Higgs Physics from the Lattice Lecture 2: Triviality and Higgs Mass Upper Bound

Higgs Physics from the Lattice Lecture 2: Triviality and Higgs Mass Upper 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 on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 1/37

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 2: Triviality and Higgs Mass Upper Bound 2/37

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 2: Triviality and Higgs Mass Upper Bound 3/37

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 2: Triviality and Higgs Mass Upper Bound 4/37

Standard Model Scales: RG Fixed Points and Triviality H = =g 2 3 2 1.5 1 0.5 0 running of λ, g t, g 3 0 1 2 3 4 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 ( 689 25 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 = 10 41 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 2: Triviality and Higgs Mass Upper Bound 5/37

Landau pole in perturbation theory: Higgs mass upper bound The RGE in the pure Higgs sector is known to three-loop order in the MS scheme β(λ) = d λ = 3 d t 2π 2 λ2 39 7176 + 4032ζ(3) 32π 4 λ3 + (16π 2 ) 3 λ 4. The RGE exhibits an IR attractive fixed point λ=0 (perturbative triviality ) with general solution in one loop order 1 λ(µ) λ(µ) = 1 λ(λ) + 3 2π 2 ln Λ, λ(λ) = 1 µ 3 2π 2 λ(µ)ln Λ µ Increasing λ(µ) at fixed λ(λ), the Landau pole is hit at 3 2π 2 λ(µ)ln Λ µ = 1 bound λ(µ) < 2π2 3ln Λ µ which is related to the upper bound on the Higgs mass. with the naive upper λ = m2 H v 2. When we try to increase λ(µ) at fixed λ(λ) beyond the Landau pole limit, something else will happen which will reveal the intrinsic non-removable cutoff in the theory. This will be illustrated in the large N limit of the Higgs-Yukawa fermion model. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 6/37

Functional integral saddle point: Large N F limit of Top-Higgs model The Lagrangian density of the continuum model after rescaling the coupling constants: L = 1 2 φ ( + m 2 )φ ) + ψ i γ µ µ ψ i y ψ i ψ i φ + 1 λ φ 4, i = 1, 2,...N F. NF 24 N F With rescaling of the scalar field φ N F and integrating out the fermion fields, the partition function is given by ( Z = Dφ exp [ N F Tr ln(γ µ µ yφ) + d 4 x[ 1 2 φ( + m2 )φ + 1 )] 24 λφ4 ]. The N F limit is a saddle point for the functional integral. The solution of the saddle point equations is equivalent to summing all Feyman diagrams with leading fermion bubbles which are proportional to N F. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 7/37

Renormalization: Large N F limit of Top-Higgs model We first consider a Higgs-Yukawa model of a single real scalar field coupled to N F massless fermions in the exact N F limit Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 8/37

Renormalization: Large N F limit of Top-Higgs model We first consider a Higgs-Yukawa model of a single real scalar field coupled to N F massless fermions in the exact N F limit This will allow us to demonstrate that the theory is trivial and the supposed vacuum instability is an artifact of ignoring the necessary cutoff in the model Bare Lagrangian of the Higgs-Yukawa theory in Euclidean space-time is defined by L = 1 2 m2 0 φ2 0 + 1 24 λ 0φ 4 0 + 1 2 ( µ φ 0 ) 2 + ψ a 0 ( γµ µ + y 0 φ 0 ) ψ a 0, where a = 1,..., N F sums over the degenerate fermion flavors and the subscript 0 denotes bare quantities. We rewrite this as L = 1 2 m2 0 Z φφ 2 + 1 24 λ 0Zφ 2 φ4 + 1 2 Z ( φ µ φ ) 2 + Zψ ψ ( a γ µ µ + y 0 Zφ φ ) ψ a = 1 2 (m2 + δm 2 )φ 2 + 1 24 (λ + δλ)φ4 + 1 2 (1 + δz φ) ( µ φ ) 2 +(1 + δz ψ ) ψ a γ µ µ ψ a + ψ a (y + δy)φψ a, Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 8/37

Renormalization: Large N F limit of Top-Higgs model We first consider a Higgs-Yukawa model of a single real scalar field coupled to N F massless fermions in the exact N F limit This will allow us to demonstrate that the theory is trivial and the supposed vacuum instability is an artifact of ignoring the necessary cutoff in the model Bare Lagrangian of the Higgs-Yukawa theory in Euclidean space-time is defined by L = 1 2 m2 0 φ2 0 + 1 24 λ 0φ 4 0 + 1 2 ( µ φ 0 ) 2 + ψ a 0 ( γµ µ + y 0 φ 0 ) ψ a 0, where a = 1,..., N F sums over the degenerate fermion flavors and the subscript 0 denotes bare quantities. We rewrite this as L = 1 2 m2 0 Z φφ 2 + 1 24 λ 0Zφ 2 φ4 + 1 2 Z ( φ µ φ ) 2 + Zψ ψ ( a γ µ µ + y 0 Zφ φ ) ψ a = 1 2 (m2 + δm 2 )φ 2 + 1 24 (λ + δλ)φ4 + 1 2 (1 + δz φ) ( µ φ ) 2 +(1 + δz ψ ) ψ a γ µ µ ψ a + ψ a (y + δy)φψ a, with wavefunction renormalization factors, renormalized parameters, and counterterms Z φ = 1 + δz φ, Z ψ = 1 + δz ψ, m 2 0 Z φ = m 2 + δm 2, λ 0 Z 2 φ = λ + δλ, Z ψ Zφ y 0 = y + δy. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 8/37

Renormalization: Large N F limit of Top-Higgs model In the limit where N F becomes large, all Feynman diagrams with Higgs loops are suppressed relative to those with fermion loops. Hence, two of the counterterms vanish, Renormalization conditions (1,2) δy = 0, δz ψ = 0 as there are no radiative corrections to the fermion propagator or to the Higgs-fermion coupling. To maintain tree level relation m 2 + 1 6 λv2 = 0 to all orders: Renormalization condition (3) δm 2 + 1 6 δλv2 4N F y 2 k 1 k 2 + y 2 v 2 = 0 Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 9/37

Renormalization: Large N F limit of Top-Higgs model In the large N F limit, the inverse propagator of the Higgs fluctuation ϕ = φ v is G 1 ϕϕ(p 2 ) = p 2 + m 2 + 1 2 λv2 + p 2 δz φ + δm 2 + 1 2 δλv2 Σ(p 2 ) Σ(p 2 ) = 4N F y 2 y 2 v 2 k.(k p) k (k 2 + y 2 v 2 )((k p) 2 + y 2 v 2 ) We impose the condition G 1 ϕϕ(p 2 0) = p 2 + m 2 H, which separates into two renormalization conditions: Renormalization condition (4) δm 2 + 1 2 δλv2 Σ(p 2 = 0) = 0 Renormalization condition (5) δz φ dσ(p2 ) dp 2 p 2 =0 = 0 Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 10/37

Counterterms: Large N F limit of Top-Higgs model Renormalization condition δm 2 + 1 2 δλv2 Σ(p 2 = 0) = 0 maintains the tree-level relation m 2 H = m2 + 1 2 λv2 = 1 3 λv2 exactly. The counterterms precisely cancel all the finite and infinite radiative contributions. The Higgs mass defined as the zero-momentum piece of G 1 ϕϕ is identical to the curvature U eff (v). True physical mass given by the pole of the propagator and the renormalized masses m H and m can be related in perturbation theory. Renormalization conditions (3) and (4) yield δm 2 = 4N F y 2 k δλ = 24N F y 4 k 2 + 2y 2 v 2 (k 2 + y 2 v 2 ) 2, k 1 (k 2 + y 2 v 2 ) 2. The closed form for δz φ is more complicated and less illuminating. To demonstrate triviality, we use some finite cutoff in the momentum integrals and examine what occurs as this cutoff is removed. We will use a simple hard-momentum cutoff k Λ. Exactly the same conclusions would be reached using instead e.g. Pauli-Villars regularization. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 11/37

Counterterms: Large N F limit of Top-Higgs model The non-zero counterterms are δm 2 N F y 2 [ 1 y 4 v 4 = 2π 2 2 Λ2 + 2(Λ 2 + y 2 v 2 ) 1 ] 2 y2 v 2 δλ = 3N Fy 4 [ y 2 v 2 π 2 2(Λ 2 + y 2 v 2 ) 1 2 1 ( 2 ln y 2 v 2 )] Λ 2 + y 2 v 2 δz φ = N Fy 2 [ ( 1 y 2 2π 2 4 ln v 2 + Λ 2 ) y 2 v 2 + 5Λ4 3Λ 2 y 2 v 2 ] 12(Λ 2 + y 2 v 2 ) 2. In the large N F limit, the fermion inverse propagator receives no radiative correction, G 1 ψψ (p) = p µγ µ + yv, so we identify the fermion mass as m T = yv. Because both δy and δz ψ vanish, we can substitutey 2 = Z φ y 2 0 Z φ = 1 + N Fy 2 0 8π 2 ln Λ2 m 2 5 1 3. T For any finite bare Yukawa coupling y 0, the Higgs wavefunction renormalization factor Z φ vanishes logarithmically as the cutoff is removed, m T /Λ 0. This same logarithmic behavior will appear in all of the renormalized couplings, for any choice of bare couplings. Triviality: a finite cutoff must be kept to maintain non-zero interactions. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 12/37

Triviality: Large N F limit of Top-Higgs model Explicitly, the renormalized Yukawa coupling is y 2 = y 2 0 Z φ = y 2 0 1 + N Fy 2 0 8π 2 ln Λ2 m 2 5 1 3 T N 1 F Λ2 ln 8π2 m 2, as m T Λ 0. T For the renormalized Higgs coupling, we have λ = λ 0 Zφ 2 δλ = λ 0Zφ 2 + 3N Fy 4 m 2 T π 2 2(Λ 2 + m 2 T ) 1 2 1 2 ln m 2 T Λ 2 + m 2 T Zφ 2 λ 0 + 3N Fy 4 0 π 2 1 2 1 2 ln m2 T Λ 2 12 N 1 F Λ2 ln 8π2 m 2, as m T Λ 0. T Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 13/37

Triviality: Large N F limit of Top-Higgs model The slow logarithmic vanishing of y and λ enables a relatively large separation of cutoff and physical scales and still maintain significant interactions. Such a theory can in some limited sense be considered physical, if the cutoff effects are sufficiently small. The ratio of the Higgs and fermion masses in the large N F is m 2 H m 2 T = λv2 3y 2 v 2 = λ 3y 2 4, as m T Λ 0. Although completely unphysical, we can also consider the limit m T /Λ 1, where the cutoff is much below the physical scale. From Equation, we see this gives δλ = 0, δz φ = 0, and hence Z φ 1. In this limit, the connection between bare and renormalized parameters is simply λ = λ 0, y = y 0. This result is not surprising: deep in the cutoff regime, we simply have the bare theory, with no separation into renormalized parameters and their counterterms. This will be relevant when we discuss the Landau pole and the vacuum instability. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 14/37

Renormalization Group flow: Large N F limit of Top-Higgs model The physical properties of the theory, at finite cutoff Λ are fixed by the choice of a complete set of bare parameters. Using the explicit cutoff dependence of y and λ, we can calculate the Callan-Symanzik flow of the renormalized couplings inn the m T /Λ 1 limit β y (y, λ) = Λ dy2 dλ = y 2 N F y 2 0 Z 2 0 φ 4π 2 = N Fy 4 4π 2 β λ (y, λ) = Λ dλ dλ = 1 16π 2 [ 8NF λy 2 + 48N F y 4] This is exactly the same RG flow one would calculate in the large N F limit using e.g. dimensional regularization, where the cutoff simply does not appear and the renormalized couplings flow with the arbitrary renormalization scale µ. The overall signs of the β functions would be opposite: increasing Λ corresponds to decreasing µ). We should expect this: when the cutoff is far above the physical scales, the finite cutoff effects are negligible and we must reproduce the unique cutoff-independent β functions. However, as the cutoff is reduced and m T /Λ increases, this cannot continue to hold indefinitely, as the renormalized couplings must eventually flow to the bare ones! Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 15/37

Renormalization Group flow: Large N F limit of Top-Higgs model Let us demonstrate an explicit example of the Callan-Symanzik RG flow. In the large N F limit, m T = yv = y 0 v 0. The bare vev is determined by the minimum of the bare effective potential U eff,0 = 1 2 m2 0 φ2 0 + 1 24 λ 0φ 4 0 2N F ln [ 1 + y 2 0 φ2 0 /k2]. k Using a hard-momentum cutoff, this gives m 2 0 + 1 6 λ 0v 2 0 N Fy 2 0 2π 2 1 2 Λ2 + 1 2 y2 0 v2 0 ln y 2 0 v2 0 Λ 2 + y 2 = 0. 0 v2 0 We express all dimensionful quantities in units of the cutoff Λ. We pick some fixed values for λ 0 and y 0. Varying the value of m 2 0 /Λ2 changes the solution v 0 /Λ of Equation and hence the ratio m T /Λ. As we said, choosing the values of the bare parameters completely determines everything in the theory. For example, to attain a very small value of m T /Λ requires m 2 0 /Λ to be precisely fine tuned! This is the origin of the so-called fine-tuning problem. The critical surface, where v 0 /Λ = 0, is the transition line m 2 0 Λ 2 N Fy 2 0 4π 2 = 0, with all counterterms and renormalized parameters expressed in terms of λ 0, y 0, m 2 0 and v 0. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 16/37

Renormalization Group flow: Large N F limit of Top-Higgs model All of the counterterms and renormalized parameters can be expressed in terms of λ 0, y 0, m 2 0 and v 0. We make an arbitrary choice λ 0 = 0.1, y 0 = 0.7 and vary the value of m 2 0 /Λ2 to explore the range 10 13 < m T /Λ < 10 2. When the cutoff is high, the exact RG flow is exactly the same as if the cutoff had been completely removed and follows precisely the continuum form of Equation. However, as the cutoff is reduced, the exact RG flow eventually breaks away from the continuum form and reaches a plateau at the value of the bare coupling. running Higgs coupling (Holland,JK) Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 17/37

Renormalization Group flow: Large N F limit of Top-Higgs model All of the counterterms and renormalized parameters can be expressed in terms of λ 0, y 0, m 2 0 and v 0. We make an arbitrary choice λ 0 = 0.1, y 0 = 0.7 and vary the value of m 2 0 /Λ2 to explore the range 10 13 < m T /Λ < 10 2. When the cutoff is high, the exact RG flow is exactly the same as if the cutoff had been completely removed and follows precisely the continuum form of Equation. However, as the cutoff is reduced, the exact RG flow eventually breaks away from the continuum form and reaches a plateau at the value of the bare coupling. running Top coupling (Holland,JK) Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 18/37

Higgs Mass Upper Bound: 1-component Higgs field 1-component lattice φ 4 model is characterized by two bare parameters κ, λ with lattice action 4 S = 2κ φ(x)φ(x + ˆµ) + φ(x) 2 + λ(φ(x) 2 1) 2. x µ=1 There are two phases separated by a line of critical points κ = κ c (λ). For κ > κ c (λ) the symmetry is spontaneously broken and the bare field φ(x) has a non-vanishing expectation value v. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 19/37

Higgs Mass Upper Bound: 1-component Higgs field Study the vacuum expectation value and the connected two-point function G(x) = φ(x)φ(0) c = φ(x)φ(0) v 2. A renormalized mass m r and a field wave function renormalization constant Z r are defined through the behavior of Fourier transform of G(x) for small momenta: { G(k) 1 = Zr 1 m 2 r + k 2 + O(k 4 ) }. We also define the normalization constant associated with the canonical bare field through ˆ Z r = 2κZ r = 2κm 2 r χ. In the framework of perturbation theory correlation functions of the multiplicatively renormalized field φ r (x) = Zr 1/2 φ(x), have at all orders finite continuum limits after mass and coupling renormalization are taken into account. Correspondingly a renormalized vacuum expectation value is defined through v r = vzr 1/2. Finally a particular renormalized coupling is defined by g r 3m 2 r /v 2 r = 3m 4 r χ/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 2: Triviality and Higgs Mass Upper Bound 20/37

Higgs Mass Upper Bound: 1-component Higgs field The renormalization group equations predict that the mass and vacuum expectation value go to zero according to m r τ 1/2 ln(τ) 1/6, v r τ 1/2 ln(τ) 1/3, for τ = κ/κ c 1 0, and correspondingly the renormalized coupling is predicted to go to zero logarithmically which is the expression of triviality. The critical behavior in the broken phase is conveniently expressed in terms of three integration constants C i, i = 1, 2, 3 appearing in the critical behaviors: m r = C 1 (β 1g r ) 17/27 e 1/β 1g r {1 + O(g r )}, β 1 = 3 16π 2, { Z r = C 2 1 7 } g r 36 16π 2 + O(g2 r ), 1 κ κ c = 2 C 3 m2 r g 1/3 r {1 + O(g r )}. These constants were estimated by relating them to the corresponding constants C i in the symmetric phase. These in turn were computed by integrating the renormalization group equations with initial data on the line κ = 0.95κ c (λ) obtained from high temperature expansions. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 21/37

Higgs Mass Upper Bound: 1-component Higgs field One massive Higgs particle analytic results: Lüscher, Weisz lattice simulations: JK,Lin,Shen m R v 3.2 at am R = 0.5 How far can we lower the cutoff? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 22/37

Higgs Mass Upper Bound: 4-component O(4) Higgs model One massive Higgs particle + 3 Goldstone particles analytic results: Lüscher, Weisz lattice simulations: JK,Lin,Shen m R v 2.6 at am R = 0.5 How far can we lower the cutoff? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 23/37

Higgs Mass Upper Bound: 4-component O(4) Higgs model δ measures cutoff effects in Goldstone-Goldstone scattering Lüscher, Weisz Ad hoc and connected with lattice artifacts But threshold of new physics is in the continuum! What do we do when cutoff is low? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 24/37

Higgs Physics and the Lattice Continuum Wilsonian RG 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 2: Triviality and Higgs Mass Upper Bound 25/37

Higgs Physics and the Lattice 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 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 2: Triviality and Higgs Mass Upper Bound 25/37

Higgs Physics and the Lattice 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 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 2: Triviality and Higgs Mass Upper Bound 25/37

Higgs Physics and the Lattice 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 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 2: Triviality and Higgs Mass Upper Bound 25/37

Example for UV Completion: Higer derivative (Lee-Wick) Higgs sector Jansen,JK,Liu, Phys. Lett. B309 (1993) p.119 and p.127 (Model was recently reintroduced by Grinstein, O Connell, Wise in arxiv:0704.1845) Represent the Higgs doublet with four real components φ a which transform in the vector representation of O(4) and include new higher derivative terms in the kinetic part of the O(4) Higgs Lagrangian, L H = 1 2 µφ a µ φ a cos(2θ) M 2 φ a φ a + 1 2M 4 µφ a µ φ a V(φ a φ a ) The Higgs potential is V(φ a φ a ) = 1 2 µ2 φ a φ a + λ(φ a φ a ) 2. The higher derivative terms of the Lagrangian lead to complex conjugate ghost pairs in the spectrum of the Hamilton operator Complex conjugate pairs of energy eigenvalues and the related complex pole pairs in the propagator are parametrized by M = Me ±iθ. Choice Θ = π/4 simplifies. The absolute value M of the complex ghost mass M will be set on the TeV scale Unitary S-matrix, macroscopic causality, Lorent invariance? λ > 0 asymptotically! Vacuum instability? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 26/37

Example for UV Completion: Gauged Lee-Wick extension Higher derivative Yang-Mills gauge Lagrangian for the SU(2) L U(1) weak gauge fields W µ, B µ follows similar construction with covariant derivative D ab µ = δ ab µ + gf abc W c µ, L W = 1 4 Ga µνg aµν 1 4M 4 D2 G a µνd 2 G aµν, L W is superrenormalizable but not finite. Full gauged Higgs sector is described by the Lagrangian L = L W + L B + L Higgs, L Higgs = (D µ Φ) D µ Φ + 1 2M 4 (D µd DΦ) (D µ D DΦ) V(Φ Φ) Gauge-covariant derivative is D µ Φ = ( µ + i g 2 σ W µ + i g 2 B µ) Φ. Similar fermion construction: L fermion = iψ D Ψ + SM particle content is doubled Logarithmic divergences only i 2M 4 Ψ D 2 D D 2 Ψ. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 27/37

Higher derivative β-function and RG: Liu s Thesis (1994) β function: Lee-Wick UV Completion Higher derivative Higgs sector is finite field theory Mass dependent β(t)-function vanishes asymptotically Grows logarithmically in gauged Higgs sector running Higgs coupling: Running Higgs coupling λ(t) freezes asymptotically The fixed line of allowed Higgs couplings must be positive! Vacuum instability? Higgs mass lower bound from λ( ) > 0 Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 28/37

S-matrix, Unitarity, and Causality: Lee-Wick UV Completion Liu,Jansen,JK Nucl.Phys. B34 (1994) p.635 cross section phase shift Equivalence theorem, Goldstone scattering Higgs mass upper bound relaxed m H =1 TeV, or higher, but ρ-parameter and other Electroweak precision? Phase shift reveals ghost, microscopic time advancement, only π/2 jump in phase shift m H =1 TeV, M = 3.6 TeV v/m=0.07, m H /M = 0.28 What about the ρ-parameter? Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 29/37

The ρ-parameter ρ = m 2 W m 2 Z cos2 θ W = 1 + Z(+) Z ( 0) cosθ W is determined independently in high precision lepton processes. The 1-loop vacuum polarization operator will shift the physical pole locations for the weak gauge bosons: Π H W ρ 1 Higgs = MW,tree 2 ΠH Z MZ,tree 2 = 3 d 4 k Σ H (k 2 ) 4 g 2 k 2 <Λ 2 (2π) 4 (k 2 + MW,tree 2 )(k2 + MZ,tree 2 )(k2 + Σ 2 H (k2 )) Heavy Higgs with acceptable ρ-parameter would be a broad resonace How to calculate 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 2: Triviality and Higgs Mass Upper Bound 30/37

Higgs resonance: Finite width in finite volume? Energy spectrum of two-particle states in a finite box with periodic boundary conditions = elastic scattering phase shifts in infinite volume Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 31/37

Higgs resonance: Finite width in finite volume? Energy spectrum of two-particle states in a finite box with periodic boundary conditions = elastic scattering phase shifts in infinite volume Two-particle energy levels are calculable by Monte Carlo techniques = extract phase shifts from numerical simulations on finite lattices Infinite bare λ limit = 4-dimensional O(4) non-linear σ-model in broken phase with unstable Higgs particle Small Goldstone mass is required by the method = add external source term to the lattice action of the 4-dimensional O(4) non-linear σ-model: 4 S = 2κ φ α x φ α x+ˆµ + J φ 4 x x µ=1 x The scalar field is represented as a four component vector φ α x of unit length: φ α x φ α x = 1 Two Goldstone bosons ( pions ) of mass m π with zero total momentum in a cubic box of size L 3 in the elastic region are characterised by centre-of-mass energies W or momenta k defined through W = 2 m 2 π + k 2, k = k with W and k in the ranges: 2m π < W < 4m π 0 < k/m π < 3 They are classified according to irreducible representations of the cubic group. Their discrete energy spectrum W j, j = 0, 1, 2,..., is related to the scattering phase shifts δ l with angular momenta l which are allowed by the cubic symmetry of the states Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 31/37

Higgs resonance: Finite width in finite volume? In the subspace of cubically invariant states only angular momenta l = 0, 4, 6,... contribute. Due to the short-range interaction it seems reasonable to neglect all l 0. Then an energy value W j belongs to the two-particle spectrum, if the corresponding momentum k j = (W j /2) 2 m 2 π is a solution of ( ) kj L δ 0 (k j ) = φ + jπ 2π The continuous function φ(q) is given by qπ 3/2 tan ( φ(q)) = Z 00 (1, q 2 ), φ(0) = 0 with Z 00 (1, q 2 ) defined by analytic continuation of Z 00 (s, q 2 ) = 1 4π n Z 3 ( n 2 q 2) s. This result holds only for 0<q 2 <9, which is fulfilled in our simulations, and provided finite volume polarisation effects and scaling violations are negligible In our Monte-Carlo investigation we calculate the momenta k j corresponding to the two-particle energy spectrum W j for a given size L, and can then read off the scattering phaseshift δ 0 (k j ) from its defining equation In order to scan the momentum dependence of the phase shift we have to vary the spatial extent L of 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 2: Triviality and Higgs Mass Upper Bound 32/37

Higgs resonance: Finite width in finite volume? The two-particle states in our model are also classified according to the remaining O(3)-symmetry: They have isospin 0, 1, 2. Since we expect the σ-resonance to be in the isospin-0 channel, we restrict ourselves to that case. Operator O 0 (t) = Φ 4 0 =,t L 3 x Φ 4 for σ-field at zero momentum is included x,t Two-particle correlation matrix function C ij (t) = ( O i (t) O i (t + 1) ) O j (0) i, j = 0, 1, 2,... Eigenvalues decaying as exp( W i t) The set of states is truncated at finite i=r to keep W r below inelastic threshold J m =2m =3m =4m Lines of constant ratios m σ /m π and m π L (dotted line) Reflection multi-cluster algorithm for finite external source J Set of 3 κ, J configurations with middle point κ = 0.315, J = 0.01 tuned to keep σ-mass in elastic region Cylindrical lattices L 3 T up to sizes of 24 3 32, 32 3 40 c Figure 3: Lines of constant mass ratios m=m and ml in the -J plane derived with the Julius help of Kuti, theuniversity scaling laws of California of ref. at [22]. San The Diegosolid lineint represents Summer School the ratio on Lattice m=m QCD = 3. andour its applications choice Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 33/37

Higgs resonance: Finite width in finite volume? The pion mass m π can be measured by a fit to the inverse propagator in four-dimensional momentum space: G 1 aa (p, p) = Zπ 1 {m 2 π + ˆp 2 }, a = 1, 2, 3 For perturbation theory checks we also want m σ and λ R, λ R = 3Z π m 2 σ m 2 π Σ 2 where Σ is the infinite volume σ field expectation value Two-parameter fit to the inverse σ and π propagators in momentum space 32 3 40 lattice at κ = 0.315, J = 0.01 [G 44(p;?p)]?1 and [G aa(p;?p)]?1 The negative intercept and slope of the inverse Goldstone propagator G 1 aa (p, p) determine the propagator mass m π The negative intercept and slope of the inverse σ propagator G 1 44 (p, p) determine the propagator mass m σ for the Higgs particle bp 2 Figure Julius Kuti, University 5: of California The upper at San (lower) Diego curve INT shows Summer a two-parameter School on Lattice t QCD to and the its inverse applications () Seattle, propagator August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 34/37

Higgs resonance: Two-particle states W m Comparison of simulation results (crosses) with perturbative prediction in isospin 0 channel at κ = 0.315, J = 0.01 Solid lines depict perturbative prediction (dashed lines perturbative estimates based on propagator masses) Dotted lines represent free pion pairs The location of the resonance energy m σ is indicated by dotted horizontal line at W 3m π m L Figure 8: Comparison of the Monte Carlo results (crosses) with the perturbatively predicted two-particle energy spectrum in the isospin-0 channel for the simulation point ( = 0:315, J =0:01). The data are in excellent agreement with the perturbative predictions (solid lines) based on the results of table 11. Dotted lines refer to the free energy spectrum, while the dashed lines show the perturbative spectrum based on the estimates of table 4. The location of the resonance energy m is symbolized by the dotted horizontal line at W 3m. { Errors are smaller than the symbols. Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 35/37

Higgs resonance: Phase shift in perturbation theory 1. We will use the linear model for the phase shift calculation S[φ; j, m 2 ] = d 4 x [ 1 2 µφ α µ φ α + 1 2 m2 φ α φ α + λ 4! (φα φ α ) 2 jφ N] with α = 1,..., N and N = 4 in our case. The mass parameter m 2 is negative in broken phase 2. Pions have isospin index a=1,...n-1 (single pion state isospin is I=1 for N=4) Two-particle state decomposes into I=0,1,2 irreducible representations Partial wave decomposition T I = 16πW k (2l + 1)P l (cosϑ)tl I l=0 Bose symmetry requires tl I = 0 if I+l is odd ( ) tl I = 1 2i e 2iδI l 1 with real phase shifts δ I l in elastic region 2m π < W < 4m π 3. Leading perturbative result: δ 0 0 = λ R N 1 48π δ 2 0 = k W λ R m 2 σ m 2 π 96π kw ( 1 m2 σ m 2 π m 2 σ W ( 1 + m2 σ 2k 2 ) ln ) + δ 2 0 ( 4k 2 + m 2 σ m 2 σ ) λ R m 2 σ m 2 π 48π kw Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 36/37

Higgs resonance: Phase shift Göckeler et al. 0 0 am σ = 0.72 m σ = 3.07 m π Γ σ = 0.18 m σ k=m Agreement with perturbation theory igure 15: Comparison of the t with respect to the perturbative ansatz (3.6) (solid curve) nd the perturbative prediction based on the estimates ~m and ~g R of table 4 (dashed curve) n the isospin-0 channel for (=0:315, J =0:01). Julius Kuti, University of California at San Diego INT Summer School on Lattice QCD and its applications Seattle, August 8-28, 2007, Lecture 2: Triviality and Higgs Mass Upper Bound 37/37