Decoupling of Higgs boson from the inflationary stage of Universe evolution
aa r X i v : . [ g r- q c ] M a y EPJ manuscript No. (will be inserted by the editor)
Decoupling of Higgs boson from the inflationary stage ofUniverse evolution
V.V.Kiselev , S.A.Timofeev Russian State Research Center “Institute for High Energy Physics”, Pobeda 1, Protvino, Moscow Region, 142281, Russia,e-mail:
[email protected] Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow Region, 141701, Russia
Abstract.
The constraint on the mass of Higgs field in the Standard Model at the minimal interaction withthe gravity is derived in the form of lower bound m H >
150 GeV by the strict requirement of decouplingthe Higgs boson from the inflation of early Universe: the inflation produced by the Higgs scalar couldcrucially destroy visible properties of large scale structure of Universe, while the large mass makes theHiggs particle not able to produce the inflation and shifts its cosmological role into the region of quantumgravity.
At present, in cosmology the inflation stage has becamethe commonly recognized model for the evolution of earlyUniverse [1,2,3,4,5]. In the simplest scenario consistentwith the observed anisotropy of cosmic microwave back-ground radiation (CMBR) [6,7], the supernova data [8,9,10] and large-scale structure of Universe (LSS) [11], ascalar field of inflaton should possess some specific proper-ties: an almost flat potential of self-action with the energydensity of the order of (10 GeV) .In this respect, recently the possibility of producingthe inflation stage by the Higgs boson with a non-minimalcoupling to the gravitation has been studied [12,13,14,15,16]. Then, in addition to the Einstein–Hilbert Lagrangianof gravitational field L EH = − πG R, the interaction Lagrangian has included the term of theform ξΦ † ΦR , where Φ denotes the Higgs field, R is theRicci scalar, G is the gravitational constant, and ξ is thecoupling constant. So, an appropriate conformal transfor-mation introduces an effective field minimally coupled tothe gravity, while a relevant effective potential has got aflat plateau with an altitude regulated by the parameter ξ . The realistic model suggests ξ ≫
1, and in this respect,a new dynamical scale is introduced by M Pl /ξ [17,18],where the Planck mass M Pl is defined by the gravitationalconstant G as M = 1 /G . The new scale determines thealtitude of potential plateau and produces the thresholdfor changing a regime of coupling at ultraviolet virtual-ities. In this way, some definite constraints on the massof Higgs boson have been derived. A lower bound for theHiggs boson mass is determined by the cosmology data (mainly, a slope of primary spectrum for the inhomogene-ity of matter distribution), while the upper bound is setby a reasonable weakness of the self-coupling.Similarly, a modified model of induced gravity withthe Higgs boson non-minimally coupled to the gravitywas considered in [19,20]. In that approach, the Einstein–Hilbert term with the bare gravitational constant is ex-cluded from the primary action. This formulation leadsto an essential change of cosmological dynamics due to avarying gravitational constant. In this way, the inflation-ary evolution in the model results in the fact that theobserved contrast of energy density in the Universe givesthe strict preference for extremely large values of Higgsboson mass though below the Planck mass by four or fiveorders of magnitude.As for the Higgs field nonminimally coupled to thegravity, it can successfully produce the inflation compat-ible with the observed properties of Universe. However,in that case one should introduce the additional couplingconstant of Higgs field with the gravity at a specific valueof such the parameter. Then, the mechanism of inflationcaused by the Higgs field nonminimally coupled to thegravity essentially differs from the mechanism with theHiggs field minimally coupled to the gravity, so that weconsider the minimal version in detail with no further ref-erence to the nonminimal version beyond the Introduc-tion.In this paper we consider conditions of developing theinflation stage produced by the Higgs field of StandardModel (SM) minimally coupled to the gravity , i.e. in thecase of ξ = 0, when the renormalizable potential of Higgsscalar is determined by two parameters: a vacuum expec- In principle, the coupling ξ could deviate from zero by asmall value, which can be neglected in the consideration, say,at | ξ | ≪
1. V.V.Kiselev, S.A.Timofeev: Decoupling of Higgs boson from the inflationary stage of Universe evolution tation value fixed by the Fermi constant G F in the weakinteraction, and a mass not yet measured experimentally.In theoretical models treating the inflation of Universedue to the scalar field, the mass of inflaton should get avalue at the scale of 10 GeV in order to agree with theobserved spectra of inhomogeneities in both CMBR andLSS. In contrast, the mass of Higgs boson is not greaterthan several hundreds GeV as follows from SM with ac-count of loop corrections including the Higgs boson (thecurrent status of Higgs particle physics see in review [21],while the probable fate of SM is discussed in [22]). In ad-dition, the observations particularly require an extremelysmall constant of self-action for the scalar field produc-ing the fluctuations transformed into the inhomogeneity ofmatter and anisotrophy of cosmic microwave backgroundradiation. So, we will use this argumentation in our mo-tivation to theoretically forbid the inflation produced bythe Higgs field minimally coupled to the gravity.Thus, the main conclusion could be drawn as follows:the scenario, when the Higgs boson alone generates theUniverse inflation consistent with the modern cosmologi-cal observations, is experimentally forbidden . However, theHiggs boson is an ordinary scalar field, hence, in the frame-work of classical gravitation theory, the field is able to pro-duce the inflation regime. In practice, no traces of such thespecified inflation have been yet observed. Then, we haveto conclude that in the very beginning at a sufficientlyhigh density of energy there could be, at least, two op-tions in producing the inflation of early Universe either bythe standard Higgs boson or by the special scalar, and thesecond scenario with the special field actually occurred .What was a reason for such the discrimination betweentwo possible ways of evolution? We argue for the situa-tion when the inflation originated by the special inflatonfield had got no alternative if the mass of Higgs scalar inSM exceeds a critical value , so that the inflation generatedby the Higgs boson could not develop in principle. Such thecritical value of Higgs boson mass defines the decouplingof Higgs boson from the inflation, at all.Definitely, in the framework of quantum field theoryin the curved spacetime, when the gravity is treated as aclassical theory, there is a critical value of the Hubble con-stant, which sets the end of inflation regime depending onthe parameters of inflaton. At lower values of the Hubbleconstant, a transition to a reheating of the Universe oc-curs due to generating various quanta of both the inflatonas well as other matter fields. The stage of reheating isproperly called the moment of “Big Bang”. However, thecritical value of Hubble constant and, hence, the corre-sponding energy density could be quite great, so that thegravity would not allow the classical description, i.e. quan- In the former case, one could imagine the scenario, whenthe inflation caused by Higgs boson at high density of energy,is further transformed to the regime of inflation driven by thescecific inflaton with the more flat potential at lower densityof energy. Then, one should suggest a fine tunning, since thechange of regime would be before the end of inflation gener-ated by the Higgs boson, otherwise one meets with the sameexperimental constraints mentioned above. tum fluctuations in a metric would be essential, and thetheory enters the scope beyond the validity of inflationarytheory. Thus, a border of quantum gravity in cosmologycould actually determine the decoupling of Higgs bosonfrom the inflation regime.Essential quantum fluctuations should be inevitablyintroduced, if the classical gravitational action S with ac-count of term due to the relevant inflaton potential be-comes comparable with the period of quatum mechanicalamplitude Ψ taken in the classical limit, Ψ ∼ exp { i S } .The action in cosmology is related to the Hubble rate H for the Universe expansion, hence, the curvature ofspace-time. The curvature of Planckian scale is beyondthe classical description . However, it turns out formally,that the Higgs boson with a rather large mass would clas-sicaly produce the inflation at the Planckian curvature ofspace-time, i.e. at the stage, when the quantum descrip-tion of gravity cannot be ignored, hence, the inflationaryregime cannot be induced.In the present paper, we estimate a lower bound forthe Higgs boson mass by requirement that the Higgs fieldcannot produce the inflation regime at early stages of theUniverse evolution. The decoupling mass of the Higgs par-ticle is quite actual for modern experimental searches ofHiggs boson at colliders [21,22].Some other aspects of Higgs particle physics as con-cerns for the inflation, basically for various fluctuations,were considered in [24].We have tried to treat rather a complex problem todistinguish between two fine possibilities, when1. the scalar field is able to produce the inflation of uni-verse, but the parameters of such the inflation wouldbe in a sharp conflict with the observed properties ofour Universe, and therefore, this fact is leading to theconclusion that such the inflation should be forbiddenexperimentally ;2. the scalar field is not able to produce the inflation ofuniverse, since such the inflation is forbidden theoreti-cally by some critical properties of field self-action.The first of above possibilities occurs if the value ofself-action coupling λ for the Higgs particle minimallycoupled to the gravity is below the critical value , whilethe second possibility occurs if the coupling λ exceeds thecritical value. Thus, we can discriminate two answers tothe question: why the observed Universe did not evolve atearly times through the inflationary stage produced by thescalar Higgs particle of Standard model? The first answeris the following: the inflation produced by the Higgs field This fact was originally recognized in [4], where A.Lindeused it to set the constraint on the scalar field self-couplingconstant λ by the order of magnitude, as was recently rederivedin [23], λ ≪ − . In the present paper we get an exact valuefor the critical value of self-coupling constant, but the order ofmagnitude result. Extremely small values of constant λ , when the inflationgenerated by the Higgs boson further develops due to switchinginto the regime driven by the specific inflaton, are excludedexperimentally: λ > .
11 (seediscussion in [23])..V.Kiselev, S.A.Timofeev: Decoupling of Higgs boson from the inflationary stage of Universe evolution 3 was occasionally missed, since the other scalar field witha specific properties has produced the different inflation.The second answer states that the inflation produced bythe Higgs field cannot exist because the Higgs particle istoo heavy to produce the inflation. The second statementis determinative, while the first answer leads to the prob-lem of preference for one of two scenarios of inflation bythe specific inflaton or Higgs particle giving very differentpost-inflationary universes.
Let us consider the model of the Higgs boson in the gaugesetting the real field Φ = φ/ √ V = λ ( φ − v ) / , where the vacuum expectation value v = 1 / p √ G F ≈ . m = 2 λv . (1)The Einstein–Hilbert action of gravity is classically de-fined by scalar curvature RS g = − πG Z d x √− g R, (2)while the cosmology in the case of spatial homogeneity isdescribed by the metric with a time-dependent scale factor a ( t ), d s = d t − a ( t ) d r . (3)In the inflation regime, the metric can be well approxi-mated by de Sitter one in the standard cosmological form,wherein the 3-dimensional space is flat and an observer isposed in the center pointd s ≈ d t − e Ht d r , (4)where H = d ln a/ d t = ˙ a/a is the Hubble constant, whichvalue to the end of inflation can be strictly related with theconstant λ defining the self-coupling for the Higgs field.The appropriate framework of quasiattractor approachis systematically simple: the motion can be straightfor-wardly treated in terms of autonomous differential equa-tions with a parametric attractor, whose critical pointsslowly drift with the Hubble constant [25,26,27].Indeed, in terms of dimensionless variables defined as x = κ √ φH , y = r λ κ φ √ κH , z = √ λ √ κH , (5) We assume, that a nonzero cosmological constant can besurely neglected during the inflation. The scale factor runs as the exponent of e-folding N bydefinition a ∼ exp {− N } , while the Hubble constant gets a slowdriftage logarithmic in the scale factor, more exactly, linear ine-folding for the quartic self-action of inflaton, H − H ∗ ∼ N . giving the fractions of kinetic energy x and potential en-ergy y for the energy budget of scalar field: x + y = 1 at κ = 8 πG , while z introduces the parametric dependencein the field equations x ′ = − x + 3 x + 2 y z, y ′ = − x y − xz, (6)wherein the evolution, i.e. the differentiation denoted byprime, is calculated with respect to e-folding defined by N = ln a end − ln a , so that the parameter z evolves ac-cording to z ′ = − x z. (7)There are stable critical points for (6) (see [26,27]) at z < . (8)The existence of critical points is caused by specific“friction term” in equations. The magnitude of friction isgiven by the Hubble constant, so that the kinetic energyof inflaton is suppressed with respect to the potential, andthe field slowly rolls down to the minimum of potential.At large amount of e-folding N ≫
1, the quasiattractoris equivalent to the slow-roll approximation in the leadingorder of 1 /N -expansion [28]. However, in contrast to theslow-rolling in the 1 /N -expansion, the quasiattractor al-lows us to get the strict description for the final stage ofinflation due to the exact determination of critical pointsfor the parametric attractor. Reasonably, the inflation fin-ishes at such value of Hubble rate, when a condition onthe existence of stable critical points invalidates and theattractor becomes unstable, i.e. it disappears. Then, fromthe condition of (8) we get2 πGH = λ. (9)To the end of inflation evolving with the parametric at-tractor we get z = 3 / x = 2 / y = 1 / H = 2 πGλφ , so that equivalently to (9) weget 2 πGφ = 1 . (10)Thus, the inflation produced by the Higgs boson stops atthe Planckian scale of field value.From (9) we see that if λ ∼
1, the value of H end isabout the Planck scale. This result repeats the argumentsof [4,23] as mentioned above. Therefore, the heavy Higgsboson formally corresponds to the inflationary Hubble rateabout the Planckian scale of energy, where effects of quan-tum gravity cannot be ignored, hence, the inflation dy-namics cannot develop, since the curvature of space-timegets the Planckian values. Critical points are defined by condition x ′ = y ′ = 0, whilethe stability takes place when linear perturbations in differen-tial equations near the critical points decline to zero. V.V.Kiselev, S.A.Timofeev: Decoupling of Higgs boson from the inflationary stage of Universe evolution De Sitter metric (4) straightforwardly determines the scalarcurvature standing in the action of classical gravitationalfield, R = − H .In the calculation of gravitational action, it is worthsto note, that the coordinate r takes values in the regionfrom zero to the horizon r H = 1 /H and the integration intime t is limited by the interval from the negative infinityto a moment, which can be put to zero with no lose of gen-erality of consideration (in fact, to a moment of inflationend). Notice, that the specified coordinate system coversonly a half of de Sitter manifold , therefore, the action canbe doubled, in principle, but this would incorporate a partof the manifold causally independent of the cosmologicalobserver. Finally, we get S g = 13 GH . (11)Similarly, we add the contribution of action for thematter approximated by the form S m ≈ − Z d x √− g V, (12)where V is the matter potential, so that in the frameworkof inflation regime we neglect the kinetic term of infla-ton in comparison with the potential. The contribution ofpotential is determined by the Einstein equations V ≈ H πG . Finally, the matter action equals S m = − GH , (13)yielding the sum S = S g + S m equal to S = 16 GH . (14)We have just got the action by making use of de Sittermetric. For the sake of generality, we have performed exactcalculations in the case of matter with a state parameter w equal to the ratio of pressure p to energy density ρ : p = wρ . In the range of − < w <
1, the integrationin time runs along a finite interval with the scale factorspanning the region from a cosmic singularity to the mo-ment defined by a = 1. Then, the action of matter andgravity takes the same value of (14) independent of thestate parameter w . This fact is important, since it pointsto the stability of cosmological action versus the mattercontent. In addition, the spacetime to the end of infla-tion produced by the Higgs boson becomes to essentiallydiffer from de Sitter spacetime: the fractions of kinetic The other half of manifold can be associated with the ex-ponentially contracting universe, in contrast to the case of ex-panding universe, we consider. and potential energies get values equal to and , corre-spondingly, that gives w = specific for a radiation, i.e. alight-like or ultra-relativistic matter. Nevertheless, the es-timate of (14) is rather universal, it does not significantlyvary with changes in the expansion regime.The quantum mechanical amplitude Ψ in the classicallimit takes the form of Ψ ∼ exp { i S } , therefore, in order toseparate the quantum regime from the classical behavior,one should compare the action with the period δS = 2 π .Then, we get the constraint on H , when the quantumgravity effects become essential12 πGH > . (15)The confidence level of such the constraint is discussed insection 5. For the case of inflation produced by the Higgs scalar, therelation of Hubble constant at the end of inflation withthe self-action constant λ results in λ > . (16)Thus, the constraint on the Higgs mass takes the form m > v √ . (17)Substituting the experimental inputs, we get the lowerbound for the Higgs mass as m min = 142 . The decoupling mass obtained from the theoretical forbid-ding the inflation produced by the Higgs boson is basedon the breaking the classical description due to quantumfluctuations, which make the inflation impossible. So, inorder to estimate the confidence level of such the lowerbound derived above, let us consider, for instance, a har-monic oscillator with the Hamiltonian H = 12 ( Q + P ) ~ ω, (18)wherein Q is the coordinate, while P is its canonicallyconjugated momentum. The state maximally close to theclassical system with the energy E = ~ ω ( n + ) is the co-herent state with the minimized fluctuation of coordinateand momentum at any time of evolution( δQ ) = ( δP ) = 12 , (19)while the stationary quantum state with the definite en-ergy of n quanta gives essential fluctuations( δQ ) = ( δP ) = n + 12 . (20) .V.Kiselev, S.A.Timofeev: Decoupling of Higgs boson from the inflationary stage of Universe evolution 5 Therefore, one could estimate the relevance of the stateassignement to the nonclassical system by evaluating χ = ( δQ ) ( δQ ) − χ = 2 n, (21)wherein we put χ = 1 in order to match the vacuum tocompletely the quantum state.The same criterium can be derived by considering thetime evolution of average coordinate in the coherent state.So, from Q ( t ) = Q cos ωt + P sin ωt, (22)one gets h Q ( t ) i = 0 , h Q ( t ) i = 12 ( Q + P ) = n, (23)so that the fluctuation is equal to( δQ ) = n, (24)that yields χ = ( δQ ) ( δQ ) = 2 n. (25)Then, we can estimate the hypothesis that the system es-sentially requires to carefully take into account importantquantum fluctuation by the χ -probability depending onthe number of quanta in the system. The evaluation of n is system-dependent. We put n = S π , (26)wherein S is the action of the cosmological system withthe inflaton. The lower bound m H > m min is equivalentto n < χ <
2, so that the system is es-sentially quantum within the 2 σ confidence level, i.e. withthe probability of 90%.Note, that changing the determination for the numberof quanta in (26) by n = S/π or n = S/ π would result inthe respective modification of confidence level for the lowercosmological bound for the Higgs boson mass: 99% or 68%,correspondingly. Analogously, the shift m min
7→ √ m min would dicrease the confidence level of such the estimatewith (26) to the value of 68%. The above consideration has been based on the leading ap-proximation of effective action, while quantum loop cor-rections would both modify the potential at large fieldsrelevant to the inflation and renormalize the physical pa-rameters of Lagrangian for the Higgs scalar, i.e. the filednormalization, mass and coupling constant λ . These ef-fects could be effectively taken into account by makinguse of renormalization group in SM [24] with an appropri-ate choice of renormalization point at the inflation stage,so that the whole effect would be reduced to the running of λ ( µ ) with the scale µ . Similar strategy has been explored in [16,15] for the Higgs field non-minimally coupled to thegravity. So, in estimates we fit a pole mass of Higgs par-ticle, determining the running mass m ( µ ) at the scale of t -quark mass, with other parameters of SM at the samescale µ = m t to reach the critical value of λ in (16) at thescale of the order of the Planck mass. Then, the variationof final result due to the renormalization group can beestimated by comparing one- and two-loop calculations,which points to the uncertainty caused by the choice offinal scale in the running. Another source of uncertain-ties is connected to the empirical accuracy in the mea-surement of SM parameters at the starting scale of renor-malization group evolution. So, the one-loop renormaliza-tion group results in the decoupling mass of Higgs particleequal to 153 GeV, while the two-loop evolution approx-imately gives the lower value of 150 GeV at m t = 171GeV. A complete analysis of uncertainties caused by vari-ation of different parameters in the calculations by meansof renormalization group will be given elsewhere.Then, the renormalization group improvement of esti-mate results in the lower bound for the Higgs boson mass m min ≈
150 GeV with uncertainty of 3 GeV. The differ-ence between estimates at the tree level and due to thetwo-loop renormalization is significant, but it is rathermoderate, so that we can draw the conclusion that thehigher order corrections are still under control.
It is worth to note, that having estimated the inflation pa-rameters we have neglected terms quadratic in the Higgsfield, which is correct, if the vacuum expectation valueis much less than the Planck mass, i.e. at v ≪ M P l (thiscondition is safely valid for the Higgs boson in SM). There-fore, the estimation of (17) is valid for any scalar Higgsboson with a small vacuum expectation value in gaugetheories including grand unified theories (GUT). In addi-tion, a grand unification could change both the running ofgauge coupling constants and set of quantum fields activein the running. Then, the estimate obtained due to therenormalization group improvement would slightly move,though the value of such displacement should not sizablyexceed the calculation uncertainty given above. In respectof GUT with the SU(5) symmetry we mention the mod-ified induced gravity scenario with the Higgs field non-minimally coupled to the gravity as studied in [20].
Acknowledgements
This work was partially supported by grants of RussianFoundations for Basic Research 09-01-12123 and 10-02-00061, Special Federal Program “Scientific and academicspersonnel” grant for the Scientific and Educational Cen-ter 2009-1.1-125-055-008, ant the work of T.S.A. was sup-ported by the Russian President grant MK-406.2010.2.
V.V.Kiselev, S.A.Timofeev: Decoupling of Higgs boson from the inflationary stage of Universe evolution
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