Table Of ContentDetermination of SU(4)TC Technicolor Gauge Group from Embedding in Extended
Technicolor
Masafumi Kurachia, Robert Shrockb, and Koichi Yamawakia
(a) Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI),
Nagoya University, Nagoya 464-8602, Japan and
(b) C. N. Yang Institute for Theoretical Physics
Stony Brook University, Stony Brook, NY 11794
In technicolor theories using an SU(NTC) gauge group, the value of NTC is not, a priori, de-
termined and is typically chosen by phenomenological criteria. Here we present a novel way to
determine NTC from the embedding of a one-family technicolor model, with fermions in the fun-
damental represention of SU(NTC), in an extended technicolor theory, and use it to deduce that
5
NTC =4 in this framework.
1
0
PACSnumbers: 11.15.-q,12.60-i,12.60.Nz
2
n
a The possibility that electroweak symmetry breaking studies extend earlier analyses using Schwinger-Dyson
J (EWSB) may occur dynamically, as in technicolor (TC) andBethe-Salpeterequations[20]. Recentlatticestudies
6 theories [1] remains, although the Higgs-like scalar bo- of (vectorial) SU(3) gauge theories with N = 8 Dirac
f
2 sondiscoveredatthe LargeHadronCollider(LHC) with fermions (which is the value of N in the one-family TC
f
mass mH ≃ 125 GeV [2, 3] is consistent with being the (1FTC)modeldiscussedhere)haveobservedwalkingbe-
]
h Standard Model (SM) Higgs. A TC theory features an havior and a light composite TD-like scalar [14, 21] (see
p asymptotically free, vectorialTC gauge symmetry and a also [22, 23]). The technidilaton in a WTC theory ap-
- set of massless TC-nonsinglet, SM-nonsinglet fermions, pears to be consistent, to within theoretical and experi-
p
{F}. The TC theory becomes strongly coupled at the mentaluncertainties,withcurrentlymeasuredproperties
e
h TeV scale, confining and producing technifermion con- of the Higgs-like scalar discovered at the LHC [18, 19],
[ densates hF¯Fi, with associated spontaneous chiral sym- [24]-[27] (although these properties are also consistent
metry breaking (SχSB) and dynamical EWSB. Three of with the SM Higgs). Future data from the LHC will
1
the resultant Nambu-Goldstone bosons (NGBs) are ab- constrain the TC/TD scenario for the Higgs-like scalar
v
sorbed to give masses to the W± and Z. To give masses further.
4
5 to SM fermions, one embeds the TC theory in a larger,
InadditiontotherequirementthatthecompositeTD-
4 extended technicolor (ETC) theory [4]. Reasonably ul-
like scalarin technicolormust be consistent with the ob-
6 traviolet (UV)-complete ETC theories have been con-
served Higgs-like scalar, TC/ETC theories are also sub-
0 structedasasymptoticallyfreechiralgaugetheoriesthat
. jecttoanumberofphenomenologicalconstraints,includ-
1 self-break [5] in stages down to the (vectorial) TC sub-
ing those from precision electroweak data [28], limits on
0 sector,yieldingthegenerationalhierarchyofSMfermion
flavor-changing neutral current (FCNC) processes, etc.
5 masses, including light neutrino masses [6]-[8].
1 Both continuum (e.g., [18, 29, 30]) and lattice studies
: [15],[21]-[23] have shown that TC corrections to the W
v Viable TC theories exhibit a squared gauge coupling
and Z propagators (in particular, the S parameter [28])
i α (µ) = g (µ)2/(4π) that grows to O(1), but runs
X TC TC can be substantially reduced in a theory with walking
very slowly (walks) over a substantial interval of Eu-
r clidean momenta µ and an associated large anomalous and, moreover, via ETC effects [30]. Further, explicit
a calculations in a reasonably UV-complete ETC theory
dimension γ ∼ O(1) for the bilinear technifermion op-
m showed that residual approximate generational symme-
erator[9]-[10]. ThesepropertiesofawalkingTC (WTC)
tries suppress FCNC processes [8].
theory follow naturally if the theory has an approximate
infrared(IR)zerointheTCbetafunctionβ atavalue The simplest embedding of the TC theory in ETC is
TC
α thatisslightlylargerthanthecriticalminimalvalue, obtained if one takes the technifermions to comprise one
IR
αcr,F¯F, for the formation of the hF¯Fi condensates [11]- SM family [31], since in this case the ETC gauge bosons
[12]. Since αcr,F¯F ∼ O(1), it is useful to calculate βTC areSM-singletsand[GETC,GSM]=∅. Acommonchoice
and α to higher-loop order [13]. Indeed, lattice stud- for the TC gauge group is SU(N ). Further, the sim-
IR TC
ies have shown that walking behavior can occur nonper- plestmodels inthisclassofTC/ETCtheorieshavetech-
turbatively even if the perturbative beta function does nifermions transforming according to the fundamental
not exhibit an IR zero [14, 15]. These hF¯Fi conden- representation, ,oftheSU(N )TCgaugegroup,since
TC
sates spontaneously break the approximate scale invari- inthiscaseonejustextendstheTCgaugeindicesonvar-
anceoftheTCtheory,givingrisetoalightpseudo-NGB iousfieldstobeETCgaugeindices(seeEq. (2)). There-
(PNGB), the technidilaton (TD), φ [9, 16, 17]. Using fore,weshallconsiderherea1FTCmodelwithTCgauge
holographic methods, it has been shown that WTC the- group SU(N ) and technifermions in the fundamental
TC
ories may yield a light TD [18, 19]. These holographic representation. (We do not consider topcolor or higher-
2
dimensional TC fermion representations.) The value of uaRi : ( , ,1)4/3 , daRi : ( , ,1)−2/3 (4)
N is typically determined by phenomenological crite-
TC
ria,suchasminimizing technifermionloopcorrectionsto
HthieggWs-liaknedscZalparro.pagatorsorfitting the properties ofthe LiL = (cid:18)νℓii (cid:19) : ( ,1, )−1 (5)
L
Here we determine N from the embedding of a
TC
1FTC theory, with technifermnions in the fundamental
representationofSU(NTC),inaspecificETCtheory. We νRi : ( ,1,1)0 , ℓiR : ( ,1,1)−2 . (6)
find N = 4 in this framework, which agrees with the
TC
The indices i = 1,...,N are SM generation indices
value preferred by the fit to the Higgs-like scalar in this gen.
andN +1≤i≤N areTCindices;thea=1,2,3
typeoftheory[18,24]. Inadditiontoitsphenomenologi- gen. ETC
are color SU(3) indices. Thus, for example, ua1 ≡ ua,
calapplication,this providesa novelexample ofhow the c
ua2 ≡ ca, ua3 ≡ ta for the charge 2/3 quarks, and so
structure of a low-energy effective field theory is deter-
forth for the other SM fermions.
mined by its ultraviolet completion [32].
The minimal set of SM-singlet, ETC-nonsinglet
The natural embedding of the TC theory in ETC is
fermions (written as right-handed) is as follows:
SU(N )⊂SU(N ) . (1)
TC ETC
ψij =ψ[ij] : ( ,1,1) (7)
The ETC theory gauges the SM generation index and R R 0
combines it with the TC gauge index in such a way that
and
each generational multiplet of SM fermions of a given
type is combinedwith the 1FTC fermions withthe same
χ : ( ,1,1) , 1≤s≤N −4 . (8)
i,s,R 0 ETC
SMquantumnumbersintoafundamentalrepresentation
of SU(NETC). Hence, with Ngen. denoting the number where s labels each of the NETC −4 copies (flavors) of
of SM fermion generations, χ . We denote the contribution of a chiral fermion
i,s,R
of representationR ofSU(N) to the triangle anomaly as
N =N +N . (2)
ETC gen. TC
Anom(R). Since
This determines N in terms of the known N =
ETC gen.
3 and a given value of NTC based on the embedding Anom( )=(N −4)Anom( ) , (9)
(1). This procedure was used with the minimal value,
N = 2, to construct SU(5) ETC theories in [6–8]. In our ETC theory is anomaly-free. One can also add a
TC
[6–8], given the value N = 2, the fermion content of vectorlikeSM-singlet,ETC-nonsingletfermionsubsector;
TC
theultravioletcompletionwaschosentoyieldtheknown here we restrict ourselves to the minimal version of the
N =3 [33]. model.
gen.
Weproceedtoshowhowtheembedding(1)ofa1FTC ForagivengaugegroupG,wedenotethebetafunction
model,withtechnifermionsinthefundamentalrepresen- β = dα /dlnµ = −2α ∞ b(G)(α /(4π))ℓ. The
G G G ℓ=1 ℓ G
tation of the TC gauge group, in a specific ETC theory ETC theory is asymptoticalPly free, with one-loop beta
with Eq. (2) determines that NTC = 4. The key to function coefficient
our result is the specification of the fermion content of
1
the ETC theory and the use of the requirement that the bSU(NETC) = (9N −10) , (10)
ETC theory must be free of chiral gauge anomalies. In- 1 3 ETC
terestingly, our result is independent of N (where it
gen. soasthescaleµdecreasesfromtheUVtowardtheIR,the
is implicitly understood that N is small enough so
gen. ETC squared coupling α (µ) grows. As µ decreases
ETC
that SU(3) and SU(N ) areasymptotically free.) To
c ETC through a scale that we denote Λ , α (µ) increases
1 ETC
make this manifest, we will keep N general in our
gen. throughthe minimal criticalvalue for the formationof a
derivation.
bilinear fermion condensate. Using a vacuum alignment
The gauge group at the ETC scale is SU(N ) ⊗
ETC argument [34], we infer that this forms in the channel
G ,whereG =SU(3) ⊗SU(2) ⊗U(1) isthe SM
SM SM c L Y
gauge group, with Q = T + (Y/2). The fermion
em 3L × → , (11)
content of the ETC theory is composed of two sec-
tors, namely SM-nonsinglets and SM-singlets. The SM- breaking SU(N ) to SU(N −1). The associated
ETC ETC
nonsingletsector is determined by our 1FTC theory and condensate is
the embedding (1) with (2). The fields are indicated be-
lowinYoungtableauxnotation,whereaisacolorSU(3) NETC
index, i = 1,...,NETC is the ETC index, and the reprec- h ψR1j TCχj,1,Ri , (12)
sentation is R = (R ,R ,R ) , with I and Y the Xj=2
ETC c I Y
weak isospin and hypercharge:
whereC is the Dirac charge-conjugationmatrix,and,by
uai convention,we have takenthe ETC gaugeindex i=1 in
Qai = : ( , , ) (3)
L (cid:18) dai (cid:19)L 1/3 ψRij andthe copyindexs=1inχj,s,R. The fermionsψR1j
3
and χ with 2 ≤ j ≤ N involved in this conden- provided that N is sufficiently small that the ETC
j,1,R ETC gen.
sategaindynamicalmassesoforderΛ andareintegrated theory is asymptotically free andbreaks in the indicated
1
out of the low-energy effective theory (LEET) applica- manner, and also that SU(3) is asymptotically free.
c
ble at scales µ < Λ . Of the fermions in Eqs. (7) and Very interestingly, our resulting value N = 4 agrees
1 TC
(8), the remainingnonsingletones inthis SU(N −1) withthe value inferredfroma fitto the propertiesofthe
ETC
LEET are ψ2j with 3 ≤ j ≤ N ; and χ with Higgs-like scalar in a 1FTC theory [18, 24, 32, 36].
R ETC j,s,R
2≤j ≤NETC and2≤s≤NETC−4. The(2NETC−1) Henceforth, we set Ngen. equal to the known value,
ETCgaugebosonsinthecosetSU(NETC)/SU(NETC−1) Ngen. = 3. Combining this with Eqs. (2) and (15),
gain masses of order gETCΛ1 ≃ Λ1. Diagrams involving we infer that the the ETC gauge group is SU(7)ETC.
exchanges of these massive ETC vector bosons connect- As discussed above, this breaks in three stages to the
ing SM fermions with technifermions produce masses for (vectorial) SU(4)TC group: SU(7)ETC → SU(6)ETC →
the first generation (i=1) of SM fermions [35]. SU(5)ETC → SU(4)TC. The theory naturally accounts
ThisSU(N −1)theoryisagainasymptoticallyfree for the mass hierarchy in the SM fermion generations,
ETC
(with bSU(NETC−1) = (1/3)(9N −19)) so the gauge sincetheSMfermionmassesinthei’thgenerationresult
1 ETC from exchange of ETC vector bosons with mass Λ and,
coupling (inherited at Λ from the SU(N ) theory) i
1 ETC in the ETC boson propagators,Λ−2 ≪Λ−2 ≪Λ−2.
grows, and we infer that at a somewhat lower scale, Λ2, 1 2 3
ThefermioncontentintheSU(4) theoryconsistsof
there is again condensation in the channel (11), break- TC
the SM-nonsinglet fermions in Eqs. (3)-(6) and the SM-
ing SU(N − 1) to SU(N − 2). The associated
ETC ETC
singlet fermions in Eqs. (7)- (8) with 4 ≤ i,j ≤ 7. This
condensate is
TCbetaisagainasymptoticallyfree(withb(TC) =26/3).
1
NETC Hence,theTCcouplingα (µ)inheritedfromthelowest
h ψ2j TCχ i, (13) TC
R j,2,R ETC theory, SU(5)ETC at Λ3, continues to grow as µ
Xj=3 decreases below Λ [37].
3
For a fermion condensation channel (Ch) R ×R →
where, by convention, we have taken the gauge index 1 2
R , a measure of the attractiveness is (∆C ) =
i = 2 in ψij and the copy index s = 2 in χ . The Ch 2 Ch
R j,s,R C (R )+C (R )−C (R ),whereC (R)isthequadratic
fermions ψ2j and χ with 3 ≤ j ≤ N involved in 2 1 2 2 2 Ch 2
R j,2,R ETC Casimir invariant. A rough estimate of αcr,Ch is
this condensate gain dynamical masses of order Λ and
2 α ≃ 2π/[3(∆C ) ]. The field is self-conjugate
areintegratedoutofthe SU(N −2)LEEToperative cr,Ch 2 Ch
ETC in SU(4) and, at a scale Λ (where A denotes the
at µ < Λ . Of the fermions in Eqs. (7) and (8), the TC AA
2
antisymmetric rank-2 tensor, ) forms a condensate in
remainingonesthatarenonsingletsintheSU(N −2)
ETC
LEET are ψ3j with 4 ≤ j ≤ N ; and χ with 3 ≤ the most attractive channel × → 1, of the form
R ETC j,s,R
j ≤NETC and3≤s≤NETC−4. The(2NETC−3)ETC h 7i,j,k,ℓ=4ǫijkℓψRij TCψRkℓi, with (∆C2)AA = 2C2( ) =
gaugebosonsin the cosetSU(NETC−1)/SU(NETC−2) 5P. This is invariantunder both SU(4)TC and GSM. The
gainmassesoforderΛ2. Diagramsinvolvingexchangesof next most attractive channel is × → 1 in TC, with
thesemassivevectorbosonsconnectingSMfermionswith (∆C2)F¯F = 15/4, forming at the scale ΛF¯F and involv-
technifermionsproducemassesforthe secondgeneration ing the condensates hF¯Fi=hF¯ F i+ hF¯ F i with the
L R R L
of SM fermions [35]. F technifermions in (3)-(6). The hF¯Fi condensates pro-
This sequential self-breaking of the SU(NETC) the- duce EWSB. One has the rough estimate
ory continues iteratively in N −4 stages, using the
ETC
Λ 2π
NETC −4 copies of χj,s,R fermions, so that the original AA ≃exp (α−1 −α−1 ) . (16)
SU(NETC) (chiral)gaugesymmetry is finally reduced to ΛF¯F (cid:20)b(1TC) cr,AA cr,F¯F (cid:21)
the (vectorial) SU(N ) subgroup, with the indices cor-
respondingtothe broTkCenETCsymmetriesbeing theSM Thisyields ΛAA/ΛF¯F ≃1.5. Ingeneral,a TD-likescalar
in this theory contains F¯F, AA, a techni-glueball com-
generation indices. Hence,
ponent,etc.; theF¯F componentplausiblydominatesbe-
Ngen. =NETC −4 . (14) cause of the ΛAA/ΛF¯F ratio and the walking behavior,
which implies that the TD mass is ≪ m , where
TC,had
Substituting this expression for Ngen. into Eq. (2), we mTC,had denotes the mass scale of techni-vector mesons
obtain the result and techni-glueballs.
NeglectingETCandSMgaugeinteractions,the1FTC
N =4 . (15) theoryhasa(non-anomalous)globalflavorsymmetryin-
TC
volving SM-nonsinglet fermions, SU(8) ⊗ SU(8) ⊗
FL FR
This is our main result. We have determined N from U(1) . This is broken to SU(8) ⊗U(1) by the hF¯Fi
TC V V V
the structure of the specific ETC theory in which our condensate. In addition to the three NGBs absorbed by
±
1FTC theory is embedded. A particularly intriguing as- the W and Z, this yields 60 PNGBs. Taking account
pect of our result is that, although Eq. (2) connects of walking and the strong ETC interactions, it appears
>
NETC and NTC, it does so in a manner that involves possible that their masses could be ∼ O(1) TeV, above
N , but our result is actually independent of N , current LHC limits [38].
gen. gen.
4
Although 1FTC with N = 4 has a large perturba- tended technicolor theory, with the SM fermions com-
TC
tivevalueofS,viz.,S =8/(3π),itiswell-knownthat bined with technifermions into fundamental representa-
pert.
theperturbativeestimateofS isnotreliablebecauseTC tions of SU(N ) as specified in Eqs. (3)-(6) and have
ETC
is strongly coupled at the scale of m and m . Here, shown that this yields the value N = 4. This value is
W Z TC
motivated by the results of [18, 29, 30], we will assume the sameasoneinferredfromafittothe 125GeVscalar
that the walking and ETC effects can suppress S suffi- bosoninTC[18,24]. Ourresultmotivateslatticestudies
ciently to obey experimental constraints. Ongoing and of SU(4) gauge theory with N = 8 Dirac fermions. Fu-
f
future lattice calculations will further test this assump- tureLHCdatawillyieldstringenttestsofthismodel. In
tion. A constraint on TC/ETC models is that the spec- additiontothisphenomenologicalapplication,ourresult
trum of technihadrons must be consistent with current is of general interest for the insight that it provides on
limits from the LHC. We have already commented on how the structure of a low-energy effective field theory -
thePNGBs. Italsoappearstobepossiblethatthe1FTC here the TC theory - is determined by its embedding in
techni-vectormesonmassesmaylieabovetheLHClimits an ultraviolet completion, the ETC theory.
of a few TeV [39]. Additional ingredients are needed to
fully explain the spectrum of quark and lepton masses, ThisresearchwaspartlysupportedbytheJSPSGrant-
in particular, t-b mass splitting. in-Aid for Scientific Research (S) No. 22224003and (C)
In summary, we have presented a novel way to deter- No. 23540300 (K. Y.) and the U.S. NSF Grant No.
mineN fromtheembeddingofaone-familySU(N ) NSF-PHY-13-16617 (R.S.). R.S. thanks the Kobayashi
TC TC
technicolor theory having technifermions in the funda- MaskawaInstitute atNagoyaUniversityforwarmhospi-
mental representation, in a particular SU(N ) ex- tality during a visit when this researchwas performed.
ETC
[1] S.Weinberg,Phys.Rev.D19,1277(1979);L.Susskind, [13] T. A. Ryttov and R. Shrock, Phys. Rev. D 83, 056011
Phys.Rev.D 20, 2619 (1979). (2011);C.PicaandF.Sannino,Phys.Rev.D83,035013
[2] ATLAS Collab. (G. Aad et al.) Phys. Lett. B 716, 1 (2011); T. A. Ryttov and R. Shrock, Phys. Rev. D 86,
(2012); ATLAS-CONF-2014-009. 085005 (2012); R. Shrock, Phys. Rev. D 87, 105005
[3] CMS Collab. (S. Chatrchyan et al.), Phys. Lett. B 716, (2013);Phys.Rev.D87,116007(2013);Phys.Rev.D90,
30 (2012); CMS PAS-HIG-14-009; arXiv:1412.8662. 045011 (2014); T. A. Ryttov, Phys. Rev. D 90, 056007
[4] S. Dimopoulos, L. Susskind, Nucl. Phys. B 155, 237 (2014); G.ChoiandR.Shrock,Phys.Rev.D90,125029
(1979);E.Eichten,K.Lane,Phys.Lett.B90,125(1980). (2014).
[5] Early papers on self-breaking of strongly coupled chiral [14] While an IR zero in the perturbative beta function can
gaugetheoriesincludeH.Georgi,Nucl.Phys.B266,274 yieldwalkingbehavior,therearecaseswherethewalking
(1978) and S. Dimopoulos, S. Raby, and L. Susskind, canoccurnonperturbativelyintheabsenceofsuchanIR
Nucl.Phys. B 169, 373 (1980). zero.Thisisshownbylatticestudiesof(vectorial)SU(3)
[6] T.Appelquist,J.Terning,Phys.Rev.D50,2116(1994). theory with N =8 Dirac fermions in the representa-
f
[7] T. Appelquist and R. Shrock, Phys. Lett. B 548, 204 tion,whichfindwalkingbehavior[15].Forthistheory,b1
(2002); Phys.Rev. Lett.90, 201801 (2003). andb2 havethesamesign,sothetwo-loopbetafunction
[8] T.Appelquist,M.Piai,andR.Shrock,Phys.Rev.D69, has no IR zero (it has an IR zero for 9≤N ≤16).
f
015002 (2004); Phys. Lett. 593, 175 (2004); Phys. Lett. [15] For SU(3) with N = 8 Dirac fermions: Y. Aoki et
f
595, 442 (2004); T. Appelquist, N. D. Christensen, M. al. (LatKMI Collab), Phys. Rev. D 87, 094511 (2013);
Piai, and R. Shrock, Phys. Rev. D 70, 093010 (2004); T. Appelquist et al. (LSD Collab.), Phys. Rev. D 90,
N. D. Christensen and R. Shrock, Phys. Rev. Lett. 94, 114502 (2014); A. Hasenfratz, D. Schaich, and A. Veer-
241801 (2005). nala, arXiv:1410.5886.
[9] K.Yamawaki, M. Bando,and K.Matumoto, Phys.Rev. [16] M. Hashimoto and K. Yamawaki, Phys. Rev. D 83,
Lett. 56, 1335 (1986); M. Bando, K. Matumoto, and K. 015008 (2011) and references therein.
Yamawaki, Phys. Lett.B 178, 308 (1986). [17] Other early discussions of dilatons include W. A.
[10] B.Holdom,Phys.Lett.B150,301(1985);T.Akibaand Bardeen, C. N. Leung, and S. T. Love, Nucl. Phys. B
T. Yanagida, Phys. Lett. B 169, 432; T. Appelquist, D. 273, 649 (1986); B. Holdom and J. Terning, Phys. Lett.
Karabali, and L. C. R. Wijewardhana, Phys. Rev. Lett. 198,535(1987);andV.A.MiranskyandV.P.Gusynin,
57, 957 (1986); T. Appelquist and L. C. R. Wijewhard- Prog. Theor. Phys. 81, 426 (1989).
hana, Phys. Rev. D 35, 774 (1987); M. Bando, T. Mo- [18] K. Haba, S. Matsuzaki, and K. Yamawaki, Phys. Rev.
rozumi, H. So, and K. Yamawaki, Phys. Rev. Lett. 59, D 82, 055007 (2010); S. Matsuzaki and K. Yamawaki,
389 (1987). Phys. Rev.D 86, 115004 (2012).
[11] W.E.Caswell,Phys.Rev.Lett.33,244(1974);D.R.T. [19] D. Elander, C. Nunez, and M. Piai, Phys. Lett. B 686,
Jones, Nucl. Phys. B 75, 531 (1974); T. Banks and A. 64(2010);L.Anguelova,P.Suranyi,andL.C.R.Wijew-
Zaks, Nucl.Phys. B 196, 189 (1982). ardhana, Nucl. Phys. B 862, 671 (2012); D. Kutasov, J.
[12] T. Appelquist, J. Terning, and L. C. R. Wijewardhana, Lin, and A. Parnachev, Nucl. Phys. B 863, 361 (2012);
Phys. Rev. Lett. 77, 1214 (1996); Phys. Rev. Lett. 79, D. Elander and M. Piai, Nucl. Phys. B 864, 241 (2012);
2767 (1997); V. A. Miransky and K. Yamawaki, Phys. Nucl. Phys. B 867, 779 (2013); B. Bellazzini, C. Csaki,
Rev.D 55, 5051 (1997). J. Hubisz, J. Serra and J. Terning, Eur. Phys. J. C 74,
5
2790 (2014); F. Coradeschi et al., JHEP 1311 057 2013. 964(1991); Phys.Rev.D46,381(1992);M.Goldenand
[20] M. Harada, M. Kurachi, and K. Yamawaki, Phys. Rev. L.Randall,Nucl.Phys.B361,3(1991);R.N.Cahnand
D 68, 076001 (2003); M. Kurachi and R. Shrock, JHEP M. Suzuki,Phys.Rev.D 44, 3641 (1991);
0612, 034 (2006). [29] T. Appelquist andF. Sannino,Phys.Rev.D 59, 067702
[21] For SU(3) with N = 8: Y. Aoki, T. Aoyama, M. Ku- (1999);M.Harada,M.Kurachi,andK.Yamawaki,Phys.
f
rachi, T. Maskawa, K. Miura, K.-i. Nagai, H. Ohki, E. Rev. D 70, 033009 (2004); Prog. Theor. Phys. 115 765
Rinaldi, A. Shibata, K. Yamawaki, and T. Yamazaki, (2006);M.KurachiandR.Shrock,Phys.Rev.74,056003
(LatKMI Collab.), Phys.Rev. D 89, 111502 (2014). (2006); M.Kurachi,R.Shrock,andK.Yamawaki,Phys.
[22] For SU(3) with N = 12 (light fermions): Y. Aoki Rev. D 76, 035003 (2007).
f
et al. (LatKMI Collab.), Phys. Rev. Lett. 111, 162001 [30] See, e.g., R. S. Chivukula, E. H. Simmons, H.-J. He, M.
(2013); Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and Kurachi, and M. Tanabashi, Phys. Rev. D 71, 015001
C. H. Wong, PoS Lattice 2013, 070 (2013); R. Brower, (2005).
A. Hasenfratz, C. Rebbi, E. Weinberg, and O. Witzel, [31] AnearlyreviewisE.FarhiandL.Susskind,Phys.Rept.
arXiv:1411.3243. A light scalar has also been observed 74, 277 (1981).
in lattice studies of theories with fermions in higher- [32] Ourresults donot preclude othertypesof TC and ETC
dimensional representations [23]. models, e.g., those with different ETC/TC fermion con-
[23] For recent reviews, see, e.g., CP3 Workshop at tent, additional strongly coupled gauge symmetries or
http://cp3-origins.dk/events/meetings/mass2013; LME- othercomposite-Higgsmodels.Futureexperimentaldata,
BSM (Lattice Meets Experiment: Beyond the Standard which at present is consistent with the minimal SM
Model) 2013 at http://www.bnl.gov/lme2013; SCGT14 Higgs,willfurtherconstrainallcomposite-Higgsmodels.
(Strongly Coupled Gauge Theories 2014) Mini at [33] Indeed, this fermion content would have been different
http://www.kmi.nagoya-u.ac.jp/workshop/SCGT14Mini; if one had assumed a hypothetical value of Ngen. differ-
andLattice2014athttp://rbc.phys.columbia.edu/Lattice2014. ent from the real-world value of 3: T. A. Ryttov and R.
[24] S.MatsuzakiandK.Yamawaki,Phys.Rev.D85,095020 Shrock, Phys.Rev.D 81, 115013 (2010).
(2012); Phys. Lett. B 719, 378 (2013); S. Matsuzaki, [34] As in [7, 8], we assume that the vacuum alignment is
arXiv:1304.4882. such that condensates donot form in thechannel ×
[25] Someotherpapersdiscussingadilatonand/orfitstothe until µ decreases to ΛF¯F. See also T. Appelquist and
Higgs-likescalarincludeW.D.Goldberger,B.Grinstein, R. Shrock, Phys. Rev. 88 105012 (2013); Y. Shi and R.
and W. Skiba, Phys. Rev. Lett. 100, 111802 (2008); J.- Shrock, Phys.Rev.D, in press [arXiv:1411.2042].
JR.eFva.n7,9W,0.3D50.1G7o(l2d0b0e9r)g;eTr,.AA.pRpoelsqsu,iasntdanWd.YS.kBibaai,,PPhhyyss.. [35] tAiptlsectawleitΛh1,2χ≤11,Rj ≤spNlitEsToCffafrnodmctohmemreusnticoafttehsewχij1t,hRtmheusle-
Rev.D82,071701(2010);B.GrinsteinandP.Uttayarat, only via exchange of ETC vector bosons with masses
JHEP 1107, 038 (2011); V. Barger, M. Ishida, and W.- ∼Λ1.Similarly, atscale Λ2,χ12,R andχ22,R split offfrom
YLe.tKt.e1u0n8g,, 1P0h1y8s0.2R(e2v0.1D2);8B5,. 0C1o5l0ep24pa(,2T01.2G);r´ePghoyirse.,Raenvd. tchome mreustniocfattehewiχthj2,Rthmesuelotinpllyetviwaietxhch3a≤ngejo≤fENTECTCvecatnodr
H. E. Logan, Phys. Rev. D 85, 055001 (2012); P. P. Gi- bosons with masses ∼Λ2, etc.
ardino, K. Kannike, M. Raidal, and A. Strumia, Phys. [36] If one were to assume a type of unification of the TC
Lett. B 718, 469 (2012); T. Corbett et al., Phys. Rev. and SM sectors in the deep UV, one might naively be
Dnes8s6y,,0P7h5y0s1.3R(e2v0.12D);8I.6L, o0w93,0J1.2L(y2k0k1e2n),;aJn.dEGll.isSahnadugTh-. ilsedemtobiendfderedaivnalEuTeCoffNarTCbellaorwgerathhyapnot3h.eHtiocwalevTeCr,-STMC
You, JHEP 1306, 103 (2013); B. Bellazzini, C. Csaki, unification; see N. D. Christensen and R. Shrock, Phys.
J. Hubisz, J. Serra and J. Terning, Eur. Phys. J. C 73, Rev. D 72, 035013 (2005).
2333 (2013); O. Antipin, E. Mølgaard, and F. Sannino, [37] AlthoughourminimalTCtheorydoesnothaveapertur-
JFH. ESaPnn1i3n0o9,,P1h2y2s.(2R0e1v3.);DR8.7F,o0a9d5i,00M1.(T20.1F3r)a;nZd.seCnh,aacnkdo bseattiovfeStwMo--slionogpletIR, (zEe)rToCin-nβoTnCsi,nognleetcfaenrmaidodnsapvreocdtourcliinkge
and R. K. Mishra, Phys. Rev. D 87, 115006 (2013); R. suchazeroandhencewalkingbehaviorfromthetwo-loop
Contino et al., JHEP 1307, 035 (2013); K. Lane, Phys. perturbativebeta function (see also [14]).
Rev. D 90, 095025 (2014); I. Brivio et al., JHEP 1403, [38] S.Matsuzaki,M.Kurachi,andK.Yamawaki,Phys.Rev.
024(2014);P.P.Giardinoetal.,JHEP1405,046(2014). D 90, 095013 (2014). Several studies showed enhance-
[26] Other physics beyond the SM besides TC/ETC also af- mentofPNGBmassesinwalkingTCtheories,including
fects Higgs properties, possibly producing, e.g., invisible [9]; B. Holdom, Phys. Lett. B 198, 535 (1987); T. Ap-
decays: R.E.ShrockandM.Suzuki,Phys.Lett.B110, pelquist and L. C. R. Wijewardhana, Phys. Rev. D 36,
250 (1982); D.Curtin,R.Essig, et al., Phys.Rev.D 90, 568 (1987); and J. Jia, S. Matsuzaki, and K. Yamawaki,
075004 (2014). Phys. Rev.D 87, 016006 (2013).
[27] See,e.g.,R.S.Chivukuka,M.Narain,andJ.Womersley, [39] S.Matsuzaki,M.Kurachi,andK.Yamawaki,Phys.Rev.
in http://pdg.lbl.gov. D 90, 055028 (2014); see also [30] and S. DeCurtis, M.
[28] B.HoldomandJ.Terning,Phys.Lett.B247,88(1990); Redi, and A. Tesi, JETP 1204, 042 (2012).
G. Altarelli and R. Barbieri, Phys. Lett. B 253, 161
(1990); M.PeskinandT.Takeuchi,Phys.Rev.Lett.65,
