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IFT-UAM/CSIC-07-01, UAB-FT/623 Novel Effects in Electroweak Breaking from a Hidden Sector José Ramoń Espinosa1 and Mariano Quiro´s2 1IFT-UAM/CSIC, Cantoblanco, 28049 Madrid, SPAIN 2 ICREA/IFAE, UAB 08193-Bellaterra Barcelona, SPAIN (Dated: February 2, 2008) The Higgs boson offers a unique window to hidden sector fields Si, singlets under the Standard 2 2 Model gauge group, via the renormalizable interactions |H| Si. We prove that such interactions can provide new patterns for electroweak breaking, including radiative breaking by dimensional transmutationconsistentwithLEPbounds,andtriggerthestrongenoughfirstorderphasetransition required byelectroweak baryogenesis. PACSnumbers: 11:30.Qc,12.60.Fr 7 0 1. Introduction. The Standard Model (SM) of elec- where α = S,Z,W,t,h,G for singlet hidden sector 0 troweak and strong interactions can not be considered fields, gauge {bosons, top, Hi}ggs and Goldstones respec- 2 as a fundamental theory, since it fails to provide an an- tively, with N = N,3,6, 12,1,3 . Inspired by the α { − } n swer to many open questions (the hierarchy, cosmologi- case of stops, we choose N =12 for our numerical work. a cal constant and flavor problems, the origin of baryons, Next,C =3/2forfermionsorscalarsand5/6forgauge α J the Dark Matter and Dark Energyof the Universe,...), bosons, and the h-dependent masses are M2 = ζ2h2, S 8 but ratherasaneffective theorywith aphysicalcutoff Λ M2 = (g2 + g′2)h2/4, M2 = g2h2/4, M2 = h2h2/2, 1 that mostlikely shallbe probedatthe LHC experiment. MZ2 = 3λh2 +m2, M2 =Wλh2 +m2. Thetrenormtaliza- h G Many SM extensions, e.g. string theory, contain hidden tion scale Q enters explicitly in the one-loop logarith- 1 sectors with a matter content transforming non-trivially mic correctionand implicitly through the dependence of v 5 under a hidden sector gauge groupbut singlet under the all couplings and fields on t = lnQ in such a way that 4 SM gauge group. It has recently been noticed that the dV/dt=0issatisfied. Fornowwesimplychoosethescale 1 SM Higgs field H plays a very special role with respect as Q = M (v) and fix the parameters (at that scale) to t 1 to such hidden sector since it can provide a window (a get h =v 246 GeV. 0 portal [1]) into it throughthe renormalizableinteraction Fohriζ2 <h≃2/2 0.65theone-looptermin(2)isdomi- 7 H 2S2 where the bosons S are SM singlets. natedbythetstan≃dardtopcontributionbutforζ2 >h2/2 0 | | i i t / This coupling to the hidden sector can have im- hidden scalars start to dominate. The structure of the h portant implications both theoretically and for LHC effectivepotentialisbestdescribedbyusingFig.1. Con- p phenomenology as has been discussed in recent litera- sider first the (ζ,λ)-plane in the upper plot. Besides the - p ture [1, 2, 3, 4, 5, 6, 7, 8]. In this letter we show that lines of constant Mh, we can distinguish four regions. i) e thepresenceofahiddensectormayhavedramaticconse- The region below the blue line [defined by V′′(v) = 0] h quencesforelectroweaksymmetrybreaking(inparticular is forbidden: there M2 < 0. The extremal at h = v is : h v it enables new patterns of electroweak symmetry break- a maximum that degenerates into an inflection point on i ing, including radiative breaking by dimensional trans- the blue line. ii) In the region above the blue line but X mutation consistent with present LEP bounds on the below the red line there is an electroweakminimum, but r Higgs mass) and for electroweak baryogenesis (it makes itisafalseminimumwithrespecttothe(true)minimum a easy to get a first order phase transition as strong as re- at the origin. The red line is defined by V(v) = V(0), quired for electroweak baryogenesis). Furthermore, un- i.e. both minima, at the origin and at h=v, are degen- dermildassumptionsthosehiddensectorfieldsarestable erate on that line. This regionii) is therefore unphysical and can constitute the Dark Matter of the Universe. without a mechanism to populate the metastable mini- 2. Electroweak breaking. We will consider a set mum (in general, the true minimum at the origin would of N fields S coupled to the SM Higgs doublet by the be preferred at high temperature and the electroweak i (tree-level) potential transition would never take place). iii) In the region above the red line but below the green line [defined by V0 = m2H†H +λ(H†H)2+ζ2H†H Si2. (1) V′′(0)=0] the electroweakminimum is stable and there Xi is a barrier separating the false minimum at the origin from the electroweak minimum (m2 >0). This region is We will assume for the moment that the fields S are i very interesting for two reasons: massless so they only will get a mass from electroweak breaking. In the background Higgs field configuration Thebarrierbetweenbothminima(atzerotempera- definedby H0 =h/√2,the one-loopeffectivepotential • h i ture) will produce an overcoolingof the Higgs field (in Landau gauge and MS scheme) is given by at the origin at finite temperature, strengthening m2 λ N M4 M2 the first order phase transition (see below). V = h2+ h4+ α α ln α C , (2) 2 4 64π2 (cid:20) Q2 − α(cid:21) Electroweak symmetry breaking is not associated Xα • 2 0.1 0.2 100 200 0.1 ~λ 0.05 ^λ M h= 50 GeV 4V) 0 e G λ 0 00 1 b h)/(-0.1 r V( -0.05 g -0.2 -0.1 -0.3 0 0.5 ζ 1 1.5 0 1 2 3 4 5 h/100 GeV FIG. 2: Green: Effective potential for the conformal case. Black: runningλ˜ and λˆ, with Q=Mt(h). 0.4 b barrierbetweentheoriginandtheelectroweakminimum 0.2 while for the red potential the two minima become de- 4eV) r generate. The next line corresponds to the potential for G 100 0 λ = −0.04 where the electroweak minimum is already h)/( g a false minimum, which becomes an inflection point at V( the blue line where Mh = 0. Finally the highest line -0.2 corresponds to λ = 0.08 and the electroweak extremal − is a maximum (the potential has a minimum somewhere else, for some h >v. If ζ2 were smaller, ζ2 < h2/2, the -0.40 1 2 3 4 potential woulhd iinstead be destabilized due t∼o λt<0.). h/100 GeV In order to have a better understanding of the phe- FIG. 1: Upperplot: In the plane (ζ,λ), the green line corre- nomenon of radiative electroweak breaking by dimen- sponds to the condition V′′(0) = 0, the red to V(v) = V(0) sional transmutation in this setting consider the confor- and the blue to V′′(v) = 0. Black solid lines correspond to mal case with m2 =0. Then improve the one-loop effec- theindicatedvaluesofMh. Lowerplot: Potential for ζ =1.0 tivepotentialofEq.(2)byincludingtherunningwiththe and different values of λ (or Mh) as marked on the vertical renormalization scale of couplings and wave functions. line in upperplot. We use for that the SM renormalizationgroupequations (RGEs) supplemented by the effects of S loops plus the i RGEsforthenewcouplingstothehiddensector(see[10] with the presence of a tachyonic mass at the ori- for details). The RGE-improved effective potential is gin, as in the SM. Instead it is triggered by radia- scale independent and we can take advantage of that to tive corrections via the mechanism of dimensional take Q = M (h) as a convenient choice to evaluate the transmutation. t potential at the field value h (with all couplings ran to The minimum at the origin becomes a maximum at the that particular renormalization scale). This results in a green line. In fact the greenline corresponds to the con- “tree-level” approximation V (1/4)λˆh4 with [11] ≃ formalcasewhere m2 =0andelectroweakbreakingpro- N κ2 κ ceeds by pure dimensional transmutation (see also [9]). λˆ λ+ α α ln α C , (3) iv) Finally, in the region above the green line the origin ≡ Xα 64π2 (cid:20) h2t − α(cid:21) is a maximum as in the SM, with m2 <0. Notice that, while λ > 0 is required in the SM case where the κα’s are coupling constants, defined by the (ζ =0 axis), nowλ<0 is accessible for sufficiently large massesasMα2 =(1/2)καh2. Thebehavioroftheone-loop ζ. The shape of the potential for the different cases is il- potentialasafunctionofhiscapturedbythe“tree-level” lustratedbythelowerplotofFig.1,whereζ =1hasbeen approximation above through the running of λˆ with the fixedandwevaryλasindicatedbytheverticallineinthe renormalization scale, linked to a running with h by the upperplotofFig.1. Frombottom-upthepotentialshave choice Q = Mt(h). To illustrate this, we show in Fig. 2 decreasing values of λ. The lowestpotential corresponds theeffectivepotentialforthisconformalcase(greenlines to λ = 0.01 and has the conventional maximum at the in Fig. 1) with m2 = 0 and ζ = 1, together with the origin. Thegreenpotentialcorrespondstotheconformal effectivequarticcouplingλˆ(h). Wecanseethatthescale case where m2 = 0 (in this particular example also λ is of dimensional transmutation is related to the scale at zero!). The next line corresponds to λ = 0.02 with a which the potential crosses through zero. The structure − 3 ofthepotentialisthendeterminedbythe evolutionofλˆ: 0.02 for smallh, λˆ <0 destabilizesthe originwhile, forlarger h, λˆ > 0 stabilizes the potential curving it upwards in 0 the usual way. We can define a different effective coupling, λ˜, by the 4V)-0.02 approximation ∂V/∂h λ˜h3, which fixes λ˜ to be given Ge tbhyro(3u)ghwiztherCoαp→recCiseαly−≃a1t/2t.hFeigm.i2nismhuowms othfatthλ˜e cprootsesnes- T)/(100 -0.04 tial. This shows then how the electroweak scale is gen- V(h,-0.06 erated by dimensional transmutation: a suitably defined -0.08 effective quartic Higgs coupling turns from positive to negative values, with v given by the implicit condition λ˜(v) = 0. Needless to say, such running of λ˜ would not -0.10 0.5 1 1.5 2 2.5 3 h/100 GeV be possible in the SM and is due to the effect of ζ in the RGEs, which counterbalances the effect of h . t 0.5 3. Electroweak phase transition. In the presence of hidden sector fields S coupled to the SM Higgs as i 0.4 in Eq. (1) the electroweak phase transition is strength- ened by: a) The thermal contribution from S , if ζ is large enough. This fact was known already [12i, 13]. b) 4V) 0.3 e The fact that, in part of the (ζ,λ)-plane, there is a bar- 00 G 0.2 rier separating the origin (energetically favored at high 1 tpeemrapteurraet.uTreh)isanedffetchteieslencetwro[w14ea].kminimumatzerotem- V(h,T)/( 0.1 To study the strength of the phase transition we con- 0 sider the effective potential at finite temperature, T. In the one-loop approximation and after resumming hard- -0.1 thermal loops for Matsubara zero modes, the thermal 0 0.5 1 1.5 2 2.5 3 h/100 GeV correction to the effective potential ∆V is given by T 2Tπ42 Xα NαZ0∞dx x2logh1−εαe−√x2+Mα2/T2i 1Ffo0Ir8G.M0.03h:a=nEdff1e12c05t5i.vG0e0epVGo.etVeUn,ptwipaieltrharpRolou≃tn:d1ζ.t3h=7e.E0L.Wo8waepnrhdpaslToet:t=rSaan1ms1i0tei.o8fon5r,, + T 1+εαN M3 M2+Π (T2) 3/2 , (4) ζ =1.365andT =50.00,40.00,30.08and0GeVwithR≃8. 12πXα 2 αn α−(cid:2) α α (cid:3) o where ε = +1( 1) for bosons (fermions) and Π (T2) upper plot we consider a case where the strength of the α α is the thermal ma−ss of the corresponding field (for more phase transition is only due to the thermal barrier from details see Ref. [10]). The considered approximation is Si fields (with ζ =0.8)with no T =0 barrier,leading to good enough for our purposes since, as we will see, the R 1.37. In the lower plot, with ζ = 1.365, the barrier ≃ phasetransitionis stronglyfirstorderandmainlydriven persists all the way down to T = 0 making the value of by the contribution to the thermal potential of the S R much larger (R 8). The dependence of R with ζ for i ≃ fields for which the thermal screening ΠS is enough to different values of Mh is displayed in Fig. 4 where the solve the infrared problem. Notice that the second term strong enhancement in the values of R produced inside inEq.(4),responsibleforthe thermalbarrier,takescare the region where the barrier between the origin and the of the thermal resummation for bosonic zero modes. electroweak minima persists at T = 0 is apparent (the We define T as the critical temperature at which the square dots mark in each case the region beyond which c origin and the non-trivial minimum at h(T ) become there is a barrier at T = 0). The answer to the general c degenerate, calling its ratio R h(T ) /hT . Thie baryo- questionofwhatistheupperboundontheHiggsmassto c c genesisconditionfornon-erasu≡rehoftheipreviouslygener- avoid baryon asymmetry washout depends on how large ated baryon asymmetry requires R > 1 [15]. In general, ζ can be, which in turn depends on the cutoff Λ. A low ∼ identifying the criticaltemperature with the realtunnel- cutoff, e.g. Λ 1 10 TeV, allows values of ζ up to ing temperature (which is smaller) underestimates R so 1.3 1.8 while∼a hig−her cutoff Λ 105 GeV would only − ∼ that our approximationprovides a conservativeestimate allow values of ζ ∼< 1. of the order parameter R. For a more detailed analysis A pending issue is how the baryon asymmetry is cre- see Ref. [10]. ated(perhaps by the hidden sector)since within the SM We illustrate inFig.3 the behaviorofthe effective po- the amount of CP violation, given by the CKM phase, tential around the critical temperature for a fixed Higgs is admittedly insufficient [16] (although a way out as- mass (M = 125 GeV) and for two typical cases. In the sociated with physics solving the flavor problem at a h 4 7 4. Conclusion. In this letter we have explored new and dramatic effects that a hidden sector, singlet under 6 M h = 100 GeV the SM gauge group, can have concerning electroweak 125 symmetry breaking and electroweak baryogenesis. Com- 5 pletelynewpatternsfortheHiggspotentialandnewways of radiative breaking by dimensional transmutation are 150 4 found, some of them indirectly leading to a very strong R EW first order phase transition. For such a strong first- 3 order phase transition the model can provide a strong 175 signature in gravitational waves [19]. Moreover if the 2 hidden sector has a global U(1) symmetry that guaran- 200 tees the stability of S -scalars (as we are assuming) and i 1 somesubsectorofithasalargeinvariantmassitcanalso 0.7 0.8 0.9 1 ζ 1.1 1.2 1.3 1.4 1.5 provide good candidates for Dark Matter [10, 20]. FIG. 4: R ≡ hh(Tc)i/Tc as a function of ζ for several values of Mh, as indicated. Acknowledgments high-scale was proposed in [17]). An interesting possi- bility from the low energy point of view is the appear- Work supported in partby CICYT, Spain, under con- anceofCP-violatingeffectiveoperators. Forinstancethe tracts FPA2004-02015 and FPA2005-02211; by a Co- dimension-six operator g2 H 2FF˜/(32π2Λ2) can gener- munidad de Madrid project (P-ESP-00346); and by | | ate the baryon-to-entropy ratio (for maximal CP viola- the European Commission under contracts MRTN-CT- tion)[18]n /s 3.1κ 10−9(T /Λ)2,whereκ 0.01 1, 2004-503369andMRTN-CT-2006-035863. 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