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Overview of the Standard Model

A brief overview is presented of the standard model of electroweak interactions, including precision tests, the role of the Cabibbo-Kobayashi-Maskawa (CKM) matrix in describing CP violation, windows on intermediate-scale physics, and the nature of the Higgs phenomenon.


Introduction
The standard model of electroweak interactions has guided experimental discoveries for more than ten years, starting with the observation of the W and Z bosons and culminating this past year with evidence for the top quark. The new particles' properties have been very close to those anticipated by theory: reassuring to some physicists, frustrating to others. In this brief overview we describe the present state of electroweak theory and indicate ways in which new insights could emerge from experiments within the next decade.
Precise electroweak tests, described in Sec. 2, show promise of shedding light on the Higgs, and restrict our ability to add large numbers of new particles to the theory, as in some models of dynamical electroweak symmetry breaking. A plausible theory of CP violation, based on phases in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, can be checked by many forthcoming experiments on rare K decays and properties of B mesons, as noted in Sec. 3.
While electroweak physics is based on physics below the TeV scale, unification of electroweak and strong interactions must take place at scales above 10 15 GeV, or protons would decay too rapidly. Although many schemes envision a "desert" between 1 TeV and the unification scale, we describe in Sec. 4 several probes of intermediate scales (such as the interesting range 10 9 − 10 12 GeV), including the study of neutrino masses, the investigation of baryogenesis in the early Universe, and the search for light pseudoscalar particles ("axions").
The mechanism of electroweak symmetry breaking is still unknown. Does it proceed via fundamental scalars in the theory, whose masses are protected from large radiative corrections by some mechanism such as supersymmetry, or is it dynamically generated through new interactions which are strong at the TeV scale? These questions are briefly considered in Sec. 5. We summarize in Sec. 6. More details on some of the topics presented here may be found in Ref. 1.

In celebration of the top quark
In April of this past year the CDF Collaboration at Fermilab presented evidence 2 for a top quark with mass m t = 174±10 +13 −12 GeV. The measured cross section is slightly higher than (though consistent with) theoretical expectations based on quantum chromodynamics (QCD). Mechanisms which could give an elevated cross section include the production of more than one flavor of new quark and schemes involving dynamical symmetry breaking. 3,4 The mass lies in the middle of the range anticipated on the basis of electroweak radiative corrections (to be discussed in this section) and analyses of the CKM matrix (to be discussed in the next). But why is the top quark so heavy?

Quark and lepton masses
The top quark is the last quark to fit into a set of three families of quarks and leptons, whose masses are shown in Fig. 1: Only the ν τ has not yet been directly observed. The fractional error on the top mass is less than that for the light quarks u, d, and s, whose masses are obscured by QCD effects. In logarithmic terms, the t mass seems not all that anomalous, but we have no theory for the pattern in Fig. 1.

Electroweak unification
The singular four-fermion beta-decay interaction does not permit calculations to higher order. In a theory where this interaction is the low-energy limit of massive charged particle (W ) exchange, however, higher-order calculations make sense. 5 The predicted W and a massive neutral particle, the Z, entailed by the simplest version of the theory, were both discovered in 1983. The theory also predicted new weak chargepreserving interactions mediated by Z exchange, first seen in neutrino interactions a decade earlier. The theory has an SU(2) × U(1) symmetry, broken by the masses of the W , Z, and fermions.
The low-energy limits of W and Z exchange are described by where G F is the Fermi constant, g = e/ sin θ and g ′ = e/ cos θ are SU(2) and U(1) coupling constants, e is the proton charge, and θ is the weak mixing angle. The parameter ρ, which can arise from effects of quark loops on W and Z self-energies, is dominated by the top: 6 Consequently, if we define θ by means of the precise measurement at LEP of M Z , then θ will depend on m t , and so will One must remember that the electric charge becomes stronger at the short distances characterizing W and Z exchanges as a consequence of vacuum polarization effects in calculating M W and M Z from these expressions.

The Higgs boson
The replacement of the Fermi interaction by massive W exchange does not cure all the problems of the electroweak theory. At high energies, W + W − scattering would violate unitarity (i.e., probability conservation) unless a spinless neutral boson (the Higgs boson) existed below about 1 TeV. 7 The search for this boson is one motivation for multi-TeV hadron colliders. Searches in e + e − → Higgs boson + . . . at LEP have set a lower limit of about 60 GeV on M H . 8 (Here and elsewhere c =h = 1.) The Higgs boson affects the parameter ρ through loop diagrams contributing to W and Z self-energies. It is convenient to express contributions to ρ in terms of deviations of the top quark and Higgs boson masses from nominal values. For m t = 175 GeV, M H = 300 GeV, the measured value of M Z leads to a nominal expected value of sin 2 θ 0 ≡ x 0 = 0.2320. Defining a parameter T of order 1 by ∆ρ ≡ αT , we find This expression is quadratic in m t , but only logarithmic in M H . The weak mixing angle θ, the W mass, and other electroweak observables depend on m t and M W .

S and T parameters
The weak charge-changing and neutral-current interactions are probed under a number of different conditions, corresponding to different values of momentum transfer. For example, muon decay occurs at momentum transfers small with respect to M W , while the decay of a Z into fermion-antifermion pairs imparts a momentum of nearly M Z /2 to each member of the pair. Although coupling constants and masses on the right-hand sides of (3) vary fairly rapidly with momentum transfer, their quotients vary much less rapidly, as first pointed out by Veltman. 6 Small "oblique" corrections, 9,10 logarithmic in m t and M H , arise from contributions of new particles to the photon, W , and Z propagators. Other (smaller) "direct" radiative corrections are important in calcuating actual values of observables.
We may then replace (3) by where S W and S Z are coefficients representing variation with momentum transfer. Together with T , they express a wide variety of electroweak observables in terms of quantities sensitive to new physics. Expressing the "new physics" effects in terms of deviations from nominal values of top quark and Higgs boson masses, we have the expression for T written above, while contributions of Higgs bosons and of possible new fermions U and D with electromagnetic charges Q U and Q D to S W and S Z are 11 The expressions for S W and S Z are written for doublets of fermions with N C colors and m U ≥ m D ≫ m Z , while Q ≡ (Q U + Q D )/2. The sums are taken over all doublets of new fermions. In the limit m U = m D , one has equal contributions to S W and S Z . For a single Higgs boson and a single heavy top quark, Eqs. (9) and (10) become (11) We shall use these expressions, together with previous ones, to express all electroweak observables as functions of m t and M H .

Electroweak experiments
Recent direct W mass measurements, in GeV, include 79 49±0.12 nb (the hadron production cross section), and R ℓ ≡ Γ hadrons /Γ leptons = 20.795±0.040, which may be combined to obtain the Z leptonic width Γ ℓℓ (Z) = 83.96± 0.18 MeV. Leptonic asymmetries include the forward-backward asymmetry parameter A ℓ F B = 0.0170 ± 0.0016, leading to a value sin 2 θ ℓ ≡ sin 2 θ eff = 0.23107 ± 0.0090, and independent determinations of sin 2 The last three values may be combined to yield sin 2 θ = 0.2317 ± 0.0007. We do not use asymmetries as measured in decays of Z to bb (which may reflect additional new-physics effects, to cc (which are of limited weight because of large errors), or to light quarks (for which interpretations are more model-dependent). This last result differs by about two standard deviations from that based on the left-right asymmetry parameter A LR measured with polarized electrons at SLC 21 : sin 2 θ = 0.2294 ± 0.0010.
Parity violation in atoms, stemming from the interference of Z and photon exchanges between the electrons and the nucleus, provides further information on electroweak couplings. The most precise constraint at present arises from the measurement of the weak charge (the coherent vector coupling of the Z to the nucleus), Q W = ρ(Z −N −4Z sin 2 θ), in atomic cesium, 22 with the result Q W (Cs) = −71.04±1.58±0.88. The first error is experimental, while the second is theoretical. 23 The prediction 24 Q W (Cs) = −73.20 ± 0.13 is insensitive to standard-model parameters 24,25 ; discrepancies are good indications of new physics (such as exchange of an extra Z boson).

Results of fits to electroweak observables
We have performed a fit to the electroweak observables listed in Table 1. The "nominal" values (including 26 sin 2 θ eff = 0.2320) are calculated for m t = 175 GeV and M H = 300 GeV. We use Γ ℓℓ (Z), even though it is a derived quantity, because it has little correlation with other variables in our fit. It is mainly sensitive to the axialvector coupling g ℓ A , while asymmetries are mainly sensitive to g ℓ V . We also omit the total width Γ tot (Z) from the fit, since it is highly correlated with Γ ℓℓ (Z) and mainly provides information on the value of the strong fine-structure constant α s . With α s = 0.12±0.01, the observed total Z width is consistent with predictions. The partial width Γ(Z → bb) will be treated separately below.  Each observable in Table 1 specifies a band in the S − T plane with different slope, as seen from the ratios of coefficients of S and T . Parity violation in atomic cesium is sensitive almost entirely to S. 24,25 The impact of sin 2 θ eff determinations on S is considerable. The leptonic width of the Z is sensitive primarily to T . The We have evaluated the overall quality of a fit to the standard model of the data in Table 1, when the constraint m t = 174 ± 17 GeV is added. 1 The result (see also

The decay Z → bb
The measured ratio 27 R b ≡ Γ(Z → bb)/Γ(Z → hadrons) lies about 2σ above the standard model prediction. 28 If this discrepancy becomes more significant, there are many possibilities for explaining it within the context of new physics. 29 Two-Higgsdoublet models 28 cannot reduce the discrepancy much without running afoul of other constraints (such as the observed rate 30 for b → sγ).

The CKM Matrix
The  Table 1; (b) based on data in Table 1 aside from last row.
links the observed effect in the neutral kaon system to a phase in this matrix, whose parameters need to be specified as precisely as possible in order to test the theory. Moreover, theories of quark masses necessarily predict the CKM elements, 33 and may thereby be tested.

Definitions and magnitudes
We use a convenient parametrization 34 of the matrix: The four parameters are measured as follows: 1. The parameter λ is measured by a comparison of strange particle decays with muon decay and nuclear beta decay, leading to λ ≈ sin θ ≈ 0.22, where θ is the Cabibbo 31 angle.
2. The dominant decays of b-flavored hadrons occur via the element V cb = Aλ 2 . The lifetimes of these hadrons and their semileptonic branching ratios then lead to an estimate A = 0.79 ± 0.06.
3. The decays of b-flavored hadrons to charmless final states allow one to measure the magnitude of the element V ub and thus to conclude that |V ub /V cb | = 0.08 ± 0.02 or √ ρ 2 + η 2 = 0.36 ± 0.09.
4. The least certain quantity is the phase of V ub : Arg (V * ub ) = arctan(η/ρ). Information on this quantity may be obtained by studying its effect on contributions of higher-order diagrams involving the top quark.
The unitarity of the matrix (up to order λ 3 ) is implicit in the parametrization; in particular, the relation V * ub + V td ≃ Aλ 3 , when normalized so that its right-hand side is unity, becomes ρ + iη + (1 − ρ − iη) = 1. We shall be concerned with the allowed region in the (ρ, η) plane.

Indirect information
Two important sources of indirect information on ρ and η are B 0 − B 0 mixing and CP-violating K 0 − K 0 mixing. The presence of "wrong-sign" leptons in semileptonic B decays provided the first evidence for B 0 − B 0 mixing. 35 Many experiments, including recent observations of time-dependent oscillations, 36 have confirmed the effect, leading to a mixing amplitude ∆m/Γ = 0.71±0.07. The dominant contribution to the mixing is expected to arise from one-loop diagrams ("box graphs") involving internal W and top quark lines, leading to ∆m ∼ f 2 B m 2 t |V td | 2 (times a slowly varying function of m t /M W ). Here the "B decay constant," f B , describes the amplitude for finding a b antiquark and a light quark at the same point in a B meson. Since |V td | ∼ |1 − ρ − iη|, the B 0 − B 0 mixing amplitude leads to a constraint in the (ρ, η) plane consisting of a circular band with center (1,0). The main contribution to the width of this band is uncertainty in f B .
A similar set of box diagrams contributes to the parameter ǫ describing CPviolating K 0 − K 0 mixing. The imaginary part of the mass matrix is proportional to f 2 K m 2 t Im(V 2 td ) times a slowly varying function of m t , with a small correction for the charmed quark contribution and an overall factor B K describing the degree to which the box graphs account for the effect. Since Im(V 2 td ) ∼ η(1 − ρ), the constraint imposed by CP-violating K 0 − K 0 mixing consists of a hyperbolic band in the (ρ, η) plane with focus at (1,0), whose width is dominated by uncertainty in the magnitude of V cb . 37

Constraints on ρ and η
The allowed region 1 in (ρ, η) is shown in Fig. 3. Parameters used, in addition to those mentioned above, include B K = 0.8 ± 0.2, f B = 180 ± 30 MeV (in units where f π = 132 MeV), η QCD = 0.6 ± 0.1 (a correction to the B − B mixing diagrams), and B B = 1 for the factor analogous to B K . The center of the allowed region lies around ρ ≃ 0, η ≃ 0.35. We may ask how our information may be improved.

Improved tests
We shall concentrate on a few key aspects of B physics, since extensive discussions of processes involving rare kaon decays and the CP-violating parameter ǫ ′ /ǫ appear elsewhere.  The WA75 Collaboration 41 reports a handful of D s → µν candidates, leading to f Ds = 232 ± 69 MeV, while the CLEO Collaboration, 42 based on several dozen such events and an average by Muheim and Stone, 43 obtains f Ds = 315 ± 46 MeV. We average these two values to obtain f Ds = 289 ± 38 MeV. The BES group also has a few candidates for D s → µν or D s → τ ν; 44 the E653 Collaboration has candidates for D s → µν for which results are expected soon. 45 The lattice determination noted above obtains f D /f Ds ≃ 0.9, while an argument based on the quark model and the observed hyperfine splittings between singlet and triplet charmed nonstrange and strange mesons 46 suggests f D /f Ds ≃ 0.8. Using this last ratio, we obtain f D = 231 ± 31 MeV, not far below the present upper limit (90% c.l.) of 290 MeV obtained by the Mark III Collaboration 47 on the basis of their search for the decay D + → µ + ν. The BES Collaboration should be able to observe this process in events obtained in the reaction e + e − → ψ(3770) → D + D − .
Heavy meson decay constants are related to the square of the quark-antiquark wave function at the origin, 48 which can be estimated 46 with the help of spin-dependent isospin mass splittings in K, D, and B systems. The end result of this exercise 1,46 is the result f B = 180 MeV employed above. The error is based on the spread in various theoretical estimates.

B Decays to CP non-eigenstates: pair of light pseudoscalars.
A difference between the rates for a process and its charge-conjugate, such as B + → π + K 0 and B − → π − K 0 , signifies CP violation. Under charge conjugation, weak phases change sign, but strong phases do not. In order for a rate difference to appear, there must be both a weak phase difference and a strong phase difference in the channels with isospins I = 1/2 and 3/2. Recently it has been shown that one may be able to measure weak phases via the rates for B decays to pairs of light pseudoscalar mesons without having any strong phase differences. 49 The presence of electroweak penguins 50 is one possible obstacle to this program, which is under further investigation.
3.4.3 Decays of neutral B mesons to CP eigenstates: π −B correlations. Although produced initially as flavor eigenstates, neutral B mesons can undergo B 0 − B 0 mixing, leading to time-dependent asymmetries in decays to CP eigenstates like J/ψK S . Time-integrated decays also can display rate asymmetries, whose interpretation is often independent of final-state effects. For example, the asymmetry in decays of B 0 or B , where x d = (∆m/Γ)| d = 0.70 ± 0.07 is the mixing parameter mentioned earlier. One has to know the flavor of the neutral B at time of production. One proposed means for "tagging" the B involves its correlation with charged pions produced nearby in phase space. 51 The existence of such a correlation is predicted both by fragmentation and resonance decay pictures.

Neutrino masses and new mass scales
The pattern of neutrino masses in Fig. 1 is at least as puzzling as the large mass of the top quark. Why are neutrinos so light?
One proposed solution to the apparent suppression of the solar neutrino flux, especially for neutrinos with energies of a few MeV, 52 is based on matter-induced oscillations between electron neutrinos and some other species. 53 An acceptable solution involves a muon neutrino mass of several millielectron volts and a much lighter electron neutrino. A large Majorana mass 54 of 10 9 − 10 12 GeV for the right-handed neutrino, combined with a Dirac mass of order 1 GeV linking the right-handed and left-handed muon neutrino, could give such a mass eigenstate.
If the corresponding tau neutrino Dirac mass is related to the top quark mass (as it is in some grand unified theories), the ν τ could then have a mass of a few eV. It might well mix with the ν µ with an angle θ ≥ m µ /m τ , leading to a mixing parameter sin 2 2θ ≥ 10 −2 .
Present limits 55 from Fermilab Experiment E531 restrict the muon and tau neutrinos to have ∆m 2 ≤ 1 eV 2 for large θ and sin 2 2θ ≤ (a few) × 10 −3 for large ∆m 2 . A wider range of parameters are accessible to CHORUS, 56 an emulsion experiment, and NOMAD, 57 a fine-grained detector, both operating at CERN, which should reach sin 2 2θ ≤ (a few) × 10 −4 for large ∆m 2 . New short-and long-baseline ν µ ↔ ν τ oscillation experiments 58,59 are also envisioned for Fermilab, while experiments on at-mospheric neutrinos 60 and low-energy reactor neutrinos 61 have recently presented tantalizing hints of oscillation phenomena. 62 A new solar-neutrino experiment, sensitive mainly to high-energy electron neutrinos but eventually to other species as well through neutral-current interactions, will begin operating in a couple of years. 63

Electroweak-strong unification
The U(1), SU(2), and SU(3) couplings of the electroweak and strong interactions approach one another at high energies, 64 but do not really meet at a single point, in contrast to the predictions of the simplest SU(5) grand unification scheme. 65 One cure 66 for this "astigmatism" is provided by the supersymmetric extension of SU (5)

Baryogenesis
In 1967 Sakharov 69 identified three key elements of any theory seeking to explain the observed baryon abundance of the Universe, n B /n γ ≃ 4 × 10 −9 : (1) C and CP violation; (2) baryon number violation, and (3) a period during which the Universe was out of thermal equilibrium. A toy model is provided by the decay of SU(5) "X" bosons, which couple both to pairs of quarks and to antiquark-lepton pairs. The total rates for X and X decays must be the same as a consequence of CPT invariance, but the branching ratios B(X → uu) and B(X → e +d ) can differ from B(X →ūū) and B(X → e − d) as a result of CP violation, leading to a baryon asymmetry once the Universe cools below the energy where these decay processes are balanced by the reverse processes. This example conserves B − L, where B is baryon number (1/3 for quarks) and L is lepton number (1 for electrons).
It was pointed out by 't Hooft 70 that the electroweak theory contains an anomaly as a result of nonperturbative effects which conserve B − L but violate B + L. If a theory leads to B − L = 0 but B + L = 0 at some primordial temperature T , the anomaly can wipe out any B + L as T sinks below the electroweak scale. 71 Thus, the toy model mentioned above and many others are unsuitable in practice.
One proposed solution is the generation of nonzero B − L at a high temperature, e.g., through the generation of nonzero lepton number L, which is then reprocessed into nonzero baryon number by the 't Hooft anomaly mechanism. 72 The existence of a baryon asymmetry, when combined with information on neutrinos, could provide a window to a new scale of particle physics. Large Majorana masses acquired by righthanded neutrinos would change lepton number by two units and thus would be ideal for generating a lepton asymmetry if Sakharov's other two conditions are met.
The question of baryogenesis is thus shifted onto the leptons: Do neutrinos indeed have masses? If so, what is their "CKM matrix"? Do the properties of heavy Majorana right-handed neutrinos allow any new and interesting natural mechanisms for violating CP at the same scale where lepton number is violated? Majorana masses for righthanded neutrinos naturally violate left-right symmetry and could be closely connected with the violation of P and C in the weak interactions. 73 An open question in this scenario, besides the form of CP violation at the lepton-number-violating scale, is the manner in which CP violation is communicated to the lower mass scale at which we see CKM phases.

The strong CP problem
The Lagrangian density of QCD in principle can contain a term (g 2 3θ /32π 2 )F a µνF µνa , whereθ ≡ θ+ Arg det M, with M the quark mass matrix and θ an angle characterizing the vacuum structure. 74 The neutron electric dipole moment receives a contribution of order d n ≃ 10 −16θ e· cm, implyingθ ≤ 10 −9 . How can we understand the small value ofθ without fine-tuning?
Proposed solutions include (1) an unconventional and still controversial interpretation of the vacuum 75 as composed of an incoherent mixture of θ and −θ, in the sense of a density matrix; (2) vanishing of one of the quark masses (e.g., m u ), which probably bends chiral symmetry beyond plausible limits; 76 (3) the promotion 77 ofθ to the status of a dynamical variable relaxing to zero, implying the existence 78 of a light Nambu-Goldstone boson, the axion. Searches for this particle by means of RF cavities 79 turn out to be uniquely sensitive to the scale of symmetry breaking in the range we have been discussing, 10 9 − 10 12 GeV!

Electroweak symmetry breaking
A key question facing the standard model of electroweak interactions is the mechanism for breaking SU(2) × U(1). We discuss two popular alternatives; Nature may turn out to be cleverer than either.

Fundamental Higgs boson(s)
If there really exists a relatively light fundamental Higgs boson in the context of a grand unified theory, one has to protect its mass from large corrections. Supersymmetry is the popular means for doing so. Then one expects a richer neutral Higgs structure, charged scalar bosons, and superpartners, all below about 1 TeV.

Strongly interacting Higgs sector
The scattering of longitudinally polarized W and Z bosons violates unitarity above a TeV or two if there does not exist a Higgs boson below this energy. 7 The behavior is similar to that of pion-pion scattering in the non-linear sigma model above a few hundred MeV, where we wouldn't trust the model. Similarly, we mistrust the present version of electroweak theory above a TeV. If the theory has a strongly interacting sector, its I = J = 0 boson (like the σ of QCD) may be its least interesting and most elusive feature. The rich spectrum of resonances in QCD is now understood in terms of the interactions of quarks and gluons. We hope that the exploration of the potentially rich TeV physics of the electroweak sector can proceed through the use of both hadron and electron-positron colliders.

Summary
Increasingly precise electroweak and top quark measurements show promise of shedding indirect light on the Higgs sector, when present accuracies are improved by about a factor of three. Electron-positron collisions at LEP and SLC, hadron collisions at Fermilab, and even table-top experiments on atomic parity violation all will play key roles.
The Cabibbo-Kobayashi-Maskawa (CKM) matrix is the leading candidate for the source of the observed CP violation in the neutral kaon system, through phases which require the existence of the third family of quarks. The observation of events consistent with a top quark has given encouragement to this scheme, but further confirmation is needed, through the study of CP-violating and other rare kaon decays and through systematic exploration of the properties of hadrons containing b quarks.
Several prospective windows exist on the "intermediate" mass scale of 10 9 − 10 12 GeV, including the study of neutrino masses, partial unification of interactions, baryogenesis, and axions. Thus, there appears to be at least a lamppost, if not an oasis, in the grand desert between the TeV scale and the unification scale.
Candidates for understanding the Higgs sector include a theory of fundamental scalars with masses protected by supersymmetry and a composite-Higgs theory which could, in principle, be merely the first hints of compositeness on a deeper level. Perhaps even quarks and leptons are composite. I, for one would not be distressed if we were simply uncovering one more layer of the onion in our journey toward the Planck scale.