Higgs Physics from the Lattice Lecture 1: Standard Model Higgs Physics

Transcription

1 Higgs Physics from the Lattice Lecture 1: Standard Model Higgs Physics Julius Kuti University of California, San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007 Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 1/34

2 Outline of Lecture Series: Higgs Physics from the Lattice 1. Standard Model Higgs Physics Outlook for the Higgs Particle Standard Model Review Expectations from the Renormalization Group Nonperturbative Lattice Issues? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 2/34

3 Outline of Lecture Series: Higgs Physics from the Lattice 1. Standard Model Higgs Physics Outlook for the Higgs Particle Standard Model Review Expectations from the Renormalization Group Nonperturbative Lattice Issues? 2. Triviality and Higgs Mass Upper Bound Renormalization Group and Triviality in Lattice Higgs and Yukawa couplings Higgs Upper Bound in 1-component φ 4 Lattice Model Higgs Upper Bound in O(4) Lattice Model Strongly Interacting Higgs Sector? Higgs Resonance on the Lattice Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 2/34

4 Outline of Lecture Series: Higgs Physics from the Lattice 1. Standard Model Higgs Physics Outlook for the Higgs Particle Standard Model Review Expectations from the Renormalization Group Nonperturbative Lattice Issues? 2. Triviality and Higgs Mass Upper Bound Renormalization Group and Triviality in Lattice Higgs and Yukawa couplings Higgs Upper Bound in 1-component φ 4 Lattice Model Higgs Upper Bound in O(4) Lattice Model Strongly Interacting Higgs Sector? Higgs Resonance on the Lattice 3. Vacuum Instability and Higgs Mass Lower Bound Vacuum Instability and Triviality in Top-Higgs Yukawa Models Chiral Lattice Fermions Top-Higgs and Top-Higgs-QCD sectors with Chiral Lattice Fermions Higgs mass lower bound Running couplings in 2-loop continuum Renormalization Group Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 2/34

5 Tuesday, May 15, 2007 A Giant Takes On Physics Biggest Questions Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 3/34

6 Tuesday, May 15, 2007 A Giant Takes On Physics Biggest Questions The physicists are scrambling like Spiderman over this assembly, appropriately named Atlas, getting ready to see the universe born again. According to the Standard Model, the Higgs can have only a limited range of masses without severe damage to the universe. If it is too light, the universe will decay. If it is too heavy, the universe would have blown up already... According to Dr. Ellis, there is a magic value between 160 billion and 180 billion electron volts that would ensure a stable universe and require no new physics at all. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 3/34

7 Tuesday, May 15, 2007 A Giant Takes On Physics Biggest Questions The physicists are scrambling like Spiderman over this assembly, appropriately named Atlas, getting ready to see the universe born again. According to the Standard Model, the Higgs can have only a limited range of masses without severe damage to the universe. If it is too light, the universe will decay. If it is too heavy, the universe would have blown up already... According to Dr. Ellis, there is a magic value between 160 billion and 180 billion electron volts that would ensure a stable universe and require no new physics at all. These lectures will explain: 1. Planck scale magic value connection Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 3/34

8 Tuesday, May 15, 2007 A Giant Takes On Physics Biggest Questions The physicists are scrambling like Spiderman over this assembly, appropriately named Atlas, getting ready to see the universe born again. According to the Standard Model, the Higgs can have only a limited range of masses without severe damage to the universe. If it is too light, the universe will decay. If it is too heavy, the universe would have blown up already... According to Dr. Ellis, there is a magic value between 160 billion and 180 billion electron volts that would ensure a stable universe and require no new physics at all. These lectures will explain: 1. Planck scale magic value connection 2. What if UV completion in TeV range? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 3/34

9 Some comments on the lectures 1. Cutoff plays a very different role in Standard Model Higgs Physics Not like QCD where cutoff is your enemy In Standard Model cutoff represents the threshold of new physics New physics is not hypercubic, cutoff is your enemy AND friend Is there a way to do this right on the lattice? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 4/34

10 Some comments on the lectures 1. Cutoff plays a very different role in Standard Model Higgs Physics Not like QCD where cutoff is your enemy In Standard Model cutoff represents the threshold of new physics New physics is not hypercubic, cutoff is your enemy AND friend Is there a way to do this right on the lattice? 2. Standard Model UV completions Only a simple model (Lee-Wick extension) will be briefly discussed Illustration only! It is crazy but probably not crazy enough Dim Reg model building has 1 ɛ models on the market With O(ɛ) chance for each to succeed (Work on all?) Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 4/34

11 Some comments on the lectures 1. Cutoff plays a very different role in Standard Model Higgs Physics Not like QCD where cutoff is your enemy In Standard Model cutoff represents the threshold of new physics New physics is not hypercubic, cutoff is your enemy AND friend Is there a way to do this right on the lattice? 2. Standard Model UV completions Only a simple model (Lee-Wick extension) will be briefly discussed Illustration only! It is crazy but probably not crazy enough Dim Reg model building has 1 ɛ models on the market With O(ɛ) chance for each to succeed (Work on all?) 3. Vacuum instability Lessons in field theory Might play a role in Cosmology Somewhat surprising role of weak gauge couplings Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 4/34

12 Some comments on the lectures 1. Cutoff plays a very different role in Standard Model Higgs Physics Not like QCD where cutoff is your enemy In Standard Model cutoff represents the threshold of new physics New physics is not hypercubic, cutoff is your enemy AND friend Is there a way to do this right on the lattice? 2. Standard Model UV completions Only a simple model (Lee-Wick extension) will be briefly discussed Illustration only! It is crazy but probably not crazy enough Dim Reg model building has 1 ɛ models on the market With O(ɛ) chance for each to succeed (Work on all?) 3. Vacuum instability Lessons in field theory Might play a role in Cosmology Somewhat surprising role of weak gauge couplings 4. Lectures, Notations Level between university lecture course and review talks Notations are entirely uniform throughout the lectures ± 1 2 m2 φ 2, m 2 µ 2, and λ 4! φ4 are examples (and should not confuse) Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 4/34

13 Standard Model: Outlook for the Higgs particle Standard Model is expected to be UV-incomplete on TeV scale (triviality?) low cut-off is favored by the hierarchy problem or, war of choice for theorists Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 5/34

14 Standard Model: Outlook for the Higgs particle Standard Model is expected to be UV-incomplete on TeV scale (triviality?) low cut-off is favored by the hierarchy problem or, war of choice for theorists Higgs particle = how Electroweak symmetry is broken in nature Higgs discovery potential Energy scale of new physics Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 5/34

15 Standard Model: Outlook for the Higgs particle Standard Model is expected to be UV-incomplete on TeV scale (triviality?) low cut-off is favored by the hierarchy problem or, war of choice for theorists Higgs particle = how Electroweak symmetry is broken in nature Higgs discovery potential Energy scale of new physics What the Particle Data Book tells us Lower bound on the Higgs mass from direct searches is GeV From Electroweak precision measurements: M H = GeV with 2σ error Ignores intrinsic cutoff! Tension in data analysis! Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 5/34

16 Standard Model: Outlook for the Higgs particle Standard Model is expected to be UV-incomplete on TeV scale (triviality?) low cut-off is favored by the hierarchy problem or, war of choice for theorists Higgs particle = how Electroweak symmetry is broken in nature Higgs discovery potential Energy scale of new physics What the Particle Data Book tells us Lower bound on the Higgs mass from direct searches is GeV From Electroweak precision measurements: M H = GeV with 2σ error Ignores intrinsic cutoff! Tension in data analysis! Landau Pole and Vacuum Instability (mis)representations of triviality? Lattice and modern view on effective field theories should be reconciled Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 5/34

17 Standard Model: Outlook for the Higgs particle Standard Model is expected to be UV-incomplete on TeV scale (triviality?) low cut-off is favored by the hierarchy problem or, war of choice for theorists Higgs particle = how Electroweak symmetry is broken in nature Higgs discovery potential Energy scale of new physics What the Particle Data Book tells us Lower bound on the Higgs mass from direct searches is GeV From Electroweak precision measurements: M H = GeV with 2σ error Ignores intrinsic cutoff! Tension in data analysis! Landau Pole and Vacuum Instability (mis)representations of triviality? Lattice and modern view on effective field theories should be reconciled Should this PDG figure be reworked nonperturbatively? for low cutoff Λ Effects of new Higgs physics from UV completion? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 5/34

18 Standard Model: Separate Higgs and Top mass fits Γ Z σ 0 had R 0 l A 0,l fb A l (P τ ) R 0 b R 0 c A 0,b fb A 0,c fb A b A c A l (SLD) sin 2 θ lept eff (Q fb ) m W * Γ W * Q W (Cs) sin 2 θ (e e MS ) sin 2 θ W (νn) g 2 L (νn) g 2 R (νn) *preliminary M H [GeV] Top-Quark Mass [GeV] CDF ± 2.2 D ± 2.4 Average ± 1.8 χ 2 /DoF: 9.2 / 10 LEP1/SLD LEP1/SLD/m W /Γ W m t [GeV] Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 6/34

19 Standard Model: Correlated Higgs and Top mass fits χ 2 m 6 Limit = 144 GeV Theory uncertainty α (5) had = ± ± incl. low Q 2 data Excluded Preliminary m H [GeV] m t [GeV] High Q 2 except m t 68% CL Excluded m t (Tevatron) m H [GeV] Significant Higgs 1-loop corrections growing like the logarithm of m H affect 6 amplitudes with external vector bosons only Two each for 2,3,4 point functions. Only 2-point function is inderictly accessible experimentally today. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 7/34

20 Standard Model: Top quark pole mass on the lattice Top quark capture in BW channel Experimentalists measure the pole mass Color backflow correction (1-2 GeV uncertainty?) Lattice work is in small finite box L Λ QCD 1 Running g QCD (L) is close to M Top scale in small QCD world (L large for Top-Higgs dynamics) In lattice simulation measure Top quark pole mass or propagator mass Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 8/34

21 Standard Model: Three families of quarks and leptons Standard Model is gauge theory with symmetry group SU(3) c SU(2) L U(1) Y of strong, weak and electromagnetic interactions Forces are exchanged via spin-1 gauge fields: eight massless gluons, one massless photon, and three massive weak bosons, W ± and Z Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 9/34

22 Standard Model: Three families of quarks and leptons Standard Model is gauge theory with symmetry group SU(3) c SU(2) L U(1) Y of strong, weak and electromagnetic interactions Forces are exchanged via spin-1 gauge fields: eight massless gluons, one massless photon, and three massive weak bosons, W ± and Z Leptons and quarks are organized in a three-fold family structure: [ ] νe u e d, [ ] νµ c µ s, [ ] ντ t τ b with SU(2) L doublets and singlets in each family (quark in three colours): [ ] νl q u l q d ( νl l ) L, ( qu q d ) L, l R, q ur, q dr Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 9/34

23 Standard Model: Three families of quarks and leptons Standard Model is gauge theory with symmetry group SU(3) c SU(2) L U(1) Y of strong, weak and electromagnetic interactions Forces are exchanged via spin-1 gauge fields: eight massless gluons, one massless photon, and three massive weak bosons, W ± and Z Leptons and quarks are organized in a three-fold family structure: [ ] νe u e d, [ ] νµ c µ s, [ ] ντ t τ b with SU(2) L doublets and singlets in each family (quark in three colours): [ ] νl q u l q d ( ) νl l, L ( ) qu, l q R, q ur, q dr d L Spontaneous symmetry breaking in vacuum from electroweak gauge group to electromagnetic gauge group generates Higgs mechanism: SU(3) c SU(2) L U(1) Y SSB SU(3) c U(1) QED Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 9/34

24 Standard Model: SU(3) c and QCD Free quark Lagrangian L 0 = f q f ( iγ µ µ m f ) qf is invariant under arbitrary global SU(3) c transformations q α (q f α f ) = U α β qβ f Require now the Lagrangian to be also invariant under local SU(3) c transformations, θ a = θ a (x) (generalization of QED) We need to change the quark derivatives by covariant objects with eight independent gauge parameters and eight different gauge bosons G µ a(x) D µ q f [ µ λ a ] + ig s 2 Gµ a(x) q f [ µ + ig s G µ (x) ] q f Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 10/34

25 Standard Model: SU(3) c and QCD Free quark Lagrangian L 0 = f q f ( iγ µ µ m f ) qf is invariant under arbitrary global SU(3) c transformations q α (q f α f ) = U α β qβ f Require now the Lagrangian to be also invariant under local SU(3) c transformations, θ a = θ a (x) (generalization of QED) We need to change the quark derivatives by covariant objects with eight independent gauge parameters and eight different gauge bosons G µ a(x) D µ q f [ µ λ a ] + ig s 2 Gµ a(x) q f [ µ + ig s G µ (x) ] q f To build gauge-invariant kinetic term for gluon fields, we introduce the corresponding field strengths: G µν a (x) = µ G ν a ν G µ a g s f abc G µ b Gν c Taking the proper normalization for the gluon kinetic term, we get the SU(3) c invariant Lagrangian of Quantum Chromodynamics: L QCD 1 4 Gµν a G a µν + f q f ( iγ µ D µ m f ) qf Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 10/34

26 Standard Model: SU(3) c and QCD detailed form of the Lagrangian: L QCD = 1 4 ( µ G ν a ν G µ a) ( µ G a ν ν G a µ) + f ( g s G µ λ a q α a ) f γ µ q β 2 f f αβ q α f ( iγ µ µ m f ) q α f + g s 2 f abc ( µ G ν a ν G µ a) G b µ G c ν g2 s 4 f abc f ade G µ b Gν c G d µ G e ν. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 11/34

27 Standard Model: SU(3) c and QCD detailed form of the Lagrangian: L QCD = 1 4 ( µ G ν a ν G µ a) ( µ G a ν ν G a µ) + f ( g s G µ λ a q α a ) f γ µ q β 2 f f αβ q α f ( iγ µ µ m f ) q α f + g s 2 f abc ( µ G ν a ν G µ a) G b µ G c ν g2 s 4 f abc f ade G µ b Gν c G d µ G e ν. First line in Lagrangian: kinetic terms = propagators Second line: color interaction between quarks and gluons with SU(3) c matrices λ a Third line: G µν a G a µν term generates cubic and quartic gluon self-interactions (same coupling g s appears in all parts of the Lagrangian) Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 11/34

28 Standard Model: SU(3) c and QCD detailed form of the Lagrangian: L QCD = 1 4 ( µ G ν a ν G µ a) ( µ G a ν ν G a µ) + f ( g s G µ λ a q α a ) f γ µ q β 2 f f αβ q α f ( iγ µ µ m f ) q α f + g s 2 f abc ( µ G ν a ν G µ a) G b µ G c ν g2 s 4 f abc f ade G µ b Gν c G d µ G e ν. First line in Lagrangian: kinetic terms = propagators Second line: color interaction between quarks and gluons with SU(3) c matrices λ a Third line: G µν a G a µν term generates cubic and quartic gluon self-interactions (same coupling g s appears in all parts of the Lagrangian) Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 11/34

29 Standard Model: SU(2) L U(1) Y sector The symmetry group to be gauged is G SU(2) L U(1) Y L refers to left-handed fields and Y is the weak hypercharge For simplicity, single family of quarks ( u ψ 1 (x) = d ), ψ 2 (x) = u R, ψ 3 (x) = d R L Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 12/34

30 Standard Model: SU(2) L U(1) Y sector The symmetry group to be gauged is G SU(2) L U(1) Y L refers to left-handed fields and Y is the weak hypercharge For simplicity, single family of quarks ( u ψ 1 (x) = d ), ψ 2 (x) = u R, ψ 3 (x) = d R L Our discussion will also be valid for the lepton sector, with the identification ( ) νe ψ 1 (x) = e, ψ 2 (x) = ν er, ψ 3 (x) = e R L Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 12/34

31 Standard Model: SU(2) L U(1) Y sector The symmetry group to be gauged is G SU(2) L U(1) Y L refers to left-handed fields and Y is the weak hypercharge For simplicity, single family of quarks ( u ψ 1 (x) = d ), ψ 2 (x) = u R, ψ 3 (x) = d R L Our discussion will also be valid for the lepton sector, with the identification ( ) νe ψ 1 (x) = e, ψ 2 (x) = ν er, ψ 3 (x) = e R L As in QCD case, we start from the free Lagrangian: L 0 = i ū(x) γ µ µ u(x) + i d(x) γ µ µ d(x) = 3 i ψ j (x) γ µ µ ψ j (x) j=1 Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 12/34

32 Standard Model: SU(2) L U(1) Y sector L 0 is invariant under global G transformations in flavour space: ψ 1 (x) ψ 2 (x) ψ 3 (x) G ψ 1 (x) exp {iy 1β} U L ψ 1 (x), G ψ 2 (x) exp {iy 2β} ψ 2 (x), G ψ 3 (x) exp {iy 3β} ψ 3 (x) The SU(2) L transformation U L exp { i σ i 2 αi } (i = 1, 2, 3) only acts on the doublet field ψ 1 The parameters y i are called hypercharges, since the U(1) Y phase transformation is analogous QED The matrix transformation U L is non-abelian as in QCD Notice that we have not included a mass term because it would mix the left- and right-handed fields spoiling symmetry considerations Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 13/34

33 Standard Model: SU(2) L U(1) Y sector We can now require the Lagrangian to be also invariant under local SU(2) L U(1) Y gauge transformations with α i = α i (x) and β = β(x) In order to satisfy this symmetry requirement, we need to change the fermion derivatives by covariant objects and for four gauge parameters, α i (x) and β(x), four different gauge bosons are needed: D µ ψ 1 (x) [ µ + i g W µ (x) + i g y 1 B µ (x) ] ψ 1 (x), D µ ψ 2 (x) [ µ + i g y 2 B µ (x) ] ψ 2 (x), D µ ψ 3 (x) [ µ + i g y 3 B µ (x) ] ψ 3 (x) where W µ (x) σ i 2 Wi µ(x) is SU(2) L matrix field Thus we have the correct number of gauge fields to describe the W ±, Z and γ gauge bosons L = 3 j=1 i ψ j (x) γ µ D µ ψ j (x) is invariant under local G transformations Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 14/34

34 Standard Model: SU(2) L U(1) Y sector In order to build the gauge-invariant kinetic term for the electroweak gauge fields, we introduce the corresponding field strengths: B µν µ B ν ν B µ, W i µν = µ W i ν ν W i µ g ɛ ijk W j µ W k ν, W µν σ i 2 Wi µν B µν remains invariant under G transformations, while W µν transforms covariantly: B µν G Bµν, W µν G UL W µν U L Properly normalized kinetic Lagrangian is given by L Kinetic = 1 4 B µν B µν 1 2 Tr [ W µν W µν] = 1 4 B µν B µν 1 4 Wi µν W µν i. Since the field strengths W i µν contain a quadratic piece, the Lagrangian L Kinetic gives rise to cubic and quartic self-interactions among the gauge fields The strength of these interactions is given by the same SU(2) L coupling g which appears in the fermionic piece of the Lagrangian The SU(2) L U(1) Y Lagrangian only contains massless fields Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 15/34

35 Standard Model: Charged currents in SU(2) L U(1) Y sector W W q d g q u l g (1 γ ) (1 γ ) 5 2 3/ /2 ν l The Electroweak Lagrangian contains interactions of the fermion fields with the gauge bosons: SU(2) L matrix W µ = σi 2 Wi µ = 1 2 with the boson field W µ (W 1 µ + i W 2 µ)/ 2 For a single family of quarks and leptons: y j ψ j γ µ ψ j L 3 g ψ 1 γ µ W µ ψ 1 g B µ j=1 W 3 µ 2 W µ 2 Wµ Wµ 3 generates charged-current interactions L CC = g 2 { W µ [ūγ µ (1 γ 5 )d + ν e γ µ (1 γ 5 )e ] + h.c. } Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 16/34

36 Standard Model: Z and γ in SU(2) L U(1) Y sector γ Z f f f e Q f 2 s e θc (v θ f a f γ 5 ) f Mixing of the neutral gauge fields Wµ 3 and B µ generates the Z boson and the photon γ: ( ) ( ) ( ) W 3 µ cos θ W sin θ W Zµ B µ sin θ W cos θ W A µ The neutral-current Lagrangian is given by { [ L NC = ψ j γ µ A µ g σ ] [ 3 2 sin θ W + g y j cos θ W + Z µ g σ ]} 3 2 cos θ W g y j sin θ W ψ j. j In order to get QED from the A µ piece, one needs to impose the conditions: g sin θ W = g cos θ W = e, Y = Q T 3, where T 3 σ 3 /2 and Q denotes the electromagnetic charge operator ( ) Qu/ν 0 Q 1, Q 0 Q 2 = Q u/ν, Q 3 = Q d/e d/e Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 17/34

37 Standard Model: Z and γ in SU(2) L U(1) Y sector Fermion hypercharges in terms of their electric charge and weak isospin: Quarks: y 1 = Q u 1 2 = Q d = 1 6, y 2 = Q u = 2 3, y 3 = Q d = 1 3 Leptons: y 1 = Q ν 1 2 = Q e = 1 2, y 2 = Q ν = 0, y 3 = Q e = 1 Neutral-current Lagrangian can be written as L NC = L QED + L Z NC, L QED = e A µ ψ j γ µ Q j ψ j e A µ J em, µ L Z NC = e J µ sin θ j W cos θ Z Z µ W J µ Z ψ j γ ( ) µ σ 3 2 sin 2 θ W Q j ψj = J µ 3 2 sin2 θ W J em µ j Weak boson cubic and quartic self-interactions are generated (like in QCD): Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 18/34

38 Standard Model: Spontaneous symmetry breaking Let us consider a complex scalar field φ(x), with the Lagrangian: L = µ φ µ φ V(φ), V(φ) = µ 2 φ φ + λ ( φ φ ) 2 L is invariant under global phase transformations of the scalar field: φ(x) φ (x) exp {iθ} φ(x) In order to have a ground state the potential should be bounded from below, i.e., h > 0 1. µ 2 > 0: The potential has only the trivial minimum φ = 0. It describes a massive scalar particle with mass µ and quartic coupling λ 2. µ 2 < 0: The minimum is obtained for those field configurations satisfying φ 0 = µ 2 2λ v 2 > 0, V(φ 0 ) = λ 4 v4 Owing to the U(1) phase-invariance of the Lagrangian, there is an infinite number of degenerate states of minimum energy φ 0 (x) = v exp {iθ} 2 Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 19/34

39 Standard Model: SSB and Goldstone theorem By choosing a particular ground state solution, θ = 0 for example, the symmetry gets spontaneously broken. If we parametrize the excitations over the ground state as φ(x) 1 2 [ v + ϕ1 (x) + i ϕ 2 (x) ], where ϕ 1 and ϕ 2 are real fields, the potential takes the form V(φ) = V(φ 0 ) µ 2 ϕ λ v ϕ ( 1 ϕ ϕ2) 2 λ ( ) + ϕ ϕ Thus, ϕ 1 describes a massive state of mass m 2 ϕ 1 = 2µ 2, while ϕ 2 is massless. Origin of a massless particle when µ 2 < 0: the field ϕ 2 describes excitations around a flat direction in the potential which do not cost any energy. The fact that there are massless excitations associated with the SSB mechanism is a completely general result: Goldstone theorem If a Lagrangian is invariant under a continuous symmetry group G, but the vacuum is only invariant under a subgroup H G, then there must exist as many massless spin-0 particles (Goldstone bosons) as broken generators Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 20/34

40 Standard Model Higgs Sector: Higgs-Kibble mechanism The basic building block of the Higgs sector is the SU(2) L doublet of complex scalar fields ( φ φ(x) (+) ) (x) φ (0) (x) Gauged Higgs Lagrangian, invariant under local SU(2) L U(1) Y : L S = ( D µ φ ) D µ φ µ 2 φ φ λ ( φ φ ) 2 (λ > 0, µ 2 < 0), D µ φ = [ µ + i g W µ + i g y φ B µ] φ, y φ = Q φ T 3 = 1 2. Scalar hypercharge is fixed by having the correct couplings between φ(x) and A µ (x); i.e., the photon does not couple to φ (0), and φ (+) has the right electric charge. There is a infinite set of degenerate states with minimum energy, satisfying 0 φ (0) 0 = µ 2 2λ v 2. Note that we have made explicit the association of the classical ground state with the quantum vacuum. Since the electric charge is a conserved quantity, only the neutral scalar field can acquire a vacuum expectation value. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 21/34

41 Standard Model Higgs Sector: Higgs-Kibble mechanism Once we choose a particular ground state, the SU(2) L U(1) Y symmetry gets spontaneously broken to the electromagnetic subgroup U(1) QED, which by construction still remains a true symmetry of the vacuum. According to the Goldstone theorem three unwanted massless states should then appear. Higgs-Kibble mechanism is the answer. Higgs-Kibble mechanism 1. Parametrize the complex scalar doublet with four real fields θ i (x) and H(x) φ(x) = exp { i σ i 2 θi (x) } ( ) v + H(x) 2. Local SU(2) L invariance allows to rotate away any dependence on θ i (x). 3. In the physical (unitary) gauge, θ i (x) = 0, ( Dµ φ ) D µ φ θ i =0 1 { g 2 µh µ H + (v + H) W µw µ g 2 } + 8 cos 2 Z µ Z µ, θ W and the eliminated θ i (x) fields are precisely the would-be massless Goldstone bosons associated with the SSB mechanism 4. Through vacuum expectation value of neutral scalar, gauge bosons acquired masses: M Z cos θ W = M W = 1 2 v g Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 22/34

42 Standard Model: The Higgs Boson After L S added to SU(2) L U(1) Y model the total Lagrangian is still gauge invariant which guarantees renormalizability. After SSB, three broken generators give rise to three massless Goldstone bosons which, owing to the underlying local gauge symmetry, can be eliminated from the Lagrangian. Going to the unitary gauge, we discover that the W ± and the Z (but not the γ, because U(1) QED is an unbroken symmetry) have acquired masses. Number of degrees of freedom conserved after SSB. The scalar Lagrangian L S has introduced a new scalar particle into the model: the Higgs H. In terms of the physical fields (unitary gauge), L S takes the form L S = 1 4 λ v4 + L H + L HG 2, M H = 2µ 2 = 2λ v. L H = 1 2 µh µ H 1 2 M2 H H2 M2 H 2v H 3 M2 H 8v 2 H 4 L HG 2 = { } { } MW 2 W µw µ v H + H2 v M2 Z Z µz µ v H + H2 v 2 Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 23/34

43 Standard Model: Yukawa couplings and fermion masses A fermionic mass term L m = m ψψ = m ( ψ L ψ R + ψ R ψ L ) is not allowed, because it breaks the gauge symmetry. However, since we have introduced an additional scalar doublet into the model, we can write the following gauge-invariant fermion-scalar coupling: L Y = c 1 (ū, d ) L ( φ (+) φ (0) ) (ū, d R c 2 d ) ( L φ (0) φ ( ) ) u R c 3 ( ν e, ē) L ( φ (+) φ (0) ) e R + h.c., where the second term involves the C-conjugate scalar field φ c i σ 2 φ. In the unitary gauge (after SSB), this Yukawa-type Lagrangian takes the simpler form L Y = 1 2 (v + H) { c 1 dd + c 2 ūu + c 3 ēe }. The SSB mechanism generates fermion masses: m d = c 1 v 2, m u = c 2 v 2, m e = c 3 v 2. Yukawa couplings are fixed by the arbitrary fermion masses: L Y ( = 1 + H ) { md dd + m u ūu + m e ēe }. v Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 24/34

44 Standard Model: Electroweak parameters We have now all the needed ingredients to describe the electroweak interaction within a well-defined Quantum Field Theory. Higgs-Kibble mechanism has produced a precise prediction for the W ± and Z masses, relating them to the vacuum expectation value of the scalar Higgs field: M Z = ± GeV, M W = ± GeV. From these experimental numbers, one obtains the electroweak mixing angle sin 2 θ W = 1 M2 W M 2 Z = The Fermi coupling M 2 W g 2 q2 g2 M 2 W = 4πα sin 2 θ W M 2 W 4 2 G F, gives a direct determination of the electroweak scale, v = ( 2 G F ) 1/2 = 246 GeV. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 25/34

45 Standard Model Scales: Running Higgs coupling 1-loop Higgs couplings: 1-loop Feynman diagrams: Higgs boson self-couplings Running Higgs coupling λ(t) is defined as the Higgs 4-point function at scale t=log p µ Higgs beta function: β λ (t) = dλ(t) dt Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 26/34

46 Standard Model Scales: Running gauge and Yukawa couplings 1-loop gauge couplings: 1-loop Feynman diagrams: gauge boson couplings to fermions Running gauge couplings g(t), g (t), g 3 (t) can be defined as the gauge-fermion 3-point function at scale t=log p µ gauge beta functions: β g (t) = dg(t) dt 1-loop Yukawa couplings: 1-loop Feynman diagrams: Higgs boson Yukawa couplings to fermions Running Top coupling g Top (t) is defined as the Higgs fermion 3-point function at scale t=log p µ Top beta function: β gtop (t) = dg Top(t) dt Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 27/34

47 Standard Model Scales: RG Fixed Points and Triviality H = =g running of λ, g t, g t = gt 2 =g2 3 Top-Higgs sector (1-loop) with notation R = λ g 2 t dg 2 t dt = 9 16π 2 g 4 t g 2 dr t dg 2 t = 1 3 (8R2 + R 2) = m2 H 4m 2 t IR fixed line at R = 1 16 ( 65 1) = 0.44 Trivial fixed point only! Is the Landau pole the upper bound? Is λ(λ) = 0 the lower bound? Top-Higgs-QCD sector (1-loop) Pendleton-Ross fixed point: 2 m t = 9 g 3(µ = m t )v/ 2 95 GeV ( m H = 72 g 3 2v 53 GeV Weak gauge couplings and 2-loop destabilize the Pendleton-Ross fixed point Landau pole only in α 1 at µ = GeV with all couplings running? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 28/34

48 Standard Model Scales: Vacuum Instability Casas,Espinoza,Quiros Effective Higgs potential Vacuum instability: m t destabilizes ground state if m H below some critical value Small running couplings λ(µ), g t (µ), g 3 (µ) in RG improved V eff (φ) Tunneling time 14 billion years across barrier is often used to set lower Higgs mass bound (Hall,Ellis,Linde,Sher,...) Higgs vacuum instability: misrepresentation of triviality? Tunneling universe from false vacuum? Critical in the argument that λ(µ) turns negative at some scale in the RG evolution of the Standard Model! Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 29/34

49 Standard Model Scales: Higgs Mass Lower Bound Altarelli,Isidori and Casas,Espinosa,Quiros Higgs mass lower bound (m top dependent): Interpretation of Higgs vacuum instability when intrinsic cutoff? Requires more precise definition of Standard Model and its extensions Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 30/34

50 Standard Model Scales: Higgs Mass Upper Bound Higgs mass upper bound: Landau pole and unitarity? Triviality? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 31/34

51 Standard Model Scales: Fate of the false Higgs vacuum? Holland,JK Lattice effective potential Staggered fermion lattice simulations clearly show the absence of vacuum instability Lattice applications which are relevent for continuum SM will require chiral fermions (demanding extreme lattice computing) Confirmed by large N using other regulators Generic behavior: φ Λ cutoff for unstable range of V eff (φ) What happens when ignoring the cutoff in the running RG equations? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 32/34

52 Standard Model Scales: Higgs Mass Upper Bound 40 Running Higgs coupling Upper bound and running coupling λ λ Planck scale At large Higgs masses growing λ requires strong coupling Call for nonperturbative calculations strong coupling? α 1 "Landau pole" Running gauge couplings are important! Is large m H consistent with EW data? What is the Standard Model beyond PT? 5 m H =160 GeV t=log(mu/gev) Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 33/34

53 Standard Model Scales: Continuum Wilsonian RG Higgs Physics and the Lattice UV Completion unknon new physics - Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 34/34

54 Standard Model Scales: Continuum Wilsonian RG Higgs Physics and the Lattice UV Completion unknon new physics - Below new scale M integrated UV completion is represented by non-local L eff with all higher dimensional operators, 1 1 M 2 φ 2 φ, M 4 φ 3 φ, λ 6 M 2 φ6,... Propagator K(p2 /M 2 ) p 2 +M 2 with analytic K thins out UV completion with exponential damping - Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 34/34

55 Standard Model Scales: Continuum Wilsonian RG Higgs Physics and the Lattice UV Completion unknon new physics - Below new scale M integrated UV completion is represented by non-local L eff with all higher dimensional operators, 1 1 M 2 φ 2 φ, M 4 φ 3 φ, λ 6 M 2 φ6,... Propagator K(p2 /M 2 ) p 2 +M 2 with analytic K thins out UV completion with exponential damping Āt the symmetry breaking scale v = 250 GeV only relevant and marginal operators survive Only 1 2 m2 H φ2 and λφ 4 terms in V Higgs (φ), in addition to ( φ) 2 operator Narrow definition of Standard Model: only relevant and marginal operators at scale M Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 34/34

56 Standard Model Scales: Continuum Wilsonian RG UV Completion unknon new physics - Below new scale M integrated UV completion is represented by non-local L eff with all higher dimensional operators, 1 1 M 2 φ 2 φ, M 4 φ 3 φ, λ 6 M 2 φ6,... Propagator K(p2 /M 2 ) p 2 +M 2 with analytic K thins out UV completion with exponential damping Āt the symmetry breaking scale v = 250 GeV only relevant and marginal operators survive Only 1 2 m2 H φ2 and λφ 4 terms in V Higgs (φ), in addition to ( φ) 2 operator Narrow definition of Standard Model: only relevant and marginal operators at scale M Higgs Physics and the Lattice Lattice Wilsonian RG Regulate with lattice at scale Λ = π/a L lattice has all higher dimensional operators a 2 φ 2 φ, a 4 φ 4 φ, a 2 λ 6 φ 6,... Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 34/34

57 Standard Model Scales: Continuum Wilsonian RG UV Completion unknon new physics - Below new scale M integrated UV completion is represented by non-local L eff with all higher dimensional operators, 1 1 M 2 φ 2 φ, M 4 φ 3 φ, λ 6 M 2 φ6,... Propagator K(p2 /M 2 ) p 2 +M 2 with analytic K thins out UV completion with exponential damping Higgs Physics and the Lattice Lattice Wilsonian RG Regulate with lattice at scale Λ = π/a L lattice has all higher dimensional operators a 2 φ 2 φ, a 4 φ 4 φ, a 2 λ 6 φ 6,... Āt the symmetry breaking scale v = 250 GeV only relevant and marginal operators survive Only 1 2 m2 H φ2 and λφ 4 terms in V Higgs (φ), in addition to ( φ) 2 operator Narrow definition of Standard Model: only relevant and marginal operators at scale M At the physical Higgs scale v = 250 GeV only relevant and marginal operators survive Only 1 2 m2 H φ2 and λφ 4 terms in V Higgs (φ), in addition to ( φ) 2 operator Choice of L lattice is irrelevant unless crossover phenomenon is required to insert intermidate M scale Two-scale problem for the lattice Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 34/34

58 Standard Model Scales: Continuum Wilsonian RG UV Completion unknon new physics - Below new scale M integrated UV completion is represented by non-local L eff with all higher dimensional operators, 1 1 M 2 φ 2 φ, M 4 φ 3 φ, λ 6 M 2 φ6,... Propagator K(p2 /M 2 ) p 2 +M 2 with analytic K thins out UV completion with exponential damping - At the symmetry breaking scale v = 250 GeV only relevant and marginal operators survive Only 2 1 m2 H φ2 and λφ 4 terms in V Higgs (φ), in addition to ( φ) 2 operator Narrow definition of Standard Model: only relevant and marginal operators at scale M Higgs Physics and the Lattice Lattice Wilsonian RG Regulate with lattice at scale Λ = π/a L lattice has all higher dimensional operators a 2 φ 2 φ, a 4 φ 4 φ, a 2 λ 6 φ 6,... Scale M missing? At the physical Higgs scale v = 250 GeV only relevant and marginal operators survive Only 1 2 m2 H φ2 and λφ 4 terms in V Higgs (φ), in addition to ( φ) 2 operator Choice of L lattice is irrelevant unless crossover phenomenon is required to insert intermidate M scale Two-scale problem for the lattice Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 34/34

59 Standard Model Scales: Continuum Wilsonian RG UV Completion unknon new physics - Below new scale M integrated UV completion is represented by non-local L eff with all higher dimensional operators, 1 1 M 2 φ 2 φ, M 4 φ 3 φ, λ 6 M 2 φ6,... Propagator K(p2 /M 2 ) p 2 +M 2 with analytic K thins out UV completion with exponential damping - At the symmetry breaking scale v = 250 GeV only relevant and marginal operators survive Only 2 1 m2 H φ2 and λφ 4 terms in V Higgs (φ), in addition to ( φ) 2 operator Narrow definition of Standard Model: only relevant and marginal operators at scale M Higgs Physics and the Lattice Lattice Wilsonian RG Regulate with lattice at scale Λ = π/a L lattice has all higher dimensional operators a 2 φ 2 φ, a 4 φ 4 φ, a 2 λ 6 φ 6,... Scale M missing? Possible to insert intermediate continuum scale M with L eff to include 1 1 M 2 φ 2 φ, M 4 φ 3 φ, λ 6 M 2 φ6,... to represent new degreese of freedom above M or, Lee-Wick and other UV completions which exist above scale M (not effective theories!) At the physical Higgs scale v = 250 GeV only relevant and marginal operators survive Only 1 2 m2 H φ2 and λφ 4 terms in V Higgs (φ), in addition to ( φ) 2 operator Choice of L lattice is irrelevant unless crossover phenomenon is required to insert intermidate M scale Two-scale problem for the lattice Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 1: Standard Model Higgs Physics 34/34