aa r X i v : . [ phy s i c s . g e n - ph ] S e p The Higgs scalar field with no massive Higgs particle
R. K. Nesbet
IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120-6099, USA (Dated: September 29, 2018)The postulate that all massless elementary fields have conformal Weyl local scaling symmetry hasremarkable consequences for both cosmology and elementary particle physics. Conformal symmetrycouples scalar and gravitational fields. Implications for the scalar field of a conformal Higgs modelare considered here. The energy-momentum tensor of a conformal Higgs scalar field determines acosmological constant. It has recently been shown that this accounts for the observed magnitude ofdark energy. The gravitational field equation forces the energy density to be finite, which precludesspontaneous destabilization of the vacuum state. Scalar field fluctuations would define a Higgstachyon rather than a massive particle, consistent with the ongoing failure to observe such a particle.
PACS numbers: 04.20.Cv,98.80.-k,14.80.Bn
INTRODUCTION
The standard model of spinor and gauge boson fieldshas higher symmetry than does Einstein gravitationaltheory[1]. For massless fields with definite conformalcharacter action integrals are invariant under local Weyl(conformal) scaling, g µν ( x ) → g µν ( x ) e α ( x ) [1]. A con-formal energy-momentum tensor is traceless, while theEinstein tensor is not.Compatibility can be imposed in gravitational the-ory by replacing the Einstein-Hilbert field action by auniquely determined action integral I g constructed us-ing the conformal Weyl tensor[1]. Conformal gravity ac-counts for anomalous galactic rotation velocities withoutinvoking dark matter[1]. Relativistic phenomenology atthe distance scale of the solar system is preserved.An inherent conflict between gravitational and elemen-tary particle theory is removed if all massless elementaryfields have conformal symmetry. Standard cosmology[2]postulates uniform, isotropic geometry, described by theRobertson-Walker (RW) metric tensor. In RW geom-etry, conformal gravitational L g vanishes identically[1],but the residual gravitational effect of a conformal scalarfield is consistent with Hubble expansion[1], dominatedin the current epoch by dark energy, with negligible spa-tial curvature[3, 4].In electroweak theory, the Higgs mechanism introducesan SU(2) doublet scalar field Φ that generates gauge bo-son mass[5, 6]. Postulating universal conformal symme-try for massless elementary fields, these two scalar fieldscan be identified[7]. Lagrangian density L Φ for conformalscalar field Φ( x ) → Φ( x ) e − α ( x ) includes a term depen-dent on Ricci scalar R = g µν R µν , where R µν is the grav-itational Ricci tensor[1]. In uniform, isotropic geometrythis determines a modified Friedmann cosmic evolutionequation[3] consistent with cosmological data back to themicrowave background epoch[4].Implications for the standard electroweak model areexamined here. The Higgs model Lagrangian densitycontains ∆ L Φ = ( w − λ Φ † Φ)Φ † Φ, where w and λ are undetermined positive constants[6]. Units here set¯ h = c = 1. Lagrangian term λ (Φ † Φ) is conformally co-variant. w Φ † Φ breaks conformal symmetry, but can begenerated dynamically[7]. Conformal symmetry requiresa term − R Φ † Φ[1]. Empirical cosmological
R > − R and w have opposite signs. A consistent theorymust include ( w − R )Φ † Φ[3].The conformal scalar field equation has exact solutionssuch that Φ † Φ = φ = ( w − R ) / λ , if this ratio is pos-itive and R is treated as a constant. Only the magni-tude of Φ is determined. For φ >
0, a modified Fried-mann cosmic evolution equation has been derived[3] andsolved to determine cosmological parameters. The resid-ual constant term in conformal energy-momentum tensorΘ µν Φ defines a cosmological constant (dark energy)[1, 3].Nonzero φ produces gauge boson masses[6].Conformal theory identifies w with the empiricallypositive cosmological constant[3], but does not specifythe algebraic sign of parameter λ . For the Higgs mecha-nism, condition φ = ( w − R ) / λ > λ to agree with w − R . The scalar field energy den-sity determined by the coupled equations derived here isnecessarily finite for any real value of λ . This precludesdestabilization of the vacuum.Fluctuations δφ → ∂ µ ∂ µ δφ → − λφ δφ . If λ > m H = 4 λφ =2( w − R ), which defines a Higgs boson[6] if R < w .In the conformal Higgs model, empirical values of param-eters w , R , and φ determine parameter λ . It is arguedhere that these parameters, now well-established fromcosmological and electroweak data, imply λ <
0, consis-tent with an earlier formal argument[1]. Hence fluctua-tions of a conformal Higgs scalar field do not satisfy aKlein-Gordon equation. This rules out a standard Higgsparticle of any real mass. Negative m H , or finite pureimaginary mass, would define a tachyon[8], if such a par-ticle or field could exist, and might justify an experimen-tal search for such a tachyon. THE MODIFIED FRIEDMANN EQUATION
In cosmological theory, a uniform, isotropic universe isdescribed by Robertson-Walker (RW) metric ds = dt − a ( t )( dr − kr + r dω ), if c = ¯ h = 1 and dω = dθ + sin θdφ . Gravitational field equations aredetermined by Ricci tensor R µν and scalar R . The RWmetric defines two independent functions ξ ( t ) = ¨ aa and ξ ( t ) = ˙ a a + ka , such that R = 3 ξ and R = 6( ξ + ξ ).The field equations reduce to Friedmann equations forscale factor a ( t ) and Hubble function h ( t ) = ˙ aa ( t ).If the scalar field required by Higgs symmetry-breakinghas conformal symmetry, its action integral I Φ must de-pend on the Ricci scalar, implying a gravitational effect.Because conformal gravitational action integral I g van-ishes identically in RW geometry[1], it is consistent toassume that uniform cosmological gravity is determinedby this scalar field.Including term ( w − R )Φ † Φ in L Φ [3], the field equa-tion for scalar Φ is ∂ µ ∂ µ Φ = ( w − R − λ Φ † Φ)Φ.Generalizing the Higgs construction, and neglecting thecosmological time derivative of R , constant Φ = φ is aglobal solution if φ = λ ( w − R ). Evaluated for thisfield solution, L Φ = φ ( w − R − λφ ) = φ ( w − R ).Variational formalism of classical field theory[9] is eas-ily extended to the context of general relativity[1]. Themetric functional derivative √− g δIδg µν of generic actionintegral I = R d x √− g L is X µν = x µν + g µν L , if δ L = x µν δg µν . The energy-momentum tensor is Θ µν = − X µν . Varying g µν for fixed scalar field solution Φ,metric functional derivative X µν Φ = 16 R µν Φ † Φ + 12 g µν L Φ = 16 φ ( R µν − Rg µν + 32 w g µν ) (1)implies modified Einstein and Friedmann equations[3].The gravitational field equation driven by energy-momentum tensor Θ µνm = − X µνm for uniform matter andradiation is X µν Φ = Θ µνm . Since Θ µνm is finite, determinedby fields independent of Φ, X µν Φ must be finite, regardlessof any parameters of the theory. This precludes sponta-neous destabilization of the conformal Higgs model.Defining ¯ κ = − /φ and ¯Λ = w , the modified Ein-stein equation is R µν − Rg µν + ¯Λ g µν = − ¯ κ Θ µνm . (2)Traceless conformal tensor R µν − Rg µν here replacesthe Einstein tensor of standard theory[3]. Cosmologicalconstant ¯Λ is determined by Higgs parameter w . Non-standard parameter ¯ κ < ρ = Θ m this implies − ( R − R ) = ξ ( t ) − ξ ( t ) = (¯ κρ + ¯Λ). Hence uniform, isotropic matter and radiation determine themodified Friedmann cosmic evolution equation[3] ξ ( t ) − ξ ( t ) = ˙ a a + ka − ¨ aa = 23 (¯ κρ + ¯Λ) . (3)Because the trace of R µν − Rg µν is identically zero,a consistent theory must satisfy the trace condition g µν ¯Λ g µν = 4 ¯Λ = − ¯ κg µν Θ µνm . From the definition of anenergy-momentum tensor, this is just the trace conditionsatisfied in conformal theory[10], g µν ( X µν Φ + X µνm ) = 0.Vanishing trace eliminates the second Friedmann equa-tion derived in standard theory. Although the w termin ∆ L Φ breaks conformal symmetry, a detailed argumentshows that the trace condition is preserved[7]. FITS TO COSMOLOGICAL DATA
The modified Friedmann equation determines dimen-sionless scale parameter a ( t ) = 1 / (1 + z ( t )), for red-shift z ( t ), and function h ( t ) = ˙ aa ( t ) in units of cur-rent Hubble constant H =70.5 km/s/Mpc[4], such that z = 0 , a = 1 , h = 1 at present time t . Distances here arein Hubble units c/H .The modified Friedmann equation depends on nom-inally constant parameters, fitted to cosmological datafor z ≤ z ∗ : α = ¯Λ = w > k ≃ β = − ¯ κρ m a > γ = 3 β/ R b ( t ). z ∗ = 1090 here characterizes thecosmic microwave background, at t ∗ , when radiation be-came decoupled from matter. R b ( t ) is the ratio ofbaryon to radiation energy densities. Empirical value R b ( t ) = 688 . ρ m a and ρ r a , for matter and radiation respectively,are constant. In the absence of dark matter, ρ m ≃ ρ b ,the baryon density.The parametrized modified Friedmann equation is˙ a a − ¨ aa = − ddt ˙ aa = ˆ α = α − ka − βa − γa . (4)Dividing this equation by h ( t ) implies dimensionlesssum ruleΩ m ( t ) + Ω r ( t ) + Ω Λ ( t ) + Ω k ( t ) + Ω q ( t ) = 1 , (5)where Ω m ( t ) =
23 ¯ κρ m ( t ) h ( t ) <
0, Ω r ( t ) =
23 ¯ κρ r ( t ) h ( t ) < Λ ( t ) = w h ( t ) >
0, Ω k ( t ) = − ka ( t ) h ( t ) , and Ω q ( t ) = ¨ aa ˙ a = − q ( t ). In contrast to the standard sum rule, Ω m and Ω r are negative, while acceleration parameter Ω q ( t )appears explicitly.Hubble expansion is characterized for type Ia super-novae by scaled luminosity distance d L as a function ofredshift z . Here d L ( z ) = (1 + z ) d z , for geodesic distance d z corresponding to r z = R cdt/a ( t ), integrated from t z to t . In curved space (for k < d z = sinh( √− kr z ) √− k . In thestandard Λ CDM model[2], radiation density and curva-ture Ω k can be neglected in the current epoch ( z ≤ Λ + Ω m = 1. Empiricalvalue Ω Λ = 0 .
726 forces Ω m to be much larger than canbe accounted for by observed matter, providing a strongargument for dark matter. Mannheim[1, 11] questionedthis implication, and showed that observed luminositiescould be fitted equally well for z ≤ m = 0,using the standard Friedmann equation. However, sumrule Ω Λ + Ω k = 1 then requires an empirically improb-able large curvature parameter Ω k . Empirical limits areΩ k ≃ ± . k, β, γ set to zero[3]. Ω q is determined by the solution. The modified sum ruleΩ Λ + Ω q = 1 then presents no problem. Computed d L ( z )agrees with Mannheim’s empirical function for z ≤ α = Ω Λ ( t ) = 0 . k ( t ) = 0. This is consistent with current empiricalvalues Ω Λ = 0 . ± . , Ω k = − . ± . m and Ω r can apparently be neglected for z ≤ t = 0 is defined by h ( t ) = 0 in the conformal model,which describes an initial inflationary epoch[3]. Themodified Friedmann equation was solved numerically for0 ≤ t ≤ t [3], with parameters fitted to d L ( z ) for z ≤
1, to shift parameter R ( z ∗ )[12], and to acousticscale ratio ℓ A ( z ∗ )[12]. This determines model parameters α = 0 . , k = − . , β = 0 . × − . Fixed at γ = 3 β/ R b ( t ), which neglects dark matter, parameter γ = 0 . × − . There is no significant inconsistencywith model-independent empirical data[4].Defining ζ = R − w , the dimensionless sum ruledetermines ζ = ξ + ξ − w = h ( t ) (2Ω q + Ω m + Ω r ). For a →
0, when both α and k can be neglected, the sum ruleimplies ζ = h ( t ) (Ω q + 1). For large a , ζ = h ( t ) (2Ω q ). ζ > q >
0. The present empirical parameters imply that ζ is positive for all z [3].Conformal symmetry is consistent with any realvalue of parameter λ . However, in electroweak theoryHiggs symmetry-breaking requires nonvanishing confor-mal scalar field Φ[5]. A positive value of ζ implies λφ = 12 ( w − R ) = − ζ < . (6)As argued above, for φ > DYNAMICAL ESTIMATE OF PARAMETER w Since term w Φ † Φ in standard parametrized ∆ L breaks conformal symmetry, it must be generated dy-namically in a consistent theory[10]. As shown above,this term accounts for dark energy. Dynamically induced w preserves the conformal trace condition[7]. The Higgs model deduces gauge boson mass froman exact solution of the parametrized scalar fieldequation[6]. For interacting fields, this logic can be ex-tended to deduce nominally constant field parametersfrom a solution of the coupled field equations. Such asolution of nonlinear equations does not depend on lin-earization or on perturbation theory.Interaction of scalar and gauge boson fields definesa quasiparticle scalar field in Landau’s sense: Φ isdressed via virtual excitation of accompanying gaugefields. The derivation summarized here considers gravi-tational field g µν interacting with scalar field Φ and U (1)gauge field B µ . Solution of the coupled semiclassical fieldequations[7] gives an order-of-magnitude estimate of pa-rameter w , in agreement with the empirical cosmologicalconstant, while confirming the Higgs formula for gaugeboson mass[5, 6].The conformal Higgs model assumes incremental La-grangian density ∆ L Φ = w Φ † Φ − λ (Φ † Φ) , with unde-termined numerical parameters w and λ . The impliedscalar field equation is ∂ µ ∂ µ Φ+ R Φ = √− g δ ∆ Iδ Φ † = ( w − λ Φ † Φ)Φ. If
R, w , λ are constant, this has an exact so-lution Φ † Φ = φ = ( w − R ) / λ , if this ratio is positive.For massive complex vector field B µ , parametrized ∆ L B implies field equation ∂ ν B µν = 2 √− g δ ∆ IδB ∗ µ = m B B µ − J µB .For interacting fields, both ∆ L Φ and ∆ L B can be iden-tified with incremental Lagrangian density ∆ L = i g b B µ ( ∂ µ Φ) † Φ − i g b B † µ Φ † ∂ µ Φ + 14 g b Φ † B † µ B µ Φ , (7)due to covariant derivatives, with coupling constant g b .Evaluated for solutions of the coupled field equations,2 1 √− g δ ∆ IδB ∗ µ = 12 g b Φ † Φ B µ − ig b Φ † ∂ µ Φ (8)implies Higgs mass formula m B = g b φ . The fieldsare coupled by current density J µB = ig b Φ † ∂ µ Φ. For thescalar field, neglecting derivatives of B µ ,1 √− g δ ∆ Iδ Φ † = 14 g b B ∗ µ B µ Φ − i g b ( B ∗ µ + B µ ) ∂ µ Φ (9)implies w = g b B ∗ µ B µ .For ζ = R − w >
0, Φ † Φ = φ = − ζ/ λ solves thescalar field equation if λ <
0. Ricci scalar R ( t ) varies ona cosmological time scale, so that ˙ φ φ =
12 ˙ RR − w = 0, forconstant w and λ . This implies small but nonvanish-ing real ˙ φ φ , hence nonzero pure imaginary source currentdensity J B = ig b φ ∗ ∂ φ = ig b ˙ φ φ φ .Derivatives due to cosmological time dependence actas a weak perturbation of SU(2) scalar field solutionΦ = (Φ + , Φ ) → (0 , φ ). Neglecting extremely smallderivatives of the induced gauge fields (but not of Φ),the gauge field equation reduces to m B B µ = J µB . Im-plied pure imaginary B µ does not affect parameter λ .The coupled field equations imply w B = g b | B | , pro-portional to ( ˙ φ φ ) . Since observable properties dependonly on | B | , a pure imaginary virtual field implies noobvious physical inconsistency. Gauge symmetry is bro-ken in any case by a fixed field solution. The scalar fieldis dressed by the induced gauge field.Numerical solution of the modified Friedmannequation[3, 7] implies ζ ( t ) = 1 . × − eV , atpresent time t . Given φ = 180 GeV [6], λ = − ζ/φ = − . × − .U(1) gauge field B µ does not affect λ . Using | B | = | J B | /m B , | J B | = g b ( ˙ φ φ ) φ and m B = g b φ , the dy-namical value of w due to B µ is w B = g b | B | = ( ˙ φ φ ) .From the solution of the modified Friedmannequation[7], ˙ φ φ ( t ) = − .
651 and w B = 7 . w B = 2 . hH = 3 . × − eV inenergy units. This can be considered only an order-of-magnitude estimate, since time dependence of the as-sumed constants, implied by the present theory, wasnot considered in fitting empirical cosmological data[3].Moreover, the SU(2) gauge field has been omitted. NOTE ON DARK MATTER
As stated in[3], interpretation of parameter Ω m mayrequire substantial revision of the standard cosmologicalmodel. Directly observed inadequacy of Newton-Einsteingravitation may imply the need for a modified theoryrather than for inherently unobservable dark matter.Mannheim has applied conformal gravity to anoma-lous galactic rotation[1], fitting observed data for a setof galaxies covering a large range of structure and lu-minosity. The role played in standard ΛCDM by darkmatter, separately parametrized for each galaxy, is takenover in conformal theory for Schwarzschild geometry byan external linear radial potential. The remarkable fitto observed data shown in[1][Sect.6.1,Fig.1] requires onlytwo universal parameters for the whole set of galaxies.As discussed by Mannheim[1][Sects.6.3,9.3], a signifi-cant conformal contribution to centripetal acceleration isindependent of total galactic luminous mass. This im-plies an external cosmological source. Such an isotropicsource would determine an inherently spherical halo ofgravitational field surrounding any galaxy. Quantita-tive results for lensing and for galactic clusters shouldbe worked out before assuming dark matter. CONCLUSIONS
This paper is concerned with determining parameters w and λ in the incremental Lagrangian density of theHiggs model, ∆ L Φ = ( w − λ Φ † Φ)Φ † Φ. Fitting the mod-ified Friedmann equation to cosmological data[3] implies dark energy parameter Ω Λ = w = 0 . w = √ . hH = 1 . × − eV .The modified Friedmann equation determines the timederivative of the cosmological Ricci scalar, which impliesnonvanishing source current density for induced U(1)gauge field B µ , treated here as a classical field in semiclas-sical coupled field equations. The resulting gauge field in-tensity estimates the U(1) contribution to w such that w B = 2 . hH = 3 . × − eV . This order-of mag-nitude agreement between computed w B and empirical w supports the conclusion that conformal theory explainsboth the existence and magnitude of dark energy[7].The present argument obtains an accurate empiri-cal value of parameter λ from the known dark energyparameter[4], from the implied current value of Ricciscalar R [3], and from scalar field amplitude φ deter-mined by gauge boson masses[6]. The mass parameterfor a fluctuation of the conformal Higgs scalar field satis-fies m H = 4 λφ . Empirical value λ = − . × − isnegative, implying finite pure imaginary parameter m H .If such a particle or field could exist or be detected, thiswould define a tachyon[8], the quantum version of a clas-sical particle that moves more rapidly than light. Ex-perimental data rule out a standard massive Higgs bosonwith mass 0 ≤ m H ≤ [1] P. D. Mannheim, Prog.Part.Nucl.Phys. , 340 (2006).[2] S. Dodelson, Modern Cosmology , (Academic Press, NewYork, 2003).[3] R. K. Nesbet,
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