light 0++ overview near-conformal pNGB summary Review on Composite Higgs Models Oliver Witzel Lattice 2018 East Lansing, MI, USA, July 24, 2018 1 / 26
overview
light 0++ overview near-conformal pNGB summary Experimental observations -1 -1 CMS s = 7 TeV, L = 5.1 fb s = 8 TeV, L = 5.3 fb ◮ Discovery of the Higgs boson in 2012 S/(S+B) Weighted Events / 1.5 GeV Events / 1.5 GeV [Atlas PLB716(2012)1] [CMS PLB716(2012)30] Unweighted 1500 1500 ◮ Higgs boson → M H 0 = 125 . 18(16) GeV [PDG 2018] 1000 → Spin 0 preferred over spin 2; spin 1 excluded ( H 0 → γγ ) 1000 120 130 m (GeV) → CP difficult to determine (mixing of e/o eigenstates) γ γ → SM decay width too small for LHC measurement Data 500 S+B Fit → Improving precision on coupling to SM particles B Fit Component 1 ± σ 2 ± σ 0 110 120 130 140 150 ◮ So far no other states found m (GeV) γ γ ⇒ No supersymmetric particles ⇒ No heavier resonances → What is the origin of the electro-weak sector? � Maybe new resonances of a few TeV? 2 / 26
light 0++ overview near-conformal pNGB summary General idea: composite Higgs models ◮ Extend the Standard Model by a new, strongly coupled gauge-fermion system ◮ The Higgs boson arises as bound state of this new sector → Mass and quantum numbers match experimental values when accounting for SM interactions/corrections ◮ System exhibits a large separation of scales → Explaining why a 125 GeV Higgs boson but no other states have been found → Indications that such a system cannot be QCD-like (e.g. quark mass generation) � near-conformal gauge theories ◮ Exhibits mechanism to generate masses for SM fermions and gauge bosons ◮ In agreement with electro-weak precision constraints (e.g. S-parameter)? 3 / 26
light 0++ overview near-conformal pNGB summary Composite Higgs models ◮ Aim: describe states of the SM as well as particles originating from new physics L UV → L SD + L SM 0 + L int → L SM + . . . 4 / 26
light 0++ overview near-conformal pNGB summary Composite Higgs models ◮ Aim: describe states of the SM as well as particles originating from new physics ◮ Start with a Higgs-less, massless SM L UV → L SD + L SM 0 + L int → L SM + . . . 4 / 26
light 0++ overview near-conformal pNGB summary Composite Higgs models ◮ Aim: describe states of the SM as well as particles originating from new physics ◮ Start with a Higgs-less, massless SM ◮ Add new strong dynamics coupled to SM L UV → L SD + L SM 0 + L int → L SM + . . . ✻ full SM + states from L SD ◮ Leads to an effective theory giving mass to → the SM gauge fields → the SM fermions fields: 4-fermion interaction or partial compositeness 4 / 26
light 0++ overview near-conformal pNGB summary Composite Higgs models ◮ Aim: describe states of the SM as well as particles originating from new physics ◮ Start with a Higgs-less, massless SM ◮ Add new strong dynamics coupled to SM L UV → L SD + L SM 0 + L int → L SM + . . . ✻ full SM + states from L SD ◮ Leads to an effective theory giving mass to → the SM gauge fields → the SM fermions fields: 4-fermion interaction or partial compositeness ◮ Does not explain mass of L SD fermions and 4-fermion interactions: L UV 4 / 26
light 0++ overview near-conformal pNGB summary Two scenarios for a composite Higgs ◮ Light iso-singlet scalar (0 ++ ) ◮ pseudo Nambu Goldstone Boson (pNGB) → “Dilaton-like” → Spontaneous breaking of flavor symmetry → Scale: F π = SM vev ∼ 246 GeV ⇒ N f ≥ 3 → ideal 2 massless flavors → Mass emerges from its interactions ⇒ giving rise to 3 Goldstone bosons → Non-trivial vacuum alignment ⇒ longitudinal components of W ± and Z 0 F π = (SM vev) / sin( χ ) > 246 GeV ◮ 2-flavor sextet [LatHC, CP3] ◮ Two-representation model by Ferretti [TACoS] (Kuti Wed 2:00 PM, Wong Wed 2:20 PM) (Jay Thu 12:20 PM) ◮ 8-flavor fundamental [LatKMI, LSD] ◮ Mass-split models [4+8, LSD] (Rebbi Thu 11:00 AM, Neil Thu 12:00 PM) ◮ SU(4)/Sp(4) composite Higgs [Bennett et al.] ◮ 2-flavor fundamental [Drach et al.] (Lee Tue 2:20 PM) see appendix see appendix 5 / 26
near-conformal gauge theories
light 0++ overview near-conformal pNGB summary Near-conformal gauge theories ◮ Gauge-fermion system with N c ≥ 2 colors and N f flavors in some representation β ✻ g 0 g FP ✲ > > > > < < g IR freedom ✦ ✻ ✦✦✦✦✦✦✦✦✦✦✦ ✛ conformal window N f ✦ ✦✦✦✦✦✦✦✦✦✦✦✦✦✦ chirally broken phase β ✻ g 0 g ren ✲ > > > > ✛ g ✲ N c 6 / 26
light 0++ overview near-conformal pNGB summary Near-conformal gauge theories ◮ Gauge-fermion system with N c ≥ 2 colors and N f flavors in some representation β ✻ g 0 g FP ✲ > > > > < < g IR freedom ✦ ✻ ✦✦✦✦✦✦✦✦✦✦✦ ✛ conformal window N f ✦ ✦✦✦✦✦✦✦✦✦✦✦✦✦✦ near-conformal chirally broken phase β ✻ g 0 g ren ✲ > > > > ✛ g ✲ N c 6 / 26
light 0++ overview near-conformal pNGB summary Conformal window fundamental ◮ Indications of the conformal window for different representations, N c , and N f [Dietrich, Sannino PRD75(2007)085018] ◮ Derived from perturbative and Schwinger-Dyson arguments ◮ Lower bonds of conformal window typically require nonperturbative two-index antisymm. calculations two-index symm. adjoint 7 / 26
light 0++ overview near-conformal pNGB summary SU(2) gauge theory with fermions in the adjoint representation ◮ Nonperturbative investigations include e.g. → Scaling of hadron masses → Mode number of Dirac operator ⇒ Determination of the anomalous dimension (Scior poster) ◮ Conclusions 25 32 3 × 64 → N f = 2 is conformal [Bergner et al. PRD96(2017)034504] 24 3 × 48 16 3 × 32 48 3 × 96 20 → N f = 1 likely conformal [Athenodorou et al. PRD91(2015)114508] Lm P S L √ σ Lm V → N f = 1 / 2 (1 Majorana fermion) is QCD-like 15 LM [Bergner et al. JHEP03(2016)080] 10 → N f = 3 / 2 (3 Majorana fermions) is conformal Mode number: γ ∗ ≈ 0 . 38(2); fit spectrum γ ∗ ≈ 0 . 37(2) 5 0 1 2 3 4 5 6 7 8 9 10 ◮ Mixed fundamental-adjoint action (Bergner Fri 5:10 PM) ( L /a )( am PCAC ) 1 / (1+0 . 38) � Investigations of supersymmetric QCD [Bergner et al. JHEP01(2018)119] 8 / 26
light 0++ overview near-conformal pNGB summary The step-scaling β -function 0 1-loop 2-loop ◮ IRFP: β function has zero for g 2 > 0 − 0 . 5 Schrödinger Functional Gradient Flow − 1 ◮ For large g 2 nonperturbative methods are required − 1 . 5 0 ◮ Calculate discretized β function (step scaling) − 2 β ( g ) − 0 . 05 − 2 . 5 → Requires calculations on a set of different volumes − 0 . 1 − 3 − 0 . 15 → Well established in QCD [L¨ uscher et al. NPB359(1991)221] − 0 . 2 − 3 . 5 − 0 . 25 − 4 − 0 . 3 0 . 08 0 . 1 0 . 12 0 . 14 0 . 16 0 . 18 0 . 2 − 4 . 5 0 0 . 2 0 . 4 0 . 6 0 . 8 1 ◮ Gradient flow step scaling [L¨ α uscher JHEP08(2010)071] [Dalla Brida et al. PRD95(2017)014507] [Fodor et al. JHEP11(2012)007][Fodor et al. JHEP09(2014)018] √ 128 π 2 c ; L ) = g 2 c ( sL ) − g 2 ( L ) 1 g 2 C ( c , L ) t 2 � E ( t ) � s ( g 2 β c c ( L ) = with 8 t = c · L ; 3( N 2 log( s 2 ) c − 1) ◮ Extrapolate L → ∞ to remove discretization effects and take the continuum limit ◮ Expect to find agreement for results based on different actions, flows, operators . . . 9 / 26
light 0++ overview near-conformal pNGB summary The challenge of establishing an IRFP (Nogradi Wed 3:00 PM) [Fodor et al. JHEP09(2015)039] [Fodor et al. PLB779(2018)230] SU(3) N f = 10 c = 3/10 s = 2 SSC s=2 c=0.25 continuum limit -function 0.8 0.3 1.2 non-perturbative staggered 1 loop Hasenfratz,Rebbi,Witzel 0.7 0.25 Ting-Wai Chiu ( g 2 (sL) - g 2 (L) ) / log(s 2 ) 2 loop 1 ( g 2 (sL) - g 2 (L) ) / log(s 2 ) 0.6 Lattice Higgs Collab 0.2 5-loop 5-loop 2 ) 0.8 0.5 (g 2 (sL)-g 2 (L))/log(s 0.15 0.4 staggered staggered 0.6 0.1 0.3 0.4 0.05 0.2 4-loop Ref. 3 IRFP 0 DWF 0.2 0.1 c =0.35, s =1.5 precision tuning and targeted interpolation combined -0.05 0 0 a 4 /L 4 cutoff effects included 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 -0.1 g 2 (L) 5.5 6 6.5 7 7.5 8 g 2 (L) g 2 (L) N f = 2 sextet N f = 12 fundamental N f = 10 fundamental 0.4 0.30 1.2 2-loop nZS c=0.25 2-loop s =2 c = 0.35; L = 12-24 0.25 4-loop 4-loop τ 0 = 0.0 0.3 1.0 5-loop 5-loop 2-loop series 0.20 nZS c=0.4 s =2 8-16 2-loop perturb. staggered Stg 0.25 4-loop MS 10-20 0.8 0.2 0.15 8-16 12-24 2 ) 10-20 β 3/2 (g 2 0.10 2 0.6 12-24 0.1 14-28 0.05 16-32 Wilson 0.4 L lin 0.00 0 L quad 0.2 0.05 DWF DWF PRELIMINARY PRELIMINARY -0.1 0.10 0 1 2 3 4 5 6 0.0 0 1 2 3 4 5 6 7 8 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 2 g 2 g c g 2 c c [Hasenfratz et al. 1507.08260] [Hasenfratz, Rebbi, Witzel 1710.11578] (Hasenfratz poster) 10 / 26
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