Conformal Higgs model of dark energy

0.12 MB·

by  R. K. Nesbet

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ModernPhysicsLetters A
(cid:13)c WorldScientificPublishingCompany
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b DARK ENERGY AND THE SCALAR FIELD IN CONFORMAL
e THEORY
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ROBERTK.NESBET

] IBM Almaden Research Center
h 650 Harry Road, San Jose, CA 95120, USA
p [email protected]
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Received(receiveddate)
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Revised(reviseddate)
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i IfconformalWeylscalingsymmetryispostulatedtobeageneralpropertyofallelemen-
s
taryfields,bothgravitationalandelectroweaktheoryaremodified.Inuniform,isotropic
y
geometry, conformal gravitational and Higgs scalar fields imply a modified Friedmann
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cosmicevolutionequationthatdefinesacosmologicalconstant(darkenergy).Thisequa-
p
tionhasbeenparametrizedtofitrelevantcosmologicaldatawithinempiricalerrorlimits.
[
The Higgs mechanism for gauge boson mass is preserved in conformal theory, but the
2 imaginarymassparameterw2oftheHiggsmodelisrequiredtobeofdynamicalorigin.It
v isshownhereinasimplifiedcalculationthatthecosmologicaltimedependenceofnomi-
7 nallyconstantparametersofaconformalHiggsmodelcouplesscalarandgaugefieldsand
9 determines parameter w2. The implied cosmological constant is in order-of-magnitude
0 agreement withitsempiricalvalue.
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Keywords: Conformaltheory;Higgsscalarfield;darkenergy
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: 1. Introduction
v
Xi ExtensionofgravitationaltheorytoincorporatelocalWeylscaling(conformal)sym-
1
metryhasbeenshownto resolvea numberoflongstandingparadoxes. Inthe uni-
r
a form,isotropicgeometryof standardcosmology,aconformalscalarfieldis required
1
in order to produce the observed Hubble expansion. The Higgs mechanism for
SU(2) symmetry breaking and gauge boson mass invokes a scalar field Φ such that
Φ Φ=φ2 isanonzeroconstant.2 Aspacetimeconstantfieldisnecessarilyacosmo-
† 0
logicalentity.ConformalsymmetrymodifiesthestandardHiggsmodelbyrequiring
the scalar field Lagrangian density to include a term proportional to gravitational
Ricci scalar R. The energy-momentum tensor of such a field contains a term that
1
acts as a cosmologicalconstant (dark energy).
Postulatinguniversalconformalsymmetryforallmasslesselementaryfields,this
suggeststhat Higgs symmetry-breakinganddark energy aretwo different effects of
auniqueconformalscalarfield.Theseassumptionshavebeenshowntobeconsistent
1
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2 R. K. Nesbet
with cosmologicaldata3,4 and with the Higgs mechanism itself. 3,5
If these assumptions are valid, a crucial parameter in the Higgs model can be
identified with the cosmological constant. The present paper tests this implication
byanapproximatecalculationofthisparameter,usingtheformalismofelementary
4
particle theory. The result agrees in magnitude with its empirical value.
2. Implications of universal conformal symmetry
Conformal symmetry is defined by local Weyl scaling such that metric tensor
g (x) g (x)Ω2(x),6 forarbitraryrealdifferentiableΩ(x). T(x) Ωd(x)T(x)+
µν µν
→ →
(x)definesweightd[T]andresidue [T]foranyLorentztensorT(x).ForLorentz
R R
scalar Lagrangiandensity , action integral I =R d4x√ g is a conformal invari-
L − L
1
ant if weight d[ ] = 4 and residue [ ] = 0 up to a 4-divergence. g here is the
L − RL
determinant of the metric tensor. For scalar field Φ, d[Φ] = 1. The fundamental
−
postulatethatallprimitivefieldshaveconformalWeylscalingsymmetryissatisfied
1
by the gauge fields, but not by the scalar field as usually assumed.
Conformal symmetry requires term −61RΦ†Φ in LΦ.1 Ricci scalar R here is
defined as contraction g Rµν of the gravitational Ricci tensor. In standard elec-
µν
2
troweak theory , the Higgs scalar field model postulates incremental Lagrangian
density ∆ =w2Φ Φ λ(Φ Φ)2. λ(Φ Φ)2 with any real λ retains conformal sym-
† † †
L −
metry, but w2Φ Φ does not, and must be generateddynamically. Some remarkable
†
implications of conformal modification of electroweak theory are explored here.
Becauseconformalgravitational vanishesidentically inuniform isotropicge-
g
L
ometry, the gravitational field equation due to −61RΦ†Φ in LΦ drives conformal
cosmology.1,3 The relevantparametersarew2, λ, andRicciscalarR.The resulting
modified Friedmann equation contains a cosmologicalconstant determined by w2.
Because Ricci scalar R varies on a cosmological time scale, it induces an ex-
tremely weak time dependence of φ2, which in turn produces source current den-
0
sities for the gauge fields. As shown here, the resulting coupled semiclassical field
equationsdetermine nonvanishingbut extremelysmallparameterw2,inagreement
with the cosmological constant deduced from empirical data. This argument de-
pends only on squared magnitudes of quantum field amplitudes.
Although the interacting field solutions break conformalsymmetry, the dynam-
ically induced w2 is shown here to preserve the conformal condition that the trace
7
of the total energy-momentum tensor should vanish.
3. The modified Friedmann equation
In cosmological theory, a uniform, isotropic universe is described by Robertson-
Walker (RW) metric ds2 =dt2 a2(t)( dr2 +r2dω2), where c=~=1 and dω2 =
− 1 kr2
dθ2+sin2θdφ2. Scale factor a(t) satisfie−s a Friedmann equation, which determines
Hubble expansion.
If δ = xµνδg , metric functional derivative 1 δI of action integral I =
L µν √−gδgµν
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Dark energy and the scalar field 3
R d4x√ g isXµν =xµν+1gµν .Theenergy-momentumtensorisΘµν = 2Xµν.
− L 2 L −
For conformal gravitational action I in RW geometry, Xµν vanishes identically.1
g g
ConformalactionintegralIΦ depends on R,suchthat XΦµν = 61RµνΦ†Φ+21LΦgµν.
InRWgeometry,this resultsinagravitationalfieldequationXµν = 1Θµν,1 driven
Φ 2 m
byenergy-momentumtensorΘµν = 2Xµν foruniformmatterandradiation.Since
m − m
Θm isfinite, determinedbyfields independent ofΦ, XΦ mustalsobe finite,regard-
less of any parameters of the theory. This precludes spontaneous destabilization of
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the conformal Higgs model.
The scalar field equation including ∆ is3 ∂ ∂µΦ = ( 1R+w2 2λΦ Φ)Φ.
L µ −6 − †
Generalizing the Higgs construction, and treating Ricci R as a constant, φ2 =
0
(w2−61R)/2λif this ratio is positive. EvaluatingXΦ for this solution, anddefining
κ¯ = 3/φ2 and Λ¯ = 3w2, the gravitationalfield equation in RW geometry is
− 0 2
1
Rµν Rgµν +Λ¯gµν = κ¯Θµν. (1)
− 4 − m
TracelesstensorRµν 1Rgµν replacesEinsteintensorGµν inthisconformaltheory.
−4
Energy density ρ=Θ00 implies the modified Friedmann equation3
m
a˙2 k a¨ 2
+ = (κ¯ρ+Λ¯). (2)
a2 a2 − a 3
Consistencyrequirestraceconditiong Λ¯gµν =4Λ¯ = κ¯g Θµν.Fromthe def-
µν − µν m 7
inition of an energy-momentum tensor, this is just the conformal trace condition,
g (Xµν +Xµν) = 0. Vanishing trace eliminates the second Friedmann equation
µν Φ m
derivedinstandardtheory.Althoughtheinducedgaugefieldconsideredherebreaks
conformal symmetry, a detailed argument given below shows that the trace condi-
tion is preserved.
4. Semiclassical coupled field equations
2
TheHiggsmodel derivesgaugebosonmassfromcouplingviacovariantderivatives
toapostulatedSU(2)doubletscalarfieldΦ.SU(2)symmetryisbrokenbyasolution
of the scalar field equation such that Φ Φ = φ2, a spacetime constant. Only one
† 0
component of doublet field Φ is nonzero.
TheessentialresultoftheHiggsmodel,generationofgaugefieldmasses,follows
2
fromasimplifiedsemiclassicaltheoryofthecoupledscalarandgaugefields. Thisis
extended here to include the gravitationalfield, metric tensor g , but greatlysim-
µν
plifiedbyassumingthestandardcosmologicalmodeldescribedbyRobertson-Walker
geometry. This is further simplified, for the purpose of establishing credibility and
orders of magnitude, by considering only the U(1) gauge field B . The nonlinear
µ
coupled field equations are solved to obtain definite numerical results.
U(1) gauge invariance replaces bare derivative ∂ by covariant derivative D =
µ µ
∂ + ig B . This retains and augments conformal 0 by coupling term ∆ =
µ 2 b µ LB LΦ L
(D Φ) DµΦ (∂ Φ) ∂µΦ= ig Bµ(∂ Φ) Φ ig B Φ ∂µΦ+1g2Φ B BµΦ. Weyl
µ † − µ † 2 b µ † −2 b µ† † 4 b † µ†
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4 R. K. Nesbet
scaling residues in ∆ cancel exactly for realgaugefields, so that the total energy-
L
momentum tensor is conformaland traceless.This trace condition must be verified
for the pure imaginary gauge field B derived here.
µ
Derivatives due to cosmologicaltime dependence act as anextremely weakper-
turbation of the Higgs scalar field. The scalar field is dressed by an induced gauge
field amplitude. Derivatives of the induced gauge field (but not of Φ) can be ne-
glected and are omitted from the derivation here.
If ∆ were equivalent to standard parametrized (w2 λΦ Φ)Φ Φ, the scalar
† †
L −
field equation would be
1 1 δ∆I
∂µ∂µΦ+ RΦ= =(w2 2λΦ†Φ)Φ. (3)
6 √ g δΦ −
− †
1 δ∆I 1 i
= g2B BµΦ g (B +B )∂µΦ (4)
√ g δΦ 4 b µ∗ − 2 b µ∗ µ
− †
implies w2 = 1g2B Bµ. Pure imaginary B does not affect λ.
4 b µ∗ µ
1 δ∆I
∂ Bµν =2 =m2Bµ Jµ, (5)
ν √ gδB B − B
− µ∗
for a massive vector field.2 ∆ given above determines parameters for field Bµ.
L
1 δ∆I 1
2 = g2Φ ΦBµ ig Φ ∂µΦ (6)
√ gδB 2 b † − b †
− µ∗
implies m2 = 1g2φ2, verifying the Higgs mass formula. Defining real parameter
B 2 b 0
φφ˙00, the fields are coupled by pure imaginary source density JB0 = igbφ∗0∂0φ0 =
igbφφ˙00φ∗0φ0. Neglecting derivatives of induced field Bµ, the gauge field equation re-
duces to m2Bµ =Jµ.
B B
The standard Higgs model omits R and assumes w2,λ > 0. For exact solution
Φ = φ0 of the scalar field equation, constant Φ†Φ = φ20 = w2/2λ. This implies
φ˙0 =0, which does not couple the fields.
φ0
In conformal theory, solving the modified Friedmann equation determines Ricci
scalarR(t),whichvariesincosmologicaltime,butretains 1R w2 >0.3 Hence,for
6 −
λ<0,Φ Φ=φ2 =(1R w2)/( 2λ).Ifw2 andλremainconstant, φ˙0 = 1 R˙ =
† 0 6 − − φ0 2R 6w2 6
0. This implies small but nonvanishing φ˙0, hence nonzero source current J−µ. The
φ0 B
scalarfieldequationimpliesw2 = 1g2 B 2,proportionalto(φ˙0)2.TheinducedU(1)
4 b| | φ0
gauge field does not affect parameter λ.
OmittingSU(2)gaugefieldWµ here,parameterλisnotdetermined.w2 andφ0
arewell-definedifλ<0,asrequiredbyempiricalR>6w2.Verifyingtheconformal
Higgs model, gauge field B acquires mass m from coupling to Φ.
µ B
The coupled fields break conformal and SU(2) symmetries. Time-dependent R
impliesnonvanishingrealφ˙0andpureimaginaryJB0.Complexsolutionsofthegauge
field equations, induced by pure imaginary current densities, exist but do not pre-
serve gauge symmetry. Imaginary gauge field amplitudes model quantum creation
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Dark energy and the scalar field 5
and annihilation operators. Only the squared magnitudes of these nonclassical en-
tities have classical analogs.
5. Numerical values
The Robertson-Walker metric for uniform, isotropic geometry is determined for
redshift z(t) by scale factor a(t) = 1/(1+z(t)). Ricci scalar R = 6(ξ0(t)+ξ1(t)),
where ξ0(t) = aa¨ and ξ1(t) = aa˙22 + ak2. Fitting the modified Friedmann equation to
3
cosmologicaldata determines dimensionless parameters at present time t0:
Ωq =a¨a/a˙2 =0.271,ΩΛ=w2 =0.717,Ωk = k/a2 =0.012. (7)
−
Here H(t) = aa˙ is defined in units of the current Hubble constant H0, so that
H(t0) = 1. Then 61R = aa˙22(1−Ωk+Ωq) and w2 = aa˙22ΩΛ. w(t0) = √ΩΛ in Hubble
energy units (~H0 =1.503 10−33eV).
×
From the scalar field equation, φ2 = ζ/2λ, where
0
−
1 a˙2
ζ = 6R−w2 = a2(1−ΩΛ−Ωq −Ωk+2Ωq). (8)
Computed ζ(t0) = 1.224 10−66eV2. If φ0 = 180GeV,2 dimensionless λ =
×
1ζ/φ2 = 0.189 10 88. Because λ < 0, the conformal model does not imply a
−2 0 − × −
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Higgs particle mass.
Given ζ(z), φφ˙00 =−12+ζzddζzaa˙. For z →0, φφ˙00 →−21ζddζz in Hubble units (H0/sec),
such that aa˙(t0) = 1. The dynamical value of w2 due to induced field Bµ is wB2 =
1g2 B 2.Usingm2 = 1g2φ2, B 2 = J 2/m4,and J 2 =g2(φ˙0)2φ4,thisreduces
4 b| | B 2 b 0 | | | B| B | B| b φ0 0
to w2 = (φ˙0)2. Numerical solution of the modified Friedmann equation3 implies
B φ0
φφ˙00(t0) = −2.651, so that wB2 = 7.027, in Hubble units. In energy units, wB =
2.651~H0=3.984 10−33eV.
×
6. Verification of the trace condition
If any covariant V is independent of metric variations, contravariant Vα =gαβV
α β
is not. δVα = gαµδg gνβV follows from the reciprocal matrix formula. This
µν β
−
implies δ(U Vα) = UµVνδg , for any scalar product in a Lagrangian density.
α µν
−
The resulting term in the functional derivative of an action integral is Xµν =
UµVν + 1gµνU Vη, whose trace is g Xµν = U Vµ. Since ∆ here is a sum
− 2 η µν µ L
of scalar products, trace ∆X = g ∆Xµν = ∆ . For B = J /m2, trace ∆X is
µν L µ µ B
∆ = 1m2 B 2 = w2φ2.
L −2 B| | − 0
Fornoninteractingfields,bothLΦandLB areconformal.DenotingtracegµνXaµν
byXaingeneral,barefieldtracesXΦandXB vanish.Timedependenceofthescalar
field adds a kinetic energy term to XΦ: (∂0Φ)†∂0Φ = φ˙20 = w2φ20. Adding this to
∆X, the total trace vanishes.
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6 R. K. Nesbet
7. Conclusions
The present results provide an explanation of the huge disparity in magnitude of
parametersrelevantto cosmologicalandelementary particle phenomena.Small pa-
rameter w2 10 66eV2 is determined by a cosmologicaltime derivative,on a time
−
≃
scale1010years,whileφ2 =3.24 1022eV2isthelargeratiooftwosmallparameters,
0
×
approximately 1w2/λ.
2 | |
Parameter ΩΛ = w2 = 0.717 is determined by fitting the modified Friedmann
cosmic evolution equation of conformal theory to well-established cosmological
data.3 Thisdeterminesparameterw =√ΩΛ~H0 =1.273 10−33eV.Thesecalcula-
×
tions also determine the time derivative of the cosmologicalRicci scalar,which im-
plies nonvanishingsourcecurrentdensityfor the U(1)gaugefield, treatedhere asa
classicalfieldinsemiclassicalcoupledfieldequations.Theresultingfieldintensityde-
terminesthe U(1)contributiontow2 suchthatwB =2.651~H0 =3.984 10−33eV.
×
Thisisonlyanorder-of-magnitudeestimate,sincethederivationomitstheSU(2)
gauge field and ignores the cosmologicaltime dependence of dynamical parameters
in solving the Friedmann equation. A quantitative test must include the SU(2)
gaugefield andevaluate time dependence ofallnominalconstants. The estimate of
w computed here justifies the conclusion that conformal theory explains both the
existence and magnitude of dark energy.
References
1. P. D.Mannheim, Prog.Part.Nucl.Phys. 56, 340 (2006).
2. W. N. Cottingham and D. A. Greenwood, An Introduction to the Standard Model of
Particle Physics (Cambridge Univ.Press, New York,1998).
3. R.K.Nesbet,Mod.Phys.Lett.Axx,xxx(2011),(arXiv:0912.0935v2[physics.gen-ph]).
4. E. Komatsu et al, ApJS 180, 330 (2009).
5. R.K. Nesbet, (arXiv:1009.1372v1 [physics.gen-ph]).
6. H.Weyl, Sitzungber.Preuss.Akad.Wiss. , 465 (1918).
7. P. D.Mannheim, (arXiv:0909.0212v3 [hep-th]).