Lattice field theory and physics beyond the Standard Model C.-J. David Lin ( 林及仁 ) National Chiao Tung University, Taiwan 01/12/2020 National Taiwan University 1
High-energy physics frontiers Energy Frontier Origin of Mass CP Dark Violation Matter Origin of Universe Proton Decay Dark Energy Intensity Cosmic Frontier Frontier 2
Paths to the origin of mass Experiments Origin of Mass Phenomenology Lattice Field Theory 3
Outline Introduction: the standard-model Higgs and why it is not enough Lattice Field Theory and strong dynamics beyond the SM The Higgs-Yukawa model Higgs as a bound state: Composite Higgs A new approach inspired by BSM physics and condensed matter theory Tensor networks 4
The standard model (SM) Higgs 5
The standard model Higgs JCMB 4417 For the standard model: complex doublet (4 real scalars) ϕ → 6
The standard model Higgs V ( ϕ ) = μ 2 ϕ * ϕ + λ ( ϕ * ϕ ) 2 for illustration Spontaneous symmetry breaking V ( ϕ ) V ( ϕ ) μ 2 ≥ 0 μ 2 < 0 Re( ) Re( ) ϕ ϕ | ⟨ 0 | ϕ | 0 ⟩ | = v Im( ) Im( ) ϕ ϕ VEV h ( x ) − μ 2 m h = 4 v λ Choose ⟨ 0 | ϕ | 0 ⟩ = v = 2 λ Goldstone boson θ ( x ) ϕ ( x ) = [ h ( x ) + v ] e i θ ( x ) 7
The standard model Higgs for the origin of masses Coupled to weak gauge bosons via (coupling ) ∂ μ ϕ → D μ ϕ g The weak gauge boson masses M W , Z ∝ gv Coupled to fermions via the Yukawa coupling + h.c. y ¯ ψ L ϕψ R The 44 corridor @ JCMB The fermion masses m ψ ∝ yv 8
The good, the bad and the ugly of the standard model 9
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What the LHC revealed to us hitherto Totally compatible with the SM 11
What the LHC revealed to us hitherto Searched up here ~2 TeV Higgs boson ~125 GeV The Higgs boson is light 12
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Running coupling in QFT Figure From Roberto Soldati Charge screening in Quantum Electrodynamics Interaction/coupling strength changes with distance (energy scale) 14
The scalar (Higgs) sector is trivial! Possible new physics appears at this scale ¯ M ¯ with the Higgs quartic self coupling λ 125 GeV M Higgs ∼ with the Higgs quartic self coupling λ 15
The scalar (Higgs) sector is trivial! ¯ Possible new physics appears at this scale M ¯ with the Higgs quartic self coupling λ 125 GeV M Higgs ∼ with the Higgs quartic self coupling λ 16
The scalar (Higgs) sector is trivial! Possible new physics appears at this scale ¯ M ¯ with the Higgs quartic self coupling λ 125 GeV M Higgs ∼ with the Higgs quartic self coupling λ 17
The scalar (Higgs) sector is trivial! Non-perturbative Possible new physics appears at this scale ¯ M ¯ λ with the Higgs quartic self coupling 125 GeV M Higgs ∼ with the Higgs quartic self coupling λ 18
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The Higgs is “unnaturally” light! V ( ϕ ) = μ 2 ϕ * ϕ + λ ( ϕ * ϕ ) 2 for illustration Possible new physics appears at this scale ¯ , M which could be as high as GeV 10 19 Non-perturbative M Higgs ∼ − μ 2 − ¯ M 2 ∼ (125 GeV) 2 Huge cancellation! 20
Lattice field theory 21
Basic ingredients Matter fields (fermions and scalars) a Gauge fields In finite 4-dimensional volume Z x ¯ ψ D ψ e − S U e − S φ e − a 4 P DUD φ D ¯ ψ x M [ U, φ ] ψ x Z = Z ψ D ψ det( M [ U, φ ]) e − S U e − S φ DUD φ D ¯ = Monte-Carlo simulations with importance sampling c.f. Partition function in statistical physics 22
The continuum limit Matter fields (fermions and scalars) a Gauge fields means or a → 0 am → 0 ξ / a → ∞ : typical low-energy mass scale m : typical long-distance length scale (correlation length) ξ The continuum limits are at 2nd-order phase transition points Critical phenomena are crucial for lattice field theory 23
New physics in the Higgs-Yukawa theory? 24
The Higgs-Yukawa sector of the SM… V ( ϕ ) = μ 2 ϕ † ϕ + λ ( ϕ † ϕ ) 2 with y ( ¯ at fixed ψ L ϕψ R + h . c.) λ 1 − 2 λ 8 + μ 2 “Let me describe a typical computer simulation: … the first thing to do is to look for phase transition.” G. Parisi in Field Theory, Disorder and Simulations y 25
is a small corner on the phase diagram V ( ϕ ) = μ 2 ϕ † ϕ + λ ( ϕ † ϕ ) 2 with y ( ¯ at fixed ψ L ϕψ R + h . c.) λ 1 − 2 λ FM FM 8 + μ 2 v ≠ 0 v ≠ 0 SYM SYM v = v s = 0 v = v s = 0 AFM AFM v s ≠ 0 v ≠ 0 v s ≠ 0 v s ≠ 0 y v = 1 v s = 1 V 4 ∑ V 4 ∑ η ( x ) ⟨ 0 | ϕ ( x ) | 0 ⟩ ⟨ 0 | ϕ ( x ) | 0 ⟩ X X 26
It is a challenging yet important task V ( ϕ ) = μ 2 ϕ † ϕ + λ ( ϕ † ϕ ) 2 with y ( ¯ at fixed ψ L ϕψ R + h . c.) λ Possible new fixed point? D.Y.-J.Chu, K.Jansen, B.Knippschild, CJDL, JHEP01 (2019) 110 J.Bulava et al. , AHEP 2013 (2013) 875612 Possible new four-fermion condensate? S.Catterall and D.Schaich, PRD96 (2017) Possible first-order phase transition? A.Hasenfratz, K.Jansen, Y.Shen, NPB394 (1993) Other directions ( e.g. 2HDM)? 27
New physics as new interactions? V ( ϕ ) = μ 2 ϕ † ϕ + λ ( ϕ † ϕ ) 2 + λ 6 ( ϕ † ϕ ) 3 with y ( ¯ ψ L ϕψ R + h . c.) 1 − 2 λ 8 + μ 2 κ λ 6 = 0.001 λ D.Y.-J.Chu, K.Jansen, B.Knippschild, CJDL, A.Nagy, PLB 744 (2015) 146 28
The Higgs boson as a bound state 29
The standard model Higgs V ( ϕ ) = μ 2 ϕ * ϕ + λ ( ϕ * ϕ ) 2 for illustration V ( ϕ ) V ( ϕ ) μ 2 ≥ 0 μ 2 < 0 SSB Re( ) Re( ) ϕ ϕ | ϕ 0 | = v Im( ) Im( ) ϕ ϕ VEV h ( x ) − μ 2 m h = 4 v λ Choose ⟨ 0 | ϕ | 0 ⟩ = v = 2 λ Goldstone boson θ ( x ) ϕ ( x ) = [ h ( x ) + v ] e i θ ( x ) 30
Different paths to electroweak symmetry breaking Self-interacting scalars replaced by strongly-interacting fermions and gauge bosons Coupling as energy O p p o s i t e t o λ i n V ( ϕ ) ! Bound state formed at low energy r 0 = 1/ Λ b Typical bound-sate mass Λ b Typical bound-sate size r 0 = 1/ Λ b Not yet seen experimentally 2 × 10 3 ≲ Λ b < 10 4 GeV GeV 31
Two issues: I. The light Higgs Q: Why is the Higgs so light, ? M Higgs ≪ Λ b A: Resort to spontaneous symmetry breaking Global symmetry breaking like QCD (Composite Higgs) Scale-invariance breaking unlike QCD (Dilaton Higgs) 32
Two issues: II. The SM fermion mass The need to suppress flavour-changing neutral-current processes… mass FCNC GeV Λ f ∼ 10 7 1 1 ψ SM ψ SM ¯ ¯ ψ SM ψ SM ¯ ¯ ff ψ SM ψ SM Λ 2 Λ 2 High to suppress FCNC f f ~ 10 - 100 GeV 33
Two issues: II. The SM fermion mass The need to suppress flavour-changing neutral-current processes… mass FCNC GeV Λ f ∼ 10 7 ~ 10 - 100 GeV 1 1 ψ SM ψ SM ¯ ¯ f f ψ SM ψ SM ¯ ¯ ψ SM ψ SM Λ 2 Λ 2 f f 34
Two issues: II. The SM fermion mass The need to suppress flavour-changing neutral-current processes… mass FCNC GeV Λ f ∼ 10 7 1 1 ψ SM ψ SM ¯ ¯ ψ SM ψ SM ¯ ¯ ff ψ SM ψ SM Λ 2 Λ 2 High to suppress FCNC f f ~ 10 - 100 GeV 35
Two issues: II. The SM fermion mass The need to suppress flavour-changing neutral-current processes… mass FCNC GeV Λ f ∼ 10 7 1 1 ψ SM ψ SM ¯ ψ SM ψ SM ¯ ¯ ψ SM ψ SM ¯ ff Λ 2 (1/ Λ 2 Λ f f ) → 1/ Λ f f ~ 10 - 100 GeV 36
Two issues: II. The SM fermion mass The need to suppress flavour-changing neutral-current processes… mass FCNC GeV Λ f ∼ 10 7 1 ψ SM ψ SM ¯ 1 ¯ ff ψ SM ψ SM ¯ ¯ ψ SM ψ SM Λ 2 Λ f f ~ 10 - 100 GeV 37
Two issues: II. The SM fermion mass The need to suppress flavour-changing neutral-current processes… mass FCNC GeV Λ f ∼ 10 7 1 ψ SM ψ SM ¯ 1 1/ Λ f → (1/ Λ f ) 2 ¯ ff ψ SM ψ SM ¯ ¯ ψ SM ψ SM Λ 2 Λ 2 f f ~ 10 - 100 GeV 38
Two issues: II. The SM fermion mass The need to suppress flavour-changing neutral-current processes… mass FCNC GeV Λ f ∼ 10 7 ~ 10 - 100 GeV 1 ψ SM ψ SM ¯ 1 ¯ ff ψ SM ψ SM ¯ ¯ ψ SM ψ SM Λ 2 Λ 2 f f 39
Two issues: II. The SM fermion masses ψ SM ψ SM ¯ Dramatically alter the suppression of from 1/ Λ 2 to 1/ Λ f ¯ ff f Need power-law scaling behaviour in ¯ ff The system is at criticality for a large range of interaction strength c.f. Berezinskii-Kosterlitz-Thouless phase transition 40
The Higgs boson as a bound state: Composite Higgs models 41
Lesson from Quantum Chromodynamics M ρ , M N , . . . ∼ Λ (QCD) ∼ 1 GeV b SSB via ⟨ 0 | ¯ qq | 0 ⟩ GeV M π = 0 42
Lesson from Quantum Chromodynamics Electroweak ~ 1000 GeV Couple M ρ , M N , . . . ∼ Λ (QCD) ∼ 1 GeV b SSB via ⟨ 0 | ¯ qq | 0 ⟩ GeV M π = 0.132 43
Lesson from Quantum Chromodynamics Planck scale 10 19 GeV Electroweak ~ 1000 GeV Couple M ρ , M N , . . . ∼ Λ (QCD) ∼ 1 GeV b SSB via ⟨ 0 | ¯ qq | 0 ⟩ GeV M π = 0.132 Higgs mass 125 GeV 44
Composite Higgs models 10 19 Planck scale GeV 10 7 Flavour scale ~ GeV Λ f Lattice studies System at criticality (for SM fermion mass) Couple s e i d u Composite Higgs scale Λ (CH) ∼ 10 3 GeV t s Lattice studies e b c (Many bound states here) i t t a L SSB via ⟨ 0 | ¯ ff | 0 ⟩ Higgs mass and VEV ~ 10 2 GeV 45
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