4 IR-conformal gauge theories and composite Higgs 1 0 2 n a J 9 1 E. T.Tomboulis ∗ Dept. ofPhysicsandAstronomy,UniversityofCalifornia,LosAngeles ] h LosAngeles,CA90095,USA p E-mail: tomboulis@physics.ucla.edu - p e h Theexistenceofnon-trivialIRandUVfixedpointsingaugetheoriesasafunctionofthenumber [ offermionflavorsandbarecouplingisdiscussedinthelightofrecentwork.Itispointedoutthat 1 infactonlyasmallsubsetofpotentialIR-conformalgaugetheories,i.e.theorieswhoseIRbehav- v 7 iorisdeterminedbyanIRfixedpoint,hassofarbeenexamined. Recentlatticecomputationsof 2 thespectruminsomecaseswhereexistenceofanIRfixedpointisreasonablyassured,however, 6 4 reveala non-QCD-likespectrumwiththelighteststatesbeingscalars. Itissuggestedthatthese . 1 naturallylightcompositescalarsprovideanaturalsettingfortheconstructionofnewcomposite 0 Higgsmodels. Aschematicoutlineofsuchamodelisgiven. 4 1 : v i 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 = 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. 3 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 ). 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

IR-conformal gauge theories and composite Higgs

Similar IR-conformal gauge theories and composite Higgs