Review on Composite Higgs Models Oliver Witzel Lattice 2018 East - - PowerPoint PPT Presentation

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near-conformal light 0++ pNGB summary

Review on Composite Higgs Models

Oliver Witzel Lattice 2018 East Lansing, MI, USA, July 24, 2018

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near-conformal light 0++ pNGB summary

Experimental observations

(GeV)

γ γ

m

110 120 130 140 150

S/(S+B) Weighted Events / 1.5 GeV

500 1000 1500

Data S+B Fit B Fit Component σ 1 ± σ 2 ±

• 1

= 8 TeV, L = 5.3 fb s

• 1

= 7 TeV, L = 5.1 fb s CMS (GeV)

γ γ

m

120 130

Events / 1.5 GeV

1000 1500 Unweighted

◮ Discovery of the Higgs boson in 2012 [Atlas PLB716(2012)1] [CMS PLB716(2012)30] ◮ Higgs boson → MH0 = 125.18(16) GeV [PDG 2018] → Spin 0 preferred over spin 2; spin 1 excluded (H0 → γγ) → CP difficult to determine (mixing of e/o eigenstates) → SM decay width too small for LHC measurement → Improving precision on coupling to SM particles ◮ So far no other states found ⇒ No supersymmetric particles ⇒ No heavier resonances → What is the origin of the electro-weak sector? Maybe new resonances of a few TeV?

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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)?

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near-conformal light 0++ pNGB summary

Composite Higgs models

◮ Aim: describe states of the SM as well as particles originating from new physics

LUV → LSD + LSM0 + Lint → LSM + . . .

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near-conformal light 0++ 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

LUV → LSD + LSM0 + Lint → LSM + . . .

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near-conformal light 0++ 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

LUV → LSD + LSM0 + Lint → LSM + . . .

✻ full SM + states from LSD

◮ Leads to an effective theory giving mass to → the SM gauge fields → the SM fermions fields: 4-fermion interaction or partial compositeness

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near-conformal light 0++ 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

LUV → LSD + LSM0 + Lint → LSM + . . .

✻ full SM + states from LSD

◮ 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 LSD fermions and 4-fermion interactions: LUV

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near-conformal light 0++ pNGB summary

Two scenarios for a composite Higgs

◮ Light iso-singlet scalar (0++) → “Dilaton-like” → Scale: Fπ = SM vev ∼ 246 GeV → ideal 2 massless flavors ⇒ giving rise to 3 Goldstone bosons ⇒ longitudinal components of W ± and Z 0 ◮ 2-flavor sextet [LatHC, CP3]

(Kuti Wed 2:00 PM, Wong Wed 2:20 PM)

◮ 8-flavor fundamental [LatKMI, LSD]

(Rebbi Thu 11:00 AM, Neil Thu 12:00 PM)

◮ 2-flavor fundamental [Drach et al.]

see appendix

◮ pseudo Nambu Goldstone Boson (pNGB) → Spontaneous breaking of flavor symmetry ⇒ Nf ≥ 3 → Mass emerges from its interactions → Non-trivial vacuum alignment

Fπ = (SM vev)/ sin(χ) > 246 GeV

◮ Two-representation model by Ferretti [TACoS]

(Jay Thu 12:20 PM)

◮ Mass-split models [4+8, LSD] ◮ SU(4)/Sp(4) composite Higgs [Bennett et al.]

(Lee Tue 2:20 PM) see appendix

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near-conformal gauge theories

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near-conformal light 0++ pNGB summary

Near-conformal gauge theories

◮ Gauge-fermion system with Nc ≥ 2 colors and Nf flavors

in some representation ✲

Nf

✻

Nc

✦✦✦✦✦✦✦✦✦✦✦ ✦ ✦✦✦✦✦✦✦✦✦✦✦✦✦✦ ✦

IR freedom conformal window chirally broken phase

✲

β ✻ g

g0 gren

> > > > ✲

β ✻ g

g0 gFP

> > > > < < ✛ ✛

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near-conformal light 0++ pNGB summary

Near-conformal gauge theories

◮ Gauge-fermion system with Nc ≥ 2 colors and Nf flavors

in some representation ✲

Nf

✻

Nc

✦✦✦✦✦✦✦✦✦✦✦ ✦ ✦✦✦✦✦✦✦✦✦✦✦✦✦✦ ✦

IR freedom conformal window chirally broken phase

✲

β ✻ g

g0 gren

> > > > ✲

β ✻ g

g0 gFP

> > > > < < ✛ ✛

near-conformal

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near-conformal light 0++ pNGB summary

Conformal window

fundamental two-index antisymm. two-index symm. adjoint

◮ Indications of the conformal window

for different representations, Nc, and Nf [Dietrich, Sannino PRD75(2007)085018]

◮ Derived from perturbative and

Schwinger-Dyson arguments

◮ Lower bonds of conformal window

typically require nonperturbative calculations

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near-conformal light 0++ 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

5 10 15 20 25 1 2 3 4 5 6 7 8 9 10 LM (L/a)(amPCAC)1/(1+0.38) 323 × 64 243 × 48 163 × 32 483 × 96 LmP S L√σ LmV

(Scior poster)

[Bergner et al. JHEP01(2018)119] ◮ Conclusions → Nf = 2 is conformal [Bergner et al. PRD96(2017)034504] → Nf = 1 likely conformal [Athenodorou et al. PRD91(2015)114508] → Nf = 1/2 (1 Majorana fermion) is QCD-like [Bergner et al. JHEP03(2016)080] → Nf = 3/2 (3 Majorana fermions) is conformal

Mode number: γ∗ ≈ 0.38(2); fit spectrum γ∗ ≈ 0.37(2)

◮ Mixed fundamental-adjoint action (Bergner Fri 5:10 PM) Investigations of supersymmetric QCD

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near-conformal light 0++ pNGB summary

The step-scaling β-function

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0.2 0.4 0.6 0.8 1 β(g) α 1-loop 2-loop Schrödinger Functional Gradient Flow 0.08 0.1 0.12 0.14 0.16 0.18 0.2 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05

[Dalla Brida et al. PRD95(2017)014507] ◮ IRFP: β function has zero for g 2 > 0 ◮ For large g 2 nonperturbative methods are required ◮ Calculate discretized β function (step scaling) → Requires calculations on a set of different volumes → Well established in QCD [L¨ uscher et al. NPB359(1991)221] ◮ Gradient flow step scaling [L¨ uscher JHEP08(2010)071] [Fodor et al. JHEP11(2012)007][Fodor et al. JHEP09(2014)018]

g 2

c (L) =

128π2 3(N2

c − 1)

1 C(c, L)t2E(t) with √ 8t = c · L; βc

s (g 2 c ; L) = g 2 c (sL) − g 2(L)

log(s2)

◮ Extrapolate L → ∞ to remove discretization effects and take the continuum limit ◮ Expect to find agreement for results based on different actions, flows, operators . . .

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The challenge of establishing an IRFP

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 3 4 5 6 7 ( g2(sL) - g2(L) ) / log(s2) g2(L) non-perturbative 1 loop 2 loop

5.5 6 6.5 7 7.5 8

g2(L)

• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.25 0.3

(g2(sL)-g2(L))/log(s

2)

SSC s=2 c=0.25 continuum limit

• function
• Ref. 3 IRFP

precision tuning and targeted interpolation combined a4/L4 cutoff effects included 4-loop 5-loop

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 9 ( g2(sL) - g2(L) ) / log(s2) g2(L) SU(3) Nf = 10 c = 3/10 s = 2 Hasenfratz,Rebbi,Witzel Ting-Wai Chiu Lattice Higgs Collab 5-loop 1 2 3 4 5 6 gc

2

• 0.1

0.1 0.2 0.3 0.4 β3/2(g

2) τ0 = 0.0 2-loop series 2-loop perturb. 4-loop MS

c = 0.35; L = 12-24

1 2 3 4 5 6 7 8

g2

c 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 2

nZS c=0.25

PRELIMINARY 2-loop 4-loop 5-loop Stg 0.25 8-16 10-20 12-24 14-28 16-32 L lin L quad

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 g2

c

0.0 0.2 0.4 0.6 0.8 1.0 1.2

2

nZS c=0.4 PRELIMINARY

2-loop 4-loop 5-loop 8-16 10-20 12-24

[Fodor et al. JHEP09(2015)039] [Fodor et al. PLB779(2018)230]

(Nogradi Wed 3:00 PM) Nf = 2 sextet Nf = 12 fundamental Nf = 10 fundamental

[Hasenfratz et al. 1507.08260]

(Hasenfratz poster)

[Hasenfratz, Rebbi, Witzel 1710.11578]

staggered

c=0.35, s=1.5

staggered staggered DWF Wilson staggered DWF

s=2

DWF

s=2 10 / 26

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near-conformal light 0++ pNGB summary

Active research

◮ Larger volumes might be required for L → ∞ extrapolation → Small c-values certainly require larger volumes than larger c-values Larger c-values have larger statistical uncertainties → Different actions have different discretization errors ◮ DWF with Symanzik gauge action feature a fully O(a2) improved set-up `

a la Symanzik

→ Zeuthen flow [Ramos, Sint EPJC76(2016)15] → Symanzik operator → Perturbative tree-level normalization [Fodor et al. JHEP09(2014)018] works for Nf = 12 and 10 Perturbative improvement breaks down for staggered with Nf = 8 [Fodor et al JHEP06(2015)19] ◮ [Rooted] Staggered Fermions: Good, Bad or Ugly? [Sharpe Plenary Lattice 2006] Are staggered and DW/Wilson fermions in conformal systems in the same universality class?

(Hasenfratz poster, Kuti Wed 2:00 PM, Holland Wed 2:40 PM, Nogradi Wed 3:00 PM)

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Higgs as a light 0++ scalar

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near-conformal light 0++ pNGB summary

SU(3) with Nf = 2 sextet flavors (two-index symmetric representation)

◮ Minimal flavor content to describe EW symmetry breaking ◮ Likely very close to the onset of the conformal window

0.01 0.02 0.03 0.04 0.05 Mπ

2

2 4 6 8 10 12 14 M / Fπ a0 π f0 β=3.25 0.01 0.02 0.03 0.04 0.05 Mπ

2

2 4 6 8 10 12 14 M / Fπ 0.5 1 1.5 2 2.5 3 M / TeV N a1 ρ β=3.20 Decreasing Mπ Decreasing Mπ

[Fodor et al. PoS LATTICE2015 219] ◮ LatHC → Chirally broken spectrum [Fodor et al. PLB718(2012)657] → 0++ (f0) is light → No IRFP in the explored range of the β-function [Fodor et al. JHEP09(2015)039]

(Kuti Wed 2:00 PM, Wong Wed 2:20 PM)

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near-conformal light 0++ pNGB summary

SU(3) with Nf = 2 sextet flavors (two-index symmetric representation)

◮ Minimal flavor content to describe EW symmetry breaking ◮ Likely very close to the onset of the conformal window ◮ Hansen, Drach, Pica → Two different phases one chirally broken,

• ne looking IR conformal [Hansen, Drach, Pica PRD96(2017)0345]

◮ Hasenfratz, Liu, Yu-Han Huang → Indications for a possible IRFP [Hasenfratz et al. 1507.08260]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

• ❱

• ✁

β = 5.1 β = 5.2 β = 5.3 β = 5.4 β = 5.5

MV /MPS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

• ❱

• ✁

β = 3.0 β = 4.0 β = 4.6 β = 4.8 β = 5.0

MV /MPS

[Hansen, Drach, Pica PRD96(2017)0345]

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near-conformal light 0++ pNGB summary

SU(3) with Nf = 8 fundamental flavors

◮ Theory considered to be chirally broken but close to the onset of the conformal window → Step scaling analysis of the discrete β function [Hasenfratz et al. JHEP06(2015)143][Fodor et al. JHEP06(2015)019] → Finite temperature phase diagram [Deuzeman et al. PLB670(2008)41][Jin, Mawhinney PoS LATTICE2010 055][Schaich et al. PoS LATTICE2012 028] → Studies of the low-lying meson spectrum [Aoki et al. PRD89(2014)111502][Appelquist et al. PRD93(2016)114514][Aoki et al. PRD96(2017)014508] [Appelquist et al. 1807.08411] ◮ Theory has 63 Goldstone boson — not an ideal candidate to explain EW symmetry breaking → Allows to investigate qualitative features of near-conformal gauge theories → Reduce number of light Goldstones by assigning e.g. mass or charge to some flavors ◮ 0++ is light, degenerate with the pion!

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near-conformal light 0++ pNGB summary

Determining the iso-singlet scalar 0++

◮ Receives quark line connected and disconnected contributions → Stochastic estimator (noisy) ◮ Its quantum numbers are the same vacuum → Large vacuum subtraction ◮ Lighter in near-conformal systems than in QCD → Easier to determine, stable particle

(energetically protected from decaying)

◮ Nevertheless most expensive state in the spectrum ◮ Idea: take advantage of correlators at non-zero momenta

to avoid the vacuum subtraction (Rebbi Thu 11:00 PM)

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 10 20 30 40 50 60 70 p=(1,0,0) p=(1,1,0) p=(1,1,1) p=(2,0,0) 14 / 26

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near-conformal light 0++ pNGB summary

Iso-singlet scalar (0++)

0.01 0.02 0.03 0.04 0.05 0.06

mf

0.1 0.2 0.3 0.4 0.5 0.6

MH

σ L=42 σ L=36 σ L=30 σ L=24 σ L=18 π ρ(PV)

LatKMI

[Aoki et al. PRD96(2017)014508]

LSD

[Appelquist et al. PRD93(2016)114514] ◮ 0++ is light, degenerate with the pion ⇒ χPT not applicable

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near-conformal light 0++ pNGB summary

Comparison of SU(3) with Nf = 4 and 8 fundamental flavors

(Neil Thu 12:00 PM, Cushman poster)

[Appelquist et al. 1807.08411]

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

amf

3.0 3.5 4.0 4.5 5.0 5.5 6.0 p

8t0 /a

Nf = 8 Nf = 4

0.05 0.10 0.15 0.20 0.25 0.30 2 4 6 8 10 12 14

MX/Fπ

Nf = 4

0.05 0.10 0.15 0.20

Nf = 8

π a1 ρ N σ

mf/Fπ

◮ Wilson flow scale √8t0 vs fermion mass amf ◮ Strong mass-dependence for Nf = 8 ⇒ Show quantities in units of √8t0 or

dimensionless ratios

◮ Spectrum in units of Fπ ◮ Nf = 8: pion and σ (0++) degenerate

rather different than in Nf = 4

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near-conformal light 0++ pNGB summary

Effective field theories (a selection of recent work)

Holdom, Koniuk: A bound state model for a light scalar [Holdom, Koniuk JHEP12(2017)102]

→ Existence of light scalar, well separated from heavier states related to

existence of near conformal gauge dynamics extending over a wide range

→ Scalar mass and form factor close to parity doubled limit → Light scalar has characteristics different from light dilaton

Appelquist, Ingoldby, Piai: Dilaton EFT [Appelquist, Ingoldby, Piai JHEP07(2017)035][JHEP03(2018)039]

→ Light singlet scalar interpreted as dilaton (spontaneous breaking of conformal symmetry) → Treat dilaton together with pions (spontaneous breaking of chiral symmetry) → Add general form for the dilaton potential to be determined from lattice data → EFT “fits” lattice data (Nf = 8 fundamental and Nf = 2 sextet)

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near-conformal light 0++ pNGB summary

Effective field theories (a selection of recent work)

Golterman, Shamir: The large-mass regime of the dilaton-pion low-energy effective theory

[Golterman, Shamir 1805.00198] ◮ Investigate dilaton-pion EFT in the Veneziano limit (Nf → ∞ for Nf /Ncfixed) ◮ Expand around nf − n∗ f , with nf = Nf /Nc and n∗ f onset of conformal window in Veneziano limit → Small mass region: dilaton decouples from pions, typical chiral behavior → Large mass region: hadron masses, decay constants scale with m1/(1+γ∗) f

(hyperscaling)

⇒ LSD Nf = 8 data is in the large mass region → Explains characteristics of LSD data → To reach small mass region: reduce mf → mf /100 Small mass region may show that Nf = 8 is indeed confining

(Golterman Thu 11:20 PM)

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near-conformal light 0++ pNGB summary

Effective field theories (a selection of recent work)

Meurice: Linear sigma model for multiflavor gauge theories

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Mπ /Mη′

0.4 0.2 0.0 0.2 0.4 0.6 0.8

R

Rσ N =12 RaO N =12 Rσ N =8 RaO N =8

[Meurice PRD96(2017)114507] → EFT describing σ, a0, η′, π → Explicit breaking of axial UA(1) symmetry depends on Nf ⇒ Effect on spectrum and onset of conformal window → Tree-level spectrum leads to dimensionless ratios:

Rσ = (M2

σ − M2 π)/M2 η′ + (1 − 2/Nf )(1 − M2 π/M2 η′)

Ra0 = (M2

a0 − M2 π)/M2 η′ − (2/Nf )(1 − M2 π/M2 η′) → LatKMI data: almost flat, no Nf dependence e.g. bound on Nfc

De Floor, Gustafson, Meurice: Mass splittings in a linear sigma model for multiflavor gauge theories

◮ Consider flavors with two masses m1 and m2 = m1 + δm

(δm small)

→ Spectrum exhibits light-light, heavy-light, and heavy-heavy mesons → If M2 πll < M2 πhl < M2 πhh, then inverse ordering for scalars M2 a0ll > M2 a0hl > M2 a0hh [Floor, Gustafson, Meurice, 1807.05047] (Gustafson poster)

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Higgs as a pseudo Nambu-Goldstone boson

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near-conformal light 0++ pNGB summary

Ferretti’s Model [Ferretti JHEP06(2014)142]

◮ SU(4) gauge theory with fermions in two representations → NW 6

= 5 Weyl massless flavors of sextet (Q) (two-index antisymmetric) fermions with EW charge

→ N4 = 3 fundamental Dirac flavors (q) with color charge ◮ Mesons → sextet QQ, Q ¯

Q, ¯ Q ¯ Q pNGBs, vectors

→ fundamental q¯

q pNGBs, vectors

◮ Baryons ◮ sextet QQQQQQ bosons ◮ fundamental qqqq bosons ◮ chimera Qqq fermions ◮ Ferretti limit (m6 → 0) Higgs is a massless sextet NGB, its potential arises from SM interactions ◮ Fermion acquire mass from quartic mixing u¯

uH → u¯ uQQ

◮ Non-anomalous superposition of UA(4)(1) and UA(6)(1) → axial singlet pNGB (ζ meson) ◮ top quark mixes linearly with chimera ⇒ large mt

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near-conformal light 0++ pNGB summary

Adaption of Ferretti’s model on the lattice

[Ayyar et al. PRD97(2018)074505][PRD97(2018)114502][PRD97(2018)114505]

8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 0.120 0.122 0.124 0.126 0.128 0.130 0.132 0.134

4

Confined & Broken Deconfined & Restored Ambiguous m4 < 0 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 0.1150 0.1175 0.1200 0.1225 0.1250 0.1275 0.1300 0.1325 0.1350

6

Confined & Broken Deconfined & Restored m6 < 0

[Ayyar et al. PRD97(2018)114502] ◮ SU(4) gauge theory → N6 = 2 Dirac flavors (N4 6 = 4 Weyl) sextet flavors → N4 = 2 fundamental Dirac flavors ◮ Finite temperature phase diagram: only two phases

Low-temperature both fermion species confined and chirally broken High-temperature both fermion species deconfined and chirally restored

◮ Single phase transition appears to be first order

as theoretically predicted

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near-conformal light 0++ pNGB summary

Results in the Ferretti limit (m6 → 0)

0.00 0.02 0.04 0.06 0.08 0.10 0.12

m4

p

t0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

M 2t0

Fundamental U(1) Goldstone

[Ayyar et al. PRD97(2018)074505] ◮ ζ meson → MPS6 = 0 (sextet pNGB exactly massless) → Mζ < MPS4 → ζ meson lightest, massive state → Reconstruct Mζ from chiral fit

(function of m4 and m6)

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near-conformal light 0++ pNGB summary

Results in the Ferretti limit (m6 → 0)

◮ Spectrum in units of F6 → (1/2,0) chimera (Qqq) is top partner

and lightest baryon

→ Experimental constraint: F6 1.1 TeV ⇒ Mass of top partner chimera M 6.5 TeV ◮ Further details (Jay Thu 12:20 PM)

5 10 15 20

(MP4/F6)2

5 10 15 20

M/F6

Sextet J=0 Fundamental J=0 Chimera (J,I)=(1/2,0) Fundamental Pseudoscalar Fundamental Vector Sextet Vector

[Ayyar et al. PRD97(2018)114505]

sextet J = 0 baryon f u n d a m e n t a l J = b a r y

• n

top partner chimera fundamental vector meson sextet vector meson fundamental pNGB

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near-conformal light 0++ pNGB summary

Mass-split models

◮ Promising candidates are chirally broken in the IR but conformal in the UV [Luty, Okui JHEP09(2006)070], [Dietrich, Sannino PRD75(2007)085018], [Vecchi 1506.00623], [Ferretti, Karateev JHEP03(2014)077]

UV ✲ IR ΛUV ΛIR conformal fermion masses chirally broken Higgs dynamics

◮ Mass-split models e.g. SU(3) gauge theory with “heavy” and “light” (massless) fundamental flavors ◮ Nℓ = 4 light flavors are chirally broken in the IR ◮ Add Nh heavy flavors to push the system

near an IRFP of a conformal theory ❄ fundamental composite 2HDM with 4 flavors in SU(3) gauge [Ma, Cacciapaglia JHEP03(2016)211] ❄ heavy flavors could be invisible to SM

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near-conformal light 0++ pNGB summary

The mass-split paradigm

◮ In QCD: g 2 → 0 (continuum limit); fermion mass mf → 0 (chiral limit) ◮ Theory with degenerate Nf = Nh + Nℓ is (mass-deformed) conformal and exhibits an IRFP ◮ All ratios of hadron masses scale with the anomalous dimension (hyperscaling) → Continuum limit is taken by sending fermion mass mf → 0 ◮ Mass-split models live in the basin of attraction of the IRFP of Nf degenerate flavors → Inherit hyperscaling of ratios of hadron masses but are chirally broken → Continuum limit: mh → 0 keeping mℓ/mh fixed → Chiral limit: mℓ → 0 i.e. mℓ/mh → 0 → Gauge coupling is irrelevant → No free parameters after taking the chiral and continuum limit,

but light-light, heavy-light, and heavy-heavy bound states

[Hasenfratz, Rebbi, Witzel PLB773(2017)86]

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near-conformal light 0++ pNGB summary

Results for four light and eight heavy flavors

0.2 0.4 0.6 0.8 2 4 6 8 10 12 14

MH/Fπ

Mπ/Fπ M̺/Fπ Ma1/Fπ

mℓ/mh

amh = 0.05, β = 4.0 amh = 0.06, β = 4.0 amh = 0.08, β = 4.0 amh = 0.10, β = 4.0 amh = 0.07, β = 4.4

0.2 0.4 0.6 0.8

Ma0/Fπ M0++/Fπ Mn/Fπ

mℓ/mh

amh = 0.05, β = 4.0 amh = 0.06, β = 4.0 amh = 0.08, β = 4.0 amh = 0.10, β = 4.0 amh = 0.07, β = 4.4

◮ Hyperscaling in the light-light sector ◮ Mn/Fπ ≈ 11 ◮ M̺/Fπ ≈ 8 ◮ M0++/Fπ ≈ 4 − 5 → taking the chiral limit is difficult

but 0++ well separated from the ̺ and degenerate with the pion

◮ Statistical errors only ◮ “Scatter” indicates corrections

to scaling

◮ Gauge coupling is irrelevant [Brower et al. PRD 93 (2016) 075028]

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near-conformal light 0++ pNGB summary

Results for four light and eight heavy flavors

pseudoscalar

PDG 4 8 12 16 20 24 28 32 36

Mps/Fπ

π ηs ηc

mℓ/

0.2 0.4 0.6 0.8 Mπ/Fπ Mhh

π /Fπ

mℓ/mh

amh = 0.05, β = 4.0 amh = 0.06, β = 4.0 amh = 0.08, β = 4.0 amh = 0.10, β = 4.0 amh = 0.07, β = 4.4

12f avg

vector

PDG 4 8 12 16 20 24 28 32 36

Mvt/Fπ

̺ φ J/ψ

mℓ/

0.2 0.4 0.6 0.8 M̺/Fπ Mhh

̺ /Fπ

mℓ/mh

amh = 0.05, β = 4.0 amh = 0.06, β = 4.0 amh = 0.08, β = 4.0 amh = 0.10, β = 4.0 amh = 0.07, β = 4.4

12f avg

axial

PDG 4 8 12 16 20 24 28 32 36

Max/Fπ

a1 f1 χc1

mℓ/

0.2 0.4 0.6 0.8 Ma1/Fπ Mhh

a1 /Fπ

mℓ/mh

amh = 0.05, β = 4.0 amh = 0.06, β = 4.0 amh = 0.08, β = 4.0 amh = 0.10, β = 4.0 amh = 0.07, β = 4.4

12f avg

◮ 4+8 heavy-heavy spectrum is not QCD-like; QCD is not hyperscaling ◮ Mhh/Fπ increases but Fπ is finite in the chiral limit ◮ Mhh ̺ ∼ 3M̺ ⇒ could be accessible at the LHC ◮ Data at β = 4.0 and 4.4: gauge coupling is irrelevant [Hasenfratz, Rebbi, Witzel PLB773(2017)86]

23 / 26

• verview

near-conformal light 0++ pNGB summary

Results for four light and eight heavy flavors

◮ The system is chirally broken

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 12 flavors

M̺/Mπ mℓ/mh

amh = 0.05, β = 4.0 amh = 0.06, β = 4.0 amh = 0.08, β = 4.0 amh = 0.10, β = 4.0 amh = 0.07, β = 4.4 0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2

a2

⋆ · M2 π

mℓ/mh

amh = 0.05, β = 4.0 amh = 0.06, β = 4.0 amh = 0.08, β = 4.0 amh = 0.10, β = 4.0 amh = 0.07, β = 4.4 0.2 0.4 0.6 0.8 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

a⋆Fπ mℓ/mh

amh = 0.05, β = 4.0 amh = 0.06, β = 4.0 amh = 0.08, β = 4.0 amh = 0.10, β = 4.0 amh = 0.07, β = 4.4

◮ All data points in a⋆ units ◮ a⋆Fπ is finite ◮ Linearity in M2 π for small mℓ ◮ QCD: md/ms = 4.7/96 ≈ 0.05 ◮ Nf = 4 (QCD-like): ratio diverges ◮ Nf = 12: almost constant ratio [Cheng at al. PRD90(2014)014509]

23 / 26

• verview

near-conformal light 0++ pNGB summary

Outlook

0.1 0.2 0.3 0.4 0.5

• mℓ/

mh

0.2 0.4 0.6 0.8

MP S/Mhh

V T

M hl

PS/M hh V T

M hh

PS/M hh V T

M ll

PS/M hh V T

amh = 0.200 amh = 0.175 amh = 0.150

pseudoscalar preliminary

◮ Mass-split models using 4 light and 6 heavy flavors of MDWF → If degenerate Nf = 10 is conformal, expect to see hyperscaling → First data with eventually large systematics look promising → Nf = 10 would have larger anomalous dimension ◮ Simpler to calculate phenomenologically interesting quantities → Generation of mass for SM fermions (partial compositeness,

four-fermion interaction

→ Baryon anomalous dimension e.g. via new gradient flow method [Carosso et al. 1806.01385] (Hasenfratz Fri 4:50 PM) → S-parameter [Appelquist et al. PRL106(2011)231601], Higgs-potential, . . . ◮ Combine two representations with mass-split model

24 / 26

summary

• verview

near-conformal light 0++ pNGB summary

Summary

◮ The experiments will tell us whether the Higgs is a composite particle → Performing nonperturbative simulations we can guide experimentalists and model builders → Even (old) QCD calculations can be useful (DeGrand Thu 11:40 AM) ◮ Proposal of a new, alternative Higgs mechanism based on dynamical mass generation

(Garofalo Mon 3:00 PM and Frezzotti Mon 3:20 PM), appendix

◮ Simulating near-conformal systems is more costly than QCD but can be as controversial → Particular challenge: identifying an IRFP at strong coupling ◮ Simulations of near-conformal systems revealed a light 0++ with mass M0++ ∼ Mπ → Different effective field theories are required/explored ◮ Models based on two representations or mass-split systems exhibit novel features → E.g. chimera baryons, hyperscaling in a chirally broken system

25 / 26

• verview

near-conformal light 0++ pNGB summary

Acknowledgments

Step-scaling

• A. Hasenfratz, K. Holland, J. Kuti, D. Nogradi

Nf = 8

• A. Hasenfratz, G.T. Fleming, E.T. Neil, E. Rinaldi, C. Rebbi, D. Schaich

EFTs

• T. Appelquist, M. Golterman, R. Koniuk, Y. Meurice

Fundamental Higgs

• A. Maas

Ferretti model

• T. DeGrand, D.C. Hackett, W.I. Jay, E.T. Neil

Mass-split models

• A. Hasenfratz, C. Rebbi

SU(2) adjoint

• G. Bergner

SU(2) fundamental

• V. Drach

Dynamical mass generation

• M. Garofalo, R. Frezzotti

SU(4)/Sp(4)

J.W. Lee

26 / 26

appendix

Nf = 12 step-scaling

[Hasenfratz, Schaich JHEP02(2018)132] ◮ Hasenfratz, Schaich

nHYP-smeared staggered, Wilson gauge w/ adjoint term Wilson flow, clover operator

◮ Lin, Ogawa, Ramos

unimproved staggered, Wilson gauge, twisted BC Wilson flow, clover operator

◮ Fodor et al.

3× stout smeared staggered Symanzik gauge Symanzik flow, clover operator Newer results discussed in a moment!

27 / 26

Nf = 12 step-scaling

1 2 3 4 5 6 7 8

g2

c

0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

2

nZS c=0.25

PRELIMINARY

2-loop 4-loop 5-loop Stg 0.25 8-16 10-20 12-24 14-28 16-32 L lin L quad

[Hasenfratz, Rebbi, Witzel 1710.11578] ◮ Hasenfratz, Rebbi, Witzel

M¨

• bius domain-wall fermions, Symanzik gauge

Zeuthen flow, Symanzik operator

→ Perturbative tree-level normalization [Fodor et al. JHEP09(2014)018]

(working for Nf = 12 and 10)

◮ Result robust → Alternative flow/operators → Without tree-level normalization → Alternative L → ∞ extrapolation → Changing scheme, e.g., c = 0.3

(Hasenfratz poster)

27 / 26

Nf = 12 step-scaling

◮ Page provided by

Julius Kuti

5.5 6 6.5 7 7.5

g2(L)

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

(g2(sL)-g2(L))/log(s

2)

s=2 c=0.20 SSC scheme continuum step

• function
• Ref. 3 IRFP

precision tuning and targeted interpolation combined a4/L4 cutoff effects included 4-loop 5-loop

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

a2/L2

10-3

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

(g2(2L)-g2(L))/log(s

2)

SSC s=2 c=0.2 target F continuum limit

(g2(sL) - g2(L))/log(s

2) = c0 + c1 a2/L2

c0= 0.143 0.038 c1= -98.1 39

2/dof= 0.39

a2/L2

• 0.4
• 0.3
• 0.2
• 0.1

(g2(2L)-g2(L))/log(s 2) SSC s=2 c=0.2 target F continuum limit (g2(sL) - g2(L))/log(s

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

a2/L2

10-3 6.9 6.92 6.94 6.96 6.98 7 7.02 7.04 7.06 7.08 7.1

g2(L) target F s=2 c=0.2 tuning g2 (tuned) = 6.9842 0.0014

2/dof = 0.3 Q = 0.91 new L=32->64

LatHC PLB B779 (2018) 230-236 arXiv:1710.09262 confirmed with new updated results:

L=32 -> L=64 step at several targets adds evidence against nf=12 IRFP Talk: J. Kuti Wed. 14:00 BSM room 104

+ a^4/L^4 term

nf=12 new nf=12 new

consistent with published 27 / 26

Dynamical generation of elementary particle masses — an alternative to the Higgs mechanism

[Dimopoulos, Frezzotti, Garofalo, Kostrzewa, Pittler, Rossi, Urbach]

◮ Non-abelian strongly interacting fermions coupled via Yukawa couplings

to a scalar field and a Wilson-like term

→ Exact symmetry acting on fermions and scalars prevents power divergent fermion mass terms → Fermionic chiral invariance broken by Yukawa and Wilson-like term,

but restored at critical Yukawa coupling

◮ Scalar field with double-well potential → Left-over breaking of chiral symmetry at cutoff scale polarizes vacuum ⇒ spontaneous chiral symmetry breaking dynamically generates PCAC fermion mass ◮ Dynamical fermion mass can be naturally “small” and fermion masses exhibit natural hierarchy ◮ Higgs boson is a composite state in WW + ZZ channel bound by new strongly interacting particles

28 / 26

Dynamical generation of elementary particle masses — an alternative to the Higgs mechanism

◮ Dynamical generation of fermion mass demonstrated by numerical simulations

(Garofalo Mon 3:00 PM)

◮ Electro-weak interactions and how electro-weak boson acquire mass by this mechanism → Dynamical EW symmetry breaking due to a super-strongly sector → No “unnatural” fine tuning of effective four-fermion coupling

(Frezzotti Mon 3:20 PM)

28 / 26

Effects of a fundamental Higgs

[Maas, T¨

• rek]

◮ Physical spectrum must be gauge invariant (in QCD guaranteed by confinement) ◮ Weak sector: perturbative description BRST-invariant, but gauge dependent → Experimental results match predictions due to the Fr¨

• hlich-Morchio-Strocchi (FMS) mechanism

[Fr¨

• hlich, Morchio, Strocchi PLB97 (1981)249][NPB160(1981)553]

→ SM: weak gauge group matches global custodial symmetry → Not guaranteed for BSM models ⇒ discrepancy between physical and elementary spectrum ◮ Investigate SU(3) gauge theory with a fundamental Higgs field [Maas, T¨

• rek 1804.04453]

→ Blue gauge invariant spectrum → Red predictions from gauge-invariant PT [Maas 1712.04721] → Standard PT fails

• ++
• ±

++

±

• +
• ++
• ()
• ()
• 29 / 26

SU(2) with Nf = 2 fundamental flavors

[Drach, Janowski, Pica, Prelovsek]

◮ Starting new investigations using Wilson-clover fermion with Symanzik gauge action ◮ Improving on earlier work with (unimproved) Wilson fermions and plaquette gauge action

1.0 1.2 1.4 1.6 1.8 2.0 2.2 −1.0 −0.8 −0.6 −0.4 −0.2

β m0 weak coupling strong coupling mPCAC < 0 unphysical unmapped

0.00 0.05 0.10 0.15 0.20 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

(amPS )2 mV mPS

0.00 0.05 0.10 0.15 0.20 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

(amPS )2 mV mPS

0.00 0.05 0.10 0.15 0.20 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

(amPS )2 mV mPS

0.00 0.05 0.10 0.15 0.20 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

(amPS )2 mV mPS β = 1.45 , L = 16 , T = 32 β = 1.45 , L = 24 , T = 48 β = 1.5 , L = 16 , T = 32 β = 1.5 , L = 24 , T = 48

◮ Little changes w.r.t. unimproved setup ◮ Approaching the chiral limit ◮ Investigate scattering and ρππ coupling

30 / 26

Fundamental composite 2HDM with four flavors

[Ma, Cacciapaglia JHEP03(2016)211] ◮ Global symmetry at low energies:

SU(4) × SU(4) broken to SU(4)diag

◮ 15 pNGB transform under custodial symmetry

SU(2)L × SU(2)R

⇒ 15SU(4)diag = (2, 2) + (2, 2) + (3, 1) + (1, 3) + (1, 1) → One doublet plays the role of the Higgs doublet field → Other doublet and triplets are stable; could play role of dark matter ◮ Vecchi: “choose the right couplings to RH top” [Edinburgh talk] ⇒ (2, 2) + (2, 2) + (3, 1) + (1, 3) + (1, 1)

✪ ✪ ✪ ✪ ✪ ✪

effectively SU(4)/Sp(4)

31 / 26

SU(4)/Sp(4) composite Higgs

[Bennett, Hong, Lee, Lin, Lucini, Piai, Vadacchino]

◮ Systematic program to investigate Sp(2N) gauge theories for Nf = 2 fund. flavors and N > 1 → Quenched results for Sp(4) published [Bennett et al. JHEP 1803(2018)185] → First dynamical results for masses and decay constants (Lee Thu 2:20 PM) → Qualitative agreement between quenched and dynamical results → Comparison between Nf = 2 fundamental and anti-symmetric

Out[7418]=

Antisymmetric Fundamental

• 0.0

0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 w0 mps2 w0 mM

• vector
• axialvector

Out[11228]=

Quenched Dynamical

• 0.0

0.5 1.0 1.5 2.0 2.5 3.0 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 w0 mps2 w0 fM

• pseudoscalar
• vector
• axialvector

Quenched Dynamical

• 0.0

0.5 1.0 1.5 2.0 2.5 3.0 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 w0 mps2 w0 fM

• pseudoscalar
• vector
• axialvector

32 / 26