By discovering or excluding a light Higgs boson, LEP II, the Tevatron, and the LHC will establish whether WW interactions become strong or not, independent of the presence or absence of supersymmetry. The triviality of the scalar Higgs theory implies that any such scalar theory is at best a low-energy effective theory valid below some higher energy scale: the lower the Higgs mass, the higher the upper bound on the new scale.  The discovery of light Higgs (say 200 or 300 GeV) and nothing else (e.g. no TeV-scale SUSY) would leave us with no definite prospect of further discoveries at accessible energies.

The absence of a light Higgs and the presence of a convincing signal of strong WW scattering would be compelling evidence for a strong EWSB sector. Supersymmetry would be established by the discovery and characterization of the superpartners of the ordinary particles.

While the discovery of strong WW scattering would be a convincing signal for a strong EWSB sector, the absence of strong WW scattering need not imply the EWSB sector is weakly coupled. It may be that there are sufficiently many other particles (typically pseudo-Goldstone bosons), and the inelastic scattering of WW to these new bosons is large instead (for an extreme case, see Phys. Lett. B267:233-239, 1991). We should also be wary of relying too heavily on QCD, since a successful technicolor theory will not be QCD-like in detail.

In the scenario where the results from the LHC are inconclusive, is there a lower energy limit for a new hadron or lepton collider that would be needed to understand the character of EWSB? (with what luminosities required?) For example, imagine that no light resonances are found at the LHC, yet precision electroweak data continues to prefer a light Higgs with mass of order 100 GeV. What possible resolutions to this puzzle could be possible? Or, what would be the impact of seeing no sparticles, but a Higgs-like object with mass 200 GeV? (500 GeV?) 

In the absence of a light Higgs or some other resonance, a nonresonant signal is likely to be observed at the LHC in the W⁺W⁺ or W^-W^- channel if WW scattering is strong (see hep-ph/9511206 and references therein) and no other channels (e.g. PGBs) are relevant.  Such a signal may not, however, uniquely identify the physics of symmetry breaking. A 4-TeV muon collider or 100 TeV hadron collider capable of accumulating 200 fb^-1 of data would be able to probe resonances up to of order 3 TeV, the mass that would be expected in a technicolor model by naively scaling from QCD.

Since the details (e.g. mass splittings and couplings) of the resonance spectrum of a walking technicolor theory could be very different from QCD, precision electroweak measurements of the S and T parameters cannot directly be used to rule out theories of dynamical EWSB. 

A Higgs-like object of a mass of 500 or 600 GeV would imply, because of triviality, that there must be new interactions  below an energy scale of 5 to 10 TeV. Such a theory could naturally yield corrections to S and T of a size which can reconcile a heavy Higgs with precision electroweak data (see hep-ph/9907414 and references therein). As mentioned above, the discovery of a 200-GeV Higgs and no other new physics would be very disappointing: there would be no definite prospect of further discoveries at accessible energies.

Are there appealing models of strongly-coupled EWSB which are derivable or motivated from string theory/GUTs? 

In principle, such theories could be embedded in a GUT,  and this may actually be easier in the context of string theory (hep-ph/9309258).

To what extent do present precision EW fits accommodate either Susy or strong dynamics theories? What would be the benefit of a very high statistics Z factory in improving our resolution of models?

Fits to precision electroweak data in the context of the standard model with no other dynamics, prefer a light Higgs boson. The effects of SUSY partners decouple as the new particles become heavy, and therefore this is also consistent with the prediction of a light Higgs boson in supersymmetric theories.

In general, further precision measurements are of limited use in distinguishing between various models since they typically provide only a few constraints in theories with many parameters.
 

Vector mesons are expected to be the lightest resonances in technicolor theories, while composite Higgs models could produce a heavy Higgs. The LHC is expected to be able to probe spin-0 isospin-0 resonances (Higgs-like) up to of order 800 GeV, and spin-1 isospin-1 resonances (technirho-like) up to of order 1.6 TeV. The LHC is also expected to be able to establish a signal for strong W⁺W⁺/W^-W^- scattering up to energies of order 2 TeV.  (See hep-ph/9704217 and references therein.) Finally, we should also keep in mind that there may be additional PGBs which can effect the properties of VV scattering.

What is the requisite energy required to explore strong dynamical models carefully with new lepton colliders or high energy hadron colliders, if some excess VV production is seen at LHC?

A 4-TeV muon or 100-TeV hadron collider which can collect of order 200 fb^-1 of data should be able to find resonances in WW or WZ up to masses of order 3 TeV. This should be sufficient in a technicolor-like theory.

Depending on the amount of "fine-tuning" one allows, the scale of topcolor or top-seesaw theory could be higher. In this case, VV production is not the best signal. A VLHC should be sensitive to topgluons with a mass scale of order 10-20 TeV, and possibly higher.

 How can one distinguish among a scalar Higgs, technirho or low energy theorem as the source of VV scattering?

A scalar Higgs and a technirho are distinguished by their spin and custodial isospin. In VV scattering, a Higgs couples to two parts W⁺W^- and one part ZZ. In contrast,  in the simplest model a charged technirho couples only to WZ  and a neutral one to W⁺W^-. In a general technicolor-like theory with a larger global symmetry, the various vector mesons will decay into pseudo-Goldstone bosons as well.

 If electroweak symmetry is broken dynamically, how likely is it that relatively light particles, such as pseudo-Goldstone bosons, exist? Could they escape detection at the LHC, and, if so, what future collider would be adequate to find them?

Given the constraints of a walking technicolor theory, perhaps with topcolor, it seems likely that there will be pseudo-Goldstone bosons or other scalars present. These particles cannot be too light or couple too strongly to top and bottom, or they will give rise to large corrections to ZÆ b bbar and b Æ s g (see, for example,   hep-ph/9505313, hep-ph/9702265, and hep-ph/9510376).

A thorough study of the reach of the LHC to detect PGBs, including the possibility of b-tagging, has yet to be done. The reach of the LHC for a neutral PGB decaying to gamma gamma should be at least of order 150 GeV, similar to the reach for a light Higgs decaying in this channel. Older studies of the pair production of colored PGBs decaying to four jets indicated a reach of order 350 GeV. 

A lepton collider is expected to have a reach for charged PGBs up to half of the center of mass  energy. A substantial reach should also be possible for a color-octet PGB produced in conjunction with a gluon.

 The basic motivation for strong coupling models based on higher scale QCD-like theories was attractive. In the face of experimental constraints, various modifications have been necessary (walking coupling constant evolution, multiple SU(3) and extra U(1) groups, extended gauge boson sectors, etc.). To what extent do these complications and loss of predictive power weaken the appeal of strong dynamical theories? Can experiments deliver a decisive rejection of strong coupling?

The basic motivation for strong-coupling models is to avoid the introduction of fundamental scalar particles, of which we have no examples in nature. This motivation remains.

Planned experiments should be able to decisively confirm or reject the existence of strong WW scattering. The presence of strong WW scattering would be compelling evidence for strong coupling EWSB.

We cannot rule out the possibility of a strongly coupled composite Higgs theory which happens to produce a light Higgs; such a scenario would seem unlikely, however, if TeV-scale SUSY were also discovered.
 

A high-energy e⁺e^- collider with a center of mass energy of 1.5 TeV and an integrated luminosity of 200 fb^-1 can provide a measurement of the form factor of the W which could be sensitive to resonance masses as high as 4 TeV. This would be a measurement of an effect, not a direct discovery of the resonance. Such a collider may also have a greater reach than the LHC for charged (but not colored) PGBs which decay into quarks.

A 500 GeV collider would likely not provide much complementarity, given the preceding Tevatron and LHC experiments. 

Helicity analysis of e+e^- Æ W⁺W^- has been proposed as a sensitive probe of strong-coupled electroweak symmetry breaking. What inhibits analogous studies at a hadron collider via q qbar to VV? How might those roadblocks be overcome? 

A helicity analysis of W Æ jj which enriched the fraction of longitudinally polarized W's could be quite useful (see Physical Review D 40, 2231-2244 (1989)). Most studies at hadron colliders have considered only the leptonic decay modes of the W/Z bosons, and therefore suffer from very limited statistics. The primary obstacle to using the decay modes in which one vector decays leptonically and the other hadronically is our limited understanding of  the Wjj background.
 

It is a general property, based on the low-energy theorems for a theory with a custodial symmetry, that longitudinal WW scattering is largest in the isospin-0 and 2 channels, and smallest in the isospin-1 channel. If the r meson is light, the isospin-1 channel becomes enhanced near the resonance. If the r meson is heavy, however, the isospin-2 channel (W⁺W⁺/W^-W^-) grows more quickly and provides a better signal for strong WW scattering. One can view isospin-2 scattering as arising from r-exchange in the t-channel. In this sense, the two channels are complementary (see hep-ph/9311336).
 

The dynamics of top-quark mass generation can give rise to corrections to the usual dimension-4 t- and b- couplings to the W and Z gauge bosons, although there are constraints from the decay width of the Z Æ b bbar. A deviation of this sort may be observable in single-top production at hadron colliders (see hep-ph/9612402). Such an effect may also be observable in the t-pair production cross section at a lepton collider.

These dynamics may also give rise to dimension-5 magnetic-moment like couplings, though such couplings are typically suppressed in theories of dynmical EWSB (see hep-ph/9305289). Larger contributions may be possible if the top-quark is a composite particle. Studies of the anomalous chromomagnetic and chromoelectric couplings have been performed (see hep-ph/9407366 and references therein).

At a 1.5-TeV NLC, with 100% polarization, it should be possible to begin the study of WW to ttbar and to probe whether or not this scattering amplitude becomes strong at energies of order 1 TeV.

 The top quark Yukawa coupling (m_t ÷2/v) is equal to one within present errors. What utility would much improved understanding of the top mass have as a tool for understanding the mechanism of EWSB?

The top-quark must, because of its large mass, be more strongly coupled to the EWSB sector. A detailed understanding of its electroweak properties (e.g. whether their are anomalous couplings as discussed above, or if the top-quark is part electroweak singlet as predicted in top-quark seesaw models) could provide light on EWSB in general. Conversely, the top is likely to be a common decay product for extra scalar states in these theories (e.g. top-pions, PGBs) if such a decay is kinematically allowed.

 Could dedicated B experiments be useful in understanding EWSB and the origin of the top mass through loop effects or direct b production?

Loop effects from top-pions or PGBs can give rise to potentially large contributions to bÆsg and bÆs l⁺ l^- (see, for example, hep-ph/9505313, hep-ph/9702265, and hep-ph/9510376). A deviation in one of these processes alone, however, would not make a convincing case for strong EWSB as there are many other possible explanations for such effects.
 

The LHC may be able to discover a Z' up to 5 TeV, if this particle is relatively narrow. However, as the boson gets wider, the signal becomes weaker.

A lepton collider can look for the effects of a heavy Z' (through its interference with the ordinary Z and photon) in the production of t tbar and b bbar. Depending on the model and the couplings, this could yield a sensitivity to a Z' with a mass from 2-10 the center of mass energy of the collider (see hep-ph/9612384). For a hadron collider, sensitivity is usually limited by the highest partonic center of mass energy for which there is reasonable (partonic) luminosity.

The LHC could find new color triplet or octet particles to roughly 5 TeV. How far could one extend the searches to higher mass with new lepton or hadron colliders?

In the case of a color octet technirho mixing (through "vector meson dominance") with the gluon, the effective reach of any hadron collider is limited by the highest partonic center of mass energy with appreciable (partonic) luminosity. A color-triplet scalar or vector would have to be pair-produced at a hadron collider, and the reach is not likely to be nearly as high as 5 TeV at the LHC. More work is needed to determine the absolute reach.

A lepton collider may have some sensitivity to color-octet scalars and vectors through production (via a loop of techniferemions or top/bottom quarks) in conjunction with a gluon (see M. Swartz, Snowmass `96). It was estimated that a 1 TeV e⁺ e^- collider had a reach up to 700 GeV for a color-octet scalar.
 

Anomalous couplings are not likely to provide an important information at a hadron collider. The estimated sizes of the effects are too small, except in those cases when a very large resonant signature (e.g. a technirho) would also be observed.  The situation is different at an e⁺ e^- collider: using a helicity analysis of e⁺ e^- Æ W⁺W^-, a precise measurement of the ZWW form factor at a 1.5 TeV NLC with 200 fb^-1 integrated luminosity would be able to establish the presence of a strong EWSB sector. (See hep-ph/9704217.)

The same question may be asked for anomalous fermion-vector-boson interactions. Is the answer different for the top quark than for lighter fermions?

Top- and bottom-quark anomalous couplings may provide insight to the EWSB sector, as discussed above.
 

Roughly speaking, the compositeness scale that a collider can reach scales with the partonic center of mass energy. Using the Eichten-Peskin-Lane normalization for a four-fermion operator arising from compositeness, current results from Run I at the Tevatron put a lower bound of order 3 TeV for quark or quark/lepton compositeness. We would then expect the LHC to reach a compositeness scale of order 20 TeV and a 100 TeV VLHC to reach a scale of order 160 TeV.

LEP II results from OPAL at 130-180 GeV place a lower bound of order 3 TeV on lepton compositeness. From this we can estimate that the reach of a 500-GeV, 1-TeV, or 1.5-TeV NLC would be of order 10 TeV, 20 TeV, and 30 TeV respectively.

These bounds will vary depending on the particular form of the compositeness interaction considered.

As a specific example, imagine the case where quarks and leptons share constituents in different combinations, and the compositeness scale is not far above the LHC energy so that some hint of contact interactions might be seen there. What are the relative merits of possible future colliders for identifying the source to be compositeness?

The reach of the LHC and a 1-TeV NLC are roughly comparable, the exploration of such a signal would require a higher energy collider.
 

Updated 12/1/99,  R.S.C.