JOHN ELLIS Theoretical Physics Division, CERN, CH-1211 Geneva 23 E-Mail: John.Ellis@Cern.Ch

For Publisher’s use

DECLINE AND FALL OF THE STANDARD MODEL?

JOHN ELLIS Theoretical Physics Division, CERN, CH-1211 Geneva 23 E-mail: John.Ellis@cern.ch

Motivations for physics beyond the Standard Model are reviewed, with particular emphasis on supersymmetry at the TeV scale. Constraints on the minimal supersyymetric extension of the Standard Model with universal soft supersymmetry-breaking terms (CMSSM) are discussed. These are also combined with the supersymmetric interpretation of the anomalous magnetic moment of the muon. The prospects for observing supersymmetry at accelerators are reviewed using benchmark scenarios to focus the discussion. Prospects for other experiments including the detection of cold dark matter, µ → eγ and related processes, as well as proton decay are also discussed.

CERN-TH/2001-275 hep-ph/yymmnnn

1 Introduction ‘constant’, which recent data suggest is non- zero 2, and may not even be constant. Talk- ing of cosmology, we would need at least one The empire of the Standard Model has re- extra parameter to produce an inflationary sisted all attacks by accelerator data. Never- potential, and at least one other to generate theless, we theorists are driven to overcome the baryon asymmetry, which cannot be ex- our ignorance of the barbarian territory be- plained within the Standard Model. yond its frontiers. In the gauge sector, the Standard Model has three independent gauge Confronted by our ignorance of so much couplings and (potentially) a CP-violating barbarian territory, we legions of theorists or- phase in QCD. In the Yukawa sector, it ganize our explorations on three main fronts: has six random-seeming quark masses, three unification – the quest for a single framework charged-lepton masses, three weak mixing for all gauge interactions, flavour – the quest angles and the Kobayashi-Maskawa phase. for explanations of the proliferation of quark Finally, the symmetry-breaking sector has at and lepton types, their mixings and CP vio- least two free parameters. Moreover, this list lating phases, and mass – the quest for the of 19 parameters in the Standard Model begs origin of particle masses and an explanation the more fundamental questions of the origins why they are so much smaller than the Planck mass m 1019 GeV. Beyond all these be- of the particle quantum numbers. As if this P ∼ were not enough, non-accelerator neutrino yonds, other scouting parties of theorists seek experiments 1 now convince us that we need a Theory of Everything that includes grav- three neutrino mass parameters, three neu- ity, reconciles it with quantum mechanics, ex- trino mixing angles and three CP-violating plains the origin of space-time and why we phases in the neutrino sector: one observ- live in four dimensions (if we do so). able in oscillation experiments and two that Physics beyond the Standard Model is affect ββ0ν experiments, without even talk- therefore a very broad subject. However, ing about the mechanism of neutrino mass many aspects are discussed here by other generation. Moreover, we should not for- speakers: electroweak flavour physics 3, CP get about gravity, with at least two parame- 4 5 6 violation , the Higgs sector , gµ 2 , ters to understand: Newton’s constant G 7 8− N ≡ searches for new particles , neutrinos , dark m−2 (1019 GeV)−2 and the cosmological P ∼

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9 10 matter , strings and extra dimensions . 5 10 15 -15 -10 -5 15 10 5

Therefore, in this talk I seek a complemen- -5 2 2 2 ε ε ε 3 3 3

-10 0 0 0

tary approach. 1 1 1

-15 For reasons that I describe in Section 2, 103ε 103ε 103ε many theorists believe that supersymmetry 1 b 3 is the inescapable framework for discussing 15 3 3 ε ε

10 3 3 physics at the TeV scale and beyond. In 0 0 1 1 the rest of this talk, I ﬁrst discuss the con- 5

3 straints imposed on (the simplest) supersym- 103ε 10 ε 1 b metric models by the available experimen- 1- generation TC -15 Standard Model b ε

tal and cosmological constraints, then ad- 3

-10 0

1 Data dress the prospects for understanding gµ 2 − -5 in supersymmetric models, the prospects for 103ε detecting sparticles directly at present and 1 future colliders, and the prospects for non- Figure 1. Predictions for the radiative corrections ²i collider experiments, including the searches in the Standard Model and a minimal one-generation 14 for dark matter, µ eγ and proton decay. model are compared with the precision electroweak → data 13.

2 The Electroweak Vacuum variant of technicolour might emerge that is The generation of particle masses requires the consistent with the data, but for now we fo- breaking of gauge symmetry in the vacuum: cus on elementary Higgs models.

mW,Z = 0 < 0 XI,I 0 >= 0 (1) Within this framework, the data favour a 6 ⇔ | 3 | 6 relatively light Higgs boson, with m 115 for some field X with isospin I and third com- H ' GeV, just above the exclusion unit provided ponent I3. The measured ratio by direct searches at LEP, being the ‘most- m2 W probable’ 15. This is one reason why many ρ 2 2 1 (2) ≡ mZ cos θW ' theorists were excited by the possible sighting tells us that X mainly has I = 1/2 11, which during the last days of LEP of a Higgs boson, is also what is needed to generate fermion with a preferred mass of 115.6 GeV 7. If this masses. The key question is the nature of were to be confirmed, it would suggest that the field X: is it elementary or compos- the Standard Model breaks down at some rel- ite? A fermion-antifermion condensate v < atively low energy < 103 TeV 16. As seen ≡ 0 X 0 >=< 0 F¯F 0 >= 0 would be anal- in Fig. 2, above this∼scale the effective Higgs | | | | 6 ogous to what we know from QCD, where potential of the Standard Model becomes un- < 0 q¯q 0 >= 0, and conventional supercon- stable as the quartic Higgs self-coupling is | | 6 ductivity, where < 0 e−e− 0 >= 0. How- driven negative by radiative corrections due | | 6 ever, analogous ‘technicolour’ models of elec- to the relatively heavy top quark 17. This is troweak symmetry breaking 12 fail to fit the not necessarily a disaster, and it is possible values of the radiative corrections ²i to ρ and that the present electroweak vacuum might other quantities extracted from the precision be metastable, provided that its lifetime is electroweak data provided by LEP and other longer than the age of the Universe 18. How- experiments, as seen in Fig. 1 13. One cannot ever, we would surely feel more secure if such exclude the possibility that some calculable instability could be avoided.

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10 M0=70.9GeV M =71.0GeV 1 0

) 0.1 µ (

H λ 0.01

0.001

0.0001 102 106 106 108 1010 1012 1014 1016 1018 µ(GeV) 5

) 3 µ ( H λ 2

Figure 2. The range allowed for the mass of the Higgs 1 boson if the Standard Model is to remain valid up to a given scale Λ. In the upper part of the plane, the 0 102 104 106 108 1010 1012 1014 1016 1018 effective potential blows up, whereas in the lower part µ(GeV) the present electroweak vacuum is unstable 17. Figure 3. (a) If the quartic coupling M0 (3) is too large, the effective potential blows up (solid line), whereas it is unstable if M0 is too small (dotted line), This may be done by introducing new indicating a need for fine tuning. (b) This occurs nat- bosons φ coupled to the Higgs field 16: urally in a supersymmetric model (solid line) but not if the H˜ are omitted (dotted line) 16. 2 2 2 2 λ22 H φ : M λ22v (3) | | | | 0 ≡ As seen in Fig. 3a, the effective potential is very sensitive to the coupling parameter electron masses? One might think naively that it would be sufficient to set mW mP M0: for M0 70.9 GeV in this example, the ¿ ≤ by hand. However, radiative corrections tend potential still collapses, whereas for M0 ≥ 71.0 GeV the potential blows up instead. to destroy this hierarchy. For example, one- Thus the bosonic coupling (3) must be finely loop diagrams generate tuned 16. This occurs naturally in supersym- α δm2 = Λ2 m2 (4) metry, in which the Higgs bosons are accom- W O π À W panied by fermionic partners H˜ . As seen in ³ ´ where Λ is a cut-off representing the appear- Fig. 3b, again the Higgs coupling blows up ance of new physics, and the inequality in in the absence of the H˜ , whereas it is well (4) applies if Λ 103 TeV, and even more so behaved in the minimal supersymmetric ex- ∼ if Λ m 1016 GeV or m 1019 tension of the Standard Model (MSSM). ∼ GUT ∼ ∼ P ∼ GeV. If the radiative corrections to a physical The avoidance of fine tuning has long quantity are much larger than its measured been the primary motivation for supersym- values, obtaining the latter requires strong metry at the TeV scale 19. This issue is nor- cancellations, which in general require fine mally formulated in connection with the hi- tuning of the bare input parameters. How- erarchy problem: why/how is mW mP , ever, the necessary cancellations are natural ¿2 in supersymmetry, where one has equal num- or equivalently why is GF 1/mW 2 ∼ À bers of bosons B and fermions F with equal GN = 1/mP , or equivalently why does the Coulomb potential in an atom dominate over couplings, so that (4) is replaced by 2 the Newton potential, e GN mpme α 2 À ∼ δm2 = m2 m2 . (5) (m/mP ) , where mp,e are the proton and W O π | B − F | ³ ´

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The residual radiative correction is naturally range, whereas composite Higgs model gen- small if erally predict heavier eﬀective Higgs masses.

m2 m2 < 1 TeV2 (6) The gauge symmetries of the MSSM | B − F | ∼ would permit the inclusion of interactions Note that this argument is logically distinct that violate baryon number and/or lepton from that in the previous paragraph. There number 24: supersymmetry was motivated by the control c 0 c ” c c c of logarithmic divergences, and here by the λLLE + λ QD L + λ U D D (7) absence of quadratic divergences. where the L(Q) are left-handed lepton (quark) doublets and the Ec(Dc, U d) are con- jugates of the right-handed lepton (quark) 3 The MSSM singlets. Their possible appearance is ignored in this talk, in which case the lightest su- The MSSM has the same gauge interactions persymmetric particle is stable, and hence a 25 as the Standard Model, and similar Yukawa candidate for dark matter . In the follow- couplings. A key difference is the necessity of ing this is assumed to be a neutralino, i.e., a ˜ ˜ two Higgs doublets, in order to give masses to mixture of the γ˜, H and Z. all the quarks and leptons, and to cancel tri- The final ingredient in the MSSM is the angle anomalies. This duplication is impor- soft supersymmetry breaking, in the form of tant for phenomenology: it means that there scalar masses m0, gaugino masses m1 2 and are five physical Higgs bosons, two charged / trilinear couplings A 26. These are presumed H§ and three neutral h, H, A. Their quartic to be inputs from physics at some high-energy self-interactions are determined by the gauge scale, e.g., from some supergravity or su- interactions, solving the vacuum instability perstring theory, which then evolve down to problem mentioned above and limiting the lower energy scale according to well-known possible mass of the lightest neutral Higgs bo- renormalization-group equations. In the case son. However, the doubling of the Higgs mul- of the Higgs multiplets, this renormalization tiplets introduces two new parameters: tan β, can drive the effective mass-squared negative, the ratio of Higgs vacuum expectation val- triggering electroweak symmetry weaking 27. ues and µ, a parameter mixing the two Higgs In this talk, it is assumed that the m0 are doublets. a universal at the input scale , as are the m1/2 There are two key experimental hints in and A parameters. In this case the free pa- favour of supersymmetry. One is provided rameters are by the LEP measurements of the gauge cou- m0, m1/2, A and tan β , (8) plings, that are in very good agreement with supersymmetric GUTs 20 if sparticles weigh with µ being determined by the electroweak 1 TeV. This agreement appears completely vacuum conditions, up to a sign. ∼ 12 fortuitous in composite Higgs models , and This constrained MSSM (CMSSM) 21 is difficult (though not impossible ) to re- serves produce accurately in models with large ex- as the basis for the subsequent discussion. tra dimensions 22. The other experimental hint is provided by the preference of the pre- aUniversality between the squarks and sleptons of different generations is motivated by upper limits on cision electroweak data for a relatively light 28 15 flavour-changing neutral interactions , but univer- Higgs boson . In the MSSM, one predicts sality between the soft masses of the L, Ec, Qc, Dc 23,5 c mh < 130 GeV , right in the preferred and U is not so well motivated. ∼

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It has the merit of being sufficiently specific that the different phenomenological constraints can be combined meaningfully. On the other hand, it is just one of the phenomenological possibilities offered by super- 200 tan β=10, µ > 0 symmetry 29.

4 Constraints on the CMSSM m χ ± =104 GeV m =113 GeV 0 h m (GeV) 100 Important constraints on the CMSSM parameter space are provided by direct searches 7 at LEP and the Tevatron collider , as seen m∼< mχ Run II τ 1 § > in Fig. 4. One of these is the limit mχ 103 ~e GeV provided by chargino searches at∼LEP, where the third signiﬁcant ﬁgure depends on other CMSSM parameters. LEP has also pro- 100 200 300 400 500 600 700 800 vided lower limits on slepton masses, of which m 1/2 (GeV) the strongest is me˜ > 99 GeV, again depend- ∼ 200 ing only sightly on the other CMSSM param- tan β=10, µ < 0 eters, as long as m˜ m > 10 GeV. The e − χ most important constraints on∼ the u, d, s, c, b m =113 GeV squarks and gluinos are provided by the Teva- h tron collider: for equal masses mq˜ = mg˜ >

300 GeV. In the case of the t˜, LEP provides∼ 0 m (GeV) 100 the most stringent limit when m˜ m is t − χ small, and the Tevatron for larger mt˜ mχ. − m ± =104 GeV m < m Their eﬀect is almost to exclude the range of χ ∼τ χ 1 parameter space where electroweak baryoge- ~ nesis is possible 30. e

Another important constraint is pro- 100 200 300 400 500 600 700 800 vided by the LEP limit on the Higgs mass: m 1/2 (GeV) mH > 114.1 GeV. This holds in the Stan- dard Model, for the lightest Higgs boson h in the general MSSM for tan β < 5, and in the Figure 4. Compilations of phenomenological con- CMSSM for all tan β, at least∼as long as CP is straints on the CMSSM for tan β = 10 and (a) µ > 0, b (b) µ < 0. Representative contours of the selectron, conserved . Since mh is sensitive to sparticle chargino and Higgs masses are indicated, as is the masses, particularly mt˜, via loop corrections: likely physics reach of Run II of the Tevatron Col- lider in (a). The dark shaded regions are excluded m4 m2 δm2 t ln t˜ + . . . (9) because the LSP is charged, whereas a neutralino h 2 2 LSP has acceptable relic density (10) in the light- ∝ mW Ãmt ! shaded regions 31. The medium-shaded region in (b) the Higgs limit also imposes important con- is excluded by b → sγ 32. bThe lower bound on the lightest MSSM Higgs boson may be relaxed signiﬁcantly if CP violation feeds into the MSSM Higgs sector 33.

Decline˙and˙Fall: submitted to World Scientiﬁc on January 21, 2002 5 For Publisher’s use straints on the CMSSM parameters, princi- 500 ∼χ pally m1/2 as seen in Fig. 4. m β µ < 0 = b) tan = 10, ∼ m τ R Also shown in Fig. 4 is the constraint im- 400 posed by measurements of b sγ 32. These → agree with the Standard Model, and there- er 300 t

0 t fore provide bounds on chargino and charged ∼χ a m m .1 M Higgs masses, for example. For moderate 1 k = r ∼ a tan β, the b sγ constraint is more impor- 200 m τ R D → ed tant for µ < 0, as seen in Fig. 4b, but it is rg ha also signiﬁcant for µ > 0 when tan β is large. 100 C Fig. 4 also displays the regions where the supersymmetric relic density ρχ = Ωχρcritical falls within the preferred range 500 1000 1500 2000 m1/2 2 0.1 < Ωχh < 0.3 (10)

The upper limit is rigorous, since astro- Figure 5. The large-m1/2 ‘tail’ of the χ − τ˜1 coannihilation region for tan β = 10 and µ < 0 35. physics and cosmology tell us that the total matter density Ωm < 0.4, and the Hubble ex- pansion rate h 1/∼√2 to within about 10 % 34 ∼ . . .) . In this way, the allowed CMSSM re- (in units of km/s/Mpc). On the other hand, gion may acquire a ‘tail’ extending to large the lower limit in (10) is optional, since there mχ, as in the case where the NLSP is the could be other important contributions to the lighter stau: τ˜1 and m˜ m as seen in τ1 ∼ χ overall matter density. Fig. 5 35. Another mechanism for extending the allowed CMSSM region to large m As is seen in Fig. 4, there are generic re- χ is rapid annihilation via a direct-channel pole gions of the CMSSM parameter space where when m 1 m 36,37. This may yield the relic density falls within the preferred χ ∼ 2 Higgs,Z a ‘funnel’ extending to large m and m range (10). What goes into the calculation 1/2 0 at large tan β, as seen in Fig. 6 37. An- of the relic density? It is controlled by the other allowed region at large m and m annihilation rate 25: 1/2 0 is the ‘focus-point’ region 38, which is ad- 1 ρ = m n : n (11) jacent to the boundary of the region where χ χ χ χ ∼ σ (χχ . . .) ann → electroweak symmetry breaking is possible, as seen in Fig. 7. However, in this region and the typical annihilation rate σann 2 ∼ mχ is not particularly large. 1/mχ. For this reason, the relic density typ- ically increases with the relic mass, and this These filaments extending the preferred combined with the upper bound in (10) then CMSSM parameter space are clearly excep- < leads to the common expectation that mχ tional, in some sense, so it is important to ∼ 1 TeV. However, there are various ways in understand the sensitivity of the relic density which the generic upper bound on mχ can be to input parameters, unknown higher-order increased along filaments in the (m1/2, m0) effects, etc. One proposal is the relic-density plane. For example, if the next-to-lightest fine-tuning measure 39 sparticle (NLSP) is not much heavier than 2 < ∂ ln(Ωχh ) χ: ∆m/mχ 0.1, the relic density may be ∆Ω (12) ∼ ≡ ∂ ln a suppressed by coannihilation: σ(χ+NLSP s i i → X

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where the sum runs over the input parameters, which might include (relatively) poorly- known Standard Model quantities such as mt and mb, as well as the CMSSM parameters m0, m1/2, etc. As seen in Fig. 7, the sensitivity ∆Ω (12) is relatively small in the ‘bulk’

region at low m1/2, m0, and tan β. However, it is somewhat higher in the χ τ˜1 coanni- − hilation ‘tail’, and at large tan β in general. The sensitivity measure ∆Ω (12) is particularly high in the rapid-annihilation ‘funnel’ and in the ‘focus-point’ region. This explains why published relic-density calculations may differ in these regions 40, whereas they agree well when ∆Ω is small: differences may arise because of small differences in the treatments of the inputs.

Figure 6. The region where the cosmological relic It is important to note that the relic- density is in the preferred range (10) for tan β = 50 density ﬁne-tuning measure (12) is distinct and µ > 0. Note the rapid-annihilation ‘funnel’ at 37 from the traditional measure of the ﬁne- intermediate m0/m1/2 . tuning of the electroweak scale 41:

∂ ln mW ∆i (13) ≡ ∂ ln ai This electroweak fine-tuning is a completely tan β = 10 , m = 171 GeV, µ > 0 2500 t different issue, and values of the ∆i are not 1000 300 necessarily related to values of ∆Ω. Elec- 100 troweak fine-tuning is sometimes used as a 2000 criterion for restricting the CMSSM param- 30 eters. However, the interpretation of the ∆i

(GeV) (13) is unclear. How large a value of ∆i is 0

m 10 tolerable? Different physicists may well have 1000 different pain thresholds. Moreover, correla- tions between input parameters may reduce 3 its value in specific models.

0 100 200 300 400 500 600 700 800 900 1000 5 Muon Anomalous Magnetic

m1/2 (GeV) Moment

Figure 7. The m1/2, m0 plane for tan β = 10 and µ > 0, including the ‘focus-point’ region 38 at large As reported at this meeting 6, the BNL E821 m0, close to the boundary of the shaded region where experiment has recently reported a 2.6-σ de- electroweak symmetry breaking occurs, and exhibit- viation of a 1 (g 2) from the Standard ing contours of the cosmological sensitivity (12) 39. µ ≡ 2 µ − Model prediction 42:

aexp ath = (43 16) 10−10 (14) µ − µ § ×

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The largest contribution to the error in (14) is the statistical error of the experiment, which will soon be significantly reduced, as many more data have already been recorded. The next-largest error is that due to strong- interaction uncertainties in the Standard Model prediction. Recent estimates converge on an estimate of about 7 10−10 for the error × in the hadronic vacuum polarization constri- bution to (14) 43, and the error in the hadron light-by-light scattering contribution is gen- erally thought to be smaller 44. Therefore, if the central value in (14) does not change substantially with the new data, this would be strong evidence for new physics at the TeV scale. Figure 8. The medium-shaded region is that compat- 45 ible with the BNL E821 measurement of gµ −2 at the As many authors have pointed out , the 2-σ level 6,42, the light-shaded region has a relic den- discrepancy (14) could well be explained by sity in the preferred range (10), and the dark-shaded supersymmetry if µ > 0 and tan β is not too region does not have a neutralino LSP. Good com- small, as exemplified in Fig. 8. Good consis- patibility is found between gµ − 2 and the other phenomenological constraints for tan β ∼ 5 or more 45. tency with all the experimental and cosmological constraints on the CMSSM is found for tan β < 10 and mχ 150 to 350 GeV. Al- ' benchmark points are consistent with gµ 2 ready before∼ the measurement (14), the LHC − (14) at the 2 σ level, but this was not im- was thought to have a good chance of discov- − posed as an absolute requirement. ering supersymmetry 46. If the result (14) were to be confirmed, this would be almost The proposed points were chosen not to guaranteed, as we now discuss. provide an ‘unbiased’ statistical sampling of the CMSSM parameter space, whatever that means in the absence of a plausible a pri- 6 Prospects for Observing ori measure, but rather are intended to illus- Supersymmetry at Accelerators trate the different possibilities that are still allowed by the present constraints 48. Five of the chosen points are in the ‘bulk’ region at As an aid to the assessment of the prospects small m1/2 and m0, four are spread along the for detecting sparticles at different accelera- coannihilation ‘tail’ at larger m1/2 for various tors, benchmark sets of supersymmetric pa- values of tan β, two are in the ‘focus-point’ rameters have often been found useful 47, region at large m0, and two are in rapid- since they provide a focus for concentrated annihilation ‘funnels’ at large m1/2 and m0. discussion. A set of post-LEP benchmark The proposed points range over the allowed scenarios in the CMSSM has recently been values of tan β between 5 and 50. Most of 48 proposed , and are illustrated schematically them have µ > 0, as favoured by g 2, but µ − in Fig. 9. They take into account the di- there are two points with µ < 0. rect searches for sparticles and Higgs bosons, b sγ and the preferred cosmological den- Various derived quantities in these su- → sity range (10). About a half of the proposed persymmetric benchmark scenarios, includ-

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35 35 30 30 25 25

→ γ 20 20 mh, b s 0 15 15

m 10 10 5 5 0 0 g-2 I L B G C J A M H E F K D I L B G C J A M H E F K D

Nb. of Observable Particles 35 35 30 30 25 25 20 20 15 15 10 10 m1/2 5 5 0 0 I L B G C J A M H E F K D I L B G C J A M H E F K D Figure 9. Schematic overview of the benchmark points proposed in 48. They were chosen to be com- patible with the indicated experimental constraints, as well as have a relic density in the preferred range (10). The points are intended to illustrate the range Figure 10. Estimates of the numbers of different of available possibilities. types of CMSSM particles that may be detectable 48 at (a) the LHC, (b) a 1-TeV linear e+e− collider 49, and (c,d) a 3(5)-TeV e+e− 50 or µ+µ− collider 51. Note the complementarity between the sparticles de- − ing the relic density, g 2, b sγ, elec- tectable at the LHC and at a 1-TeV linear e+e col- µ − → troweak fine-tuning ∆ and the relic-density lider. sensitivity ∆Ω, are given in 48. These enable the reader to see at a glance which models would be excluded by which refinement of the colliders are amply documented in various 49 experimental value of g 2. Likewise, if you design studies . Not only is the lightest µ − find some amount of fine-tuning uncomfort- MSSM Higgs boson observed, but its major ably large, then you are free to discard the decay modes can be measured with high ac- corresponding models. curacy, as seen in Fig. 11. Moreover, if sparticles are light enough to be produced, their The LHC collaborations have analyzed masses and other properties can be measured their reach for sparticle detection in both very precisely, enabling models of supersym- generic studies and specific benchmark sce- metry breaking to be tested 52. narios proposed previously 46. Based on these studies, Fig. 10 displays estimates how As seen in Fig. 10, the sparticles visible + − many different sparticles may be seen at the at an e e collider largely complement those 48 LHC in each of the newly-proposed bench- visible at the LHC . In most of benchmark scenarios 48. The lightest Higgs boson mark scenarios proposed, a 1-TeV linear col- is always found, and squarks and gluinos are lider would be able to discover and measure usually found, though there are some sce- precisely several weakly-interacting sparticles narios where no sparticles are found at the that are invisible or difficult to detect at the LHC. The LHC often misses heavier weakly- LHC. However, there are some benchmark interacting sparticles such as charginos, neu- scenarios where the linear collider (as well tralinos, sleptons and the other Higgs bosons. as the LHC) fails to discover supersymmetry. Only a linear collider with a higher The physics capabilities of linear e+e− centre-of-mass energy appears sure to cover

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-1 10 SM Higgs Branching Ratio

-2 10

-3 10 90 100 110 120 130 140 150 160 2 MH (GeV/c )

Figure 11. Analysis of the accuracy with which Higgs decay branching ratios may be measured with a linear e+e− collider 53. all the allowed CMSSM parameter space, as seen in the lower panels of Fig. 12, which illustrate the physics reach of a higher-energy lepton collider, such as CLIC 50 or a multi- TeV muon collider 51.

7 Prospects for Other Experiments

Detection of cold dark matter

Fig. 12 shows rates for the elastic spin- independent scattering of supersymmetric Figure 12. Rates for the elastic spin-independent 54 relics , including upper limits from the scattering of supersymmetric relics on (a) protons UKDMC, CDMS and Heidelberg experi- and (b) neutrons calculated in benchmark scenar- 9 ios 54, compared with upper limits from the UKDMC, ments , as well as the range suggested by the 9 55 CDMS and Heidelberg experiments , as well as the DAMA collaboration . Also shown are the range suggested by the DAMA collaboration 55. rates calculated in the proposed benchmark scenarios discussed in the previous section, which are considerably below the DAMA range, but may be within reach of future projects. Indirect searches for supersymmetric dark matter via the products of annihi- lations in the galactic halo or inside the Sun also have prospects in some of the benchmark scenarios 54.

µ eγ and related processes →

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nitude of B(µ eγ), and it is in principle → possible to measure CP violation in µ 3e → decay. This may provide another interesting interface with neutrino physics and cosmology 58. The minimal supersymmetric seesaw model has six CP-violating phases: the MNS phase δ, two light-neutrino Majorana phases, and three phases arising from neutrino Dirac Yukawa couplings, which may be responsi- ble for our existence via leptogenesis in the early Universe 8. The CP-violating neutrino phases induce phases in slepton mass ma- trices, which may show up in µ 3e de- → cay, τ 3e/µ decays and leptonic electric → dipole moments. In principle, the leptogenesis phases might be obtainable by comparing Figure 13. Illustration in one particular lepton- CP-violating measurements in the charged- 58 flavour texture model, for tan β = 30 and µ > 0, indi- lepton and neutrino sectors . cating that µ → eγ decay may occur at a rate close to the present experimental upper limit, in the CMSSM Proton decay with parameters chosen 57 to match the measured 6,42 value of gµ − 2 . This could be within reach, with τ(p → e+π0) via a dimension-six operator possibly 1035y if m 1016 GeV as expected ∼ GUT ∼ in a minimal supersymmetric GUT. Such a The BNL E821 report of a possible de- model also suggests that τ(p ν¯K+) < viation from the Standard Model suggests → 1032y via dimension-five operators 59, unless that a non-trivial µ µ γ vertex is gen- − − measures are taken to suppress them 60. This erated at a scale < 1 TeV. Neutrino oscilla- provides motivation for a next-generation tions indicate that∼ there are ∆L = 0 pro- µ 6 megaton experiment that could detect processes 8, so it is natural to expect that there ton decay as well as explore new horizons in might also be a non-trivial µ e γ vertex. − − neutrino physics 61. This is indeed the case in a generic supersymmetric GUT, where neutrino mixing induces slepton mixing 56. Within this framework, the measurement of g 2 fixes the sparticle 8 Conclusions µ − scale, and Γ(µ eγ) may then be calculated → within any given flavour texture. Very ap- proximately, if g 2 is within one or two σ As we have seen, future colliders such as µ − + − of the present central value, one may expect the LHC and a TeV-scale linear e e col- B(µ eγ) with one or two orders of magni- lider have good prospects of discovering su- → tude of the present experimental upper limit, persymmetry and making detailed measure- as illustrated in Fig. 13 57. ments. In parallel, B and ν factories have good prospects of making inroads on the The decay µ 3e and µ e conversion flavour and unification problems. Searches → → on nuclei are expected to occur with branch- for dark matter, stopped-muon experiments ing ratios within two or three orders of mag- and searches for proton decay also have in-

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