Higgs Physics from the Lattice Lecture 3: Vacuum Instability and Higgs Mass Lower Bound

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 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 1/25 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 φ4 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 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 2/25 Large NF Effective Higgs Potential

The continuum RG predicts that λ < 0 as the flow continues and this has been used as an indication that the ground state of the theory is unstable.

Where λ vanishes is then used as a prediction of the energy scale of new physics.

The true RG flow with the full cutoff dependence saturates at λ0 and never turns negative.

Using the continuum RG flow when mT /Λ is not small simply gives the wrong prediction. The apparent instability is an artifact of neglecting the necessary finite cutoff. Not only is that shown in the RG flow of λ, it can also be demonstrated directly in the Higgs effective potential. running Higgs coupling (Holland,JK)

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 3/25 Effective Higgs potential

Let us review the argument that a light Higgs leads to an unstable vacuum:

For Higgs-Yukawa model with NF fermions the 1-loop renormalized effective potential, including now the Higgs-loop contribution is Z 1 2 2 1 4 1 2 2 1 4 2 2 2 Ueff = m φ + λφ + δm φ + δλφ − 2NF ln[1 + y φ /k ] 2 24 2 24 k 1 Z   + ln[k2 + V00(φ)] − ln[k2 + V00(0)] , 2 k 1 1 V = m2φ2 + λφ4 . 2 24

Because δy and δzψ are non-zero (we no longer impose the large NF limit), we specify two additional renormalization conditions. The fermion inverse propagator is

−1 Gψψ(p) = pµγµ + yv + δzψpµγµ + δyv − ΣF(p), Z −kµγµ + yv Σ (p) = y2 , F 2 2 2 2 2 k (k + y v )((k − p) + mH) the radiative correction coming from a single Higgs-loop diagram, and we require that

−1 Gψψ(p → 0) = pµγµ + yv.

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 4/25 Effective Higgs potential

−1 The requirement G (p → 0) = pµγµ + yv gives two renormalization conditions ψψ dΣF δyv − ΣF(p → 0) = 0 , δzψ − = 0 . d(pµγµ) p→0 We regulate the momentum integrals using e.g. a hard-momentum cutoff. The counterterms and the renormalized effective potential are calculated exactly using a finite cutoff. We then take the naive limit φ/Λ → 0 to remove all cutoff dependence. This ignores the fact that a finite and possibly low cutoff is required to maintain λ, y , 0. This will be the crucial point why the instability does not occur. The continuum 1-loop renormalized effective potential is then

4 " 2 # 1 2 1 4 NFy 3 4 2 2 4 φ Ueff = mφ + λφ − − φ + 2v φ + φ ln 2 24 16π2 2 v2  1  1 m2 + λφ2/2 +  (λ2φ4 − 2λφ2m2 )ln 2  16 H 2 16π mH # 1 m2 + λφ2/2 3 7 + m4 ln − λ2φ4 + λφ2m2 , 16 H m2 32 16 H q 2 2 2 where mH = λv /3 and the tree-level vev v = 3mH/λ is not shifted. The large NF limit can be recovered by omitting the Higgs-loop terms.

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 5/25 Effective Higgs potential Vacuum instability

The stability of the ground state is determined by Ueff for large φ with the dominant terms 2 4 2 2 4 4 2 2 2 4 λ φ ln(φ /v ) and −NFy φ ln(φ /v ) from the relative values of λ and y (related to mH and mT ).

If the fermionic term dominates at large φ, the minimum at v is only a local one and will decay. If we believe that the vacuum is absolutely stable, then new degrees of freedom must enter at the scale where Ueff (φ) first becomes unstable. For given values of mH and mT , this predicts the emergence of new physics.

Turning this around, let us fix mT and ask that no new stabilizing degrees of freedom are needed for φ ≤ E. Then we obtain a lower bound mH(E): if the Higgs is lighter than this, Ueff is already unstable for φ below E because the fermion term dominates even earlier.

Vacuum instability is improved using the 1-loop RG equations

dy 1 4 µ = (3 + 2NF)y , dµ 8π2

dλ 1 2 2 4 µ = (3λ + 8NFλy − 48NFy ). dµ 16π2

2 2 We can set the initial conditions λ(µ = v) = 3mH/v and y(µ = v) = mT /v. If mT is sufficiently heavy relative to mH, the Yukawa coupling dominates the RG flow and dλ/dµ < 0. The renormalized Higgs coupling eventually becomes negative at some µ = E.

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 6/25 Effective Higgs potential RG improved Vacuum instability

If the instability occurs at very large φ/v, large logarithmic terms ln(φ/v) in Ueff might spoil the perturbative expansion. This can be reduced using renormalization group improvement to resum the leading large logarithms.

1 2 2 λ(t) 4 Φ 1-loop RG improved effective potential is Ueff ≈ − 4 mH(t)Φ + 4! Φ , with t = ln µ .

RG improved effective Higgs potential

4 The dominant terms of Ueff at large φ then become λ(µ)φ (µ). Hence λ(E) = 0 indicates that the ground state is just about to become unstable.

We can calculate the effective potential non-perturbatively, using lattice simulations and compare with the RG predictions of vacuum instability

Casas,Espinoza,Quiros

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 7/25 Effective Higgs potential Lattice simulations

After the fermion field is integrated out in Top-Higgs Yukawa lattice field theory, the constraint effective potential in a finite euclidean volume Ω is defined by   Y Z  1 X  −  −  − exp( ΩUΩ(Φ)) = dφ(x)δ Φ φ(x) exp( Seff [φ]). x Ω x The delta function enforces the constraint that the scalar field φ fluctuates around a fixed average Φ. The constraint effective potential UΩ(Φ) has a very physical interpretation. If the constraint is not imposed, the probability that the system generates a configuration where the average field takes the value Φ is

1 Z P(Φ) = exp(−ΩU (Φ)), Z = dΦ0 exp(−ΩU (Φ0)). Z Ω Ω This is in close analogy to the probability distribution for the magnetization in a spin system. The scalar expectation value vev = hφi is the value of Φ for which UΩ has an absolute minimum. In a finite volume, the constraint effective potential is non-convex and can have multiple local minima. In a finite volume, it is convenient to work with the constraint effective potential, where multiple minima can be observed and the transition between the Higgs and symmetric phases is clear. It is also more natural, as the probability distribution P(Φ) can be directly observed in lattice simulations. In what follows, we drop the subscript Ω.

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 8/25 Effective Higgs potential Lattice simulations An accurate simulation method is available to calculate the derivative of the effective potential. For the Top-Higgs Yukawa model with NF degenerate fermions, the derivative is dU 1 eff = m2Φ + λhφ3i − N yhψψ¯ i , hψψ¯ i = hTr(D[φ]−1)i . dΦ 6 Φ F Φ Φ Φ

The expectation values h...iΦ mean that, in the lattice simulations, the scalar field fluctuates around some fixed average value Φ. A separate lattice simulation has to be run for every value of Φ. This is the method we propose to use in our investigation of the vacuum instability. Configurations are generated using the Hybrid Monte Carlo algorithm, where a fictitious time t and momenta π(x, t) are introduced. New configurations are generated from the equations of motion which we will write (for the scalar field only) as

φ˙(x, t) = π(x, t)    ∂Seff 1 X ∂Seff  π˙(x, t) = −  −  ,   ∂φ(x, t) Ω y ∂φ(y, t)

where the effective action Seff is obtained after fermion integration. The only difference from the usual HMC algorithm is the second term inπ ˙(x, t). This is a Lagrange multiplier which is present to enforce the constraints

1 X X φ(y, t) = Φ, π(y, t) = 0. Ω y y

This method requires extreme lattice computing if Ginsbarg-Wilson chiral fermions are used!

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 9/25 Effective Higgs potential Fate of the vacuum instability

0.15

continuum PT lattice PT 0.1 simulations The derivative of the effective potential dUeff /dΦ for the bare 2 couplings y0 = 0.5, λ0 = 0.1, m = 0.1, for which the vev is 0.05 0 av = 2.035(1). The upper plot is a close-up of the behavior near 0 the origin. The circles are the results of the simulations and the

d e r i v a t curves are given by continuum and lattice renormalized -0.05 e f

U perturbation theory.

-0.1 1-loop RG improved effective potential: 0 0.5 1 1.5 2 Φ 1 2 2 λ(t) 4 Ueff ≈ − m (t)Φ + Φ 40 4 H 4! continuum PT lattice PT simulations Φ 30 with t = ln µ .

20 2-loop RG improved effective potential is the

d e r i v a t most accurate state of the art today

e f 10 U

0

024 6810 12 Φ

effective Higgs potential (Holland,JK)

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 10/25 Ginsparg-Wilson overlap fermions The Ginsparg-Wilson overlap fermion operator was adopted to represent the chiral Yukawa coupling between the Top quark fermion field and the Higgs scalar field. Staggered and Wilson fermions would not work. Lattice Dirac operator D is required to satisfy the Ginsparg-Wilson (GW) algebraic relation

2 γ5(γ5D) + (γ5D)γ5 = 2Ra(γ5D)

and R = 1 will be chosen. The corresponding Wilson-Dirac operator is given by

1 1 D = γ (∇ + ∇∗ ) − a∇∗ ∇ , w 2 µ µ µ 2 µ µ ∗ where ∇µ (∇µ) is the forward (backward) lattice derivative with or without the SU(3) gauge link † matrices from QCD (note that γ5Dwγ5 = Dw ). Define now 1 A = − aD , 2 w

where Dw is the standard massless Wilson-Dirac operator. Then define ! 1 1 D = 1 − A √ . 2a A†A With R = 1, D satisfies the GW relation

γ5D + Dγ5 = 2aDγ5D .

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 11/25 Ginsparg-Wilson overlap fermions

Withγ ˆ5 = γ5(1 − 2D) we define two projection operators 1 1 P± = (1 ± γ ) , Pˆ ± = (1 ± γˆ ) , 2 5 2 5 and chiral components ψ¯ L,R = ψ¯P±, ψR,L = Pˆ ±ψ.

With Γ5 = γ5(1 − D) the scalar and pseudscalar densities are defined as

S(x) = ψ¯ LψR + ψ¯ RψL = ψγ¯ 5Γ5ψ,

P(x) = ψ¯ LψR − ψ¯ RψL = ψ¯Γ5ψ.

The most natural Lagrangian consistent with lattice U(1) chiral symmetry δψ = iγˆ5ψ, δψ¯ = ψ¯iγ5 and δφ = −2iφ, is defined by

† Lfermion = ψ¯Dψ + +2gyψ¯(P+φPˆ + + P−φ Pˆ −)ψ † = ψ¯ RDψR + ψ¯ LDψL + 2gy[ψ¯ LφψR + ψ¯ Rφ ψL].

The full Lagrangian for a real scalar filed has an exact discrete chiral symmetry when the fermion field is rotated by π under overlap chiral transformation and simultaneously φ is flipped into −φ. The fermion scalar density will change sign under this rotation wich is compensated by the φ → −φ reflection in the real scalar field. This symmetry is spontaneously broken in the Higgs vacuum. It is straightforward to incorporate the four-component Higgs field with O(4) chiral symmetric Yukawa coupling.

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 12/25 Ginsparg-Wilson overlap fermions Top-Bottom Doublet Coupling

Although difficult in simulations, it is possible to include the third, heaviest generation of quarks consisting of the left-handed SU(2) top-bottom doublet

tL
QL = , bL

and the corresponding right-handed SU(2) singlets tR, bR, . The complex SU(2) doublet Higgs field Φ(x) with U(1) hypercharge Y = 1 is

 φ+  Φ = , φ0

where the suffixes +,0 characterize the electric charge +1, 0 of the components. Since φ+ and φ0 are complex, we can introduce four real components,

 φ + iφ  Φ = 1 2 , iφ3 + φ4

and the Higgs potentil will have O(4) symmetry, with broken custodial O(3) symmetry, if yt and yb are chosen to be different.

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 13/25 Ginsparg-Wilson overlap fermions Top-Bottom Doublet Coupling

The Higgs potential in the complex doublet notation has the form,

V(Φ) = −m2Φ†Φ + λ(Φ†Φ)2.

It is parametrized such that the Higgs field acquires a vacuum expectation value responsible for the spontaneous electroweak symmetry breakdown

1  0  m < Φ >= √ , v = √ . 2 v λ √ −1/2 The numerical value of v is given in terms of the Fermi constant v = ( 2GF) = 246.2 GeV .

Of the four Higgs degrees of freedom three are Goldstone degrees of freedom, furnishing the longitudinal degrees of freedom for the massive weak gauge bosons, thus providing the W boson mass mW = vg2/2. The remaining one corresponds to the physical Higgs boson field √ √ h = 2Reφ0 − v/ 2).

We do not use the Higgs mechanism in the limit of zero weak gauge couplings and keep all four Higgs field components where the φ1, φ2, φ3 fluctuations represent Goldstone particles with the symmetry breaking in the φ4 direction.

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 14/25 Ginsparg-Wilson overlap fermions Top-Bottom Doublet Coupling

In the Lagrangian all 4 components are treated on equal footing where LYukawa describes the interactions of the SU(2)L doublet Higgs field with the quark fields

c LYukawa = yt · QLΦ tR + yb · QLΦbR + h.c.

c ∗ Φ = iτ2Φ is the charge conjugate of Φ, τ2 the second Pauli matrix, yt, yb are the top and bottom Yukawa couplings, respectively. When they are equal, the O(3) custodial symmetry of the Higgs potential is preserved after symmetry breaking. For unequal couplings, only the SU(2)L symmetry of the Lagrangian is maintained.

It is easy to write out the Yukawa couplings in components:

LYukawa = yt{tL(φ4 − iφ3)tR + bL(iφ2 − φ1)tR} +

yb{tL(φ1 + iφ2)bR + bL(iφ3 + φ4)bR} + h.c.

All masses are are proportional to v as they are induced by spontaneous symmetry breaking. The weak gauge boson masses allow to determine v which was already introduced. The tree level top, bottom, and Higgs masses are given in terms of the√ vacuum expectation value v and v v their respective couplings mt = gt √ , mb = gb √ , mH = 2λv . 2 2 It is esay to readjust coupling constant and field normalizationswith right and left handed quark fields as chiral overlap components.

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 15/25 Chiral Top-Higgs Model: Phase diagram Fodor,Holland,JK,Nogradi,Schroder Top-Higgs phase diagram: nf = 1, yukawa = 0.35 12 0.35 10 1-component model 8

6

vev 4 Large N hints full 2 phase diagram

0

-2 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 kappa hopping parameter notation for lattice Lagrangian:

4 2 2 2 L = −2κΣµ=1φ(x)φ(x + aµˆ) + φ(x) + λ[φ(x) − 1] , 1 − 2λ 6λ aφ (x) = (2κ)1/2φ(x) , a2m2 = − 8 , g = . 0 0 κ 0 κ2

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 16/25

4 Chiral Top-Higgs Model: Phase diagram

Gerhold and Jansen Top-Higgs doublet phase diagram:

0.2 L =8 L =6

0.1 Fermion doublet model 0 SYM FM N ˜ κ -0.1 Compared with large N

-0.2 AFM AFM Very similar to -0.3 1-component model

0 2 4 6 8 10

y˜N

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 17/25 Chiral Top-Higgs Model: Large N test

Fodor,Holland,JK,Nogradi,Schroder 1 vev with subleading N corrections: a + b / nf fit - mass^2 0.0637110 - yukawa sqrt( nf ) 0.35777 - no sqrt rescaling 1.25

1.24

1.23

1.22 Higgs vacuum expectation value

1.21

1.2

vev / sqrt( nf ) continuum notation Subleading 1/N correction extrapolates correctly 1.19

1.18 2 a4 + b / nf fit 6- mass^2 0.1959020 8 10- yukawa sqrt( 12 nf ) 0.35777 14 - no sqrt 16 rescaling 18 20 0.3 nf

0.28

0.26

0.24

0.22

0.2

0.18 vev / sqrt( nf ) continuum notation 0.16

0.14

0.12 2 4 6 8 10 12 14 16 18 20 nf

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 18/25 Chiral Top-Higgs Model: Large N test

Fodor,Holland,JK,Nogradi,Schroder 1 subleading mTop N corrections: a + b / nf fit - mass^2 0.0637110 - yukawa sqrt( nf ) 0.35777 - no sqrt rescaling 0.97

0.96

0.95 0.94 Top quark mass 0.93 m_top

0.92 Subleading 1/N correction 0.91 extrapolates correctly 0.9

0.89 2 a4 + b / nf fit 6- mass^2 0.1292680 8 -10 yukawa sqrt( 12 nf ) 0.35777 14 - no sqrt 16 rescaling 18 20 0.64 nf

0.63

0.62

0.61

0.6 m_top

0.59

0.58

0.57

0.56 2 4 6 8 10 12 nf

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 19/25 Chiral Top-Higgs Model: Higgs mass lower bound

Fodor,Holland,JK,Nogradi,Schroder Cutoff dependence of Higgs lower bound: 16^3 x 20 - nf 1 - lambda 0.0001 - yukawa 0.38 - no sqrt rescaling 220 m_top m_h 200

180

160

140

[GeV] 120

100

80

60

40 0.5 1 1.5 2 2.5 3 cut-off (pi/a) [TeV] Preliminary! Extreme lattice computing: USQCD, Wuppertal(GPU) Higgs physics as a video game

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 20/25 2-loop continuum SM RG: Running gauge couplings 2-loop RG with 1-loop matching 120

Runing gauge couplings µ = MZ initial conditions:

100 α1(MZ ) = 0.0102 α2(MZ ) = 0.0338 α3(MZ ) = 0.123 80 Planck scale m =500 GeV H Higgs and Yukawa couplings: m =80 GeV −1 H 60 α1 mf = yf v 1/2 mH = λ v v = 246.22 GeV 40 −1 α 2 α1 is U(1)Y coupling

20 "Landau pole" “Landau pole” in α1 −1 α1 α3 41 t=log(mu/GeV) ≈ 10 GeV for mH = 80GeV 0 0 5 10 15 20 25 30 35 40 45 inherited from QED shifted to ≈ 1048GeV for Gauge sector (2-loop): mH = 500GeV (4π)4g−3β(2) = 199 g2 + 27 g2 + 44 g2 − P ( 17 y2 + y2 + 3y2 ), 1 g1 50 1 10 2 5 3 g 5 ug dg eg (4π)4g−3β(2) = 9 g2 + 35 g2 + 12g2 − P (3y2 + 3y2 + y2 ), 2 g2 10 1 6 2 3 g ug dg eg (4π)4g−3β(2) = 11 g2 + 9 g2 − 26g2 − 4 P (y2 + y2 ). 3 g3 10 1 2 2 3 g ug dg

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 21/25 2-loop continuum SM RG: Running Top coupling

Top quark yukawa coupling 6 Top yukawa coupling yt

Planck scale m =500 GeV For mH < 160GeV monotone 5 H Y decrease in yt top Lower Higgs mass bound (and 4 300 250 vacuum instability?) might remain 200

180 perturbative with running λ for 3 mH < 160GeV

yt ≈ 5 pseudo-fixed point 2 strong coupling ? At mH = 500GeV strong yukawa α "Landau pole" 1 coupling below the Planck scale 1 170 mH = 500GeV Higgs mass range is 160 80 becoming strongly interacting 0 0 5 10 15 20 25 30 35 40 45 t=log(mu/GeV) Upper bound with heavy Higgs is strong coupling problem! Yukawa sector (1-loop): (4π)2y−1β(1) = 3y2 + 2 P (3y2 + 3y2 + y2 ) − 9 g2 − 9 g2, τ yτ τ g ug dg eg 4 1 4 2 (4π)2y−1β(1) = 3y2 − 3y2 + 2 P (3y2 + 3y2 + y2 ) − 17 g2 − 9 g2 − 8g2, t yt t b g ug dg eg 20 1 4 2 3 (4π)2y−1β(1) = 3y2 − 3y2 + 2 P (3y2 + 3y2 + y2 ) − 1 g2 − 9 g2 − 8g2. b yb b t g ug dg eg 4 1 4 2 3

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 22/25 2-loop continuum SM RG: Running Higgs coupling for light Higgs

0.4

λ Running Higgs coupling Lower bound and running λcoupling 0.3 At low Higgs masses negative λ would Planck scale 0.2 suggest vacuum instability

Large changes in Ueff (φ) from 1-loop to 0.1 2-loop would call for nonperturbative 140 calculations 135 0 130 Running gauge couplings are important!

110 α "Landau pole" −0.1 1 80 Problem is on borderline of perturbation

m =40 GeV H theory, at best −0.2

−0.3 0 5 10 15 20 25 30 35 40 45 t=log(mu/GeV) Higgs sector (1-loop): (4π)2β(1) = 12λ2 + 8λ P (3y2 + 3y2 + y2 ) − 9λ( 1 g2 + g2) λ g ug dg eg 5 1 2 −16 P (3y4 + 3y4 + y4 ) + 9 ( 3 g4 + g4 + 2 g2g2), g ug dg eg 4 25 1 2 5 1 2

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 23/25 2-loop continuum SM RG: Running Higgs coupling for heavy Higgs

Running Higgs coupling 40 Upper bound and running coupling λ

35 Planck scale At large Higgs masses growing λ λ requires strong coupling 30 Call for nonperturbative calculations 25

500 Running gauge couplings are important! 20 300 α "Landau pole" 250 1 200 Is large mH consistent with EW data? 15 180 170

strong coupling ? What is the Standard Model beyond PT? 10

m =160 GeV H 5

t=log(mu/GeV) 0 0 5 10 15 20 25 30 35 40 45

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 24/25 I With Higgs, Top, and QCD gauge couplings in few percent range precision in Higgs mass lower and upper bounds

I Higgs physics from the lattice is most interesting if mH is “too heavy” compared to magic range 160-180 GeV (low MUVcompletion threshold)

I The finite lattice cutoff keeps us honest, but separation of new π physics scale of MUVcompletion and a lattice cutoff remains a difficult challenge

I Don’t ask what the higher dimensional operators can do for you, ask what you can do for the higher dimensional operators (Lee-Wick example)

I Role of weak gauge couplings beyond their leading perturbative effects remains unclear

I Urgency of LHC: We do not have the luxury of many years to explore

Conclusions

I For low lattice cutoff, it is feasible to do high precision Higgs physics calculations

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 25/25 I Higgs physics from the lattice is most interesting if mH is “too heavy” compared to magic range 160-180 GeV (low MUVcompletion threshold)

I The finite lattice cutoff keeps us honest, but separation of new π physics scale of MUVcompletion and a lattice cutoff remains a difficult challenge

I Don’t ask what the higher dimensional operators can do for you, ask what you can do for the higher dimensional operators (Lee-Wick example)

I Role of weak gauge couplings beyond their leading perturbative effects remains unclear

I Urgency of LHC: We do not have the luxury of many years to explore

Conclusions

I For low lattice cutoff, it is feasible to do high precision Higgs physics calculations

I With Higgs, Top, and QCD gauge couplings in few percent range precision in Higgs mass lower and upper bounds

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 25/25 I The finite lattice cutoff keeps us honest, but separation of new π physics scale of MUVcompletion and a lattice cutoff remains a difficult challenge

I Don’t ask what the higher dimensional operators can do for you, ask what you can do for the higher dimensional operators (Lee-Wick example)

I Role of weak gauge couplings beyond their leading perturbative effects remains unclear

I Urgency of LHC: We do not have the luxury of many years to explore

Conclusions

I For low lattice cutoff, it is feasible to do high precision Higgs physics calculations

I With Higgs, Top, and QCD gauge couplings in few percent range precision in Higgs mass lower and upper bounds

I Higgs physics from the lattice is most interesting if mH is “too heavy” compared to magic range 160-180 GeV (low MUVcompletion threshold)

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 25/25 I Don’t ask what the higher dimensional operators can do for you, ask what you can do for the higher dimensional operators (Lee-Wick example)

I Role of weak gauge couplings beyond their leading perturbative effects remains unclear

I Urgency of LHC: We do not have the luxury of many years to explore

Conclusions

I For low lattice cutoff, it is feasible to do high precision Higgs physics calculations

I With Higgs, Top, and QCD gauge couplings in few percent range precision in Higgs mass lower and upper bounds

I Higgs physics from the lattice is most interesting if mH is “too heavy” compared to magic range 160-180 GeV (low MUVcompletion threshold)

I The finite lattice cutoff keeps us honest, but separation of new π physics scale of MUVcompletion and a lattice cutoff remains a difficult challenge

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 25/25 I Role of weak gauge couplings beyond their leading perturbative effects remains unclear

I Urgency of LHC: We do not have the luxury of many years to explore

Conclusions

I For low lattice cutoff, it is feasible to do high precision Higgs physics calculations

I With Higgs, Top, and QCD gauge couplings in few percent range precision in Higgs mass lower and upper bounds

I Higgs physics from the lattice is most interesting if mH is “too heavy” compared to magic range 160-180 GeV (low MUVcompletion threshold)

I The finite lattice cutoff keeps us honest, but separation of new π physics scale of MUVcompletion and a lattice cutoff remains a difficult challenge

I Don’t ask what the higher dimensional operators can do for you, ask what you can do for the higher dimensional operators (Lee-Wick example)

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 25/25 I Urgency of LHC: We do not have the luxury of many years to explore

Conclusions

I For low lattice cutoff, it is feasible to do high precision Higgs physics calculations

I With Higgs, Top, and QCD gauge couplings in few percent range precision in Higgs mass lower and upper bounds

I Higgs physics from the lattice is most interesting if mH is “too heavy” compared to magic range 160-180 GeV (low MUVcompletion threshold)

I The finite lattice cutoff keeps us honest, but separation of new π physics scale of MUVcompletion and a lattice cutoff remains a difficult challenge

I Don’t ask what the higher dimensional operators can do for you, ask what you can do for the higher dimensional operators (Lee-Wick example)

I Role of weak gauge couplings beyond their leading perturbative effects remains unclear

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 25/25 Conclusions

I For low lattice cutoff, it is feasible to do high precision Higgs physics calculations

I With Higgs, Top, and QCD gauge couplings in few percent range precision in Higgs mass lower and upper bounds

I Higgs physics from the lattice is most interesting if mH is “too heavy” compared to magic range 160-180 GeV (low MUVcompletion threshold)

I The finite lattice cutoff keeps us honest, but separation of new π physics scale of MUVcompletion and a lattice cutoff remains a difficult challenge

I Don’t ask what the higher dimensional operators can do for you, ask what you can do for the higher dimensional operators (Lee-Wick example)

I Role of weak gauge couplings beyond their leading perturbative effects remains unclear

I Urgency of LHC: We do not have the luxury of many years to explore

Julius Kuti, University of California at San Diego INT Summer School 2007, Seattle, Lecture 3: Vacuum Instability and Higgs Mass Lower Bound 25/25