The Shadow of a Scale-Invariant Hidden U(1)

Transcription

1 The Shadow of a Scale-Invariant Hidden U1) We-Fu Chang Feb. 26, 2008 at APCTP 2008 with John Ng and Jackson Wu) PRD74:095005,75: Motivation Model Coleman-Weinberg Phenomenology and U1) We-Fu Chang, NTHU p. 1/23

2 Not so hidden U1) Recently, the physics involved hidden sector have been very popular. In order to study the phenomenology of hidden sectors), we consider the simplest case, a models with a hidden U1) sector, to capture the physics. Extra U1) sector is also well motivated in many physics beyond SM. By hidden we mean this extra U1) is SM singlet, also all SM fields are not charged under this extra U1). In this simple model, there are two ways to commute between the hidden sector and the visible world: The kinematic mixing of the two U1) s, Note that the field strength, F µν, of any U1) is gauge invariant by itself. And this is true only for U1). A modified scalar potential. The general Lagrangian can be spelled out: L = L SM 1 4 Xµν X µν ǫ 2 Bµν X µν + µ 12 ) 2 g sx µ φ s V φ s, Φ) We call this not so hidden sector the Shadow World. We-Fu Chang, NTHU p. 2/23

3 Global fit to EWPT A priori, there is NO any physical reason for it to be small. χ 2 s ǫ, M 3 ) i i exp i s δ i exp ) 2. χ SM 2 = χ 2 0, M 3 ) = χ 2 s ǫ, ) = s ǫ 1.0 χ 2 > 2χ SM % EW fit 0% EW fit M 3 GeV) The EWPT data favors those models in which s ǫ is radiatively generated. However, if χ 2 /χ SM 2 ) 0.01 this model can be probed at LHC and ILC. We-Fu Chang, NTHU p. 3/23

4 How about the Scalar sector? The most general renormalizable GSM U1) s invariant scalar potential is: V Φ, φ s ) = µ 2 sφ sφ s + λ s φ sφ s ) 2 + 2κ ) Φ Φ φ sφ s ) +µ 2 Φ Φ + λφ Φ) 2 After SSB the scalars acquire nonzero VEV, Φ = v 0 ), φ p = v s 2, with v 2 0 = λ sµ 2 κµ 2 s λλ s κ 2, v 2 s = λµ2 s κµ 2 λλ s κ 2. For the potential to be bounded from below, and the above values global minimum, λ, λ s > 0, κ > 0 The shadow world gives us one extra neutral scalar but the Higgs phenomenology is trivial. We-Fu Chang, NTHU p. 4/23

5 Let s try Coleman-Weinberg Why?: 1) Less free parameters 2) provides a naturally light scalar. Following Gildener and Weinberg PRD13,3333), let s consider a classically scale-invariant theory to start up: V 0 Φ, φ s ) = λ s φ sφ s ) 2 + λφ Φ) 2 + 2κ ) Φ Φ φ sφ s ) where κ = λλ s. One can normalize the scalar fields by ρ, the radial distance from the origin of the field space, V 0 N, ρ) = ρ4 4 λn4 1 + λ sn κN2 1 N2 2 ) = ρ4 4 λn 2 1 λ 2 N 2 2 ) We-Fu Chang, NTHU p. 5/23

6 At 1-loop level, along the flat direction, it receives a correction: V 1L = A ρ 4 + B ρ 4 log ρ2 A = B = 1 64π 2 v 4 Λ 2 W { 6m 4 W log m2 W v 2 + 3m4 Z 1 log m2 Z 1 v 2 + 3m 4 Z 2 log m2 Z 2 v 2 } +m 4 H, 0 log m2 H, 0 v 2 12m 4 t log m2 t v π 2 v 4 6m 4 W + 3m4 Z 1 + 3m 4 Z 2 + m 4 H, 0 12m4 t For v = 300 GeV, m H0 = 130 GeV, M Z2 = 250GeV, A and B The quantum correction curves up the flat direction a little bit. It picks up a ρ and gives small mass to scalon, shadow Higgs. ) We-Fu Chang, NTHU p. 6/23

7 2 remarks The extra local U1) gauge helps evade the light top problem in usual Coleman-Weinberg for Standard Model You actually have two ways to identify the SM Higgs. Foot, Kobakhidze, McDonald and Volkas: PRD76,075014, arxiv: , PLB665,156. global U1), SM Higgs is the pseudo-goldstone of broken scale inv. Another one is a very heavy one.) We-Fu Chang, NTHU p. 7/23

8 Shadow Higgs, Scalon After spontaneous breaking of conformal symmetry by the CW, in Unitary gauge, ) Φ = n 1 v + h, φ s = 1 2 n 2 v + s) we have defined r λ/λ s, n 1 = 1/ 1 + r, n 2 = n/ 1 + r Tree level vector boson mass: M 2 W = g 2 W v2 r/4, M 2 Z s = rv 2 rg 2 s/4, and v r = 246 GeV = n 1 v At tree level m 2 h,0 = 2 λλ s v , m 2 s = 0 + 8Bv 2 and it relates to physical mass basis: h s ) = n 1 n 2 n 2 n 1 ) H 1 H 2 ) H 2 is identified as the SM Higgs, and one gets a light scalon H 1, the corresponding pseudo-goldstone boson to the broken scale invariant symmetry. We-Fu Chang, NTHU p. 8/23

9 Shadow Higgs mass Shadow Higgs mass depends on few independent parameters: m 2 H 1 = 3v 2 r 64π r) [ 3g 4 W 2 ] + gy 2 g2 W + g4 Y 2 + g4 sr 2 + v2 r 2 2π r) κ2 3m 4 t 2π 2 v 2 r1 + r) where κ = λλ s m H1 GeV) m Zs = 250 GeV) g s Shadow Higgs mass as function of g s for M H2 = 200GeV and various M Zs. We-Fu Chang, NTHU p. 9/23

10 Scalar Potential in mass basis In the mass basis, the scalar potential looks like: V 0 = m2 H, 0 2 H 2 2 λ ) m H, 0 H2 3 + λ r 4 κ H 2 1H 2 2 2κ 1 + r v r H 1 H 2 2 λ κ 1 1 ) 2 H2 4 r 1 1 ) H 1 H2 3 r Note that κ < 0 since λ, λ s > 0. The Feynman rules in the scalar sector of our model can be readily read from it. Notice the quartic terms contain no more than two scalon H 1 ) fields, and in cubic terms no more than one. This is a general feature of the Gildener-Weinberg1976) framework that follows from the stationary point condition We-Fu Chang, NTHU p. 10/23

11 How to see it? Let s exam a general form of the potential in terms of ˆθ and ˆr. V = ˆθ 4 + a 1ˆθ3ˆr + a 2ˆθ2ˆr 2 + a 3ˆθˆr 3 + a 4ˆr 4 and do a rotation to adjust the flat direction such that ˆθ = 0 and ˆr = v 0 V ˆr V ˆθ = a 1ˆθ3 + 2a 2ˆθ2ˆr + 3a 3ˆθˆr 2 + 4a 4ˆr 3 = 0 a 4 = 0 = 4ˆθ 3 + 3a 1ˆθ2ˆr + 2a 2ˆθˆr 2 + a 3ˆr 3 = 0 a 3 = 0 V = ˆθ 4 + a 1ˆθ3ˆr + a 2ˆθ2ˆr 2 Now, we rotate it back to the original direction and do the usual small perturbation around the minimum, Therefore, it s easy to see why the potential can t have a H 2 H 2 1 coupling. We-Fu Chang, NTHU p. 11/23

12 Higgs direct search at LEP From previous page, we know how both Higgs couple to SM fields. ξ 2 1 = gh1 ZZ ghzz SM ) 2 = r, ξ2 2 = gh2 ZZ ghzz SM ) 2 = r 1 + r, It s easy to escape from LEP direct search. 95% CL limit on ξ a) LEP s = GeV Observed Expected for background m H GeV/c 2 ) We-Fu Chang, NTHU p. 12/23

13 Search in B Meson decay? For m H1 1 GeV, the most stringent constraint comes from the B meson decays. ΓB H 1 X) ΓB eνx) r mt M W ) 4 1 m2 H 1 m 2 b ) 2 Vts V tb V cb 2, Taking BrB eνx) = and m H1 m b, we get BrB H 1 X) r From PDG, we have BrB µ + µ X) < Suppose the shadow Higgs decays only into µ + µ, then a rather) conservative lower bound on r is given by r > m2 H 1 m 2 b ) 2, for m H1 m b) Given this bound, the quarkonium decays branching ratios BrJ/Ψ H 1 γ) < 10 9 and BrΥ H 1 γ) = /1 + r) 10 8, which involve tree-level processes, become insignificant in comparison with BrB µ + µ X), which involves a one-loop process. We-Fu Chang, NTHU p. 13/23

14 Extremely Light Shadow Higgs? < 2m e ) It decays almost completely into two photons. The corresponding effective interaction can be derived by summing up the contributions from having the t-quark and the W-boson running in the loop, L = g H 1 γγ 4 F µν F µν H 1, g H1 γγ = 7α 3πv r 1 + r) GeV) r, where α is the fine-structure constant. In this case, its life time is: ) 68 kev 3 ) 0.1eV 3 τ H1 1 + r) sec 1 + r) yr m H1 m H1 CAST Collaboration: gh1 γγ < GeV 1, or r > Stellar energy lost via eγ eh1 put another bound for even lighter shadow Higgs, r > for MH1 < 0.1r 1/3 ev, 2.15, 215) ev for r = 10 4, ) its life time is cosmologically interestingwithout a Z 2 parity). We-Fu Chang, NTHU p. 14/23

15 See it at LHC? There is no tree-level H2 2H 1 decay, also we have checked that H 2 H 1 f f doesn t help at all. The only hope left is Top Physics! t W k ր W t t q b P b q l H 1 P H 1 l k ց W a) b) This decay width is evaluated to be Γt H 1 b W + ) {16.85, 4.78, 0.221} 2GF 256π 3 m 3 t 1 + r {18, 5, 0.2} r GeV for m H1 = 0.001, 1, 30 GeV respectively. This is to be compared with the SM top decay width 1.37 GeV For Mb, r > 10 4 ; For < 2m e, astrophysical constraints push it to higher values; Too heavy, suppressed by phase space. Works only for M H1 a few tens of GeV. We-Fu Chang, NTHU p. 15/23

16 Conclusion There are two renormalizable ways for us to probe the hidden U1) gauged sector. It s possible to see an extra Z boson with a very narrow width at LHC. With ILC, it can be distinguished from other Z-prime model. In a special case where our model is classically scale invariant, a light shadow Higgs can be generated from the SSB of the scale-invariant through Coleman-Weinberg mechanism. It s viable under the current experimental limits, and a shadow Higgs with a mass a few tens of GeV can be searched for at LHC in the t H 1 bw decay. When the shadow Higgs is very light, it could pay some contribution to the dark matter relic density. We-Fu Chang, NTHU p. 16/23

17 A simple hidden U1) model 2 Typically, for a visible extra Z this mixing term is expected to be only induced at the loop level. and thus ǫ 1 is generally assumed. This is not necessary the case here nor in brane world models. Hence, we shall leave ǫ as a free parameter to be constrained by experiments in particular the electroweak precision tests EWPTs). The kinetic terms including the mixing 1 4 B2 1 4 X2 ǫ 2 BX can be recast into canonical form through a GL2) transformation. Explicitly, this is given by ) ) ) X c ǫ 0 X = B s ǫ 1 B, where s ǫ ǫ 1 ǫ 2, c ǫ 1 1 ǫ 2. Note they are not the trigonometric sine and cosine. The above works for any ǫ 1. We-Fu Chang, NTHU p. 17/23

18 ElectroWeak precision test - part 1 Quantity ) ) Exp exp 1 SM s Model 1 SM Γ Z ) s 2 η s ηc η s ǫ σ had ) s η c η s ǫ Γhad) ) s 2 η s η c η s ǫ Γinv) ) s 2 η s η c η s ǫ Γl l) ) s 2 η s η c η s ǫ R e ) s η c η s ǫ R µ ) s η c η s ǫ R τ ) s η c η s ǫ R b ) s η c η s ǫ R c ) s η c η s ǫ A 0,e) FB ) 38.67s η c η s ǫ A 0,µ) FB ) 38.67s η c η s ǫ A 0,τ) FB ) 38.67s η c η s ǫ A 0,b) FB ) 19.59s η c η s ǫ A 0,c) FB ) 21.24s η c η s ǫ A 0,s) FB ) 19.59s η c η s ǫ We-Fu Chang, NTHU p. 18/23

19 ElectroWeak precision test - part 2 Quantity ) ) Exp exp 1 SM s Model 1 SM A e ) s η c η s ǫ ) s η c η s ǫ ) s η c η s ǫ A µ ) s η c η s ǫ A τ ) s η c η s ǫ ) s η c η s ǫ A b ) 0.251s η c η s ǫ A c ) 1.909s η c η s ǫ A s ) 0.251s η c η s ǫ NuTeV gl N) ) 2s 2 η s η c η s ǫ gr N) ) 2s 2 η s η c η s ǫ SLAC Møller sin 2 θ eff ) 0.019s 2 η s η c η s ǫ Q W Cs 133 ) ) s 2 η c η s η s ǫ Q W Tl 205 ) ) s 2 η c ηs η s ǫ We-Fu Chang, NTHU p. 19/23

20 LHC signal Maximal Event number l l χ 2 /χ SM 2 ) = uū t t W + W Zh 0 1 c α = 1) Maximal Event number l l χ 2 /χ SM 2 ) = uū t t W + W Zh 0 1 c α = 1) M Zs TeV) M Zs TeV) The maximal expected number of Z s events at the LHC for integrated luminosity of 100fb 1 For a fixed M Zs, we have used the largest allowed s ǫ comes from the global fit studied in previous section. The left and right panes are for 0%EWPT and 1%EWPT respectively. We-Fu Chang, NTHU p. 20/23

21 ILC signal-1 σ[pb] σpb) q q µµ 1 t t µµ sgev) 1 M Zs = 500 GeV s ǫ = t t s TeV) The cross section for e + e f f with a 500 GeV Z s and s ǫ = In the main frame, the spike tips have been chopped to avoid overlapping among curves.) We-Fu Chang, NTHU p. 21/23

22 ILC signal-2 Be + e q q)/be + e µ + µ ) SM GeV) M Zs = 500 GeV s ǫ = s TeV) The ratio of e + e q t q q over e+ e µ + µ with a 500 GeV Z s and s ǫ = We-Fu Chang, NTHU p. 22/23

23 ILC signal A µ FB SM Shadow M Zs = 500 GeV s ǫ = stev) The forward-backward asymmetry of muon v.s. s with a 500 GeV Z s and s ǫ = We-Fu Chang, NTHU p. 23/23