Higgs Parti le Mass in Cosmology A.B. Arbuzov, L.A. Glinka, and V.N. Pervushin Joint Institute for Nu lear Resear h, 141980 Dubna, Russia (Dated: February 1, 2008) A version of the Standard Model is onsidered, where the ele troweak symmetry breaking is hφi provided by osmologi al initial data given for the zeroth Fourier harmoni of the Higgs (cid:28)eld . Theinitialdatasymmetrybreakingme hanismremovestheHiggs(cid:28)eld ontributiontotheva uum energydensity,possible reationofmonopoles,andta hionbehaviorathighenergies,ifoneimposes Higgs(hφi) = 0 8 an (cid:16)inertial(cid:17) ondition on the Higgs potential V . The requirement of zero radiative 0 orre tionstothisinertial ondition oin ides withthelimitingpointoftheva uumstabilityinthe 20 SmtaasnsdoafrdthMeoHdieglg.sTbhoesolnatttoerbteoginetthheerrwainthgeth1e14dan=d0the se ondary one → E with the va uum postulate | give us the tra(cid:30) rules in the (cid:28)eld spa e of events P 0, a a loga I loga I ≥ ≤ (29) η 0 P = loga U and the arrow of time ≥ is given by Eq. (12) for both values of the energy ±E [23℄. Thus, the time arrowproblemissolvedby both the primaryquantizationof the energy onstraint(23) and the se ondary one in the spirit of QFT anomalies arising with the onstru tion of va uum as a state with η 0 minimal energy [23℄. One an say that the arrowof time ≥ is the eviden e of the quantum nature of our Universe. As it was dis ussed yet by Friedmann more than 80 years ago [16℄ with a referen e to the Weyl idea of the onformal symmetry [24℄, the Einstein General Relativity (6) admits two types of osmologi al variables and oordinates that an be identi(cid:28)ed with observable quantities. These two types are marked F,ds F,ds (F,ds) on theleft andrighthandsidesof(6)as and . Nowboth thesevariablesthe standard, , (F,ds) and onformal, are well-known in urrenteliteerature [25℄ as two di(cid:27)erent types of Cosmology: T = T /a(t) R = the Standard Cosemoelogy (SC) with a hot temperature SC 0 , expanded distan es SC ra(t) m = m SC 0 , and onstant masses , and the Conformal Cosmology (CC) with onstant onformal T = T R = r m = m a(η) CC 0 CC CC 0 temperature , oordinate distan es , and running masses de(cid:28)ned by φ = a φ R,t h i h i, respe tively [26, 27, 28℄. Standard variables are used as a mathemati al tool to solve e 5 Ψ(k)(η,r) the S hr(cid:4)odinger wave equation A with the running mass and size. It gives equidistant spe trum i(d/dη)Ψ(n)(η,r) = [α2m /(2n2)]Ψ(n)(η,r) − A 0 e A for any wave lengths of osmi photons remembering the size of theeatom at the moment of teheir emission [18℄. Inthe(cid:28)rst ase(SC)wehavethetemperaturehistoryoftheUniverse;whereasinse ond ase(CC),we have the mass evolution, where the onstant old Early Universe looks like the hot one for any parti les be ause their masses are disappearing. The best (cid:28)t to 186 high-redshift Type Ia supernovae and SN1997(cid:27) data [29, 30℄ requires osmologi al Ω = 0.7 Ω = 0.3 Λ ColdDarkMatter onstants and in the ase of the osmologi al evolution of lengths (SC). In the ase of the osmologi al evolution of masses (CC) these data are onsistent with the rigid state Ω 0.85 0.10 rigid regime of inertial motion ≈ ± . In both the ases the Friedmann equation takes the same form ρ(a) H2[Ω +a2Ω +a3Ω +a6Ω ]=a′2, ≡ 0 rigid radiation M Λ (30) ρ(a) H 0 where is the onformal density and is the Hubble parameterin units (1). In ontrast to the SC, Ω ColdDarkMatter the (cid:28)t in the CC almost does not depend on the value [26, 27, 28℄. Ω 4 10−2 b Cal ulation of the primordial helium abundan e [11, 27℄ takes into a ount ≃ · , weak e△m/T 1/6 m intera tions,theBoltzmannfa tor,(n/p) ∼ ,where△ istheneutron-protonmassdi(cid:27)eren e, m /T = m /T = (1+z)−1m /T SC SC CC CC 0 0 whi h is the same for both SC and CC, △ △ , and the square (1+z)−1 √t measurable root dependen e of the z-fa tor on the measurable time-interval ∼ (see Eq. (27)). Thus, in CC the rigid state regime initiated by the inertial evolution of the s alar (cid:28)eld zeroth modes without any potentials is the dominant regime for all epo h in luding the va uum reation of parti les. COSMOLOGICAL CREATION OF SM PARTICLES Re allthatinQFTobservableparti lesareidenti(cid:28)edwithholomorphi representationofthe onformal (cid:28)eld variables 1 ei 2π F(η, )= c (a,ω ) kx F+(η)+F−(η) , = , e x V0 l,Xl26=0 F F,l p2ωF,l (cid:2) l −l (cid:3) k V01/3l (31) 2 ds c (a,ω ) F F, in the (cid:29)at spa e-time . Here l is the normalized fa tor that provides the free parti le Hamiltonian e A F (am F)= n + ω (a) free F0 F, F, H | F,X,26=0(cid:20) l 2 (cid:21) l (32) e ll n =F+F− F, intheforωmof(ath)e=sumko2v+ermm2omae2ntaofprodu tsofo upationnumbers2 =l0 l -l andtheone-parti le energies F,l p l T F0 [33, 34, 35℄. The zeroth harmoni l in the sum (32) is ex luded be ause the transverse ( ) ve tor and tensor (cid:28)elds are onstru ted by means of the inverse Beltrami- Fd3x = 0 Lapla e operator a ting in the lass of fun tions of nonzero harmoni s with the onstraint . AFR= +1 The free parti le Hamiltonian ontains the Casimir energies [31℄, positive for bosons and A = 1 F negative for fermions − , vanishing in the large volume limit. The similar transformation (31) of the linear di(cid:27)erential form i i Z d3xXF PFe∂0F = 2XF, (cid:0)F−+l∂0Fl−−Fl−∂0F−+l(cid:1)+ 2XF, (cid:0)F−+lFl+−Fl−F−−l(cid:1)∂0△F (33) e l l e in a tion (8) is not anoni al. Therefore, the transition from (cid:28)eld variables to the observable quantities ( onformalo upationnumberandone-parti leenergy)hasphysi al onsequen esinthelinearform(33). They are sour es of reation of pairs from the stable va uum: a △F=vT,f = log√ωF, △F=v|| =log√ωF; (34) e e △F=h,q = loga√ωF, △F=Q,hTT =loga; (35) e e 6 v = v|| +vT W Z f hTT h here are (cid:28)elds of and ve tor bosons, are fermions, is graviton, is a massive s alar (Higgs) parti le (see the massive ve tor theory in detail in [22, 32℄). log√ωf The equation (6) shows us that the onformal fermion sour e (34) di(cid:27)ers from the standard (3/2)loga one by the term whi h an lead in SC to intensive reation of massless fermions forbidden by observational data and general theorem of (cid:28)eld theory [35℄. In omparisonwiththe lassi al(cid:28)eld theorby−w0ith>a=rb0itraryo bu−pationnumbers onsideredbefore,the new element of QFT is the stable va uum F,l| , where F,l is the operatorFo+f a=nnαibh+ila+tioβn∗bo−f a quasi-parti le de(cid:28)ned by the Bogoliubov transformation of the operator of parti ∂leb±l = ωFb,l± F,l, sωothattheequationsofmotionoftheBogoliubovquasi-parti lebe omediagonal η F,l ± b F,l,where b is the quasi-parti le energy [33, 34℄. A ording to these formulae (33) (cid:21) (35) massless parti les, photons and neutrinos, annot be reated in homogeneous Universe (see [33℄). There is an estimate in [33℄ that fermions and transverse ve tor bosons (34) are not su(cid:30) ient, in order des ribe the present-day ontent of the Universe. The reation of gravitons is suppressed by the isotropization pro esses dis ussed in [33℄. It was shown [22℄ that just the W Z (Ω ) radiation longitudinal , ve torbosonsarethe best andidatesin SMto form theradiation and the (Ω ) b baryon matter ontributions to the Universe energy budget in the Conformal Cosmologi al model. The Higgs parti le reation is similar to the one of the longitudinal omponents of the ve tor bosons ( ompare (35) and (34)). The reationofve torbosonsstartedatthemoment,whentheirwavelength oin idedwiththehorizon M−1 = (a M )−1 = H−1 = a2(H )−1 length v v 0W v v 0 . This follows from the un ertainty prin iple that gives the instan e of reation of primordial parti les H a3 = 0 27 10−45 =(3 10−15)3 a 3 10−15 =(1+z )−1. v M ≃ · · → v ≃ · v (36) 0W As it was shown in [36℄ using the s alar (cid:28)eld model that taking into a ount intera tions ∂ v±( ,η)= iω v±( ,η)+∂ (η)v±( ,η)+i[H ,v±( ,η)] η k ± v k η△v|| k int k (37) an lead to the ollision integral and the Boltzmann-type distribution. As a model of su h a statisti al system, a degenerate Bos−e1-Einstein gas was onsidered in [22℄, whose distribution fun tion has the form exp ωv(η)−Mv(η) 1 T n h kBTv i− o where v is the boson temperature treated as the measurable parameter of the parti le distribution fun tion in the kineti equation with the ollision integral. The value of the ve torboson temperature dire tly follows from the analysis of the numeri al al ula- n(T ) T3 tionsin [22℄, fromthe dominan e ofloηngitu=di[nna(Tl v)eσ torb]o−s1onswith highmomenta v ∼ v and from v the fa t that the relaxation time [37℄ rel s att isn(eTqu)al toTt3he invσerse Hubb1/leMp2arameter, if initial data (36) is hosenT. In t(hMe 2aHse)o1/f3re=lat(iMvis2ti Hbo)1s/o3ns 3 v ∼ v and s att ∼ v the ve tor boson temperature value v ∼ v v 0W 0 ∼ K, is lose to the observed temperature of the osmi mi rowave ba kground radiation. So the temperature arises in this ase after reation of parti les and it is des ribed in the usual way [36℄. Note that the massesof those parti les is provided by the standard me hanism of the absorbtion of the extra Higgs (cid:28)eld omponents. The latter happens due η =0 to the nonzero Higgs (cid:28)eld va uum expe tation value, whi h already existed at the initial moment , when there were no any parti les and hen e no temperature. Ω M2 a−2 = In this way CMB inherits the primordial ve tor boson temperature and density, rad ≃ W · I 10−341029 10−5 ∼ . In the early epo h with the dominant abundan e of weak bosons (due to the Bogoliubov ondensation), their Bell-Ja kiw-Adler triangle anomaly and the SM CKM mixing in the environment of the Universe evolution lead to the non- onservation of the sum onf lepton and baryon b X 10−9 CP numbers and to the baryon-antibaryon asymmetry of matter in the Universe n ∼ ∼ . γ The present-day baryon density is al ulated by the evolution of the baryon density from the early Ωstage,1w0−h3e4n10it−9w1a0s43d(aire/ atly)3relaαted =toαthe p/hositno2nθdensit0y..03So that its presen(ta(cid:21)d/aay v)a3lueαis equal to b v L W QED W v L W ≃ ≃ ≃ , where the fa tor ≃ arise as a retardation aused by the life-time of the W-boson [22℄. Q Thus we gave a set of argument in favor of that the GR and SM a ompanied by a s alar (cid:28)eld an des ribe osmologi al reation of the Universe with its matter ontent <0 ˆ [P,F]0> = 4V2a2[a′2 ρ(a)]=0, |CU | 0 − (38) e 7 Ω = 0.85 0.10 Ω 4 10−5 rigid radiation in agreement with the observational data in (30), where ± , ≃ · , and Ω 3 10−2 b ≃ · , if observables (one-parti le energy, o upation number, temperature, distan e, time, et .) Ω 0.3 CDM are identi(cid:28)ed with onformal variables with inertial initial data [22, 28℄. ≃ an be onsidered Q as the input parameter for (cid:28)tting -parti le potential parameters. In order to pose the problem of a more a urate al ulation that an be done in this model in future, one needs to establish the parameters of the Higgs potential. It is the topi of the next se tion. HIGGS FIELD CONTRIBUTION TO ENERGY DENSITY ThenonzeroaverageoftheHiggs(cid:28)eldgivenbytheinitial onditionsprovidestheele troweaksymmetry breaking required by SM. In the Standard Model embedded in the osmology equations (24), the values φ =M /g I W W of the initial data (15) are dire tly de(cid:28)ned by the other parameters of the model: . On the lassi al level the introdu tion of the initial data for the Higgs (cid:28)eld allows us to onsider the c φ 0 situation, when the parameter ≡h i in the Higgs potential, so that (φ)=λ φ2 φ 2 2, ( φ ) 0. Higgs Higgs V −h i V h i ≡ (39) (cid:2) (cid:3) Inthiswaywe anremove ontributionoftheHiggs(cid:28)eldzerothharmoni intotheenergydensitytogether with possible reation of monopoles and ta hions. In the perturbation theory loop diagrams lead to the Coleman(cid:21)Weinberg potential [38℄, whi h an substantially modify the initial lassi potential leading to the (cid:28)ne tuning problem in the Standard Model. Contrary to the ase of the SM, loop orre tions an not shift the position of the minimum in (39) be ause of the symmetry in the potential. In our ase the ondition ( φ )=0 eff V h i (40) is the natural onstraint of the unit va uum-va uum transition amplitude at the point of the potential extremum: ( φ )= iTrlog(<00>[ φ ]), <00>[ φ ]=1= ( φ )=0. eff eff V h i − | h i | h i ⇒V h i (41) Inotherwords,the onditionismotivated by the prin iple ofminimization of theva uum energyand by the very de(cid:28)nition of the lassi al potential. φ= φ So we should havethe zero value of the Coleman(cid:21)Weinbergpotential and of its derivative for h i. These onditions orrespond to the va uum stability boundary in the Standard Model as dis ussed in Ref. [39℄. The boundary has been extensively studied in the literature (see review [40℄ and referen es therein). The orrespondingequation anberesolvedwithrespe ttotheHiggsmass,whi hthandepends Z W on the masses of top-quark, and bosons, on the EW oupling onstants, and on the value of the Λ ut-o(cid:27) parameter , whi h regularizes divergent loop integrals. The modern studies [41, 42, 43℄ whi h in lude omplete one-loop with a ertain resummation for the running masses and oupling onstants and the dominant two-loop EW ontributions. They are in a reasonableagreement with ea h other and Λ=1 give in SM the following range of the lowerHiggs mass limit. For TeV, the improved lowerbound reads [43℄: α (M ) 0.118 m [GeV]>mbound. =52+0.64(m (GeV) 175) 0.5 s Z − . h h t − − 0.006 (42) Λ 1019 m > mbound. 134 For very high values of the ut-o(cid:27) → GeV one gets h h ≈ GeV. These values mbound. h in the Standard Model orrespond to the limiting ase, where the model breaks down. On the m h ontrary, in our ase these values are just our predi tions for : 52 GeV < m <134 GeV. h ∼ ∼ (43) Numerous expemrim(SenMtafiltd)a=ta1i2n9d+ir7e4 GtlyeVsupport the existen e of a SM-like Hmigg>s p1a1r4t.i4 le of a relatively lowmass[44℄: h −49 with thedire texperimentallimit h GeVatthe95% CL [45℄. 8 Soone anseethattheStandardModeldeservesnewphysi s ontributionsparameterizedbythe ut-o(cid:27) 100 not lower than at a rather high energy s ale ∼ TeV. The domain of Higgs masses below 134 GeV (and higher) will be studied soon experimentally at the b LargeHardonCollider(LHC).Higgsbosonswithsu hmassesde aymainlyintopairsof -quarks[2, 40℄. m h As on ernstheprodu tionme hanism,forthegivenrangeof thesub-pro esswithgluon-gluonfusion dominates[46℄andthe orresponding rossse tionsprovideagoodpossibilitytodis overtheHiggsboson at the high-luminosity LHC ma hine. RealHiggsparti les reatedintheEarlyUniversewereimportantfortheenergybudgetoftheUniverse as des ribed above. The present-day ontribution of Higgs parti les is vanishing, sin e the produ tion rate des ribed by Eq. (37) is suppressed for the present-day value of the Hubble parameter. The initial data s enario removesthe in(cid:28)nite potential va uum energy density, reation of monopoles, and ta hion behavior at high energies, be ause the Higgs potential has form (39) whi h an be ast as h h2 λ h4 g2 m2 V φ= φ + =m2 + m h3+λ , λ= W h 0.2 0.3. Higss(cid:18) h i √2(cid:19) h 2 r2 h 4 4 M2 ∼ ÷ (44) W CONCLUSION TheHiggse(cid:27)e twasstudiedinthe osmologi almodelfollowingfromtheemergingGR&SMa tion(8) Q supplementedby the additional (cid:28)eldunder the assumptionof thepotential-free(inertial) zerothmode ( φ )=0, ( Q )=0 Higgs U dynami sofboths alar(cid:28)eldsV h i V h i . Sothatthepotentialva uumenergydensity, possible reationofmonopoles,ata hionbehaviorathighenergiesareex ludedfromtheverybeginning. The spontaneoussymmetrybreaking anbe providedby initialdataofthezerothharmoni ofthes alar φ = M /g , φ ′ = 0 W W Higgs (cid:28)eld h i h i without its ontribution to the energy density. The latter an be P = 2V H Ω hQi 0 0 rigid formed by an inertial motion of the zeroth harmoni of an additional s alar (cid:28)eld . p Intheneighborhoodofthepointof osmologi alsingularity,thismotion orrespondstothemostsingular primaryenergeti regimeoftherigidstate. The resear hofthe onstraint-shelldynami s in terms ofthe onformalvariablesshowsusthatatthepointof osmologi alsingularitythereisnoanyphysi alsour es of the in(cid:29)ation me hanism. a = 0 In the limit of osmologi al singularity , GR and SM ontain the pro ess of va uum parti le reation. This va uum parti le reation is des ribed as the Bogoliubov va uum expe tation value of the energy onstraint operator. The estimation of this va uum expe tation value is in agreement with the observational data, if observable quantities are identi(cid:28)ed with the onformal variables [22℄. These variables are distinguished by both the observational Cosmology and parti le reation tool. This Con- formal Cosmology is not ex luded by modern observational data in luding hemi al evolution and SN Ω 1 rigid data [47, 48℄, if at all these epo hs the primordial rigid state dominates ∼ . p InthenewInertials enariotheCMB onformaltemperatureispredi tedbythe ollisionintegralkineti W Z equationoflongitudinalve torbosons and togetherwiththeHiggsparti les. Thetemperaturearises as the onsequen e of the primordial parti le ollisions after their reations in the old Universe (cid:28)lled in Q by the zeroth harmoni energy density. In order to pose a problem of more a urate al ulation that an be done in this model in future, we establishedtheparametersoftheHiggspotentialthatfollowfromtheLEP/SLC experimentaldata. The present (cid:28)t of the LEP/SLC experimental data indire tly supports rather low values of the Higgs mass, 114

Higgs Particle Mass in Cosmology

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