Supersymmetry. Theory, Experiment, and Cosmology

160 Phenomenology of supersymmetric models: supersymmetry at the quantum level

where g(x) = 1−8x2+8x6−x8−24x4 ln x is the phase space factor, and the last factor is the next-to-leading QCD correction to the semileptonic decay (f (mc/mb) 2.41) [62].

where the corrections (of the order of 10 %) can be computed in the context of the Heavy Quark E ective Theory (HQET). Thus, one obtains

We now allow physics beyond the Standard Model. There is a significant contribution from charged Higgs exchange (Fig. 7.1b):

This, for example, puts a stringent lower bound on the charged Higgs mass in the context of the two Higgs doublet model [174]. However, in a supersymmetric context,

162 Phenomenology of supersymmetric models: supersymmetry at the quantum level

The anomalous moment of the muon has been measured by the experiment E821 at Brookhaven [30]

This has to be compared with the prediction of the Standard Model, which depends mostly, at this level of precision, on the correct evaluation of the hadronic vacuum polarization and on the hadronic light-by-light contribution5. Depending on this one finds

which represents a departure of 1–2.8 standard deviation.

If there is indeed a deviation, is it a sign of new physics? There are many possible candidates: anomalous couplings, lepton flavor violations, muon substructure, etc. But one should stress that, starting with [150], such a deviation has been heralded by many as a possible signature of supersymmetry.

Indeed, in the case of supersymmetry, there are new loop corrections involving neutralino–smuon or chargino–sneutrino mu. To have an idea of these new contributions, one may consider the simple situation where all superpartners have the same mass MSUSY, in which case, at one loop, one has approximately

One thus expects significant corrections for large tan β or a light supersymmetric spectrum. Moreover, a positive value of the supersymmetric parameter µ is strongly favored by the (gµ − 2) data.

One should however note that the simplicity of the rule of thumb (7.23) is somewhat misleading: it does not apply, even approximately, in the case where supersymmetric partners are nondegenerate.

7.1.5Constraints on minimal supergravity

We now illustrate on a specific model the constraints discussed in the preceding subsections. We follow the standard choice of the minimal supergravity model described in Section 6.8 of Chapter 6, not so much because it is well motivated but because it has a small number of parameters: m0, M1/2, tan β, A0 and the sign of µ. In fact, we choose µ > 0 in order to satisfy the constraint imposed by data on the muon magnetic moment.

We give in Figures 7.3 and 7.4 the di erent constraints in the plane (M = M1/2, m = m0) for A0 = 0 and three values of tan β : 5, 35 and 50. On each plot, the region to the left of the light grey dashed line is excluded by the lower bound on the lightest Higgs h0 mass(se next section). The region to the left of the dotted line is excluded by the lower bound on the chargino mass mχ±1 > 103 GeV. The region to the left of the solid line is excluded by b → sγ. The region at the bottom (Stau LSP) is excluded because the lightest stau is the LSP.

sγ¯ ρs)/3 is the light quark electromagnetic current. It took some time before its sign was determined correctly [253].

Why does the MSSM survive the electroweak precision tests? 163

Stau LSP

Stau LSP

Fig. 7.3 Constraints on the mSUGRA parameter space for tan β = 5 (left) and tan β = 35 (right).

Stau LSP

Fig. 7.4 Constraints on the mSUGRA parameter space for tan β = 50.

The region between grey contours fulfils 0.1 ≤ Ωχh2 ≤ 0.3, whereas that between black contours indicates the WMAP range 0.094 < ωχh20 ≤ 0.129 [141]. We see that the region favored by dark matter relic density is one of light scalars and gauginos, which is excluded by the negative searches of light supersymmetric particles. The bulk of the remaining parameter space yields too large a neutralino relic density. One remains

The Higgs sector 165

where we have made profit of the assumption of soft supersymmetry breaking to disregard field-independent contributions proportional to STrM 2.

We recall, from Appendix Appendix A, equation (A.247) that V is the non-

¯ e

derivative term in the e ective action Γ φ . We consider the proper Green’s function

(2)2 ¯2

Γ(p) = δ Γ/δφ , which is the inverse propagator,

The physical mass is the zero of this function:

If the physical mass is much smaller than the masses of particles running in the loops,

one may take Σ(m2phys) Σ(0).

Thus, at the one-loop level, masses are given by the matrix of double derivatives of the e ective potential, evaluated at its minimum. Let us check that one thus recovers the preliminary results obtained in Section 5.2.1 of Chapter 5. We keep only in the supertrace (7.26) the top quark (of mass mt = λtH20) and the stop squarks (assumed to be degenerate in mass: m˜ 2t = m2t + m˜ 2, where m˜ arises form the soft breaking of supersymmetry):

gives a contribution to the Higgs mass which coincides with the one obtained in (5.36) of Chapter 5).

If we take into account the mixing in the stop sector (see (5.53) of Chapter 5),

where Xt ≡ At − µ cot β.

166 Phenomenology of supersymmetric models: supersymmetry at the quantum level

one may obtain, in the e ective potential approximation, the mass matrices from the second derivatives of the e ective potential (7.24) evaluated at the minimum:

As stressed earlier, Ve is expressed in terms of renormalized fields and couplings. One traditionally uses the DR scheme (see Appendix E): the corresponding quantities will

vanishing momentum. The e ective potential approximation thus misses some possibly important momentum corrections. This is why a purely diagrammatic approach has

Whereas, at one loop [134,216,299], the full corrections are known, only the leading contributions at two loops have been computed. These include corrections of order λ4t ,

The Higgs sector 167

λ2b α3 and λ2b α3. Let us take this opportunity to answer a question which is often asked: if the one-loop corrections to the lightest Higgs are so large, should we not expect large corrections at even higher orders? The answer is no: two-loop and higher corrections are expected to be small. The reason is that, in contrary to the one-loop level where new terms with a dependence in λ4t appear, we do not expect any surprise at higher orders, as can be seen for example from the list of the leading two-loop corrections just given.

7.2.2Upper limit on the mass of the lightest Higgs

In the late 1990s, a significant amount of theoretical activity has been spent in trying to derive a precise upper limit for the lightest Higgs mass in the context of the MSSM. The main reason was that the available energy at the LEP collider allowed searches of the lightest Higgs only in a restricted range. In the present decade, that should see the first runs of the LHC collider, this question is of less importance since LHC should cover the whole range of mass for h0 in the MSSM.

We have seen that the one-loop radiative corrections to the lightest Higgs mass depend mostly on the precise value of the top mass, on tan β and on the masses of the stops (and sbottoms if tan β is large). Obviously, if one increases the soft scalar mass parameter, one increases the squark masses and thus the corrections to the Higgs mass. Pushing to the extreme case where this mass parameter is 10 TeV, one obtains mh0 < 150 GeV in the MSSM. Taking values of this soft mass scale more in line with a reasonable fine tuning of the parameters significantly decreases this upper limit: one finds mh0 < 110 GeV if the stops have a common mass of 1 TeV (often referred in the literature as the “no mixing” scenario) and mh0 < 130 GeV if there is a large splitting between the two (the “maximal mixing” scenario).

Let us note that, in the context of the MSSM model, the Higgs is light at tree level because the quartic coupling λ is of order g2 + g 2 (see 7.25): as in the Standard Model, the mass is of order λv2. But this property is lost in extensions of the MSSM. For example, in the NMSSM model described in Section 5.6 of Chapter 5, the superpotential coupling λS SH2 · H1 induces in the potential a quartic term of order λ2S which is not necessarily small. Thus, in principle the lightest Higgs could be heavy.

However, if we put the low energy supersymmetric theory in the context of a more fundamental theory with a typical scale in the MU or MP range, then the triviality arguments presented in Section 1.2.1 of Chapter 1 apply and one does not expect the lighter scalar mass to be above 200 GeV. Let us illustrate this on the NMSSM model.

In the framework of the NMSSM model, the lightest Higgs mass satisfies the upper

where λS is one of the cubic couplings in the superpotential:

We have assumed a discrete symmetry forbidding any quadratic term and we have kept only the top Yukawa coupling. We see that, if λS is allowed to be as large as possible, there is no absolute upper bound to the Higgs mass.