IR-conformal gauge theories and composite Higgs

0.16 MB·

by  E. T. Tomboulis

4 IR-conformal gauge theories and composite Higgs
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1 E. T.Tomboulis
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Dept. ofPhysicsandAstronomy,UniversityofCalifornia,LosAngeles
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h LosAngeles,CA90095,USA
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E-mail: [email protected]
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p
e
h Theexistenceofnon-trivialIRandUVfixedpointsingaugetheoriesasafunctionofthenumber
[
offermionflavorsandbarecouplingisdiscussedinthelightofrecentwork.Itispointedoutthat

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infactonlyasmallsubsetofpotentialIR-conformalgaugetheories,i.e.theorieswhoseIRbehav-
v
7 iorisdeterminedbyanIRfixedpoint,hassofarbeenexamined. Recentlatticecomputationsof
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thespectruminsomecaseswhereexistenceofanIRfixedpointisreasonablyassured,however,
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4 reveala non-QCD-likespectrumwiththelighteststatesbeingscalars. Itissuggestedthatthese
.
1 naturallylightcompositescalarsprovideanaturalsettingfortheconstructionofnewcomposite
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Higgsmodels. Aschematicoutlineofsuchamodelisgiven.
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1
:
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X
r
a
QCD-TNT-III-Fromquarksandgluonstohadronicmatter: Abridgetoofar?,
2-6September,2013
EuropeanCentreforTheoreticalStudiesinNuclearPhysicsandRelatedAreas(ECT*),Villazzano,Trento
(Italy)
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
1. Introduction
The search for non-trivial IR or UV fixed points (FP) in gauge theories as a function of the
fermion flavor number and gauge group representation has been the focus of considerable effort
overthelastseveralyears[1]. Apartfromtheirintrinsicquantumfieldtheoryinterest,suchstudies
are motivated by their potential application to physics beyond the Standard Model (BSM). One
such proposal is that ofwalking TC,where the number offlavors issuch as toplace asystem just
outside and below the lowerend of aconformal window. Many other possibilities, however, exist
forBSMphysicsinvolvingnon-trivialIRFP’s. Infact,aswewillnoteinthefollowing,sofar,only
asmallsubsetofpossible IR-conformalgaugetheories, i.e. theories whoselongdistance behavior
isgovernedbyanIRFP,hasbeen,evenpartially, explored.
Herewewillfirstreviewanddiscusswhatisknownandthemanyopenissuesconcerning the
phase diagram, existence ofFT’sandthespectrum ofstates asafunction ofthenumber offlavors
andthegaugecoupling. LatticecomputationsofthespectrumincaseswheretheexistenceofanIR
FPisreasonably certain reveal (atany finite conformality deformation) anon-QCD-like spectrum
of composites with the scalar states as the lowest states. We suggest that the appearance of these
naturallylight(massless)scalarstatesinthespectrumofIR-conformaltheoriesallowsconstruction
of a wide class of composite Higgs models by the direct coupling of the electroweak and other
gauge interactions totheIR-conformal theory. Suchmodels, whichdonorelyonanywalking TC
mechanism or a composite Higgs as a NG boson due to the formation of some strong dynamics
condensate, may evade some of the usual fine-tuning problems. The possibility, in particular, of
obtainingIRFT’sat(relatively)weakcouplingsbyappropriatechoiceofN andN maybecrucial
f c
fornaturally obtaining weakly-coupled SMHiggsanddarkmattersectors.
2. Phasediagram ing,N and FP’s
f
Recall that for a theory with N fermion flavors in representation R of a simple color group
f f
G the perturbative beta function b (g) = [b g3/(4p )2+b g5/(4p )4+ ] possesses, to 2-loop
0 1
− ···
order, a non-trivial zero g = (4p )2b /b for a range of number of flavors N <N <N . The
∗ − 0 1 ∗f∗ f ∗f
upper end of this range, at which b reverses sign, is given by N = 11C2(G) where k =1(1/2)
0 ∗f 4k T(Rf)
for 4 (2 ) - component fermions. For G=SU(N ) with fundamental representation fermions this
c
is the well-known result N =11N /2k . If (N N )<<1 this zero is within the perturbative
∗ c ∗f − f
validity regime, and its existence can be trusted (Banks-Zaks IR FT) [2]. The perturbative value
of the lower end N , however, given by the point where b firstreverses sign, cannot by trusted.
∗f∗ 1
Determining its actual (non-perturbative) value, i.e. the true extent of the “conformal window”
(CW), is a question that has been intensively investigated in recent years for a variety of fermion
representations R andmostlyG=SU(3)orSU(2)[1]. Inthisconnection, ithasbeencommonly
f
assumed, e.g. [3], that chiral symmetry will eventually be broken regardless of the number of
fermion flavors provided the coupling is taken strong enough. In fact, it was recently found that
thisisincorrect. MCsimulationsforN =3atvanishingorsmallinversegaugecouplingb showed
c
thatchiralsymmetryisrestoredviaafirst-ordertransition aboveacriticalnumberofflavors( 52
∼
inthecontinuum) [4]. Thesameresultwasarrivedatbyresummationofthehopping expansionin
thestrongcouplinglimit: thefamiliarchiralsymmetrybreakingsolutionabruptlydisappearsabove
2
IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
acriticalN /N [5]. Puttingtheavailableinformationtogethersuggestsagrosspictureofthephase
f c
diagram of N versus bare coupling g at fixed N shown in Fig.1 below; the case of fundamental
f c
representation SU(N )fermionsischosen fordefiniteness, othercasesbeingqualitatively similar.
c
Asjust noted, the boundary separating the chirally broken phase from the chirally symmetric
phaseterminatesatinfinitecouplingatafinitecriticalN [4],[5]. Attheotherextremeatvanishing
f
coupling it terminates at a critical N ( 12 for N = 3) whose exact value remains somewhat
∗f∗ ∼ c
controversial [1]. Theboundaryisknowntobeafirstorderphasetransitionatleastforsomerange
starting from the strong coupling limit end (g ¥ ). Also, several studies over the years [6] at
→
fixedN insidetheCWhaveobservedafirstorderphasetransitiontoachirallybrokenphaseasthe
f
coupling isincreased,1 whichisconsistent withthepictureinFig. 1. Thesimplest scenario would
then be that the transition line is first order everywhere, though this might in fact depend on the
fermionrepresentationandgaugegroup. (Incidentally,afirstordertransitionwouldbeproblematic
forstandard walkingTC.)
The region enclosed by the boundary between the chirally broken and unbroken phases and
the horizontal broken line in Fig. 1is then the putative CW.Thestandard picture ofthe RGflows
intheplaneofirrelevantcouplings insidetheCWisshowninFig. 2-left.
∞
N
f
¯
< ψψ >= 0
11
N
2 c
¯
< ψψ >6= 0
∞
0 g
Figure1: PhasediagramofN vs. barecouplingg.
f
AsN N frombelowthenon-trivialIRFTg insidetheCWmovesandeventuallymerges
f → ∗f ∗IR
with the UV FT at the origin. For any N >N perturbation theory (PT) gives a trivial IR fixed
f ∗f
point at g=0. The simplest scenario then would be that no other FT is encountered at non-zero
coupling (Fig. 2-right).
Life in the region N >N , however, could turn out to be more exciting. Simulation stud-
f ∗f
ies measuring toleron mass, Dirac spectrum and hadron spectrum for N = 3 at zero or small b
c
performed in[4]findevidence for anontrivial IRFPin theregion N >N . Based on these mea-
f ∗f
surements the conjecture wasmade in[4]that the FPlocation varies continuously withb , as well
as N , reaching the value zero for b ¥ , N > N , and for N ¥ . This would amount to a
f → f ∗f f →
1Additional, likelyspurious, transitionsmay appear though atintermediatecoupling depending ontheparticular
fermionlatticeactionbeingused.
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IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
m
fixed N ,N
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) f c
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) m
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) fixedNf,Nc
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)g (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
gU∗V<ψ¯ψ>=0gI∗R (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)<(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)ψ¯ψ>6=0 gI∗R (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)<(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)ψ¯gψ>6=0
{g′} (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) <ψ¯ψ>=0 (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
{g′} <ψ¯ψ>=(cid:0)(cid:0)(cid:1)(cid:1)0(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
β(g) .
β(g)
g g
Figure 2: Left: Standard picture within the putative CW (N <N ). g denotes the set of irrelevant
f F∗ { ′}
couplingsandmassmistherelevantdirection.Right: LonetrivialIRFTabovetheCW(N >N ).
f F∗
line of IR fixed points as depicted in Fig. 3-left. Such a line, however, would seem to contradict
weak coupling PTwhere no FPline ending at the FT at g=0 is seen. If such non-trivial IR FP’s
actuallyexist,theirexistencecanbereconciledwithPTifanevenzeroofthebetafunctionobtains
as depicted in Fig. 3-right. Actually, one would expect such a zero to be unstable under changes
in N or other parameters unless perhaps it is an infinite order zero. More generally, though, this
f
zero could appear asthelimiting case ofthesituation shown inFig. 4-left. Herethepossibility of
other relevant directions (in addition to mass) isconsidered. These can arise from operators, such
aschirallysymmetric4-fermiinteractions, e.g.,G(yg¯ m y )2,whoseanomalousdimensionsatsome
intermediate couplings are such that they become relevant (marginal). Massless quenched QED
4
provides an example [7]. As N or other parameters are varied the non-trivial IR FT eventually
f
merges with the non-trivial UV FT leading to the even degree zero in Fig. 3-right. Upon further
increaseofN thiszerodisappears resultinginthesituation inFig. 2-right,andconsistentwiththe
f
fact [4], [5] that for N ¥ at fixedN the theory becomes trivial. The situation depicted in Fig.
f c
→
4-leftprovidesonepossible scenarioforreconciling theFT’sfoundin[4],comingfromthestrong
coupling side,withweakcoupling PT.Itis,however,nottheonlyone(cf. below).
The analog of the FP structure in Fig. 4-left could actually arise also within the CW but, of
course, withthe sign ofthe beta-function reversed. This wasdiscussed in[8]. Itwould amount to
theoccurrenceofanUVFTbeyondthenon-trivialIRFTofFig. 2-left,asdepictedinFig. 4-right.
Itshould be emphasized inthis connection that existing lattice simulations exploring IR con-
formalbehaviorareallatfixedlowN ,typically2,3. TheydonotexploretheregimeofbothN ,N
c f c
large, as, for example, when they are adjusted so that a BZ-like IR FT occurs at large N . This
f
is precisely one of the regimes of interest for some of these new possibilities. Some indications
already appear in considering the zeros of the beta-function in PT. A general exploration of the
non-trivialIRandUVzerosofthe4-loopbetafunctionforavarietyoftheoriesandrepresentations
is given in [9]. As always the question is whether such zeros can persist in the full theory. Typi-
cally, givenazerofoundatafixedorder, theneglected subsequent ordertermsinthebetafunction
expansion, when evaluated at the location of the zero, can be as large as the retained terms. This
4
IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
m
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) fixedNf,Nc
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)g
m(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) fixedNf,Nc gI∗R gI∗R (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)<(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)ψ¯ψ>6=0
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) <ψ¯ψ>=0 (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) {g′} <ψ¯ψ>=(cid:0)(cid:0)(cid:1)(cid:1)0(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) g
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
gI∗R (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)<ψ¯ψ>6=0 β(g) .
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
{g′} <ψ¯ψ>=0 (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) g
Figure 3: Here N >N . Left: Line of IR fixed points with trivial end-point. Right: Non-trivial IR FT
f F∗
correspondingtoabeta-functionevenzero.
G m
fixedNf,Nc fixedNf,Nc
m=0
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)<(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)ψ¯gψ>6=0 gU∗V gI∗R gU∗V g
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) <ψ¯ψ>=0
{g′} <ψ¯ψ>=0 <ψ¯ψ>=(cid:0)(cid:0)(cid:1)(cid:1)0(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) {g′}
β(g) β(g)
g g
Figure4: Left:AdditionalrelevantdirectionGresultinginnontrivialUVandIRfixedpointsforN >N .
f F∗
Right: Analogous situation withanon-trivial UVFTforN <N .
f F∗
can happen even when this zero location goes as N 1 for large N . Thus the unknown higher
∼ −f f
order corrections cannot be a priory neglected, and so, in contrast to the BZ FT, the existence of
suchzeros,eventhoughperhaps suggestive, isnotassured.
An alternative approach to computation of the beta-function in the continuum theory going
beyond standard weak coupling PT is by means of an 1/N expansion at fixed ’tHooft coupling
f
N g2 [10]. The expansion amounts to a loop expansion with a modified gauge boson propagator
f
dressed by the fermion bubble contribution to the self-energy, and modified n-point gauge boson
vertices whichinclude thecontribution ofthefermionloopwithnexternalbosons. Theexpansion
ofthebetafunction hastheform
b (l )= 8l 2 1+(cid:229)¥ Hk(l ) , (2.1)
3 Nk
" k=1 f #
where l k T(R )N g2/(4p )2. Each H (l ) represents the contribution of a class of graphs,
f f k
≡
consisting of all graphs of the same, fixed N dependence but including all orders in l . Be-
f
5
IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
cause of the non-local nature of vertices and inverse propagators in this expansion such compu-
tations generally cannot be performed exactly beyond leading order. For H , however, one has
1
H (l )= 11C (G)/(3k T(R ))+H˜ (l ),where, remarkably, anexactintegral representation can
1 2 f 1
−
be given for H˜ (l ) (for SU(N ) and also U(1)) so that its singularity structure can be deduced.
1 c
Furthermore,thefirstseveraltermsintheexpansionofH (l ),fork 4,inpowersofl areknown
k
≤
[10]. If all H (l ) are bounded at a given fixed l convergence follows for sufficiently large N .
k f
| |
TheH ,however, appear togenerally possess (pole orlog)singularities asfunctions ofl . Onone
k
sideofapolesingularity(oreithersideofalogsingularity)ofsomeH thebetafunction(2.1)pos-
k
sesses aIRorUVzero(depending onthepoleresidue sign). b (l )thusexhibits distinctbranches,
each branch delineated by a pair of singularities of the set of the H ’s in a manner completely
k
analogous tothatfoundinthecaseofthesupersymmetric pureSU(N )beta-function [11]. Within
c
each such branch the beta function is given, forsufficiently large N , by aconvergent series (2.1).
f
ForH (l )thesingularitystructurecanbededucedfromtheavailableexactintegralrepresentation,
1
but for higher H ’s only partial information is available. [10] gives a summary of what is known
k
or can be reasonably conjectured, see also [9], [12]. For many purposes it would be sufficient to
justknowthelocation ofthefirst,orfirsttwo(i.e.,lowestinl )singularities in(2.1). Inparticular,
for SU(3) theories, H is surmised to possess a pole singularity lower than the H log singularity,
2 1
and resulting in an IR FT just above it. This would be exactly the structure needed to provide an
explanation of the IR FT’sfound in [4] coming from the strong coupling side. Again, however, it
ishardtoassessthereliabilityofthesefindingfortheexacttheoryasthelocationandnatureofthe
singularities delineating branches can be drastically altered by higher omitted contributions. The
density ofpoles fromthecomplete setoftheH ’sisnotknown,andthepossibility alsoexiststhat
k
aseriesofpolesmightsumuptoanessential singularity.
At this point it is worth remarking that all cases investigated so far form only a small subset
ofpossibleIR-conformalgaugetheories. Inparticular, onlysimplecolorgroupshavebeenconsid-
ered. If the color group is semi-simple, e.g., SU(N ) SU(N ) SU(N ), there are n gauge
1 2 n
× ×···×
couplings resulting in a coupled set of equations for their beta functions. There are now corre-
spondingly manychoices forthe coupling of fermions. Different fermion subsets maybecoupled
to different subsets of group factors and indifferent representations. Depending onthe number of
suchparameters available, manymorepossibilities fornon-trivial IRandUVFT’smaynowarise.
Investigations of FT’s of such coupled beta function equations can easily become quite involved
and,sofar,havenotbeencarried outininteresting casesevenwithinweakcoupling. Inparticular,
existenceofanynon-trivialFT’sat(relatively)weakcouplingsinsuchmoregeneraltheoriescould
turnouttobeimportantforelectroweakphenomenology.
3. Spectrum inIR-conformal theories
Lattice simulations in IR-conformal or near conformal theories, and spectrum computations
in particular, are very challenging. Asymptotic scale invariance is explicitly broken by the finite
lattice size and non-vanishing mass for the fermions at which the simulations are necessarily car-
ried out, as well as, in the case of Wilson fermions, by the non-chiral discretization of the Dirac
operator. If the putative IR FP has additional relevant directions the finite lattice spacing may be
another source of explicit deformation. Extrapolation to the chiral/conformal limits from aregion
6
IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
of sufficiently small fermion masses and large enough lattices must be guided by some analytical
picture: chiralPTinthecaseofQCD-liketheories,andonsetofsomehyper-scaling regimeforIR-
conformaltheories. Distinguishingbetweenthetwocasesconvincinglycanbeverytricky. Keeping
systematic errors and in particular finite size effects under control, especially in the computation
of gluonic spectra, can be very expensive. Progress, however, has been achieved in recent years
instate-of-the-art large-scalecomputationsallowingsomepictureofthespectruminIR-conformal
theories toemerge[13],[14],[15].
Relevant parameters away from conformality are a quark mass mˆ =m/m =am, and the lat-
tice size L. (Coupling to other, extraneous gauge interactions may of course provide other ex-
plicit deformations.) The basic picture one has of IR-conformal behavior is as follows. There
is a "locking" scale M (which depends on the specific theory dynamics) below which a scaling
l
regime obtains where physical mass ratios remain essentially constant. Hadron masses scale as
MH m mˆ1/(1+gm∗) [16], with mass anomalous dimension gm∗ at the FP. The actual ordering of the
∼
spectrum, which is theory-dependent in its exact detail, is essentially set at this scale. If M is
l
relatively high, one expects a spectrum ordering similar to that of QCD with heavy fermions, i.e.
its lower part consisting of anearly degenerate meson spectrum above a still lower set ofglueball
stateswith0++ beingthelowest. IfM islow,onemayexpectaspectrumwherethepseudoscalars
l
are lower than the vector mesons and roughly at the same level as the low gluonic states. As m
is lowered below the locking scale M then, the spectrum (including the string tension which is
l
always the lowest level) scales toward zero according to the above relation. Simulations in the
case of SU(2) with two flavors of adjoint Wilson fermions [14], and for SU(3) with 12 flavors
(three degenerate staggered fermions) [15] favor the first scenario. The lowest states are the glu-
onicstates, with0++ beingthelowest,andthemesonstatesclearlyseparated above. IntheSU(3)
case the flavor singlet scalar 0++ meson is lowest and found to be very close to the gluonic 0++
state, thesetwothenformingtheloweststates. (Flavors singletswherenotcomputedintheSU(2)
case.) Thispicturewasinfactfirstsuggested fromanalyticconsiderations aroundaBZFTin[17],
and, atleastinthecasesconsidered, according tothese latticesimulations appears toholdalsofor
anon-perturbative IRFP.
In a phase governed by an IR FP, as all relevant deformation parameters are switched off
(m 0, L ¥ ), the spectrum collapses to only massless states (“unparticles"). But at any small
→ →
non-zero deformation such as a non-vanishing m or finite box size one has a particle spectrum,
with mass gap as sketched above, and, according to the above findings, containing a light scalar
0++ meson and a scalar 0++ glueball state plus the (somewhat heavier) rest of the glueball and
meson/baryon spectrum.
Thepresence ofnaturally light scalar composite states inthespectrum ofIR-conformal theo-
riesprovides anaturalsettingforconsideration ofcomposite Higgsmodels.
4. CompositeHiggsinIR conformaltheories
Consider a theory with N ‘techniflavors’ and N ‘technicolors’ such that its IR behavior is
f c
controlled by an IR FP. This IR FT may arise from any of the situations reviewed above, either
inside the so-called CW or above it. For phenomenological reasons we would generally prefer
it to be at weak coupling. Note, however, that, as seen from our previous discussion, this does
7
IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
not necessarily mean that the formation of the composites in the spectrum originated in a weak
coupling regime. As we also saw, this may imply that N and N , must be suitably adjusted and
f c
be large. It is expedient to consider a non-simple color group of at least two factors as this most
naturallycanaccommodateadarksector-seebelow. Thebasicideawewanttosuggesthereisthe
following.
Inthepresence ofasmallrelevantdeformation (e.g.,asmallquarkmassmorlarge finitebox
size L) one has a well-defined discrete spectrum of composite states. We make the assumption
that, in accordance with the available computations, the scalar states are the lightest states in the
spectrum. Coupling next other gauge interactions, in particular electroweak interactions, renders
this system of (arbitrarily) light scalar states unstable under the Coleman-Weinberg mechanism.
Theresultingmassgapisnowineffectthedynamicallygeneratedconformalitydeformation,which
persists in the limit where the original explicit deformation is removed (m 0 or L ¥ ). This
→ →
coupling of electroweak interactions directly to the naturally light (massless) states present in an
IR-conformal theoryallowsforawideclassofpotential compositeHiggsmodels.
Tosketch anexample ofthis type ofmodel, consider anIR-conformal theory withN flavors
f
in the fundamental representation of the color gauge group. Single out just two of these flavors
Q=(U,D). There are now composite states formed by Q and the remaining fermion flavors y ,
a
a=1,...,N 2,suchasyy¯ ,y¯Q,Q¯Q,.... The“mixed"sectorcanbeeliminatedbytakingsemi-
f
−
simple color gauge group, e.g., SU(N ) SU(N ) with y charged under both factors, and the Q
1 2
×
charged under only one factor. ‘Mixed’ composites such as y¯Q, y QQ, ... no longer form. The
only possible color singlet mixedstates that could form arehighly unstable multi-quark (tetra and
higher) statesiftheyformatall.
Now consider coupling the electroweak interactions. This may of course be done in various
waysdependinghowtheyaretobealignedrelativetoourIR-conformaltheory. Herewejustcouple
tothesingled-out Qfermionsasfollows. ThescalarmesonQ¯Qgivesrisetothefourfields:
h+ = D¯U, h =U¯D, h =(U¯U D¯D)/√2, h =(U¯U+D¯D)/√2. These may be taken
− 3 0
− −
toformtheweakscalardoublet
h+ (h ih )/√2
H = (h +ih )/√2 , H˜ =it2H∗= 0−h 3 (4.1)
0 3 ! − !
after giving Q ordinary quark elw charges. In addition one has, of course, the other pseudoscalar
P=Q¯g Q, vectorV =Q¯g Q, etc., meson states, as well as baryon states. The glueball states are
5 k k
allweaksinglets. Inparticular onehasthe0++ glueball state, which, together withtheh ,areex-
0
pected tobethelightest states. Thesetwoscalar states mayingeneral mix. Ifthemixingissmall,
thepredominantlyfermioniccompositecanserveasthephysicalHiggs,whereasthepredominantly
gluoniccomponentcanserveasanearly‘invisible’weaklyinteractingparticleofcomparablemass
(WIMP).Anotherphenomenologically interestingscenarioisthecaseofthefermionicscalarcom-
posite being not toodifferent inmassfrom the pseudoscalars and other mesons, whilethegluonic
scalar mass is rather lower (more like the SU(2) spectrum case). Mixing then can result into a
lower mass scalar (Higgs) and a second heavier physical scalar among the other massive meson
states. Such detailed dynamical questions as the exact mass splittings among the light states and
the amountofmixing arespecific theory dependent and canonly beanswered byactual computa-
tion.
8
IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
Inadditiononehasofcoursethe‘dark’sectorcontainingthescalarY =yy¯ andtheotherme-
sonandbaryonstatesformedamongtheremainingflavorsy whichcarrynoelectroweakcharges.
a
ItwouldbeinterestingtoconsiderreplacingoneorbothfactorsinSU(N ) SU(N )byU(N ),
1 2 1
×
U(N )factorswhichwouldeliminatethebaryonstatesinoneorbothsectors. Forsufficientlylarge
2
numberofcolorsandflavorsthiswouldbeexpectedtootherwisemakelittlequalitativedifference,
butagaintherearenoavailablesimulations ofIR-conformaltheories withU(N)colorgroups.
The effective theory of the IR-conformal interactions at low energies is in principle obtained
by matching the composites to interpolating fields. Schematically, this will result in an effective
potential oftheform
l (H†H)2+l (P†P)2+l P†H 2+ l V†V 2+ +l (Y †Y )2+l (Y †Y )(H†H)+ (4.2)
1 2| | ··· V| k k| ··· d d′ ···
If the IR FPoccurs at relatively weak coupling, all effective couplings in this long distance effec-
tive potential are correspondingly weak. Note, however, that, though desirable for computational
orotherreasons,suchweakcouplingsarenotafundamentalrequirement. Alsonoteinthisconnec-
tion that all available simulations in various IR-(nearly) conformal models give small anomalous
dimensions. Thisimpliesthattheircontributions through theinterpolating fieldmatchingslumped
intotheeffectivecouplingsl in(4.2)aresmall. Thecoupling totheelectroweakgaugefieldsthen
i
rendersthissystemof(nearly)masslessscalarfieldsunstableundertheColeman-Weinbergmecha-
nism. WithparityandLorentzsymmetryassumedpreserved,asusual,onlythescalarH condenses
in the resulting breaking which will give the usual electroweak breaking pattern. From the avail-
abledeformedspectracomputationsitisnothardtoenvisionthatagapofafactorof,say,5-10can
develop between the lightest massive scalar (physical Higgs), plus possibly aWIMP-like particle,
and the higher massive states resulting from this breaking. Interactions with the dark sector enter
naturally herethroughthe‘Higgsportal’.
Forthedarksectoritselfonemayintroduceother‘dark’gaugeinteractionsamongthey ’sso
a
that,throughColeman-Weinberg, fermioncondensatesorothermechanisms,itisrenderedmassive
orconsists ofmassiveandunparticle sectors.
MassesfortheordinarySMquarkcouldbeintroducedbytheusualmechanismofeffective4-
fermiinteractionsbetweentheQfermionsandtheSMquarksatahighscale. Theinterpolatingfield
matchingtotheQ-compositesconvertssuchinteractions toeffectiveYukawacouplingsfortheSM
quarks. TheseprovidethequarkmassesasintheSMafterthecompositeH fieldcondensesviathe
Coleman-Weinberg mechanism. (Notethatnotechniquark condensate, asinextendedTC,atsome
intermediate scale, withitsattending fine-tuning problems, needhere beinvoked.) Thenumberof
boson degrees of freedom present generally suffices to allow the Coleman-Weinberg mechanism
to be implemented also after the introduction of the quarks. The effective theory describing such
models would infact besimilar tomanymodelsinthe literature withelementary scalar fieldsand
electroweak breaking àlaColeman-Weinberg inthepresence ofadarksector driven bypotentials
ofthetype(4.2)[18].
5. Conclusions
Thesearchfornon-trivialFT’singaugetheoriesisstillatanearlystage. Despiteconsiderable
efforts already devoted to it, there are, as we saw, many unanswered questions concerning the
9
IR-conformalgaugetheoriesandcompositeHiggs E.T.Tomboulis
occurrence of IR, but also UV, nontrivial FT’s both inside and outside the putative "conformal
window" in non-abelian theories with simple color group. More general gauge theories coupled
to large number of fermion flavors have in fact hardly been considered. In few cases where the
existenceofanIRFTcanbereasonablyascertained,however,latticecomputationsofthespectrum
haveprovidedsignificantfindings,inparticular, thenon-QCD-likemassorderingofstateswiththe
scalar states being the lightest. As suggested here, these naturally light composite scalar states
(massless in the limit of removing any relevant conformal deformation) provide the setting for
modelsofelectroweakbreakingwheretheHiggsisone(oralinearcombination)ofthesecomposite
scalars. These then are composite Higgs models very different from the Higgs as NG boson or
walking TC models. As noted above their effective description would be more similar to that of
models with scalar (but now composite as opposed to elementary) fields and Coleman-Weinberg
breaking inthepresenceofadarksector.
Insummary,itshouldbeclearthatexistenceofIR-conformaltheoriesandtheassociatedpres-
enceoflightcompositescalarsopensupthepossibilityforavarietyofnewcompositeHiggsmod-
els. The real difficulty is identifying the FT structure and spectrum of a candidate IR-conformal
theoryandextractingthequantitativeinformationfromitneededformodelbuildingineachpartic-
ularinstance. Withpresently available techniques, thisgenerally requires extensive computational
effort.
Theauthorwouldliketoacknowledge theAspenCenterforPhysicsforhospitality andthank
theparticipantsoftheoftheworkshop"LGTattheLHCera",inparticular,F.Sannino,Y.Meurice,
A.Hasenfratz, M.P.Lombarto,B.Lucini,E.Pallante,G.FlemingandE.Neilfordiscussions.
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10