Electroweak Effective Theory

Transcription

1 Electroweak Effective Theory Antonio Pich IFIC, Univ. Valencia - CSIC Schladming 2014, Styria, Austria, 1 8 March 2014

2 Coset Space Coordinates: G SU(2) L SU(2) R H SU(2) L+R H ξ(ϕ) g h ϕ ϕ G/H EWET A. Pich Schladming

3 Coset Space Coordinates: G SU(2) L SU(2) R H SU(2) L+R H ξ(ϕ) ξ(ϕ) (ξ L (ϕ),ξ R (ϕ)) G g h ξ L (ϕ) ξ R (ϕ) G g L ξ L (ϕ)h (ϕ,g) G g R ξ R (ϕ)h (ϕ,g) ϕ ϕ G/H EWET A. Pich Schladming

4 Coset Space Coordinates: G SU(2) L SU(2) R H SU(2) L+R H ξ(ϕ) ξ(ϕ) (ξ L (ϕ),ξ R (ϕ)) G g h ξ L (ϕ) ξ R (ϕ) G g L ξ L (ϕ)h (ϕ,g) G g R ξ R (ϕ)h (ϕ,g) ϕ ϕ G/H U(ϕ) ξ L (ϕ)ξ R (ϕ) G g L U(ϕ)g R EWET A. Pich Schladming

5 Coset Space Coordinates: G SU(2) L SU(2) R H SU(2) L+R H ξ(ϕ) ξ(ϕ) (ξ L (ϕ),ξ R (ϕ)) G g h ξ L (ϕ) ξ R (ϕ) G g L ξ L (ϕ)h (ϕ,g) G g R ξ R (ϕ)h (ϕ,g) ϕ ϕ G/H U(ϕ) ξ L (ϕ)ξ R (ϕ) G g L U(ϕ)g R Canonical choice: ξ L (ϕ) = ξ R (ϕ) G u(ϕ) g L u(ϕ)h (ϕ,g) = h(ϕ,g)u(ϕ)g R { U(ϕ) = u(ϕ) 2 } = exp i 2 f Φ EWET A. Pich Schladming

6 CCWZ Formalism: Callan Coleman Wess Zumino u(ϕ) G g L u(ϕ) h (ϕ,g) = h(ϕ,g) u(ϕ) g R SU(2) L+R triplets: X G h(ϕ,g) X h(ϕ,g) R 1 2 σ R, µ R = µ R +[Γ µ,r] u µ i ud µ U u = u µ, h µν = µ u ν + ν u µ Γ µ = 1 2 f µν ± = u Ŵ µν u ±u ˆB µν u { } u( µ i ˆB µ)u + u ( µ i Ŵ µ)u EWET A. Pich Schladming

7 CCWZ Formalism: Callan Coleman Wess Zumino u(ϕ) G g L u(ϕ) h (ϕ,g) = h(ϕ,g) u(ϕ) g R SU(2) L+R triplets: X G h(ϕ,g) X h(ϕ,g) R 1 2 σ R, µ R = µ R +[Γ µ,r] u µ i ud µ U u = u µ, h µν = µ u ν + ν u µ Γ µ = 1 2 f µν ± = u Ŵ µν u ±u ˆB µν u { } u( µ i ˆB µ)u + u ( µ i Ŵ µ)u L (2) EW = v2 4 uµ u µ EWET A. Pich Schladming

8 Resonance Effective Theory Towers of heavy states are usually present in strongly-coupled models of EWSB: Technicolour, Walking TC... The low-energy constants (LECs) of the Goldstone Lagrangian contain information on the heavier states. The lightest states not included in the Lagrangian dominate EWET A. Pich Schladming

9 Resonance Effective Theory Towers of heavy states are usually present in strongly-coupled models of EWSB: Technicolour, Walking TC... The low-energy constants (LECs) of the Goldstone Lagrangian contain information on the heavier states. The lightest states not included in the Lagrangian dominate 1 Build L eff (ϕ i,r k ) with the lightest R k coupled to the ϕ i 2 Require a good UV behaviour Low # of derivatives 3 Match the two effective Lagrangians LECs EWET A. Pich Schladming

10 Resonance Effective Theory Towers of heavy states are usually present in strongly-coupled models of EWSB: Technicolour, Walking TC... The low-energy constants (LECs) of the Goldstone Lagrangian contain information on the heavier states. The lightest states not included in the Lagrangian dominate 1 Build L eff (ϕ i,r k ) with the lightest R k coupled to the ϕ i 2 Require a good UV behaviour Low # of derivatives 3 Match the two effective Lagrangians LECs This program works in QCD: RχT (Ecker Gasser Leutwyler Pich de Rafael) Good dynamical understanding at large N C EWET A. Pich Schladming

11 LO Resonance EW Lagrangian Pich Rosell Sanz-Cillero L eff = L ϕ + R L R + R,R L RR + Triplets: V(1 ), A(1 ++ ) ; Singlet: S 1 (0 ++ ) L S1 +L A +L V = v 2 κ W S 1 u µ u µ + F A 2 2 A µνf µν L S1A = 2 λ SA 1 µ S 1 A µν u ν + F V 2 2 V µν f µν + + i G V 2 2 V µν [u µ u ν ] Antisymmetric V µν and A µν fields (better UV properties): L Kin = 1 λ R λµ νr νµ M2 R RµνRµν µ S 1 µs M2 S S R=V,A EWET A. Pich Schladming

12 Resonance Exchange G V V G V G V 2 2 M V V F V G V F V G V 2 q << 2 M V,A 2 M V V F 2 V 2 F V F V M V F A A F A F 2 A 2 M A a 1 = F2 V 4M 2 V + F2 A 4M 2 A, a 2 = a 3 = F VG V 4M 2 V, a 4 = a 5 = G2 V 4M 2 V EWET A. Pich Schladming

13 Left (Ŵ µ ) and Right (ˆB µ ) Sources: U(ϕ) g L U(ϕ)g R Ŵ µ g L Ŵ µ g L +i g L µ g L, ˆBµ g R ˆBµ g R +i g R µ g R Left / Right (Vector / Axial) Currents EWET A. Pich Schladming

14 Left (Ŵ µ ) and Right (ˆB µ ) Sources: U(ϕ) g L U(ϕ)g R Ŵ µ g L Ŵ µ g L +i g L µ g L, ˆBµ g R ˆBµ g R +i g R µ g R Left / Right (Vector / Axial) Currents Vector Form Factor: ϕ(p 1)ϕ(p 2) J µ V 0 = (p1 p2)µ F v ϕϕ(s) κ W F V V G V F v ϕϕ(s) = 1 + F V G V v 2 s M 2 V s EWET A. Pich Schladming

15 Left (Ŵ µ ) and Right (ˆB µ ) Sources: U(ϕ) g L U(ϕ)g R Ŵ µ g L Ŵ µ g L +i g L µ g L, ˆBµ g R ˆBµ g R +i g R µ g R Left / Right (Vector / Axial) Currents Vector Form Factor: ϕ(p 1)ϕ(p 2) J µ V 0 = (p1 p2)µ F v ϕϕ(s) κ W F V V G V F v ϕϕ(s) = 1 + F V G V v 2 s M 2 V s Short-distance constraint: lim F v s ϕϕ(s) = 0 F V G V = v 2 EWET A. Pich Schladming

16 Axial Form Factor: S 1(p 1)ϕ(p 2) J µ A 0 = (p1 p2)µ F a S 1 ϕ(s) κ W F A A λ 1 SA F a S 1 ϕ (s) = κ W (1 + F Aλ SA 1 κ W v 2 ) s MA 2 s EWET A. Pich Schladming

17 Axial Form Factor: S 1(p 1)ϕ(p 2) J µ A 0 = (p1 p2)µ F a S 1 ϕ(s) κ W F A A λ 1 SA F a S 1 ϕ (s) = κ W (1 + F Aλ SA 1 κ W v 2 ) s MA 2 s Short-distance constraint: lim F a s S 1 ϕ (s) = 0 F A λ SA 1 = κ W v 2 EWET A. Pich Schladming

18 Weinberg Sum Rules Chiral Symmetry: Π µν LR (q) d 4 x e iqx 0 T(J µ L (x)jν R (0) ) 0 = ( g µν q 2 +q µ q ν ) Π LR (q 2 ) = 0 EWET A. Pich Schladming

19 Weinberg Sum Rules Chiral Symmetry: Π µν LR (q) d 4 x e iqx 0 T(J µ L (x)jν R (0) ) 0 = ( g µν q 2 +q µ q ν ) Π LR (q 2 ) = 0 OPE: Π µν LR (q) 0 only through order parameters of EWSB (operators invariant under H but not under G) EWET A. Pich Schladming

20 Weinberg Sum Rules Chiral Symmetry: Π µν LR (q) d 4 x e iqx 0 T(J µ L (x)jν R (0) ) 0 = ( g µν q 2 +q µ q ν ) Π LR (q 2 ) = 0 OPE: Π µν LR (q) 0 only through order parameters of EWSB (operators invariant under H but not under G) Asymptotically-Free Theories: lim s s2 Π LR (s) = 0 (Bernard et al.) EWET A. Pich Schladming

21 Weinberg Sum Rules Chiral Symmetry: Π µν LR (q) d 4 x e iqx 0 T(J µ L (x)jν R (0) ) 0 = ( g µν q 2 +q µ q ν ) Π LR (q 2 ) = 0 OPE: Π µν LR (q) 0 only through order parameters of EWSB (operators invariant under H but not under G) Asymptotically-Free Theories: lim s s2 Π LR (s) = 0 (Bernard et al.) 1 π 1 π 0 0 ds [ImΠ VV (s) ImΠ AA (s)] = v 2 ds s [ImΠ VV (s) ImΠ AA (s)] = 0 (1 st WSR) (2 nd WSR) EWET A. Pich Schladming

22 LO V, A Π LR (s) = v2 s + F 2 V M 2 V s F 2 A M 2 A s 1 st WSR: F 2 V F2 A = v2 2 nd WSR: F 2 V M2 V F2 A M2 A = 0 EWET A. Pich Schladming

23 LO V, A Π LR (s) = v2 s + F 2 V M 2 V s F 2 A M 2 A s 1 st WSR: F 2 V F2 A = v2 2 nd WSR: F 2 V M2 V F2 A M2 A = 0 M 2 A FV 2 = v2 MA 2 M2 V, F 2 A = v2 M 2 V M 2 A M2 V, M A > M V EWET A. Pich Schladming

24 LO V, A Π LR (s) = v2 s + F 2 V M 2 V s F 2 A M 2 A s 1 st WSR: F 2 V F2 A = v2 2 nd WSR: F 2 V M2 V F2 A M2 A = 0 M 2 A FV 2 = v2 MA 2 M2 V, F 2 A = v2 M 2 V M 2 A M2 V, M A > M V 1 st WSR likely valid also in gauge theories with non-trivial UV fixed points 2 nd WSR questionable (not valid) in walking (conformal) TC scenarios Appelquist Sannino, Orgogozo Rychkov EWET A. Pich Schladming

25 Gauge Boson Self-Energies: S, T q L = 1 2 W3 µ Πµν 33 (q2 )W 3 ν 1 2 Bµ Πµν 00 (q2 )B ν W 3 µ Πµν 30 (q2 )B ν W + µ Πµν WW (q2 )W ν ) Landau gauge (ξ = 0): Π µν ij (q 2 ) = ( g µν + qµ q ν q 2 Π ij (q 2 ) There is no mixing with the Goldstones T S 16π ( e3 g 2 e3 SM ) = 0.03±0.10 4π ( g 2 sin 2 e1 e SM ) 1 = 0.05±0.12 θ W e 3 = g g Π30(0) e 1 = Π33(0) Π WW(0) M 2 W Π 30(q 2 ) = g2 tanθ W 4 s [Π VV (s) Π AA (s)] = q 2 Π30(q 2 ) + g2 tanθ W 4 v 2 e SM 1,3 defined at M H = 126 GeV EWET A. Pich Schladming

26 Dispersive Representation S = 16 g 2 tanθ W 0 ds s [ Im Π 30 (s) Im Π SM 30 (s) ] (Peskin-Takeuchi) Im Π SM 30 (s) = g2 tanθ W 192π 2 [ θ(s) ( 1 M2 H s ) 3 ( θ s MH) ] 2 (1 loop) EWET A. Pich Schladming

27 Dispersive Representation S = 16 g 2 tanθ W 0 ds s [ Im Π 30 (s) Im Π SM 30 (s) ] (Peskin-Takeuchi) Im Π SM 30 (s) = g2 tanθ W 192π 2 [ θ(s) ( 1 M2 H s ) 3 ( θ s MH) ] 2 (1 loop) T = 4π g 2 cos 2 θ W 0 ds s 2 [ ρt (s) ρ SM T (s)] (Pich Rosell Sanz-Cillero) ρ T (s) = 1 π Im[Σ(0) (s) Σ (+) (s)] (Goldstone self-energies) ρ SM T (s) = 3g 2 s 64π 2 [ θ(s) + ( 1 M4 H s 2 )θ(s M 2H ) ] (1 loop) SM contribution defined at M H = 126 GeV EWET A. Pich Schladming

28 Gauge Boson LO V, A S LO = 4π( F 2 V M 2 V ) F2 A MA 2, T LO = 0 Sensitive to vector and axial states M A > M V > 1.5 TeV MA TeV 4 SLO MV TeV MV TeV σ bounds AP Rosell Sanz-Cillero ( ) 1 st +2 nd WSR: S LO = 4πv2 MV 2 1+ M2 V MA 2 1 st { ( WSR (M A > M V ): v 2 S LO = 4π M V 2 + FA 2 1 M V 2 1 )} M A 2 > 4πv2 M V 2 > 4πv2 M A 2 EWET A. Pich Schladming

29 Gauge Boson NLO Sensitive to the light scalar S 1 (125) AP, Rosell, Sanz-Cillero V V V V S A A A A S S S 0.4 M V [1.5, 6.0] TeV 0 κ W V V V V M V T 0.0 A A A A S S S S 0.2 Κ W 0.4 κ W g SWW g SM HWW = M2 V M 2 A [0.94, 1] S M A M V > 4 TeV (95% CL) 1 st +2 nd WSRs EWET A. Pich Schladming

30 Gauge Boson NLO Weaker assumptions: 1 st WSR only, M A > M V > 0.4 TeV AP, Rosell, Sanz-Cillero ΚW M V TeV 0.2 < M V M A < < M V M A < 0.2 κ W g SWW /g SM very different from one HWW requires large (unnatural) mass splittings EWET A. Pich Schladming

31 Linear Realization: SU(2) L U(1) Y Assumes that H(126) and ϕ combine into an SU(2) L doublet: Φ = ( Φ + Φ 0 ) = 1 2 (v +H) U( ϕ) ( 0 1 ) The SM Lagrangian is the low-energy effective theory with D = 4 L eff = L SM + D>4 i c (D) i O(D) D 4 i Λ 1 operator with D = 5: O (5) = L L Φ Φ T L c L 59 independent O (6) i 5 independent O (6) i (violates L by 2 units) preserving B and L (for 1 generation) Weinberg Buchmuller Wyler, Grzadkowski Iskrzynski Misiak Rosiek violating B and L (for 1 generation) Weinberg, Wilczek Zee, Abbott Wise, EWET A. Pich Schladming

32 O G O G O W O W D = 6 Operators (other than 4-fermion ones) Grzadkowski Iskrzynski Misiak Rosiek X 3 Φ 6 and Φ 4 D 2 ψ 2 Φ 3 f ABC Gµ AνGBρ ν Gρ Cµ O Φ (Φ Φ) 3 O eφ (Φ Φ)( l pe rφ) f ABC G Aν µ Gν Bρ Gρ Cµ O Φ (Φ Φ) (Φ Φ) O uφ (Φ Φ)( q pu r Φ) ε IJK Wµ IνWJρ ν Wρ Kµ O ΦD ( Φ D µ Φ ) ( Φ D ) µφ O dφ (Φ Φ)( q pd rφ) εijk W Iν µ Wν Jρ Wρ Kµ X 2 Φ 2 ψ 2 XΦ ψ 2 Φ 2 D O ΦG Φ Φ G A µν GAµν O ew ( l pσ µν e r) τ I ΦW I µν O (1) Φl (Φ i D µφ)( l pγ µ l r) O Φ G Φ Φ G A µνg Aµν O eb ( l pσ µν e r) ΦB µν O (3) Φl (Φ i D I µ Φ)( l pτ I γ µ l r) O ΦW Φ Φ W I µν WIµν O ug ( q pσ µν T A u r) ΦG A µν O Φe (Φ i D µφ)(ē pγ µ e r) O Φ W Φ Φ W µνw I Iµν O uw ( q pσ µν u r) τ I ΦW µν I O (1) Φq (Φ i D µφ)( q pγ µ q r) O ΦB Φ Φ B µνb µν O ub ( q pσ µν u r) ΦB µν O (3) Φq (Φ i Dµ I Φ)( qpτi γ µ q r) O Φ B Φ Φ B µνb µν O dg ( q pσ µν T A d r) ΦG A µν O Φu (Φ i D µφ)(ū pγ µ u r) O ΦWB Φ τ I Φ W I µν Bµν O dw ( q pσ µν d r) τ I ΦW I µν O Φd (Φ i D µφ)( d pγ µ d r) O Φ WB Φ τ I Φ W I µν Bµν O db ( q pσ µν d r) ΦB µν O Φud i( Φ D µφ)(ū pγ µ d r) q = q L, l = l L, u = u R, d = d R, e = e R, p,r = generation indices EWET A. Pich Schladming

33 D = 6 Four-Fermion Operators Grzadkowski Iskrzynski Misiak Rosiek ( LL)( LL) ( RR)( RR) ( LL)( RR) O ll ( l pγ µl r)( l sγ µ l t) O ee (ē pγ µe r)(ē sγ µ e t) O le ( l pγ µl r)(ē sγ µ e t) O (1) qq ( q pγ µq r)( q sγ µ q t) O uu (ū pγ µu r)(ū sγ µ u t) O lu ( l pγ µl r)(ū sγ µ u t) O (3) qq ( q pγ µτ I q r)( q sγ µ τ I q t) O dd ( d pγ µd r)( d sγ µ d t) O ld ( l pγ µl r)( d sγ µ d t) O (1) lq ( l pγ µl r)( q sγ µ q t) O eu (ē pγ µe r)(ū sγ µ u t) O qe ( q pγ µq r)(ē sγ µ e t) O (3) lq ( l pγ µτ I l r)( q sγ µ τ I q t) O ed (ē pγ µe r)( d sγ µ d t) O (1) qu ( q pγ µq r)(ū sγ µ u t) O (1) ud (ū pγ µu r)( d sγ µ d t) O (8) qu ( q pγ µt A q r)(ū sγ µ T A u t) O (8) ud (ū pγ µt A u r)( d sγ µ T A d t) O (1) qd ( q pγ µq r)( d sγ µ d t) O (8) qd ( q pγ µt A q r)( d sγ µ T A d t) ( LR)( RL) and ( LR)( LR) B-violating O ledq ( l p j er)( d sq j t ) O [ duq ε αβγ ε jk (d α p ) T ][ Cur β (q γj s ) T Clt k ] O (1) ( q j quqd p ur)ε jk ( q s k dt) [ Oqqu εαβγ ε jk (q αj p )T Cqr βk ][ (u γ s ) T ] Ce t O (8) quqd ( q p j TA u r)ε jk ( q s k [ TA d t) O qqq (1) ε αβγ ε jk ε mn (q αj p )T Cqr βk ][ (q γm s ) T Clt n ] O (1) lequ ( l p j er)ε jk ( q s k ut) O(3) qqq ε αβγ (τ I ε) jk (τ I [ ε) mn (q αj p )T Cqr βk ][ (q γm s ) T Clt n ] O (3) lequ ( l p j σµνer)ε jk ( q s k σµν u t) O duu ε [ ][ αβγ (dp α )T Cur β (u γ s ) T ] Ce t q = q L, l = l L, u = u R, d = d R, e = e R, p,r,s,t = generation indices EWET A. Pich Schladming

34 Shifts of SM parameters: L (D=6) eff = i C i Λ 2 O i V(H) = λ ) 2 (Φ Φ v2 C ( ( 3 Φ Φ Φ) v 2 Λ 2 T = 1+ 3C Φ 8λ ) v 2 v Λ 2 EWET A. Pich Schladming

35 Shifts of SM parameters: L (D=6) eff = i C i Λ 2 O i V(H) = λ ) 2 (Φ Φ v2 C ( ( 3 Φ Φ Φ) v 2 Λ 2 T = 1+ 3C Φ 8λ ) v 2 v Λ 2 ( Canonical Normalization: H ch kin H, ch kin = C Φ 1 ) v 2 4 C ΦD Λ 2 EWET A. Pich Schladming

36 Shifts of SM parameters: L (D=6) eff = i C i Λ 2 O i V(H) = λ ) 2 (Φ Φ v2 C ( ( 3 Φ Φ Φ) v 2 Λ 2 T = 1+ 3C Φ 8λ ) v 2 v Λ 2 ( Canonical Normalization: H ch kin H, ch kin = C Φ 1 ) v 2 4 C ΦD Λ 2 M 2 H = 2λv 2 T ( 1 3C Φ 2λ ) v 2 Λ 2 +2ckin H EWET A. Pich Schladming

37 Shifts of SM parameters: L (D=6) eff = i C i Λ 2 O i V(H) = λ ) 2 (Φ Φ v2 C ( ( 3 Φ Φ Φ) v 2 Λ 2 T = 1+ 3C Φ 8λ ) v 2 v Λ 2 ( Canonical Normalization: H ch kin H, ch kin = C Φ 1 ) v 2 4 C ΦD Λ 2 M 2 H = 2λv 2 T ( 1 3C Φ 2λ ) v 2 Λ 2 +2ckin H Flavour: Y f M f L = Q r L[Y d ] rs Φd s R + [M f ] rs = v ( ) T [Y ψ ] rs v2 2 2Λ [C fφ] 2 rs [Y f ] rs = 1 v T [M f ] rs ( 1+cH kin ) v2 Λ 2 [C fφ] rs EWET A. Pich Schladming

38 Shifts of SM parameters: L (D=6) eff = i C i Λ 2 O i V(H) = λ ) 2 (Φ Φ v2 C ( ( 3 Φ Φ Φ) v 2 Λ 2 T = 1+ 3C Φ 8λ ) v 2 v Λ 2 ( Canonical Normalization: H ch kin H, ch kin = C Φ 1 ) v 2 4 C ΦD Λ 2 Flavour: Y f M f L = Q r L[Y d ] rs Φd s R + 4G F 2 = 2 v 2 T M 2 H = 2λv 2 T ( 1 3C Φ 2λ ) v 2 Λ 2 +2ckin H [M f ] rs = v ( ) T [Y ψ ] rs v2 2 2Λ [C fφ] 2 rs [Y f ] rs = 1 v T [M f ] rs ( 1+cH kin + 1 ( 2 [ C (3) Λ ]ee +2[ C (3) ]µµ [ C 2 ll ]µe,eµ [ ] C ll Φl Φl ) v2 Λ 2 [C fφ] rs eµ,µe EWET A. Pich Schladming )

39 Gauge Fields: L (D=6) eff = i C i Λ 2 O i Alonso et al ḡ = g ( 1+C ΦB v 2 T Λ 2 ), ḡ = g ( 1+C ΦW v 2 T Λ 2 ), ḡ s = g s ( 1+C ΦG v 2 T Λ 2 ) tan θ W = ḡ ḡ + v2 T 2Λ 2 ) (1 ḡ 2 ḡ 2 C ΦWB MW 2 = v2 T 4 ḡ2, MZ 2 = v2 T (ḡ2 +ḡ 2) + v4 T {(ḡ2 4 8Λ 2 +ḡ 2) C ΦD +4ḡ ḡ } C ΦWB ē = ḡ sin θ W { 1 cot θ W v 2 T 2Λ 2 C ΦWB }, ḡ Z = ē sin θ W cos θ W { 1+ ḡ2 +ḡ 2 2ḡḡ v 2 T Λ 2 C ΦWB } ρ ḡ2 M 2 Z ḡ 2 Z M2 W = 1+ v2 T 2Λ 2 C ΦD EWET A. Pich Schladming

40 Equivalent Bases of CP-even Operators (EW bosons only) BW HISZ GGPR Willenbrock Zhang O W = ǫ IJK W Iν µ WJρ ν WKµ ρ O WWW = Tr[ŴµνŴνρ Ŵ µ ρ ] O 3W = 1 3! g ǫ abcw aν µ Wb νρ Wcρµ O ΦW = Φ Φ W I µν WIµν O WW = Φ Ŵ µν Ŵ µν Φ O ΦB = Φ Φ B µνb µν O BB = Φ ˆBµν ˆBµν Φ O BB = g 2 H 2 B µνb µν O ΦWB = Φ σ I Φ W I µν Bµν O BW = Φ ˆBµν Ŵ µν Φ O W = (D µφ) Ŵ µν (D νφ) O HW = ig (D µ H) σ a (D ν H)W a µν O B = (D µφ) ˆBµν (D νφ) ( ) O HB = ig (D µ H) (D ν H)B µν O DW = Tr [D µ,ŵνρ][dµ,ŵνρ ] O 2W = 1 2 (Dµ W a µν )2 O DB = g 2 2 ( µb νρ)( µ B νρ ) O 2B = 1 2 ( µ B µν) 2 O W = ig 2 (H σ a D µ H)D ν W a µν O B = ig 2 (H D µ H) ν B µν O ΦD = (Φ D µ Φ) (Φ D µφ) O Φ,1 = (D µφ) ΦΦ (D µ Φ) O T = 1 2 (H D µ H) 2 O Φ = (Φ Φ) (Φ Φ) O Φ,2 = 1 2 µ(φ Φ) µ (Φ Φ) O H = 1 2 ( µ H 2 ) 2 O Φ = (Φ Φ) 3 O Φ,3 = 1 3 (Φ Φ) 3 O 6 = λ H 6 O Φ,4 = (D µφ) (D µ Φ)(Φ Φ) BW = Buchmuller Wyler, Grzadkowski et al. HISZ = Hagiwara et al. GGPR = Giudice et al. EWET A. Pich Schladming

41 Anomalous Triple Gauge Couplings L WWV = i g WWV {g V 1 ( W + µν W µ V ν W + µ W µν V ν ) +κ V W + µ W ν V µν + λ V M 2 W } W µνw + νρ V ρ µ g WWγ = e = g cosθ W, g WWZ = g cosθ W, κ V = 1+ κ V, g V 1 = 1+ gv 1 Relevant operators: O W, O ΦWB, O (3) φl (BW) O WWW, O W, O B, O BW, O DW O 3W, O HW, O HB, O 2W, O W, O B (HISZ) (GGPR) D = 6 Relations: λ γ = λ Z, g Z 1 = κ Z +tan2 θ W κ γ EWET A. Pich Schladming

42 Anomalous Triple Gauge Couplings HISZ basis: L eff = n f n Λ 2 O n Corbett Eboli González-Fraile González-García λ γ = λ Z = 3g2 M 2 W 2Λ 2 f WWW, κ γ = g2 v 2 8Λ 2 (f W +f B ) κ Z = g2 v 2 8Λ 2 ( f W tan 2 θ W f B ), g Z 1 = g 2 v 2 8cos 2 θ W Λ 2 f W (95% CL) Higgs data: (90% CL) g Z , 0.19 κγ κ Z Combined: (90% CL) g Z , κγ κ Z LEP, D0, ATLAS assume λγ = λ Z = 0 EWET A. Pich Schladming

43 Global Fit to L (D=6) eff Pomarol Riva GGPR basis: O H = 1 2 ( µ H 2 ) 2 O T = 1 2 (H D µh) 2 O 6 = λ H 6 O W = ig 2 (H σ a D µh)d νw aµν O B = ig 2 (H D µh) νb µν O BB = g 2 H 2 B µνb µν O GG = g 2 s H 2 G A µνg Aµν O HW = ig (D µ H) σ a (D ν H)W a µν O HB = ig (D µ H) (D ν H)B µν O 3W = 1 3! g ǫ abcwµ aνwb νρwc ρµ O yu = y u H 2 Q L HuR + h.c. O yd = y d H 2 Q L Hd R + h.c. O ye = y e H 2 L L He R + h.c. O u R = i (H D µh)(ū R γ µ u R ) O d R = i (H D µh)( d R γ µ d R ) O e R = i (H D µh)(ē R γ µ e R ) O q L = i (H D µh)( Q L γ µ Q L ) O (3)q L = i (H σ a D µh)( Q L σ a γ µ Q L ) O (3)ql LL = ( Q L σ a γ µq L )( L L σ a γ µ L L ) O (3)l LL = ( L L σ a γ µ L L )( L L σ a γ µl L ) EWET A. Pich Schladming

44 LEP-I + SLC + KLOE (CKM univ) + pp l ν (LHC) Pomarol Riva c T (95% CL) c L q c R e c L 3 q c V c R u c LL 3 l c R d c LL 3 ql c L q3 c L 3 q3 All ; --- without KLOE ; include O t L + O(3)t L ; All other O i set to zero L = c T v 2 O T + c+ V M 2 W + cq L v 2 Oq L + c (3) q L O (3)q v 2 L (O W + O B ) + c(3)l LL O (3)l v 2 LL + ce R v 2 Oe R + cu R v 2 Ou R + cd R v 2 Od R + c (3) ql LL v 2 O (3)ql LL c ± V = 1 2 (c W ± c B ) EWET A. Pich Schladming

45 Pomarol Riva LEP-II + H Zγ (LHC) H VV,f f (LHC) c V Κ HV (95% CL) Κ GG Κ HV Κ BB All ; All other O i set to zero ; c H and c yf effects constrained to be below 50% of SM L TGC = κ+ HV m 2 W L Higgs = c H v 2 O H + (O HW + O HB ) + c V + κ HV 2m 2 W f=t,b,τ O + + κ 3W mw 2 O 3W c yf v 2 Oy + c6 f v 2 O6 + κ BB MW 2 O BB + κ GG MW 2 O GG + c V κ HV 2MW 2 O O ± (O W O B ) ± (O HW O HB ), O WW = 4O + O BB, κ ± HV = 1 2 (κ HW ± κ HB ) EWET A. Pich Schladming

46 Renormalization Group Equations L (D=6) eff = i C i (µ) Λ 2 O i (µ) Loop Corrections µ dci dµ = j γ ij 16π 2 Cj Ci(µ) = Ci(Λ) j ( ) γ ij Λ Cj(Λ) log 16π2 µ Anomalous Dimensions: γ ij EWET A. Pich Schladming

47 Anomalous Dimensions γ ij Complete matrix computed in the BW basis: Jenkins et al. γ O ( 1,g 2,λ,y 2,g 4,g 2 λ,g 2 y 2,λ 2,λy 2,y 4,g 6,g 4 λ,g 6 λ ) g 3 X 3 H 6 H 4 D 2 g 2 X 2 H 2 yψ 2 H 3 gyψ 2 XH ψ 2 H 2 D ψ 4 g 3 X 3 g H 6 g 6 λ λ,g 2,y 2 g 4,g 2 λ,λ 2 g 6,g 4 λ λy 2,y 4 0 λy 2,y 4 0 H 4 D 2 g 6 0 g 2,λ,y 2 g 4 y 2 g 2 y 2 g 2,y 2 0 g 2 X 2 H 2 g g 2,λ,y 2 0 y yψ 2 H 3 g 6 0 g 2,λ,y 2 g 4 g 2,λ,y 2 g 2 λ,g 4,g 2 y 2 g 2,λ,y 2 λ,y 2 gyψ 2 XH g g 2 1 g 2,y ψ 2 H 2 D g 6 0 g 2,y 2 g 4 y 2 g 2 y 2 g 2,λ,y 2 y 2 ψ 4 g g 2 y 2 g 2,y 2 g 2,y 2 (X 2 H 2,X 2 H 2 ) submatrix computed in the HISZ basis Chen Dawson Zhang A submatrix computed in the GGPR basis J. Elias-Miro et al. EWET A. Pich Schladming

48 Oblique Corrections (NLO) g 2 16π S M2 W Λ 2 [ ] {C [ ( )]} BW (µ)+ g2 M C 32π 2 WW +tan 2 2 θ W C BB 1 2 log H µ 2 g 2 sin 2 θ W 4π T v2 2Λ 2 C Φ,1 (HISZ basis) Mebane et al., Chen Dawson Zhang c,1 2 TeV Tree-level Bounds c BW 2 TeV 2 EWET A. Pich Schladming

49 OUTLOOK Effective Field Theory: powerful low-energy tool Mass Gap: E,m light Λ NP Assumption: relevant symmetries (breakings) & light fields Most general L eff (φ light ) allowed by symmetry Short-distance dynamics encoded in LECs LECs constrained phenomenologically Goal: get hints on the underlying fundamental dynamics New Physics EWET A. Pich Schladming

50 Learning from QCD experience. EW problem more difficult Fundamental Underlying Theory unknown QCD? χpt Standard Model Additional dynamical input (fresh ideas!) needed EWET A. Pich Schladming

51 A new boson discovered It behaves as the SM Higgs particle The SM appears to be the right theory at the EW scale New physics needed to explain many pending questions Flavour, CP, baryogenesis, cosmology... How far is the Scale of New-Physics Λ NP? Which symmetry keeps Å À away from Λ NP? Supersymmetry, scale/conformal symmetry... Eagerly awaiting more LHC data... EWET A. Pich Schladming