Theories of EW symmetry breaking and precision data after LEP2 3) - PowerPoint PPT Presentation

Theories of EW symmetry breaking and precision data after LEP2 3) Universal models: ˆ S, ˆ 1) The little hierarchy ‘problem’ T, W, Y 2) Relevance of LEP2 4) Extra d, little Higgs, SUSY,. . . アレサンナロ ��� スツルミア �� - �� 14 ㈰2㈪05 From works with Barbieri, Marandella, Pomarol, Rattazzi, Schappacher

The higgs hierarchy ‘problem’... Recent experimental progress ? 1) Direct and indirect data showed that the top is heavy , m t ≈ 180 GeV 2) Indirect data suggest the existence of a light higgs , m h < ∼ 200 GeV This shifts ‘solutions’ to the hierarchy problem towards lower energies. In the SM, cut-offing top loop at E < Λ UV ≈ 12 λ 2 δm 2 h ≈ δm 2 (4 π ) 2 Λ 2 t h (top) = UV δm 2 ∼ m 2 h < Λ UV < if ∼ 400 GeV no longer few TeV! h But at the same time m > 3) direct: no new detectable particles, ˜ ∼ 100 GeV. 4) indirect: no new non-renormalizable-operators, Λ > ∼ 10 TeV.

...and its ‘solutions’ A lot of activity before LHC... Two types of solutions: I New symmetry implies m h = 0; its breaking gives the EW scale – Supersymmetry : t → ˜ t stop stops top. – Attempts with scale symmetry, 5d gauge invariance, little Higgs . II h becomes an extended object 1 / Λ ∼ TeV – Technicolor : h = hadron of a QCD-like group with Λ TC < ∼ TeV > – Large extra dimensions : h = string with length ∼ 1 / TeV? – Warped extra dimension (AdS dual to CFT ‘walking technicolor’) Precision data disfavour type II solutions

Extended particles ↔ form factors LEP finds that SM fermions, gauge bosons and also the Higgs are point-like. (Although the Higgs has not been discovered, its properties have been tested because the 3 massive SM vector boson acquired a longitudinal polarization eating 3 components of the Higgs doublet). Form-factors in QFT are introduced as higher dimensional operators, that encode the low energy effects of new physics too heavy to be directly seen. Even restricting to SU(2) L ⊗ U(1) Y , B, L, B i , L i , CP symmetric operators...

L eff = L SM + O / Λ 2 operator O affects constraint on Λ 1 Lγ µ τ a L ) 2 2 (¯ µ -decay 10 TeV 1 Lγ µ L ) 2 2 (¯ LEP 2 5 TeV | H † D µ H | 2 θ W in M W /M Z 5 TeV ( H † τ a H ) W a θ W in Z couplings 8 TeV µν B µν i ( H † D µ τ a H )(¯ Lγ µ τ a L ) Z couplings 10 TeV i ( H † D µ H )(¯ Lγ µ L ) Z couplings 8 TeV Dλ D λ U λ † H † ( ¯ U γ µν Q ) F µν b → sγ 10 TeV Qλ U λ † 1 U γ µ Q ) 2 2 ( ¯ B mixing 6 TeV Cut-off above 10 TeV leaves δm 2 h ∼ 500 m 2 h : ‘little hierarchy problem’

Heavy universal new physics

Kinds of new physics ◦ Generic: p -decay, ν masses. see-saw, GUT ◦ B, L conserving: EDM, µ → eγ , ε K ,. . . ◦ Minimal Flavour Violation: b → sγ , B -mixing,... only SUSY? • ‘Universal’ (i.e. not coupled to fermions): ˆ S, ˆ T, W, Y .Little Higgs, extra d ◦ Effects only in Higgs: S, T . some technicolor

Heavy universal new physics ‘ Universal ’: affects only inverse propagators of vectors: p 2 − M 2 + SM loops + Π( p 2 ) ‘ Heavy ’: expand new physics corrections Π( p 2 ) as Π( p 2 ) = Π(0) + p 2 Π ′ (0) + p 4 2 Π ′′ (0) + · · · 3 coefficients for each kinetic term: Π W + W − , Π W 3 W 3 , Π W 3 B , Π BB . 3 · 4 = 12 coefficients. 3 are just redefinitions of the SM parameters g, g ′ , v . 2 combinations vanish because γ must be massless and coupled to Q = T 3 + Y . Adimensional form factors custodial SU(2) L ( g ′ /g ) � Π ′ = W 3 B (0) + S − M 2 W � T = Π W 3 W 3 (0) − Π W + W − (0) − − Π ′ W 3 W 3 (0) − Π ′ − � U = W + W − (0) − − 2 M − 2 Π ′′ W 3 W 3 (0) − Π ′′ W V = W + W − (0) − − 2 M − 2 Π ′′ = W 3 B (0) + W X − 2 M − 2 Π ′′ = BB (0) + + W Y 2 M − 2 Π ′′ W W = W 3 W 3 (0) + + 2 M − 2 Π ′′ W Z = GG (0) + + 3 are suppressed ( V ≪ ˆ U ≪ ˆ T and X ≪ ˆ S ). 5 remain: ˆ S, ˆ T, W, Y and Z

Final result Adimensional form factor operator effect ( g ′ /g ) � Π ′ ( H † τ a H ) W a S = W 3 B (0) µν B µν correction to s W M 2 | H † D µ H | 2 W � T = Π W 3 W 3 (0) − Π W + W − (0) correction to M W /M Z 2 M − 2 Π ′′ ( ∂ ρ B µν ) 2 / 2 = BB (0) anomalous g 1 ( E ) W Y 2 M − 2 Π ′′ ( D ρ W a µν ) 2 / 2 W W = W 3 W 3 (0) anomalous g 2 ( E ) 2 M − 2 Π ′′ ( D ρ G A µν ) 2 / 2 W Z = GG (0) anomalous g 3 ( E ) (We here use canonically normalized vectors) If the higgs exists, ˆ S, ˆ T, Y, W, Z correspond to dimension 6 operators. Neglected form factors (e.g. U ) correspond to dimension 8 operators. Extended H gives ˆ S, ˆ T . Extended vector bosons give W, Y, Z . T probed by comparing s W ( M W , M Z ) with s W ( α, G F , M Z ) with s W ( g Z V , g Z • ˆ S, ˆ A ) • W, Y probed by comparing (low E with Z -pole) and ( Z -pole with LEP2).

S, ˆ ˆ T, W, Y from data

Observables: at and below the Z pole Corrections to Z -pole observables from universal new physics are condensed in T − W − Y s 2 W δε 1 = � δε 3 = � , δε 2 = − W , S − W − Y . c 2 W A few observables ( M Z , M W , α, G F , g Z V ℓ , g Z Aℓ ) measured with per-mille accuracy. Corrections to any low-energy observable are easily computed from √ d L γ µ u L ] + h.c. + ν L γ µ ℓ L ][¯ L eff ( E ≪ M Z ) = L QED , QCD − 2 2 G F [¯   2 √ � ψ ( T 3 − s 2 ¯ 2 G F (1 + �   − 4 T ) W k Q ) γ µ ψ ψ = Q,L S − c 2 T + W ) − s 2 � W ( � W Y k = 1 + . c 2 W − s 2 W Measured with per-cent accuracy: little impact.

Global fit Large effects still allowed χ 2 (ˆ W,Y χ 2 (ˆ χ 2 ( W, Y ) = min χ 2 (ˆ S, ˆ S, ˆ S, ˆ T ) = min T, W, Y ) T, W, Y ) S, ˆ ˆ T 10 10 90, 99% CL (2 dof) 90, 99% CL (2 dof) 5 5 1000 T ^ 1000 W 0 0 -5 -5 -10 -10 -10 -5 0 5 10 -10 -5 0 5 10 ^ 1000 Y 1000 S 4 − 3 = 1: ε 1 , 2 , 3 give bands; adding low-energy transforms into long ellipses.

Observables: above the Z -pole e → f ¯ Corrections to LEP2 e ¯ f cross sections: use modified propagators Z γ   − c 2 W W + s 2 W Y 1 δε 1 + = Z   p 2 − M 2 p 2 − M 2  M 2    Z Z W   − c 2 W ( δε 1 − δε 2 ) − s 2 p 2 − s 2 W W + c 2   W δε 3 − s W c W ( W − Y ) 1 W Y   γ s W c W ( p 2 − M 2 M 2 M 2 Z ) W W e f Z, γ _ _ e f Measured with per-cent accuracy, but effects of W, Y enhanced by p 2 /M 2 Z ∼ 5 (Measurements of s W below the Z -peak are well emphasized: APV, Møller, NuTeV. LEP2 has comparable accuracy above the Z -peak, and is missed)

Global fit after LEP2 T , W , Y parameters must vanish within few · 10 − 3 All ˆ S , ˆ χ 2 (ˆ W,Y χ 2 (ˆ χ 2 ( W, Y ) = min χ 2 (ˆ S, ˆ S, ˆ S, ˆ T ) = min T, W, Y ) T, W, Y ) S, ˆ ˆ T 10 10 90, 99% CL (2 dof) 90, 99% CL (2 dof) 5 5 1000 T 1000 W ^ 0 0 -5 -5 -10 -10 -10 -5 0 5 10 -10 -5 0 5 10 ^ 1000 Y 1000 S Hard time for Higgsless, little-Higgs etc.

LEP1 vs LEP2: χ 2 (ˆ S = 0 , ˆ T = 0 , W, Y ) 10 5 EWPT Models 1000 W 0 -5 LEP2 90, 99% CL (2 dof) -10 -10 -5 0 5 10 1000 Y LEP2 relevant and shifts towards W < 0. Models give W, Y ≥ 0 Precise analysis by LEP2 experimentalists would be welcome

Universal models

Hidden universal models Popular models (vectors in extra dimensions, little-Higgs,. . . ) have extra heavy vectors coupled to SM fermionic gauge currents J F . Integrating out heavy vector mass eigenstates gives additional operators:  4-fermion operators   O ∼ ( J F + J H ) 2 = ( ¯ ψγ µ ψ + iH † D µ H ) 2 = corrections to Z, W, γ couplings   corrections to Z, W, γ masses Looks non-universal, so analyses performed by computing all observables. A posteriori can be rewritten as universal: e.g. on shell J B µ J B µ = ( ∂ α B µν ) 2 / 2 → Y . How to skip superfluous steps putting a priori effects in SM vector propagators? 1 1 1 → + p 2 − m 2 p 2 − m 2 p 2 − m 2 new SM SM

How to compute ˆ S, ˆ T, W, Y Do not integrate out the heavy vector mass eigenstates. Integrate out vector bosons not coupled to SM fermions. ⋆ Apparently non-universal operators involving fermions not generated. ⋆ No need of diagonalizing and indentifying heavy mass eigenstates. It works like a pig → sausage machine: 1) Choose a model; write kinetic matrix Π full of neutral ( W 3 , B, . . . ) and charged ij ( W ± , . . . ) vectors. ( . . . indicates extra vectors not coupled to SM fermions) 2) Π − 1 = (Π full ) − 1 restricted to SM vectors 3) Extract ˆ S, ˆ T, W, Y from Π 4) Compare with χ 2 (ˆ S, ˆ T, W, Y )

A simple example: U(1) Y ⊗ U(1) ′ Y Extra hypercharge vector with mass M not mixed with SM. 1) Kinetic matrix: W 3 B ′ B µ µ µ   p 2 − M 2 Z s 2 M 2 0 B µ Z s W c W W   Π full = p 2 − M 2 W 3 M 2 Z c 2   0 Z s W c W µ   W p 2 − M 2 B ′ 0 0 µ 2) Integrate out B µ − B ′ µ not coupled to fermions: go to the basis ( B, W 3 , B − B ′ ), invert Π full and restrict to B, W 3 : the result of course is � � − 1 � � p 2 − M 2 1 / ( p 2 − M 2 ) Z s 2 M 2 Z s W c W 0 Π − 1 = W + p 2 − M 2 M 2 Z c 2 Z s W c W 0 0 W T/t 2 W = M 2 W /M 2 , W = 0. LEP1 not affected: δε 1 , 2 , 3 = 0 3) Extract ˆ S = Y = ˆ