Supersymmetry. Theory, Experiment, and Cosmology

High-energy vs. low-energy supersymmetry breaking 179

Brignole, Ib´a˜nez and Mu˜noz [51] have proposed to parametrize the supersymmetry breaking e ects as follows:

Thus, θ = π/2 corresponds to gaugino mass universality (the so-called dilaton dominated scenario) whereas θ 1 (moduli dominated) leads to large nonuniversalities.

Universality remains in any case a preferred option from a phenomenological perspective because of the dangerous flavor-changing neutral currents (FCNC) that nonuniversalities generate. As mentioned in Section 6.8 of Chapter 6, even in the universal case, some care should be taken since renormalization to the electroweak scale induces some nonuniversalities.

7.4.3Anomaly mediation

In the context of a locally supersymmetric theory, attention must be paid to the regularization procedure. It induces a generic contribution to the gaugino masses and the A-terms, known as the (conformal) anomaly mediated contribution. Such a contribution is dominant in models where there are no singlet fields present in the hidden sector (such as the S field of the preceding section) to generate a significant mass to gauginos. In such cases, this source of supersymmetry breaking in the observable sector is called anomaly mediation [189, 320]. But it must be stressed that this contribution is always present, although often nonleading. Let us now see more precisely how it arises.

The low energy theory may be considered as an e ective theory of a deeper theory which sets boundary values at a fundamental scale Λ. Care must be taken at the cut-o scale Λ because the procedure of cutting o momenta in an integral is not consistent with local supersymmetry. Inspiration may be found in the case where the scale Λ is related to the expectation value of a scalar field, i.e. the superheavy masses are determined by a scalar vev (as in the Higgs mechanism). In a supersymmetric context, the scalar field is merged into a chiral superfield and the supersymmetry breaking e ects trigger a F -term of order m3/2 . In other words, to the cut-o scale Λ must be associated an e ective F -term FΛ = Λm3/2 .

If we now consider the observable sector gauge kinetic term

where we have included one-loop renormalization e ects, the preceding considerations lead us to introduce for supersymmetric completion, according to (6.65),

Taking into account the factor g2 in front of the kinetic term for gauginos, this gives a universal gaugino mass:

180 Phenomenology of supersymmetric models: supersymmetry at the quantum level

As discussed in Section A.5.4 of Appendix Appendix A, the type of renormalization e ects that we have taken into account here are directly related to the (super)conformal anomaly. Hence the name anomaly mediation14.

Similarly [189, 320], A-terms are generated:

where the γ’s are the anomalous dimensions (see Section E.3 of Appendix E) of the corresponding cubic term in the superpotential: λijkφiφj φk. Finally, scalar masses receive a contribution

fundamental theory .

In the situation where the anomaly mediated contribution dominates, the prediction for gaugino masses is strikingly di erent from gravity mediation. For example, in the case of the minimal supersymmetric model, one obtains from (7.88), using

where the right-hand sides give the values at the electroweak scale (we have neglected here for M1 and M2 finite contributions coming from the Higgs sector).

˜3

This set of values implies an unexpected feature: the wino A is lighter than the

˜ ˜3 ˜ ±

bino B and becomes the LSP. Moreover, neutral (A ) and charged (W ) winos are almost degenerate. This is due to the fact that the splitting between the two arises from electroweak breaking and is of order MW4 . For example, in the limit µ M1,2, MW , one finds respectively for the neutral and charged wino mass

14In more technical terms, the contribution given here is related to the super-Weyl anomaly. It should be stressed [19, 172] that there exists other contributions associated with the K¨ahler and

sigma model anomalies. They depend explicitly on the form of the K¨ahler potential.

High-energy vs. low-energy supersymmetry breaking 181

where the notation makes it clear that these are the lightest neutralino and chargino. Because wino annihilation is very e cient, the thermal relic density of such a LSP is insu cient to account for dark matter. One may appeal to nonthermal production mechanisms.

Regarding scalar masses, one obtains from (7.90) and the expressions of anomalous dimensions given in Appendix E:

and similarly for squarks and sleptons of the first two families. We note that, because the lepton Yukawa couplings are small, this contribution to slepton squared masses are negative. This would indicate an instability but we have noted above that that the hidden sector may give extra contributions. For phenomenological purposes, one usually introduces a universal contribution m0 to represent these contributions.

7.4.4Gauge mediation

The potential problems of gravity-messenger models with FCNC together with developments in dynamical supersymmetry breaking have led a certain number of authors to reconsider models with “low energy” messengers. These are the gauge-mediated models [112,113] which rely on standard gauge interactions as the mediator [109,111]: since “standard gauge interactions are flavor-blind”, soft masses are universal.

One can discuss the nature of messenger fields on general grounds. These fields are charged under SU (3) × SU (2) × U (1) and since their masses are larger than the electroweak scale they must appear in vectorlike pairs. One can parametrize this through a (renormalizable) coupling to a singlet field N , fundamental or composite:

FN = 0.In other words, the Goldstino field overlaps with the supersymmetric partner

184 Phenomenology of supersymmetric models: supersymmetry at the quantum level

The worst problem faced by gauge-mediated models is however the µ/Bµ problem: since all soft supersymmetry-breaking terms scale as Λ, it is very di cult to avoid the relation:

Given the value of Λ, this is obviously incompatible with a relation such as (5.25) of Chapter 5. There has been attempts to decouple the origin of µ and of Bµ [112,114]. For instance, one may introduce a new singlet N with coupling λ N H2 · H1: µ = λ N . But it is then extremely di cult to decouple N and N . There does not seem to be a completely satisfactory solution to this µ/Bµ problem.

[An interesting development is the elaboration of gauge-mediated models without messengers. These models take more advantage of the developments in dynamical symmetry breaking that we will describe in the next chapter (the corresponding DSB sector was often a black box in the previous models): the rˆole of the messengers is played by e ective degrees of freedom of the DSB sector. This implies that the gauge symmetry of the Standard Model is a subgroup of the flavor symmetry group of the DSB sector. This group is therefore rather large and there are many e ective messengers. Unless they are heavy, this spoils the perturbative unification of gauge couplings. How then to make e ective messengers heavy? Remember that the messenger mass is λ N whereas FN / N is fixed by supersymmetry breaking; since N itself is not fixed, the idea is to require N FN 1/2. Explicit realizations involve nonrenormalisable terms [10, 314] or inverted hierarchy [291].]

7.5Limits on supersymmetric particles

Generally speaking, in the R-parity conserving scenarios that we will consider in this section, the typical supersymmetric signature is missing energy carried away by the decay LSP (which is assumed to be here the lightest neutralino).

7.5.1Sleptons and squarks

Supersymmetric particles have been searched at the LEP collider and not been found, which allows us to put a limit on their mass. Since the limit is somewhat less model dependent for smuons than for others, we will first detail the procedure in this case.

Smuon mass limits

At LEP, the production of a pair of right-handed17 smuons e+e− → µ˜+R µ˜−R goes through the exchange of a photon or a Z: thus the production cross-section depends only on the smuon mass. Then, the smuon decays into a muon and the LSP: µ˜R → µχ. Thus, besides the missing energy, one is searching for an observable final state with muons which are acoplanar with the beam (because of the momentum taken away by the LSP). As the mass of the LSP decreases, the smuons become less acoplanar and the signal becomes similar to e+e− → W +W − → µνµν (although it would provide an anomalously large W → µν branching ratio). Thus the limit is expected to degrade for smaller LSP masses.

17For a given smuon mass, the µ˜L production cross-section is larger than the µ˜R one: the µ˜R mass limit provides a conservative smuon mass limit.

Limits on supersymmetric particles 185

This is what is shown in Fig. 7.10 which gives the domain excluded experimentally by the search for acoplanar muons in the plane (mµ˜R , mχ) assuming 100% branching ratio for the decay µ˜−R → µχ: the dotted line gives the expected limit, i.e. the limit which would a priori be obtained if no signal was found in the data; it becomes less stringent as one goes to lower mχ. Thanks to a welcome statistical fluctuation, the actual limit does not follow this trend however. As the mass di erence mµ˜R − mχ becomes very small, the limit also degrades rapidly: there is not enough final state visible energy to allow a good detection e ciency.

If the branching ratio µ˜−R → µχ turns out not to be 100%, one must allow for decay chains, such as µ˜−R → µ(χ → γχ).This reduces the e ciency of the acoplanar muon search in the region of light χ.

Other slepton limits are more model dependent. In the case of stau, mixing may be important as for the stop and the most conservative limit corresponds to a vanishing coupling of the lightest stau to Z. For the selectron, there is an extra contribution (neutralino exchange in the t channel) which leads to a higher limit as can be seen from Figure 7.10.

Stop mass limit

As emphasized in Section 5.3.3 of Chapter 5, the large mixing term in the stop mass matrix (5.53) allows for light stop mass eigenstates. Light stops have been searched

˜ →

for at LEP: their decay t cχ arises through a loop and the decay time may be longer than the stop hadronization time, in which case stop-hadrons are first produced.

Fig. 7.11 gives the exclusion plot in the (mt˜, mχ) plane coming from LEP experiments as well as the Tevatron Run I.

Generic squarks and gluinos have also been searched for at the Tevatron where there is a strong production of q˜q˜ or g˜g˜ pairs. In the case where squarks are lighter than gluinos, squarks decay predominantly as q˜ → qχ and the final state is a pair of acoplanar jets plus missing transverse energy. However, other decay modes such

Fig. 7.10 Domain excluded at 95% CL by the four LEP experiments in the plane (m˜ , mχ)

R

resp. from left to right for τ˜R , µ˜R and e˜R (LEP SUSY working group).

186 Phenomenology of supersymmetric models: supersymmetry at the quantum level

˜ → Fig. 7.11 Domain excluded by the LEP experiments in the (mt˜, mχ) plane from t cχ

˜˜

decays: inner (resp. outer) contour corresponds to vanishing (resp. maximal) Ztt couplings. Also indicated is the domain excluded by CDF at Tevatron Run I [211].

as cascade decays (˜q → qχ , q χ± with subsequent decay of χ or χ±) are also possible which lead to di erent topologies. Since the decay branching ratios are model dependent, it is di cult to obtain generic mass limits.

7.5.2Neutralinos and charginos

Charginos (resp. neutralinos) are pair produced at LEP through s-channel γ/Z (resp. Z) exchange and t-channel sneutrino (resp. selectron) exchange: the two channels interfere destructively (resp. constructively). If sleptons are too heavy to be produced in the subsequent decays, the relevant parameters are M2, µ and tan β (plus M1 in the case of neutralinos). Typical decays are χ+ → χW and χ → χZ , which make the signatures rather straightforward: the kinematic limit of 104 GeV is basically reached for the chargino at LEP [380]. For lighter sleptons, invisible final states such as χ → νν˜ open up.

In the case where the chargino χ+ and the neutralino χ become almost degenerate (Higgsino-like for large M2 or wino-like in the case of anomaly mediation, see (7.92)), the above mentioned search loses its sensitivity. One has to resort to more elaborate techniques such as tagging on a low energy photon radiated from the initial state.

7.5.3Constraints on the LSP mass

The limits on the LSP arising from LEP searches are more intricate because they result from di erent channels: the direct production e+e− → χ˜0χ˜0 cannot be used since it gives an invisible final state. The mass limit in terms of tan β is given in Fig. 7.12. We have already seen that negative Higgs boson searches give a lower limit to the value of tan β. For values within this bond, the main constraint comes from the chargino channel: e+e− → χ˜+χ˜−. However, when chargino and sneutrino become almost degenerate, this channel looses its sensitivity because the decay χ˜± → ν becomes invisible. Such a configuration is called the corridor in Fig. 7.12. This is where slepton searches take over, giving an absolute mass lower limit of 45 GeV/c2.

R-parity breaking 187

Fig. 7.12 LSP mass limit as a function of tan β obtained by a combination of the four LEP experiments.

In the more constrained case of the minimal SUGRA model, there are fewer pathological mass configurations, which yields a better limit of 60 GeV/c2.

7.6R-parity breaking

Violations of R-parity are often associated with violations of baryon or lepton number and are thus bounded by experimental results on baryon or lepton violating interactions. The weakest constraints are obtained when one assumes that a single R-violating coupling dominates, typically 10−1 to 10−2 times (m/˜ 100 GeV), where m˜ is the mass of the superpartner involved. Much more stringent constraints may arise on the product of some specific couplings.

The R-parity violating renormalizable superpotential reads (see equation (5.4) of Chapter 5)

where UicDjcDkc ≡ αβγ Uiαc Djβc Dkγc (α, β, γ color indices). It is straightforward to show that

At this level, it is possible to redefine the fields in order to absorb the quadratic term into the µ term: H1 → H1 µH1 + µiLi. However, this is no longer equivalent once one includes soft supersymmetry breaking terms.

Let us illustrate how constraints arise on specific couplings λijk.

We first assume that a single coupling λ12k dominates. The corresponding R-parity violating operator yields an extra contribution to the muon lifetime as can be seen from Fig. 7.13.