5. Fingerprints of Heavy Scales

Transcription

1 5. Fingerprints of Heavy Scales Electroweak Resonance Effective Theory Integration of Heavy States Short-Distance Constraints Low-Energy Constants Outlook EFT. Pich 017 1

2 Higher-Order Goldstone Interactions L (4) EW Bosonic = i F i (h/v) O i F i (h/v) = n=0 ( ) h n F i,n v ppelquist-bernard, Longhitano, Buchalla et al, lonso et al, Pich et al... O(p 4 ) P-even bosonic operators O 1 = 1 4 O = 1 O 3 = i.p., Rosell, Santos, Sanz-Cillero f µν + f + µν f µν f µν O 6 = 1 v ( µ h)( µ h) u ν u ν f µν + f + µν +fµν f µν O 7 = 1 v ( µ h)( ν h) u µ u ν f µν + [u µ,u ν ] O 8 = 1 v 4 ( µ h)( µ h)( ν h)( ν h) O 4 = u µ u ν u µ u ν O 9 = 1 v ( µh) f µν u ν O 5 = u µ u µ U = u = exp { i v σ ϕ}, u µ i u(d µu) u = u µ, f µν ± = u Ŵ µν u ± u ˆB µν u Custodial symmetry assumed EFT. Pich 017

3 EW 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 EFT. Pich 017 3

4 EW 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 Require a good U behaviour Low # of derivatives 3 Match the two effective Lagrangians LECs EFT. Pich 017 3

5 EW 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 Require a good U 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 EFT. Pich 017 3

6 Coset Space Coordinates: G SU() L SU() R H SU() H g g h ξ(ϕ) ξ(ϕ) (ξl (ϕ),ξ R (ϕ)) G ξ L (ϕ) ξ R (ϕ) G g L ξ L (ϕ)g h (ϕ,g) G g R ξ R (ϕ)g h (ϕ,g) ϕ ϕ G/H U(ϕ) ξ L (ϕ)ξ R (ϕ) G g L U(ϕ)g R EFT. Pich 017 4

7 Coset Space Coordinates: G SU() L SU() R H SU() H g g h ξ(ϕ) ξ(ϕ) (ξl (ϕ),ξ R (ϕ)) G ξ L (ϕ) ξ R (ϕ) G g L ξ L (ϕ)g h (ϕ,g) G g R ξ R (ϕ)g 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) u(ϕ) G g L u(ϕ)g h (ϕ,g) = g h (ϕ,g)u(ϕ)g R U(ϕ) = u(ϕ) = exp { i v σ ϕ} EFT. Pich 017 4

8 Coset Space Coordinates: G SU() L SU() R H SU() H g g h ξ(ϕ) ξ(ϕ) (ξl (ϕ),ξ R (ϕ)) G ξ L (ϕ) ξ R (ϕ) G g L ξ L (ϕ)g h (ϕ,g) G g R ξ R (ϕ)g 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) u(ϕ) G g L u(ϕ)g h (ϕ,g) = g h (ϕ,g)u(ϕ)g R U(ϕ) = u(ϕ) = exp { i v σ ϕ} SU() triplets: X 1 σa X a G g h (ϕ,g) X g h (ϕ,g) µ X = µ X +[Γ µ,x ], Γ µ = 1 {u ( µ i Ŵ µ)u + u( µ i ˆB µ)u } u µ i ud µ U u = u µ, f µν ± = u Ŵ µν u ±u ˆB µν u EFT. Pich 017 4

9 LO Resonance EW Lagrangian:.P., Rosell, Santos, Sanz-Cillero L eff = L EWET + R L R + R,R L RR + Heavy Triplets: (1 ), (1 ++ ), P(1 ++ ) ; Heavy Singlet: S 1 (0 ++ ) R L R = v κ W h uµ u µ + F µν f µν + + i G µν [u µ u ν ] + F µνf µν + λ h 1 µ h µν u ν + d P v µh P u µ + c d S 1 u µ u µ + λ hs1 v h S 1 U = u = exp { i v σ ϕ}, u µ i u(d µu) u = u µ, f µν ± = u Ŵ µν u ± u ˆB µν u ntisymmetric µν and µν fields (better U properties): L Kin = 1 λ R λµ νr νµ 1 M R RµνRµν R=, EFT. Pich 017 5

10 Resonance Exchange F G G G q << M, G M F G M F F F M F F F M Pich, Rosell, Santos, Sanz-Cillero F 1 = F F 4M 4M F 4 = G 4M F 7 = d P M P, F = F F 8M 8M, F 5 = c d 4M S 1 G 4M + (λh 1 ) v M, F 3 = F G M, F 6 = (λh 1 ) v M, F 8 = 0, F 9 = F λ h 1 v M EFT. Pich 017 6

11 Short-Distance Constraints ector Form Factor: ϕ(p 1)ϕ(p ) J µ 0 = (p1 p)µ F ϕϕ(s) κ W F G F ϕϕ(s) = 1 + F G v s M s lim s F ϕϕ (s) = 0 F G = v EFT. Pich 017 7

12 Short-Distance Constraints ector Form Factor: ϕ(p 1)ϕ(p ) J µ 0 = (p1 p)µ F ϕϕ(s) κ W F G F ϕϕ(s) = 1 + F G v s M s lim s F ϕϕ (s) = 0 F G = v xial Form Factor: h κ W λ 1 F h(p 1)ϕ(p ) J µ 0 = (p1 p)µ F hϕ(s) F hϕ (s) = κ W (1 + F λ h 1 κ W v ) s M s lim s F hϕ (s) = 0 F λ h 1 = κ W v EFT. Pich 017 7

13 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 +q µ q ν ) Π LR (q ) = 0 EFT. Pich 017 8

14 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 +q µ q ν ) Π LR (q ) = 0 OPE: Π µν LR (q) 0 only through order parameters of EWSB (operators invariant under H but not under G) EFT. Pich 017 8

15 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 +q µ q ν ) Π LR (q ) = 0 OPE: Π µν LR (q) 0 only through order parameters of EWSB (operators invariant under H but not under G) symptotically-free Theories: lim s s Π LR (s) = 0 Bernard et al. EFT. Pich 017 8

16 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 +q µ q ν ) Π LR (q ) = 0 OPE: Π µν LR (q) 0 only through order parameters of EWSB (operators invariant under H but not under G) symptotically-free Theories: lim s s Π LR (s) = 0 Bernard et al. 1 π 1 π 0 0 ds [ImΠ (s) ImΠ (s)] = v ds s [ImΠ (s) ImΠ (s)] = 0 (1 st WSR) ( nd WSR) EFT. Pich 017 8

17 LO: Π LR (s) = v s + F M s F M s, 1 st WSR: F F = v nd WSR: F M F M = 0 M F = v M M, F = v M M M, M > M EFT. Pich 017 9

18 LO: Π LR (s) = v s + F M s F M s, 1 st WSR: F F = v nd WSR: F M F M = 0 M F = v M M, F = v M M M, M > M NLO: κ W g hww g SM hww = M M.P., Rosell, Sanz-Cillero EFT. Pich 017 9

19 LO: Π LR (s) = v s + F M s F M s, 1 st WSR: F F = v nd WSR: F M F M = 0 M F = v M M, F = v M M M, M > M NLO: κ W g hww g SM hww = M M.P., Rosell, Sanz-Cillero 1 st WSR likely valid also in gauge theories with non-trivial U fixed points nd WSR questionable (not valid) in walking (conformal) TC scenarios ppelquist Sannino, Orgogozo Rychkov EFT. Pich 017 9

20 Short-distance constraints bring sharper predictions F 1 = F 4M F 4M F = F F 8M 8M F 3 = F 4 = F 5 = F 6 = F G M G 4M cd G 4MS 4M 1 (λh 1 ) v M + (λh 1 ) v M F 7 = d P MP F 8 = 0 = v 4 Pich, Rosell, Santos, Sanz-Cillero ( 1 M + 1 M ) = v (M 4 +M4 ) 8M M (M M ) = v M = = (M M )v 4M M c d 4M S 1 (M M )v 4M M = M (M M )v M 6 = d P M P + M (M M )v M 6 F 9 = F λ h 1 v M = M v M 4 EFT. Pich

21 symptotically-free Theories.P., Rosell, Santos, Sanz-Cillero, arxiv: M Te M Te κ W = M /M κ W = 0.8 κ W = 0.9 κ W = 0.95 S,T constraints M Te M Te M Te M S1 c d M P d P EFT. Pich

22 Gauge Boson Self-Energies: S, T q L = 1 W3 µ Πµν 33 (q )W 3 ν 1 Bµ Πµν 00 (q )B ν W 3 µ Πµν 30 (q )B ν W + µ Πµν WW (q )W ν ) Landau gauge (ξ = 0): Π µν ij (q ) = ( g µν + qµ q ν q Π ij (q ) There is no mixing with the Goldstones T S 16π ( e3 g e3 SM ) = 0.05±0.11 4π ( g sin e1 e SM ) 1 = 0.09±0.13 θ W e 3 = g g Π30(0) e 1 = Π33(0) Π WW(0) M W Π 30(q ) = g tanθ W 4 s [Π (s) Π (s)] = q Π30(q ) + g tanθ W 4 v e SM 1,3 defined at M h = 15 Ge EFT. Pich 017 1

23 Gauge Boson LO.P., Rosell, Sanz-Cillero, S LO = 4π( F M ) F M, T LO = 0 1 st + nd WSR: S LO = 4πv M 1 st WSR (M > M ): S LO = 4π { v ( ) 1+ M M M + F ( 1 M 1 )} M > 4πv M > 4πv M M Te 4 3σ bounds at LO SLO.5 3 WSR 1 WSR M Te M Te Sensitive to vector and axial states M > M > 1.5 Te EFT. Pich

24 Gauge Boson NLO Sensitive to the light scalar h(15) P, Rosell, Sanz-Cillero S S S S 0.4 M [1.5, 6.0] Te 0 κ W 1 0. M T 0.0 S S S S 0. Κ W 0.4 κ W g SWW g SM HWW = M M [0.94, 1] S M M > 4 Te (95% CL) 1 st + nd WSRs EFT. Pich

25 symptotically-free Theories.P., Rosell, Santos, Sanz-Cillero, arxiv: M Te M Te κ W = M /M κ W = 0.8 κ W = 0.9 κ W = 0.95 S,T constraints M Te M Te M Te M S1 c d M P d P EFT. Pich

26 Ongoing work:.p., Rosell, Santos, Sanz-Cillero, arxiv: C. Krause,.P., Rosell, Santos, Sanz-Cillero CP-odd operators Couplings with SM fermions Equivalence of different spin 1 field formalisms Proca, Hidden-Gauge, ntisymmetric Short-distance constraints Green functions, additional states,... Heavy fermion fields Coloured heavy fields Flavour dynamics... EFT. Pich

27 OUTLOOK Effective Field Theory: powerful low-energy tool Mass Gap: E,m light Λ NP ssumption: 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 EFT. Pich

28 Learning from QCD experience. EW problem more difficult Fundamental Underlying Theory unknown QCD? χpt Standard Model dditional dynamical input (fresh ideas!) needed EFT. Pich

29 Backup Slides

30 Proca Spin 1 Fields: ˆRµ = ˆ µ,  µ LˆR = 1 4 ˆR µν ˆRµν M R ˆR µˆrµ + ˆR µν ˆχ µν ˆR }{{} O(p 3 ) ˆχ µν ˆ = fˆ f µν + + i gˆ [uµ,u ν ], ˆχ µν  = f  f µν + λhâ 1 [ ( µ h)u ν ( ν h)u µ] L O(p4 ) ˆR = 0 F (P) i = 0 EFT. Pich 017 0

31 Proca Spin 1 Fields: ˆRµ = ˆ µ,  µ LˆR = 1 4 ˆR µν ˆRµν M R ˆR µˆrµ + ˆR µν ˆχ µν ˆR }{{} O(p 3 ) ˆχ µν ˆ = fˆ f µν + + i gˆ [uµ,u ν ], ˆχ µν  = f  f µν + λhâ 1 [ ( µ h)u ν ( ν h)u µ] L O(p4 ) ˆR = 0 F (P) i = 0 But there are also terms without ˆR µ : F ϕϕ (s) = High-energy behaviour 1 + F G v fˆ gˆ 1 + s SD s M F3 s v s SDP s v M F3 s v L (4,P/) non-r = 9 i=1 F SDP/ i (SDET-) (SDET-P) F SD 3 = 0, F SDP 3 = fˆ gˆ O i F G M EFT. Pich 017 0

32 Gauge Boson NLO Weaker assumptions: 1 st WSR only, M > M > 0.4 Te P, Rosell, Sanz-Cillero ΚW M Te 0. < M M < < M M < 0. κ W g SWW /g SM very different from one HWW requires large (unnatural) mass splittings EFT. Pich 017 1

33 Resonance Contributions to Bosonic LECs i F i Fi.P., Rosell, Santos, Sanz-Cillero F F 4M F + F 8M 3 F G M + F F 4M F + F 8M F G M F G M F G M F F 4M F h λ 1 v M F F 4M F λ h 1 v M 4 5 G 4M + G 4M cd 4M S G G 4M 4M 1 λ h 1 v 6 M λh 1 v M dp 7 M P + λh 1 v λ h 1 v M + M F λ h 1 v M F λh 1 v M 10 ( c ˆ 1 T ) M (c  1 T ) M 1 1 F F 11 1 M 1 M 1 1 EFT. Pich 017

34 .P., Rosell, Santos, Sanz-Cillero Resonance Contributions to Two-Fermion LECs i F ψ i F ψ i 1 G C 0 M G C 0 M c d c S 1 1 M S 1 F C 0 M F C 0 M v λ h 1 C 0 M vλ h 1 C 0 M 3 F C0 F C 0 c ˆ 1 1 câ1 T câ1 1 M M M 1 M 1 4 F1 C 1 0 M 1 T c ˆ 1 F1 C 1 0 M 1 5 d P c P 1 M P 6 c ˆ 1 T c ˆ 1 1 câ1 T câ1 1 M 1 M EFT. Pich 017 3

35 .P., Rosell, Santos, Sanz-Cillero Resonance Contributions to Four-Fermion LECs i 1 3 (cs 1 ) 4M S F ψ4 i (c S 1 ) M S (c P 1 ) M P c ˆ 1 c ˆ 1 M c ˆ 1 c ˆ 1 M câ1 câ1 + M F ψ4 i + (cs 1 1 ) 4M S 1 4 (cp 1 ) 4M P + (cp 1 1 ) 4M P 1 5 (c ˆ 1 ) ( câ1 ) M M 6 ( c ˆ 1 ) (câ1 ) M M (c ˆ 7 1 ) ( câ1 ) 4M + 4M (c ˆ 1 1 ) 4M ( c  1 1 ) 4M ( c ˆ 1 ) (câ1 ) 4M + 4M 4M (c  1 1 ) 4M 1 1 ( c ˆ 1 1 ) 9 (C 0 ) M ( C 0 ) M (C0 10 ) M (C 1 0 ) M + ( C 0 ) M ( C 1 0 ) M 1 1 câ1 câ1 M c ˆ 1 1 c ˆ 1 1 M c  1 1 câ1 1 M 1 1 EFT. Pich 017 4