Radiative corrections to the Higgs potential in the Littlest Higgs model

Transcription

1 Little. results a Little. Radiative corrections to the potential in the Littlest model Cheluci In collaboration with Dr. Antonio Dobado González and Dra. Siannah Peñaranda Rivas Departamento de Física Teórica I Universidad Complutense de Madrid

2 Little. results a Outline Little. 1 2 Little

3 Little. results a Outline Little. 1 2 Little

4 Little. results a Little In the Standard Model (SM) the fundamental scalar suffer from radiative correction instability in their masses.. Problem: Little hierarchy problem The electroweak precision data require Λ 10 TeV BUT Large fine-tuning m H = m tree + Λ 2 loops However, if Λ 2 TeV no large fine-tuning New physics at this scale.

5 Little. results a Little In the Standard Model (SM) the fundamental scalar suffer from radiative correction instability in their masses.. Problem: Little hierarchy problem The electroweak precision data require Λ 10 TeV BUT Large fine-tuning m H = m tree + Λ 2 loops However, if Λ 2 TeV no large fine-tuning New physics at this scale.

6 Little. results a Little In the Standard Model (SM) the fundamental scalar suffer from radiative correction instability in their masses.. Problem: Little hierarchy problem The electroweak precision data require Λ 10 TeV BUT Large fine-tuning m H = m tree + Λ 2 loops However, if Λ 2 TeV no large fine-tuning New physics at this scale.

7 Little. results a Outline Little. 1 2 Little

8 Little. results a Little. Little The is interpreted as a pseudo-nambu-goldstone boson (GB) corresponding to a spontaneously broken global symmetry G to a subset H via a vev f. Three different scales: Λ: UV physics f: new O(TeV) v: EWSB and SM

9 Little. results a Main characteristics Little. The low energy dynamics of the GB is described by a G/H non-linear sigma model (NLSM). The mass is protected by new approximate global symmetries and by the collective symmetry breaking of these global symmetries by gauge and Yukawa interactions. New states at the TeV scale.

10 Little. results a Main characteristics Little. The low energy dynamics of the GB is described by a G/H non-linear sigma model (NLSM). The mass is protected by new approximate global symmetries and by the collective symmetry breaking of these global symmetries by gauge and Yukawa interactions. New states at the TeV scale.

11 Little. results a Main characteristics Little. The low energy dynamics of the GB is described by a G/H non-linear sigma model (NLSM). The mass is protected by new approximate global symmetries and by the collective symmetry breaking of these global symmetries by gauge and Yukawa interactions. New states at the TeV scale.

12 Little. results a Main characteristics Little. The low energy dynamics of the GB is described by a G/H non-linear sigma model (NLSM). The mass is protected by new approximate global symmetries and by the collective symmetry breaking of these global symmetries by gauge and Yukawa interactions. New states at the TeV scale.

13 Little. results a Little. Littlest model The Littlest model (LH), which is a NLSM based on SU(5)/SO(5), is one of the most economical and attractive. N. Arkani-Hamed, A.G. Cohen, E. Katz and A. Nelson, JHEP 07, 034(2002). SU(5) SO(5) Can be parameterised as: (14 Goldstone bosons) where Λ 10 TeV and f 1 TeV Σ = e iπ/f Σ 0 e iπt /f = e 2iΠ/f Σ 0, the Π matrix represents the fluctuations around the vacuum: 0 14X Π = π a X a = a=1 ξ i 2 H 0 φ i 2 H φ i H T 2 i 2 H ξ T 1 C A η diag(1, 1, 4, 1, 1), The doublet: H = (H 0, H + ), The complex tripet: φ = φ0 1 2 φ φ + φ ++ 1 A.

14 Little. results a Gauge sector: The Lagrangian. L 0 = f 2 8 tr[(d µσ)(d µ Σ) ] Little Model I (SU(2) U(1)) 2 (SU(2) U(1)) SM Model II SU(2) 2 U(1) (SU(2) U(1)) SM. When the symmetry is spontaneously broken W a = s ψ W a 1 c ψw a 2, M2 W = f 2 g 2 W a = c ψ W a 1 + s ψw a 2, m2 W = 0 4c 2 ψ s2 ψ B = s ψ B 1 c ψ B 2, M 2 B = f 2 g 2 20c 2 ψ s2 ψ B, m B = c ψ B 1 + s ψ B 2, mb 2 B 2 = 0 = 0 q c ψ = g/g 1, s ψ = g/g 2 are the mixing angles and g = g 1 g 2 / g g2, idem for U(1) group 2

15 Little. results a Gauge sector: Collective symmetry breaking Little. The model possesses two SU(3) global symmetries. Part of the SU(5) is gauged, for example by (SU(2) U(1)) 2 groups The gauge generators are embedded in the SU(5) in such a way that any given generator commutes with an SU(3) subgroup but breaks the other SU(3) symmetry. If one of the gauge couplings are off then the SU(3) global symmetry is recovered and it will break down to an SU(2) subgroup by the v.e.v Σ 0, becoming the doublet an exact GB. Any quantum contribution to the mass must be proportional to the product of all gauge coupling constants.

16 Little. results a Little. Fermion sector: The Lagrangian and Collective symmetry breaking. L Yuk = λ 1 2 f u R ɛ mn ɛ ijk Σ im Σ jn χ Lk λ 2 f U R U L + h.c. where m, n = 4, 5, i, j = 1, 2, 3, and χ L = (u, b, U) L Collective symmetry breaking If the coupling λ 1 is set to zero the is completely decoupled. If λ 2 = 0 the SU(3) 1 is unbroken and the is an exact GB. When the symmetry is spontaneously broken t R = c θ u R 0 s θ U R T R = s θ u R + c θ U R χ L u 1 0 b A U L t b T 1 A L m t = m b = 0 m 2 T = f 2 (λ λ2 2 ) λ being c θ = q 2 λ 2, s θ = 1 +λ2 2 λ q 1 λ 2 1 +λ2 2

17 Little. results a Little. Summary. Lagrangian and Spectrum The is given by: L 0 + L Yuk New : New gauge bosons One top T All of them are massive and their masses TeV. s fields Their masses are not protected by any symmetry. SM : Four gauge bosons related to the electroweak symmetry The quarks top and bottom All of them are massless. The doublet Protected by collective symmetry breaking. Would be GB : η and ξ Eaten by the new gauge bosons

18 Little. results a Outline Little. 1 2 Little

19 Little. results a Little. One loop We compute the fermionic, gauge boson and the GB radiative to the potential V at one loop level V = µ 2 HH +λ H 4(HH ) 2 +λ φ 2f 2 tr(φφ )+iλ H 2 φf (Hφ H T H φh ) where M 2 φ = f 2 λ φ 2. Integrating out the fields Γ eff [H, φ]. Taking into account that we are computing the effective potential Γ eff H/φ=const = R d x 4 V eff (H, φ) µh = µφ = 0. The divergent integrals are regularized by the NLSM scale ultraviolet Λ.

20 Little. results a Little. Fermionic radiative The Lagrangian for the quarks: L χ = χ R (i / M + Î)χ L + h.c. with M =diag(0, 0, m T ), χ L/R = (t, b, T ) T L/R and Î the interaction matrix. Integrating out the quarks...

21 Little. results a Little. Gauge boson radiative Working in the Landau gauge, then the Lagrangian for the gauge bosons are L Ω = 1 2 Ωµ (( + M 2 Ω )g µν µ ν + 2Ĩ g µν)ω ν with Model I: M Ω = (M W 1 3 3, 0 3 3, M B, 0), Ω µ = (W µa, W µa, B µ, B µ ). Model II: without B. And Ĩ is the interaction matrix. Integrating out the gauge bosons...

22 Little. results a Results I: for µ 2 f µ 2 f = N c m 2 T λ2 t 4π 2 ( ) log 1 + Λ2 mt 2 λ t = q λ 1 λ 2 λ 2 and λ T = q 1. λ 2 1 +λ2 2 λ 2 1 +λ2 2 Little Observe The three diagrams which contribute to this parameter. The µ 2 f log(λ 2 /M 2 ). Relationship: λ 2 t + λ 2 T = m T λ T f µ 2 f > 0. The cancellations occur between with same statistic M.Peresltein, M.Peskin and A.Pierce. Phys. Rev. D69, (2004).

23 Little. results a Little Results II: for µ 2 g Model I: µ I 2 g " 3 64π 2 3g 2 M 2 W log 1 + Λ2 M 2!# W +g 2 M 2 B log 1 + Λ2 M 2 B! or Model II: µ II 2 g 3 64π 2 3g 2 M 2 W log 1 + Λ2 M 2 W!! + g 2 Λ 2. Observe The two diagrams which contribute to this parameter are: The µ I 2 g log(λ 2 /M 2 ). But µ II 2 g Λ 2 + log(λ 2 /M 2 ). µ 2 g < 0. The cancellations occur between with same statistic. M.Peresltein. Prog. Part. Nucl. Phys. 58, (2007).

24 Little. results a Little. Results for λ H 4, λ φ 2, λ H2 φ For the radiative In generic ( ) λ = A Λ2 + B m2 log 1 + Λ (4π f ) 2 (4π f ) 2 m 2 They depend on the mixing angles λ T, c ψ ( ), f, the cutoff Λ and the couplings of the SM. Relationship: RC λ Λ2 H 4 = 1 4 λλ2 φ 2. In the Model II these relations are not exact The terms related to the U(1) gauge group do not satisfy this relationship. A.Dobado, L., S.Peñaranda. Phys. Rev. D75, (2007). A.Dobado, L., S.Peñaranda. Eur.Phys.J. C58, (2008).

25 Little. results a GB Little. To start with we will turn off the gauge interaction... L 0 = f 2 8 tr [ ( µ Σ)( µ Σ) ] L 0 = 1 2 g αβ µ π α µ π β where π α (GB) are the Gaussian coordinates on K The metric on the coset space K g αβ tr Σ Σ π = δ α π β αβ + (π 2 ) + O(π 4 ) Classical Lagrangian NLSM based on the coset K = G/H

26 Little. results a GB Little. To start with we will turn off the gauge interaction... L 0 = f 2 8 tr [ ( µ Σ)( µ Σ) ] L 0 = 1 2 g αβ µ π α µ π β where π α (GB) are the Gaussian coordinates on K The metric on the coset space K g αβ tr Σ Σ π = δ α π β αβ + (π 2 ) + O(π 4 ) Classical Lagrangian NLSM based on the coset K = G/H

27 Little. results a Little. We have to take into account the geometrical nature of the NSLM. e iγ[π] = [dπ g] e is[π,π] Finally, we obtain: Conclusion e iγ[π] = [dπ] e i(s[π,π]+ Γ[π,π]) The measure factor can be exponential Γ eff [π] + Γ[π] = 0 The GB do not contribute to the effective potential V in any NSLM at one loop level and in particular for the SU(5)/SO(5). A.Dobado, L., S.Peñaranda. Phys. Rev. D75, (2007).

28 Little. results a Little. We have to take into account the geometrical nature of the NSLM. e iγ[π] = [dπ g] e is[π,π] Finally, we obtain: Conclusion e iγ[π] = [dπ] e i(s[π,π]+ Γ[π,π]) The measure factor can be exponential Γ eff [π] + Γ[π] = 0 The GB do not contribute to the effective potential V in any NSLM at one loop level and in particular for the SU(5)/SO(5). A.Dobado, L., S.Peñaranda. Phys. Rev. D75, (2007).

29 Little. results a Little. We have to take into account the geometrical nature of the NSLM. e iγ[π] = [dπ g] e is[π,π] Finally, we obtain: Conclusion e iγ[π] = [dπ] e i(s[π,π]+ Γ[π,π]) The measure factor can be exponential Γ eff [π] + Γ[π] = 0 The GB do not contribute to the effective potential V in any NSLM at one loop level and in particular for the SU(5)/SO(5). A.Dobado, L., S.Peñaranda. Phys. Rev. D75, (2007).

30 Little. results a The theory also receives from additional operators coming from the ultraviolet (UV) completion of the. Little. The lowest order operators are: O f = a 1 4 λ2 1 f 4 ɛ wx ɛ yz ɛ ijk ɛ kmn Σ iw Σ jx Σ my Σ nz, where i, j, k, m, n run over 1,2,3 and w, x, y, z run over 4,5. Model I: O gb = 1 2 af 4 n g 2 j Model II: O gb = 1 2 af 4 n g 2 j P 3a=1 Tr h(qj a Σ)(Qj a Σ) i + g h 2 j Tr (Y j Σ)(Y j Σ) io, P 3a=1 Tr h(q a j Σ)(Q a j Σ) i + g 2 Tr ˆ(Y Σ)(Y Σ) o, with j = 1, 2 and Qj a and Y j being the generators of the SU(2) j and U(1) j groups, respectively. Where a and a are two unknown coefficients O(1) sensitive to the cutoff Λ. These oparators contribute to the potential. λ EO H 4 = 1 4 λeo φ 2. T.Han, H.E.Logan, B.McElrath and L.T.Wang. Phys. Rev. D67, (2003).

31 Little. results a Summary The one-loop potential Little. V (H, φ) = µ 2 HH + λ H 4 (HH ) 2 + λ φ 2 f 2 tr(φφ ) + iλ H 2 φ f (Hφ H T H φh ) The λ s coefficients, λ H 4, λ φ 2 and λ H 2 φ, are the sum of the effective operators and the fermionic and gauge bosons radiative. The GB do not contribute to the effective potential V in any NSLM at one loop level and in particular for the SU(5)/SO(5). The µ 2 parameter, in principle, receives one-loop coming from the fermionic and gauge boson sectors. µ 2 = µ 2 f + µ 2 I(II) g (+µ 2EO ) The fermionic and gauge boson have the opposite sign. Then, µ 2 f > µ2 g µ2 > 0 If λ H 4 > 0 and λ φ 2 > 0, we recover the EWSB sector: Minimum condition HH = v 2 /2 v 2 = µ 2 ; φφ = v 2 v λ = H 2 λ H 4 λ2 H 2 φ /λ φ v 2 φ 2 2λφ f 2

32 Little. results a Outline Little. 1 2 Little

33 Little. results a Little. EWSB has taken place: Four sectors: scalar sector(ss) (Up to order v 2 /f 2 ) HH = v 2 /2 and φφ = v 2. H, m 2 H = 2 µ2 Φ 0, m 2 Φ 0 M 2 φ pseudoscalar sector(ps) G 0, m 2 G 0 = 0 Φ P, m 2 Φ P M2 φ double charged sector Φ++, m 2 Φ ++ M2 φ charged sector (cs) G +, m 2 G + = 0, Φ +, m 2 Φ + M2 φ H identified with the particle. G + and G 0 are the Goldstone bosons corresponding to the SM W and Z gauge fields. The doubly charged sector remains uncoupled.

34 Little. results a Little. L GB (H, G s, Φ s) = f 2 8 tr [( µσ) ( µ Σ)] V eff (H, G s, Φ s) In order to compute the radiative, we split the field as H = H + H The kinectic terms are not properly normalized: L Kin = H2 f 2 «( µ H)( µ H) H2 2f 2 «( µφ 0 )( µ Φ 0 ) +... We write the fields in terms of a new set of properly normalized fields up to order 1/f 2. For example: ««H = 1 H2 f 2 H, Φ 0 = 1 H2 4f 2 Φ 0 Consequence 1 2 M2 φ Φ2 0 1 λ 2 M2 φ Φ2 0 + φ 2 4 H 2 Φ 2 0 This last term describes a new interaction which comes from the new normalization of the fields and the fact that M 2 φ O(f 2 ). These interactions play a decisive role in canceling the quadratic divergences that come from the GB loops.

35 Little. results a Little. L GB (H, G s, Φ s) = f 2 8 tr [( µσ) ( µ Σ)] V eff (H, G s, Φ s) In order to compute the radiative, we split the field as H = H + H The kinectic terms are not properly normalized: L Kin = H2 f 2 «( µ H)( µ H) H2 2f 2 «( µφ 0 )( µ Φ 0 ) +... We write the fields in terms of a new set of properly normalized fields up to order 1/f 2. For example: ««H = 1 H2 f 2 H, Φ 0 = 1 H2 4f 2 Φ 0 Consequence 1 2 M2 φ Φ2 0 1 λ 2 M2 φ Φ2 0 + φ 2 4 H 2 Φ 2 0 This last term describes a new interaction which comes from the new normalization of the fields and the fact that M 2 φ O(f 2 ). These interactions play a decisive role in canceling the quadratic divergences that come from the GB loops.

36 Little. results a Little. L GB (H, G s, Φ s) = f 2 8 tr [( µσ) ( µ Σ)] V eff (H, G s, Φ s) In order to compute the radiative, we split the field as H = H + H The kinectic terms are not properly normalized: L Kin = H2 f 2 «( µ H)( µ H) H2 2f 2 «( µφ 0 )( µ Φ 0 ) +... We write the fields in terms of a new set of properly normalized fields up to order 1/f 2. For example: ««H = 1 H2 f 2 H, Φ 0 = 1 H2 4f 2 Φ 0 Consequence 1 2 M2 φ Φ2 0 1 λ 2 M2 φ Φ2 0 + φ 2 4 H 2 Φ 2 0 This last term describes a new interaction which comes from the new normalization of the fields and the fact that M 2 φ O(f 2 ). These interactions play a decisive role in canceling the quadratic divergences that come from the GB loops.

37 Little. results a Little Finally... L GB = 1 2 m2 H H2 λ H 4 4 H4 + L Kin V ss eff V ps eff V cs eff +... Integrating out the scalar :. Γ L = H, G 0 or G ± and Γ H = Φ 0, ΦP or Φ ±. The last figure corresponds to the contribution to the quartic coupling from the Φ 0 propagator. Here we have expanded the Φ 0 propagator in powers of k 2 /m 2 Φ 0 and kept just the first term.

38 Little. results a Little. Results 8! < mgb 2 = 3 λ (4π) 2 : φ 2 + λ 4 H 4!) 2 1 λ H 4 mh 2 log 1 + Λ2 m H 2 λ GB = Λ 2 + λ φ 2 4 λ 2 H 2 φ λ φ 2 1 contribution to the mass λ 2 H 2 φ contribution to the quartic coupling λ φ 2 The diagrams which contribute to the mass A M 2 φ log 1 + Λ2 M 2 φ! Remember: λ Λ2 H 4 = 1 4 λλ2 φ 2, Model I: m 2 GB O(Λ2 log(λ 2 /M 2 ). λeo H 4 = 1 4 λeo φ 2. Model II: we have O(Λ 4 ) and O(Λ 2 ).

39 Little. results a Little. Summary. mass and quartic coupling We have considered all sectors: fermionic, gauge boson and scalar ones. All dangerous diagrams to the mass have been cancelled. The effective potential for the boson is given by: where V H = 1 2 m2 H H λ H H4 mass m 2 H = 2(µ2 m 2 GB ) quartic coupling λ H = λ H 4 λ GB A.Dobado, L., S.Peñaranda (2009), arxiv: Eur. Phys. J. Accepted.

40 Little. results a Outline Little. 1 2 Little

41 Little. results a Little. Conditions and Constraints Set of new parameters (λ T, c ψ, c ψ, f, Λ, a, a ) Conditions Constraints µ 2 > < c ψ < 0.9 µ 2 = v 2 (λ H 4 λ 2 H 2 φ /λ φ 2 ) 114 GeV < m H < 200 GeV The top-quark mass is known and m T 2 TeV 0.5 λ T 2 and f 1 TeV Λ 4π f < 1 10 TeV Λ 12 TeV 0.8 TeV f 1 TeV a and a O(1) Model I: The existence of a B gauge boson leads to large corrections and some problems with the direct observational bounds on Z boson from Tevatron. C. Csaki, J. Hubisz, G.D. Kribs, P. Meade, and J. Ternin, Phys. ReV. D67, (2003) and Phys. ReV. D68, (2003)

42 Little. results a Results Little. In the RC we have solutions only for Λ < 6 TeV. For Λ 6 TeV m 2 GB > µ2!!! In RC+EO case, the a parameter can take values which help to compensate the big effect for the GB radiative thus allowing larger cutoff values. For EO both parameters, a and a are positive.

43 Little. results a Results Masses of the new Little. When the model includes only radiative corrections the M Φ is about 2 TeV. In the other two cases it is 5 TeV. m T < 2 TeV and M W > 0.6 TeV

44 Little. results a Results Little. The lowest values for the mass found for the three cases: RC, RC+EO and EO. Parameters RC RC+EO EO m H GeV GeV GeV µ GeV GeV GeV mgb GeV GeV GeV λ H f 0.86 TeV 0.91 TeV 0.80 TeV Λ 5 TeV TeV TeV λ T c ψ a a

45 Little. results a Outline Little. 1 2 Little

46 Little. results a Little. Analytically All dangerous diagrams to the mass have been cancelled. We have computed the radiative corrections to the potential by integrating out fermion, gauge boson and GB at one-level. The effective potential has the right symmetry to reproduce the EWSB sector of the SM. We have also demostrated that the GB do not contribute to the effective potential V in any NSLM at one loop level and in particular for the SU(5)/SO(5) LH model.

47 Little. results a Little. ly Model I is not phenomenologically viable. Analysis done for Model II We have imposed: ESSB v 2 µ = 2 λ H 4 λ2 H 2 φ /λ φ GeV < m H < 200 GeV Results: RC results only for Λ < 6 TeV incompatibility with the precision electroweak test. However, this situation change when the RC+EO and EO cases are analyzed. The range of the masses for the new obtained are: 1.15 TeV m T 1.5 TeV 0.6 TeV M W 2 TeV 2 TeV M φ 6 TeV The lowest numerical result obtained corresponds to RC+EO case: m H GeV

48 Little. results a Little. Summarizing, we have arrived to the conclusion that the SU(5)/SO(5) Little model with gauge group [SU(2) 2 U(1)] is phenomenologically viable through some tuning in the parameter space, assuming a careful inclusion of fermions, gauge bosons, scalar loops and effective operators. In any case it will be the LHC, whose main goal is to disentangle the mechanism of the electroweak symmetry breaking, which will decide if Little are appropriated for describing this mechanism or not.

49 Little. results a Little. Thanks for your attention!

50 Little. results a λ s. Fermion sector Little. λ s!! λ H 4 = 2Nc f (4πf ) 2 (λ 2 t + λ 2 T )Λ2 + mt 2 53 λ 2 t + λ 2 T log Λ 2 m T !! λ φ 2 = 2Nc f (4πf ) 2 4(λ 2 t + λ 2 T )Λ2 (λ 2 t + λ 2 T )m2 T log Λ 2 m T 2 + 1!! λ H 2 φf = 4Nc (4πf ) 2 (λ 2 t + λ 2 T )Λ2 λ 2 T m2 T log Λ 2 m T 2 + 1

51 Little. results a λ s. Gauge sector Little. λ s: Model I 0 0 λ I H 4 = 3 g (4πf ) g2 4 c ψ 2 s2 ψ 0 1 c A Λ 2 g ψ s 2 c 2 ψ ψ s2 ψ! 1 + g 2 g c 2 A M 2 ψ s2 ψ B log 1 + Λ2 M 2 B 00 1 λ I φ 2 g = 3 4(4πf ) 2 g 2 M 2 B log Λ 2 M 2 B + 1 λ I H 2 φ g = 3 8(4πf ) g2 c ψ 2 + g 2 s2 c ψ 2 A Λ 2 g 2 M 2 ψ s2 ψ W log Λ 2 M W! 1 (s 2 ψ c2 ψ )2 c 2 A ψ s2 ψ g 2 s 2 ψ c2 ψ c ψ 2 s2 ψ!!!! Λ 2 M 2 W log Λ 2 M W M 2 W log 1 + Λ2 M 2 W! (s ψ 2! c2 ψ )2 c ψ 2 4 s2 ψ!! 1 +g 2 s2 ψ c2 ψ c 2 Λ 2 M 2 ψ s2 ψ B log Λ 2 M A B

52 Little. results a λ s. Gauge sector Little. λ s: Model II λ II H 4 g = 3 4(4πf ) 2 λ II φ 2 g " 1 g 2 4 c ψ 2 s2 ψ! = 3 g 2 4(4πf ) 2 c ψ 2 Λ 2 g 2 M 2 s2 W log Λ 2 M ψ W λ II H 2 = 3g2 s 2 ψ c2 ψ φ g 8(4πf ) 2 c ψ 2 s2 ψ! Λ g 2 Λ 2 g 2 M 2 W log Λ 2 M W!! Λ 2 M 2 W log Λ 2 M W (s ψ 2 c2 φ )2 c ψ 2 4 s2 ψ c ψ 2 s2 ψ!!! +... # + 3g 2 (4πf ) 2 Λ2,

53 Little. results a Effective operators Little. O f = a 1 4 λ2 1 f 4 ɛ wx ɛ yz ɛ ijk ɛ kmn Σ iw Σ jx Σ my Σ nz, where i, j, k, m, n run over 1,2,3 and w, x, y, z run over 4,5. Model I O gb = 1 2 af 4 {g 2 j 3 a=1 Tr [(Q a j Σ)(Q a j Σ) ] + g 2 j Tr [ (Y j Σ)(Y j Σ) ]}, with j = 1, 2 and Qj a and Y j being the generators of the SU(2) j and U(1) j groups, respectively. Model II O gb = 1 2 cf 4 {g 2 j ] } 3 a=1 [(Q Tr j a Σ)(Qj a Σ) + g 2 Tr [(Y Σ)(Y Σ) ], where j = 1, 2 and Y is the generator of the unique U(1) group.

54 Little. results a λ s. Effective operators Little. The coefficients of the potential coming from the effective operators are: 0 Model I 1 Model II! λ EO H 4 g2 g 8 s ψ c2 s ψ 2 ψ c ψ 2 A + a (λ λ2 2 ) a g 2 8 s ψ 2 a c2 6 g 2 + a (λ λ2 2 ) ψ 0 1! λ EO φ 2 g2 2 s ψ 2 + g 2 c2 s ψ 2 A + 4a (λ 2 ψ c2 ψ 1 + λ2 2 ) a g 2 2 s ψ 2 + 2ag 2 + 4a (λ 2 c2 1 + λ2 2 ) ψ 0 1 λ EO φh 2 a 4 µ 2 2 c2 ψ s2 ψ s 2 ψ c2 ψ + g 2 c2 ψ s2 ψ s 2 ψ c2 ψ A 2a (λ λ2 2 ) a 4 g 2 c2 ψ s2 ψ s 2 ψ c2 ψ af 2 g 2 2a (λ λ2 2 )

55 Little. results a Little. SSB and mass eigenstates SSB has taken place: HH = v 2 /2 and φφ = v 2 0 H = (w +, 2 1 (v + h + iw 0 )) and φ v + 1 (ξ + iρ) 2 1 φ φ + φ ++ A 2 The new fields describe the fluctuations alround the vacuum. Four sectors: the scalar (h, ξ), the pseudoscalar (ρ, ω 0 ), the charged (φ +, ω + ) and the doubly charged (φ ++ ). By diagonalizing the corresponding matrices we obtain the mass eigenstates in each sector: scalar sector(ss) h = c 0 H + s 0 Φ 0, m 2 H = 2 µ2 ξ = c 0 Φ 0 s 0 H, m 2 Φ 0 M 2 φ charged sector (cs) pseudoscalar sector(ps) w + = c +G + + s +Φ +, m 2 G + = 0, w 0 = c P G 0 + s P Φ P, m 2 G 0 = 0 φ + = c +Φ + + s +G +, m 2 Φ + M2 φ where c 0,p,+ 1 + O(v 2 /f 2 ) and s 0,p,+ O(v 2 /f 2 ). ρ = c P Φ P s P G 0, m 2 Φ P M2 φ The doubly charged sector remains unchanged with a mass M φ and uncoupled. The notation introduced for the mass eigenstates is the following: H and Φ 0 are the neutral scalars; being the first one identified with the particle, Φ P is a neutral pseudoscalar, Φ + and Φ ++ are the charged and doubly charged scalars, and G + and G 0 are the Goldstone bosons corresponding to the SM W and Z gauge fields.

56 Little. results a Masses m T and M W In order to avoid a large fine-tuning: m T 2 TeV 0.5 λ T 2 and f 1 TeV Little m T = f λ2 t +λ2 T λ T.

57 Little. results a Little Masses m T and M W From the W mass and taking into account the previous constraint on f M W = g f 2c ψ s ψ > 0.6 TeV 0.1 < c ψ < 0.5 and 0.8 < c ψ < 0.9.

58 Little. results a Little Discussion The results depend strongly on the fermionic For λ T > 1 it is very hard to satisfy the minimum condition of the potential. Low values of c ψ are preferred. Then low µ values are obtained

59 Little. results a Phenomelogical results at one-loop level Strongly dependent on a but not on a Contours of the viable regions in λ T c ψ plane Little. Some numerical results The minimum possible values are found for f = 0.8 TeV and Λ = 10 TeV µ m H λ T c ψ a a M φ RC 390 GeV 552 GeV TeV