Abstract

Dynamical electroweak symmetry breaking is an appealing, strongly coupled alternative to the weakly coupled models based on an elementary scalar field developing a vacuum expectation value. In Sections 2 and 3 of this set of lectures, I summarize the arguments, based on low-energy phenomenology, supporting walking technicolor as a realistic realization of this idea. This pedagogical introduction to walking technicolor, and more generally to the physics of extensions of the standard model, makes extensive use of effective field theory arguments, symmetries, and counting rules. The strongly coupled nature of the underlying interactions, and the peculiar quasiconformal behavior of the theory, requires to use nonperturbative methods in order to address many fundamental questions within this framework. The recent development of gauge/gravity dualities provides an ideal set of such nonperturbative instruments. Sections 4 and 5 illustrate the potential of these techniques with two technical examples, one within the bottom-up phenomenological approach to holography in five dimensions, the other within a more systematic top-down construction derived from ten-dimensional type-IIB supergravity.

1. Introduction and Outline

This set of notes on walking technicolor is based in part on sets of lectures delivered to string theory students in the context of their Ph.D. programs at Swansea University and at the University of Barcelona. As such, the content is intended for readers who are familiar with the ideas of gauge/gravity dualities [1–4], but not with technicolor (TC) [5–7], walking technicolor (WTC) [8–10], extended technicolor (ETC) [11, 12], and more in general with the details of the physics of the standard model (SM) and the problematics arising when dealing with its extensions.

This is not a complete and exhaustive review of the subject of technicolor, which can be found elsewhere, but rather a collection of exercises and arguments aimed at singling out a subset of problems that emerge in the context of physics beyond the standard model (BSM), and that are particularly important when, as is the case for WTC, the BSM new physics is strongly coupled. This set of problems admits a natural formulation (if not yet a natural solution) in the context of gauge/gravity dualities, which is the main message these lectures want to convey. The reader who wants to have a complete overview of the subject of strongly coupled dynamical electroweak symmetry breaking (EWSB) should complement the reading of these notes with other reviews, which develop, in more details, many aspects and ideas that are only glanced at in the present set of lectures [13–18].

The first two parts of these notes are very general, and should be easily accessible to anybody who has a working knowledge of quantum field theory and a basic knowledge of the standard model. For pedagogical reasons, before talking about WTC, I start by reminding the reader about some classical results that are at the core of our modern understanding of the Minimal version of the Standard Model (MSM) (I will use the MSM acronym to mean the version of the standard model with one Higgs scalar doublet, ultimately responsible for EWSB, in such a way as to distinguish it from the acronym SM, by which I mean a generic theory with the fermions and gauge bosons of the Standard Model, without committing to the details of the symmetry-breaking sector itself), and of its weakly coupled extensions. These include the Coleman-Weinberg results for the loop generated effective Higgs potential [19], a discussion of the big and little hierarchy problems, the definition of the electroweak chiral Lagrangian [20–24] and of the oblique precision parameters [25, 26] (and their use in precision electroweak physics), the GIM mechanism and the resulting suppression of flavor-changing neutral current (FCNC) processes [27], the Froggatt-Nielsen mechanism for the generation of fermion mass hierarchy [28], the see-saw mechanism [29–32], and the analysis of perturbative unitarity of longitudinally-polarized gauge-boson elastic-scattering amplitudes [33]. As such, this first half of the paper should also provide a useful introduction for any reader who is interested in BSM physics in general.

Section 2 paper contains some generic introductory material, and part of the discussion is only semiquantitative (more precise statements will be developed in later sections). It starts from a reminder about the basic building blocks of the SM, the introduction of the concept of naturalness, and a discussion of the big hierarchy problem. I also provide an example of technicolor model, for pedagogical reasons chosen to be peculiarly simple [34]. Unconventionally, in comparing three popular scenarios for solving the big hierarchy problem (supersymmetry, technicolor and warped extra-dimensions), rather than the specific differences I highlight the similarities between these three approaches. In particular the fact that at low energies they all yield to the little hierarchy problem. In this way, it should be clear to the reader that the tension between the absence of fine-tuning (for which a new physics scale  TeV would be a natural expectation) and the absence of indirect effects of new physics in precision measurements (which would naturally require a larger new physics scale  TeV) is a rather general problem of all BSM scenarios, not just a problem of strongly coupled extensions of the SM. After all, the very fact that, thanks to the work of our colleagues in many successful experimental programs, we have now an unprecedented body of high-precision information about nature, per se implies that not every generic new physics model can be allowed by the data. Hence, the presence of what appears to be a fine-tuning in the low energy effective Lagrangian extending the SM is not necessarily a surprising and bad thing, but rather seems to suggest to us that something very special is going to be discovered at the LHC, hence explaining this very special low-energy results. What singles out the strongly coupled scenarios as particularly problematic is the strong coupling itself, which severely limits the possibility of exploring the parameter-space of models, in the search of regions that are compatible with the current data, in the same way as is done in the weakly coupled context, and hence explaining the origin of the aforementioned 5% tuning in terms of specific, special detail properties of the new physics.

Section 3 contains a more systematic and precise pedagogical discussion, and, besides a preliminary introduction of the idea of walking dynamics, consists of four main subsections. In Section 3.1, precision electroweak physics is discussed, starting from the effective field theory (EFT) treatment, introducing the oblique parameters, and summarizing some classical perturbative results. In comparing to strongly coupled scenarios, I introduce the rules of naive dimensional analysis (NDA) [35] and hidden local symmetry (HLS) [36–43]. Section 3.2 is devoted to the physics of flavor. Starting again from the weakly coupled case, the GIM mechanism is introduced, and the basic logic behind the construction of weakly coupled models of flavor generation explained. ETC and WTC are then introduced and their specific physical implications are explained, and contrasted with what happens in the weakly coupled case. I do not discuss CP violation, the treatment of which to large extent follows the same logic as the physics of FCNC, up to subtleties in dealing with dipole moments that, while important phenomenologically, are not very important for the present purposes. Section 3.3 is a brief reminder about a classical result in the MSM; the scattering amplitude of longitudinally polarized gauge bosons is shown to be manifestly unitary in perturbation theory, at the diagrammatic level, provided all the couplings (including the Higgs self-coupling) are perturbative. This is obviously not the case in TC, where showing explicitly that the scattering amplitudes never grow beyond the unitarity bounds would require a full nonperturbative analysis. many ideas have been proposed addressing this problem, which is actively studied, but a definite answer does not exist, and hence I will limit the discussion to the well-understood perturbative case. In Section 3.4 I summarize, by means of examples, a few very general expectations about the spectrum of TC, focusing on those bound states (technipions, technirhos and technidilaton) the physics of which is controlled by strong symmetry arguments. I conclude with a brief subsection containing a wish list, to some extent personal and incomplete. This is a list of specific questions about strongly coupled EWSB, all of which emerge from the previous discussion, all of which have crucial phenomenological implications, and answering to any of which requires a very high level of understanding of nonperturbative physics. Gauge/gravity dualities represent a remarkable opportunity for addressing some of these questions in a new context, which is complementary to what is known from traditional techniques.

The second half of this paper contains a set of examples of applications of the ideas and techniques derived from the context of gauge/gravity dualities, in order to address some of the questions in the wish list. This is not done systematically, rather I explain in some details two very special such applications, one in the context of effective five-dimensional theories (bottom-up approach), and one in the context of ten-dimensional supergravity and string theory (top-down approach). This may be understood as a proof of principle, showing that indeed some of the questions in the wish list can be addressed within specific holographic models, and, furthermost, that the technology necessary for this type of studies to be carried out does exist and is well understood. It must be stressed that many more examples exist in the literature, that are not even mentioned in these lectures. Since the early papers on the AdS/CFT correspondence [1–4] and by Randall and Sundrum [44], a huge body of literature on these and related subjects appeared, reviewing which goes very far beyond the aims of these lecture notes.

Section 4 is devoted to one specific five-dimensional effective theory that most resembles what might be the dual of a four-dimensional walking technicolor theory [45–55]. I show how calculability is strongly improved within this approach, in respect to what happens in four-dimensional EFT and HLS approaches, yielding useful phenomenological correlations between observable quantities such as the oblique precision parameters and the spectrum and couplings of techni-rho mesons. I also show that some fundamental questions cannot be answered within this approach, but require more information about the underlying dynamical properties of the models.

In Section 5, a more ambitious approach is discussed [56–58]. The proposal is to study the properties of walking dynamics in isolation, separating them from the specific embedding of walking into a detailed model of electroweak symmetry breaking and fermion mass generation. This is done by constructing a 10-dimensional string theory model such that, in the supergravity limit, walking, or some feature that resembles it, emerges dynamically, and for which the technology exists allowing to perform a set of calculations that can yield a quantitative answer to some of the questions about walking dynamics that are left open by bottom-up approaches. This is a rather novel research program, and hence only few such calculations exist. I will limit myself to one specific model, which has its own advantages and disadvantages. A few other possible models and other interesting calculations are suggested, but the main aim of this section is to encourage the reader to use his/her own experience, knowledge, and creativity to devise alternative scenarios, and use them to address some of the problems listed earlier on.

2. Electroweak Symmetry Breaking, within and beyond the Standard Model

2.1. The Minimal Standard Model versus Technicolor

The standard model (SM) is a quantum field theory in which weak, strong and electromagnetic interactions are described by a gauge theory with group . All the known fundamental matter particles are described by (chiral) fermions transforming in some representation of (see Table 1). If the gauge symmetry was unbroken, all of the resulting particles would be massless. However, the electroweak gauge symmetry is spontaneously broken as , by some condensate that provides a nontrivial structure for the vacuum of the theory. The unbroken is the gauge symmetry of electromagnetism. Hence all the fermions, together with the and bosons, acquire a mass, proportional to the symmetry-breaking condensate. The proportionality constant is the coupling of the field to the condensate, and depends on the specific field. All of this is well established, and can be found nowadays even in popular science books.

It is also known that none of the SM fermions nor any of the SM gauge interactions are responsible for electroweak symmetry breaking itself. (The chiral condensate of QCD does, in fact, spontaneously break the SM gauge symmetry. But the scale is so small that this fact can be ignored for most phenomenological purposes.) Some new fields and new interactions must be present, in order to implement in the theory a mechanism yielding spontaneous electroweak symmetry breaking. Identifying these new fields and, equally important, new interactions is the main goal of large hadron collider (LHC) program at CERN. In the case of the minimal version of the standard model (MSM), the sector responsible for electroweak symmetry breaking is constituted by one scalar field, transforming as under . This is assumed to have a weakly coupled description in terms of a scalar potential with a negative quadratic interaction and a quartic interaction, the latter being the only free parameter to be measured at LHC. The classical minimum of the potential is non-trivial, the Higgs field acquires a vacuum expectation value (VEV), and hence yields electroweak symmetry breaking. The complete Lagrangian of the MSM contains five types of terms, which depend on the fields and on their covariant derivatives. We will discuss in more detail some of these terms later on, for the time being it suffices to write them out schematically (suppressing all the indexes). (i)Gauge-boson Lagrangian (ii)Fermion kinetic terms (iii)Scalar kinetic term (iv)Yukawa couplings (v)Scalar potential

The MSM is a very successful model. It agrees to very high accuracy with a number of precision electroweak tests, carried on in particular at SLAC, LEP, and TeVatron. It also successfully reproduces most of the flavor physics studied experimentally in rare processes involving -mesons, kaons, pions, and muons. One important reason for this success is that the MSM automatically implements the GIM mechanism, so that all sources of flavor changing processes and of CP-violation are encoded in the CKM matrix, hence naturally suppressing flavor changing neutral current (FCNC) processes. It is also a remarkably satisfactory model from a theoretical point of view, in the sense that it automatically yields results that are very non-trivial. For example, thanks to the cancellation of all gauge anomalies, and to the perturbative nature of all the couplings, including the Higgs potential, it is possible to show that unitarity of certain scattering amplitudes is manifest at the diagrammatic level up to very high scales.

Yet, there is no experimental evidence supporting the idea that symmetry breaking be due to the weakly coupled Higgs sector of the MSM, with LEP and TeVatron yielding exclusion regions and bounds on the mass, but no unambiguous positive discovery signal. It is hence important to consider alternatives. In particular, there is no real reason why the sector responsible for electroweak symmetry be weakly coupled. Actually, it is a remarkable fact that in the two most celebrated examples of spontaneous symmetry breaking (the theory of superconductors, and the description of chiral symmetry breaking in QCD) the condensate does not arise from an elementary scalar developing a VEV, but rather a strong interaction produces the formation of a composite condensate.

Besides this simple fact, a set of serious reasons for concern arises when studying the UV behavior of the Higgs sector itself. The Higgs potential does not satisfy the naturalness condition: switching off any of the couplings in the Higgs potential does not result in enhancing the symmetry of the system. We will discuss this in more details later. As a result, both the quadratic and quartic operator receive additive renormalization from (divergent) loop diagrams involving all the couplings of the model. In a sense, the dynamics encoded in the potential is not fundamental, and as a result it is affected by fine-tuning problems. Finally, the beta-function of the quartic coupling of the Higgs and of the Yukawa couplings is not asymptotically free (which is related to the triviality problem). All of this suggests that the Higgs sector might provide a very good description of low-energy physics, but needs to be UV completed.

Technicolor models are a radical alternative to the Higgs of the MSM. There is no weakly coupled scalar, but rather a completely new non-Abelian gauge interaction is assumed to be present, morally analog to QCD. At low-energies (of the order of the electroweak scale), the interaction becomes strong enough to trigger confinement and chiral symmetry breaking, via the formation of a condensate of fermions. As a simple example, consider for instance a gauge theory, in which fermions , , , , and with the same quantum numbers as a family of SM fermions transform on the fundamental of (Here and in several other points of Sections 2 and 3 I chose to refer to this specific model mainly for its striking simplicity and elegance. The reader however should not be mislead into thinking that this is always what a TC model has to look like, but should simply think of TC as a strongly coupled theory in which a dynamically generated condensate induces EWSB. For example, lots of attention has been given in recent years to small- models with techniquarks in 2-index representations of the gauge group [59], rather than on the fundamental representation, and nothing prevents from much more exotic scenarios to be considered.) At the electroweak scale, condensates form This condensate breaks electroweak symmetry in exactly the same way as the Higgs of the MSM, because each of the bilinears entering the condensates has the same quantum numbers as .

The properties of models of this type are the opposite of those of the MSM Higgs sector. All the couplings are natural, there is no fine-tuning nor triviality problem, all the scales are generated dynamically and naturally stabilized. However, calculability (and hence predictivity) becomes problematic, because of the strong coupling. Back-of-the-envolope estimates based on naive dimensional analysis (NDA) suggest that it is difficult to reconcile these models with electroweak precision measurements. Unitarity of scattering amplitudes is not manifest at the diagrammatic level, but relies on uncalculable contributions from the strong dynamics. And, contrary to what happens in the MSM, the generation of the mass of the SM fermions cannot proceed via marginal (Yukawa) couplings, but requires the introduction of a whole new sector of interactions, via what is referred to as extended technicolor (ETC). As a result, there is reason of concern regarding the contribution of this new ETC sector to FCNC processes, because an analog to the GIM mechanism is not automatically present.

One very plausible way to alleviate the shortcomings of technicolor, while preserving its good features, is provided by walking technicolor. The basic idea is that if the new strongly coupled theory is not some variation of QCD, but rather it has very different dynamics, such that it be strongly coupled over a large range of energies above the electroweak scale, then large anomalous dimensions are likely to be generated nonperturbatively. If so, the counting rules of NDA do not hold, and large enough masses for the SM fermions (the top quark is particularly problematic) might be generated dynamically, without necessarily running into troubles with precision electroweak parameters and FCNC processes. The dynamical justification for this scenario is the assumption that an approximate fixed point governs the physics in the IR, just above the electroweak scale, hence slowing down the running of the couplings in the strongly coupled regime, whence the name walking. While this adds to the calculability problems of traditional approaches to technicolor, it also provides an ideal arena in which the techniques of the gauge/string duality can be applied. Indeed, in the walking regime the theory is very strongly coupled and approximately conformal.

2.2. Naturalness, Hierarchy Problem(s), and Their Proposed Solutions

All known interactions can be reduced, at the microscopic level, to three fundamental ones: electroweak, strong, and gravitational interactions. The strength of these interactions can be characterized in terms of a typical scale, that can be identified, respectively, with three dimensionful coupling constants: where is the Fermi constant, is the pion decay constant, and the Newton constant.

At present, the SM does not incorporate a universally accepted theory of quantum gravity. However, it is conceivable that if such a formulation exists, the SM should be thought of as containing the leading-order terms a low-energy effective field theory (EFT) description of all fundamental interactions. In this spirit, quantum gravity should amount to two main effects (at low-energies): the renormalization of the SM Lagrangian itself, and the generation of higher-order corrections, encoded in higher-dimensional operators to be added to . Both effects would be controlled by the typical scale of gravitational interactions . Here is one way of seeing the emergence of a technical problem, the big hierarchy problem. On the one hand, the fact that justifies phenomenologically the fact that higher-order terms can be safely neglected, because of being strongly suppressed. On the other hand, the very fact that these two scales be so far apart, in a context in which quantum corrections are present, needs to be explained. In the absence of such an explanation, one would expect either that the electroweak scale and the Planck scale should be of the same order of magnitude, , or that their hierarchy is due to accidental cancellations between interactions of very different nature, coming from physics taking place above and below the Planck scale, hence requiring an implausible amount of fine-tuning.

This formulation of the big hierarchy problem shows explicitly that it has a very general origin. The only explicit assumptions used in order to highlight the problem itself being the existence of a quantum theory of gravity (or more general of any new interaction beyond the SM), and of a sensible low-energy EFT description, characterized by the Planck scale (or by a new physics scale ). There is another hidden assumption in all of the above, that the space-time symmetries be described by the four-dimensional Lorentz (or Poincaré) group. A vast literature exists on possible solutions to the big hierarchy problem, that propose to modify the fundamental theory in very different ways. It is remarkable that the three main classes of such solutions (based upon supersymmetry, extra-dimension, and technicolor, resp.) all rely on the idea of extending the group of space-time symmetries.

The reason why this is a good idea has to do with the relation between the concept of naturalness and renormalization in a general field theory. Using a definition often attributed to 't Hooft: A coupling is natural if, in the limit in which it vanishes, the theory has an enlarged symmetry.

In field theory, quantum corrections renormalize the bare Lagrangian in two possible ways: some couplings renormalize multiplicatively, while others renormalize additively (due to operator mixing).

Couplings that are not natural may in general renormalize additively. Hence, they may be affected by fine-tuning problems. Setting to a very small value a parameter that is renormalized additively does not ensure that quantum corrections will preserve its smallness. A simple illustration of this is given within the MSM by the Higgs potential in (2.5), with neither nor being a natural parameter. (It should be noticed that setting would render the MSM Lagrangian scale-invariant at the classical level. However, such invariance is not preserved by quantum correction, due to diagrams involving the quartic, gauge, and Yukawa couplings, and even allowing for fine-tuning in the quadratically divergent part of quantum corrections one ends up with the appearance of a scale in the 1-loop generated Higgs potential, which ultimately might lead to electroweak symmetry breaking itself [19].) As a consequence, nothing prevents them from receiving additive renormalization. A way to compute the (divergent) part of the (perturbative) 1-loop correction makes use of the results from Coleman and Weinberg [19] for the quantum effective action where is the UV cut-off, is the mass matrix of all fields as a function of the classical external scalar fields, and is the supertrace, a trace in which fermion and boson degrees of freedom enter with opposite signs. For instance, in the MSM, the coefficient of the quadratic operator receives, among others, (quadratically divergent) positive contributions from loops of gauge bosons and negative from loops of the top as following: with and numerical coefficients with mild 1-loop suppressions, the gauge coupling, and the top Yukawa coupling. If the cut-off is at the Planck scale , in order for the resulting quadratic coupling to be at the electroweak scale, a very fine-tuned cancellation would be needed between this loop correction and an appropriately chosen counter-term.

Conversely, a natural coupling (barring the possibility of anomalies) renormalizes multiplicatively: in the limit in which the coupling is set to zero, the theory has an extended symmetry, that must be preserved also by quantum corrections. The fact that quantum corrections are themselves proportional to the natural coupling ensures that this symmetry property be automatically satisfied. As such, if one could find a way to render naturally the parameter in the Lagrangian setting the electroweak scale, this would not receive dangerously large additive renormalization via (2.9), and the coexistence of the widely separated electroweak scale and Planck scale would not require fine-tuning, making the EFT useful and predictive. This is what is done in the most popular extensions of the standard model.

A most popular class of solutions proposes to enlarge the symmetry by incorporating some form of supersymmetry, that is, by effectively adding fermionic directions to the space-time (superspace formulation), and assuming that perturbation theory is a good tool up to the Planck scale. The standard example is the Minimal Supersymmetric Standard Model (MSSM) [60, 61]. The basic idea is to enlarge the spectrum by adding a super-partner with different spin and statistics to each of the MSM fields, and to adjust the couplings so that the new Lagrangian is supersymmetric. Details about the construction require to introduce two Higgs doublets instead of one. The technical way in which supersymmetry solves the hierarchy problem is that, when computing quantum corrections to the relevant operator controlling the electroweak scale, loops involving bosons and fermions in the same supersymmetry multiplets cancel against each other. In other words, in an supersymmetric theory in a generic vacuum (and under a few rather soft conditions such as the absence of mixed gauge-gravity anomalies), As a result, the quadratically divergent corrections to the supersymmetric generalization of (2.5) drop, in favor of a milder logarithmic divergence. In the MSSM, supersymmetry is broken explicitly, by adding to the supersymmetric Lagrangian a restricted set of mass terms and couplings the effect of which are soft, they preserve the logarithmic dependence of the divergence on the cut-off scale. Interestingly, because the Higgs potential, in the supersymmetric limit, does not admit an electroweak symmetry breaking minimum, the latter is controlled by the scale of supersymmetry breaking itself , hence linking electroweak scale and supersymmetry-breaking scale.

A second class of solutions to the big hierarchy problem involves the introduction of space-like extra-dimensions. There is a vast number of variations on this idea. One most appealing such realization is the Randall-Sundrum scenario [44], possibly assisted by some implementation of the Goldberger-Wise mechanism [62, 63]. The basic idea is to introduce a fifth dimension , and assume that the space-time has the geometry of a slice of AdS space, with the curvature scale, and with two 4-dimensional boundaries so that . The radial direction is related to the scale of the 4-dimensional theory living on a probe localized at , with corresponding to the UV and corresponding to the IR. While the 4-dimensional Planck scale is related to the five-dimensional fundamental scale as and one chooses (up to factors) , the typical scale of the physics taking place in the IR is suppressed as In the Goldberger-Wise mechanism, the hierarchy between the position of the two boundaries is determined dynamically to be where are the boundary values of some dynamical bulk scalar, and is a natural parameter, related to the coefficient of the only relevant coupling of the five-dimensional scalar (i.e., its five-dimensional mass). If the physics of electroweak symmetry breaking is to take place near the IR boundary, its scale is going to be exponentially suppressed in respect to the Planck scale, hence solving the big hierarchy problem, provided a modest hierarchy (e.g., ) is chosen, together with a large value of the exponent (e.g., ). For discussions about the holographic interpretation of this picture see also [64, 65].

A simple way of introducing the idea behind the third class of solutions to the hierarchy problem (technicolor) is to go back to QCD, observing that in what we said up to now, we never mentioned a strong-interaction fine-tuning problem related to the fact that . For a good reason, there is no such a problem! The Lagrangian of a QCD-like theory with massless fermions transforming on the fundamental representation of is where the covariant derivative is The beta-function computed at 1-loop is and accordingly where is the renormalization scale and is defined at some reference scale . The expression for diverges for defined by If , and for choices of and such that , this means that defined in this way is exponentially suppressed, provided that the theory is asymptotically free, and the coupling measured at the scale is perturbatively small. Because is the scale at which the theory becomes strongly coupled, then one has .

The symmetry principle behind the mechanism that makes small compared to the Planck scale can be read from (2.16). At the classical level, this Lagrangian is scale invariant because all the operators are exactly marginal. Quantum effects break conformal invariance, the gauge coupling representing a marginally relevant deformation of the massless free theory living at the UV fixed point. However, the breaking is controlled by the coupling itself, setting the coupling (as defined at a given scale ) to very tiny values, implies that , because in the limit the theory reduces back to a trivial conformal theory of noninteracting, massless gluons and quarks. Again, a space-time symmetry is present, broken only by a natural parameter (the gauge coupling) which renormalizes multiplicatively (all renormalizations of the gauge coupling are proportional to the gauge coupling itself). As a result, no fine-tuning is needed in order to parametrically separate the scale of the strong dynamics from the Planck scale.

The basic mechanism behind technicolor is the same: it relies on removing the Higgs sector from the MSM, and replacing it with a new strong sector with a new gauge symmetry (technicolor) acting on new degrees of freedom (technifermions). The electroweak gauge group is defined by gauging a subgroup of the global symmetry of such fermions (techni-flavor). Once the theory becomes strongly coupled at scale , a chiral condensate will form dynamically, generating and spontaneously breaking the electroweak symmetry, in analogy to what happens in QCD. Appropriately choosing the new gauge coupling, it is possible to arrange for , without any fine-tuning, because in the limit one recovers a conformal theory, with space-time symmetry described by .

Summarizing, the three examples discussed share the same basic idea of enlarging the space-time symmetry. While the way in which this is done is very different, it yields very similar results, turning the dangerous quadratic divergences relating the electroweak and Planck scales into logarithms, and linking the electroweak scale to a new scale characterizing the breaking of the enlarged space-time symmetry itself. Namely, below , supersymmetry is lost, the theory is four-dimensional, and scale-invariance is lost, respectively, in the three cases. Provided  TeV, no fine-tuning is present in the electroweak symmetry breaking sector. The scale is also the scale at and above which a new physics sector (fields and interactions) appears. It controls the masses of the superpartners of the SM fields, the masses of the KK modes coming from the compact extra-dimension, and the masses of the resonances and composite states in technicolor (technimesons, technibaryons, ). But also controls indirect contributions to low-energy precision physics, which we will discuss at length in these lectures.

2.2.1. Little Hierarchy Problem

Whichever your favorite solution to the big hierarchy problem is, provided it is based on the idea of enlarging the space-time symmetries as in the three cases discussed earlier, the net result is that there are now four physical scales, characterizing four distinct interactions: , , , and . The more the two scales and are close to each other, the more satisfactory is the solution to the hierarchy problem.

Below , the process of integrating out all of the sectors responsible for the breaking of the enlarged symmetry (and for the arising of itself) means that corrections to the low-energy description of electroweak and strong interactions will be induced, controlled by . In other words, the EFT whose leading-order terms are given by the standard model will have corrections suppressed by inverse powers of , besides those suppressed by the Planck scale . These corrections are tightly bounded experimentally. The precision electroweak tests can be summarized in the precision parameters of the electroweak chiral Lagrangian (or equivalently in the oblique parameters such as and ), and they indicate that the physics of the electroweak gauge bosons can receive correction coming from higher-dimensional operators that are at most at the per mille level. Naively, this suggests that  TeV. The physics of flavor changing neutral currents, tested in great details by studying rare processes involving kaons, mesons, charmed mesons, pions, and charged leptons also indicates that, unless has nothing to do with flavor, must be large.

In the specific case of the MSSM, analogous source of tension can be shown to have a more subtle origin. Some of the scalars from the Higgs multiplets complete, in the supersymmetric limit, the heavy vector multiplets containing the gauge bosons and the gauginos. In this limit, the mass of such scalars is closely linked to the mass of the electroweak gauge bosons. In particular, for the mass of the lightest Higgs scalar in the MSSM the bound holds in complete generality at the tree-level. (Technically, this originates from the fact that the quartic couplings in the Higgs potential are not free parameters, but actual gauge couplings derived from the -terms within the supersymmetric gauge theory.). Loop effects involving (finite) diagrams affected by supersymmetry breaking partially remove this bound, and one can show, for example, that the corrections to the physical mass of the lightest scalar particle in the spectrum, very schematically look like [66–69] where is again 1-loop suppressed. Because is perturbatively small (loop-suppressed), evading the experimental bounds from direct searches GeV requires, again, that should be somewhere in the few-TeV range (or at least the mass of one of the scalar partners of the top should). The logarithmic dependence also means that if the lower bound on the Higgs mass raises, evading it would require a very significant amount of fine-tuning.

Concluding, solving the big hierarchy problem requires linking the electroweak scale to a new scale which is kept naturally smaller than the Planck scale, and at which some new enlarged space-time symmetry be broken. Absence of any fine-tuning would be achieved if  TeV. However, indirect searches strongly suggest that  TeV, both in weakly and strongly coupled scenarios, substantially above the scale dictated by . This tension goes under the name of little hierarchy problem. There are several ways of addressing the little hierarchy problem, which may require the introduction of entirely new fields and symmetries (such as in the case of composite Higgs [70–77] models and Little Higgs models [78–82]), hence again naturally separating . Or, one might modify the solution to the big hierarchy problem in such a way as to implement some discrete symmetry, in such a way as to enforce cancellations and suppressing unwanted processes. The literature on the subject is vast, and these proposals are very interesting phenomenologically, both for the LHC and for astrophysics and cosmology (e.g., models with parities might play an important role in the physics of cold dark matter), but are not the subject of these lectures.

It must be stressed that numerically the little hierarchy problem is, indeed, little. In a generic low-energy theory it amounts to a fine-tuning at most . The arguments presented here are strongly affected by model-dependent coefficients that in this discussion have been taken to be , but that in special models might just occur to be small enough to evade the experimental bounds. Particularly in the case of technicolor, which naively is the one example in which the little hierarchy problem is most worrisome, the strongly coupled nature of the theory at the scale makes the calculation of such coefficients quite delicate. Accurate and efficient ways of computing such observables are needed. Part of these lectures will be devoted to this specific topic, in the context of walking technicolor (which itself involves the introduction of a very special symmetry), and we will see that this possibility is not excluded. While doing so, the arguments presented in this short subsection will be made more rigorous and quantitative, and concrete examples will be developed.

2.2.2. Final Remarks

Before we move away from these general introductory remarks, and start studying more concrete examples and perform more specific calculations, a final set of comments are due to the reader. First of all, one might argue that hierarchy problems as defined in the previous section are not severe failures of the theory that require its complete rewriting, they do not indicate a fatal contradiction between theory and experiment. After all, once properly renormalized, the MSM and its parameters can be used to reproduce correctly a vast amount of experimental data, with no clear indication of a contradiction (barring the fact the the Higgs particle has not been observed yet, and hence might not exist in nature, which is a different problem). In general, the emergence of a hierarchy problem can be looked at as the emergence of an opportunity for research. A fine-tuned parameter is an indication given to us by nature of a possible avenue for discovery of actually new, interesting physical phenomena, by highlighting a sector of the theory which we do not really understand at the fundamental level, and which is probably incomplete. It would be regrettable not to pursue this avenue. This is one reason why so much attention is paid by model-builders to the hierarchy problems. Besides the electroweak hierarchy problem(s) summarized here, similar attitude is applied towards other hierarchy (fine-tuning) problems, such as the strong-CP problem in QCD, and, most important, the problem of the cosmological constant.

Second, it must be stressed that the smallness of a parameter does not necessarily constitute a hierarchy problem. At least, not in the sense of naturalness used here. For instance, one might legitimately ask why there exists such a large hierarchy between the numerical values of the masses of the SM fermions, comparing, for instance, the neutrinos to the electron, and to the top quark. However, all of these masses are controlled by parameters that are natural, and hence their values are stable against radiative corrections, without the technical problem of fine-tuning. It would certainly be very interesting and useful to have a theory of flavor, in which these masses and their hierarchical structures are generated dynamically from first principles, and a vast literature exists on the subject, with beautiful and very interesting results having been collected over the years. Yet, the argument in favor of such a research program is somewhat less compelling, at least within weakly coupled theories. However, this kind of research program becomes compelling in strongly coupled extensions of the standard model, for reasons that will be explained later in these notes.

Finally, the ultimate reason why fine-tuned theories are unsatisfactory is deeply connected with the concept of effective field theory. All progress in physics up to recent times can be thought of in terms of a chain of EFTs, and the vast majority of physical phenomena that we believe we understand are actually understood in terms of their EFT description. EFTs, as opposed to would-be fundamental theories, are constructed on the basis of very few and simple general symmetry rules. These rules cannot, by themselves, single out a Lagrangian (or action) which contains only a finite number of local interactions. In general, an EFT contains an infinite number of such interactions, and an infinite number of free parameters. In the absence of anything else, these theories cannot be calculable and predictive, because no matter how many independent experimental quantities are measured and used as input, no unique prediction can be formulated. What makes EFTs work (and they do in real situations) is that one can implement in the EFT also counting rules, ensuring that within the regime of validity of the EFT, and requiring a given accuracy from the calculations to be performed, only a finite number of such parameters are actually important. The (infinite) others are, on the basis of the counting rules, expected to give contributions that, while uncalculable, are safely smaller than the required accuracy. Hence, the EFT Lagrangian can be truncated to contain only a finite number of terms.

The problem comes from the fact that the counting rules do not yield actual predictions for the values of the (unknown) parameters, but provide only order-of-magnitude estimates. Such estimates are hence useful only provided no fine-tuning turns out to be necessary. In the case where one of the parameters that are kept in the truncated EFT turns out to be severely fine-tuned, such a parameter is explicitly violating the counting rules. It becomes hence impossible to justify the truncation itself, and predictivity is lost. Hence, the hierarchy problem(s) in the standard model highlight a contradiction in the procedure that would allow to use it as (the leading order part of) an EFT description. The history of successes of EFTs itself provides a very compelling reason to look for solutions to the hierarchy problem, beyond the SM.

3. From Weak to Strong Coupling: Walking Technicolor

We have already anticipated that technicolor provides a solution to the big hierarchy problem based on the same mechanism at work in QCD, namely, at the Planck scale the theory is very close to a trivial fixed point of the renormalization group equations. The coupling of a marginally relevant operator (the gauge coupling) drives the theory away from this fixed point towards the IR. The theory becomes strongly coupled at a scale . An exponential hierarchy is natural, because controlled by the (perturbative) value of the gauge coupling in the UV. Finally, the strong coupling triggers the formation of a condensate that induces EWSB, hence linking and . We also anticipated that the phenomenological viability of such a scenario is heavily questioned by the fact that the little hierarchy problem is exacerbated by the strong coupling at the scale , particularly in a QCD-like theory, and that walking technicolor is one proposal that might soften such a problem. Let us be more specific.

Walking technicolor is based on a gauge theory in which the running of the gauge coupling is completely different from what is expected in a QCD-like theory. Figure 1 illustrates the basic idea. In Figure 1(a) is the QCD-like running as obtained for instance from (2.19). One dynamical scale exists, identified by the divergence of the running coupling . Figure 1(b) presents a theory that in the IR flows to a non-trivial fixed point. Again, one dynamical scale exists , separating the two energy regimes at which the theory is well approximated by the UV and IR fixed points, respectively. There exists two well-established examples of such behavior. One can be seen to exist by looking at the 2-loop perturbative -function of a theory with vectorial fermions on the fundamental representation [83]. If is very close to the limit beyond which asymptotic freedom is lost, the -function is suppressed by anomalously small coefficients. One then finds that a fixed point exists in the IR, and that provided this is perturbative (so that neglecting higher-loop contributions is justified) the theory flows into a weakly coupled non-Abelian Coulomb phase.

(a)

(b)

(c)

(a)
(b)
(c)

Figure 1 

Schematic depiction of the evolution of the gauge coupling as a function of the renormalization scale in three different asymptotically free theories: a QCD-like theory, a theory with an IR fixed point, and a walking theory (left to right).

Another example emerges for the supersymmetric version of the same gauge theory, in which case the beta function is [84–86] where the anomalous dimension of the mass is Again, when is very close to , a fixed point in the spirit of [83] exists. However, because of supersymmetry the theory possesses a Seiberg duality [87–89], which in turns means that such a fixed point exists in the whole conformal window, The interesting fact is that towards the lower end of the conformal window, the fixed point is strongly coupled. Again thanks to supersymmetry, one can compute the anomalous dimension at the fixed point as which agrees with the intuition by vanishing when , but towards the other boundary of the conformal window yields with the dimension of the (super-)quark bilinear changing continuously in the range as a function of , by assuming values that are and hence completely nonperturbative.

The dynamics of a walking theory is a combination of these confining and conformal theories. In the IR, below the scale , the theory is approaching an approximate fixed point, governed by a strongly coupled conformal field theory, described by operators that have large anomalous dimensions. The fixed point is approximate in the sense that some dynamical feature prevents the theory from actually reaching such a fixed point. Deep into the IR, the theory actually confines and yields a symmetry breaking condensate at the electroweak scale . The gauge coupling evolves as in Figure 1(c), in a way that is somewhat a hybrid of the previous two cases.

Notice that in the walking case there are at least three dynamical scales. A scale analogous to the conformal case, at which the theory enters a quasiconformal phase, a scale deep in the IR in which the gauge coupling diverges in the same sense as in the QCD-like case. and an additional intermediate scale , at which the theory stops walking, and enters the fast running leading ultimately to confinement at . The word walking refers to the fact that the coupling is approximately constant in the region , where the beta function is parametrically small in comparison to the coupling itself. Somewhere between the two lower scales (which might coincide) the symmetry breaking scale should be dynamically generated too.

A second feature, of utmost phenomenological relevance, is that because the physics in the walking region is governed by the strongly coupled approximate IR fixed point, its operators acquire in general very large, nonperturbative anomalous dimensions, which drastically change the power-counting rules expected from NDA and implemented in the low-energy effective description of electroweak physics in the standard model.

In principle, one would like to a have a nonperturbative, nonsupersymmetric theory with the properties mentioned above. We have no rigorous proof that such a thing exists, but it is usually believed that some critical exists above which the theory flows into an IR fixed point, and below which it confines and condensates form. A walking theory is supposed to be given by the limit in which is close to this critical value. In this lecture, we will be more precise about the phenomenological implications of the previous statements, and discuss what are the challenges of this proposal, in direct comparison with what happens in perturbative theories such as the MSM.

3.1. Electroweak Chiral Lagrangian and Oblique Precision Parameters

A convenient way of parameterizing the low-energy effects of the strongly coupled sector responsible for electroweak symmetry breaking is provided by the electroweak chiral Lagrangian [20–24]. The basic idea is that at very low energy the only relevant degrees of freedom are given by the Goldstone bosons and the electroweak gauge bosons. (In these lectures, it is assumed that the SM fermions do not carry quantum numbers of the technicolor sector, and are essentially treated as spectators.) An effective description is then obtained by writing (infinite numbers of) possible local interactions involving pions and gauge bosons, compatibly with the symmetries of the problem, and organizing the expansion as a powers-series.

One can think of the resulting effective action as a systematic approximation of the quantum effective action that would be obtained by integrating out all the heavy degrees of freedom in the theory (technimesons, technibaryons, techniglueballs, ). In principle, this is a nonlocal function of the momentum . Provided is much smaller than a scale related to the lightest among the masses of all the degrees of freedom that are integrated out, one can expand in powers of , and truncate at a given order in the expansion. The truncation reduces the number of independent couplings to be finite. At the same time, the truncation determines the level of accuracy of the results of calculations of physical observables, which is itself controlled by a power of .

The electroweak chiral Lagrangian [20–24] is defined from the two-sites model in Figure 2. One defines a global symmetry, gauges the subgroup, with generated by , and introduces the dimensionless with , , and the Pauli matrices, so that transforms as a bifundamental under . The decay constant is the electroweak VEV, . The pion fields are the Goldstone bosons of the spontaneous breaking . In unitary gauge , and the pions become the longitudinal components of the electroweak gauge bosons. The covariant derivative is with the weakly coupled gauge bosons of , the gauge boson associated with the diagonal , and and the (weak) gauge couplings. In terms of the field strength and , of the covariant derivative , of the SM currents (which contain the SM fermions), and of the custodial-symmetry breaking operator , the Lagrangian reads where

Figure 2 

Diagram of the 2-site model, the electroweak chiral Lagrangian.

The notation introduced here is somewhat unconventional, and demands for an explanation. In the terms , the index indicates the dimension of the operators, obtained by counting , , and . The index counts the explicit appearances of , the operator introducing explicit breaking of custodial symmetry. In practice, there are three expansions being performed, and used in classifying corrections beyond . One is the aforementioned expansion in derivatives (). But there is also an expansion in a custodial-symmetry breaking parameter , implicit within the definition of many of the . And finally we are making use of the perturbative expansion in . The way in which the Lagrangian is written is such that the small parameters are all implicit in the definition of the coefficients , which are hence expected to be small.

In order to understand the origin of all these couplings, and how the counting works, let us start by assuming that custodial symmetry be exact. The only terms surviving in this limit are the ones in and . (Some insight might be obtained from the comparison to the chiral Lagrangian, and the classification by Gasser and Leutwyler [90] of the coefficients appearing at the . corresponds to , and to , to , and to .) One could start from alone. The fact that the symmetry is nonlinearly realized is reflected in the fact that contains only the three degrees of freedom of the pion (the constraint is applied). Expanding in powers of , at the leading order one gets the (canonically normalized) kinetic terms for the pions, while the next-to-leading order yields a (derivative) quartic coupling proportional to , which, for instance, yields the leading-order contribution to the elastic scattering amplitude. This coupling can also be used to construct 1-loop diagrams. Because the EFT is nonrenormalizable, the divergences of the 1-loop diagrams cannot be easily removed; there are logarithmically divergent amplitudes proportional to . In order to cure these divergences, one has to introduce new couplings in the Lagrangian, which are those in . This means that one should think of the expansion in as an expansion in . Going to higher loops will involve introducing new couplings with higher number of derivatives. Most important, in comparing with experimental data, one has to be careful to use the Lagrangian appropriately; if one wants to extract a leading-order value for, say , the comparison should be done by including 1-loop diagrams built with the couplings extracted from .

Analogously, one can think of custodial-symmetry breaking effects as arising from loop diagrams involving fundamental degrees of freedom (techniquarks) in the underlying theory, whose masses and couplings violate custodial symmetry. And hence we associate also with corrections. The power-counting is hence determined by assuming that

Looking back at one can ask what is the physical meaning of the couplings in terms of the physics of electroweak gauge bosons. One immediately sees that , , and produce corrections to the gauge boson propagators, while , , , and correct the cubic couplings and , , , , and the quartic couplings. Precision studies of the electroweak interactions bound mostly the corrections to the vacuum polarizations. Following [26], the oblique parameters can be defined starting from the transverse vacuum polarization amplitudes of the gauge bosons of the standard model, expanding as and defining, for , The parameters , and are hence directly related to , , and of [25], and to the , , and in [26]: (Notice that in the definition of , , and the corrections are neglected, and hence the identification of and with is valid only up to corrections.) Bounds on the other exist, but are much less stringent, because much less data is available for the cubic and quartic vertexes. The parameters , , and are related to terms in .

The strategy for extracting bounds on new physics from the experimental data requires computing measurable amplitudes from the MSM at 1-loop level, and include and (equivalently, introducing the corrections from and in the propagators) at the tree level. In this way, all UV and IR divergences cancel, and a consistent set of amplitudes computed at can be compared to the data (the reader should remember that at the tree-level the MSM itself does not provide a satisfactory fit of all the electroweak precision observables, for which the inclusion of 1-loop corrections is necessary). One then performs a fit to the data using as free parameters the values of the electroweak MSM parameters , , and , of the masses of top and Higgs , and the precision parameters and . A list of the observables used can be found for instance in [26, 91].

Here comes one subtlety, which is going to be important also later, in computing the MSM 1-loop contributions to the physical observables, an explicit logarithmic dependence on the mass of the Higgs is present. Borrowing the results from [25]: with some reference value for the Higgs mass. The result is that the MSM provides a good fit to all the existing data, for values of the Higgs mass  TeV, provided all the precision parameters such as and are at most of order few . One may arbitrarily decide to do the fit while setting all the oblique parameters to zero. In doing so, one finds a bound on the mass of the Higgs to be GeV ( c.l.) [92, 93], because of the logarithmic dependence on of the 1-loop corrections. This bound amounts to assuming that no new physics be present at any scale.

The comparison of the above procedure to a generic extension of the SM can be done rigorously only provided the only light degrees of freedom include all the SM fermions and gauge bosons, together with one Higgs scalar, while all new physics can be integrated out and reabsorbed in the higher-order coefficients in . This means that some caution must be used when comparing to models which predict the presence of new light degrees of freedom, for which the 1-loop perturbative part of the calculations should be redone.

But also highlights a difficulty with higgsless models, in which strictly speaking the precision parameters diverge, and are hence not calculable. A physical cut-off should be kept in the calculation, entering logarithmically in place of the logarithmic dependence on the mass of the Higgs. In practice, phenomenological studies usually assume that Higgsless models (including TC) are well described by the limit of the MSM case in which  TeV. This is motivated mostly by the pragmatic observation that, because the dependence on the mass of the Higgs is logarithmic, by varying the Higgs mass between  GeV and GeV (which covers most of the regime over which the Higgs can be treated perturbatively), the bound on changes only by , which is smaller that the error bar on itself. Imposing, for instance, on the calculable part of the new physics contribution is hence going to yield sensible quantitative constraints. Notice also that the result of the fit of the data yields some ellipsoid in the space of the complete set of precision parameters, and the resulting bounds on and in particular are correlated. Because of the intrinsic uncertainty in the treatment of strongly coupled systems, in particular due to what we just said about the role of scalars, in the following I will use the bounds as uncorrelated, with the understanding that these are somewhat conservative bounds; models that end up outside the bounds are very unlikely to be fixed by improved calculations, and are hence excluded, while for models that fit into the bounds, detailed strong-coupling calculations would be needed to decide if they can or cannot fit the data.

The nondecoupling of the Higgs degrees of freedom is one of the many subtle, phenomenologically important, aspects of the MSM, which sources theoretical problems in its extensions. In the case of precision electroweak tests, this feature still allows for useful quantitative information to be extracted, thanks to the mild logarithmic dependence on the Higgs mass. In short, generic new physics contributions are larger than the uncertainty introduced by the Higgs-mass dependence, and hence useful (and quite restrictive) constraints can be extracted. We will see that this is not the case for other observables, such as elastic-scattering amplitudes of longitudinally polarized gauge bosons, in which the non-decoupling manifests itself with a strong, quadratic dependence on . In Higgless theories one is going to face quadratically divergent contributions to these scattering amplitudes, that are removed by completely uncalculable contributions from the scalar sector, hence limiting greatly the amount of information about new physics that can be extracted. We will be more precise on this topic in a later subsection.

Before going any further, the reader should be reminded of the role played by precision studies in the discovery of the top, which is a nice illustration of how powerful simple power-counting is, and how important precision studies are. The electroweak chiral Lagrangian highlights a very special feature: custodial symmetry is broken by an operator of dimension-2, in . Very interestingly, in the SM there is one very large, explicit source of breaking of , namely, the mass of the top is much larger that the mass of the bottom. Dimensional analysis suggests, hence, that (or, equivalently, ). (We will see later that indeed this is the result of perturbative calculations with a heavy fermion.) Which means that the precision tests of the SM are very sensitive to the precise value of the top mass. And indeed, a clear indication that the top is heavy, in the  GeV region, was obtained by analyzing LEP data before the actual TeVatron discovery. As we already saw, similar arguments yield much more limited information about the Higgs mass, that enters only logarithmically in the precision tests.

3.1.1. Perturbative Estimates of and

Following Peskin and Takeuchi, we can start from computing the contribution to the precision parameters from a new family of weakly coupled, heavy fermions. This perturbative calculation will be useful in showing us how the corrections are generated, and we will later discuss what to expect in the nonperturbative case.

Consider a fermion left-handed doublet , with hypercharge , and right-handed , such that the physical states have masses, respectively, and . Assuming that , one can compute the correction to the vacuum polarizations induced at 1-loop from diagrams involving loops of such fermions, with SM gauge bosons on the external lines. Assuming also, for simplicity, that , the expressions simplify to First of all, this confirms that is doubly suppressed, as anticipated in speaking about the electroweak chiral Lagrangian, and hence can be ignored in phenomenological analysis. Second, it explicitly shows that is proportional to the amount of custodial-symmetry breaking in the new physics sector, and is hence negligibly small provided . For these reasons, we focus on the parameter. Finally, if one has different fermions, their contributions sum is which, taking as a reference the bound , implies .

If one could borrow this result and use it as an estimate of the TC contribution, this would imply a strong bound on the possible choice of field content of the theory. For example, reconsidering the vectorial gauge theory introduced earlier on, with families of techniquarks, each family including fermions on the fundamental of that carry the quantum numbers of a family of SM fermions, one obtains which effectively restricts to one point of parameter space with and . And still, this single point would lie on the upper bound.

The use of this perturbative estimate is, of course, very hard to motivate: by definition technicolor is strongly coupled. The numerical coefficients have to be taken for what they are. The robust messages that can be extracted from this exercise are that some form of custodial symmetry better be present, to suppress , that is potentially problematic, and that the precision parameters are expected to grow with the number of degrees of freedom coupled to the gauge bosons.

3.1.2. Nonperturbative Estimates of : NDA and Hidden Local Symmetry

One might want to get a feeling for how big the coefficients of are expected to be, in particular (equivalent to ), which is the most severe test of dynamical electroweak symmetry breaking. A set of very simple rules for obtaining such an estimate exists, and goes under the name of Naive Dimensional Analysis (NDA). A summary and discussion of such rules can be found in [35]. Here are the rules, applied to , restricted by setting to zero all the custodial-symmetry violating terms. First of all, one has to write out all possible local operators compatible with the symmetries, and include them in the effective Lagrangian. An estimate of the coefficients is then given by the following three rules: (i)include a factor of overall, (ii)include a factor of for any occurrence of strongly coupled fields (such as the pions ), (iii)include appropriate factors of to get the dimension to 4.

Hence, the coefficient of the terms in is , and for all the coefficients are . The scale is associated with the mass of the lightest degree of freedom that has been integrated out of the theory, which typically is the mass of the technirho meson . Hence, one expects that Identifying with the electroweak scale, taken literally this result would mean that  TeV. We will see that much more sophisticated techniques agree to an impressive degree with this NDA estimate. It must be stressed that additional symmetry arguments provide systematic ways of including exceptions to this rules, as is illustrated by the fact that on the basis of this counting only one would conclude that , while promoting to an approximate symmetry makes .

The relation between the parameter and the physics of spin-1 resonances in a strongl coupled theory can be made more precise and model-independent with the use of dispersion relations to rewrite (3.13) by saturating on the vector resonances () and axial resonances (), which results in where the index identifies the tower of resonances with mass and decay constant . In the limit in which were unbroken, axial and vector states would be exactly degenerate and connected by a symmetry, and hence . The pion decay constant measures how broken this symmetry is. This expression is useful because it has shown that a suppression of not necessarily implies taking large values of , but might be achieved by suppressing the decay constants, or by ensuring a cancellation between vectorial and axial contributions.

One way of drawing a closer relation between and the heavy composite states, in particular the lightest spin-1 meson, the techni-rho, is provided by the inclusion of the latter in the low-energy effective theory. This is done with the help of so-called hidden local symmetry (HLS) [36–43]. The basic trick is to extend the global symmetry of the chiral Lagrangian (truncated to the leading order), by introducing an additional, spontaneously-broken gauged as in Figure 3. The in the middle site is gauged with strength , and there are now two sigma-model fields, so that in unitary gauge the new gauge bosons are massive.

Figure 3 

Diagram of the 3-site model.

The important terms in the Lagrangian are, at the leading-order, with and the covariant derivatives In unitary gauge, , and the mass of the neutral gauge bosons is, in the basis, Assuming that , one can integrate out the heavy techni-rho, of mass and finally obtain where . This estimate implies that the bound on requires  TeV, which is a factor of milder that what derived from the NDA estimate (3.24), but consistent with it.