EWSB by strongly-coupled dynamics: an EFT approach

Transcription

1 EWSB by strongly-coupled dynamics: an EFT approach EWSB by strongly-coupled dynamics: an EFT approach (page 1)

2 Outline Motivation Linear vs non-linear realization of EWSB Organizing the expansion: Power-counting criteria Operators at NLO Back to the linear case: SM at NLO Conclusions and future prospects EWSB by strongly-coupled dynamics: an EFT approach (page 2)

3 Motivation (Why bother about strongly-coupled EWSB?) Higgs detection? CMS and ATLAS recently reported discovery of a light particle around 126 GeV. For many, this is evidence that the Higgs is there and the SM is ratified in its most mysterious aspect: the mechanism of EWSB. EWSB by strongly-coupled dynamics: an EFT approach (page 3)

4 Motivation (Why bother about strongly-coupled EWSB?) Higgs detection? CMS and ATLAS recently reported discovery of a light particle around 126 GeV. For many, this is evidence that the Higgs is there and the SM is ratified in its most mysterious aspect: the mechanism of EWSB. Case 1: the new state is a light scalar responsible for EWSB Information on the TeV is necessary to ascertain whether it is a manifestation of weakly-coupled or strongly-coupled dynamics. EWSB by strongly-coupled dynamics: an EFT approach (page 4)

5 Motivation (Why bother about strongly-coupled EWSB?) Higgs detection? CMS and ATLAS recently reported discovery of a light particle around 126 GeV. For many, this is evidence that the Higgs is there and the SM is ratified in its most mysterious aspect: the mechanism of EWSB. Case 1: the new state is a light scalar responsible for EWSB Information on the TeV is necessary to ascertain whether it is a manifestation of weakly-coupled or strongly-coupled dynamics. (i) Case 1.1: weakly-coupled (fundamental) scalar Some mechanism has to be invoked to stabilize its mass. The best candidate is SUSY. Signature: states at the TeV scale. EWSB by strongly-coupled dynamics: an EFT approach (page 5)

6 Motivation (Why bother about strongly-coupled EWSB?) Higgs detection? CMS and ATLAS recently reported discovery of a light particle around 126 GeV. For many, this is evidence that the Higgs is there and the SM is ratified in its most mysterious aspect: the mechanism of EWSB. Case 1: the new state is a light scalar responsible for EWSB Information on the TeV is necessary to ascertain whether it is a manifestation of weakly-coupled or strongly-coupled dynamics. (i) Case 1.1: weakly-coupled (fundamental) scalar Some mechanism has to be invoked to stabilize its mass. The best candidate is SUSY. Signature: states at the TeV scale. (ii) Case 1.2: strongly-coupled (composite) scalar No mass stabilization mechanism needed: naturally at the EW scale if the scalar is a PGB (but hard to compute in any model!). Signature: states at the TeV scale. EWSB by strongly-coupled dynamics: an EFT approach (page 6)

7 Motivation (Why bother about strongly-coupled EWSB?) Higgs detection? CMS and ATLAS recently reported discovery of a light particle around 126 GeV. For many, this is evidence that the Higgs is there and the SM is ratified in its most mysterious aspect: the mechanism of EWSB. Case 1: the new state is a light scalar responsible for EWSB Information on the TeV is necessary to ascertain whether it is a manifestation of weakly-coupled or strongly-coupled dynamics. (i) Case 1.1: weakly-coupled (fundamental) scalar Some mechanism has to be invoked to stabilize its mass. The best candidate is SUSY. Signature: states at the TeV scale. (ii) Case 1.2: strongly-coupled (composite) scalar No mass stabilization mechanism needed: naturally at the EW scale if the scalar is a PGB (but hard to compute in any model!). Signature: states at the TeV scale. Case 2: the new state is not responsible for EWSB Strong evidence for strongly-coupled scenarios. EWSB by strongly-coupled dynamics: an EFT approach (page 7)

8 Motivation (Why bother about strongly-coupled EWSB?) Higgs detection? CMS and ATLAS recently reported discovery of a light particle around 126 GeV. For many, this is evidence that the Higgs is there and the SM is ratified in its most mysterious aspect: the mechanism of EWSB. Case 1: the new state is a light scalar responsible for EWSB Information on the TeV is necessary to ascertain whether it is a manifestation of weakly-coupled or strongly-coupled dynamics. (i) Case 1.1: weakly-coupled (fundamental) scalar Some mechanism has to be invoked to stabilize its mass. The best candidate is SUSY. Signature: states at the TeV scale. (ii) Case 1.2: strongly-coupled (composite) scalar No mass stabilization mechanism needed: naturally at the EW scale if the scalar is a PGB (but hard to compute in any model!). Signature: states at the TeV scale. Case 2: the new state is not responsible for EWSB Strong evidence for strongly-coupled scenarios. Conclusion: Identification of the scalar (Higgs or not) can take many years... If new states at the TeV, signatures can hardly distinguish SUSY particles from strongly-coupled resonances. Strongly-coupled scenarios have a lot to say. But so far no evidence of TeV physics... EWSB by strongly-coupled dynamics: an EFT approach (page 8)

9 Linear vs non-linear EWSB Massless gauge and fermion sector of the SM [SU(3) C, SU(2) L, U(1) Y ]: q organized as: with G µ [8,1,0], W µ [1,2,0], B µ [1,1,0] [ 3,2, 1 ] [, l 1,2, 1 ] [, u 3,1, 2 ] [, d 3,1, 1 ], e[1,1, 1] L 4 = 1 2 G µνg µν 1 2 W µνw µν 1 4 B µνb µν + i q Dq + i l Dl + iū Du + i d Dd + iē De D µ ψ L = ( µ + igw µ + ig Y L B µ )ψ L D µ ψ R = ( µ + ig Y R B µ )ψ R Masses to gauge bosons and fermions through Higgs mechanism: SSB with the pattern SU(2) L U(1) Y U(1) Q. But in which precise way EW symmetry is broken? EWSB by strongly-coupled dynamics: an EFT approach (page 9)

10 Linear sigma model: the Higgs boson Masses to the gauge bosons are given by a fundamental (complex) scalar doublet in a renormalizable interaction: ( ) L H = D µ Φ D µ Φ + µ 2 Φ Φ λ(φ Φ) 2 φ +, Φ = φ 0 SSB is achieved through picking a nontrivial vacuum Φ 0 = 1 ( ) ( ) 0 µ 2 1/2 ( ) 1/2 v = = 2GF 246 GeV 2 v + H λ Goldstone bosons are converted into longitudinal components of the gauge boson, i.e., they generate masses: m W = v 2 g m Z = v 2 g2 + g 2 m A = 0 Price to pay: appearance of a fundamental scalar H with arbitrary mass m H = 2λv 2 Custodial symmetry: accidental global SU(2) L SU(2) R symmetry that ensures that ρ = g2 +g 2 g 2 m 2 W m 2 Z 1. EWSB by strongly-coupled dynamics: an EFT approach (page 10)

11 Non-linear sigma model: Higgsless scenario SSB à la CCWZ: Assume that the Goldstone bosons arise from the spontaneous breaking of a global SU(2) L SU(2) R symmetry to the diagonal subgroup SU(2) V (minimal scenario). The Goldstone modes can be collected in a SU(2) matrix U, transforming as U g L Ug R, g L,R SU(2) L,R Convenient realization: U = exp(2iφ/v), Φ = ϕ a τa 2 = 1 ( ϕ 0 2 ϕ + 2 ϕ ϕ0 2 ) with τ a = τ a the generators of SU(2). Gauge the SU(2) L U(1) Y subgroup: D µ U = µ U + igw µ U ig B µ U τ 3 2 Collect the right-handed quark and lepton fields in spurion doublets r = (u, d) T and η = (ν, e) T. EWSB by strongly-coupled dynamics: an EFT approach (page 11)

12 List down to most general Lagrangian at leading order: L U = v2 4 D µu D µ U v ( qy u UP + r + qy d UP r + ly e UP η + h.c. ) where P ± are SU(2) L projectors: P ± 1 ± τ 3 2, P 12 τ 1 + iτ 2 2, P 21 τ 1 iτ 2 2 Moving to unitary gauge it is clear that masses for the gauge bosons and fermions are also generated: L U = v2 8 [ 2g 2 W + µ W µ + (g 2 + g 2 )Z µ Z µ] v ( qy u u + qy d d + ly e e + h.c. ) No fundamental scalar is requested. The NσM at leading order is equivalent to the LσM with the scalar integrated out. However, composite scalars can be accommodated, with natural masses at the electroweak scale. [Georgi, Kaplan, Galison 84] The theory is no longer renormalizable, but if phrased in an EFT language, renormalizability order by order can be recovered and the framework can become predictive. Power-counting essential. EWSB by strongly-coupled dynamics: an EFT approach (page 12)

13 A survey on strongly-coupled EWSB First works in early 80 s [Longhitano; Appelquist, Bernard] (before NLO χpt!). LO and NLO bosonic sector with comments on the power-counting. Size of the operators estimated with technicolor models. Revisited in [Appelquist et al 93,95; Nyffeler et al 99; Grojean et al 06] with redundancies eventually eliminated. Fermion bilinears [Appelquist et al 85], completed in [Peccei et al 90]. Four-fermion operators partially discussed in [Bagan et al 99]. Mostly all work oriented to phenomenological studies with technicolor in mind. From early 90 s, phenomenological studies with LEP physics. However, (i) not systematic and power-counting discussions absent; (ii) some of the technical developments ignored. EWSB by strongly-coupled dynamics: an EFT approach (page 13)

14 Organizing the expansion: power-counting Canonical dimension of operators clearly not the right expansion. L 4 + L U inhomogeneous... Landau gauge is especially suited: ghosts and Goldstones decoupled (ghosts decoupled from EWSB dynamics), so no ghost fields needed to build the effective operators. In the absence of fermions and gauge bosons the power-counting should reduce to the familiar χpt formula: D ( v2 ϕ ) B Λ 2L p2l+2 v The addition of fermions leads to D (yv)ν v F L+F R 2 p d Λ 2L ψ F 1 L L ψf2 L L ψ F1 R R ψf2 R R ( ϕ v ) B, d = 2L + 2 ν (FL + F R )/2 Divergences, i.e., d 0 exhaust the list of counterterms. EWSB by strongly-coupled dynamics: an EFT approach (page 14)

15 A more general formula can be written down to include gauge fields. D (yv)ν (gv) m+2r+2x+u+z v F L+F R 2 where the power of p is p d Λ 2L ψ F 1 L L ψf2 L L ψ F1 R R ψf2 R R ( Xµν v ) V ( ϕ ) B v Bounded from above. d 2L + 2 ν F L + F R 2 V m 2r 2x u z The power-counting formula justifies that the naively LO custodial symmetry breaking operator is actually NLO. β 1 v 2 τ L L µ 2 EWSB by strongly-coupled dynamics: an EFT approach (page 15)

16 Operators at NLO The general effective Lagrangian reads: L = L 4 + L U + L β1 + i Classes of operators O i : c i v 6 d i Λ 2 O i UD 4 : D µ U D µ U (5) XUD 2 : U W µν D µ U U D ν Uτ 3 (2) X 2 U : B µν U W µν Uτ 3 (4) ψ 2 UD : iēγ µ e U D µ Uτ 3 (10) ψ 2 UD 2 lup η D µ U D µ U (15) ψ 4 U lγµ Uτ 3 U lēγ µ e (64) Potential classes [UD 6, XUD 4, X 2 UD 2, X 3 U, ψ 2 UD 3, ψ 2 UXD, ψ 2 UX] can be shown to be subleading (NNLO). Notation: to simplify the operators I will work with L µ = UD µ U and τ L = Uτ 3 U. EWSB by strongly-coupled dynamics: an EFT approach (page 16)

17 Operators at NLO: UD 4 Pure Goldstone sector [Longhitano 80-81] O D1 = L µ L µ L ν L ν O D2 = L µ L ν L µ L ν O D3 = τ L L µ 2 τ L L ν 2 O D4 = τ L L µ τ L L µ L ν L ν O D5 = τ L L µ τ L L ν L µ L ν In unitary gauge they correspond to all the possible quartic contractions of gauge bosons: Z µ Z µ Z ν Z ν, W + µ W µ+ W ν W ν, W + µ W µ W + ν W ν, Z µ Z µ W + µ W µ, Z µ Z ν W + ν W µ They correspond to the usual χpt operators at NLO. EWSB by strongly-coupled dynamics: an EFT approach (page 17)

18 Operators at NLO: XUD 2 and X 2 U CP even and CP odd operators: [Longhitano 80-81, Appelquist et al 93] O XU1 = g gb µν W µν τ L O XU4 = g gǫ µνλρ τ L W µν B λρ O XU2 = g 2 W µν τ L 2 O XU5 = g 2 ǫ µνλρ τ L W µν τ L W λρ O XU3 = gǫ µνλρ W µν L λ τ L L ρ O XU6 = g W µν L µ τ L L ν O XU7 = ig B µν τ L [L µ, L ν ] O XU10 = ig ǫ µνλρ B µν τ L [L λ, L ρ ] O XU8 = ig W µν [L µ, L ν ] O XU11 = igǫ µνλρ W µν [L λ, L ρ ] O XU9 = ig W µν τ L τ L [L µ, L ν ] O XU12 = igǫ µνλρ W µν τ L τ L [L λ, L ρ ] Relevant for EWPT: oblique parameters and triple gauge vertices. Caveat for phenomenology: redundancies such as W µν [L µ, L ν ] = g ψ L γ µ L µ ψ L ig 2 v2 L µ L µ + ig B µν τ L W µν ig W µν W µν ε µνλρ W µν [L λ, L ρ ] = ig ε µνλρ B µν τ L W λρ not taken into account in phenomenological studies! EWSB by strongly-coupled dynamics: an EFT approach (page 18)

19 Operators at NLO: ψ 2 UD Operators involving a fermionic vector current: [Appelquist et al 93] O ψv 1 = i qγ µ q τ L L µ, O ψv 4 = iūγ µ u τ L L µ O ψv 2 = i qγ µ τ L q τ L L µ, O ψv 5 = i dγ µ d τ L L µ O ψv 3 = i qγ µ P12q L L µ P21 L (h.c.), O ψv 6 = iūγ µ d L µ P21 L (h.c.) O ψv 7 = i lγ µ l τ L L µ, O ψv 10 = iēγ µ e τ L L µ O ψv 8 = i lγ µ τ L l τ L L µ O ψv 9 = i lγ µ P L 12l L µ P L 21 (h.c.) EWSB by strongly-coupled dynamics: an EFT approach (page 19)

20 Operators at NLO: ψ 2 UD 2 Operators involving a fermionic scalar and tensor current: Buchalla, O.C. 12] [Peccei et al 90; O ψs1,2 = qup ± r L µ L µ O ψs3,4 = qup ± r τ L L µ 2 O ψs5 = qup 12 r L µ P21 L τ L L µ O ψs6 = qup 21 r L µ P12 L τ L L µ O ψs7 = lup η L µ L µ O ψs8 = lup η τ L L µ 2 O ψs9 = lup 12 η L µ P21 L τ L L µ O ψt1 = qσ µν UP 12 r L µ P21 τ L L L ν O ψt2 = qσ µν UP 21 r L µ P12 τ L L L ν O ψt3,4 = qσ µν UP ± r L µ P12 L L ν P21 L O ψt5 = lσ µν UP 12 η L µ P21 τ L L L ν O ψt6 = lσ µν UP η L µ P12 L L ν P21 L EWSB by strongly-coupled dynamics: an EFT approach (page 20)

21 Operators at NLO: ψ 4 U LL LL operators (16): [Buchmüller et al 86; Bagan et al 99; Buchalla, O.C. 12] O LL6 = qγ µ τ L q qγ µ τ L q, O LL8 = q α γ µ τ L q β q β γ µ τ L q α, O LL7 = qγ µ τ L q qγ µ q O LL9 = q α γ µ τ L q β q β γ µ q α O LL10 = qγ µ τ L q lγ µ τ L l O LL11 = qγ µ τ L q lγ µ l, O LL13 = qγ µ τ L l lγ µ τ L q, O LL15 = lγ µ τ L l lγ µ τ L l, O LL12 = qγ µ q lγ µ τ L l O LL14 = qγ µ τ L l lγ µ q O LL16 = lγ µ τ L l lγ µ l RR RR operators without U fields (7). LL RR operators (18): O LR10 = qγ µ UT 3 U q ūγ µ u, O LR12 = qγ µ τ L q dγ µ d, O LR14 = ūγ µ u lγ µ τ L l, O LR16 = qγ µ τ L q ēγ µ e, O LR11 = qγ µ T A τ L q ūγ µ T A u O LR13 = qγ µ T A τ L q dγ µ T A d O LR15 = dγ µ d lγ µ τ L l O LR17 = lγ µ τ L l ēγ µ e O LR18 = qγ µ τ L l ēγ µ d EWSB by strongly-coupled dynamics: an EFT approach (page 21)

22 LR LR operators (12): O ST5 = qup + r qup r, O ST7 = qup + T A r qup T A r, O ST9 = qup + r lup η, O ST11 = qσ µν UP + r lσ µν UP η, O ST6 = qup 21 r qup 12 r O ST8 = qup 21 T A r qup 12 T A r O ST10 = qup 21 r lup 12 η O ST12 = qσ µν UP 21 r lσ µν UP 12 η Four-quark operators with Y f 0 (11): O FY 1 = qup + r qup + r, O FY 3 = qup r qup r, O FY 5 = qup r rp + U q, O FY 7 = qup r lup η, O FY 2 = qup + T A r qup + T A r O FY 4 = qup T A r qup T A r O FY 6 = qup T A r rp + U T A q O FY 8 = qσ µν UP r lσ µν UP η O FY 9 = lup η rp + U q O FY 10 = lup η lup η O FY 11 = lup r rp + U l EWSB by strongly-coupled dynamics: an EFT approach (page 22)

23 Dynamical scales The power-counting formula is useful to identify the classes of operators associated with physics of electroweak symmetry breaking. These are counterterms needed to renormalize the theory order by order. However, some operators are unrelated to EWSB processes. A full renormalization program is in any case needed. Electroweak scale: Λ EW = 4πv (2 : 3) TeV Lepton number violating scale: Q LV = l T CU P + U l Operator of dimension-3 giving Majorana mass to the neutrinos. Λ ν TeV-GUT scale (?). Baryon number breaking scale: ǫ abc [d T a Cu b ][u T c Ce] Dimension-6 operators responsible for proton decay. Λ BV GUT scale (?). Flavor mass scale: Λ Y... Generic new physics: ψl ψ L ψl ψ L can be mediated with with heavy vector exchange (Z ). Λ NP TeV-GUT scale (?) EWSB by strongly-coupled dynamics: an EFT approach (page 23)

24 Back to the linear case: SM at NLO Replace U H/v = 1/v( φ, φ) in the basis operators. The theory will become renormalizable and so the electroweak and new physics scale decouple: v/λ 0. The power-counting becomes the naive dimensional one. H H 1, so those factors have to be added in all places possible. This trivially generates d = 4 terms (φ φ) 3, (φ φ) 2 (φ φ) (φ φ) leφ, (φ φ) qdφ, (φ φ) qu φ (φ φ)x µν X µν, (φ φ) X µν X µν (D µ φ φ)(φ D µ φ), B µν φ W µν φ Some dimensional reshuffling (different power-counting): some classes get relegated to NNLO (UD 4, ψ 4 U, ψ 2 UD 2 ). We recover the SM at NLO. [Buchmüller et al 86; Grzadkowski et al 10] EWSB by strongly-coupled dynamics: an EFT approach (page 24)

25 Applications Realistic models that accommodate existing experimental data hard to find. EFT approach can be a common ground where possible UV completions can be easily tested, for instance for EWPT or top physics. Oblique parameters (S, T, U). They test deviations in the gauge sector for two-point functions. L V P = 1 2 W µπ 3 µν 33 (q2 )Wν B µπ µν 00 (q2 )B ν WµΠ 3 µν 30 (q2 )B ν W µ + Π µν WW (q2 )Wν ) ( g g Ŝ = Π Π 33 (0) Π WW (0) 30(0) ˆT = m 2 W and involve only 3 operators of our basis, namely Then, O β = v 2 L µ τ L 2 O XU1 = g g B µν W µν τ L O XU2 = g 2 W µν τ L W µν τ L Ŝ = 16πα 1 ˆT = 2β1 Û = 16πα 2 Û = Π 33(0) Π WW(0) The coefficients can be determined from experimental fits and then used to constrain UV completion scenarios. EWSB by strongly-coupled dynamics: an EFT approach (page 25)

26 Triple gauge vertices (revisited) e, q W + e +, q γ, Z W Our effective operator basis can be matched onto the general vertex parametrization L WWV = iκ V W + µ W ν V µν + i κ V W + µ W ν Ṽ µν + ig V 1 (W + µνw µ W µνw µ+ )V ν + g V 4 (W + µνw µ + W µνw µ+ )V ν g V 5 ( W + µνw µ + W µνw µ+ )V ν where κ V = κ 0 V + κ V (α 1, α 2,..., β 1 ). Triple gauge vertices loosely constrained (LEP). However, in combination with the oblique parameters (and LHC data?) bounds might get tighter, especially when redundancies are eliminated. EFT language really fits that purpose. EWSB by strongly-coupled dynamics: an EFT approach (page 26)

27 Conclusions An EFT description of a strongly-coupled EWSB scenario in general is useful even in the presence of a light scalar. For the first time we have a systematic and complete basis of operators at NLO. Future prospects for phenomenological applications: EWPT, top physics, etc. So far only the minimal setting worked out (several phenomenological applications are Higgs-insensitive). However, a light scalar can (should!) be added with a consistent power-counting. EWSB by strongly-coupled dynamics: an EFT approach (page 27)