arXiv:hep-ph/0611025 v1 2 Nov 20062006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer InstituteRM3-TH/06-21CERN-PH-TH/2006-228Introduction to the TerascaleGuido AltarelliDipartimento di Fisica ‘E. Amaldi’, Universita` di Roma Tre and INFN, Sezione di Roma Tre, I-00146 Rome, Italy andCERN, Department of Physics, Theory Division, CH–1211 Geneva 23, SwitzerlandWe briefly review the status of the electroweak theory, in the Standard Model and beyond, on the brink of the LHCstart1. THE PROGRAMME OF LHC PHYSICSThe first collisions at the LHC are expected near the end of ’07. The physics run at 14 TeV will start in the springof ’08. The particle physics community eagerly waits for the answers to the big questions that one expects to getfrom the LHC. The top physics issues at the LHC, addressed by the ATLAS and CMS collaborations, will be: 1) theexperimental clarification of the Higgs sector of the electroweak (EW) theory, 2) the search for new physics at theweak scale that, on conceptual grounds, one predicts should be in the LHC discovery range, and 3) the identificationof the particle(s) that make the dark matter in the Universe. In addition the LHCb detector will be devoted to thestudy of precision B physics, with the aim of going deeper in the knowledge of the Cabibbo-Kobayashi-Maskawa(CKM) matrix and of CP violation. The LHC will also devote a number of runs to accelerate heavy ions and theALICE collaboration will study their collisions for an experimental exploration of the QCD phase diagram.2. THE HIGGS PROBLEMThe Higgs problem is really central in particle physics today. On the one hand, the experimental verification ofthe Standard Model (SM) cannot be considered complete until the structure of the Higgs sector is not established byexperiment. On the other hand, the Higgs is also related to most of the major problems of particle physics, like theflavour problem and the hierarchy problem, which in turn strongly suggest the need for new physics near the weakscale. In turn the discovery of new physics could clarify the dark matter identity. It is clear that the fact that somesort of Higgs mechanism is at work has already been established. The W or the Z with longitudinal polarization thatwe observe are not present in an unbroken gauge theory (massless spin-1 particles, like the photon, are transverselypolarized). The longitudinal degree of freedom for the W or the Z is borrowed from the Higgs sector and is anevidence for it. Also, the couplings of quarks and leptons to the weak gauge bosons W± and Z are indeed preciselythose prescribed by the gauge symmetry. The accuracy of a few per mil in the precision tests implies that, not onlythe tree level, but also the structure of quantum corrections has been verified. To a lesser accuracy the triple gaugevertices γWW and ZWW have also been found in agreement with the specific predictions of the SU(2)⊗U(1) gaugetheory. This means that it has been verified that the gauge symmetry is unbroken in the vertices of the theory:all currents and charges are indeed symmetric. Yet there is obvious evidence that the symmetry is instead badlybroken in the masses. Not only the W and the Z have large masses, but the large splitting of, for example, the t-bdoublet shows that even a global weak SU(2) is not at all respected by the fermion spectrum. This is a clear signal ofspontaneous symmetry breaking and the implementation of spontaneous symmetry breaking in a gauge theory is viathe Higgs mechanism. The big remaining questions are about the nature and the properties of the Higgs particle(s).The present experimental information on the Higgs sector, mainly obtained from LEP as described in section 4, isL0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Institutesurprisingly limited. It can be summarized in a few lines, as follows. First, the relationM2W =M2Z cos2 θW , modifiedby small, computable radiative corrections, has been experimentally proven. This relation means that the effectiveHiggs (be it fundamental or composite) is indeed a weak isospin doublet. The Higgs particle has not been found but,in the SM, its mass can well be larger than the present direct lower limit mH >∼ 114 GeV (at 95% c.l.) obtainedfrom searches at LEP-2. As we shall see, the radiative corrections computed in the SM when compared to the dataon precision electroweak tests lead to a clear indication for a light Higgs, not too far from the present lower bound.The exact experimental upper limit for mH in the SM depends on the value of the top quark mass mt (the one-loopradiative corrections are quadratic in mt and logarithmic in mH). The CDF and D0 combined value after Run II isat present [1] mt = 171.4± 2.1 GeV (it went down with respect to the value mt = 178 ± 4.3 GeV from Run I andalso the experimental error is now sizably reduced). As a consequence the present limit on mH is more stringent[2]: mH < 199 GeV (at 95% c.l., after including the information from the 114 GeV direct bound). On the Higgsthe LHC will address the following questions : one doublet, more doublets, additional singlets? SM Higgs or SUSYHiggses? Fundamental or composite (of fermions, of WW...)? Pseudo-Goldstone boson of an enlarged symmetry?A manifestation of large extra dimensions (5th component of a gauge boson, an effect of orbifolding or of boundaryconditions...)? Or some combination of the above or something so far unthought of?3. THEORETICAL BOUNDS ON THE SM HIGGSIt is well known [3], [4], [5] that in the SM with only one Higgs doublet a lower limit on mH can be derived fromthe requirement of vacuum stability (or, in milder form, from a moderate instability, compatible with the lifetimeof the Universe [6]). The limit is a function of mt and of the energy scale Λ where the model breaks down andnew physics appears. The Higgs mass enters because it fixes the initial value of the quartic Higgs coupling λ forits running up to the large scale Λ. Similarly an upper bound on mH (with mild dependence on mt) is obtained[7] from the requirement that for λ, up to the scale Λ, no Landau pole appears, or in more explicit terms, that theperturbative description of the theory remains valid. We now briefly recall the derivation of these limits.The possible instability of the Higgs potential V [φ] is generated by the quantum loop corrections to the classicalexpression of V [φ]. At large φ the derivative V ′[φ] could become negative and the potential would become unboundfrom below. The one-loop corrections to V [φ] in the SM are well known and change the dominant term at largeφ according to λφ4 → (λ + γ log φ2/Λ2)φ4. The one-loop approximation is not enough in this case, because itfails at large enough φ, when γ log φ2/Λ2 becomes of order 1. The renormalization group improved version of thecorrected potential leads to the replacement λφ4 → λ(Λ)φ′4(Λ) where λ(Λ) is the running coupling and φ′(µ) =φexp∫ tγ(t′)dt′, with γ(t) being an anomalous dimension function and t = logΛ/v (v is the vacuum expectationvalue v = (2√2GF )−1/2). As a result, the positivity condition for the potential amounts to the requirement thatthe running coupling λ(Λ) never becomes negative. A more precise calculation, which also takes into account thequadratic term in the potential, confirms that the requirements of positive λ(Λ) leads to the correct bound down toscales Λ as low as ∼ 1 TeV. The running of λ(Λ) at one loop is given by:dλdt=34π2[λ2 + 3λh2t − 9h4t + small gauge and Yukawa terms] , (1)with the normalization such that at t = 0, λ = λ0 = m2H/2v2 and the top Yukawa coupling h0t = mt/v. We see that,for mH small and mt fixed at its measured value, λ decreases with t and can become negative. If one requires that λremains positive up to Λ = 1015–1019 GeV, then the resulting bound on mH in the SM with only one Higgs doubletis given by [5]:mH(GeV) > 129.5 + 2.1 [mt − 171.4]− 4.5 αs(mZ)− 0.1180.006. (2)Note that this limit is evaded in models with more Higgs doublets. In this case the limit applies to some averagemass but the lightest Higgs particle can well be below, as it is the case in the minimal SUSY extension of the SM(MSSM).L0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer InstituteThe upper limit on the Higgs mass in the SM is clearly important for assessing the chances of success of the LHCas an accelerator designed to solve the Higgs problem. The upper limit [7] arises from the requirement that theLandau pole associated with the non asymptotically free behaviour of the λφ4 theory does not occur below the scaleΛ. The initial value of λ at the weak scale increases with mH and the derivative is positive at large λ (because ofthe positive λ2 term - the λϕ4 theory is not asymptotically free - overwhelms the negative top-Yukawa term). Thusif mH is too large the point where λ computed from the perturbative beta function becomes infinite (the Landaupole) occurs at too low an energy. Of course in the vicinity of the Landau pole the 2-loop evaluation of the betafunction is not reliable. Indeed the limit indicates the frontier of the domain where the theory is well described bythe perturbative expansion. Thus the quantitative evaluation of the limit is only indicative, although it has been tosome extent supported by simulations of the Higgs sector of the EW theory on the lattice. For the upper limit onmH one finds [7] mH <∼ 180 GeV for Λ ∼ MGUT −MPl and mH <∼ 0.5 − 0.8 TeV for Λ ∼ 1 TeV . Actually, formt ∼ 171 GeV, only a small range of values for mH is allowed, 130 < mH < ∼ 200 GeV, if the SM holds up toΛ ∼MGUT or MPl. This upper limit implies that the SM Higgs cannot escape detection at the LHC.4. PRECISION TESTS OF THE STANDARD ELECTROWEAK THEORYThe results of the electroweak precision tests as well as of the searches for the Higgs boson and for new particlesperformed at LEP and SLC are now available in final form [2] [8]. Taken together with the measurements of mt,mW and the searches for new physics at the Tevatron, and with some other data from low energy experiments, theyform a very stringent set of precise constraints to be compared with the SM or with any of its conceivable extensions.All high energy precision tests of the SM are summarized in fig. 1 [2]. For the analysis of electroweak data in theSM one starts from the input parameters: as in any renormalizable theory masses and couplings have to be specifiedfrom outside. One can trade one parameter for another and this freedom is used to select the best measured ones asinput parameters. Some of them, α, GF and mZ , are very precisely known, some other ones, mflight , mt and αs(mZ)are far less well determined while mH is largely unknown. Among the light fermions, the quark masses are badlyknown, but fortunately, for the calculation of radiative corrections, they can be replaced by α(mZ), the value of theQED running coupling at the Z mass scale. The value of the hadronic contribution to the running, ∆α(5)had(mZ),reported in Fig. 1, is obtained through dispersion relations from the data on e+e− → hadrons at low centre-of-massenergies [2]. From the input parameters one computes the radiative corrections to a sufficient precision to match theexperimental accuracy. Then one compares the theoretical predictions with the data for the numerous observableswhich have been measured, checks the consistency of the theory and derives constraints on mt, αs(mZ) and mH .The computed radiative corrections include the complete set of one-loop diagrams, plus some selected large subsetsof two-loop diagrams and some sequences of resummed large terms of all orders (large logarithms and Dyson resum-mations). In particular large logarithms, e.g., terms of the form (α/π ln (mZ/mfℓ))n where fℓ is a light fermion,are resummed by well-known and consolidated techniques based on the renormalisation group. For example, largelogarithms dominate the running of α from me, the electron mass, up to mZ , which is a 6% effect, much larger thanthe few per mil contributions of purely weak loops. Also, large logs from initial state radiation dramatically distortthe line shape of the Z resonance observed at LEP-1 and SLC and have been accurately taken into account in themeasurement of the Z mass and total width.Among the one loop EW radiative corrections a remarkable class of contributions are those terms that increasequadratically with the top mass. The large sensitivity of radiative corrections to mt arises from the existence ofthese terms. The quadratic dependence on mt (and possibly on other widely broken isospin multiplets from newphysics) arises because, in spontaneously broken gauge theories, heavy loops do not decouple. On the contrary, inQED or QCD, the running of α and αs at a scale Q is not affected by heavy quarks with mass M ≫ Q. Accordingto an intuitive decoupling theorem [9], diagrams with heavy virtual particles of mass M can be ignored for Q≪Mprovided that the couplings do not grow with M and that the theory with no heavy particles is still renormalizable.In the spontaneously broken EW gauge theories both requirements are violated. First, one important difference withrespect to unbroken gauge theories is in the longitudinal modes of weak gauge bosons. These modes are generatedL0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer InstituteMeasurement Fit |Omeas−Ofit|/σmeas0 1 2 30 1 2 3∆αhad(mZ)(5) 0.02758 ± 0.00035 0.02766mZ [GeV] 91.1875 ± 0.0021 91.1874ΓZ [GeV] 2.4952 ± 0.0023 2.4957σhad [nb]0 41.540 ± 0.037 41.477Rl 20.767 ± 0.025 20.744Afb0,l 0.01714 ± 0.00095 0.01640Al(Pτ) 0.1465 ± 0.0032 0.1479Rb 0.21629 ± 0.00066 0.21585Rc 0.1721 ± 0.0030 0.1722Afb0,b 0.0992 ± 0.0016 0.1037Afb0,c 0.0707 ± 0.0035 0.0741Ab 0.923 ± 0.020 0.935Ac 0.670 ± 0.027 0.668Al(SLD) 0.1513 ± 0.0021 0.1479sin2θefflept(Qfb) 0.2324 ± 0.0012 0.2314mW [GeV] 80.392 ± 0.029 80.371ΓW [GeV] 2.147 ± 0.060 2.091mt [GeV] 171.4 ± 2.1 171.7Figure 1: Pulls, Summer’06by the Higgs mechanism, and their couplings grow with masses (as is also the case for the physical Higgs couplings).Second, the theory without the top quark is no more renormalizable because the gauge symmetry is broken if theb quark is left with no partner (while its couplings show that the weak isospin is 1/2). Because of non decouplingprecision tests of the electroweak theory may be sensitive to new physics even if the new particles are too heavy fortheir direct production.While radiative corrections are quite sensitive to the top mass, they are unfortunately much less dependent on theHiggs mass. If they were sufficiently sensitive, by now we would precisely know the mass of the Higgs. However, thedependence of one loop diagrams on mH is only logarithmic: ∼ GFm2W log(m2H/m2W ). Quadratic terms ∼ G2Fm2Honly appear at two loops and are too small to be important. The difference with the top case is that m2t −m2b is adirect breaking of the gauge symmetry that already affects the relevant one loop diagrams, while the Higgs couplingsto gauge bosons are ”custodial-SU(2)” symmetric in lowest order.The various asymmetries determine the effective electroweak mixing angle for leptons with highest sensitivity. Theresults on sin2 θeff are compared in Figure 2. The weighted average of these six results, including small correlations,is:sin2 θeff = 0.23153± 0.00016 . (3)Note, however, that this average has a χ2 of 11.8 for 5 degrees of freedom, corresponding to a probability of 3.7%.The χ2 is pushed up by the two most precise measurements of sin2 θeff , namely those derived from the measurementsof Al by SLD, dominated by the left-right asymmetry ALR, and of the forward-backward asymmetry measured in bb¯L0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Institute10 210 30.23 0.232 0.234sin2θlepteffmH [GeV]χ2/d.o.f.: 11.8 / 5A0,lfb 0.23099 ± 0.00053Al(Pτ) 0.23159 ± 0.00041Al(SLD) 0.23098 ± 0.00026A0,bfb 0.23221 ± 0.00029A0,cfb 0.23220 ± 0.00081Qhadfb 0.2324 ± 0.0012Average 0.23153 ± 0.00016∆αhad= 0.02758 ± 0.00035(5)mt= 172.7 ± 2.9 GeVFigure 2: Effective electroweak mixing angle sin2 θeff derived from measurement results depending on lepton couplings only(above) and also quark couplings (below).production at LEP,AbFB, which differ by about 3.2 σ’s. In general, there appears to be a discrepancy between sin2 θeffmeasured from leptonic asymmetries ((sin2 θeff)l) and from hadronic asymmetries ((sin2 θeff)h), as seen from Figure 2.In fact, the result from ALR is in good agreement with the leptonic asymmetries measured at LEP, while all hadronicasymmetries, though their errors are large, are better compatible with the result of AbFB. This very unfortunate factmakes the interpretation of precision tests less sharp and some perplexity remains: is it an experimental error or asignal of some new physics? The situation is shown in Figure 3 [10]. The values of (sin2 θeff)l, (sin2 θeff)h and theirformal combination are shown each at the mH value that would correspond to it given the central value of mt. Ofcourse, the value for mH indicated by each sin2 θeff has an horizontal ambiguity determined by the measurementerror and the width of the ±1σ band for mt. Even taking this spread into account it is clear that the implicationson mH are sizably different.One might imagine that some new physics effect could be hidden in the Zbb¯ vertex. Like for the top quark massthere could be other non decoupling effects from new heavy states or a mixing of the b quark with some other heavyquark. However, it is well known that this discrepancy is not easily explained in terms of some new physics effect inthe Zbb¯ vertex. In fact, AbFB is the product of lepton- and b-asymmetry factors: AbFB = (3/4)AeAb. The sensitivityof AbFB to Ab is limited, because the Ae factor is small, so that a rather large change of the b-quark couplingswith respect to the SM is needed in order to reproduce the measured discrepancy (precisely a ∼ 30% change inthe right-handed coupling gbR, an effect too large to be a loop effect but which could be produced at the tree level,e.g., by mixing of the b quark with a new heavy vectorlike quark [11]). But this effect is not confirmed by thedirect measurement of Ab performed at SLD using the left-right polarized b asymmetry, even taking into account themoderate precision of this result. No deviation is manifest in the accurate measurement of Rb ∝ g2Rb + g2Lb. Thus,even introducing an ad hoc mixing the overall fit of AbFB, Ab and Rb is not terribly good, but we cannot excludethis possibility completely. Alternatively, the observed discrepancy could be due to a large statistical fluctuation orL0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Institute 0.231 0.2312 0.2314 0.2316 0.2318 0.232 0.2322 0.2324 500 400 300 200 100 50sin2θ effleptMH [GeV]Mt=171.4 GeVMt +1σMt -1σhadr. asymmlept. asymmsin2θefflept world av.Figure 3: The data for sin2 θlepteff are plotted vs mH . For presentation purposes the measured points are shown each at the mHvalue that would ideally correspond to it given the central value of mt (updated from [10]).an unknown experimental problem. In any case the effective ambiguity in the measured value of sin2 θeff is actuallylarger than the nominal error, reported in Eq. 3, obtained from averaging all the existing determinations.We now discuss fitting the data in the SM. One can think of different types of fit, depending on which experimentalresults are included or which answers one wants to obtain. For example, in Table I we present in column 1 a fit of allZ pole data plus mW (this is interesting as it shows the value of mt obtained indirectly from radiative corrections,to be compared with the value of mt measured in production experiments), in column 2 a fit of all Z pole data plusmt (here it is mW which is indirectly determined), and, finally, in column 3 a fit of all the data listed in Fig. 1(which is the most relevant fit for constraining mH). From the fit in column 1 of Table I we see that the extractedvalue of mt is in good agreement with the direct measurement (see the value reported in Fig. 1). Similarly we seethat the direct determination of mW reported in Fig. 1 is still a bit larger with respect to the value from the fit incolumn 2 (although the direct value ofmW went down recently). We have seen that quantum corrections depend onlylogarithmically on mH . In spite of this small sensitivity, the measurements are precise enough that one still obtainsa quantitative indication of the mass range. From the fit in column 3 we obtain: log10mH(GeV) = 1.93 ± 0.17 (ormH = 85+39−28 GeV). We see that the central value of mH from the fit is below the lower limit on the SM Higgs massfrom direct searches mH >∼ 114 GeV, but within 1σ from this bound. If we had reasons to remove the result on AbFBfrom the fit, the fitted value of mH would move down: mH = 55+31−21 GeV [12], further away from the lower limit.We have already observed that the experimental value ofmW (with good agreement between LEP and the Tevatron)is a bit high compared to the SM prediction (see Figure 4). The value of mH indicated by mW is on the low side, justin the same interval as for sin2 θlepteff measured from leptonic asymmetries. The recent decrease of the experimentalvalue of mt maintains the tension between the experimental values of mW and sin2 θlepteff measured from leptonicasymmetries on one side and the lower limit on mH from direct searches on the other side [13], [14].With all these words of caution in mind it remains true that on the whole the SM performs rather well, so that itis fair to say that no clear indication for new physics emerges from the data. Actually the result of precision tests onthe Higgs mass is particularly remarkable. The value of log10mH(GeV) is, within errors, within the small windowbetween ∼ 2 and ∼ 3 which is allowed, on the one side, by the direct search limit (mH >∼ 114 GeV from LEP-2 [2]),and, on the other side, by the theoretical upper limit on the Higgs mass in the minimal SM [7], mH <∼ 600−800 GeV.L0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer InstituteFit 1 2 3Measurements mW mt mt, mWmt (GeV) 177.6+12−9 171.4 ± 2.1 171.7 ± 2.0mH (GeV) 137+228−76 103+54−37 85+39−28log [mH(GeV)] 2.14 ±+0.39 2.01 ± 0.19 1.93± 0.17αs(mZ) 0.1190 ± 0.0028 0.1190 ± 0.0027 0.1186 ± 0.0026mW (MeV) 80380 ± 21 80361 ± 20 80371 ± 16Table I: Standard Model fits of electroweak data. All fits use the Z pole results and ∆α(5)had(mZ) as listed in Fig. 1. Inaddition, the measurements listed on top of each column are included as well. The fitted W mass is also shown [2] (the directlymeasured value is mW = 80392 ± 29MeV). 80.25 80.3 80.35 80.4 80.45 500 400 300 200 100 50MW [GeV]MH [GeV]MW world averageMt +1σMt=171.4 GeVMt -1σFigure 4: The world average for mW is plotted vs mH (updated from [10]).Thus the whole picture of a perturbative theory with a fundamental Higgs is well supported by the data on radiativecorrections. It is important that there is a clear indication for a particularly light Higgs: at 95% c.l. mH <∼ 199 GeV.This is quite encouraging for the ongoing search for the Higgs particle. More in general, if the Higgs couplings areremoved from the Lagrangian the resulting theory is non renormalizable. A cutoff Λ must be introduced. In thequantum corrections logmH is then replaced by log Λ plus a constant. The precise determination of the associatedfinite terms would be lost (that is, the value of the mass in the denominator in the argument of the logarithm). Aheavy Higgs would need some conspiracy: the finite terms, different in the new theory from those of the SM, shouldaccidentally compensate for the heavy Higgs in a few key parameters of the radiative corrections (mainly ǫ1 and ǫ3,see, for example, [15]). Alternatively, additional new physics, for example in the form of effective contact terms addedto the minimal SM lagrangian, should accidentally do the compensation, which again needs some sort of conspiracy,although this possibility is not so unlikely to be apriori discarded.L0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Instituteρ-1 -0.5 0 0.5 1 1.5 2η-1.5-1-0.500.511.5αβγηγγααdm∆KεKεdm∆ & sm∆cb/VubVβsin2 < 0βsol. w/ cos2(excl. at CL > 0.95)excluded area has CL > 0.95excluded at CL > 0.95BEAUTY 2006CKMf i t t e rFigure 5: Constraints in the ρ¯, η¯ plane including the most recent α, γ and ∆Ms inputs in the global CKM fit [17]).5. THE PHYSICS OF FLAVOURIn the last decade great progress in different areas of flavour physics has also been achieved. In the quark sector,the important results of a generation of frontier experiments have become available, obtained at B factories and ataccelerators. The hope of these experiments was to detect departures from the CKM picture of mixing and of CPviolation as signals of new physics. But so far the observed B mixing and CP violation agree very well with the SMpredictions based on the CKM matrix [18]. The recent measurement of ∆ms by CDF and D0, in fair agreementwith the SM expectation, has closed another door for new physics. In quantitative terms all measurements are inagreement with the CKM description of mixing and CP violation as shown by Fig. 5. It is only in channels thatare forbidden at tree level and occur through penguin loops (as is the case for B → πK modes) that some deviationcould still be hidden. The amazing performance of the SM in flavour changing transitions and for CP violation in Kand B decays poses a strong constraint on all proposed models of new physics.In the leptonic sector the study of neutrino oscillations has led to the discovery that at least two neutrinos are notmassless and to the determination of the mixing matrix [19]. Neutrinos are not all massless but their masses are verysmall. Probably masses are small because νs are Majorana particles, and, by the see-saw mechanism, their massesare inversely proportional to the large scale M where lepton number (L) violation occurs (as expected in GUT’s).Indeed the value of M ∼ mνR from experiment is compatible with being close to MGUT ∼ 1014 − 1015 GeV , so thatneutrino masses fit well in the GUT picture and actually support it. It was realized that decays of heavy νR with CPand L violation can produce a B-L asymmetry. The range of neutrino masses indicated by neutrino phenomenologyturns out to be perfectly compatible with the idea of baryogenesis via leptogenesis [20]. This elegant model forbaryogenesis has by now replaced the idea of baryogenesis near the weak scale, which has been strongly disfavouredby LEP. It is remarkable that we now know the neutrino mixing matrix with good accuracy. Two mixing angles arelarge and one is small. The atmospheric angle θ23 is large, actually compatible with maximal but not necessarily so:at 3σ: 0.31 ≤ sin2 θ23 ≤ 0.72 with central value around 0.5. The solar angle θ12 is large, sin2 θ12 ∼ 0.3, but certainlynot maximal (by more than 5σ). The third angle θ13, strongly limited mainly by the CHOOZ experiment, has atL0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Institutepresent a 3σ upper limit given by about sin2 θ13 ≤ 0.08. While these discoveries are truly remarkable, it is somewhatdepressing that the detailed knowledge of both the quark and the neutrino mixings has not led so far to a compellingsolution of the dynamics of fermion masses and mixings: our models can reproduce, actually along different paths,the observed values, but we do not really understand their mysterious pattern.The only plausible indication of new physics could be the apparent discrepancy from the SM prediction of themuon (g-2) measurement by the BNL experiment [21]. The deviation is reported at the level of 3.3σ’s [22]. I said thatthe anomaly is plausible. In fact this observable is sufficiently sensitive, so that it is reasonable that the detectionof new physics could occur here. There are reasonable models that could accomodate the observed discrepancy, forexample SUSY with light electroweak s-particles and moderately large tanβ [14]. There are however some wordsof caution. The dominant error is the theoretical error associated with the hadronic contributions to the vacuumpolarization and to the light by light scattering 4-point function [23]. As usual for theoretical errors, they are tosome extent debatable and authors tend to emphasize the importance of each progress, e.g. in the data on hadroniccross sections, by decreasing the theoretical error more and more. Also there is a difference between the predictionson vacuum polarization based on e+e− crosssections versus hadronic τ decays (the g-2 discrepancy is much less ifestimated from the τ data). This difference is not understood.6. PROBLEMS OF THE STANDARD MODELNo signal of new physics has been found neither in electroweak precision tests nor in flavour physics. Given thesuccess of the SM why are we not satisfied with this beautiful theory? Why not just find the Higgs particle, forcompleteness, and declare that particle physics is closed? The reason is that there are both conceptual problems andphenomenological indications for physics beyond the SM. On the conceptual side the most obvious problems are thatquantum gravity is not included in the SM and the related hierarchy problem. Among the main phenomenologicalhints for new physics we can list coupling unification, dark matter, neutrino masses, baryogenesis and the cosmologicalvacuum energy.The computed evolution with energy of the effective SM gauge couplings clearly points towards the unification ofthe electro-weak and strong forces (Grand Unified Theories: GUT’s) at scales of energy MGUT ∼ 1015 − 1016 GeVwhich are close to the scale of quantum gravity, MPl ∼ 1019 GeV . One is led to imagine a unified theory of allinteractions also including gravity (at present superstrings provide the best attempt at such a theory). Thus GUT’sand the realm of quantum gravity set a very distant energy horizon that modern particle theory cannot ignore.Can the SM without new physics be valid up to such large energies? One can imagine that some obvious problemscould be postponed to the more fundamental theory at the Planck mass. For example, the explanation of the threegenerations of fermions and the understanding of fermion masses and mixing angles can be postponed. But otherproblems must find their solution in the low energy theory. In particular, the structure of the SM could not naturallyexplain the relative smallness of the weak scale of mass, set by the Higgs mechanism at µ ∼ 1/√GF ∼ 250 GeVwith GF being the Fermi coupling constant. This so-called hierarchy problem is due to the instability of the SMwith respect to quantum corrections. This is related to the presence of fundamental scalar fields in the theory withquadratic mass divergences and no protective extra symmetry at µ = 0. For fermion masses, first, the divergencesare logarithmic and, second, they are forbidden by the SU(2)⊗U(1) gauge symmetry plus the fact that at m = 0an additional symmetry, i.e. chiral symmetry, is restored. Here, when talking of divergences, we are not worriedof actual infinities. The theory is renormalizable and finite once the dependence on the cut off Λ is absorbed in aredefinition of masses and couplings. Rather the hierarchy problem is one of naturalness. We can look at the cut offas a parameterization of our ignorance on the new physics that will modify the theory at large energy scales. Thenit is relevant to look at the dependence of physical quantities on the cut off and to demand that no unexplainedenormously accurate cancellations arise.The hierarchy problem can be put in very practical terms (the ”little hierarchy problem”): loop corrections to theHiggs mass squared are quadratic in Λ. The most pressing problem is from the top loop. With m2h = m2bare + δm2hL0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Institutethe top loop givesδm2h|top ∼ −3GF2√2π2m2tΛ2 ∼ −(0.2Λ)2 (4)If we demand that the correction does not exceed the light Higgs mass indicated by the precision tests, Λ mustbe close, Λ ∼ o(1 TeV ). Similar constraints arise from the quadratic Λ dependence of loops with gauge bosons andscalars, which, however, lead to less pressing bounds. So the hierarchy problem demands new physics to be veryclose (in particular the mechanism that quenches the top loop). Actually, this new physics must be rather special,because it must be very close, yet its effects are not clearly visible (the ”LEP Paradox” [24]).7. AVENUES BEYOND THE STANDARD MODELExamples of proposed classes of solutions for the hierarchy problem are:Supersymmetry. In the limit of exact boson-fermion symmetry the quadratic divergences of bosons cancelso that only log divergences remain. However, exact SUSY is clearly unrealistic. For approximate SUSY (withsoft breaking terms), which is the basis for all practical models, Λ is replaced by the splitting of SUSY multiplets,Λ ∼ mSUSY −mord. In particular, the top loop is quenched by partial cancellation with s-top exchange, so the s-topcannot be too heavy.Technicolor. The Higgs system is a condensate of new fermions. There is no fundamental scalar Higgs sector,hence no quadratic divergences associated to the µ2 mass in the scalar potential. This mechanism needs a verystrong binding force, ΛTC ∼ 103 ΛQCD. It is difficult to arrange that such nearby strong force is not showing up inprecision tests. Hence this class of models has been disfavoured by LEP, although some special class of models havebeen devised aposteriori, like walking TC, top-color assisted TC etc (for recent reviews, see, for example, [25]).Large extra dimensions. The idea is that MPl appears very large, or equivalently that gravity appears veryweak, because we are fooled by hidden extra dimensions so that the real gravity scale is reduced down to a lowerscale, even possibly down to o(1 TeV ). This possibility is very exciting in itself and it is really remarkable that it iscompatible with experiment.”Little Higgs”models. In these models the Higgs is a pseudo-Goldstone boson and extra symmetries allowmh 6= 0 only at two-loop level, so that Λ can be as large as o(10 TeV ) with the Higgs within present bounds (thetop loop is quenched by exchange of heavy vectorlike new quarks with charge 2/3).In the following we briefly comment in turn on these possibilities.8. SUPERSYMMETRYSUSY models are the most developed and most widely accepted. Many theorists consider SUSY as establishedat the Planck scale MPl. So why not to use it also at low energy to fix the hierarchy problem, if at all possible?It is interesting that viable models exist. The necessary SUSY breaking can be introduced through soft terms thatdo not spoil the good convergence properties of the theory. Precisely those terms arise from supergravity when it isspontaneoulsly broken in a hidden sector. This is the case of the MSSM [26] with minimal particle content. Of course,minimality is only a simplicity assumption that could possibly be relaxed. For example, adding an additional Higgssinglet S considerably helps in addressing naturalness constraints [27], [28]. Minimal versions or, even more, versionswith additional constraints like the constrained MSSM (CMSSM) (where simple conditions at the GUT scale are inaddition assumed) are economic in terms of new parameters but could be to some extent misleading. Still, the MSSMis a completely specified, consistent and computable theory which is compatible with all precision electroweak tests.In this most traditional approach SUSY is broken in a hidden sector and the scale of SUSY breaking is very large oforder Λ ∼√G−1/2F MPl. But since the hidden sector only communicates with the visible sector through gravitationalinteractions the splitting of the SUSY multiplets is much smaller, in the TeV energy domain, and the Goldstino isL0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Institutepractically decoupled. But alternative mechanisms of SUSY breaking are also being considered. In one alternativescenario [29] the (not so much) hidden sector is connected to the visible one by ordinary gauge interactions. As theseare much stronger than the gravitational interactions, Λ can be much smaller, as low as 10-100 TeV. It follows that theGoldstino is very light in these models (with mass of order or below 1 eV typically) and is the lightest, stable SUSYparticle, but its couplings are observably large. The radiative decay of the lightest neutralino into the Goldstinoleads to detectable photons. The signature of photons comes out naturally in this SUSY breaking pattern: withrespect to the MSSM, in the gauge mediated model there are typically more photons and less missing energy. Themain appeal of gauge mediated models is a better protection against flavour changing neutral currents but naturalityproblems tend to increase. As another possibility it has been pointed out that there are pure gravity contributionsto soft masses that arise from anomalies in gravity theory [30]. In the assumption that these terms are dominant theassociated spectrum and phenomenology have been studied. In this case gaugino masses are proportional to gaugecoupling beta functions, so that the gluino is much heavier than the electroweak gauginos.What is really unique to SUSY with respect to all other extensions of the SM listed above is that the MSSM or othernon minimal SUSY models are well defined and computable up to MPl and, moreover, are not only compatible butactually quantitatively supported by coupling unification and GUT’s. At present the most direct phenomenologicalevidence in favour of supersymmetry is obtained from the unification of couplings in GUTs. Precise LEP data onαs(mZ) and sin2 θW show that standard one-scale GUTs fail in predicting αs(mZ) given sin2 θW and α(mZ) whileSUSY GUTs are compatible with the present, very precise, experimental results (of course, the ambiguities in theprediction are larger because of our ignorance of the SUSY spectrum). If one starts from the known values of sin2 θWand α(mZ), one finds [31] for αs(mZ) the results: αs(mZ) = 0.073 ± 0.002 for Standard GUTs and αs(mZ) =0.129± 0.010 for SUSY GUTs to be compared with the world average experimental value αs(mZ) = 0.118± 0.002.Another great asset of SUSY GUT’s is that proton decay is much slowed down with respect to the non SUSY case.First, the unification mass MGUT ∼ few 1016 GeV, in typical SUSY GUT’s, is about 20-30 times larger than forordinary GUT’s. This makes p decay via gauge boson exchange negligible and the main decay amplitude arisesfrom dim-5 operators with higgsino exchange, leading to a rate close but still compatible with existing bounds (see,for example,[32]). It is also important that SUSY provides an excellent dark matter candidate, the neutralino. Wefinally recall that the range of neutrino masses as indicated by oscillation experiments, when interpreted in thesee-saw mechanism, point to MGUT and give additional support to GUTs [19].In spite of all these virtues it is true that the lack of SUSY signals at LEP and the lower limit on mH poseproblems for the MSSM. The lightest Higgs particle is predicted in the MSSM to be below mh <∼ 130 GeV (with theesperimental value ofmt going down the upper limit is slightly decreased). The limit on the SM HiggsmH >∼ 114GeVconsiderably restricts the available parameter space of the MSSM requiring relatively large tanβ (tanβ >∼ 2− 3: attree level m2h = m2Z cos2 2β) and rather heavy s-top (the loop corrections increase with m4t log m˜2t ). But we have seenthat a heavy s-top is unnatural, because it enters quadratically in the radiative corrections to δm2h|top. Stringentnaturality constraints also follow from imposing that the electroweak symmetry breaking occurs at the right energyscale: in SUSY models the breaking is induced by the running of the Hu mass starting from a common scalar massm0 at MGUT . The squared Z mass m2Z can be expressed as a linear combination of the SUSY parameters m20, m21/2,A2t , µ2,... with known coefficients. Barring cancellations that need fine tuning, the SUSY parameters, hence theSUSY s-partners cannot be too heavy. The LEP limits, in particular the chargino lower bound mχ+ >∼ 100 GeV , aresufficient to eliminate an important region of the parameter space, depending on the amount of allowed fine tuning.For example, models based on gaugino universality at the GUT scale, like the CMSSM, are discarded unless a finetuning by at least a factor of 20 is not allowed. Without gaugino universality [33] the strongest limit remains on thegluino mass: m2Z ∼ 0.7 m2gluino + . . . which is still compatible with the present limit mgluino >∼ 240 GeV from theTeVatron.L0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Institute9. LARGE EXTRA DIMENSIONSThe non discovery of SUSY at LEP has given further impulse to the quest for new ideas on physics beyond the SM.Large extra dimensions [34] models are among the most interesting new directions in model building. Large extradimension models propose to solve the hierarchy problem by bringing gravity down fromMPl tom ∼ o(1 TeV ) wherem is the string scale. Inspired by string theory one assumes that some compactified extra dimensions are sufficientlylarge and that the SM fields are confined to a 4-dimensional brane immersed in a d-dimensional bulk while gravity,which feels the whole geometry, propagates in the bulk. We know that the Planck mass is large because gravity isweak: in fact GN ∼ 1/M2Pl, where GN is Newton constant. The idea is that gravity appears so weak because a lotof lines of force escape in extra dimensions. Assume you have n = d − 4 extra dimensions with compactificationradius R. For large distances, r >> R, the ordinary Newton law applies for gravity: in natural units, the forcebetween two units of mass is F ∼ GN/r2 ∼ 1/(M2Plr2). At short distances, r <∼ R, the flow of lines of force inextra dimensions modifies Gauss law and F−1 ∼ m2(mr)d−4r2. By matching the two formulas at r = R one obtains(MPl/m)2 = (Rm)d−4. For m ∼ 1 TeV and n = d− 4 one finds that n = 1 is excluded (R ∼ 1015cm), for n = 2 Ris at the edge of present bounds R ∼ 1 mm, while for n = 4, 6, R ∼ 10−9, 10−12 cm. In all these models a genericfeature is the occurrence of Kaluza-Klein (KK) modes. Compactified dimensions with periodic boundary conditions,as for quantization in a box, imply a discrete spectrum with momentum p = n/R and mass squared m2 = n2/R2.There are many versions of these models. The SM brane can itself have a thickness r with r <∼ 10−17 cm or1/r >∼ 1 TeV , because we know that quarks and leptons are pointlike down to these distances, while for gravityin the bulk there is no experimental counter-evidence down to R <∼ 0.1 mm or 1/R >∼ 10−3 eV . In case of athickness for the SM brane there would be KK recurrences for SM fields, like Wn, Zn and so on in the TeV regionand above. There are models with factorized metric (ds2 = ηµνdxµdxν+hij(y)dyidyj , where y (i,j) denotes the extradimension coordinates (and indices), or models with warped metric (ds2 = e−2kR|φ|ηµνdxµdxν − R2φ2 [36]. In anycase there are the towers of KK recurrences of the graviton. They are gravitationally coupled but there are a lot ofthem that sizably couple, so that the net result is a modification of cross-sections and the presence of missing energy.Large extra dimensions provide a very exciting scenario [37]. Already it is remarkable that this possibility iscompatible with experiment. However, there are a number of criticisms that can be brought up. First, the hierarchyproblem is more translated in new terms rather than solved. In fact the basic relation Rm = (MPl/m)2/n shows thatRm, which one would apriori expect to be 0(1), is instead ad hoc related to the large ratio MPl/m. In this respectthe Randall-Sundrum variety is more appealing because the hierarchy suppression mW /MPl could arise from thewarping factor e−2kR|φ|, for not too large values of kR. The question of whether these values of kR are reasonablehas been discussed in ref. [38], which offer the best support to the solution of the hierarchy problem in this context.Also it is not clear how extra dimensions can by themselves solve the LEP paradox (the large top loop correctionsshould be controlled by the opening of the new dimensions and the onset of gravity): since mH is light Λ ∼ 1/Rmust be relatively close. But precision tests put very strong limits on Λ. In fact in typical models of this classthere is no mechanism to sufficiently quench the corrections. While no simple, realistic model has yet emerged asa benchmark, it is attractive to imagine that large extra dimensions could be a part of the truth, perhaps coupledwith some additional symmetry or even SUSY. The Randall-Sundrum warped geometry has become the commonframework for many attempts in this direction.In the general context of extra dimensions an interesting direction of development is the study of symmetrybreaking by orbifolding and/or boundary conditions. These are models where a larger gauge symmetry (with orwithout SUSY) holds in the bulk. The symmetry is reduced in the 4 dimensional brane, where the physics thatwe observe is located, as an effect of symmetry breaking induced geometrically by suitable boundary conditions.There are models where SUSY, valid in n > 4 dimensions is broken by boundary conditions [39], in particular themodel of ref.[40], where the mass of the Higgs is computable and can be estimated with good accuracy. Then thereare ”Higgsless models” where it is the SM electroweak gauge symmetry which is broken at the boundaries [41]. Ormodels where the Higgs is the 5th component of a gauge boson of an extended symmetry valid in n > 4 [42]. Ingeneral all these alternative models for the Higgs mechanism face severe problems and constraints from electroweakL0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer Instituteprecision tests [43]. At the GUT scale, symmetry breaking by orbifolding can be applied to obtain a reformulation ofSUSY GUT’s where many problematic features of ordinary GUT’s (e.g. a baroque Higgs sector, the doublet-tripletsplitting problem, fast proton decay etc) are improved [44], [37].10. LITTLE HIGGS MODELSIn ”little Higgs” models the symmetry of the SM is extended to a suitable global group G that also containssome gauge enlargement of SU(2)⊗U(1), for example G ⊃ [SU(2)⊗U(1)]2 ⊃ SU(2)⊗U(1). The Higgs particleis a pseudo-Goldstone boson of G that only takes mass at 2-loop level, because two distinct symmetries must besimultaneously broken for it to take mass, which requires the action of two different couplings in the same diagram.Then in the relation between δm2h and Λ2 there is an additional coupling and an additional loop factor that allowfor a bigger separation between the Higgs mass and the cut-off. Typically, in these models one has one or moreHiggs doublets at mh ∼ 0.2 TeV , and a cut-off at Λ ∼ 10 TeV . The top loop quadratic cut-off dependence ispartially canceled, in a natural way guaranteed by the symmetries of the model, by a new coloured, charge-2/3,vectorial quark χ of mass around 1 TeV (a fermion not a scalar like the s-top of SUSY models). Certainly thesemodels involve a remarkable level of group theoretic virtuosity. However, in the simplest versions one is faced withproblems with precision tests of the SM [45]. Even with vectorlike new fermions, large corrections to the epsilonparameters [15] arise from exchanges of the new gauge bosonsW ′ and Z ′ (due to lack of custodial SU(2) symmetry).In order to comply with these constraints the cut-off must be pushed towards large energy and the amount of finetuning needed to keep the Higgs light is still quite large. Probably these bad features can be fixed by some suitablecomplication of the model (see for example, [46]). But, in my opinion, the real limit of this approach is that itonly offers a postponement of the main problem by a few TeV, paid by a complete loss of predictivity at higherenergies. In particular all connections to GUT’s are lost. An interesting model that combines the idea of the Higgsas a Goldstone boson and warped extra dimensions was proposed and studied in refs.[47].11. DARK MATTER AND DARK ENERGYWe know by now [49] that the Universe is flat and most of it is not made up of known forms of matter: Ωtot ∼ 1,Ωbaryonic ∼ 0.044, Ωmatter ∼ 0.3, where Ω is the ratio of the density to the critical density. Most is Dark Matter (DM)and Dark Energy (DE) with ΩΛ ∼ 0.7. We also know that most of DM must be cold (non relativistic at freeze-out)and that significant fractions of hot DM are excluded. Neutrinos are hot DM (because they are ultrarelativisticat freeze-out) and indeed are not much cosmo-relevant: Ων <∼ 0.015. Identification of DM is a task of enormousimportance for both particle physics and cosmology. If really neutralinos are the main component of DM they will bediscovered at the LHC and this will be a great service of particle physics to cosmology. More in general, the LHC issensitive to a large variety of WIMP’s (Weekly Interacting Massive Particles). WIMP’s with masses in the 10 GeV-1TeV range with typical electroweak crosssections contribute to Ω terms of o(1). Also, these results on cosmologicalparameters have shown that vacuum energy accounts for about 2/3 of the critical density: ΩΛ ∼ 0.7, Translatedinto familiar units this means for the energy density ρΛ ∼ (2 10−3 eV )4 or (0.1 mm)−4. It is really interesting (andnot at all understood) that ρ1/4Λ ∼ Λ2EW /MPl (close to the range of neutrino masses). It is well known that in fieldtheory we expect ρΛ ∼ Λ4cutoff . If the cut off is set at MPl or even at 0(1 TeV ) there would an enormous mismatch.In exact SUSY ρΛ = 0, but SUSY is broken and in presence of breaking ρ1/4Λ is in general not smaller than thetypical SUSY multiplet splitting. Another closely related problem is ”why now?”: the time evolution of the matteror radiation density is quite rapid, while the density for a cosmological constant term would be flat. If so, then howcomes that precisely now the two density sources are comparable? This suggests that the vacuum energy is not acosmological constant term, buth rather the vacuum expectation value of some field (quintessence) and that the ”whynow?” problem is solved by some dynamical coupling of the quintessence field with gauge singlet fields (perhaps RHneutrinos) [50].L0012006 SLAC Summer Institute 2006 SLAC Summer Institute 2006 SLAC Summer InstituteClearly the cosmological constant problem poses a big question mark on the relevance of naturalness as a relevantcriterion also for the hierarchy problem: how we can trust that we need new physics close to the weak scale out ofnaturalness if we have no idea on the solution of the cosmological constant huge naturalness problem? The commonanswer is that the hierarchy problem is formulated within a well defined field theory context while the cosmologicalconstant problem makes only sense within a theory of quantum gravity, that there could be modification of gravity atthe sub-eV scale, that the vacuum energy could flow in extra dimensions or in different Universes and so on. At theother extreme is the possibility that naturalness is misleading. Weinberg [51] has pointed out that the observed orderof magnitude of Λ can be successfully reproduced as near the maximal value necessary to allow galaxy formation inthe Universe. In a scenario where new Universes are continuously produced we might be living in a very special one(largely fine-tuned) but the only one to allow the development of an observer (anthropic principle). One might thenargue that the same could in principle be true also for the Higgs sector. Recently it was suggested [52] to abandonthe no-fine-tuning assumption for the electro-weak theory, but require correct coupling unification, presence of darkmatter with weak couplings and a single scale of evolution from the EW to the GUT scale. A ”split SUSY” modelarises as a solution with a fine-tuned light Higgs and all SUSY particles heavy except for gauginos, higgsinos andneutralinos, protected by chiral symmetry. But, then, we could also have a two-scale non-SUSY GUT with axionsas dark matter. In conclusion, it is clear that naturalness can be a good heuristic principle but you cannot prove itsnecessity. The anthropic approach to the hierarchy problem is discussed in ref.s [53].12. SUMMARY AND CONCLUSIONSupersymmetry remains the standard way beyond the SM. What is unique to SUSY, beyond leading to a set ofconsistent and completely formulated models, as, for example, the MSSM, is that this theory can potentially work upto the GUT energy scale. In this respect it is the most ambitious model because it describes a computable frameworkthat could be valid all the way up to the vicinity of the Planck mass. The SUSY models are perfectly compatiblewith GUT’s and are actually quantitatively supported by coupling unification and also by what we have recentlylearned on neutrino masses. All other main ideas for going beyond the SM do not share this synthesis with GUT’s.The SUSY way is testable, for example at the LHC, and the issue of its validity will be decided by experiment. It istrue that we could have expected the first signals of SUSY already at LEP, based on naturality arguments appliedto the most minimal models (for example, those with gaugino universality at asymptotic scales). The absence ofsignals has stimulated the development of new ideas like those of large extra dimensions and ”little Higgs” models.These ideas are very interesting and provide an important reference for the preparation of LHC experiments. 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Introduction to the Terascale

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