Some Progress on the Construction of Technicolor andExtended Technicolor Models
Robert Shrock
C. N. Yang Institute for Theoretical Physics, Stony Brook University

SCGT12 Mini-Workshop, KMI, Nagoya University, March, 2012Outline• Motivations for considering dynamical electroweak symmetry breaking• Basics of technicolor (TC) and extended technicolor (ETC)• Mass generation mechanism for fermions• UV to IR evolution and walking TC• Some constraints on TC/ETC models• Collider signals for TC/ETC theories and constraints from LHC data
• Some further model-building results• Conclusions

Motivations for Dynamical Electroweak Symmetry
Breaking
Recall the motivations for considering dynamical electroweak symmetry breaking(EWSB). Standard Model (SM) Higgs mechanism for EWSB works but leaves somequestions:
To get EWSB, one sets µ2 < 0 in the scalar potentialof the SM Lagrangian,V (φ) = µ2φ†φ + λ(φ†φ)2, yielding hφi = v/0√2 . But why should µ2 be negative
rather than positive?
µ2 and hence m2H = −2µ2 = 2λv 2 with v = (2/g)mW = 246 GeV are unstable
to large radiative corrections from much higher energy scales - gauge hierarchy problem,
fine-tuning needed to keep the scalar light.
What is the origin of the large range of SM Yukawa couplings, from O(1) for top quarkto 10−5 for electron mass (with additional inputs needed for neutrino masses)?

Furthermore, in two major previous cases where fundamental scalar fields were used inphenomenologically modelling SSB, the underlying physics involved bilinear fermion
condensates:
Superconductivity: the Ginzburg-Landau free energy functional was a successful
phenomenological description, using complex scalar field φ with V = c2|φ|2 + c4|φ|4,with c2 ∝ (T − Tc), so for T < Tc, c2 < 0 and hφi 6= 0. But the underlying origin
of superconductivity is the dynamical formation of a condensate of Cooper pairs heei in
BCS theory.
σ model for spontaneous chiral symmetry breaking (SχSB) in hadronic physics, with
~ 2 + (λ/4)φ~ 4, where φ~ = (σ, ~V = (µ2/2)φπ ). Here, one produces SχSB by the
choice µ2 < 0, leading to hσi = fπ 6= 0. But the underlying origin of SχSB in QCD
is the dynamical formation of a hq̄qi condensate.
These examples suggest the possibility that the underlying physics responsible for
EWSB may also be a dynamically induced fermion condensate.

Indeed, there is one known source of dynamical EWSB via a fermion condensate: thehq̄qi condensate in QCD breaks electroweak symmetry.
Consider, e.g., QCD with Nf = 2 massless quarks, u, d. This theory has a global
SU(2)L × SU(2)R chiral sym. quark condensate hq̄qi = hq̄LqRi + hq̄RqLi
transforms as an Iw = 1/2, |Y | = 1 operator and breaks this symmetry to the
diagonal, vectorial isospin SU(2)V . Resultant Nambu-Goldstone bosons (NGB’s) - π ±and π 0 - are absorbed to become the longitudinal components of W ± and Z, giving
them masses:

m2W =g 2fπ24,

m2Z =(g 2 + g ′2)fπ2
4With fπ ∼ 93 MeV, this yields mW ≃ 30 MeV, mZ ≃ 33 MeV, satisfying tree-level
relation ρ = 1, where ρ = m2W /[m2Z cos2 θW ]. (A gedanken world where this is the
only source of EWSB is discussed in Quigg and RS, Phys. Rev. D79, 096002 (2009)).
Scale here is too small by ∼ 103 to explain the observed W and Z masses, butsuggests a more realistic model for dynamical EWSB.

Basics of Technicolor
Technicolor (TC) is an asymptotically free vectorial gauge theory with gauge group thatcan be taken as SU(NT C ) and a set of fermions {F } with zero Lagrangian masses,
transforming according to some representation(s) of G. The TC interaction becomesstrong at a scale ΛT C of order the electroweak scale, confining and producing a chiralsymmetry breaking technifermion condensate (Weinberg, Susskind, 1979); recent
review: Sannino, Acta Phys. Polon., arXiv:0911.0931).
Assign technifermions so L (R) components form SU(2)L doublets (singlets). Minimal
choice: “one-doublet” (1DTC) model with fund. rep. for technifermions uses
 τ
F1
τ
τ
F
,
F
1R
2R
F2τ L
with TC indices τ and Y = 0 (Y = ±1) for SU(2)L doublet (singlets).
The SU(NT C ) TC theory is asymp. free, so as energy scale decreases, αT C increases,
eventually producing condensates hF̄1F1i and hF̄2F2i transforming as Iw = 1/2,
|Y | = 1, breaking EW symmetry at ΛT C .
The W and Z pick up masses

m2W ≃

g 2 FT2C ND
4

,

m2Z ≃

(g 2 + g ′2) FT2C ND
4again satisfying ρ = 1 because of the Iw and Y of hF̄iFii, i = 1, 2. HereFT C ∼ ΛT C is the TC analogue to fπ ∼ ΛQCD and ND = number of SU(2)Ltechnidoublets. For 1DTC, ND = 1, so FT C = 250 GeV. One can add SM-singlet
technifermions to get walking.
Another class of TC models that was studied in the past (but is now disfavored) used
one SM family of technifermions (1FTC) aτ 
U
aτ
aτ
U
,
D
R
R
D aτ L
 τ
N
τ
τ
N
,
E
R
R
Eτ L
(where a, τ are color, TC indices). For 1FTC, ND = Nc + 1 = 4, so FT C ≃ 125
GeV.

Some appealing properties of TC:• Given the asymp. freedom of TC theory, the condensate formation and hence EWSB
are automatic, as in QCD, and do not require a specific parameter choice likeµ2 < 0 in SM.
• TC has no fundamental Higgs, so no hierarchy problem.
• Because hF̄ F i = hF̄LFRi + hF̄RFLi, technicolor explains why the chiral part of
GSM is broken and the residual exact gauge symmetry, SU(3)c × U(1)em, is
vectorial (also explained in SM).
To give masses to quarks and leptons, embed TC in a larger, extended technicolor(ETC) gauge theory with ETC gauge bosons transforming SM fermions into
technifermions and back (Dimopoulos and Susskind; Eichten and Lane, 1979-80).
ETC gauges SM fermion generation index and combines it with TC gauge indices in thefull ETC symmetry group.

To satisfy constraints on flavor-changing neutral current (FCNC) processes, ETC gaugebosons must have large masses. These masses naturally arise from sequential breaking
of a strongly coupled chiral ETC gauge symmetry.
Diagrams for generating SM fermion masses involve virtual exchanges of ETC gaugebosons, so resultant fermion masses depend on inverse powers of these ETC gaugeboson masses. To account for the hierarchy in the three generations of SM fermion
masses, the ETC theory should break sequentially at three corresponding scales,Λ1 > Λ2 > Λ3, e.g., Λ1 ≃ 103 TeV, Λ2 ≃ 50 − 100 TeV, Λ3 ≃ few TeV.
The ETC theory is constructed to be asymptotically free, so as energy decreases from ahigh scale, ETC coupling αET C grows, eventually becomes large enough to form
condensates that sequentially break the ETC symmetry to a residual exact subgroup,
which is the TC gauge group; so GET C ⊃ GT C .
An ETC theory is much more ambitious than the SM or MSSM because a successful
ETC model would predict the entries in the SM fermion mass matrices and theresultant values of the quark and lepton masses and mixings. It would explain
longstanding mysteries like the mass ratios me/mµ, mu/md, md/ms, etc. Not
surprisingly, no fully realistic ETC model has yet been constructed, and TC/ETC
models face many stringent constraints.

Mass Generation Mechanism for Fermions
The ETC gauge bosons enable SM fermions, which are TC singlets, to transform into
technifermions and back, communicating the EWSB in the TC sector to these SM
fermions and producing masses for them. The figure shows a one-loop graphcontributing to diagonal entries in mass matrix for SM fermion f i, where i =generation index. Basic ETC vertex is f i → f j + Vji, with Vji = ETC gauge boson,
1 ≤ i, j ≤ 5; here we distinguish first three ETC indices, which refer to SM fermion
generations, from other ETC indices that are TC indices, by denoting the latter as τ(with any color indices suppressed):
Vτi

fRi

FRτ

×

FLτ

fLi

Rough estimate:
(f )
Mii ≃

2αET C Cf
π

Z

dk2

k2ΣT C (k)[k2 + ΣT C (k)2][k2 + Mi2]

where Mi ≃ (gET C /2)Λi ≃ Λi is the mass of the ETC gauge bosons that gain massat scale Λi, Cf = quadratic Casimir invariant. For Euclidean k ≫ ΛT C ,ΣT C (k) ≃ ΣT C (0)[ΣT C (0)/k]2−γ . In walking TC (WTC), γ may be ∼ O(1), soΣT C (k) ≃ ΣT C (0)2/k; contrast with QCD, where Σ(k) ≃ Σ(0)3/k2 for(f )k ≫ ΛQCD . In general, the TC/ETC calculation of Mii givesκ Cf η Λ3T C
(f )
Mii ≃
Λ2i
where κ ≃ O(10) is a numerical factor from the integral and η is a RG factor thatenhances the mass. This is only a rough estimate, since ETC coupling is strong, so
higher-order diagrams are also important.
The sequential breaking of the ETC symmetry at the highest scale, Λ1, theintermediate scale, Λ2, and the lowest scale, Λ3, thus produces the generational
hierarchy in the SM fermion masses. Since these ETC scales enter as inverse powers in
the resultant SM fermion masses and since Λ1 is the largest ETC scale, it follows thatfirst-generation fermion masses are the smallest, and since Λ3 is the smallest ETC
scale, third-generation fermion masses are the largest.

There are mixings among the interaction eigenstates of the ETC gauge bosons to form
mass eigenstates. Insertions of these on ETC gauge boson lines lead to CKM and
lepton mixing ( Appelquist and RS, Phys. Lett. B 548, 204 (2002); Appelquist and RS,
Phys. Rev. Lett. 90, 201801 (2003); Appelquist, Piai, RS, Phys. Rev. D 69, 015002(2004); Christensen and RS, Phys. Rev. D 74, 015004 (2006)).
Since SM fermion masses arise dynamically, the running mass mfi (p) of a SM fermionof generation i is constant up to the ETC scale Λi and has the power-law decay(Christensen and RS, Phys. Rev. Lett. 94, 241801 (2005)):mfi (p) ∼ mfi(0)Λ2ip2

for Euclidean momenta p ≫ Λi (neglect subdominant logarithmic factors).
Thus, e.g., the third-generation quark masses mt(p) and mb(p) decay like Λ23/p2 forp ≫ Λ3, while the first-generation quark masses mu(p) and md(p) are hard up to
the much higher scale Λ1, eventually decaying like Λ21/p2 for p ≫ Λ1.

UV to IR Evolution and Walking (Quasi-Conformal) TC
TC models that behaved simply as scaled-up versions of QCD were excluded by theirinability to produce sufficiently large fermion masses without having ETC scales so low
as to cause excessively large FCNC.
Modern TC theories are constructed to have a coupling gT C that gets large, but runs
slowly (“walks”) over an extended interval of energy (WTC) (Holdom, Yamawaki et al.,Appelquist, Wijewardhana...).
This walking (quasi-conformal) behavior arises naturally from an approximate IR zero of
the perturbative beta function:β(αT C ) =

dαT C
dt

=−

α2T C
2π




b2 αT Cb1 ++ O(α2T C )4π

where t = ln µ, with b1 > 0 (Nf < Nf,max) for asymptotic freedom. For
sufficiently many technifermions, b2 < 0, so β has a second zero, i.e., approx. IR fixed
point (IRFP) of RG, at αT C = −4πb1/b2 ≡ αIR .

If Nf < Nf,cr (depending on technifermion rep. of GT C , R), as the theory evolves
from the UV to IR, αT C gets large, but runs slowly because β approaches this zero atαIR . For TC, we want to choose Nf so that αIR is slightly greater than the minimalvalue αcr for technifermion condensation. Then the TC theory has quasi-conformal
behavior, with a large αT C (µ) over an extended interval of energies µ.
As αT C (µ) eventually exceeds αcr at µ ∼ ΛT C , the technifermion condensate hF̄ F iforms, the technifermions gain dynamical masses, and in the low-energy theory at
smaller µ, they are integrated out, so the TC beta function changes, and αT C evolves
away from αIR which is thus an approximate IR fixed point.
Because WTC has approx. dilatational invariance, which is dynamically broken by thehF̄ F i condensate, it has been suggested that this could lead to a light approx.
Nambu-Goldstone boson (NGB), the techidilaton (Yamawaki..Goldberger, Grinstein,
Skiba; Sannino...; Appelquist and Bai; Logan, Barger, Ellis...; see also Bardeen et al.;Holdom and Terning). This might be as light as 125 GeV.
The initial ATLAS and CMS indications, seen in several channels, of a possible state at
125 GeV, if confirmed with more data, might be a Higgs but instead might be atechnidilaton, as discussed at this conf. Further experimental and theoretical studies arenecessary to decide this.

For Nf > Nf,cr , the theory would evolve from the UV to the IR in a chirally
symmetric manner, without ever producing hF̄ F i, so the (initially massless)technifermions remain massless, and the IRFP is exact. This IR-conformal phase is of
basic field-theoretic interest, although for TC, we should choose the technifermion
content so that we are in the phase with SχSB, as is necessary for EWSB.
In Walking TC, SM fermion masses are enhanced by the factor Z Λidµγ(αT C (µ))ηi = exp
ΛT C µwhere γ = anomalous dimension of F̄ F operator. If γ is approximately constant overthis range of µ, then ηi = (Λi/ΛT C )γ , which can be substantially larger than 1. So
one can increase ETC scales Λi for a fixed mfi , reducing FCNC effects.
One method (βDS) for studying quasi-conformal TC: use 2-loop β function to calculaterunning α; combine with sol. of Dyson-Schwinger (DS) eq. for technifermion
propagator in the improved ladder (one-gluon exchange) approx. (Yamawaki,
Miransky..., Appelquist et al., Lane..). DS eq. yields dynamical technifermion mass
generation for α > αcr , with αcr Cf ∼ O(1), where R is fermion rep.

As number of technifermions, Nf , increases, αIR decreases, and Nf ր Nf,cr asαIR ց αcr . This yielded the βDS estimate Nf,cr ≃ 4NT C for fund. rep.
The DS eq. captures some of the relevant physics but does not directly include effectsof confinement or instantons, both of which are important for SχSB.
Because of confinement, the technifermions and gluons have maximum wavelengths
λ ∼ 1/ΛT C and minimum momenta kmin ∼ ΛT C . The conventional DS eq. takes
the lower end of the Euclidean loop integration to k = 0, but confinement raises thisto kmin. This correction decreases the integration region, decreases tendency to SχSB
(Brodsky and RS, Phys. Lett. B666, 95 (2008)). The correction for the neglect of
instantons goes in the opposite direction; inclusion of instantons increases the tendency
to SχSB.
Since these corrections are in opposite directions, the net shift in Nf,cr may not be too
great, which helps to explain the rough agreement, to within the uncertainties of theβDS Nf,cr and general lattice data.
Other approaches to estimating Nf,cr are also of interest, e.g., analysis of gluon
propagation (Oehme-Zimmermann, Frandsen et al..)Higher-loop corrections to UV → IR evolution of gauge
theories
Because of the strong-coupling nature of the physics at an approximate IRFP of interest
to TC theories, there are generically significant higher-order corrections to results
obtained from the two-loop β function.
This motivates the calculation of the location of the IR zero in β and the value ofγ = γ(α) for an SU(N ) gauge thy. evaluated at α = αIR to higher-loop order. We
have done this to 3-loop and 4-loop order (Ryttov and RS, PRD 83, 056011 (2011),
arXiv:1011.4542; see also Pica and Sannino, PRD 83,035013 (2011), arXiv:1011.5917).
This is of general field-theoretic interest, beyond the specific application to technicolor.
We have extended this analysis to an N = 1 supersymmetric SU(N ) theory in Ryttov
and RS, arXiv:1202.1297. First discuss non-supersymmetric theory.
Although the coefficients in the beta function at 3-loop and higher-loop order are
scheme-dependent, the results give a measure of the accuracy of the 2-loop calculationof the IR zero, and similarly with the value of γ evaluated at this IR zero. For QCD,
the value of such higher-loop calculations is shown by the increased accuracy in fits todata on αs(µ) (Bethke).

We use the M S scheme, for which the coefficients of β and γ have been calculated
up to 4-loop order (highest-order from Vermaseren, Larin,Pand van Ritbergen). Withℓα ≡ αT C and a ≡ α/(4π), one has dα/dt = −2α ∞ℓ=1 bℓ a , where bℓ is theℓ-loop coefficient. Recall
1
b1 = (11CA − 4Tf Nf )
3

1
2
b2 =
34CA − 4(5CA + 3Cf )Tf Nf
3


1415
2857 3205
2
CA + Tf Nf 2Cf2 −CA Cf −
CA
b3 =
54
9
27


158
2 44
Cf +
CA
+(Tf Nf )
9
27
and
a
j DR (Ta )ij DR (Ta )jk = Cf δik andP so forth for b4, wherej,k DR (Ta )jk DR (Tb )kj = Tf δab so CA ≡ CAdj = N for SU(N ), and, e.g., forfund. rep. Cf = (N 2 − 1)/(2N ), Tf (f und.) = 1/2, etc.
P P

Similarly, for the anomalous dimension γ of F̄ F , γ =P∞ℓ=1 cℓ aℓwith

c1 = 6Cf

c2 = 2Cf

c3 = 2Cf



129
2

h3

2

Cf +

6

CA −

10
3

i

Tf N f .

129
11413 22
Cf −Cf CA +CA + Cf Tf Nf (−46 + 48ζ(3))
4−CATf Nf (and so forth for c4.

97

108

556
27

+ 48ζ(3)) −140
27(Tf Nf )2



The two-loop zero of β away from the origin is given by a = −b1/b2 and is physicalfor b2 < 0.
At the three-loop level, we identify the physical IR zero of β as the relevant physicalsolution of the quadratic eq. b1 + b2a + b3a2 = 0 (non-negative solution closest to
origin).
Similarly, at the four-loop level, we pick out the relevant root from the cubic
b1 + b2a + b3a2 + b4a3 = 0. Behavior of the higher-order coefficients bj as
functions of Nf and rep. R is studied in paper.
We then evaluate the n-loop (ℓ) expression for γ, denoted γnℓ(αIR,nℓ) at the n-loop
zero of β, αIR,nℓ.
Detailed analytic and numerical results are presented in our paper; here we give only
some illustrative numerical results for low values of N ≡ NT C .
For abstract field-theoretic purposes (not for TC application), we also give analytic
results on the approach to the large-N limit with fixed coupling λ = αN . For thefund. rep. we combine this with the limit Nf → ∞ with r = Nf /N fixed.

We find that for given SU(N ) and fermion content for which there is an IR zero of β,
the 3-loop and 4-loop values of αIR are smaller than the 2-loop value.
Results for Nf technifermions in the fundamental rep. of SU(N ) for N = 2, 3:
N Nf αIR,2ℓ αIR,3ℓ αIR,4ℓ
2 7
2.83
1.05
1.21
2 8
1.26 0.688 0.760
2 9 0.595 0.418 0.444
2 10 0.231 0.196 0.200
3 10 2.21 0.764 0.815
3 11 1.23 0.578 0.626
3 12 0.754 0.435 0.470
3 13 0.468 0.317 0.337
3 14 0.278 0.215 0.224
3 15 0.143 0.123 0.126
3 16 0.0416 0.0397 0.0398

Similarly, we find that for given N , R, and Nf , the value of γ calculated to 3-loop and4-loop order and evaluated at the value of αIR calculated to the same order issomewhat smaller than the 2-loop value:
For Nf technifermions in R = fundamental rep. of SU(N ) for N = 2, 3:
N Nf γ2ℓ(αIR,2ℓ) γ3ℓ(αIR,3ℓ ) γ4ℓ(αIR,4ℓ )2 7(2.67)
0.457
0.0325
2 8
0.752
0.272
0.204
2 9
0.275
0.161
0.157
2 10
0.09100.07380.07483 10(4.19)
0.647
0.156
3 11
1.61
0.439
0.250
3 12
0.773
0.312
0.253
3 13
0.404
0.220
0.210
3 14
0.212
0.146
0.147
3 15
0.09970.08260.0836
3 16
0.02720.02580.0259Anomalous dimension γ for SU(2) for Nf fermions in the fundamental representation; (i) blue:
2-loop; (ii) red: 3-loop; (iii) brown: 4-loop calculation (Nf,max = 11).
Figure 1:

Anomalous dimension γ for SU(3) for Nf fermions in the fundamental representation; (i) blue:
2-loop; (ii) red: 3-loop; (iii) brown: 4-loop calculation (Nf,max = 16.5).
Figure 2:

A necessary condition for a perturbative calculation to be reliable: higher-ordercontributions do not modify the result very much. Our result show that for a given N
and Nf , there is a substantial decrease in αIR and γ when one goes from 2-loop to
3-loop order, but for a reasonable range of Nf , the 3-loop and 4-loop results are close
to each other all the way down to the βDS estimate of Nf,cr .
Thus, these higher-loop calculations of αIR and γ allow one to probe the theory morereliably down to smaller values of Nf nearer to estimated Nf,cr . With the increase in
αIR as Nf decreases, perturbative calcs. of αIR and γ eventually get less reliable.
(Values of γ in parentheses are unphysically large.)
In phase with confinement and SχSB, αIR is only an approximate IRFP and γ is only
an effective quantity describing the theory at scales µ where α is near to αIR . In theconformal phase, an IRFP is exact (although our perturbative calculation of it is only
approximate), and γ describes the scaling of the bilinear F̄ F at this IRFP.
Lattice gauge simulations provide a fully nonperturbative determination of Nf,cr and γ,
motivating intensive lattice studies. Although it is difficult to determine the precise
value of Nf,cr for a given SU(N ) and rep. R, these studies have shown definite
walking behavior for Nf near to the βDS estimate of Nf,cr .

Some examples of comparison with lattice measurements:
For SU(3) with Nf = 12, from the table above,
γIR,2ℓ = 0.77,

γIR,3ℓ = 0.31,

γIR,4ℓ = 0.25

Lattice results:
γ = 0.414 ± 0.016 (Appelquist, Fleming, Lin, Neil, Schaich, PRD 84, 054501(2011), arXiv:1106.2148, analyzing data of Kuti et al., PLB 703, 348 (2011),
arXiv:1104.3124, inferring consistency with conformality)γ ∼ 0.35 (DeGrand, arXiv:1109.1237, also analyzing Kuti et al. data ).
So here the 2-loop value is slightly larger than, and the 3-loop and 4-loop values closer
to, these lattice measurements.
Thus, our higher-loop calculations improve the agreement with these latticecomputations. Lattice measurements are making good progress, with many talks here.

We have also carried out these higher-loop calculations for fermions in larger
representations. For fermions in the adjoint representation, Nf ≤ 2 to maintain
asymptotic freedom. For Nf = 2 we findN αIR,2ℓ,adj αIR,3ℓ,adj αIR,4ℓ,adj
2
0.628
0.459
0.493
3
0.419
0.306
0.323

N γ2ℓ,adj (αIR,2ℓ,adj ) γ3ℓ,adj (αIR,3ℓ,adj ) γ4ℓ,adj (αIR,4ℓ,adj )
2
0.820
0.543
0.571
3
0.820
0.543
0.561
For SU(2) with Nf = 2 fermions in the adjoint rep., lattice results include (caution:
various groups quote uncertainties differently):γ = 0.49 ± 0.13 (Catterall, Del Debbio et al., arXiv:1010.5909, PoS(Lat2010) 057)
γ = 0.31 ± 0.06 (DeGrand, Shamir, Svetitsky, PRD 83, 074507 (2011)
γ = 0.17 ± 0.05 (Appelquist et al., PRD 84, 054501 (2011), arXiv:1106.2148)−0.6 < γ < 0.6 (Catterall, Del Debbio, et al., arXiv:1108.3794)

It is of interest to carry out a similar analysis in an asymptotically free N = 1
supersymmetric gauge theory with vectorial chiral superfield content Φ, Φ̃ in theR, R̄ reps. for various R, since here Nf,cr is known (Seiberg for F = R; Ryttov andSannino for higher R).
We have done this for an SU(N ) gauge theory in Ryttov and RS, arXiv:1202.1297. We
find that, as in the non-susy theory, for a given N , Nf , and rep. R, the IR zero of βdecreases when one goes from the 2-loop to the 3-loop level; e.g., for N = 2, 3:
N Nf αIR,2ℓ αIR,3ℓ
2 4
6.28
2.65
2 5
1.14 0.898
3 5 18.85 3.05
3 6
2.69
1.40
3 7 0.992 0.734
3 8 0.343 0.308(where the entries with excessively large values of α are not reliable).

In the susy case, there is a bound γ ≤ 1 from unitarity for a theory in the IR conformal
phase. Insofar as perturbative calculations are reliable, they indicate that γ increases(from 0) as Nf decreases from Nf,max. So we get a perturbative estimate for Nf,cr
by setting the perturbatively calculated γ = 1 and solving for Nf .
For example, for R = F , fundamental rep.,Nf,max = 3N,

Nf,max3NNf,cr ==
2
2

Our perturbative estimates are approx. 1.3 to 1.4 times larger than exact result. Similar
results for higher-dim. reps.
So this comparison suggests that, in this susy case at least, perturbative results slightly
overestimate the value of Nf,cr compared with the exact results, i.e., slightly
underestimate the size of the IR-conformal phase.
We are continuing our studies of higher-loop and nonperturbative effects on αIR and γ.

Some Constraints on TC/ETC Models
Early studies of ETC considered the TC theory as an effective low-energy theory andadded various plausible four-fermion operators linking SM fermions and technifermions.
Part of our work has focused on constructing reasonably UV-complete ETC models thatpredict the forms and coefficients of the four-fermion operators in the effective
low-energy technicolor theory.
Typically, ETC is arranged to be an asymptotically free chiral gauge theory, and includes
a set of SM-singlet, ETC-nonsinglet fermions chosen so that as the scale decreases
from the deep UV, the ETC gauge coupling becomes large enough to produce
condensates of these SM-singlet fermions, which break the ETC gauge symmetry.
Since this involves strongly coupled gauge interactions, it is not precisely calculable, but
the pattern of condensate formation can be plausibly determined by the most attractive
channel (MAC) criterion. Some studies include Appelquist and Terning, PRD 50, 2116
(1994); Appelquist and RS, PLB 548, 204 (2002); PRL 90, 201801 (2003); Appelquist,
Piai, RS, PRD 69, 015002 (2004); Christensen and RS, PRD 74, 015004 (2006); Ryttov
and RS PRD 81, 115013 (2010); Ryttov and RS, PRD 84, 056009 (2011).

To account for the three generations of SM fermion masses, there is a sequential
breaking of the ETC gauge symmetry, at the three scales Λi, i = 1, 2, 3. Although
the full ETC theory is chiral, we focus here on ETC models with vectorial couplings toquarks and charged leptons, denoted VSM ETC models.
At the highest scale, Λ1, GET C breaks to HET C , and the gauge bosons in the coset
GET C /HET C gain masses ∼ gET C Λ1 ∼ Λ1, and so forth for the breakings at the
two lower scales Λ2 and Λ3.
Studies of reasonably UV-complete models showed how not just diagonal, but also
off-diagonal, elements of SM fermion mass matrices could be produced, via nondiagonal
propagator corrections to ETC gauge bosons, Vτi → Vτj , where i, j are generation
indices and τ is a TC index (Appelquist, Piai, RS, PRD 69, 015002 (2004)).
A feature that was found in these studies of reasonably UV-complete ETC models was
the presence of approximate residual generational symmetries that naturally suppress
these ETC gauge boson propagator corrections and hence also off-diagonal elements of
SM fermion mass matrices.
Further, a possible mechanism to account for the very small neutrino masses waspresented. This made use of suppressed Dirac and Majorana neutrino masses leading toa low-scale seesaw (Appelquist and RS, PLB 548, 204 (2002); PRL 90, 201801 (2003)).

TC/ETC theories are constrained by FCNC processes. These can be suppressed bymaking the ETC breaking scales Λi sufficiently large, but this is restricted by therequirement that one not cause excessive suppression of SM fermion masses.
One insight from studies of reasonably UV-complete ETC models was that theapproximate residual generational symmetries suppress the FCNC effects.
For example, consider K 0 − K̄ 0 mixing and resultant KL − KS mass difference∆mKLKS . SM contribution consistent with experimental value∆mKLKS /mK ≃ 0.7 × 10−14.
Simple effective Lagrangian used in early studies without a UV-complete ETC theory:
Lef f = c[sγµd]2 with coefficient c ∼ 1/Λ2ET C , usually with just a single generic
ETC scale.
Now in terms of ETC eigenstates, an sd¯ in a K̄ 0 produces a V12 ETC gauge boson,but this cannot directly yield a ds̄ in the final-state K 0; the latter is produced by aV21. So this requires either the ETC gauge boson mixing V12 → V21 or the related
mixing of ETC quark eigenstates to produce mass eigenstates.
The ETC gauge boson propagator insertion 12Π21 required for this breaks thegenerational symmetries associated with the i = 1 and i = 2 generations, and hence2|12Π21| <∼ Λ2Therefore, the contribution to K̄ 0 → K 0 transition from V12 → V21:
2
Λ
1
1
1
2 1
1 2
<
|c| ∼ 2 2Π1 2 ∼ 2 2 ≪ 2
Λ1
Λ1
Λ1 Λ1
Λ1

With above values for Λ1 and Λ2, the suppression factor is (Λ2/Λ1)2 ≃ 10−2. So
rather than the naive result ∆mKLKS /mK ∼ Λ2QCD /Λ21, this yields the considerably
smaller resultΛ22 Λ2QCD∆mKLKS
−15
∼
10
∼
mK
Λ41
which agrees with experimental limits on new-physics contributions.

Similar comments applies to ETC contributions to a number of other FCNC processes.
Some studies of FCNC constraints that take account of these approximate generational
symmetries include Appelquist, Piai, RS, PLB 593, 175 (2004); PLB 595, 442 (2004);
Appelquist, Christensen, Piai, RS, PRD 70, 093010 (2004). Recent boundBr(Bs → µ+µ−) < 4.5 × 10−9 from LHCb is a new constraint. Other phenom.
aspects include muon g − 2 constraint, dark matter candidates, etc.
It remains a challenge to construct a TC/ETC model that does everything that isdemanded of it, including sufficient suppression of FCNC effects and accounting forrealistic quark, charged lepton, and neutrino masses and quark and lepton mixing.
A particular challenge is to get splitting of mt and mb without excessive contributions
to ρ. One cannot do this via the dynamically generated technifermion masses
ΣT C,U > ΣT C,D because ΣT C,U ≃ ΣT C,D , and anyway, this would violate custodial
symmetry too much.
One approach: topcolor assisted TC, i.e. TC2 (Hill, Chivukula + Simmons talks here;
Eichten, Lane..). TC2 models get splitting between mt and mb by using separate,
asymp. free SU(3) gauge interactions acting on the third generation of quarks and on
the first two generations, denoted SU(3)1 and SU(3)2, respectively. The SU(3)1 isarranged to become sufficiently strong, at a scale Λt ≃ O(1) TeV, to produce a ht̄ticondensate and hence a dynamically generated mt.

As a third-generation symmetry, the SU(3)1 treats the t and b quarks in the same way.
To avoid producing hb̄bi ≃ ht̄ti, and hence mb ≃ mt, one uses an additional set of
hypercharge-type interactions, U(1)1 ⊗ U(1)2. The U(1)1 is attractive in the t̄tchannel and repulsive in the b̄b channel.
The SU(3)1 ⊗ SU(3)2 and U(1)1 ⊗ U(1)2 symmetries are assumed to break totheir respective diagonal subgroups, color SU(3)c and hypercharge U(1)Y . A UVcompletion in which one can show that these breakings plausibly occur is Ryttov andRS, Phys. Rev. D82, 055012 (2010).
The scale Λt is fixed in TC2 models by mt, and the scale at which SU(3)1 ⊗ SU(3)2
breaks to SU(3)c cannot be larger than this, or else the SU(3)1 interaction would break
before it could produce the desired ht̄ti condensate. This yields an upper bound on themasses of the eight vector bosons in the coset SU(3)1 ⊗ SU(3)2/SU(3)c of order
TeV.
The current ATLAS/CMS lower bounds of ∼ 2.5 TeV on coloron/axigluons can causesome tension with certain TC2 models.

Constraints from precision electroweak data: ∆ρ = αem(mZ )T and S, where
αemS
2sin (2θW )=ΠZZ (m2Z ) − ΠZZ (0)m2Z

S is sensitive to heavy fermion loop contributions to Z propagator.
From experimental data, SM fits obtain allowed regions in S and T , depending on anassumed value of mH mass; in general, S <∼ 0.2 (90 % CL).

Naive perturbative estimate (which is not applicable, since TC is nonperturbative atscale mZ ):(∆S)T C,pert. ≃dim(RT C ) ND
6π

where dim(RT C ) is the dimension of the TC fermion rep., e.g., dim(RT C ) = NT C
for fundamental. If TC were QCD-like, nonperturbative effects would yield(∆S)T C ≃ 2(∆S)T C,pert. (Peskin-Takeuchi, 1990), which, together with too-small
SM fermion masses, showed that TC could not be a scaled-up QCD-like theory.

A viable TC model must have a reduction in (∆S)T C wrt. its QCD-like value. This
motivates building TC models with the minimal content of SU(2)L-nonsinglet
technifermions.
Studies of Dyson-Schwinger equations have shown that (∆S)T C (per EW doublet) issomewhat reduced in walking TC as compared with its QCD-like value: Harada,
Kurachi and Yamawaki, Prog. Theor. Phys. 115, 765 (2006); Kurachi and RS, Phys.
Rev. D 74, 056003 (2006); see also Sannino, Phys. Rev. D82, 081701 (2010).
It is also useful to calculate S taking into account not just TC, but also ETC effects (Kurachi, RS, Yamawaki, Phys. Rev. D 76, 035003 (2007)). We found that ETC effects
do not increase S significantly wrt. the value for the pure TC theory.
Lattice simulations (Appelquist et al., (LSD Collab.), with SU(3) with Nf = 6, fund.
rep., have also found a reduction in (∆S)T C (per EW doublet) wrt. its QCD-like value(PRL 106, 231601 (2011)).
In general, the constraint from the S parameter remains an important one for TC/ETC
theories. The T constraint is easier to satisfy, since ΣT C,U ≃ ΣT C,D .

Collider signals for TC/ETC theories and constraints fromearly LHC data
Although TC/ETC theories have been constrained indirectly from flavor physics,
precision electroweak quantities, and Tevatron searches, key tests are now forthcomingwith LHC data.
Some collider signals for TC depend on the type of model. A general signature thatapplies to all technicolor models results from the property that the technihadrons
include a technivector mesons, in particular, techni-ρ, denoted ρT C . In QCD the ρ
couples strongly to ππ and decays to ππ with a large width, so also in technicolor.
In TC, the technipions are absorbed to become the longitudinal components of the
W ± and Z. Hence at sufficiently high energy the scattering of longitudinally polarized
W and Z’s will be enhanced by resonant s-channel contributions:
WL+WL− → ρ0T C → WL+WL−+
WL+ZL → ρ+
→
W
TC
L ZL

The ρT C mass, mρT C mass can be roughly estimated from
ΛT C
fT C Nc 1/2
mρT C
≃
≃
mρ
ΛQCD
fπ N T C
where fT C ≃ 250 GeV for a one-doublet TC theory. With fπ = 93 MeV andmρ = 775 MeV, this yields

1/2
Nc
mρT C ≃ (2.0 TeV)NT C
Studies of meson masses in WTC (Kurachi and RS, JHEP 12, 034 (2006)) obtained an
approx. 30 % increase in mρT C /mρ relative to this QCD-like estimate, suggesting thatmρT C ≃ (2.6 TeV)

 N 1/2c

NT C

By analogy with ρ → ππ in QCD, the ρT C would decay as ρ0T C → W +W − and+ρ+→W
Z. For the width of such a technihadron, a rough estimate is
TC
ΓρT C
Γρ

∼

ΛT C
ΛQCD

so, with Γρ ≃ 150 MeV, one has ΓρT C ∼ 250 GeV. Similar for other technihadrons.
LHC can search for this resonant behavior,require substantially more
R but this will
integrated luminosity than the present Ldt = 5 fb−1 per experiment. Many studiesof this overa clearR the years agree that−1√observation of this resonant behavior may
require Ldt ∼ 50 − 100 fb at s = 14 TeV.
For an example of how current LHC data are useful in constraining technicolor models,consider the one-family TC model. This is already in some tension with precisionelectroweak constraints, since it yields (∆S)T C,pert. = NT C ND /(6π). Now
ND = Nc + 1 = 4 in this one-family model, so even if one takes the minimum value,
NT C = 2, this is (∆S)T C,pert. = 4/(3π) = 0.4.
The one-family TC model makes two predictions for techni-hadrons that are tested withcurrent LHC data.
The first is a large number of pseudo-NGB’s (PNGB’s). If one neglects ETC effects anduses the fact that the SM gauge interactions are weak at the EW scale, then a generic
1FTC model has an SU(8)L × SU(8)R global chiral symmetry (where
8 = Nw (Nc + 1)). The technifermion condensates break this to SU(8)V , yielding 63(P)NGB’s, of which 3 NGB’s are eaten. The PNGB’s gain masses from color and ETC

interactions that break the above global chiral symmetry, but some of them include
color-nonsinglet states and could have masses of order several 100 GeV. There is noevidence for these at the LHC.
In particular, the one-family TC model predicts color-octet Q̄a(Tα)abQb pseudoscalarand vector states, where Q are techniquarks and Tα,α = 1, ..., 8 are SU(3)c(8)
generators. The mass of the color-octet ρp
T C can be estimated as above, withfT C ≃ 125 GeV, yielding mρ(8) ≃ 1.3 Nc/NT C TeV. Walking might raise thisTC
mass slightly, as noted above.
CMS and ATLAS have set lower bounds on color-octet resonances of 2.5 TeV. Although
the mass estimates for TC theories have significant uncertainties owing to the strongly
coupled nature of the TC physics, this causes tension with the one-family TC model.
Minimal technicolor models with only color-singlet technifermions are consistent withthese LHC data. They have no color-nonsinglet technihadrons, and they have only one
SU(2)L doublet of left-handed technifermions (with corresponding SU(2)L-singlet
right-handed technifermions), so the three SM-nonsinglet NGB’s are all eaten by the
W ± and Z and there are no residual SM-nonsinglet (P)NGB’s. With their minimal
SM-nonsinglet technifermion content, they may also yield an acceptably small S.

Some Further Model-Building Results
LHC results thus motivate further study of TC (1DTC) models with minimal
technifermion content, consisting of one color-singlet SU(2)L doublet withcorresponding right-handed components. This is also desirable to minimize TC
contributions to S.
A recent study of a model with this minimal content and technifermions in thefundamental rep. of the TC group is Ryttov and RS, PRD 84, 056009 (2011) with

FLτ =

 τ
F1

with hypercharge YFL
F2τ L
τ
τ
F1R
, F2R
, YfiR , i = 1, 2Electric charge is vectorial, so YF1R = YFL + 1 and YF2R = YFL − 1.
In order to get approx. IR zero of β and walking behavior, we add Nf,cr − 2 additional
SM-singlet technifermions. These make no contribution to S or T .

If one took the simplest choice, GT C = SU(2) and YFL = 0, the EW theory would
be free of gauge anomalies, but the most attractive channel would lead to the Majorana
condensates (α, β are SU(2)L indices)
βτ ′
ατ T
hǫαβ ǫτ τ ′ FL
CFL i,′τ Tτi,hǫτ τ ′ F1R
CF2R

which would not break EW symmetry.
So consider the next step up, GT C = SU(3). By itself, this theory would have an odd
number of SU(2)L doublets, global SU(2) anomaly. Thus, add another SU(2)L doublet(which is singlet under color and TC) to avoid this:
 
ψ1
ψL =
, ψ1R , ψ2R
ψ2 L
One requires NT C YFL + YψL = 0; a reasonable choice is YFL = 1/3 andYψ = −1. ETC gives heavy masses to these leptons.
One can get ST C,pert. <∼ 0.2 here.

A general feature is that the embedding of TC in ETC is more complicated in TC
models with color-singlet technifermions, because now the ETC gauge bosons carrycolor and charge, in contrast, in 1FTC models, where the ETC gauge bosons are
SM-singlets.
Further, ETC symmetry breaking is more complicated than in 1FTC models.
Also studies of models with color-singlet technifermions in higher-dim. reps. of TC
group (Sannino, Dietrich, Ryttov, Tuominen...)Another question concerns the extent to which one can embed TC, ETC in a theory
having higher gauge unification, using dynamical symmetry breaking. This would bedesirable in order to explain features not explained by the standard model:• unification of quarks and leptons• charge quantization
We have shown how, in principle, this is possible, using an extended strong-EW gauge
group SU(4)P S × SU(2)L × SU(2)R (Appelquist and RS, Phys. Rev. Lett. 90,
201801 (2003)). We have also studied prospects for possible higher unification of both
TC and SM symmetries and explored contrasts between Higgs-type GUT breaking anddynamical GUT breaking; e.g., in Chen, Ryttov, and RS, Phys. Rev. D82, 116006(2010).

Conclusions
Dynamical EWSB via technicolor is an interesting and well-motivated possibility which
will be decisively tested by the ATLAS and CMS experiments at the LHC. Thisapproach has shown• how a new gauge interaction that becomes strongly coupled on the TeV scale
naturally produces EWSB, W and Z masses.
• how an associated large but slowly running gauge coupling can result from anapproximate IR fixed point, enhancing fermion mass generation; higher-loop
calculations give further insight into these quasi-conformal theories.
• how fermion masses and generations could arise, by sequential breaking of ETC
symmetry.
• Dynamical EWSB has distinctive experimental signatures that can be probed at the
LHC, including resonant scattering of longitudinally polarized W and Z, and also a
possible light technidilaton.
• Future LHC data should answer the question of the origin of EWSB.