The Big Deal with the Little Higgs

Transcription

1 The Big Deal with the Little Higgs Jürgen Reuter DESY Theory Group, Hamburg Seminar Dresden, 16.Dec.2005

2 Outline Hierarchy Problem, Goldstone-Bosons and Little Higgs Higgs as Pseudo-Goldstone Boson Nambu-Goldstone Bosons The Little Higgs mechanism Examples of Models Phenomenology For example: Littlest Higgs Neutrino masses Effective Field Theories Electroweak Precision Observables Direct Searches Reconstruction of Little Higgs Models Pseudo Axions in LHM T parity and Dark Matter Conclusions

3 Outline Hierarchy Problem, Goldstone-Bosons and Little Higgs Higgs as Pseudo-Goldstone Boson Nambu-Goldstone Bosons The Little Higgs mechanism Examples of Models Phenomenology For example: Littlest Higgs Neutrino masses Effective Field Theories Electroweak Precision Observables Direct Searches Reconstruction of Little Higgs Models Pseudo Axions in LHM T parity and Dark Matter Conclusions

4 The Higgs Mechanism in the Standard Model Electroweak Theory: SU(2) U(1) gauge theory Q L U = exp[igθ a τ a /2], V = exp[ig θ Y Y/2] ( ) u UQ L, f R f R, W µ UW µ U + i d g ( µu)u L

5 The Higgs Mechanism in the Standard Model Electroweak Theory: SU(2) U(1) gauge theory Q L U = exp[igθ a τ a /2], V = exp[ig θ Y Y/2] ( ) u UQ L, f R f R, W µ UW µ U + i d g ( µu)u L Problem: Mass terms for W, Z and fermions not gauge invariant

6 The Higgs Mechanism in the Standard Model Electroweak Theory: SU(2) U(1) gauge theory Q L U = exp[igθ a τ a /2], V = exp[ig θ Y Y/2] ( ) u UQ L, f R f R, W µ UW µ U + i d g ( µu)u L Problem: Mass terms for W, Z and fermions not gauge invariant Solution: Introduction of a field which absorbs the mismatch of transformation laws: Higgs field

7 The Higgs Mechanism in the Standard Model Electroweak Theory: SU(2) U(1) gauge theory Q L U = exp[igθ a τ a /2], V = exp[ig θ Y Y/2] ( ) u UQ L, f R f R, W µ UW µ U + i d g ( µu)u L Problem: Mass terms for W, Z and fermions not gauge invariant Solution: Introduction of a field which absorbs the mismatch of transformation laws: Higgs field Spontaneous symmetry breaking: Higgs gets a Vacuum Expectation value (VEV): ( ) 0 V(Φ) = µ 2 Φ Φ + λ(φ Φ) 2, Φ exp[iπ/v] v + H D µ Φ M 2 W W 2 a, Y d Q L Φd R m d d L d R

8 Hierarchy Problem M h [GeV] Λ[GeV]

9 Hierarchy Problem M h [GeV] Λ[GeV] Motivation: Hierarchy Problem Effective theories below a scale Λ

10 Hierarchy Problem M h [GeV] Λ[GeV] Motivation: Hierarchy Problem Effective theories below a scale Λ Loop integration cut off at order Λ: Λ 2

11 Hierarchy Problem M h [GeV] Λ[GeV] Motivation: Hierarchy Problem Effective theories below a scale Λ Loop integration cut off at order Λ: Λ 2 Problem: Naturally, m h O(Λ 2 ): m 2 h = m Λ 2 (loop factors) Light Higgs favoured by EW precision observables (m h < 0.5 TeV) m h Λ Fine-Tuning!? Solution: Mechanism for eliminating loop contributions

12 Higgs as Pseudo-Goldstone Boson Invent (approximate) symmetry to protect particle mass Traditional (SUSY): Spin-Statistics = Loops of bosons and fermions

13 Higgs as Pseudo-Goldstone Boson Invent (approximate) symmetry to protect particle mass Traditional (SUSY): Spin-Statistics = Loops of bosons and fermions Little Higgs: Gauge group structure/global Symmetries = Loops of particles of like statistics

14 Higgs as Pseudo-Goldstone Boson Invent (approximate) symmetry to protect particle mass Traditional (SUSY): Spin-Statistics = Loops of bosons and fermions Little Higgs: Gauge group structure/global Symmetries = Loops of particles of like statistics Old Idea: Georgi/Pais, 1974; Georgi/Dimopoulos/Kaplan, 1984 Light Higgs as Pseudo-Goldstone boson spontaneously broken (approximate) global symmetry; non-linear sigma model w/o Fine-Tuning: v Λ/4π

15 Higgs as Pseudo-Goldstone Boson Invent (approximate) symmetry to protect particle mass Traditional (SUSY): Spin-Statistics = Loops of bosons and fermions Little Higgs: Gauge group structure/global Symmetries = Loops of particles of like statistics Old Idea: Georgi/Pais, 1974; Georgi/Dimopoulos/Kaplan, 1984 Light Higgs as Pseudo-Goldstone boson spontaneously broken (approximate) global symmetry; non-linear sigma model w/o Fine-Tuning: v Λ/4π New Ingredience: Arkani-Hamed/Cohen/Georgi/..., 2001 Collective Symmetry Breaking eliminates quadratic 1-loop level = 3-scale model

16 The Nambu-Goldstone-Theorem Nambu-Goldstone Theorem: For each spontaneously broken global symmetry generator there is a massless boson in the spectrum.

17 The Nambu-Goldstone-Theorem Nambu-Goldstone Theorem: For each spontaneously broken global symmetry generator there is a massless boson in the spectrum. π i iθ a T a ikπ k V π i T a ijπ j = 0 2 V π i π j Tjkf a k + V f π j Tji a = 0 v }{{}}{{} =(m 2 ) ij =0

18 The Nambu-Goldstone-Theorem Nambu-Goldstone Theorem: For each spontaneously broken global symmetry generator there is a massless boson in the spectrum. π i iθ a T a ikπ k V π i T a ijπ j = 0 Nonlinear Realization (Example SU(3) SU(2)): V(Φ) = ( f 2 (Φ Φ) ) [ ( 2 i Φ = exp f 2 V π i π j Tjkf a k + V f π j Tji a = 0 v }{{}}{{} =(m 2 ) ij =0 0 π π π 0 )] ( ) 0 e iπ Φ 0 f + σ

19 The Nambu-Goldstone-Theorem Nambu-Goldstone Theorem: For each spontaneously broken global symmetry generator there is a massless boson in the spectrum. π i iθ a T a ikπ k V π i T a ijπ j = 0 Nonlinear Realization (Example SU(3) SU(2)): V(Φ) = ( f 2 (Φ Φ) ) [ ( 2 i Φ = exp f 2 V π i π j Tjkf a k + V f π j Tji a = 0 v }{{}}{{} =(m 2 ) ij =0 0 π π π 0 Φ U 2 Φ = (U 2 ΦU 2 )U 2Φ 0 = e i(u2πu 2 ) Φ 0 U 2 = )] ( ) 0 e iπ Φ 0 f + σ ) (Û π Û2 π, π 0 π 0 π fundamental SU(2) rep., π 0 singlet

20 Construction of a Little Higgs model π h??

21 Construction of a Little Higgs model π h?? Let s try!

22 Construction of a Little Higgs model π h?? Let s try! Lagrangian has translational symmetry: π π + a (exact) Goldstones have only derivative interactions

23 Construction of a Little Higgs model π h?? Let s try! Lagrangian has translational symmetry: π π + a (exact) Goldstones have only derivative interactions Gauge and Yukawa interactions? Expanding the kinetic term: f 2 Φ 2 = h f 2 (h h) h h h 1 f 2 Λ 2 16π

24 Construction of a Little Higgs model π h?? Let s try! Lagrangian has translational symmetry: π π + a (exact) Goldstones have only derivative interactions Gauge and Yukawa interactions? Expanding the kinetic term: f 2 Φ 2 = h f 2 (h h) h h Theory becomes stronlgy interacting at Λ = 4πf. h 1 f 2 Λ 2 16π Bad news Easy attempts: no potential or quadratic divergences again Collective Symmetry breaking: Two ways of model building: 1. simple Higgs representation, doubled gauge group 2. simple gauge group, doubled Higgs representation

25 Prime Example: Simple Group Model enlarged gauge group: SU(3) U(1); globally U(3) U(2) Two nonlinear Φ representations L = D µ Φ D µ Φ 2 2 [ Φ 1/2 = exp ±i f ] 2/1 Θ f 1/2 0 0 f 1/2 Θ = 1 f f2 2 η 0 0 η h T h η

26 Prime Example: Simple Group Model enlarged gauge group: SU(3) U(1); globally U(3) U(2) Two nonlinear Φ representations L = D µ Φ D µ Φ 2 2 [ Φ 1/2 = exp ±i f ] 2/1 Θ f 1/2 0 0 f 1/2 Θ = 1 f f2 2 η 0 0 η h T h η Coleman-Weinberg mechanism: Radiative generation of potential + = g2 ( 16π 2 Λ2 Φ Φ 2 2) g2 16π 2 f 2

27 Prime Example: Simple Group Model enlarged gauge group: SU(3) U(1); globally U(3) U(2) Two nonlinear Φ representations L = D µ Φ D µ Φ 2 2 [ Φ 1/2 = exp ±i f ] 2/1 Θ f 1/2 0 0 f 1/2 Θ = 1 f f2 2 η 0 0 η h T h η Coleman-Weinberg mechanism: Radiative generation of potential + = g2 ( 16π 2 Λ2 Φ Φ 2 2) g2 16π 2 f 2 but: Φ 1 Φ 1 Φ 2 Φ 2 = g4 16π 2 log ( Λ 2 µ 2 ) Φ 1 Φ 2 2 g4 16π 2 log ( Λ 2 µ 2 ) f 2 (h h)

28 Yukawa interactions and heavy Top Simplest Little Higgs ( µ Model ) Schmaltz (2004), Kilian/Rainwater/JR (2004) Field content (SU(3) c SU(3) w U(1) X quantum numbers) Φ 1,2 : (1, 3) 1 3 Ψ Q : (3, 3) 1 3 Ψ l : (1, 3) 1 3 d c : ( 3, 1) 1 3 c u 1,2 : ( 3, 1) 2 3 e c, n c : (1, 1) 1,0

29 Yukawa interactions and heavy Top Simplest Little Higgs ( µ Model ) Schmaltz (2004), Kilian/Rainwater/JR (2004) Field content (SU(3) c SU(3) w U(1) X quantum numbers) Φ 1,2 : (1, 3) 1 3 Ψ Q : (3, 3) 1 3 Ψ l : (1, 3) 1 3 d c : ( 3, 1) 1 3 c u 1,2 : ( 3, 1) 2 3 e c, n c : (1, 1) 1,0 Lagrangian L = L kin. + L Yuk. + L pot. Ψ Q,L = (u, d, U) L, Ψ l = (ν, l, N) L : L Yuk. = λ u 1u 1,R Φ 1 Ψ T,L λ u 2u 2,R Φ 2 Ψ T,L λd Λ ɛijk d b RΦ i 1Φ j 2 Ψk T,L λ n n 1,R Φ 1 Ψ Q,L λe Λ ɛijk e R Φ i 1Φ j 2 Ψk Q,L + h.c.,

30 Yukawa interactions and heavy Top Simplest Little Higgs ( µ Model ) Schmaltz (2004), Kilian/Rainwater/JR (2004) Field content (SU(3) c SU(3) w U(1) X quantum numbers) Φ 1,2 : (1, 3) 1 3 Ψ Q : (3, 3) 1 3 Ψ l : (1, 3) 1 3 d c : ( 3, 1) 1 3 c u 1,2 : ( 3, 1) 2 3 e c, n c : (1, 1) 1,0 Lagrangian L = L kin. + L Yuk. + L pot. Ψ Q,L = (u, d, U) L, Ψ l = (ν, l, N) L : L Yuk. = λ u 1u 1,R Φ 1 Ψ T,L λ u 2u 2,R Φ 2 Ψ T,L λd Λ ɛijk d b RΦ i 1Φ j 2 Ψk T,L λ n n 1,R Φ 1 Ψ Q,L λe Λ ɛijk e R Φ i 1Φ j 2 Ψk Q,L + h.c., L pot. = µ 2 Φ 1 Φ 2 + h.c.

31 Yukawa interactions and heavy Top Simplest Little Higgs ( µ Model ) Schmaltz (2004), Kilian/Rainwater/JR (2004) Field content (SU(3) c SU(3) w U(1) X quantum numbers) Φ 1,2 : (1, 3) 1 3 Ψ Q : (3, 3) 1 3 Ψ l : (1, 3) 1 3 d c : ( 3, 1) 1 3 c u 1,2 : ( 3, 1) 2 3 e c, n c : (1, 1) 1,0 Lagrangian L = L kin. + L Yuk. + L pot. Ψ Q,L = (u, d, U) L, Ψ l = (ν, l, N) L : L Yuk. = λ u 1u 1,R Φ 1 Ψ T,L λ u 2u 2,R Φ 2 Ψ T,L λd Λ ɛijk d b RΦ i 1Φ j 2 Ψk T,L λ n n 1,R Φ 1 Ψ Q,L λe Λ ɛijk e R Φ i 1Φ j 2 Ψk Q,L + h.c., L pot. = µ 2 Φ 1 Φ 2 + h.c. Hypercharge embedding (remember: diag(1, 1, 2)/(2 3)): Y = X T 8 / 3 D µ Φ = ( µ 1 3 g XB X µ Φ + igw w µ )Φ

32 Cancellations of Divergencies in Yukawa sector λ t λ t + λt λ T + λ T /f

33 Cancellations of Divergencies in Yukawa sector + + λ t λ t λt λ T λ T /f d 4 k 1 { (2π) 4 k 2 (k 2 m 2 T ) λ 2 t (k 2 m 2 T ) + k 2 λ 2 T m T F λ T k 2}

34 Cancellations of Divergencies in Yukawa sector + + λ t λ t λt λ T λ T /f d 4 k 1 { (2π) 4 k 2 (k 2 m 2 T ) λ 2 t (k 2 m 2 T ) + k 2 λ 2 T m T F λ T k 2} Little Higgs global symmetry imposes relation m T F = λ2 t + λ 2 T λ T

35 Cancellations of Divergencies in Yukawa sector + + λ t λ t λt λ T λ T /f d 4 k 1 { (2π) 4 k 2 (k 2 m 2 T ) λ 2 t (k 2 m 2 T ) + k 2 λ 2 T m T F λ T k 2} Little Higgs global symmetry imposes relation m T F = λ2 t + λ 2 T λ T = Quadratic divergence cancels

36 Cancellations of Divergencies in Yukawa sector + + λ t λ t λt λ T λ T /f d 4 k 1 { (2π) 4 k 2 (k 2 m 2 T ) λ 2 t (k 2 m 2 T ) + k 2 λ 2 T m T F λ T k 2} Little Higgs global symmetry imposes relation m T F = λ2 t + λ 2 T λ T = Quadratic divergence cancels Collective Symm. breaking: λ t λ 1 λ 2, λ 1 = 0 Φ 1 Φ 2 or λ 2 = 0 SU(3) Ψ Q Ψ Q λ2 1λ 2 2 [SU(3)] 2 16π 2 log Φ 1 Φ 1 u c 1 u c 2 ( Λ 2 µ 2 ) Φ 1 Φ 2 2

37 Scales and Masses µ O(10 TeV) O(1 TeV) O(250 GeV) F v Λ Scale Λ: global SB, new dynamics, UV embedding Scale F : Pseudo-Goldstone bosons, new vector bosons and fermions Scale v: Higgs, W ±, Z, l ±,...

38 Scales and Masses µ O(10 TeV) O(1 TeV) O(250 GeV) F v Λ Scale Λ: global SB, new dynamics, UV embedding Scale F : Pseudo-Goldstone bosons, new vector bosons and fermions Scale v: Higgs, W ±, Z, l ±,... Boson masses radiative (Coleman-Weinberg), but: Higgs protected by symmetries against quadratic 1-loop level

39 Scales and Masses µ O(10 TeV) O(1 TeV) O(250 GeV) F v Λ Scale Λ: global SB, new dynamics, UV embedding Scale F : Pseudo-Goldstone bosons, new vector bosons and fermions Scale v: Higgs, W ±, Z, l ±,... H H 0 Boson masses radiative (Coleman-Weinberg), but: Higgs protected by symmetries against quadratic 1-loop level G 1 G 1 [H G 2 G 1 1, H 2 ] 0 g 1 0 g 2 0 /H 1 H /H 2 H M H g 1 g 2 Λ/16π 2

40 Generic properties Extended scalar (Higgs-) sector Extended global symmetry

41 Generic properties Extended scalar (Higgs-) sector Extended global symmetry Specific form of the potential: V(h, Φ) = C 1 F 2 Φ + i 2 F h h + C 2 F 2 Φ i 2 F h h h 1-loop level, h = v, M h v, M Φ F, i Φ v 2 /F h and Φ mix

42 Generic properties Extended scalar (Higgs-) sector Extended global symmetry Specific form of the potential: V(h, Φ) = C 1 F 2 Φ + i 2 F h h + C 2 F 2 Φ i 2 F h h h 1-loop level, h = v, M h v, M Φ F, i Φ v 2 /F h and Φ mix Extended Gauge Sector : B, Z, W ± (Hypercharge singlets & triplets)

43 Generic properties Extended scalar (Higgs-) sector Extended global symmetry Specific form of the potential: V(h, Φ) = C 1 F 2 Φ + i 2 F h h + C 2 F 2 Φ i 2 F h h h 1-loop level, h = v, M h v, M Φ F, i Φ v 2 /F h and Φ mix Extended Gauge Sector : B, Z, W ± (Hypercharge singlets & triplets) Extended top sector: new heavy quarks, t, t loops M 2 h < 0 EWSB

44 Outline Hierarchy Problem, Goldstone-Bosons and Little Higgs Higgs as Pseudo-Goldstone Boson Nambu-Goldstone Bosons The Little Higgs mechanism Examples of Models Phenomenology For example: Littlest Higgs Neutrino masses Effective Field Theories Electroweak Precision Observables Direct Searches Reconstruction of Little Higgs Models Pseudo Axions in LHM T parity and Dark Matter Conclusions

45 Little Higgs Models Plethora of Little Higgs Models in 3 categories: Moose Models Orig. Moose (Arkani-Hamed/Cohen/Georgi, ) Simple Moose (Arkani-Hamed/Cohen/Katz/Nelson/Gregoire/Wacker, ) Linear Moose (Casalbuoni/De Curtis/Dominici, ) Simple (Goldstone) Representation Models Littlest Higgs (Arkani-Hamed/Cohen/Katz/Nelson, ) Antisymmetric Little Higgs (Low/Skiba/Smith, ) Custodial SU(2) Little Higgs (Chang/Wacker, ) Littlest Custodial Higgs (Chang, ) Little SUSY (Birkedal/Chacko/Gaillard, ) Simple (Gauge) Group Models Orig. Simple Group Model (Kaplan/Schmaltz, ) Holographic Little Higgs (Contino/Nomura/Pomarol, ) Simplest Little Higgs (Schmaltz, ) Simplest Little SUSY (Roy/Schmaltz, ) Simplest T parity (Kilian/Rainwater/JR/Schmaltz,...)

46 Varieties of Particle spectra H = SU(5) [SU(2) U(1)]2, G = SO(5) SU(2) U(1) Arkani-Hamed/Cohen/Katz/Nelson, 2002 m Φ Φ ± Φ ±± Z W ± T Φ P B h W ± Z t

47 Varieties of Particle spectra H = SU(5) [SU(2) U(1)]2, G = SO(5) SU(2) U(1) Arkani-Hamed/Cohen/Katz/Nelson, 2002 H = SO(6) [SU(2) U(1)]2, G = Sp (6) SU(2) U(1) Low/Skiba/Smith, 2002 m Φ Φ ± Φ ±± Z W ± T m Φ Φ P Z W ± B T h Φ P B W ± Z t H ± h A H W ± Z t

48 Varieties of Particle spectra H = SU(5) [SU(2) U(1)]2, G = SO(5) SU(2) U(1) Arkani-Hamed/Cohen/Katz/Nelson, 2002 H = SO(6) [SU(2) U(1)]2, G = Sp (6) SU(2) U(1) Low/Skiba/Smith, 2002 m Φ Φ ± Φ ±± Z W ± T m Φ Φ P Z B W ± T h Φ P B W ± H = [SU(3)]2 SU(3) U(1), G = [SU(2)] 2 SU(2) U(1) Schmaltz, 2004 [SU(4)] 4 [SU(3)] 4 Z t m = H ± h A H Z X 0 /Y 0 W ± Z t ± W T U, C Kaplan/Schmaltz, HDM, h 1/2, Φ 1,2,3, Φ P 1,2,3, Z 1,...,8, W ± 1,2, q, l h η W ± Z t

49 Outline Hierarchy Problem, Goldstone-Bosons and Little Higgs Higgs as Pseudo-Goldstone Boson Nambu-Goldstone Bosons The Little Higgs mechanism Examples of Models Phenomenology For example: Littlest Higgs Neutrino masses Effective Field Theories Electroweak Precision Observables Direct Searches Reconstruction of Little Higgs Models Pseudo Axions in LHM T parity and Dark Matter Conclusions

50 The Littlest Higgs Model Setup Symmetry breaking: SU(5) SO(5) (global) [SU(2) U(1)] 2 SU(2) L U(1) Y (local) 1 heavy triplet X µ, 1 heavy singlet Y µ

51 The Littlest Higgs Model Setup Symmetry breaking: SU(5) SO(5) (global) [SU(2) U(1)] 2 SU(2) L U(1) Y (local) The unbroken Lagrangian: L = L (3) 0 + L (1) 0 + L G 0. 1 heavy triplet X µ, 1 heavy singlet Y µ L (3) 0 = 1 2g1 2 Tr A 1µν A µν 1 1 2g2 2 Tr A 2µν A µν 2 2 tr A µ 1 J (3) µ L (1) 0 = 1 2g 2 1 Gauge group generators: T1 a = 1 τ a 0, T2 a = Tr B 1µν B µν 1 1 2g 2 2 Tr B 2µν B µν 2 B µ 1 J (1) µ. 0 τ a, Y 1 = 1 10 diag(3, 3, 2, 2, 2) Y 2 = 1 10 diag(2, 2, 2, 3, 3)

52 The Littlest Higgs Model Setup Symmetry breaking: SU(5) SO(5) (global) [SU(2) U(1)] 2 SU(2) L U(1) Y (local) The unbroken Lagrangian: L = L (3) 0 + L (1) 0 + L G 0. 1 heavy triplet X µ, 1 heavy singlet Y µ L (3) 0 = 1 2g1 2 Tr A 1µν A µν 1 1 2g2 2 Tr A 2µν A µν 2 2 tr A µ 1 J (3) µ L (1) 0 = 1 2g 2 1 Gauge group generators: T1 a = 1 τ a 0, T2 a = Tr B 1µν B µν 1 1 2g 2 2 Tr B 2µν B µν 2 B µ 1 J (1) µ. 0 τ a, Y 1 = 1 10 diag(3, 3, 2, 2, 2) Y 2 = 1 10 diag(2, 2, 2, 3, 3) Triplet current: J (3) µ = J (3),a τ a µ 2 Singlet current: J (1) µ Couplings not unique, but: (Anomaly cancellation!)

53 L G 0 = F 2 ( 8 Tr(D µξ)(d µ Ξ), Ξ = exp 2i ) F Π Ξ 0, Ξ 0 = Π = 1 0 h φ ( ) h 0 h T 2 Φ ++ Φ +, φ = 2 φ h Φ +. Φ 0 + iφ 1 0

54 L G 0 = F 2 ( 8 Tr(D µξ)(d µ Ξ), Ξ = exp 2i ) F Π Ξ 0, Ξ 0 = Π = 1 0 h φ ( ) h 0 h T 2 Φ ++ Φ +, φ = 2 φ h Φ +. Φ 0 + iφ 1 0 Covariant derivative: D µ Ξ = µ Ξ + i k=1,2 [ ] (A µ k Ξ + Ξ(A µ k ) T ) + (B µ k Ξ + Ξ(B µ k ) T )

55 L G 0 = F 2 ( 8 Tr(D µξ)(d µ Ξ), Ξ = exp 2i ) F Π Ξ 0, Ξ 0 = Π = 1 0 h φ ( ) h 0 h T 2 Φ ++ Φ +, φ = 2 φ h Φ +. Φ 0 + iφ 1 0 Covariant derivative: D µ Ξ = µ Ξ + i k=1,2 [ ] (A µ k Ξ + Ξ(A µ k ) T ) + (B µ k Ξ + Ξ(B µ k ) T ) Chiral fields: Q R : b R, t R, T R, and Q L : q L = ( tl b L ), T L, L t = Q Qi /DQ + L Y λ 2 F ( T L T R + h.c.).

56 L G 0 = F 2 ( 8 Tr(D µξ)(d µ Ξ), Ξ = exp 2i ) F Π Ξ 0, Ξ 0 = Π = 1 0 h φ ( ) h 0 h T 2 Φ ++ Φ +, φ = 2 φ h Φ +. Φ 0 + iφ 1 0 Covariant derivative: D µ Ξ = µ Ξ + i k=1,2 [ ] (A µ k Ξ + Ξ(A µ k ) T ) + (B µ k Ξ + Ξ(B µ k ) T ) Chiral fields: Q R : b R, t R, T R, and Q L : q L = ( tl b L ), T L, L t = Q Qi /DQ + L Y λ 2 F ( T L T R + h.c.). Use a global SU(3) subsymmetry iτ 2 T L iq L 0 χ L = iq T L 0 0 L Y = λ 1 F t R Tr [ Ξ (it2 2 )Ξ ] ˆχ L + h.c

57 Neutrino masses Kilian/JR, 2003; del Aguila et al., 2004; Han/Logan/Wang, 2005 Naturalness does not require cancellation mechanism for light fermions Lepton-number violating interactions can generate neutrino masses (due to presence of triplet scalars)

58 Neutrino masses Kilian/JR, 2003; del Aguila et al., 2004; Han/Logan/Wang, 2005 Naturalness does not require cancellation mechanism for light fermions Lepton-number violating interactions can generate neutrino masses (due to presence of triplet scalars) Lagrangian invariant under full gauge symmetry L N = g N F ( L c ) T ΞL with L = (iτ 2 l L, 0, 0) T EWSB: Generation of neutrino masses m ν g N v 2 /F

59 Neutrino masses Kilian/JR, 2003; del Aguila et al., 2004; Han/Logan/Wang, 2005 Naturalness does not require cancellation mechanism for light fermions Lepton-number violating interactions can generate neutrino masses (due to presence of triplet scalars) Lagrangian invariant under full gauge symmetry L N = g N F ( L c ) T ΞL with L = (iτ 2 l L, 0, 0) T EWSB: Generation of neutrino masses m ν g N v 2 /F Caveat: m ν too large compared to observations g N small, e.g. F /Λ, where Λ : scale of lepton number breaking

60 Heavy Vector Fields Mixing of the gauge fields: A µ 1 = W µ + g X c 2 X µ, B µ 1 = B µ + g Y c 2 Y µ, A µ 2 = W µ g X s 2 X µ, B µ 2 = B µ g Y s 2 Y µ,

61 Heavy Vector Fields Mixing of the gauge fields: A µ 1 = W µ + g X c 2 X µ, B µ 1 = B µ + g Y c 2 Y µ, A µ 2 = W µ g X s 2 X µ, B µ 2 = B µ g Y s 2 Y µ, Expand the Goldstone Lagrangian L G 0 = MX 2 c 2 s 2 tr X X + g X tr[x V (3) ] M Y 2 c 2 s 2 Y Y + g Y Y V (1) tr(d µφ) (D µ φ) + (D µ h) (D µ h) 1 [ 6F 2 tr V (3) V (3)] +...,

62 Heavy Vector Fields Mixing of the gauge fields: A µ 1 = W µ + g X c 2 X µ, B µ 1 = B µ + g Y c 2 Y µ, A µ 2 = W µ g X s 2 X µ, B µ 2 = B µ g Y s 2 Y µ, Expand the Goldstone Lagrangian L G 0 = MX 2 c 2 s 2 tr X X + g X tr[x V (3) ] M Y 2 c 2 s 2 Y Y + g Y Y V (1) tr(d µφ) (D µ φ) + (D µ h) (D µ h) 1 [ 6F 2 tr V (3) V (3)] +..., Heavy vector masses: M X = g X F /2 M Y = g Y F /(2 5) Higgs current V µ = i [ h(d µ h) (D µ h)h ] 1 0 : V (1) = tr V, 3 0 : V (3) = V 1 2 tr V.

63 Effective Field Theories How to clearly separate effects of heavy degrees of freedom?

64 Effective Field Theories How to clearly separate effects of heavy degrees of freedom? Toy model: Two interacting scalar fields ϕ, Φ [ ( ) ] 1 Z[j, J] = D[Φ] D[ϕ] exp i dx 2 ( ϕ)2 1 2 Φ( +M 2 )Φ λϕ 2 Φ...+JΦ+jϕ Low-energy effective theory integrating out heavy degrees of freedom (DOF) in path integrals, set up Power Counting

65 Effective Field Theories How to clearly separate effects of heavy degrees of freedom? Toy model: Two interacting scalar fields ϕ, Φ [ ( ) ] 1 Z[j, J] = D[Φ] D[ϕ] exp i dx 2 ( ϕ)2 1 2 Φ( +M 2 )Φ λϕ 2 Φ...+JΦ+jϕ Low-energy effective theory integrating out heavy degrees of freedom (DOF) in path integrals, set up Power Counting Completing the square: Φ = Φ + λ ( ) 1 M M 2 ϕ ( Φ)2 1 2 M 2 Φ 2 λϕ 2 Φ = 1 ) 1 2 Φ (M )Φ + (1 λ2 2M 2 ϕ2 + 2 M 2 ϕ 2.

66 Integrating out Integrating out the X and Y vector fields L (1) + L (3) = L EW g,gf + f (3) JJ tr[j (3) J (3) ] + f (3) V J tr[v (3) J (3) ] + f (1) V V tr[v (3) V (3) ] + f (1) JJ J (1) J (1) + f (1) V J V (1) J (1) + f (1) V V V (1) V (1) In the Littlest Higgs e.g. f (3) V V = 1 (1 6F ) (c2 s 2 ) 2

67 Integrating out Integrating out the X and Y vector fields L (1) + L (3) = L EW g,gf + f (3) JJ tr[j (3) J (3) ] + f (3) V J tr[v (3) J (3) ] + f (1) V V tr[v (3) V (3) ] + f (1) JJ J (1) J (1) + f (1) V J V (1) J (1) + f (1) V V V (1) V (1) In the Littlest Higgs e.g. f (3) V V = 1 (1 6F ) (c2 s 2 ) 2 Coleman-Weinberg potential of the scalar 1-loop: L CW 0 = 1 2 M 2 φ tr[φφ ] + µ 2 (h h) λ 4 (h h) 2 λ 2φ i ( h φh h T φ h ) λ 2φφ tr[(φφ )(hh )] λ 4φ i(h h) ( h φh h T φ h ) λ 6 (h h) 3 Sensitive to UV completion, dimensionless parameters k and k. EWSB Constraints on k, k

68 Integrate out the heavy scalar (Power counting!) φ = φ 2iλ 2φ M 2 φ ( 1 + D2 M 2 φ ) 1 + 2λ ( 2φφ Mφ 2 hh 1 + λ ) 4φ h h hh T λ 2φ Higgs mass up to order v 4 /F 2 m 2 H = 2λeff 4 v2! = 2v 2 e 2 s + e2 k 2 w c2 c 2 w c 2 e 2 s 2 w s2 + e2 c 2 w s 2! e 2 s 2 w s2 c + e 2 2 c 2 w s 2 c 2 k + λ2 t c 2 k t! k + λ2 t c 2 k t (Remember µ 2 = m 2 H /2) EWSB λ eff 4 > 0 λ2 2φ M 4 φ < 1 8F 2

69 Equations of Motion (EOM) Couplings V J induce after SSB anom. couplings of W, Z to fermions Applying EOM eliminates corrections from field redefinitions

70 Equations of Motion (EOM) Couplings V J induce after SSB anom. couplings of W, Z to fermions Applying EOM eliminates corrections from field redefinitions Custodial-SU(2) Conserving Terms 0 = tr[v (3) δl µ ] = tr[v (3) V (3) ] 2 δw µ g 2 tr[v (3) µd ν W µν ] 2 tr[v (3) J (3) ] L (3) = L EW g,gf + f (3) JJ O(3) JJ + f V W O V W + f (3) V V O(3) V V O V W = tr V (3) µνw µν, O (3) JJ = tr J (3) J (3), O (3) V V = tr V (3) V (3)

71 Equations of Motion (EOM) Couplings V J induce after SSB anom. couplings of W, Z to fermions Applying EOM eliminates corrections from field redefinitions Custodial-SU(2) Conserving Terms 0 = tr[v (3) δl µ ] = tr[v (3) V (3) ] 2 δw µ g 2 tr[v (3) µd ν W µν ] 2 tr[v (3) J (3) ] L (3) = L EW g,gf + f (3) JJ O(3) JJ + f V W O V W + f (3) V V O(3) V V O V W = tr V (3) µνw µν, O (3) JJ = tr J (3) J (3), O (3) V V = tr V (3) V (3) Custodial-SU(2) Violating Terms L (1) = L EW g,gf + f (1) JJ O(1) JJ + f V B O V B + f (1) V V O(1) V V

72 Effective Dim. 6 Operators O (I) JJ = 1 F 2 tr[j (I) J (I) ] O h,1 = 1 F 2 ( (Dh) h ) (h (D h ) ) v2 2 Dh 2 O hh = 1 F 2 (h h v 2 /2) (Dh) (Dh) O h,3 = 1 F (h h v 2 /2) 3

73 O W W = 1 F (h h v 2 /2) tr W µν W µν O B = 1 F 2 i 2 (D µh) (D ν h)b µν O BB = 1 F (h h v 2 /2)B µν B µν O V q = 1 qh( /Dh)q F 2

74 Oblique Corrections: S, T, U All low-energy effects order v 2 /F 2 Low-energy observables parameterized by S, T, 2 parameters for contact interactions (no U here) O V W, O V B Change in gauge couplings g and g g = e s w [ 1 + M 2 W (f V W + 2f V B ) ] g = e c w [ 1 + [M 2 Z M 2 W ](f V W + 2f V B ) ]

75 Oblique Corrections: S, T, U All low-energy effects order v 2 /F 2 Low-energy observables parameterized by S, T, 2 parameters for contact interactions (no U here) O V W, O V B Change in gauge couplings g and g g = e s w [ 1 + M 2 W (f V W + 2f V B ) ] g = e c w [ 1 + [M 2 Z M 2 W ](f V W + 2f V B ) ] S parameter (O V W, O V B ) SU(2) c -violating sector (O h,1 ) S = 8πv 2 (f V W + 2f V B ) α T = M 2 W /M 2 W = 2v 2 f (1) V V 2v2 λ 2 2φ M 4 φ

76 Shift in physical vector masses: ( ) 2 ( ) 2 ev ev MW 2 = (1 + x) MZ 2 = (1 + y) 2s w 2s w c w x = α ( S/(4s 2 w) + T ) y = α S/(4s 2 wc 2 w)

77 Shift in physical vector masses: ( ) 2 ( ) 2 ev ev MW 2 = (1 + x) MZ 2 = (1 + y) 2s w 2s w c w x = α ( S/(4s 2 w) + T ) y = α S/(4s 2 wc 2 w) Very low energies Fermi theory Four-Fermion Interactions 2 GF = 1 v2 (1 + z) with z = α T v2 4 f (3) JJ.

78 Shift in physical vector masses: ( ) 2 ( ) 2 ev ev MW 2 = (1 + x) MZ 2 = (1 + y) 2s w 2s w c w x = α ( S/(4s 2 w) + T ) y = α S/(4s 2 wc 2 w) Very low energies Fermi theory Four-Fermion Interactions 2 GF = 1 v2 (1 + z) with z = α T v2 4 f (3) JJ. G F -M Z -α scheme (muon decay, LEP I, Bhabha), define the parameters ˆv 0 and ŝ 0 by s 2 w = ŝ 2 0 ˆv 0 = ( 2 G F ) 1/2 and M Z = eˆv 0 = 2ŝ 0 ĉ 0 ( ) ( ) 1 + ĉ2 0 ĉ 2 0 (y + z) c 2 ŝ2 w = ĉ ŝ2 0 0 ĉ 2 0 (y + z) ŝ2 0

79 Constraints on LHM Constraints from contact IA: ( f (3) JJ, f (1) JJ ) c2 F /4.5 TeV c 2 F /10 TeV Constraints evaded c, c 1 B, Z, W, ± superheavy (O(Λ)) decouple from fermions

80 Constraints on LHM Constraints from contact IA: ( f (3) JJ, f (1) JJ ) c2 F /4.5 TeV c 2 F /10 TeV Constraints evaded c, c 1 B, Z, W, ± superheavy (O(Λ)) decouple from fermions S, T in the Littlest Higgs model, violation of Custodial SU(2): Csáki et al., 2002; Hewett et al., 2002; Han et al., 2003; Kilian/JR, 2003 Mixing of (Z, B, Z ) and (W ±, W ± ) S [ 8π = c 2 (c 2 s 2 ) g +5 c 2 (c 2 s 2 ) 2 ] v 2 g 2 F 2 0 α T 5 4 v 2 F 2v2 λ 2 2φ 2 Mφ 4 v2 F 2

81 Constraints on LHM Constraints from contact IA: ( f (3) JJ, f (1) JJ ) c2 F /4.5 TeV c 2 F /10 TeV Constraints evaded c, c 1 B, Z, W, ± superheavy (O(Λ)) decouple from fermions S, T in the Littlest Higgs model, violation of Custodial SU(2): Csáki et al., 2002; Hewett et al., 2002; Han et al., 2003; Kilian/JR, 2003 Mixing of (Z, B, Z ) and (W ±, W ± ) S [ 8π = c 2 (c 2 s 2 ) g +5 c 2 (c 2 s 2 ) 2 ] v 2 g 2 F 2 0 α T 5 4 v 2 F 2v2 λ 2 2φ 2 Mφ 4 v2 F 2 General models Triplet sector: (almost) identical to Littlest Higgs ( S only) More freedom in U(1) sector: ( T )

82 EW Precision Observables T 0.6 LHM F = 3.5 TeV Higgs mass variable (Coleman-Weinberg, UV completion) m H = 700 GeV 400 GeV 250 GeV 120 GeV SM LHM F = 4.5 TeV S S = 1 12π ln m2 H m 2 0 T = 3 16πc 2 ln m2 H w m 2 0 Peskin/Takeuchi, 1992; Hagiwara et al., 1992 Making the Higgs heavier reduces amount of fine-tuning

83 Direct Searches Heavy Gauge Bosons: Detection: Resonance in e + e f f, Drell-Yan Tevatron: M B 650 GeV

84 Direct Searches Heavy Gauge Bosons: Detection: Resonance in e + e f f, Drell-Yan Tevatron: M B 650 GeV Branching ratios Z BR(Z ->ll) BR(Z ->qq) BR(Z ->WW,Zh) Total width Z, fixed M Z : Γ Z = g2 ( cot 2 2φ + 24 tan 2 φ ) M Z 96π Determination of F, c, s O(10 2 ) lepton LHC w. 300 fb 1

85 Heavy Scalars: Φ P : e + e, q q Φ P h, Φ P hz L Φ ± : e + e, q q Φ + W L, Φ± W LZ L Φ ±± : e e ννφ, Φ ±± W L W L Φ 0 : e + e, q q Z LΦ, Φ Z LZ L, hh, not W + W

86 Heavy Scalars: Φ P : e + e, q q Φ P h, Φ P hz L Φ ± : e + e, q q Φ + W L, Φ± W LZ L Φ ±± : e e ννφ, Φ ±± W L W L Φ 0 : e + e, q q Z LΦ, Φ Z LZ L, hh, not W + W Heavy Quarks: T LHC: bq T q Decay T W + L b, th, tz Perelstein/Peskin/Pierce, 2003 Total cross section BRs (limit g 0): Γ T = m T λ 2 T /(16π) Γ(T th) Γ(T tz) 1 2 Γ(T bw + ) m T λ 2 T 64π Determination of m T, λ T

87 Reconstruction of LHM Kilian/JR, 2003; Han et al., 2005 How to unravel the structure of colliders? Symmetry structure Quadr. Div. Cancell. Nonlinear Goldstone boson structure

88 Reconstruction of LHM Kilian/JR, 2003; Han et al., 2005 How to unravel the structure of colliders? Symmetry structure Quadr. Div. Cancell. Nonlinear Goldstone boson structure SIGNALS: Anom. Triple Gauge Couplings: W W Z, W W γ Anom. Higgs Coupl.: H(H)W W, H(H)ZZ Anom. Top Couplings: ttz, tbw

89 Reconstruction of LHM Kilian/JR, 2003; Han et al., 2005 How to unravel the structure of colliders? Symmetry structure Quadr. Div. Cancell. Nonlinear Goldstone boson structure SIGNALS: Anom. Triple Gauge Couplings: W W Z, W W γ Anom. Higgs Coupl.: H(H)W W, H(H)ZZ Anom. Top Couplings: ttz, tbw Direct Search (LHC) M V, F, c, c Vectors: ILC: Contact Terms e + e l + l, [ν νγ] M B 10[5] TeV Higgsstr., WW fusion: HZff, HW ff angular distr./energy dependence f (1/3) V J Check from TGC (ILC: per mil precision), GigaZ f (3) JJ Combining Determination of all coefficients in the gauge sector

90 Scalars: Affected by scalars and vectors T, f (1) V V, B known (λ 2φ /M 2 φ) 2 Higgsstr., W W fusion Higgs coupl., f (3) V V Higgs BRs f (1) V V, f (3) V V ; (take care of t) Goldstone contr. Evidence for nonlinear nature f (3) V V HH production f h,3 (difficult!) LHC ILC 1-2 % Higgs measurements Reconstruction of scalar sector up to F 2 TeV

91 Scalars: Affected by scalars and vectors T, f (1) V V, B known (λ 2φ /M 2 φ) 2 Higgsstr., W W fusion Higgs coupl., f (3) V V Higgs BRs f (1) V V, f (3) V V ; (take care of t) Goldstone contr. Evidence for nonlinear nature f (3) V V HH production f h,3 (difficult!) LHC ILC 1-2 % Higgs measurements Reconstruction of scalar sector up to F 2 TeV Direct LHC Top: t t production f V q, v t, a t; accuracy 1-2 % tbw from t decays, single t production g tth/g bbh anom. Yukawa coupl. f hq, nonlin. structure w. 2.5% accuracy

92 Scalars: Affected by scalars and vectors T, f (1) V V, B known (λ 2φ /M 2 φ) 2 Higgsstr., W W fusion Higgs coupl., f (3) V V Higgs BRs f (1) V V, f (3) V V ; (take care of t) Goldstone contr. Evidence for nonlinear nature f (3) V V HH production f h,3 (difficult!) LHC ILC 1-2 % Higgs measurements Reconstruction of scalar sector up to F 2 TeV Direct LHC Top: t t production f V q, v t, a t; accuracy 1-2 % tbw from t decays, single t production g tth/g bbh anom. Yukawa coupl. f hq, nonlin. structure w. 2.5% accuracy Include all observables in a combined fit if Little Higgs signals are found (sufficient data from LHC and ILC)

93 Pseudo Axions in LHM Kilian/Rainwater/JR, 2004 broken diagonal generator: η in QCD; couples to fermions as a pseudoscalar, behaves as a axion analogous particles: techni-axion, topcolor-axion, (N)MSSM-axion

94 Pseudo Axions in LHM Kilian/Rainwater/JR, 2004 broken diagonal generator: η in QCD; couples to fermions as a pseudoscalar, behaves as a axion analogous particles: techni-axion, topcolor-axion, (N)MSSM-axion QCD-(PQ) axion: α s L Ax. = 1 Λ 8π A 2 g ηg Gµν µν, Gµν = 1 2 ɛµνρσ G ρσ Anomalous U(1) η :

95 Pseudo Axions in LHM Kilian/Rainwater/JR, 2004 broken diagonal generator: η in QCD; couples to fermions as a pseudoscalar, behaves as a axion analogous particles: techni-axion, topcolor-axion, (N)MSSM-axion QCD-(PQ) axion: α s L Ax. = 1 Λ 8π A 2 g ηg Gµν µν, Gµν = 1 2 ɛµνρσ G ρσ Anomalous U(1) η : explicit symmetry breaking m η and g ηγγ independent axion bounds not applicable no new hierarchy problem m η v 250 GeV η EW singlet, couplings an to SM particles v/f suppressed

96 Example: Simple Group Model Scalar Potential: µφ 1 Φ 2 + h.c. + Coleman-Weinberg pot.: m η = κ µ 2 µ m 2 H = 2(δm 2 + m 2 η)

97 Example: Simple Group Model Scalar Potential: µφ 1 Φ 2 + h.c. + Coleman-Weinberg pot.: m η = κ µ 2 µ m 2 H = 2(δm 2 + m 2 η) 1 b b BR [η] new Higgs decays (H Zη, H ηη) 0.1 τ + τ BR(H ηη) < 10 4 [ 5 10% OSG] c c gg γγ µ + µ Zh m η [GeV] m H [GeV] m η [GeV] BR(Zη) % % % % % %

98 Pseudo Axions at LHC and ILC LHC: Gluon Fusion (axial U(1) η anomaly), Peak in diphoton spectrum

99 Pseudo Axions at LHC and ILC LHC: Gluon Fusion (axial U(1) η anomaly), Peak in diphoton spectrum ILC: associated production σ tot [fb] e + e t tη s = 1 TeV Problem: Cross section vs. bkgd e e + t tb b #evt/2 GeV s = 800 GeV L = 1 ab 1 g ttη = g ttη = m η = 50 GeV m η [GeV] M inv (b b) [GeV] Possibility: Z Hη (analogous to A in 2HDM) Kilian/JR/Rainwater (in prep.)

100 Pseudo Axions at the Photon Collider Photon Collider as precision machine for Higgs physics (s channel resonance, anomaly coupling) S/B analogous to LC η in the µ model with (almost) identical parameters as A in MSSM ( Mühlleitner et al. (2001) )

101 Pseudo Axions at the Photon Collider Photon Collider as precision machine for Higgs physics (s channel resonance, anomaly coupling) 10 1 σ eff (γγ η, H) [pb] 0.1 S/B analogous to LC η in the µ model with (almost) identical parameters as A in MSSM ( Mühlleitner et al. (2001) ) H (SM) H (with T) η [T] η [1 complete heavy gen.] η [3 gen. of heavy quarks] η [3 complete heavy gen.] m H, m η [GeV]

102 g bbη = 0.4 g bbh m η Γ γγ [kev] γγ (H, η) b b #evt/2 GeV m η = 100 GeV see = 200 GeV Lee = 400 fb 1 cos(θ) T < 0.76 p z / s ee < M inv (b b) [GeV] γγ (H, η) b b #evt/2 GeV see = 500 GeV Lee = 1 ab 1 cos(θ) T < 0.76 p z / s ee < 0.04 m η = 200 GeV M inv (b b) [GeV]

103 T parity and Dark Matter Cheng/Low, 2003; Hubisz/Meade, 2005 T parity: T a T a, X a X a, automorphism of coset space analogous to R parity in SUSY, KK parity Bounds on f relaxed, but: pair production! Lightest T -odd particle (LTP) Candidate for Cold Dark Matter

104 T parity and Dark Matter Cheng/Low, 2003; Hubisz/Meade, 2005 T parity: T a T a, X a X a, automorphism of coset space analogous to R parity in SUSY, KK parity Bounds on f relaxed, but: pair production! Lightest T -odd particle (LTP) Candidate for Cold Dark Matter Littlest Higgs: A LTP W, Z 650 GeV Φ 1 TeV T, T TeV Annihilation: A A h W W, ZZ, hh M H (GeV) f M A (GeV) H /10/50/70/ f (GeV)

105 T parity and Dark Matter Cheng/Low, 2003; Hubisz/Meade, 2005 T parity: T a T a, X a X a, automorphism of coset space analogous to R parity in SUSY, KK parity Bounds on f relaxed, but: pair production! Lightest T -odd particle (LTP) Candidate for Cold Dark Matter Littlest Higgs: A LTP W, Z 650 GeV Φ 1 TeV T, T TeV Annihilation: A A h W W, ZZ, hh M H (GeV) f M A (GeV) H /10/50/70/ f (GeV) T parity Simple Group model: Pseudo-Axion η LTP Kilian/Rainwater/JR/Schmaltz

106 Outline Hierarchy Problem, Goldstone-Bosons and Little Higgs Higgs as Pseudo-Goldstone Boson Nambu-Goldstone Bosons The Little Higgs mechanism Examples of Models Phenomenology For example: Littlest Higgs Neutrino masses Effective Field Theories Electroweak Precision Observables Direct Searches Reconstruction of Little Higgs Models Pseudo Axions in LHM T parity and Dark Matter Conclusions

107 Conclusions Little Higgs elegant alternative to SUSY structure stabilizes EW scale Gauge/Global Symmetry Generics: new heavy gauge bosons, scalars, quarks Little Higgs in accord w EW precision observ. w/o Fine Tuning (M H!) New developments: Pseudo-Axions, T -parity, LH Dark Matter UV embedding, GUT, Flavor? Clear experimental signatures: direct search [Gauge & Top sector, LHC (ILC)] precision observables [Gauge, Scalar, Top sector ILC (LHC)] Strategy for Reconstruction by Complementarity of ILC & LHC