMSSM)

In a more phenomenological approach the SUSY parameters are specified directly at the EW scale. In this way, the phenomenology of the full MSSM parameter space can be explored, while constrained high-scale models typically cover only very specific corners of the parameter space. This approach is therefore advantageous from an experimental point of view, where we aim for a full exploration and coverage of possible SUSY signatures at e.g. collider experiments.

Moreover, it contains only a minimal amount of theoretical prejudice about the high-scale behavior of the theory and the underlying SUSY breaking mechanism.

One possible choice is to assume the universality of soft-breaking masses and trilinear coup-lings, Eqs. (2.56) and (2.57), at the EW scale. However, since the third generation sfermion masses are less constrained by flavor processes and, as we have seen above, also can feature a quite different RGE running in high-scale models due to their large Yukawa couplings, we will here only assume the flavor universality, Eq. (2.56) and (2.57), for the first and second generation (i, j = 1,2) and keep the third generation sfermion masses and trilinear couplings separate:

M_τ˜L, M_˜τR, M_q˜3(=M_˜t

L =M_˜b

L), M_˜t

R, M_˜b

R, A_t, A_b, A_τ. (2.64) Additional parameters are needed to specify the MSSM Higgs sector, which are typically chosen to be

tanβ, m_A, µ, (2.65)

where m_A is the mass of the pseudoscalar Higgs bosonA, see Section 2.4.3.

We will study the implications of the Higgs boson discovery for the phenomenological MSSM in Chapter 6, also taking into account low energy observables and limits from direct collider searches for Higgs bosons and SUSY particles.

As it will be relevant for the following discussion, we briefly give the formulae for the left-right mixing of the sfermions. This mixing is particularly important for the sfermions of the third generation, while it is usually negligible for the first and second generation. In the basis of current eigenstates ( ˜f_L,f˜_R) within each fermion generation, the sfermion mass matrices read

M²_f˜= m²_f +m²_LL mfXf

m_fX_f m²_f +m²_RR

!

. (2.66)

Here, we defined

m²_LL ≡M_f²_˜

L+ (¹₂ −Q_fsin²θ_W)M_Z²cos 2β, (2.67) m²_RR ≡M_f²_˜

R +Q_fsin²θ_WM_Z² cos 2β, (2.68)

X_t≡A_f −µcotβ. (2.69)

Q_f denotes the electric charge of the (s)fermion, Eq. (2.1). The mass matrices are diagonalized by 2×2 rotation matrices of angleθ_f,

Rf = cosθf sinθf

−sinθ_f cosθ_f

!

, (2.70)

where the mixing angle is given by

sin(2θf) = 2mfXf

m²_˜

f1

−m²_˜

f2

, (2.71)

The resulting sfermion masses read m²_f˜

1,2 =m²_f +1 2

hm²_LL+m²_RR+^q(m²_LL−m²_RR)²+ 4m²_fX_f²ⁱ. (2.72) In the stop sector, the mixing is very strong for large values of the stop mixing parameter X_t=A_t−µcotβ and generates a significant mass splitting between the two mass eigenstates.

Im Dokument Higgs Couplings and Supersymmetry in the Light of early LHC Results (Seite 56-60)