View source for Technicolor (physics)

{{Short description|Hypothetical model through which W and Z bosons acquire mass}}
{{Use American English|date=August 2020}}
{{for|the film process and trademark|Technicolor}}
{{distinct|Topcolor}}
{{Beyond the Standard Model|expanded=Theories}}

'''Technicolor''' theories are models of [[physics beyond the Standard Model]] that address [[electroweak symmetry breaking|electroweak gauge symmetry breaking]], the mechanism through which [[W and Z bosons]] acquire masses.  Early technicolor theories were modelled on [[quantum chromodynamics]] (QCD), the "color" theory of the [[Strong interaction|strong nuclear force]], which inspired their name.

Instead of introducing elementary [[Higgs boson]]s to explain observed phenomena, technicolor models were introduced to dynamically generate masses for the [[W and Z bosons]] through new [[gauge theory|gauge interactions]].  Although [[asymptotic freedom|asymptotically free]] at very high energies, these interactions must become strong and [[color confinement|confining]] (and hence unobservable) at lower energies that have been experimentally probed. This dynamical approach is natural and avoids issues of [[quantum triviality]] and the [[hierarchy problem]] of the Standard Model.   

However, since the Higgs boson discovery at the 
CERN LHC in 2012, the original models are largely ruled out. Nonetheless, it remains a possibility that the Higgs boson is a composite state.<ref>For introductions to and reviews of technicolor and strong dynamics, see the following:{{br}}{{cite journal |author=Christopher T. Hill |author2=Elizabeth H. Simmons |name-list-style=amp |title=Strong Dynamics and Electroweak Symmetry Breaking |journal=Physics Reports |volume=381 |issue=4–6 |date=2003 |pages=235–402 |doi=10.1016/S0370-1573(03)00140-6 |arxiv=hep-ph/0203079 |bibcode=2003PhR...381..235H|s2cid=118933166 }}{{br}}{{cite conference |author=Kenneth Lane |title=Two Lectures on Technicolor |conference=l'Ecole de GIF at LAPP, Annecy-le-Vieux, France |date=2002 |arxiv=hep-ph/0202255 |bibcode=2002hep.ph....2255L }}{{br}}{{cite conference |author=Robert Shrock |title=Some Recent Results on Models of Dynamical Electroweak Symmetry Breaking |conference=International Workshop on Strongly Coupled Gauge Theories |book-title=Nagoya 2006: The Origin of Mass and Strong Coupling Gauge Theories |editor=M. Tanabashi |editor2=M. Harada |editor3=K. Yamawaki |date=2007 |pages=227–241 |arxiv=hep-ph/0703050| bibcode=2008omsc.conf..227S |doi=10.1142/9789812790750_0023 }}{{br}}{{cite conference |author=Adam Martin |title=Technicolor Signals at the LHC |conference=The 46th Course at the International School of Subnuclear Physics: Predicted and Totally Unexpected in the Energy Frontier Opened by LHC |date=2008 |arxiv=0812.1841 |bibcode=2008arXiv0812.1841M }}{{br}}{{cite journal |author=Francesco Sannino |title=Conformal Dynamics for TeV Physics and Cosmology |journal=Acta Physica Polonica |arxiv=0911.0931 |volume=B40 | pages=3533–3745 |date=2009 |bibcode=2009arXiv0911.0931S}}</ref>

In order to produce [[quark]] and [[lepton]] masses, technicolor or composite Higgs models have to be "extended" by additional gauge interactions. Particularly when modelled on QCD, extended technicolor was challenged by experimental constraints on [[flavor-changing neutral current]] and [[Peskin–Takeuchi parameter|precision electroweak measurements]]. The specific extensions of particle dynamics for technicolor 
or composite Higgs bosons are unknown.

Much technicolor research focuses on exploring strongly interacting gauge theories other than QCD, in order to evade some of these challenges. A particularly active framework is "walking" technicolor, which exhibits nearly [[conformal field theory|conformal]] behavior caused by an [[infrared fixed point]] with strength just above that necessary for spontaneous [[chiral symmetry]] breaking. Whether walking can occur and lead to agreement with precision electroweak measurements is being studied through [[non-perturbative]] [[lattice gauge theory|lattice]] simulations.<ref name=lattice>{{cite journal |author=George Fleming |title=Strong Interactions for the LHC |journal=Proceedings of Science |volume=LATTICE 2008 |date=2008 |page=21 |arxiv=0812.2035 |bibcode=2008arXiv0812.2035F }}</ref>

Experiments at the [[Large Hadron Collider]] have discovered the mechanism responsible for electroweak symmetry breaking, i.e., the [[Higgs boson]], with mass approximately {{val|125|ul=GeV/c2}};<ref name=CERN1207>{{cite news |url=http://press.cern/press-releases/2012/07/cern-experiments-observe-particle-consistent-long-sought-higgs-boson |title=CERN experiments observe particle consistent with long-sought Higgs boson |publisher=CERN press release |date=4 July 2012 |access-date=4 July 2012}}</ref><ref name=CMS1207>{{cite web | url = http://cms.web.cern.ch/news/observation-new-particle-mass-125-gev |title=Observation of a New Particle with a Mass of 125&nbsp;GeV |first=Lucas |last=Taylor |date=4 July 2012 |work = CMS Public Web site | publisher = CERN}}</ref><ref name=ATLAS1207>{{cite web |title=Latest Results from ATLAS Higgs Search |publisher=ATLAS |url=http://www.atlas.ch/news/2012/latest-results-from-higgs-search.html |date=4 July 2012 |access-date=4 July 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120707132113/http://www.atlas.ch/news/2012/latest-results-from-higgs-search.html |archive-date=7 July 2012}}</ref> such a particle is not generically predicted by technicolor models. However,
the Higgs boson may be a composite state, e.g., built of top and anti-top quarks
as in the Bardeen–Hill–Lindner theory.<ref>{{cite journal | author=William A. Bardeen | author2=Christopher T. Hill | author3=Manfred Lindner | name-list-style=amp | title=Minimal dynamical symmetry breaking of the standard model | journal=Physical Review | volume=D41 | date=1990 | issue=5 | pages=1647–1660 | doi=10.1103/PhysRevD.41.1647|bibcode = 1990PhRvD..41.1647B | pmid=10012522 }}.</ref>
Composite Higgs models are generally solved by the top quark [[infrared fixed point]],
and may require a new dynamics at extremely high energies such as [[topcolor]].

== Introduction ==
The mechanism for the breaking of [[electroweak interaction|electroweak]] [[gauge theory|gauge symmetry]] in the [[Standard Model]] of elementary particle interactions remains unknown. The breaking must be [[spontaneous symmetry breaking|spontaneous]], meaning that the underlying theory manifests the symmetry exactly (the gauge-boson fields are massless in the equations of motion), but the solutions (the ground state and the excited states) do not.  In particular, the physical [[W and Z bosons|''W'' and ''Z'' gauge bosons]] become massive. This phenomenon, in which the ''W'' and ''Z'' bosons also acquire an extra polarization state, is called the "Higgs mechanism". Despite the precise agreement of the electroweak theory with experiment at energies accessible so far, the necessary ingredients for the symmetry breaking remain hidden, yet to be revealed at higher energies.

The simplest mechanism of [[electroweak]] symmetry breaking introduces a single complex field and predicts the existence of the [[Higgs boson]]. Typically, the Higgs boson is "unnatural" in the sense that quantum mechanical fluctuations produce corrections to its mass that lift it to such high values that it cannot play the role for which it was introduced. Unless the Standard Model breaks down at energies less than a few TeV, the Higgs mass can be kept small only by a delicate [[Fine-tuning (physics)|fine-tuning]] of parameters.

Technicolor avoids this problem by hypothesizing a new gauge interaction coupled to new massless fermions. This interaction is [[asymptotic freedom|asymptotically free]] at very high energies and becomes strong and confining as the energy decreases to the [[electroweak scale]] of 246&nbsp;GeV. These strong forces spontaneously break the massless fermions' chiral symmetries, some of which are weakly gauged as part of the Standard Model. This is the dynamical version of the Higgs mechanism. The electroweak gauge symmetry is thus broken, producing masses for the W and Z bosons.

The new strong interaction leads to a host of new composite, short-lived particles at energies accessible at the [[Large Hadron Collider]] (LHC). This framework is natural because there are no elementary Higgs bosons and, hence, no fine-tuning of parameters. Quark and lepton masses also break the electroweak gauge symmetries, so they, too, must arise spontaneously. A mechanism for incorporating this feature is known as extended technicolor. Technicolor and extended technicolor face a number of [[phenomenology (particle physics)|phenomenological]] challenges, in particular issues of [[flavor-changing neutral current]]s, [[Peskin–Takeuchi parameter|precision electroweak tests]], and the [[top quark]] mass. Technicolor models also do not generically predict Higgs-like bosons as light as {{val|125|ul=GeV/c2}}; such a particle was discovered by experiments at the Large Hadron Collider in 2012.<ref name=CERN1207 /><ref name=CMS1207/><ref name=ATLAS1207/>  Some of these issues can be addressed with a class of theories known as "walking technicolor".

== Early technicolor ==
Technicolor is the name given to the theory of electroweak symmetry breaking by new strong gauge-interactions whose characteristic energy scale {{math|Λ}}{{sub|TC}} is the weak scale itself, {{nowrap|{{math|Λ}}{{sub|TC}} ≈ {{math|F}}{{sub|EW}} ≡ 246 GeV }}. The guiding principle of technicolor is "naturalness": basic physical phenomena should not require fine-tuning of the parameters in the Lagrangian that describes them. What constitutes fine-tuning is to some extent a subjective matter, but a theory with elementary scalar particles typically is very finely tuned (unless it is [[supersymmetry|supersymmetric]]). The quadratic divergence in the scalar's mass requires adjustments of a part in <math>\mathcal{O}\left(\frac{M^2_{\mathrm{bare}}}{M^2_{\mathrm{physical}}}\right)</math>, where {{mvar|M}}<sub>bare</sub> is the cutoff of the theory, the energy scale at which the theory changes in some essential way. In the standard electroweak model with {{nowrap|{{mvar|M}}{{sub|bare}} ~ 10{{sup|15}} GeV}} (the grand-unification mass scale), and with the [[Higgs boson]] mass {{nowrap|{{mvar|M}}{{sub|physical}} {{=}} 100–500 GeV}}, the mass is tuned to at least a part in 10{{sup|25}}.

By contrast, a natural theory of electroweak symmetry breaking is an asymptotically free gauge theory with fermions as the only matter fields. The technicolor gauge group G{{sub|TC}} is often assumed to be SU({{mvar|N}}{{sub|TC}}). Based on analogy with quantum chromodynamics (QCD), it is assumed that there are one or more doublets of massless Dirac "technifermions" transforming vectorially under the same [[complex representation]] of G{{sub|TC}}, <math>T_{i\,\mathrm{L,R}} = ( U_i, D_i)_\mathrm{L,R}\,, \text{  for } i = 1, 2, ..., \tfrac{1}{2} N_\mathrm{f}</math>. Thus, there is a [[chiral symmetry]] of these fermions, e.g., {{nowrap|SU({{mvar|N}}<sub>f</sub>)<sub>L</sub> ⊗ SU({{mvar|N}}<sub>f</sub>)<sub>R</sub>}}, if they all transform according to the same complex representation of G{{sub|TC}}. Continuing the analogy with QCD, the running gauge coupling {{mvar|α}}{{sub|TC}}({{mvar|μ}}) triggers spontaneous chiral symmetry breaking, the technifermions acquire a dynamical mass, and a number of massless [[Goldstone boson]]s result. If the technifermions transform under {{nowrap|[SU(2) ⊗ U(1)]{{sub|EW}} }} as left-handed doublets and right-handed singlets, three linear combinations of these Goldstone bosons couple to three of the electroweak gauge currents.

In 1973 Jackiw and Johnson<ref>{{cite journal |author1=Jackiw, R. |author2=Johnson, K. |name-list-style=amp |title=Dynamical model of spontaneously broken gauge symmetries |journal=Physical Review |volume=D8 |date=1973 |issue=8 |pages=2386–2398 |doi=10.1103/PhysRevD.8.2386 |bibcode = 1973PhRvD...8.2386J}}</ref> and Cornwall and Norton<ref>{{cite journal |author1=Cornwall, John M. |author2=Norton, Richard E. |name-list-style=amp |title=Spontaneous Symmetry Breaking Without Scalar Mesons |journal=Physical Review |volume=D8 |date=1973 |issue=10 |pages=3338–3346 |doi= 10.1103/PhysRevD.8.3338 |bibcode=1973PhRvD...8.3338C}}</ref> studied the possibility that a (non-vectorial) gauge interaction of fermions can break itself; i.e., is strong enough to form a Goldstone boson coupled to the gauge current. Using Abelian gauge models, they showed that, ''if'' such a Goldstone boson is formed, it is "eaten" by the Higgs mechanism, becoming the longitudinal component of the now massive gauge boson. Technically, the polarization function {{math|Π}}({{mvar|p}}{{sup|2}}) appearing in the gauge boson propagator, 
: <math>\Delta_{\mu \nu} = \frac{ \left[ \frac{ p_{\mu} p_{\nu} }{p^2} - g_{\mu \nu} \right] }{~p^2 \left[ 1 - g^2 \Pi\left( p^2 \right)\right]~} </math> 
develops a pole at {{nowrap|{{mvar|p}}{{sup|2}} {{=}} 0}} with residue {{mvar|F}}{{sup|2}}, the square of the Goldstone boson's decay constant, and the gauge boson acquires mass {{nowrap|{{mvar|M}} ≈ {{mvar|g F}} }}. In 1973, Weinstein<ref>{{cite journal |author=Marvin Weinstein |title=Conserved Currents, their Commutators, and the Symmetry Structure of Renormalizable Theories of Electromagnetic, Weak, and Strong Interactions |journal=Physical Review |volume=D8 |date=1973 |issue=8 |pages=2511–2524 |doi=10.1103/PhysRevD.8.2511 |bibcode=1973PhRvD...8.2511W |citeseerx=10.1.1.412.3345}}</ref> showed that composite Goldstone bosons whose constituent fermions transform in the "standard" way under SU(2) ⊗ U(1) generate the weak boson masses
: <math>(1)\qquad M_\mathrm{W^\pm} = \frac{1}{2} g\,F_\mathrm{EW} \quad \text{ and } \quad M_\mathrm{Z} = \frac{1}{2}\sqrt{g^2 + {g'}^{2}}F_\mathrm{EW} \equiv \frac{M_\mathrm{W}}{\cos\theta_\mathrm{W}}. </math>

This standard-model relation is achieved with elementary Higgs bosons in electroweak doublets; it is verified experimentally to better than 1%. Here, {{mvar|g}} and {{mvar|g}}{{prime}} are SU(2) and U(1) gauge couplings and <math>\tan \theta_\mathrm{W} = \frac{g'}{g}</math> defines the weak mixing angle.

The important idea of a ''new'' strong gauge interaction of massless fermions at the electroweak scale {{mvar|F}}{{sub|EW}} driving the spontaneous breakdown of its global chiral symmetry, of which an SU(2) ⊗ U(1) subgroup is weakly gauged, was first proposed in 1979 by [[Steven Weinberg|Weinberg]].<ref>{{cite journal |author=Weinberg, Steven |author-link=Steven Weinberg |title=Implications of dynamical symmetry breaking |journal=Physical Review |volume=D13 |date=1976 |issue=4 |pages=974–996 |doi=10.1103/PhysRevD.13.974 |bibcode=1976PhRvD..13..974W}}</ref><ref>{{cite journal |author1=Weinberg, S. |author1-link=Steven Weinberg |author2=Susskind, L. |title=Implications of dynamical symmetry breaking: An addendum |journal=Physical Review |volume=D19 |date=1979 |issue=4 |pages=1277–1280 |doi=10.1103/PhysRevD.19.1277 |bibcode=1979PhRvD..19.1277W }}</ref><ref>{{cite journal |author=Susskind, Leonard |title=Dynamics of spontaneous symmetry breaking in the Weinberg-Salam theory |journal=Physical Review |volume=D20 |date=1979 |issue=10 |pages=2619–2625 |doi=10.1103/PhysRevD.20.2619 |bibcode= 1979PhRvD..20.2619S|osti=1446928 |s2cid=17294645 }}</ref> This "technicolor" mechanism is natural in that no [[Fine-tuning (physics)|fine-tuning]] of parameters is necessary.

== Extended technicolor ==
Elementary [[Higgs boson]]s perform another important task. In the [[Standard Model]], [[quark]]s and [[lepton]]s are necessarily massless because they transform under {{nowrap|SU(2) ⊗ U(1)}} as left-handed doublets and right-handed singlets. The Higgs doublet couples to these fermions. When it develops its vacuum expectation value, it transmits this [[electroweak interaction|electroweak]] breaking to the quarks and leptons, giving them their observed masses. (In general, electroweak-eigenstate fermions are not mass eigenstates, so this process also induces the mixing matrices observed in charged-current weak interactions.)

In technicolor, something else must generate the quark and lepton masses. The only natural possibility, one avoiding the introduction of elementary scalars, is to enlarge {{mvar|G}}<sub>TC</sub> to allow technifermions to couple to quarks and leptons. This coupling is induced by gauge bosons of the enlarged group. The picture, then, is that there is a large "extended technicolor" (ETC) gauge group {{nowrap|{{mvar|G}}<sub>ETC</sub> ⊃ {{mvar|G}}<sub>TC</sub>}} in which technifermions, quarks, and leptons live in the same [[representation of a Lie group|representations]]. At one or more high scales {{math|Λ}}<sub>ETC</sub>, {{mvar|G}}<sub>ETC</sub> is broken down to {{mvar|G}}<sub>TC</sub>, and quarks and leptons emerge as the TC-singlet fermions. When {{mvar|α}}<sub>TC</sub>({{mvar|μ}}) becomes strong at scale {{math|Λ}}<sub>TC</sub> ≈ {{mvar|F}}<sub>EW</sub>, the [[fermionic condensate]] <math>\langle \bar{T}T\rangle_\text{TC} \approx 4 \pi F_\text{EW}^3</math> forms. (The condensate is the [[vacuum expectation value]] of the technifermion bilinear <math>\bar{T}T</math>. The estimate here is based on [[naive dimensional analysis]] of the quark condensate in [[Quantum chromodynamics|QCD]], expected to be correct as an order of magnitude.) Then, the transitions <math>q_\text{L} (\mathrm{ or }\,\,\ell_\text{L}) \rightarrow T_\text{L} \rightarrow T_\text{R} \rightarrow q_\text{R}\,(\mathrm{ or }\,\,\ell_\text{R})</math> can proceed through the technifermion's dynamical mass by the emission and reabsorption of ETC bosons whose masses {{mvar|M}}<sub>ETC</sub> ≈ {{mvar|g}}<sub>ETC</sub> {{math|Λ}}<sub>ETC</sub> are much greater than {{math|Λ}}<sub>TC</sub>. The quarks and leptons develop masses given approximately by
: <math>(2)\qquad m_{q,\ell}(M_\text{ETC}) \approx \frac{g_\text{ETC}^2 \langle \bar T T\rangle_\text{ETC}}{M_\text{ETC}^2} \approx \frac{4 \pi F_\text{EW}^3}{\Lambda_\text{ETC}^2}\,.</math>

Here, <math>\langle \bar T T\rangle_\text{ETC}</math> is the technifermion condensate renormalized at the ETC boson mass scale,
: <math>(3)\qquad \langle \bar T T\rangle_\text{ETC} = \exp{\left(\int_{\Lambda_\text{TC}}^{M_\text{ETC}} \frac{d\mu}{\mu}\gamma_m(\mu)\right)}\,\langle \bar T T\rangle_\text{TC}\,,</math>
where {{mvar|γ}}<sub>m</sub>({{mvar|μ}}) is the [[anomalous scaling dimension|anomalous dimension]] of the technifermion bilinear <math>\bar T T</math> at the scale&nbsp;{{mvar|μ}}. The second estimate in Eq.&nbsp;(2) depends on the assumption that, as happens in QCD, {{mvar|α}}<sub>TC</sub>({{mvar|μ}}) becomes weak not far above {{math|Λ}}<sub>TC</sub>, so that the anomalous dimension {{mvar|γ}}<sub>m</sub> of <math>\bar T T</math> is small there. Extended technicolor was introduced in 1979 by Dimopoulos and Susskind,<ref>{{cite journal |author=Savas Dimopoulos |author2=Leonard Susskind |name-list-style=amp |title=Mass without scalars |journal=Nuclear Physics |volume=B155 |date=1979 |issue=1 |pages=237–252 |doi=10.1016/0550-3213(79)90364-X |bibcode=1979NuPhB.155..237D}}</ref> and by Eichten and Lane.<ref name=ETC>{{cite journal |author=Estia Eichten |author2=Kenneth Lane |name-list-style=amp |title=Dynamical breaking of weak interaction symmetries |journal=Physics Letters B |volume=90 |date=1980 |issue=1–2 |pages=125–130 |doi=10.1016/0370-2693(80)90065-9 |bibcode=1980PhLB...90..125E}}</ref> For a quark of mass {{mvar|m}}<sub>{{mvar|q}}</sub>&nbsp;≈&nbsp;1&nbsp;GeV, and with {{math|Λ}}<sub>TC</sub> ≈ 246&nbsp;GeV, one estimates {{math|Λ}}<sub>ETC</sub>&nbsp;≈&nbsp;15&nbsp;TeV. Therefore, assuming that <math>g^2_\text{ETC} \gtrsim 1</math>, {{mvar|M}}<sub>ETC</sub> will be at least this large.

In addition to the ETC proposal for quark and lepton masses, Eichten and Lane observed that the size of the ETC representations required to generate all quark and lepton masses suggests that there will be more than one electroweak doublet of technifermions.<ref name=ETC/> If so, there will be more (spontaneously broken) chiral symmetries and therefore more [[Goldstone boson]]s than are eaten by the Higgs mechanism.  These must acquire mass by virtue of the fact that the extra chiral symmetries are also explicitly broken, by the standard-model interactions and the ETC interactions. These "pseudo-Goldstone bosons" are called technipions, {{math|π}}<sub>T</sub>. An application of Dashen's theorem<ref>{{cite journal |author=Roger Dashen |title=Chiral SU(3)⊗SU(3) as a Symmetry of the Strong Interactions |journal=Physical Review |volume=183 |date=1969 |issue=5 |pages=1245–1260 |doi=10.1103/PhysRev.183.1245 |bibcode=1969PhRv..183.1245D}}{{br}}{{cite journal |author=Roger Dashen |title=Some Features of Chiral Symmetry Breaking |journal=Physical Review |volume=D3 |date=1971 |issue=8 |pages=1879–1889 |doi=10.1103/PhysRevD.3.1879 |bibcode=1971PhRvD...3.1879D}}</ref> gives for the ETC contribution to their mass
: <math>(4)\qquad F_\text{EW}^2 M_{\pi T}^2 \approx \frac{g_\text{ETC}^2 \langle \bar{T}T \bar{T}T\rangle_\text{ETC}}{M_\text{ETC}^2} \approx \frac{16\pi^2 F _{EW}^6}{\Lambda_\text{ETC}^2}\,.</math>

The second approximation in Eq.&nbsp;(4) assumes that <math>\langle \bar{T}T \bar{T}T\rangle_{ETC} \approx \langle \bar{T}T\rangle^2_{ETC}</math>. For {{mvar|F}}<sub>EW</sub> ≈ {{math|Λ}}<sub>TC</sub> ≈ 246&nbsp;GeV and {{math|Λ}}<sub>ETC</sub> ≈ 15&nbsp;TeV, this contribution to {{mvar|M}}<sub>{{math|π}}T</sub> is about 50&nbsp;GeV. Since ETC interactions generate <math>m_{q,\ell}</math> ''and'' the coupling of technipions to quark and lepton pairs, one expects the couplings to be Higgs-like; i.e., roughly proportional to the masses of the quarks and leptons. This means that technipions are expected to predominately decay to the heaviest possible <math>\bar{q}q</math> and <math>\bar{\ell}\ell</math> pairs.

Perhaps the most important restriction on the ETC framework for quark mass generation is that ETC interactions are likely to induce [[flavor-changing neutral current]] processes such as  {{math|μ → e + γ}},  {{math|K<sub>L</sub> → μ + e}},  and <math>\left|\,\operatorname\Delta S \,\right| = 2 \text{  and  } \left|\,\operatorname\Delta B' \,\right| = 2  </math> interactions that induce <math>\text{K}^0 \leftrightarrow \bar{\text{K}}^0</math> and <math>\text{B}^0 \leftrightarrow \bar{\text{B}}^0</math> mixing.<ref name=ETC/> The reason is that the algebra of the ETC currents involved in <math>m_{q,\ell}</math> generation imply <math>\bar{q}q^\prime</math> and <math>\bar{\ell}\ell^\prime</math> ETC currents which, when written in terms of fermion mass eigenstates, have no reason to conserve flavor. The strongest constraint comes from requiring that ETC interactions mediating <math>\text{K} \leftrightarrow \bar{\text{K}}</math> mixing contribute less than the Standard Model. This implies an effective {{math|Λ}}<sub>ETC</sub> greater than 1000&nbsp;TeV. The actual {{math|Λ}}<sub>ETC</sub> may be reduced somewhat if CKM-like mixing angle factors are present. If these interactions are CP-violating, as they well may be, the constraint from the {{mvar|ε}}-parameter is that the effective {{math|Λ}}<sub>ETC</sub> > 10<sup>4</sup>&nbsp;TeV. Such huge ETC mass scales imply tiny quark and lepton masses and ETC contributions to {{mvar|M}}<sub>{{math|π}}T</sub> of at most a few GeV, in conflict with [[Large Electron–Positron Collider|LEP]] searches for {{math|π}}<sub>T</sub> at the {{math|Z}}<sup>0</sup>.{{clarify|date=May 2020|reason=The Z° what?}}

Extended technicolor is a very ambitious proposal, requiring that quark and lepton masses and mixing angles arise from experimentally accessible interactions. ''If'' there exists a successful model, it would not only predict the masses and mixings of quarks and leptons (and technipions), it would explain why there are three families of each: they are the ones that fit into the ETC representations of {{mvar|q}}, <math>\ell</math>, and {{mvar|T}}. It should not be surprising that the construction of a successful model has proven to be very difficult.

== Walking technicolor ==
Since quark and lepton masses are proportional to the bilinear technifermion [[fermion condensate|condensate]] divided by the ETC mass scale squared, their tiny values can be avoided if the condensate is enhanced above the weak-{{mvar|α}}<sub>TC</sub> estimate in Eq.&nbsp;(2), <math>\langle\bar{T}T\rangle_\text{ETC} \approx \langle\bar{T}T \rangle_\text{TC} \approx 4 \pi F _\text{EW}^3</math>.

During the 1980s, several dynamical mechanisms were advanced to do this. In 1981 Holdom suggested that, if the {{mvar|α}}<sub>TC</sub>({{mvar|μ}}) evolves to a nontrivial fixed point in the ultraviolet, with a large positive [[anomalous scaling dimension|anomalous dimension]] {{mvar|γ}}<sub>m</sub> for <math>\bar{T}T</math>, realistic quark and lepton masses could arise with {{math|Λ}}<sub>ETC</sub> large enough to suppress ETC-induced <math>K \leftrightarrow \bar K</math> mixing.<ref>{{cite journal |author=Holdom, Bob |year=1981 |title=Raising the sideways scale |journal=[[Physical Review D]] |volume=24 |issue=5 |pages=1441–1444 |doi=10.1103/PhysRevD.24.1441 |bibcode=1981PhRvD..24.1441H}}</ref> However, no example of a nontrivial [[ultraviolet fixed point]] in a four-dimensional gauge theory has been constructed. In 1985 Holdom analyzed a technicolor theory in which a "slowly varying" {{mvar|α}}<sub>TC</sub>({{mvar|μ}}) was envisioned.<ref>{{cite journal |author=Holdom, Bob |year=1985 |title=Techniodor |journal=[[Physics Letters B]] |volume=150 |issue=4 |pages=301–305 |doi=10.1016/0370-2693(85)91015-9 |bibcode=1985PhLB..150..301H}}</ref> His focus was to separate the chiral breaking and [[color confinement|confinement]] scales, but he also noted that such a theory could enhance <math>\langle \bar{T}T\rangle_\text{ETC}</math> and thus allow the ETC scale to be raised. In 1986 Akiba and Yanagida also considered enhancing quark and lepton masses, by simply assuming that {{mvar|α}}<sub>TC</sub> is constant and strong all the way up to the ETC scale.<ref>{{cite journal |author=Akiba, T. |author2=Yanagida, T. |name-list-style=amp |year=1986 |title=Hierarchic chiral condensate |journal=[[Physics Letters B]] |volume=169 |issue=4 |pages=432–435 |doi=10.1016/0370-2693(86)90385-0 |bibcode=1986PhLB..169..432A}}</ref> In the same year Yamawaki, Bando, and Matumoto again imagined an ultraviolet fixed point in a non-[[asymptotic freedom|asymptotically free]] theory to enhance the technifermion condensate.<ref>{{cite journal |author1=Yamawaki, Koichi |author2=Bando, Masako |author3=Matumoto, Ken-iti |name-list-style=amp |year=1986 |title=Scale-Invariant Hypercolor Model and a Dilaton |journal=Physical Review Letters |volume=56 |issue=13 |pmid=10032641 |pages=1335–1338 |doi=10.1103/PhysRevLett.56.1335 |bibcode=1986PhRvL..56.1335Y}}</ref>

In 1986 Appelquist, Karabali and Wijewardhana discussed the enhancement of fermion masses in an asymptotically free technicolor theory with a slowly running, or "walking", gauge coupling.<ref>{{cite journal |author1=Appelquist, Thomas |author2=Karabali, Dimitra |author3=Wijewardhana, L.C.R. |name-list-style=amp |year=1986 |title=Chiral hierarchies and flavor-changing neutral currents in hypercolor |journal=Physical Review Letters |volume=57 |issue=8 |pmid=10034209 |pages=957–960 |doi=10.1103/PhysRevLett.57.957 |bibcode=1986PhRvL..57..957A}}</ref> The slowness arose from the screening effect of a large number of technifermions, with the analysis carried out through two-loop perturbation theory. In 1987 Appelquist and Wijewardhana explored this walking scenario further.<ref>{{cite journal |author1=Appelquist, Thomas |author2=Wijewardhana, L.C.R. |name-list-style=amp |year=1987 |title=Chiral hierarchies from slowly running couplings in technicolor theories |journal=[[Physical Review D]] |volume=36 |issue=2 |pages=568–580 |doi=10.1103/PhysRevD.36.568 |pmid=9958201 |bibcode=1987PhRvD..36..568A}}</ref> They took the analysis to three loops, noted that the walking can lead to a power law enhancement of the technifermion condensate, and estimated the resultant quark, lepton, and technipion masses. The condensate enhancement arises because the associated technifermion mass decreases slowly, roughly linearly, as a function of its renormalization scale. This corresponds to the condensate anomalous dimension {{mvar|γ}}<sub>m</sub> in Eq.&nbsp;(3) approaching unity (see below).<ref name=CohenGeorgi>{{cite journal |author1=Cohen, Andrew |author2=Georgi, Howard |name-list-style=amp |year=1989 |title=Walking beyond the rainbow |journal=[[Nuclear Physics B]] |volume=314 |issue=1 |pages=7–24 |doi=10.1016/0550-3213(89)90109-0 |bibcode=1989NuPhB.314....7C}}</ref>

In the 1990s, the idea emerged more clearly that walking is naturally described by asymptotically free gauge theories dominated in the infrared by an approximate fixed point. Unlike the speculative proposal of ultraviolet fixed points, fixed points in the infrared are known to exist in asymptotically free theories, arising at two loops in the beta function providing that the fermion count {{mvar|N}}<sub>f</sub> is large enough. This has been known since the first two-loop computation in 1974 by Caswell.<ref>{{cite journal |author=Caswell, William E. |year=1974 |title=Asymptotic behavior of non-Abelian gauge theories to two-loop order |journal=Physical Review Letters |volume=33 |issue=4 |pages=244–246 |doi=10.1103/PhysRevLett.33.244 |bibcode=1974PhRvL..33..244C}}</ref> If {{mvar|N}}<sub>f</sub> is close to the value <math>\hat{N}_\text{f}</math> at which asymptotic freedom is lost, the resultant infrared fixed point is weak, of parametric order <math>\hat{N}_\text{f} - N_\text{f}</math>, and reliably accessible in perturbation theory. This weak-coupling limit was explored by Banks and Zaks in 1982.<ref name="Banks-Zaks">{{cite journal |author1=Banks, T. |author2=Zaks, A. |name-list-style=amp |year=1982 |title=On the phase structure of vector-like gauge theories with massless fermions |journal=Nuclear Physics B |volume=196 |issue=2 |pages=189–204 |doi=10.1016/0550-3213(82)90035-9 |bibcode=1982NuPhB.196..189B}}</ref>

The fixed-point coupling {{mvar|α}}<sub>IR</sub> becomes stronger as {{mvar|N}}<sub>f</sub> is reduced from <math>\hat{N}_\text{f}</math>. Below some critical value {{mvar|N}}<sub>fc</sub> the coupling becomes strong enough (> {{mvar|α}}<sub>{{math|χ}}&nbsp;SB</sub>) to break spontaneously the massless technifermions' [[chiral symmetry]]. Since the analysis must typically go beyond two-loop perturbation theory, the definition of the running coupling {{mvar|α}}<sub>TC</sub>({{mvar|μ}}), its fixed point value {{mvar|α}}<sub>IR</sub>, and the strength {{mvar|α}}<sub>{{math|χ}}&nbsp;SB</sub> necessary for chiral symmetry breaking depend on the particular renormalization scheme adopted. For <math>0 < \frac{\alpha_\text{IR} - \alpha_{\chi \text{SB}}}{\alpha_\text{IR}} \ll 1</math>; i.e., for {{mvar|N}}<sub>f</sub> just below {{mvar|N}}<sub>fc</sub>, the evolution of {{mvar|α}}<sub>TC</sub>(μ) is governed by the [[infrared fixed point]] and it will evolve slowly (walk) for a range of momenta above the breaking scale {{math|Λ}}<sub>TC</sub>. To overcome the <math>M_{ETC}^2</math>-suppression of the masses of first and second generation quarks involved in <math>K \leftrightarrow \bar{K}</math> mixing, this range must extend almost to their ETC scale, of <math>\mathcal{O}(10^3\hbox{ TeV})</math>. Cohen and Georgi argued that {{mvar|γ}}<sub>m</sub> = 1 is the signal of spontaneous chiral symmetry breaking, i.e., that {{mvar|γ}}<sub>m</sub>({{mvar|α}}<sub>{{math|χ}}&nbsp;SB</sub>) = 1.<ref name=CohenGeorgi/>  Therefore, in the walking-{{mvar|α}}<sub>TC</sub> region, {{mvar|γ}}<sub>m</sub> ≈ 1 and, from Eqs.&nbsp;(2) and (3), the light quark masses are enhanced approximately by <math>\frac{M_\text{ETC}}{\Lambda_\text{TC}}</math>.

The idea that {{mvar|α}}<sub>TC</sub>({{mvar|μ}}) walks for a large range of momenta when {{mvar|α}}<sub>IR</sub> lies just above {{mvar|α}}<sub>{{math|χ}}&nbsp;SB</sub> was suggested by Lane and Ramana.<ref name=LaneRamana>{{cite journal |author1=Lane, Kenneth |author2=Ramana, M.V. |name-list-style=amp |year=1991 |title=Walking technicolor signatures at hadron colliders |journal=[[Physical Review D]] |volume=44 |issue=9 |pages=2678–2700 |doi=10.1103/PhysRevD.44.2678 |pmid=10014158 |bibcode=1991PhRvD..44.2678L}}</ref> They made an explicit model, discussed the walking that ensued, and used it in their discussion of walking technicolor phenomenology at hadron colliders. This idea was developed in some detail by Appelquist, Terning, and Wijewardhana.<ref name=ATW>{{cite journal |author1=Appelquist, Thomas |author2=Terning, John |author3=Wijewardhana, L.C.R. |name-list-style=amp |year=1997 |title=Postmodern Technicolor |journal=Physical Review Letters |volume=79 |issue=15 |pages=2767–2770 |doi=10.1103/PhysRevLett.79.2767 |arxiv=hep-ph/9706238 |bibcode=1997PhRvL..79.2767A|s2cid=14292948 }}</ref> Combining a perturbative computation of the infrared fixed point with an approximation of {{mvar|α}}<sub>{{math|χ}}&nbsp;SB</sub> based on the [[Schwinger–Dyson equation]], they estimated the critical value {{mvar|N}}<sub>fc</sub> and explored the resultant [[electroweak interaction|electroweak]] physics. Since the 1990s, most discussions of walking technicolor are in the framework of theories assumed to be dominated in the infrared by an approximate fixed point. Various models have been explored, some with the technifermions in the [[fundamental representation]] of the gauge group and some employing higher representations.<ref name=LaneEichten>{{cite journal |author1=Lane, Kenneth |author2=Eichten, Estia |name-list-style=amp |year=1989 |title=Two-scale technicolor |journal=[[Physics Letters B]] |volume=222 |issue=2 |pages=274–280 |doi=10.1016/0370-2693(89)91265-3 |bibcode=1989PhLB..222..274L}}</ref><ref name="Sannino-Tuominen">{{cite journal |author1=Sannino, Francesco |author2=Tuominen, Kimmo |name-list-style=amp |year=2005 |title=Orientifold theory dynamics and symmetry breaking |journal=[[Physical Review D]] |volume=71 |issue=5 |page=051901 |doi=10.1103/PhysRevD.71.051901 |arxiv=hep-ph/0405209 |bibcode=2005PhRvD..71e1901S|s2cid=119388493 }}</ref><ref name="Sannino-lots">{{cite journal |author1=Dietrich, Dennis D. |author2=Sannino, Francesco |author3=Tuominen, Kimmo |name-list-style=amp |year=2005 |title=Light composite Higgs boson from higher representations versus electroweak precision measurements: Predictions for CERN LHC |journal=[[Physical Review D]] |volume=72 |issue=5 |page=055001 |arxiv=hep-ph/0505059 |doi=10.1103/PhysRevD.72.055001 |bibcode=2005PhRvD..72e5001D|s2cid=117871614 }}{{br}}{{cite journal |author1=Dietrich, Dennis D. |author2=Sannino, Francesco |author3=Tuominen, Kimmo |name-list-style=amp |year=2006 |title=Light composite Higgs and precision electroweak measurements on the Z resonance: An update |journal=[[Physical Review D]] |volume=73 |issue=3 |page=037701 |doi=10.1103/PhysRevD.73.037701 |arxiv=hep-ph/0510217 |bibcode =2006PhRvD..73c7701D|s2cid=119377085 }}</ref><ref>{{cite journal |author1=Dietrich, Dennis D. |author2=Sannino, Francesco |name-list-style=amp |year=2007 |title=Conformal window of SU(N) gauge theories with fermions in higher dimensional representations |journal=[[Physical Review D]] |volume=75 |issue=8 |page=085018 |arxiv=hep-ph/0611341 |doi=10.1103/PhysRevD.75.085018 |bibcode=2007PhRvD..75h5018D|s2cid=122605099 }}</ref><ref>{{cite journal |author1=Ryttov, Thomas A. |author2=Sannino, Francesco |name-list-style=amp |year=2007 |title=Conformal windows of SU(N) gauge theories, higher dimensional representations, and the size of the unparticle world |journal=[[Physical Review D]] |volume=76 |issue=10 |page=105004 |arxiv=0707.3166 |doi=10.1103/PhysRevD.76.105004 |bibcode=2007PhRvD..76j5004R|s2cid=119152612 }}</ref><ref>{{cite journal |author1=Thomas A. Ryttov |author2=Francesco Sannino |name-list-style=amp |title=Supersymmetry inspired QCD beta function |journal=[[Physical Review D]] |volume=78 |issue=6 |year=2008 |page=065001 |doi=10.1103/PhysRevD.78.065001 |arxiv=0711.3745 |bibcode=2008PhRvD..78f5001R|s2cid=17535403 }}</ref>

The possibility that the technicolor condensate can be enhanced beyond that discussed in the walking literature, has also been considered recently by Luty and Okui under the name "conformal technicolor".<ref>{{cite journal |author1=Luty, Markus A. |author2=Okui, Takemichi |name-list-style=amp |year=2006 |title=Conformal technicolor |journal=Journal of High Energy Physics |volume=0609 |issue=9 |page=070 |doi=10.1088/1126-6708/2006/09/070 |arxiv=hep-ph/0409274 |bibcode=2006JHEP...09..070L |s2cid=14173746 }}</ref><ref>{{cite journal |author=Luty, Markus A. |year=2009 |title=Strong conformal dynamics at the LHC and on the lattice |journal=Journal of High Energy Physics |volume=0904 |issue=4 |page=050 |doi=10.1088/1126-6708/2009/04/050 |arxiv=0806.1235 |bibcode=2009JHEP...04..050L|s2cid=9846381 }}</ref><ref>{{cite journal |author1=Evans, Jared A. |author2=Galloway, Jamison |author3=Luty, Markus A. |author4=Tacchi, Ruggero Altair |year=2010 |name-list-style=amp |title=Minimal conformal technicolor and precision electroweak tests |journal=Journal of High Energy Physics |volume=1010 |issue=10 |page=086 |doi=10.1007/JHEP10(2010)086 |arxiv=1001.1361 |bibcode=2010JHEP...10..086E|s2cid=118637173 }}</ref> They envision an infrared stable fixed point, but with a very large [[anomalous scaling dimension|anomalous dimension]] for the operator <math>\bar{T}T</math>. It remains to be seen whether this can be realized, for example, in the class of theories currently being examined using lattice techniques.

== Top quark mass ==
The enhancement described above for walking technicolor may not be sufficient to generate the measured [[top quark]] mass, even for an ETC scale as low as a few TeV. However, this problem could be addressed if the effective four-technifermion coupling resulting from ETC gauge boson exchange is strong and tuned just above a critical value.<ref>{{cite journal |author1=Appelquist, Thomas |author2=Takeuchi, T. |author3=Einhorn, Martin |author4=Wijewardhana, L.C.R. |name-list-style=amp |title=Higher mass scales and mass hierarchies |journal=Physics Letters |volume=B220 |date=1989 |issue=1–2 |pages=223–228 |doi=10.1016/0370-2693(89)90041-5 |bibcode=1989PhLB..220..223A |hdl=2027.42/28007 |url=https://deepblue.lib.umich.edu/bitstream/2027.42/28007/1/0000443.pdf|hdl-access=free }}</ref> The analysis of this strong-ETC possibility is that of a [[Nambu–Jona–Lasinio model]] with an additional (technicolor) gauge interaction. The technifermion masses are small compared to the ETC scale (the cutoff on the effective theory), but nearly constant out to this scale, leading to a large top quark mass.  No fully realistic ETC theory for all quark masses has yet been developed incorporating these ideas. A related study was carried out by Miransky and Yamawaki.<ref>{{cite journal |author1=Miransky, V.A. |author2=Yamawaki, K. |name-list-style=amp |title=On gauge theories with additional four-fermion interaction |journal=Modern Physics Letters A |volume=4 |date=1989 |issue=2 |pages=129–135 |doi=10.1142/S0217732389000186 |bibcode=1989MPLA....4..129M}}</ref> A problem with this approach is that it involves some degree of parameter [[Fine-tuning (physics)|fine-tuning]], in conflict with technicolor's guiding principle of naturalness.

A large body of closely related work in which the Higgs is a composite state, composed of top and anti-top quarks, is the [[top quark condensate]],<ref>{{cite conference |author=Nambu, Y. |title=BCS mechanism, quasi supersymmetry, and fermion masses |conference=XI International Symposium on Elementary Particle Physics |book-title=Proceedings of the Kazimierz 1988 Conference on New Theories in Physics |editor1=Adjduk, Z. |editor2=Pokorski, S. |editor3=Trautman, A. |date=1989 |pages=406–415}}{{br}}{{cite journal |author1=Miransky, V.A. |author2=Tanabashi, Masaharu |author3=Yamawaki, Koichi |name-list-style=amp |title=Is the t quark responsible for the mass of W and Z bosons? |journal=Modern Physics Letters A |volume=4 |date=1989 |issue=11 |pages=1043–1053 | doi=10.1142/S0217732389001210 |bibcode=1989MPLA....4.1043M}}{{br}}{{cite journal |author1=Miransky, V.A. |author2=Tanabashi, Masaharu |author3=Yamawaki, Koichi |name-list-style=amp |title=Dynamical electroweak symmetry breaking with large anomalous dimension and t quark condensate |journal=Physics Letters B |volume=221 |date=1989 |issue=2 |pages=177–183 |doi=10.1016/0370-2693(89)91494-9 |bibcode=1989PhLB..221..177M}}{{br}}{{cite journal |author1=Bardeen, William A. |author2=Hill, Christopher T. |author3=Lindner, Manfred |name-list-style=amp |title=Minimal dynamical symmetry breaking of the standard model |journal=Physical Review D |volume=41 |date=1990 |issue=5 |pages=1647–1660 |doi=10.1103/PhysRevD.41.1647 |bibcode=1990PhRvD..41.1647B |pmid=10012522}}</ref> [[topcolor]] and top-color-assisted technicolor models,<ref>{{cite journal |author=Hill, Christopher T. |title=Topcolor: top quark condensation in a gauge extension of the standard model |journal=Physics Letters B |volume=266 |date=1991 |issue=3–4 |pages=419–424 |doi=10.1016/0370-2693(91)91061-Y |bibcode=1991PhLB..266..419H|s2cid=121635635 }}{{br}}{{cite journal |author=Hill, Christopher T. |title=Topcolor assisted technicolor |journal=Physics Letters B |volume=345 |date=1995 |issue=4 |pages=483–489 |doi=10.1016/0370-2693(94)01660-5 |arxiv=hep-ph/9411426 |bibcode=1995PhLB..345..483H|s2cid=15093335 }}</ref> in which new strong interactions are ascribed to the top quark and other third-generation fermions.

== Technicolor on the lattice ==
[[Lattice gauge theory]] is a [[non-perturbative]] method applicable to strongly interacting technicolor theories, allowing first-principles exploration of walking and conformal dynamics. In 2007, Catterall and Sannino used lattice gauge theory to study ''SU''(2) gauge theories with two flavors of Dirac fermions in the symmetric representation,<ref>{{cite journal | author=Simon Catterall | author2=Francesco Sannino | name-list-style=amp | title=Minimal Walking on the Lattice | journal=Physical Review | volume=D76 | issue=3 | date=2007 | page=034504 | doi=10.1103/PhysRevD.76.034504 | arxiv= 0705.1664|bibcode = 2007PhRvD..76c4504C | s2cid=358936 }}</ref> finding evidence of conformality that has been confirmed by subsequent studies.<ref>{{cite journal | author=Simon Catterall | author2=Joel Giedt | author3=Francesco Sannino | author4=Joe Schneible | name-list-style=amp | title=Phase diagram of SU(2) with 2 flavors of dynamical adjoint quarks | journal=Journal of High Energy Physics | volume=0811 | issue=11 | date=2008 | page=009 | doi=10.1088/1126-6708/2008/11/009 | arxiv=0807.0792|bibcode = 2008JHEP...11..009C | s2cid=16246998 }}{{br}}{{cite journal | author=Ari J. Hietanen | author2=Kari Rummukainen | author3=Kimmo Tuominen | name-list-style=amp | title=Evolution of the coupling constant in SU(2) lattice gauge theory with two adjoint fermions | journal=Physical Review | volume=D80 | issue=9 | date=2009 | page=094504 | doi=10.1103/PhysRevD.80.094504 | arxiv=0904.0864|bibcode = 2009PhRvD..80i4504H | s2cid=119297303 }}</ref>

As of 2010, the situation for ''SU''(3) gauge theory with fermions in the fundamental representation is not as clear-cut. In 2007, Appelquist, Fleming, and Neil reported evidence that a non-trivial infrared fixed point develops in such theories when there are twelve flavors, but not when there are eight.<ref>{{cite journal | author=Thomas Appelquist | author2=George T. Fleming | author3=Ethan T. Neil | name-list-style=amp | title=Lattice Study of the Conformal Window in QCD-like Theories | journal=Physical Review Letters | volume=100 | date=2008 | issue=17 | pmid=18518277 | page=171607 | doi=10.1103/PhysRevLett.100.171607 | arxiv=0712.0609 | bibcode=2008PhRvL.100q1607A| s2cid=32180869 }}</ref>  While some subsequent studies confirmed these results, others reported different conclusions, depending on the lattice methods used, and there is not yet consensus.<ref>{{cite journal | author=Albert Deuzeman | author2=Maria Paola Lombardo | author3=Elisabetta Pallante | name-list-style=amp | title=The physics of eight flavours | journal=Physics Letters | volume=B670 | issue=1 | date=2008 | pages=41–48 | doi=10.1016/j.physletb.2008.10.039 | arxiv=0804.2905|bibcode = 2008PhLB..670...41D | s2cid=14791603 }}{{br}}{{cite journal | author=Thomas Appelquist | author2=George T. Fleming | author3=Ethan T. Neil | name-list-style=amp | title=Lattice study of conformal behavior in SU(3) Yang-Mills theories | journal=Physical Review | volume=D79 | issue=7 | date=2009 | page=076010 | doi=10.1103/PhysRevD.79.076010 | arxiv=0901.3766|bibcode = 2009PhRvD..79g6010A | s2cid=119190610 }}{{br}}{{cite journal | author=Erek Bilgici | display-authors=etal | title=New scheme for the running coupling constant in gauge theories using Wilson loops | journal=Physical Review | volume=D80 | issue=3 | date=2009 | page=034507 | doi=10.1103/PhysRevD.80.034507 | arxiv=0902.3768|bibcode = 2009PhRvD..80c4507B | s2cid=119306998 }}{{br}}{{cite journal | author=Xiao-Yong Jin | author2=Robert D. Mawhinney | name-list-style=amp | title=Lattice QCD with 8 and 12 degenerate quark flavors | journal=Proceedings of Science | volume=LAT2009 | date=2009 | page=049 | doi=10.22323/1.091.0049 | arxiv=0910.3216 | bibcode=2009slft.confE..49J | s2cid=115941621 | url=http://pos.sissa.it/archive/conferences/091/049/LAT2009_049.pdf | doi-access=free }}{{br}}{{cite journal | display-authors=4 | author=Zoltan Fodor | author2=Kieran Holland | author3=Julius Kuti| author4=Daniel Nogradi | author5=Chris Schroeder | name-list-style=amp | title=Chiral symmetry breaking in nearly conformal gauge theories | journal=Proceedings of Science | volume=LAT2009 | date=2009 | page=058 | url=http://pos.sissa.it/archive/conferences/091/058/LAT2009_058.pdf | bibcode=2009arXiv0911.2463F | arxiv=0911.2463}}{{br}}{{cite journal | author=Anna Hasenfratz |author-link= Anna Hasenfratz |title= Conformal or Walking? Monte Carlo renormalization group studies of SU(3) gauge models with fundamental fermions | journal=Physical Review | volume=D82 | issue=1 | date=2010 | page=014506 | doi=10.1103/PhysRevD.82.014506 | arxiv=1004.1004|bibcode = 2010PhRvD..82a4506H | s2cid= 118609076 }}</ref>

Further lattice studies exploring these issues, as well as considering the consequences of these theories for [[Peskin–Takeuchi parameter|precision electroweak measurements]], are underway by several research groups.<ref>{{cite journal | author=Thomas DeGrand | author2=Yigal Shamir | author3=Benjamin Svetitsky | name-list-style=amp | title=Phase structure of SU(3) gauge theory with two flavors of symmetric-representation fermions | journal=Physical Review | volume=D79 | issue=3 | date=2009 | page=034501 | doi=10.1103/PhysRevD.79.034501 | arxiv=0812.1427|bibcode = 2009PhRvD..79c4501D | s2cid=17730114 }}{{br}}{{cite journal | author=Thomas Appelquist | display-authors=etal | title=Toward TeV Conformality | journal=Physical Review Letters | volume=104 | issue=7 | year=2010 | page=071601 | pmid=20366870 | doi=10.1103/PhysRevLett.104.071601 | arxiv=0910.2224 | bibcode=2010PhRvL.104g1601A| s2cid=20474941 }}</ref>

== Technicolor phenomenology ==
Any framework for physics beyond the [[Standard Model]] must conform with precision measurements of the electroweak parameters. Its consequences for physics at existing and future high-energy hadron colliders, and for the dark matter of the universe must also be explored.

=== Precision electroweak tests ===
In 1990, the phenomenological [[Peskin–Takeuchi parameter|parameters {{mvar|S}}, {{mvar|T}}, and {{mvar|U}}]] were introduced by Peskin and Takeuchi to quantify contributions to electroweak radiative corrections from physics beyond the Standard Model.<ref name=PesTak>{{cite journal | author=Michael E. Peskin | author2=Tatsu Takeuchi | name-list-style=amp | title=New constraint on a strongly interacting Higgs sector | journal=Physical Review Letters | volume=65 | date=1990 | issue=8 | pmid=10043071 | pages=964–967 | doi=10.1103/PhysRevLett.65.964 | bibcode=1990PhRvL..65..964P| osti=1449235 }}{{br}}{{cite journal | author=Michael E. Peskin | author2=Tatsu Takeuchi | name-list-style=amp | title=Estimation of oblique electroweak corrections | journal=[[Physical Review D]] | volume=46 | date=1992 | issue=1 | pages=381–409 | doi=10.1103/PhysRevD.46.381| pmid=10014770 |bibcode = 1992PhRvD..46..381P | citeseerx=10.1.1.382.2460 }}</ref> They have a simple relation to the parameters of the electroweak chiral Lagrangian.<ref>{{cite journal | author=Thomas Appelquist | author2=Claude Bernard | name-list-style=amp | title=Strongly interacting Higgs bosons | journal=Physical Review | volume=D22 | date=1980 | issue=1 | pages=200–213 | doi=10.1103/PhysRevD.22.200|bibcode = 1980PhRvD..22..200A }}</ref><ref>{{cite journal | author=Anthony C. Longhitano | title=Heavy Higgs bosons in the Weinberg-Salam model | journal=[[Physical Review D]] | volume=22 | date=1980 | issue=5 | pages=1166–1175 | doi=10.1103/PhysRevD.22.1166|bibcode = 1980PhRvD..22.1166L }}{{br}}{{cite journal | author=Anthony C. Longhitano | title=Low-energy impact of a heavy Higgs boson sector | journal=Nuclear Physics B | volume=188 | date=1981 | issue=1 | pages=118–154 | doi=10.1016/0550-3213(81)90109-7|bibcode = 1981NuPhB.188..118L }}</ref>  The Peskin–Takeuchi analysis was based on the general formalism for weak radiative corrections developed by Kennedy, Lynn, Peskin and Stuart,<ref>{{cite conference | author=B. W. Lynn | author2=Michael Edward Peskin | author3=R. G. Stuart | name-list-style=amp | title=Radiative Corrections in SU(2) × U(1): LEP / SLC |conference=Second Conference on Tests of Electroweak Theories |place=Trieste, Italy |orig-year=10–12 June 1985 | book-title=Tests of electroweak theories: polarized processes and other phenomena | editor=Bryan W. Lynn | editor2=Claudio Verzegnassi | date=1985 | page=213}}{{br}}{{cite journal | author=D. C. Kennedy | author2=B. W. Lynn | name-list-style=amp | title=Electroweak radiative corrections with an effective lagrangian: Four-fermions processes | journal=Nuclear Physics B | volume=322 | date=1989 | issue=1 | pages=1–54 | doi=10.1016/0550-3213(89)90483-5 |bibcode = 1989NuPhB.322....1K | doi-access=free }}</ref> and alternate formulations also exist.<ref name=GoldenRandall>{{cite journal | author=Mitchell Golden | author2=Lisa Randall | name-list-style=amp | title=Radiative corrections to electroweak parameters in technicolor theories | journal=Nuclear Physics B | volume=361 | date=1991 | issue=1 | pages=3–23 | doi=10.1016/0550-3213(91)90614-4 |bibcode = 1991NuPhB.361....3G | osti=1879518 }}{{br}}{{cite journal | author=B. Holdom | author2=J. Terning | name-list-style=amp | title=Large corrections to electroweak parameters in technicolor theories | journal=Physics Letters B | volume=247 | date=1990 | issue=1 | pages=88–92 | doi=10.1016/0370-2693(90)91054-F|bibcode = 1990PhLB..247...88H }}{{br}}{{cite journal | author=G. Altarelli | author2=R. Barbieri | author3=S. Jadach | name-list-style=amp | title=Toward a model-independent analysis of electroweak data | journal=Nuclear Physics B | volume=369 | date=1992 | issue=1–2 | pages=3–32 | doi=10.1016/0550-3213(92)90376-M |bibcode = 1992NuPhB.369....3A | url=https://cds.cern.ch/record/222003 }}</ref>

The {{mvar|S}}, {{mvar|T}}, and {{mvar|U}}-parameters describe corrections to the electroweak gauge boson propagators from [[physics beyond the Standard Model]]. They can be written in terms of polarization functions of electroweak currents and their spectral representation as follows:
: <math>\begin{align}
(5)\qquad S &= 16\pi \frac{d}{d q^2} \left[\Pi_{33}^{\mathbf{new}} (q^2) - \Pi_{3Q}^{\mathbf{new}}(q^2)\right]_{q^2=0}\\
&= 4\pi \int\frac{dm^2}{m^4}\left[\sigma^3_V(m^2) - \sigma^3_A(m^2)\right]^{\mathbf{new}};\\
\\
(6)\qquad T &= \frac{16\pi}{M^2_Z \sin^2 2\theta_W}\; \left[\Pi_{11}^{\mathbf{new}}(0) - \Pi_{33}^{\mathbf{new}}(0) \right]\\
&= \frac{4\pi}{M^2_Z \sin^2 2\theta_W}\int_0^\infty\frac{dm^2}{m^2}\left[\sigma_V^1(m^2) + \sigma_A^1(m^2) - \sigma_V^3(m^2) - \sigma_A^3(m^2)\right]^{\mathbf{new}},\end{align} </math>
where only new, beyond-standard-model physics is included. The quantities are calculated relative to a minimal Standard Model with some chosen reference mass of the [[Higgs boson]], taken to range from the experimental lower bound of 117&nbsp;GeV to 1000&nbsp;GeV where its width becomes very large.<ref name=RPP>{{cite journal | author=Particle Data Group (C. Amsler ''et al.'') | title=Review of Particle Physics | journal=Physics Letters B | volume=667 | issue=1–5 | date=2008 | page=1 | doi=10.1016/j.physletb.2008.07.018 | url=http://pdg.lbl.gov | bibcode=2008PhLB..667....1A| hdl=1854/LU-685594 | s2cid=227119789 | hdl-access=free }}</ref> For these parameters to describe the dominant corrections to the Standard Model, the mass scale of the new physics must be much greater than {{mvar|M}}<sub>{{math|W}}</sub> and {{mvar|M}}<sub>{{math|Z}}</sub>, and the coupling of [[quark]]s and [[lepton]]s to the new particles must be suppressed relative to their coupling to the gauge bosons. This is the case with technicolor, so long as the lightest technivector mesons, {{math|ρ}}<sub>T</sub> and {{math|a}}<sub>T</sub>, are heavier than 200–300&nbsp;GeV. The {{mvar|S}}-parameter is sensitive to all new physics at the TeV scale, while {{mvar|T}} is a measure of weak-isospin breaking effects. The {{mvar|U}}-parameter is generally not useful; most new-physics theories, including technicolor theories, give negligible contributions to it.

The {{mvar|S}} and {{mvar|T}}-parameters are determined by global fit to experimental data including ''Z''-pole data from [[Large Electron–Positron Collider|LEP]] at [[CERN]], top quark and {{math|W}}-mass measurements at Fermilab, and measured levels of atomic parity violation. The resultant bounds on these parameters are given in the Review of Particle Properties.<ref name=RPP /> Assuming {{mvar|U}} = 0, the {{mvar|S}} and {{mvar|T}} parameters are small and, in fact, consistent with zero:
: <math>(7)\qquad\begin{align}
S &= -0.04 \pm 0.09\,(-0.07), \\
T &= 0.02 \pm 0.09\,( +0.09), 
\end{align}</math>
where the central value corresponds to a Higgs mass of 117&nbsp;GeV and the correction to the central value when the Higgs mass is increased to 300&nbsp;GeV is given in parentheses. These values place tight restrictions on beyond-standard-model theories – when the relevant corrections can be reliably computed.

The {{mvar|S}} parameter estimated in [[quantum chromodynamics|QCD]]-like technicolor theories is significantly greater than the experimentally allowed value.<ref name=PesTak/><ref name=GoldenRandall/> The computation was done assuming that the spectral integral for {{mvar|S}} is dominated by the lightest {{math|ρ}}<sub>T</sub> and {{math|a}}<sub>T</sub> resonances, or by scaling effective Lagrangian parameters from QCD. In walking technicolor, however, the physics at the TeV scale and beyond must be quite different from that of QCD-like theories. In particular, the vector and axial-vector spectral functions cannot be dominated by just the lowest-lying resonances.<ref>{{cite conference |author=Kenneth Lane |title=An introduction to technicolor |conference=Theoretical Advanced Study Institute (TASI 93) in Elementary Particle Physics: The Building Blocks of Creation - From Microfermis to Megaparsecs |place=Boulder, Colorado |orig-year=6 June - 2 July 1993 |book-title=Boulder 1993 Proceedings: The building blocks of creation |editor=K. T. Mahantappa |date=1994 |pages=381–408 |arxiv=hep-ph/9401324 |bibcode=1994bbc..conf..381L |doi=10.1142/9789814503785_0010 }}</ref><ref>{{cite conference |author=Kenneth Lane |title=Technicolor and precision tests of the electroweak interactions |conference=27th International Conference on High Energy Physics (ICHEP) |place=Glasgow, Scotland |orig-year=20–27 July 1994 |book-title=High energy physics: Proceedings |editor=P. J. Bussey |editor2=I. G. Knowles |date=1995 |volume=II |page=543 |arxiv=hep-ph/9409304 |bibcode=1995hep..conf..543L}}</ref> It is unknown whether higher energy contributions to <math>\sigma_\text{V,A}^3</math> are a tower of identifiable {{math|ρ}}<sub>T</sub> and {{math|a}}<sub>T</sub> states or a smooth continuum. It has been conjectured that {{math|ρ}}<sub>T</sub> and {{math|a}}<sub>T</sub> partners could be more nearly degenerate in walking theories (approximate parity doubling), reducing their contribution to {{mvar|S}}.<ref>{{cite journal | author=Thomas Appelquist | author2=Francesco Sannino | name-list-style=amp | title=Physical spectrum of conformal SU(N) gauge theories | journal=[[Physical Review D]] | volume=59 | issue=6 | date=1999 | page=067702 | doi=10.1103/PhysRevD.59.067702 | arxiv=hep-ph/9806409|bibcode = 1999PhRvD..59f7702A | s2cid=14365571 }}{{br}}{{cite journal | author=Johannes Hirn | author2=Verónica Sanz | name-list-style=amp | title=Negative {{mvar|S}} Parameter from Holographic Technicolor | journal=Physical Review Letters | volume=97 | date=2006 | issue=12 | pmid=17025952 | page=121803 | doi=10.1103/PhysRevLett.97.121803 | arxiv=hep-ph/0606086 | bibcode=2006PhRvL..97l1803H| s2cid=25483021 }}{{br}}{{cite journal | display-authors=4 | author=R. Casalbuoni | author2=D. Dominici | author3=A. Deandrea| author4=R. Gatto | author5=S. De Curtis | author6=M. Grazzini | name-list-style=amp | title=Low energy strong electroweak sector with decoupling | journal=[[Physical Review D]] | volume=53 | issue=9 | date=1996 | pages=5201–5221 | doi=10.1103/PhysRevD.53.5201 | pmid=10020517 | arxiv=hep-ph/9510431|bibcode = 1996PhRvD..53.5201C | s2cid=16253919 }}</ref>  [[Lattice gauge theory|Lattice]] calculations are underway or planned to test these ideas and obtain reliable estimates of {{mvar|S}} in walking theories.<ref name=lattice /><ref>{{cite web |url=http://www.yale.edu/LSD |title=Lattice Strong Dynamics Collaboration |publisher=Yale University}}</ref>

The restriction on the {{mvar|T}}-parameter poses a problem for the generation of the top-quark mass in the ETC framework. The enhancement from walking can allow the associated ETC scale to be as large as a few TeV,<ref name=ATW /> but – since the ETC interactions must be strongly weak-isospin breaking to allow for the large top-bottom mass splitting – the contribution to the {{mvar|T}} parameter,<ref>{{cite journal | author=Thomas Appelquist | author2=Mark J. Bowick | author3=Eugene Cohler | author4=Avi I. Hauser | name-list-style=amp | title=Breaking of isospin symmetry in theories with a dynamical Higgs mechanism | journal=[[Physical Review D]] | volume=31 | date=1985 | issue=7 | pages=1676–1684 | doi=10.1103/PhysRevD.31.1676| pmid=9955884 |bibcode = 1985PhRvD..31.1676A }}{{br}}{{cite journal | author=R. S. Chivukula | author2=B. A. Dobrescu | author3=J. Terning | name-list-style=amp | title=Isospin breaking and fine-tuning in top-color assisted technicolor | journal=Physics Letters B | volume=353 | issue=2–3 | date=1995 | pages=289–294 | doi=10.1016/0370-2693(95)00569-7 | arxiv=hep-ph/9503203 |bibcode = 1995PhLB..353..289C | s2cid=119385932 }}</ref> as well as the rate for the decay <math>\mathrm{Z^0 \rightarrow \bar{b}b}</math>,<ref>{{cite journal | author=R. Sekhar Chivukula | author2=Stephen B. Selipsky | author3=Elizabeth H. Simmons | name-list-style=amp | title=Nonoblique effects in the Zb{{overline|b}} vertex from extended technicolor dynamics | journal=Physical Review Letters | volume=69 | date=1992 | issue=4 | pmid=10046976 | pages=575–577 | doi=10.1103/PhysRevLett.69.575 | arxiv=hep-ph/9204214 | bibcode=1992PhRvL..69..575C| s2cid=44375068 }}{{br}}{{cite journal | author=Elizabeth H. Simmons | author2=R.S. Chivukula | author3=J. Terning | name-list-style=amp | title=Testing extended technicolor with R(b) | journal=Progress of Theoretical Physics Supplement | volume=123 | date=1996 | pages=87–96 | doi=10.1143/PTPS.123.87 | arxiv=hep-ph/9509392|bibcode = 1996PThPS.123...87S | s2cid=14420340 }}</ref> could be too large.

=== Hadron collider phenomenology ===
Early studies generally assumed the existence of just one [[electroweak interaction|electroweak]] doublet of technifermions, or of one techni-family including one doublet each of color-triplet techniquarks and color-singlet technileptons (four electroweak doublets in total).<ref>{{cite journal | author=E. Eichten | author2=I. Hinchliffe | author3=K. Lane | author4=C. Quigg | name-list-style=amp | title=Supercollider physics | journal=Reviews of Modern Physics | volume=56 | date=1984 | issue=4 | pages=579–707 | doi=10.1103/RevModPhys.56.579 | bibcode=1984RvMP...56..579E}}</ref><ref>{{cite journal | author=E. Eichten | author2=I. Hinchliffe | author3=K. Lane | author4=C. Quigg | name-list-style=amp | title=Erratum: Supercollider physics | journal=Reviews of Modern Physics | volume=58 | date=1986 | issue=4 | pages=1065–1073 | doi=10.1103/RevModPhys.58.1065 | bibcode=1986RvMP...58.1065E| doi-access=free }}</ref>  The number {{mvar|N}}<sub>D</sub> of electroweak doublets determines the decay constant {{mvar|F}} needed to produce the correct electroweak scale, as {{mvar|F}} = {{frac|{{mvar|F}}<sub>EW</sub>|{{radical|{{mvar|N}}<sub>D</sub>&nbsp;}} }} = {{frac|246&nbsp;GeV|{{radical|{{mvar|N}}<sub>D</sub>&nbsp;}} }}. In the minimal, one-doublet model, three [[Goldstone boson]]s (technipions, {{math|π}}<sub>T</sub>) have decay constant {{mvar|F}} = {{mvar|F}}<sub>EW</sub> = 246&nbsp;GeV and are eaten by the electroweak gauge bosons. The most accessible collider signal is the production through <math>\bar{q}q</math> annihilation in a hadron collider of spin-one <math>\mathrm \rho_\text{T}^{\pm,0}</math>, and their subsequent decay into a pair of longitudinally polarized weak bosons, <math>\mathrm W_\text{LP}^\pm \mathrm Z_\text{LP}^0</math> and <math>\mathrm W_\text{LP}^+ \mathrm W_\text{LP}^-</math>. At an expected mass of 1.5–2.0&nbsp;TeV and width of 300–400&nbsp;GeV, such {{math|ρ}}<sub>T</sub>'s would be difficult to discover at the LHC. A one-family model has a large number of physical technipions, with {{mvar|F}} = {{frac|{{mvar|F}}<sub>EW</sub>|{{radical|4 }} }}= 123&nbsp;GeV.<ref>{{cite journal | author=E. Farhi | author2=L. Susskind | name-list-style=amp | title=Grand unified theory with heavy color | journal=[[Physical Review D]] | volume=20 | date=1979 | issue=12 | pages=3404–3411 | doi=10.1103/PhysRevD.20.3404 |bibcode = 1979PhRvD..20.3404F }}</ref> There is a collection of correspondingly lower-mass color-singlet and octet technivectors decaying into technipion pairs. The {{math|π}}<sub>T</sub>'s are expected to decay to the heaviest possible quark and lepton pairs. Despite their lower masses, the {{math|ρ}}<sub>T</sub>'s are wider than in the minimal model and the backgrounds to the {{math|π}}<sub>T</sub> decays are likely to be insurmountable at a hadron collider.

This picture changed with the advent of walking technicolor. A walking gauge coupling occurs if {{mvar|α}}<sub>{{math|χ}}&nbsp;SB</sub> lies just below the IR fixed point value {{mvar|α}}<sub>IR</sub>, which requires either a large number of electroweak doublets in the [[fundamental representation]] of the gauge group, e.g., or a few doublets in higher-dimensional TC representations.<ref name=LaneEichten /><ref>{{cite journal | author=Dennis D. Dietrich | author2=Francesco Sannino | author3=Kimmo Tuominen | name-list-style=amp  | title=Light composite Higgs boson from higher representations versus electroweak precision measurements: Predictions for CERN LHC | journal=[[Physical Review D]] | volume=72 | issue=5 | date=2005 | page=055001 | doi=10.1103/PhysRevD.72.055001 | arxiv=hep-ph/0505059|bibcode = 2005PhRvD..72e5001D | s2cid=117871614 }}</ref> In the latter case, the constraints on ETC representations generally imply other technifermions in the fundamental representation as well.<ref name=ETC /><ref name=LaneRamana /> In either case, there are technipions {{math|π}}<sub>T</sub> with decay constant <math>F \ll F_{EW}</math>. This implies <math>\Lambda_{TC} \ll F_{EW}</math> so that the lightest technivectors accessible at the LHC – {{math|ρ}}<sub>T</sub>, {{math|ω}}<sub>T</sub>, {{math|a}}<sub>T</sub> (with {{mvar|I}}<sup>{{mvar|G}}</sup> {{mvar|J}}<sup>{{mvar|P C}}</sup> = 1<sup>+</sup> 1<sup>−−</sup>, 0<sup>−</sup> 1<sup>−−</sup>, 1<sup>−</sup> 1<sup>++</sup>) – have masses well below a TeV. The class of theories with many technifermions and thus <math>F \ll F_{EW}</math> is called low-scale technicolor.<ref>{{cite journal | author=Kenneth Lane | author2=Estia Eichten | name-list-style=amp | title=Natural topcolor-assisted technicolor | journal=Physics Letters B | volume=352 | issue=3–4 | date=1995 | pages=382–387 | doi=10.1016/0370-2693(95)00482-Z | arxiv=hep-ph/9503433|bibcode = 1995PhLB..352..382L | s2cid=15753846 }}{{br}}{{cite journal | author=Estia Eichten | author2=Kenneth Lane | name-list-style=amp | title=Low-scale technicolor at the Tevatron | journal=Physics Letters B | volume=388 | issue=4 | date=1996 | pages=803–807 | doi=10.1016/S0370-2693(96)01211-7 | arxiv=hep-ph/9607213 |bibcode = 1996PhLB..388..803E | s2cid=277661 }}{{br}}{{cite journal | author=Estia Eichten | author2=Kenneth Lane | author3=John Womersley | name-list-style=amp | title=Finding low-scale technicolor at hadron colliders | journal=Physics Letters B | volume=405 | issue=3–4 | date=1997 | pages=305–311 | doi=10.1016/S0370-2693(97)00637-0 | arxiv=hep-ph/9704455 |bibcode = 1997PhLB..405..305E | s2cid=8600506 }}</ref>

A second consequence of walking technicolor concerns the decays of the spin-one technihadrons. Since technipion masses <math>M_{\pi_T}^2 \propto \langle\bar{T}T \bar{T}T\rangle_{M_{ETC}}</math> (see Eq. (4)), walking enhances them much more than it does other technihadron masses. Thus, it is very likely that the lightest {{mvar|M}}<sub>{{math|ρ}}<sub>T</sub></sub> &lt; 2{{mvar|M}}<sub>{{math|π}}<sub>T</sub></sub> and that the two and three-{{math|π}}<sub>T</sub> decay channels of the light technivectors are closed.<ref name=LaneEichten /> This further implies that these technivectors are very narrow. Their most probable two-body channels are <math>\mathrm W^{\pm,0}_\mathrm L \mathrm \pi_T</math>, {{math|W}}<sub>L</sub> {{math|W}}<sub>L</sub>, {{math|γ}} {{math|π}}<sub>T</sub> and {{math|γ}} {{math|W}}<sub>L</sub>. The coupling of the lightest technivectors to {{math|W}}<sub>L</sub> is proportional to {{frac|{{mvar|F}}|{{mvar|F}}<sub>EW</sub>}}.<ref name=coupling>{{cite journal | author=Kenneth Lane | title=Technihadron production and decay in low-scale technicolor | journal=[[Physical Review D]] | volume=60 | issue=7 | date=1999 | page=075007 | doi=10.1103/PhysRevD.60.075007 | arxiv=hep-ph/9903369|bibcode = 1999PhRvD..60g5007L | s2cid=2772521 }}{{br}}{{cite journal | author=Estia Eichten | author2=Kenneth Lane | name-list-style=amp | title=Low-scale technicolor at the Tevatron and LHC  | journal=Physics Letters | volume=B669 | issue=3–4 | date=2008 | pages=235–238 | doi=10.1016/j.physletb.2008.09.047 | arxiv=0706.2339|bibcode = 2008PhLB..669..235E | s2cid=14102461 }}</ref> Thus, all their decay rates are suppressed by powers of <math>\left[\frac{F}{F_{EW}}\right]^2 \ll 1</math> or the fine-structure constant, giving total widths of a few GeV (for {{math|ρ}}<sub>T</sub>) to a few tenths of a GeV (for {{math|ω}}<sub>T</sub> and <sub>T</sub>).

A more speculative consequence of walking technicolor is motivated by consideration of its contribution to the {{mvar|S}}-parameter. As noted above, the usual assumptions made to estimate {{mvar|S}}<sub>TC</sub> are invalid in a walking theory. In particular, the spectral integrals used to evaluate {{mvar|S}}<sub>TC</sub> cannot be dominated by just the lowest-lying {{math|ρ}}<sub>T</sub> and {{math|a}}<sub>T</sub> and, if {{mvar|S}}<sub>TC</sub> is to be small, the masses and weak-current couplings of the {{math|ρ}}<sub>T</sub> and {{math|a}}<sub>T</sub> could be more nearly equal than they are in QCD.

Low-scale technicolor phenomenology, including the possibility of a more parity-doubled spectrum, has been developed into a set of rules and decay amplitudes.<ref name=coupling /> An April 2011 announcement of an excess in jet pairs produced in association with a {{math|W}} boson measured at the [[Tevatron]]<ref>{{cite journal |author=C.D.F. Collaboration (T. Aaltonen ''et al.'') |arxiv=1104.0699 |title=Invariant Mass Distribution of Jet Pairs Produced in Association with a W boson in ppbar Collisions at sqrt(s) = 1.96 TeV |date=2011 |doi=10.1103/PhysRevLett.106.171801 |pmid=21635027 |volume=106 |issue=17 |pages=171801 |journal=Physical Review Letters |bibcode=2011PhRvL.106q1801A|s2cid=38531871 }}</ref> has been interpreted by Eichten, Lane and Martin as a possible signal of the technipion of low-scale technicolor.<ref>{{cite journal |author=Estia J. Eichten |author2=Kenneth Lane |author3=Adam Martin |name-list-style=amp |arxiv=1104.0976 |title=Technicolor at the Tevatron |date=2011 |doi=10.1103/PhysRevLett.106.251803 |pmid=21770631 |volume=106 |issue=25 |pages=251803 |journal=Physical Review Letters |bibcode=2011PhRvL.106y1803E|s2cid=119193886 }}</ref>

The general scheme of low-scale technicolor makes little sense if the limit on <math>M_{\rho_{T}}</math> is pushed past about 700&nbsp;GeV. The LHC should be able to discover it or rule it out. Searches there involving decays to technipions and thence to heavy quark jets are hampered by backgrounds from <math>\bar{t}t</math> production; its rate is 100&nbsp;times larger than that at the Tevatron. Consequently, the discovery of low-scale technicolor at the LHC relies on all-leptonic final-state channels with favorable signal-to-background ratios: <math>\rho_{T}^{\pm} \rightarrow W_L^\pm Z_L^0</math>, <math>a_{T}^{\pm} \rightarrow \gamma W_L^\pm</math> and <math>\omega_{T} \rightarrow \gamma Z_L^0</math>.<ref>{{cite conference | author=Gustaaf H. Brooijmans | author2=New Physics Working Group | title=New Physics at the LHC: A Les Houches Report | conference= 5th Les Houches Workshop on Physics at TeV Colliders |orig-year=11–29 June 2007 |place=Les Houches, France | book-title=Les Houches 2007: Physics at TeV Colliders | date=2008 | pages=363–489 | arxiv=0802.3715| bibcode=2008arXiv0802.3715B }}</ref>

=== Dark matter ===
Technicolor theories naturally contain [[dark matter]] candidates. Almost certainly, models can be built in which the lowest-lying technibaryon, a technicolor-singlet bound state of technifermions, is stable enough to survive the evolution of the universe.<ref name=RPP /><ref>{{cite journal | author=S. Nussinov | title=Technocosmology – could a technibaryon excess provide a "natural" missing mass candidate? | journal=Physics Letters | volume=B165 | date=1985 | issue=1–3 | pages=55–58 | doi=10.1016/0370-2693(85)90689-6 |bibcode = 1985PhLB..165...55N }}</ref><ref>{{cite journal | author=R. S. Chivukula | author2=Terry P. Walker | name-list-style=amp | title=Technicolor cosmology | journal=Nuclear Physics B | volume=329 | date=1990 | issue=2 | pages=445–463 | doi=10.1016/0550-3213(90)90151-3 |bibcode = 1990NuPhB.329..445C | doi-access=free }}</ref><ref>{{cite journal | author=John Bagnasco | author2=Michael Dine | author3=Scott Thomas | name-list-style=amp | title=Detecting technibaryon dark matter | journal=Physics Letters B | volume=320 | issue=1–2 | date=1994 | pages=99–104 | doi=10.1016/0370-2693(94)90830-3 | arxiv=hep-ph/9310290|bibcode = 1994PhLB..320...99B | s2cid=569339 }}</ref><ref>{{cite journal | author=Sven Bjarke Gudnason | author2=Chris Kouvaris | author3=Francesco Sannino | name-list-style=amp | title=Dark matter from new technicolor theories | journal=[[Physical Review D]] | volume=74 | issue=9 | date=2006 | page=095008 | doi=10.1103/PhysRevD.74.095008 | arxiv=hep-ph/0608055|bibcode = 2006PhRvD..74i5008G | s2cid=119021709 }}</ref> If the technicolor theory is low-scale (<math>F \ll F_{EW}</math>), the baryon's mass should be no more than 1–2&nbsp;TeV. If not, it could be much heavier. The technibaryon must be electrically neutral and satisfy constraints on its abundance. Given the limits on spin-independent dark-matter-nucleon cross sections from dark-matter search experiments (<math>\lesssim 10^{-42}\,\mathrm{cm}^2</math> for the masses of interest<ref>{{cite web |first=D. |last=McKinsey |url=http://www.sns.ias.edu/~tomerv/Workshop/McKinsey.pdf |title=Direct dark matter detection using noble liquids |year=2009 |publisher=Institute for Advanced Study }}{{Dead link|date=October 2023 |bot=InternetArchiveBot |fix-attempted=yes }} alternate: {{cite web |url=https://www.sns.ias.edu/twiki/bin/view/Main/DarkMatterWorkshop |title=Workshop on Current Trends in Dark Matter |archive-url=https://web.archive.org/web/20110615162218/https://www.sns.ias.edu/twiki/bin/view/Main/DarkMatterWorkshop |archive-date=2011-06-15 |df=dmy-all }}</ref>), it may have to be electroweak neutral (weak isospin {{mvar|T}}{{sub|3}}&nbsp;=&nbsp;0) as well. These considerations suggest that the "old" technicolor dark matter candidates may be difficult to produce at the LHC.

A different class of technicolor dark matter candidates light enough to be accessible at the LHC was introduced by [[Francesco Sannino]] and his collaborators.<ref>{{cite journal | author=Sven Bjarke Gudnason | author2=Chris Kouvaris | author3=Francesco Sannino | name-list-style=amp | title=Towards working technicolor: Effective theories and dark matter | journal=[[Physical Review D]] | volume=73 | issue=11 | date=2006 | page=115003 | doi=10.1103/PhysRevD.73.115003 | arxiv=hep-ph/0603014|bibcode = 2006PhRvD..73k5003G | s2cid=119333119 }}</ref><ref>{{cite journal | author=Sven Bjarke Gudnason | author2=Chris Kouvaris | author3=Francesco Sannino | name-list-style=amp | title=Dark matter from new technicolor theories | journal=[[Physical Review D]] | volume=74 | issue=9 | date=2006 | page=095008 | doi=10.1103/PhysRevD.74.095008 | arxiv=hep-ph/0608055|bibcode = 2006PhRvD..74i5008G | s2cid=119021709 }}</ref><ref>{{cite journal | author=Thomas A. Ryttov | author2=Francesco Sannino | name-list-style=amp | title=Ultraminimal technicolor and its dark matter technicolor interacting massive particles | journal=[[Physical Review D]] | volume=78 | issue=11 | date=2008 | page=115010 | doi=10.1103/PhysRevD.78.115010 | arxiv=0809.0713|bibcode = 2008PhRvD..78k5010R | s2cid=118853550 }}</ref><ref>{{cite journal | author=Enrico Nardi | author2=Francesco Sannino | author3=Alessandro Strumia | name-list-style=amp | title=Decaying Dark Matter can explain the e<sup>±</sup> excesses | journal=Journal of Cosmology and Astroparticle Physics | volume=0901 | issue=1 | date=2009 | page=043 | doi=10.1088/1475-7516/2009/01/043 | arxiv=0811.4153 |bibcode = 2009JCAP...01..043N | s2cid=15711899 }}</ref><ref>{{cite journal | author= Roshan Foadi | author2= Mads T. Frandsen | author3= Francesco Sannino | name-list-style= amp | title=Technicolor dark matter | journal=[[Physical Review D]] | volume=80 | issue= 3 | date=2009 | page=037702 | doi=10.1103/PhysRevD.80.037702 | arxiv=0812.3406 |bibcode = 2009PhRvD..80c7702F | s2cid= 119111212 }}</ref><ref>{{cite journal | author=Mads T. Frandsen | author2=Francesco Sannino | name-list-style=amp | title= Isotriplet technicolor interacting massive particle as dark matter | journal=[[Physical Review D]] | volume=81 | issue=9 | date=2010 | page=097704 | doi=10.1103/PhysRevD.81.097704 | arxiv=0911.1570|bibcode = 2010PhRvD..81i7704F | s2cid=118661650 }}</ref> These states are pseudo Goldstone bosons possessing a global charge that makes them stable against decay.

== See also ==
* [[Higgsless model]]
* [[Topcolor]]
* [[Top quark condensate]]
* [[Infrared fixed point]]

== References ==
{{reflist|25em}}

{{Standard model of physics}}

[[Category:Mass]]
[[Category:Electroweak theory]]
[[Category:Physics beyond the Standard Model]]