Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Tsinghua University, Peking

Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Effective Chiral Lagrangian from Strong Dynamical Models Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Feng-Jun Ge Tsinghua University, Peking April 1, 2011 Outline Effective Chiral Lagrangian from Strong Dynamical Models 1 Strong Dynamical Models 2 Electroweak Chiral Lagrangian (EWCL) 3 Former work of our group 4 EWCL from a TC2 Model with Non-trivial Condensation 5 Conclusion Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Standard Model and its Difficulties Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion The standard modelµ Fermions: physical world Bosonsµforce carriers fits experiments Problemsµ triviality, unnatrualness Higgs boson SU (3)C ⊗ SU (2)L ⊗ U (1)Y ⇓ v = 246GeV SU (3)C ⊗ U (1)em New Physics Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion New physics beyond the standard model(BSM)µ include Higgsµ Supper Symmetry Little Higgs L-R Models ··· exclude Higgsµ Higgsless Model Strong Dynamics ··· Solve the problemsµ asymptotically free→triviality Fermion condensation → unnaturalness Technicolor & Extended TC Effective Chiral Lagrangian from Strong Dynamical Models TechnicolorµQCD-like GT C ⊗ SU (3)C ⊗ SU (2)L ⊗ U (1)Y Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Introduce techni-Fermion condensation: hψ̄ψi = 6 0 Ordinary quarks: merely dynamical masses from QCD ordinary leptons: massless requireµETC gives particles ”hard masses” Problemsµ large top mass wrong FCNC large S Topcolor Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Top quark mass: 173.1GeV near EWSB scale top condensation leads to EWSBº SU (3)1 ⊗ SU (3)2 ⊗ U (1)1 ⊗ U (1)2 ⊗ SU (2)L Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Pagels-Stokar formulaµ fπ2 = Nc 2 Λ2 m (ln + k) t 16π 2 m2t Λ ∼ 1015 GeV,composite Higgs boson mass: m2H (µ) = Λ2 2Nc 2 2 − (Λ − µ2 ) g2 4π fine-tuning: 1:10−30 , unnatural topcolor assisted technicolor(TC2 Model) Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Typical Gauge groupµ GT C ⊗ SU (1)1 ⊗ SU (2)2 ⊗ U (1)1 ⊗ U (1)2 ⊗ SU (2)L Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Top condensation gives top quark mass ETC gives other quarks masses Technicolor interaction breaks EW symmetry TC2 Models Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion C.T.Hill’s schematic TC2 model:(PLB,1995,345:483-489) schematic, no detail ρ ∼ 1 needs fine-tuning TC2 Models Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion K.Lane’s natural TC2 model:(PLB,1995,352:382-387) GT C ⊗ SU (3)1 ⊗ SU (3)2 ⊗ SU (2)L ⊗ U (1)1 ⊗ U (1)2 TC2 Models Effective Chiral Lagrangian from Strong Dynamical Models K.Lane’s generation mixing TC2 model:(PRD,1996,54:2204-2212) Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion hT̄ i T j i non-diagonal(i, j=1,2) ψ guarantees the theory walking EWCL Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Low energy effective theory for EW interactionµ universalµcontain all possible terms allowed by the symmetry economicµonly contain the particles below the EW scale contributions from heavy particles hidden in the coefficients = + SEM Z 1 1 2 µ 2 3 µ 2 4 d x(− f tr[X Xµ ] + β1 f (tr[τ X ]) 4 4 1 µν µν 3 3 µ ν α1 g1 Bµν tr[τ W ] + iα2 g1 Bµν tr[τ X X ] + 2iα3 tr[W Xµ Xν ] 2 µ µ 2 µ 2 ν 3 3 + α4 (tr[X Xν ]) + α5 (tr[X Xµ ]) + α6 tr[X X ]tr[τ Xµ ]tr[τ Xν ] + α7 tr[X Xµ ]tr[τ X ]tr[τ Xν ] + + iα9 tr[τ W µν ]tr[τ X X ] + µ 3 µνρλ + α11 + α13 µνρλ ν 3 3 µ 3 ν 1 2 1 4 3 2 α8 (tr[τ ]W µν ) 3 µ 3 ν α10 (tr[τ X ]tr[τ X ]) 3 3 tr[τ Xµ ]tr[Xν W ρλ ] + α12 tr[τ Xµ ]tr[Xν W 3 µνρλ g1 Bµν tr[τ W ρλ ] + α14 † Xµ = U (Dµ U ) 3 2 µν 3 ] tr[τ W µν ]tr[τ W ρλ ] W µν = U † τa 2 a Wµν U EWCL from Dynamical EWSB models Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion EWCL & LECs Effective Chiral Lagrangian from Strong Dynamical Models e.g.: Oblique correction parameters: S, T, U : Feng-Jun Ge S = −16πα1 αT Outline U Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion = 2β1 = −16πα8 expremental: S = 0.01 ± 0.10 T = 0.03 ± 0.11 U = 0.06 ± 0.10 Electroweak Chiral Lagrangian Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models QCD →Pseudo-scalar meson effective Lagrangian Wang Q, Kuang Y P, Xiao M, et al (PRD,2000,61:054011) Yang H, Wang Q, Kuang Y P, et al (PRD,2002,66:014019) Jiang S Z, Wang Q, (PRD 2010, 81:094037) Jiang S Z, Zhang Y, et al (PRD, 2010, 81:014001) Strong dynamical models→EWCL Electroweak Chiral Lagrangian (EWCL) C.T.Hill’s schematic TC2 model and one-dobulet model: HH Zhang, et.al, Phys.Rev.D,2008,77:055003 Former work of our group K.Lane’s natural TC2 model: JY Lang,et.al, Phys.Rev.D,2009,79:015002 EWCL from a TC2 Model with Non-trivial Condensation R.S.Chivukula’s hypercharge-universal topcolor model: JY Lang,et.al, Phys.Lett.B,2009,673: 63-67 Conclusion EWCL from C.T. Hill’s schematic TC2 Model Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion EWCL from K.Lane’s natural TC2 Model Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Effective Chiral Lagrangian from Strong Dynamical Models Trln expansion of non-diagonal condensation Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion walking effects more models ··· Calculating Method Effective Chiral Lagrangian from Strong Dynamical Models Goalµ Z DZµ0 DGαµ Dψ̄DψeiS[Zµ ,Gµ ,ψ̄,ψ,Wµ ,Bµ ] Z Dµ (U )eiSEW [U,Wµ ,Bµ ] Feng-Jun Ge = Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion 0 α key steps: Integrate in Goldstone boson fields Integrate out BSM gauge bosons Integrate out BSM fermions Integrate out Z 0 α α Integrate in U Field Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Introduce operator i O(x) ≡ trlc ψLi (x)ψ̄R (x) decomposed as O(x) = ζL† (x)σ(x)ζR (x) Hermite matrix σ(x)µmodular degree of freedom; Unitary matrix ζL† (x), ζR (x)µSU (2)L ÚU (1)Y phase degree of freedom. U (x) ≡ ζL† (x)ζR (x) Insert a constant Z † Dµ (U )F[O]δ(ζL OζR − ζR O† ζL† ) = const. Integrate in U Field Effective Chiral Lagrangian from Strong Dynamical Models Do special chiral rotation SU (2)L × U (1) : VL = ζL , VR = ζR 0 Seff [U, W, B, Z ] Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion 0 /+V / + Aγ / 5) SGauge [W, B, Z ] − iTr ln(i∂ = Feng-Jun Ge R − i ln † iS[G,W ,B ,ψ̄ζ ,ψζ ] ζ ζ DGD ψ̄ζ Dψζ F [Oζ ]δ(Oζ − Oζ )e R R / V / ζ +A / ζ γ 5 )ψζ i d4 xψ̄ζ (i∂+ D ψ̄ζ Dψζ e Finally 0 Seff [U, W, B, Z ] 0 0 0 =SGauge [W, B, Z ] + Snorm [U, W, B, Z ] + Sanom [U, W, B, Z ] where 0 = Snorm [U, W, B, Z ] Z † iS[G,Wζ ,Bζ ,ψ̄ζ ,ψζ ] −i ln DGD ψ̄ζ Dψζ F [Oζ ]δ(Oζ − Oζ )e 0 Sanom [U, W, B, Z ] = 5 0 = 5 /+V /ζ + V / + Aγ / ) − Tr ln(i∂ /ζ + A /ζ γ ) Tr ln(i∂ SGauge [W, B, Z ] Z 1 1 1 0 1 2 0 0µ α α,µν µν 0µν 4 d x(− Wµν F − Bµν B − Zµν Z + M0 Zµ Z ) 4 4 4 2 Integrate out BSM gauge bosons Effective Chiral Lagrangian from Strong Dynamical Models Use formula Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Z Z α DGµ D ψ̄Dψ exp{i Z Z D ψ̄Dψ exp{i = + ∞ Z X n=2 4 4 d x[− 1 4 α Fµν F 4 α,µν / − gT C t α G /α + V / + Aγ / 5 )ψ]} + ψ̄(i∂ 5 /+V / + Aγ / )ψ d x[ψ̄(i∂ 4 d x1 · · · d xn (−igT C )n n! α ···α 1 µ 1 µ n n Gµ 1 ···µn (x1 , · · · , xn )Jα 1 (x1 ) · · · Jα n (xn )]} where J αµ ≡ ψ̄tα γ µ ψ We can integrate out technigluons. Integrate out BSM fermions Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Using some approximations, we can obtain ¯ 2 )] S[U, W, B] = −iTr ln[i∂/ + v/ + a /γ 5 − Σ(∇ ¯ 2 ) is diagonal , we can use If Σ(∇ 5 ¯ 2 )] /+v − iTr ln[i∂ /+a /γ − Σ(∇ Z 4 2 2 µ 2 2 2 2 2 = d xtrf [(F0 ) a − K1 (dµ a ) − K2 (dµ aν − dν aµ ) + K3 (a ) + K4 (aµ aν ) − K13 Vµν V µν + iK14 aµ aν V µν 6 ] + O(p ) where K’s are function of self-energy Σ(p2 ) . (PRD 2002,66:014019). a contains no colron field⇒ massless coloron œ Introduce non-diagonal Σ5 Effective Chiral Lagrangian from Strong Dynamical Models New self energy form: Σ + iτ 2 γ 5 Σ5 Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) non-diagonal condensation massive coloron SD equations: 2 Z iΣ(−p ) = d4 q 12πC2 (N )αT C (−(p − q)2 ) (2π)4 (p − q)2 Σ(−q 2 ) 2 2 2 q 2 + Σ2 5 (−q ) + Σ (−q ) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion 2 iΣ5 (−p ) = Z d4 q 12πC2 (N )αT C (−(p − q)2 ) (2π)4 (p − q)2 Σ5 (−q 2 ) 2 2 2 q 2 + Σ2 5 (−q ) + Σ (−q ) Some results about Self-energy Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) After complex computation, we getµ Σ = Σ̂ cos Θ, Σ5 = Σ̂ sin Θ vacuum energy only depends on Σ̂ ˆ 2) − new expansion formulaµ−iTr ln[i∂/ + v/ + a /γ 5 − Σ(∇ 2 5 2 ˆ iτ γ Σ5 (∇ )] Θ-dependent parts: keep the same§only replace Σ by Σ̂¶ Θ-independent parts : EWCL coefficients change little§coloron masses change largely Former work of our group 0.4 Σ(p2/Λ2TC)/ΛTC EWCL from a TC2 Model with Non-trivial Condensation 0.5 0.3 0.2 0.1 Conclusion 0 Σ̂w Σ̃w Σ̂1.02αc Σ̃1.02αc Σ̂r /20 Σ̃r /20 Integrate out Colorons and Z’ Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge colorons only interact with ordinary quarks Only kinetic terms and mass termsµ C 2 Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Mcoloron = Σ6=0 Ê + 2(K + K̂13 Σ6=0 2 − 8K̂ ) + (2/g3 13 )/(cot θ 0 + tan θ 0 )2 Jackiw’s classic field method§intergrate out Z 0 : Z 0 DZ 0 eiS[U,W,B,Z ] = eiS[U,W,B,Zc ] We can get the relations between K’s and the effective Lagrangian coefficients. Analytic Results Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion f 2 2 = 5F̂0 β1 = 2 10a2 3 F̂0 M̄ 2 0 Z α1 = 5 2 (1 − Σ6=0 2β1 )(K̂2 α3 = (β1 − 1 2 1 2 α4 = (2β1 + α6 = − )+ β1 f 2 − 2M 2 0 Z α2 = (β1 − α5 = − Σ6=0 − K̂13 5 2 )( )( 1 4 5 2 5 2 )( (4β1 + β1 f 2 2M 2 0 + Z 2 Σ6=0 K̂13 Σ6=0 K̂13 5 2 1 4 − 2 β1 8 5 − Σ6=0 K̂13 8 − Σ6=0 )K̂13 5 Σ6=0 )+ 2M 2 0 2a3 − Z γβ1 2a3 Σ6=0 K̂14 ) 5 8 + 2 [(2a0 4a2 3 +10a3 K̂13 Σ6=0 K̂14 β1 f 2 γβ1 Σ6=0 K̂14 5 4 )+( (3β1 + 1 4 5 K̂4 Σ6=0 )K̂14 + Σ6=0 2 â0 )K̂3 + 2 Σ6=0 2 − 5a3 K̂14 Σ6=0 16 2 (2a0 − 2a0 D̂4 ] − + + 5 − 32 5 32 2 â0 Σ6=0 K̂14 )+ Σ6=0 − K̂4 Z (K̂3 + β1 f 2 2M 2 0 Σ6=0 )− β1 f 2 2M 2 0 Z Σ6=0 2 5a3 )K̂4 β1 5 Σ6=0 Σ6=0 Σ6=0 ( K̂ + 15K̂13 − 5K̂14 ) 2 2 4 Analytic Results Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models α7 = β1 f 2 − 2M 2 0 Z 2 β1 4a2 3 Σ6=0 2 −10a3 K̂13 α8 = − β1 f 2 2M 2 0 β1 f 2M 2 0 5 2 1 2 1 2 5 2 Σ6=0 Σ6=0 2 2 a3 + a0 + â0 )K̂3 + (a0 + â0 − a3 )K̂4 2 2 2 2 2 Σ6=0 + 5a3 K̂14 Σ6=0 + 10β1 (K̂2 Z 2 α9 = − [( Σ6=0 2 + a0 D̂3 ] − Σ6=0 − K̂13 − 10K̂13 Σ6=0 )+ Electroweak Chiral Lagrangian (EWCL) α10 = Former work of our group α11 = α12 = α13 = α14 = 0 2 5β1 4 (K̂3 Σ6=0 + K̂4 ) Σ6=0 + β1 (5K̂2 Z 4 β1 8a4 3 β1 5 Σ6=0 5 Σ6=0 Σ6=0 Σ6=0 ( K̂ − K̂4 − 15K̂13 + 5K̂14 ) 2 4 3 4 + 5 4 g4Z − Σ6=0 K̂14 3 β1 2a3 3 ) 3 2 EWCL from a TC2 Model with Non-trivial Condensation Conclusion 2 2 Σ6=0 [(2a3 + 6a0 a3 + 3â0 a3 )(K̂3 +2a0 a3 D̂2 ] InputsµMZ 0 , T γ, Di non-leading order (β1 expansion) Σ6=0 + K̂4 ) Walking Technicolor Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge increase techni-quark condensationµ large techni-quark mass large techni-NGB mass correct FCNC Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion In the region ΛT C < µ < ΛET C : near constant α∗ (ultraviolet fixed point) In infrared region:near constant α∗ (UVFP) 2-loop diagram: β(α) = −β0 3 5 gTC gTC − β 1 (4π)2 (4π)4 Walking Technicolor Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge β0 > 0 β1 < 0 Banks-Zaks IRFP α∗ = − Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion 2N β0 2 (2N ) β1 = = 11 3 C2 (SU (N )TC ) − 4 3 4πβ0 β1 [T (R1 ) + T (R2 ) + T (R3 )] 3 X 34 2 20 C2 (SU (N )TC ) − C2 (SU (N )TC )T (Ri ) + 4C2 (Ri )T (Ri )] . [ 3 i=1 3 critical coupling to form condensation: αc = 2πN C2 (R) Phase Diagram Effective Chiral Lagrangian from Strong Dynamical Models TechniColor gauge group SU(N)§N≥6 2 Feng-Jun Ge α* 1.8 α c,1,2 1.6 α 1.4 β0=0 c,3 Outline Strong Dynamical Models α Electroweak Chiral Lagrangian (EWCL) 1.2 NAF 1 0.8 0.6 Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion AF 0.4 0.2 0 2 3 4 5 6 N 7 8 9 10 Coupling Constant Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion running  7 4π  7 − 45 (2 + ln x)2 × α(x) = β0  1 ln x ln x ≤ −2 −2 ≤ ln x ≤ 0.5 ln x ≥ 0.5 walking 1 β0 1 α(x) = ln x + ln α(x) 2π α∗ α∗ − α(x) ideal walking α∗ = 1.02αc Coupling Constant Effective Chiral Lagrangian from Strong Dynamical Models αw 1.02αc 0.6 αr /20 Feng-Jun Ge 0.5 Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) α(p2/Λ2TC) Outline 0.4 0.3 0.2 Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion 0.1 0 −6 10 −4 10 −2 0 10 10 2 p /Λ2TC 2 10 4 10 6 10 Hypercharges Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) 26(+ordinary fermions 8)hypercharges§23 restraints Inputs:x1 , y1 , y1 + y2 particle 1 TL 1 UR 1 DR 2 TL 2 UR 2 DR SU (N ) N N N N N N SU (3)1 3 3 3 1 1 1 SU (3)2 1 1 1 3 3 3 SU (2)W 2 1 1 2 1 1 U (1)1 u1 v1 v1 v1 u1 u1 TL N 1 1 2 x(y, z)1 x(y, z)2 UR N 1 1 1 x(y, z)01 1 1 1 1 1 1 1 1 1 x(y, z)01 ξ ξ0 x(y, z)02 + 1 2 1 x(y, z)02 − 2 −ξ −ξ0 l(t,b) l(t,b) l(t,b) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion DR ψL ψR N 1 N (N 2 1 N (N 2 − 1) − 1) U (1)2 u2 v2 + 1 2 1 v2 − 2 v2 u2 + 1 2 u2 − 1 2 Results Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge a = −39.4015, a0 = −45.5381, b = 14.3188, b0 = 8.1822, c = −39.4015, c0 = −45.5381, d = −12.2733, d0 = −14.3188, ξ = 4.6025, ξ0 = −4.6025, x1 = 25, x01 = 18.8634, 0 0 x2 = −25.5528, x02 = −19.4161, y1 = 25, y1 = 22.9545, y2 = −25, y2 = −22.9545, 0 0 z1 = −7.7287, z1 = 18.8634, z2 = 7.7287, z2 = −18.8634, u1 = −4.1027, v1 = −6.1482, u2 = 4.1948, v2 = 6.2404. Outline 2.06 M ’=0.5TeV Strong Dynamical Models Z 2.055 M ’=1TeV 2.05 MZ’=2TeV Z MZ’=5TeV Electroweak Chiral Lagrangian (EWCL) 2.045 S 2.04 Former work of our group 2.035 EWCL from a TC2 Model with Non-trivial Condensation 2.025 Conclusion 2.03 2.02 2.015 0 0.005 0.01 0.015 0.02 T 0.025 0.03 0.035 0.04 Results Effective Chiral Lagrangian from Strong Dynamical Models −3 2 x 10 MZ’=0.5TeV 1.8 Feng-Jun Ge M ’=1TeV Z 1.6 M ’=2TeV Z 1.4 Strong Dynamical Models 1.2 Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion U Outline MZ’=5TeV 1 0.8 0.6 0.4 0.2 0 0 0.005 0.01 0.015 0.02 T 0.025 0.03 0.035 0.04 results Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group a = −19.1970, a0 = −22.1970, b = 7, b0 = 4, c = −19.1970, c0 = −22.1970, d = −6, d0 = −7, ξ = 2.25, ξ0 = −2.25, x1 = −50, x01 = −53, x2 = 2.7393, x02 = 5.7393,y1 = 0 0 0 36.0643,y1 = 35.0643,y2 = −12.4356, y2 = −11.4356, z1 = 20.0643, z1 = 33.0643, 0 z2 = 3.5644, z2 = −9.4356, u1 = 0.4116, v1 = −0.5884, u2 = −0.4111, v2 = 0.5889 4 S parameter U parameter 3 2 1 0 −1 EWCL from a TC2 Model with Non-trivial Condensation Conclusion −2 0 0.02 0.04 0.06 0.08 T 0.1 0.12 0.14 0.16 Results Effective Chiral Lagrangian from Strong Dynamical Models Compare: running and walking running 4.01 4 S Feng-Jun Ge 3.99 3.98 Outline Conclusion 0.01 0.015 0.02 0.025 T ideal walking 0.03 0.035 0.04 M ’=0.5TeV Z S 1.94 Electroweak Chiral Lagrangian (EWCL) EWCL from a TC2 Model with Non-trivial Condensation 0.005 1.96 Strong Dynamical Models Former work of our group 0 MZ’=1TeV 1.92 M ’=2TeV 1.9 MZ’=5TeV 1.88 Z 0 0.005 0.01 0.015 0.02 T 0.025 0.03 0.035 0.04 TC scale ΛTC   0.21TeV 4.5TeV =  73TeV running walking ideal walking Results Effective Chiral Lagrangian from Strong Dynamical Models Coloron mass is not sensitive to walking effect: 0.55 MZ’=0.5TeV 0.54 Feng-Jun Ge M ’=1TeV Z M ’=2TeV 0.53 Z Outline Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion 0.52 Mcoloron/TeV Strong Dynamical Models MZ’=5TeV 0.51 0.5 0.49 0.48 0.47 0.46 0 0.005 0.01 0.015 0.02 T 0.025 0.03 0.035 0.04 Conclusion Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Conclusion: walking effect can reduce S to half of running¶ In typical parameter space of this TC2 model: S∼2¶ Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion there exits the parameter space: T a little bigger§S < 0¶ α2 , α3 , α4 , α5 order 10−2 α6 , α7 , α9 order 10−5 α10 order 10−10 ¶ α6 , α7 , α9 , especially α10 , sensitive to walking effect. cover all the typical TC2 models" Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Outline Strong Dynamical Models Electroweak Chiral Lagrangian (EWCL) Former work of our group EWCL from a TC2 Model with Non-trivial Condensation Conclusion Thank you!

Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Tsinghua University, Peking

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Effective Chiral Lagrangian from Strong Dynamical Models Feng-Jun Ge Tsinghua University, Peking

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