# Implications of the conformal Higgs model

R. K. Nesbet  IBM Almaden Research Center, 650 Harry Road, San Jose, CA
95120-6099, USA<EMAIL_ADDRESS>
###### Abstract

The postulate of universal local Weyl scaling (conformal) symmetry modifies
both general relativity and the Higgs scalar field model. The conformal Higgs
model (CHM) acquires a cosmological effect that fits observed accelerating
Hubble expansion for redshifts $z\leq 1$ (7.33 Gyr) accurately with only one
free constant parameter. Conformal gravity (CG) has recently been fitted to
anomalous rotation data for 138 galaxies. Conformal theory explains dark
energy and does not require dark matter, providing a viable alternative to the
$\Lambda$CDM standard paradigm. The theory precludes a massive Higgs particle
but validates a composite gauge field $W_{2}$ with mass 125GeV.

###### pacs:

04.20.Cv,98.80.-k,11.15.-q

##  I Introduction

Gravitational phenomena that cannot be explained by general relativity as
formulated by Einstein are attributed to cold dark matter in the consensus
$\Lambda$CDM paradigm for cosmology. The search for tangible dark matter has
continued without result for many yearsSAN10 . Dark energy $\Lambda$ remains
without explanation.

Universal conformal symmetry offers an alternative paradigm, motivated by this
situation. This requires local Weyl scaling covariance WEY18 ; MAN06 ; MAN07
for all massless elementary physical fields, without dark matterNES13 . With
no novel elementary fields, this extends conformal symmetry, valid for fermion
and gauge boson fieldsDEW64 , to both the metric tensor field of general
relativity and the Higgs scalar field of elementary-particle theory HIG64 ;
CAG98 . This postulate implies conformal gravity CGMAK89 ; MAK94 ; MAN90 ;
MAN91 ; MAN06 ; MAN12  and the conformal Higgs model CHMNESM1 ; NESM2 ; NESM3
; NESM5 .

The CHM determines model parameters that retain the Higgs mechanism but
preclude a massive Higgs particle, replacing mass by implied dark energy in
agreement with observed Hubble expansion NESM1 ; NESM2 , while validating a
novel composite gauge field $W_{2}$ of mass 125GeV NESM4 . This paper reviews
the logic of the CHM and its qualitative and quantitative justification by
observed cosmological, gravitational, and particle physics data.

Conformal gravity (CG)MAN06  retains the logical structure of general
relativity, replacing the Einstein-Hilbert Lagrangian density by a quadratic
contraction of the conformal Weyl tensorWEY18 . CG ensures consistency of the
gravitational field equations, while preserving subgalactic phenomenologyMAK89
; MAN06 . The conformal Higgs model (CHM) introduces a gravitational term
confirmed by observed accelerating Hubble expansionMAN06 ; NESM1 ; NESM2 ;
NESM5 .

Substantial empirical support for this proposed break with convention is
provided by recent applications of CG to galactic rotation velocitiesMAN97 ;
OCM18 ; MAO11 ; MAO12 ; OAM12 ; OAM15  and of the CHM to Hubble expansion
NESM1 ; NESM2 , in a consistent model of extended dark galactic halosNESM3 ,
without dark matter. Gravitational lensing by a galactic halo is attributed to
the boundary condition of continuous gravitational acceleration across the
outer boundary radius of the large spherical halo depleted of primordial
matter fallen into the visible galaxyNESM3 . This accounts for the
nonclassical acceleration parameter $\gamma$ deduced from anomalous orbital
rotation velocities observed in the outer regions of galaxiesMAN06 . As
recently reviewed MAN06 ; MAN12 ; NESM5 , conformal theory fits observed data
for an isolated galaxy without invoking dark matter, resolving several
longstanding paradoxes. Higgs biquadratic parameter $\lambda$ is found to
determine $W_{2}$ mass 125GeV NESM4 . Derivations of these results are
summarized here.

##  II Depleted halo model

CG and the CHM are consistent but interdependentNESM5  in the context of a
depleted dark halo model NESM3  of an isolated galaxy. A galaxy of mass $M$ is
modeled by spherically averaged mass density $\rho_{G}/c^{2}$ within $r_{G}$,
formed by condensation of primordial uniform, isotropic matter of uniform mass
density $\rho_{m}/c^{2}$ from a sphere of large radius $r_{H}$ NESM3 . The
dark halo inferred from gravitational lensing and centripetal acceleration is
identified with this depleted sphere NESM3 . A model valid for nonclassical
gravitation can take advantage of spherical symmetry at large galactic radii,
assuming classical gravitation within an effective galactic radius $r_{G}$.
Non-spherical gravitation is neglected outside $r_{G}$. Given mean mass
density ${\bar{\rho}}_{G}/c^{2}$ within $r_{G}$, this implies large empty halo
radius $r_{H}=r_{G}({\bar{\rho}}_{G}/\rho_{m})^{\frac{1}{3}}$.

The unique Lagrangian density ${\cal L}_{g}$ of conformal gravity theory,
constructed from the conformal Weyl tensor WEY18 ; MAK89 ; MAN06 , determines
source-free Schwarzschild gravitational potential

$\displaystyle B(r)=-2\beta/r+\alpha+\gamma r-\kappa r^{2},$ (1)

valid outside a spherically symmetric mass/energy source density MAK89 ; MAN91
. This adds two constants of integration to the classical external potential:
nonclassical radial acceleration $\gamma$ and halo cutoff parameter $\kappa$
NESM3 .

The physically relevant particular solution for $B(r)$NESM5  incorporates
nonclassical radial acceleration $\gamma$ as a free parameter. Its value is
determined by the halo model. Gravitational lensing by a spherical halo is
observed as centripetal deflection of a photon geodesic passing into the empty
halo sphere from the external intergalactic space with postulated universal
isotropic mass-energy density $\rho_{m}$. The conformal Friedmann cosmic
evolution equation implies dimensionless cosmic acceleration parameters
$\Omega_{q}(\rho)$NESM3  which are locally constant but differ across the halo
boundary $r_{H}$. Smooth evolution of the cosmos implies observable particle
acceleration $\gamma$ within $r_{H}$ proportional to
$\Omega_{q}(in)-\Omega_{q}(out)=\Omega_{q}(0)-\Omega_{q}(\rho_{m})$. Uniform
cosmological $\rho_{m}$ implies constant $\gamma$ for $r\leq r_{H}$,
independent of galactic mass NESM5 . This surprising result is consistent with
recent observations of galactic rotational velocities for galaxies with
directly measured massMLS16 ; NESM7 .

##  III Variational field equations

Gravitational field equations are determined by metric functional derivative
$X^{\mu\nu}=\frac{1}{\sqrt{-g}}\frac{\delta I}{\delta g_{\mu\nu}}$, where $g$
is the determinant of $g_{\mu\nu}$, and action integral $I=\int
d^{4}x\sqrt{-g}\sum_{a}{\cal L}_{a}$ WEI72 ; NES03 ; MAN06 . Given
$\delta{\cal L}=x^{\mu\nu}\delta g_{\mu\nu}$,
$X^{\mu\nu}=x^{\mu\nu}+\frac{1}{2}{\cal L}g^{\mu\nu}$. Scalar Lagrangian
density ${\cal L}_{a}$ determines variational energy-momentum tensor
$\Theta_{a}^{\mu\nu}=-2X_{a}^{\mu\nu}$, evaluated for a solution of the field
equations.

For fixed coordinates $x^{\mu}$, local Weyl scaling is defined by
$g_{\mu\nu}(x)\to g_{\mu\nu}(x)\Omega^{2}(x)$WEY18  for arbitrary real
differentiable $\Omega(x)$. Conformal symmetry is defined by invariant action
integral $I=\int d^{4}x\sqrt{-g}{\cal L}$. For any Riemannian tensor $T(x)$,
$T(x)\to\Omega^{d}(x)T(x)+{\cal R}(x)$ defines weight $d[T]$ and residue
${\cal R}[T]$. For a scalar field, $\Phi(x)\to\Phi(x)\Omega^{-1}(x)$, so that
$d[\Phi]=-1$. Conformal Lagrangian density ${\cal L}$ must have weight
$d[{\cal L}]=-4$ and residue ${\cal R}[{\cal L}]=0$ up to a 4-divergenceMAN06
. For a bare conformal field, trace $g_{\mu\nu}X_{a}^{\mu\nu}=0$ MAN06 .
Generalized Einstein equation $\sum_{a}X_{a}^{\mu\nu}=0$ is expressed as
$X_{g}^{\mu\nu}=\frac{1}{2}\sum_{a\neq g}\Theta_{a}^{\mu\nu}$. Summed trace
$\sum_{a}g_{\mu\nu}X_{a}^{\mu\nu}$ vanishes for exact field solutions.

The action integral defined by Lagrangian density ${\cal
L}_{W}=-\alpha_{g}C_{\lambda}^{\mu\kappa\nu}C^{\lambda}_{\mu\kappa\nu}$ for
Weyl tensor $C_{\lambda}^{\mu\kappa\nu}$, a traceless projection of the
Riemann tensor WEY18 , is conformally invariant. After removing a 4-divergence
MAN06 , ${\cal L}_{g}=-2\alpha_{g}(R^{\mu\nu}R_{\mu\nu}-\frac{1}{3}R^{2})$,
where $R=g_{\mu\nu}R^{\mu\nu}$. Conformal symmetry fixes the relative
coefficient of the two quadratic terms. For uniform mass/energy density
${\bar{\rho}}$ the Weyl tensor vanishes identically. Hence
$X_{g}^{\mu\nu}\equiv 0$ for a uniform cosmos.

##  IV The conformal Higgs model

Higgs
$V(\Phi^{\dagger}\Phi)=-(w^{2}-\lambda\Phi^{\dagger}\Phi)\Phi^{\dagger}\Phi$
depends on two assumed constants $w^{2}$ and $\lambda$HIG64 ; CAG98 . Nonzero
$w^{2}$ and $\lambda$ are not determined by standard theory. The conformal
Higgs model (CHM) introduces a gravitational term confirmed by observed Hubble
expansionNESM1 ; NESM2 . The CHM supports the Higgs mechanism, spontaneous
SU(2) symmetry-breaking, which also breaks conformal symmetryNESM2  and
invalidates a transformation connecting the two distinct metrics invoked for
successful fits to galactic rotation and Hubble expansionMAN06 .

In the conformal Higgs model, unique Lagrangian density ${\cal L}_{\Phi}$MAN06
; NESM1 ; NESM2  of Higgs scalar field $\Phi$ adds a gravitational term to
$-V$, so that $(\partial_{\mu}\Phi)^{\dagger}\partial^{\mu}\Phi$ is augmented
by

$\displaystyle\Delta{\cal
L}_{\Phi}=(w^{2}-\frac{1}{6}R-\lambda\Phi^{\dagger}\Phi)\Phi^{\dagger}\Phi.$
(2)

Scalar $R=g_{\mu\nu}R^{\mu\nu}$ is the trace of the Ricci tensor.

In uniform, isotropic geometry with uniform mass/energy density
${\bar{\rho}}$, ${\cal L}_{\Phi}$ implies a modified Friedmann equation FRI22
; NESM1 ; NES13  for cosmic distance scale factor $a(t)$, with $a(t_{0})=1$ at
present time $t_{0}$:

$\displaystyle\frac{{\dot{a}}^{2}}{a^{2}}+\frac{k}{a^{2}}-\frac{\ddot{a}}{a}=\frac{2}{3}({\bar{\Lambda}}+{\bar{\tau}}c^{2}{\bar{\rho}}(t)).$
(3)

${\bar{\Lambda}}=\frac{3}{2}w^{2}\geq 0$ and
${\bar{\tau}}\sim-3/\phi_{0}^{2}\leq 0$ are determined by parameters of the
Higgs modelNESM1 .

Neglecting cosmic curvature $k$ in accord with observed data, sum rule
$\Omega_{\Lambda}(t)+\Omega_{m}(t)+\Omega_{q}(t)=1$ follows if conformal
Eq.(3) is divided by ${\dot{a}}^{2}/a^{2}$, defining dimensionless Friedmann
weights
$\Omega_{\Lambda}=\frac{2}{3}\frac{{\bar{\Lambda}}a^{2}}{{\dot{a}}^{2}}$,
$\Omega_{m}=\frac{2}{3}\frac{{\bar{\tau}}c^{2}{\bar{\rho}}a^{2}}{{\dot{a}}^{2}}$,
and acceleration weight $\Omega_{q}=\frac{{\ddot{a}}a}{{\dot{a}}^{2}}$ NESM1 ;
NESM2 . Matter and radiation are combined in $\Omega_{m}$ here while Hubble
function $H(t)=\frac{{\dot{a}}}{a}(t)\sim[T^{-1}]=h(t)H_{0}$ for Hubble
constant $H_{0}=H(t_{0})$. Setting $a(t_{0})=1$,
$h(t_{0})=\frac{{\dot{a}}}{a}(t_{0})=1$ in Hubble units of time $1/H_{0}$,
length $c/H_{0}$, and acceleration $H_{0}^{2}c/H_{0}=cH_{0}$. Evaluated at
time $t$, Ricci scalar
$R(t)=\frac{{\ddot{a}}}{a}+\frac{{\dot{a}}^{2}}{a^{2}}+\frac{k}{{\dot{a}}^{2}}$.

For $k=0$ the standard Friedmann equation, divided by $\frac{{\dot{a}}}{a}$,
produces dimensionless sum rule $\Omega_{\Lambda}(t)+\Omega_{m}(t)=1$.
$\Omega_{m}=1-\Omega_{\Lambda}$ requires mass density far greater than
observed baryonic mass. This has been considered to be a strong argument for
dark matter. Omitting $\Omega_{m}$ completely, with $k=0$, conformal sum rule
$\Omega_{\Lambda}(t)+\Omega_{q}(t)=1$ fits observed data accurately for
redshifts $z\leq 1$ (7.33Gyr) NESM1 ; NESM2 ; NESM5 . This eliminates any need
for dark matter to explain Hubble expansion.

Table 1: Scaled luminosity distance fit to Hubble data |  |  | Theory | Observed
---|---|---|---|---
z | $\Omega_{\Lambda}$ | $\Omega_{q}$ |  $H_{0}d_{L}/c$Eq.(IV) |  $H_{0}d_{L}/c$MAN03
0.0 | 0.732 | 0.268 | 0.0000 | 0.0000
0.2 | 0.578 | 0.422 | 0.2254 | 0.2265
0.4 | 0.490 | 0.510 | 0.5013 | 0.5039
0.6 | 0.434 | 0.566 | 0.8267 | 0.8297
0.8 | 0.393 | 0.607 | 1.2003 | 1.2026
1.0 | 0.363 | 0.637 | 1.6209 | 1.6216

Luminosity distance $d_{L}(z)=(1+z)\chi(z)$, for $\Omega_{k}=0$, is shown in
Table(1) for $\alpha=\Omega_{\Lambda}(t_{0})=0.732$, whereNESM2

$\displaystyle\chi(z)=\int_{t(z)}^{t_{0}}\frac{dt}{a(t)}=\int_{z}^{0}dz(1+z)\frac{dt}{dz}=$
$\displaystyle\int_{0}^{z}\frac{dz}{\sqrt{2\alpha\ln(1+z)+1}}.$ (4)

Observed redshifts have been fitted to an analytic functionMAN03  with
statistical accuracy comparable to the best standard $\Lambda$CDM fit, with
$\Omega_{m}=0$. Table(1) compares CHM $d_{L}(z)$ to this Mannheim function.
Because ${\bar{\tau}}$ is negativeMAN06 ; NESM1 , cosmic acceleration
$\Omega_{q}$ remains positive (centrifugal) back to the earliest timeNESM1 .

Conformal Friedmann Eq.(3)NESM1 ; NESM2  determines cosmic acceleration weight
$\Omega_{q}$. With both weight parameters $\Omega_{k}$ and $\Omega_{m}$ set to
zero, Eq.(3) fits scaled Hubble function $h(t)=H(t)/H_{0}$ for redshifts
$z\leq 1$, Table(1), as accurately as standard LCDM, with only one free
constant. This determines Friedmann weights, at present time $t_{0}$,
$\Omega_{\Lambda}=0.732,\Omega_{q}=0.268$ NESM1 . Hubble constant
$H(t_{0})=H_{0}=2.197\times 10^{-18}/s$ PLC15  is independent of these data.
The dimensionless sum ruleNESM1  with $\Omega_{k}=0$ determines
$\Omega_{q}(\rho_{m})=1-\Omega_{\Lambda}-\Omega_{m}$ in the cosmic background,
and $\Omega_{q}(0)=1-\Omega_{\Lambda}$ in the depleted haloNESM3 .

##  V Conformal gravity

The unique Lagrangian density ${\cal L}_{g}$ of conformal gravity, constructed
from the conformal Weyl tensor WEY18 ; MAK89 ; MAN06 , determines
Schwarzschild gravitational potential

$\displaystyle B(r)=-2\beta/r+\alpha+\gamma r-\kappa r^{2},$ (5)

valid outside a spherically symmetric mass/energy source density MAK89 ; MAN91
. Classical gravitation is retained at subgalactic distances by setting
$\beta=Gm/c^{2}$ for a spherical source of mass $m$. CG adds two constants of
integration to the classical external potential: nonclassical radial
acceleration $\gamma$ and halo cutoff parameter $\kappa$ NESM3 , with
negligible effects at subgalactic distancesMAN06 .

From the CHM, observed nonclassical gravitational acceleration
$\frac{1}{2}\gamma c^{2}$ in the halo is proportional to
$\Delta\Omega_{q}=\Omega_{q}(0)-\Omega_{q}(\rho_{m})=\Omega_{m}(\rho_{m})$NESM3
, where, given $\rho_{m}$ and $H_{0}$,
$\Omega_{m}(\rho_{m})=\frac{2}{3}\frac{{\bar{\tau}}c^{2}\rho_{m}}{H^{2}_{0}}$NESM1
. Thus the depleted halo model determines constant $\gamma$ from uniform
universal cosmic baryonic mass density $\rho_{m}/c^{2}$, which includes
radiation energy density here.

Converted from Hubble units, this implies centripetal acceleration
$\frac{1}{2}\gamma c^{2}=-cH_{0}\Omega_{m}(\rho_{m})$NESM3 . Positive
$\rho_{m}$ implies $\Omega_{m}<0$ because coefficient ${\bar{\tau}}<0$ MAN06 ;
NESM1 . Hence $\Delta\Omega_{q}=\Omega_{m}<0$ is consistent with nonclassical
centripetal acceleration $\frac{1}{2}\gamma c^{2}$, confirmed by inward
deflection of photon geodesics observed in gravitational lensing NESM3 . This
logic is equivalent to requiring radial acceleration to be continuous across
halo boundary $r_{H}$:

$\displaystyle\frac{1}{2}\gamma_{H}c^{2}-cH_{0}\Omega_{q}(0)=-cH_{0}\Omega_{q}(\rho_{m}).$
(6)

Signs here follow from the definition of $\Omega_{q}$ as centrifugal
acceleration weight.

Given mass/energy source density $f(r)$ enclosed within ${\bar{r}}$, the field
equation in the ES metric is MAK89 ; MAN91

$\displaystyle\partial_{r}^{4}(rB(r))=rf(r),$ (7)

for $f(r)\sim(\Theta_{0}^{0}-\Theta_{r}^{r})_{m}$ determined by source energy-
momentum tensor $\Theta^{\mu\nu}_{m}$MAN06 .

For constants related by $\alpha^{2}=1-6\beta\gamma$ MAN91 ,

$\displaystyle y_{0}(r)=rB(r)=-2\beta+\alpha r+\gamma r^{2}-\kappa r^{3},$ (8)

is a solution of the tensorial field equation for source-free $r\geq{\bar{r}}$
MAK89 ; MAN91 . Derivative functions $y_{i}(r)=\partial^{i}_{r}(rB(r)),0\leq
i\leq 3$ satisfy differential equations

$\displaystyle\partial_{r}y_{i}=y_{i+1},0\leq i\leq 2,$
$\displaystyle\partial_{r}y_{3}=rf(r).$ (9)

The general solution, for independent constants $c_{i}=y_{i}(0)$, determines
coefficients $\beta,\alpha,\gamma,\kappa$ such that at endpoint ${\bar{r}}$

$\displaystyle
y_{0}({\bar{r}})=-2\beta+\alpha{\bar{r}}+\gamma{\bar{r}}^{2}-\kappa{\bar{r}}^{3},$
$\displaystyle y_{1}({\bar{r}})=\alpha+2\gamma{\bar{r}}-3\kappa{\bar{r}}^{2},$
$\displaystyle y_{2}({\bar{r}})=2\gamma-6\kappa{\bar{r}},$ $\displaystyle
y_{3}({\bar{r}})=-6\kappa.$ (10)

Gravitational potential $B(r)$ is required to be differentiable and free of
singularities. $c_{0}=0$ prevents a singularity at the origin. Values of
$c_{1},c_{2},c_{3}$ can be chosen to match outer boundary conditions
$\alpha=1,\gamma=0,\kappa=0$ at $r={\bar{r}}$. This determines parameter
$\beta$. Specific values of $\gamma$ and $\kappa$, consistent with Hubble
expansion and the observed galactic dark halo NESM1 ; NESM3 , are fitted by
adjusting $c_{1},c_{2},c_{3}$, subject to $c_{0}=0,\alpha^{2}=1-6\beta\gamma$.

A particular solution for $B(r)$ MAK89 ; MAN91 , assumed by subsequent
authors, derives an integral for $\gamma$ that vanishes for residual source
density ${\hat{\rho}}$. This is replaced here by an alternative solution for
which $\gamma$ is a free parameterNESM5 .

For a single spherical solar mass isolated in a galactic halo, mean internal
mass density ${\bar{\rho}}_{\odot}$ within $r_{\odot}$ determines an exact
solution of the conformal Higgs gravitational equation, giving internal
acceleration $\Omega_{q}({\bar{\rho}}_{\odot})$.

Given $\gamma$ outside $r_{\odot}$, continuous acceleration across boundary
$r_{\odot}$,

$\displaystyle\frac{1}{2}\gamma_{\odot,in}c^{2}-cH_{0}\Omega_{q}({\bar{\rho}}_{\odot})=\frac{1}{2}\gamma
c^{2}-cH_{0}\Omega_{q}(0),$ (11)

determines constant $\gamma_{\odot,in}$ valid inside $r_{\odot}$.
$\gamma_{\odot,in}$ is determined by local mean source density
${\bar{\rho}}_{\odot}$. $\gamma$ in the halo is not changed. Its value is a
constant of integration that cannot vary in the source-free haloNESM3 ; NESM5
. Hence there is no way to determine a mass-dependent increment to $\gamma$.
This replaces the usually assumed $\gamma=\gamma_{0}+N^{*}\gamma^{*}$ by
$\gamma=\gamma_{H}$, determined at halo boundary $r_{H}$.

Anomalous rotation velocities for 138 galaxies have been fitted to Eq.(1)
using four assumed universal parameters
$\beta^{*},\gamma^{*},\gamma_{0},\kappa$ MAN06 ; MAN12 ; MAO11 ; OAM15  such
that $\beta=N^{*}\beta^{*}=GM/c^{2},\gamma=\gamma_{0}+N^{*}\gamma^{*}$.
$N^{*}$ is galactic baryonic mass $M$ in solar mass units. Inferred parameter
values MAN06 ; MAO12 ,

$\displaystyle\beta^{*}=1.475\times 10^{3}m,\gamma_{0}=3.06\times 10^{-28}/m,$
$\displaystyle\gamma^{*}=5.42\times 10^{-39}/m,\kappa=9.54\times
10^{-50}/m^{2},$ (12)

fit conformal gravity to galactic rotation velocities. The depleted halo model
removes the galactic mass dependence of nonclassical acceleration parameter
$\gamma$. For our Milky Way galaxy, with $N^{*}=6.07\times 10^{10}$ MCG08 ;
OAM15 , implied $\gamma_{H}=\gamma_{0}+N^{*}\gamma^{*}=6.35\times 10^{-28}/m$.
Parameter $\kappa$ provides a cutoff of modified radial acceleration at a
large halo radiusNESM5 .

##  VI Reconciliation of the two distinct gravitational models

CG and CHM must be consistent for an isolated galaxy and its dark halo,
observed by gravitational lensing. CG is valid for anomalous outer galactic
rotation velocities in the static spherical Schwarzschild metric, solving a
differential equation for Schwarzschild gravitational potential $B(r)$ MAK89 ;
MAN06 . The CHM is valid for cosmic Hubble expansion in the uniform, isotropic
FLRW metric, solving a differential equation for Friedmann scale factor $a(t)$
NESM1 . Concurrent validity is achieved by introducing a common hybrid metric.
The two resulting gravitational equations are decoupled by separating
mass/energy source density $\rho$ into uniform isotropic mean density
${\bar{\rho}}$ and residual ${\hat{\rho}}=\rho-{\bar{\rho}}$, which extends
only to galactic radius $r_{G}$ and integrates to zero over the defining
volume.

The conformally invariant action integral of conformal gravity is defined by a
quadratic contraction of the Weyl tensor, simplified by removing a
4-divergence MAN06 ; MAN12 . For uniform density ${\bar{\rho}}$ the Weyl
tensor vanishes identically, so that $X_{g}^{\mu\nu}\equiv 0$ for a uniform,
isotropic cosmos.

Observed excessive galactic rotational velocities have been studied and
parametrized using conformal Weyl Lagrangian density ${\cal L}_{g}$ MAN06 ;
MAN12 . The generalized Einstein equation exactly cancels any vacuum energy
density. Hubble expansion has been parametrized using conformal Higgs scalar
field Lagrangian density ${\cal L}_{\Phi}$ NESM1 .

Metric tensor $g_{\mu\nu}$ is determined by conformal field equations derived
from ${\cal L}_{g}+{\cal L}_{\Phi}$ NESM3 , driven by energy-momentum tensor
$\Theta_{m}^{\mu\nu}$, where subscript $m$ refers to conventional matter and
radiation. The gravitational field equation within halo radius $r_{H}$ is

$\displaystyle
X_{g}^{\mu\nu}+X_{\Phi}^{\mu\nu}=\frac{1}{2}\Theta_{m}^{\mu\nu}.$ (13)

Defining mean density ${\bar{\rho}}_{G}$ and residual density
${\hat{\rho}}_{G}=\rho_{G}-{\bar{\rho}}_{G}$, and assuming
$\Theta_{m}^{\mu\nu}(\rho)\simeq\Theta_{m}^{\mu\nu}({\bar{\rho}})+\Theta_{m}^{\mu\nu}({\hat{\rho}})$,
solutions for $r\leq r_{G}$ of the two equations

$\displaystyle
X_{g}^{\mu\nu}=\frac{1}{2}\Theta_{m}^{\mu\nu}({\hat{\rho}}_{G}),X_{\Phi}^{\mu\nu}=\frac{1}{2}\Theta_{m}^{\mu\nu}({\bar{\rho}}_{G})$
(14)

decouple and imply a solution of the full equation. This removes any mean
density source from the $X_{g}$ equation, leaving only the residual density,
which integrates to zero over a closed volume and cancels vacuum energy.

The decoupled solutions require a composite hybrid metric such as NESM5

$\displaystyle
ds^{2}=-B(r)dt^{2}+a^{2}(t)(\frac{dr^{2}}{B(r)}+r^{2}d\omega^{2}).$ (15)

Solutions in the two distinct primitive metrics are made consistent by fitting
parameters to boundary conditions and setting cosmic curvature constant $k=0$,
justified by currently observed data.

##  VII Baryonic Tully-Fisher and radial acceleration relations

Static spherical geometry defines Schwarzschild potential $B(r)$. For a test
particle in a stable exterior circular orbit with velocity $v$ the centripetal
acceleration is $a=v^{2}(r)/r=\frac{1}{2}B^{\prime}(r)c^{2}$. Newtonian
$B(r)=1-2\beta/r$, where $\beta=GM/c^{2}$, so that $a_{N}=\beta
c^{2}/r^{2}=GM/r^{2}$.

CG adds nonclassical $\Delta a$ to $a_{N}$, so that orbital velocity squared
is the sum of $v^{2}(a_{N};r)$ and $v^{2}(\Delta a;r)$, which cross with equal
and opposite slope at some $r=r_{TF}$. This defines a flat range of $v(r)$
centered at stationary point $r_{TF}$, without constraining behavior at large
$r$.

MONDMIL83 ; SAN10 ; FAM12  modifies the Newtonian force law for acceleration
below an empirical scale $a_{0}$. Using $y=a_{N}/a_{0}$ as independent
variable, for assumed universal constant $a_{0}\simeq 10^{-10}m/s^{2}$, MOND
postulates an interpolation function $\nu(y)$ such that observed radial
acceleration $a=f(a_{N})=a_{N}\nu(y)$. A flat velocity range approached
asymptotically requires $a^{2}\to a_{N}a_{0}$ as $a_{N}\to 0$. For $a_{N}\gg
a_{0}$, $\nu\to 1$ and for $a_{N}\ll a_{0}$, $\nu^{2}\to 1/y$. This implies
asymptotic limit $a^{2}\to a_{0}a_{N}$ for small $a_{N}$, which translates
into an asymptotically flat galactic velocity function $v(r)$ for large
orbital radius $r$MIL83 . For $a_{N}\ll a_{0}$, MOND $v^{4}=a^{2}r^{2}\to
GMa_{0}$, the empirical baryonic Tully-Fisher relationTAF77 ; MCG05 ; MCG11 ;
OCM18 .

In conformal gravity (CG), centripetal acceleration $a=v^{2}/r$ determines
exterior orbital velocity $v^{2}/c^{2}=ra/c^{2}=\beta/r+\frac{1}{2}\gamma
r-\kappa r^{2}$, compared with asymptotic $ra_{N}/c^{2}=\beta/r$. If the
asymptotic Newtonian function is valid at $r$ and $2\kappa r/\gamma$ can be
neglected, the slope of $v^{2}(r)$ vanishes at $r^{2}_{TF}=2\beta/\gamma$.
This implies that $v^{4}(r_{TF})/c^{4}=(\beta/r_{TF}+\frac{1}{2}\gamma
r_{TF})^{2}=2\beta\gamma$MAN97 ; OCM18 . This is the Tully-Fisher relation,
exact at stationary point $r_{TF}$ of the $v(r)$ function. Given
$\beta=GM/c^{2}$, $v^{4}=2GM\gamma c^{2}$, for relatively constant $v(r)$
centered at $r_{TF}$.

McGaugh et alMLS16  have recently shown for 153 disk galaxies that observed
radial acceleration $a$ is effectively a universal function of the expected
classical Newtonian acceleration $a_{N}$, computed for the observed baryonic
distribution. Galactic mass is determined directly by observation, removing
uncertainty due to adjustment of mass-to-light ratios for individual galaxies
in earlier studies. The existence of such a universal correlation function,
$a(a_{N})=a_{N}\nu(a_{N}/a_{0})$ is a basic postulate of MONDMIL83 ; FAM12 .

CG implies a similar correlation function if nonclassical parameter $\gamma$
is mass-independentNESM7 . Outside an assumed spherical source mass, conformal
Schwarzschild potential $B(r)$ determines circular geodesics such that
$v^{2}/c^{2}=ra/c^{2}=\frac{1}{2}rB^{\prime}(r)=\beta/r+\frac{1}{2}\gamma
r-\kappa r^{2}$. The Kepler formula is $ra_{N}/c^{2}=\beta/r$. Well inside a
galactic halo boundary, $2\kappa r/\gamma$ can be neglected. This defines
correlation function $a(a_{N})=a_{N}+\Delta a$ if $\Delta a=\frac{1}{2}\gamma
c^{2}$ is a universal constant NESM7 , which is implied by the depleted halo
modelNESM3 . This requires reconsideration of the definition
$\gamma=\gamma_{0}+N^{*}\gamma^{*}$ used in fitting rotation data for 138
galaxies to CG MAN06 ; MAO12 . For comparison with CG for the Tully-Fisher
relation, CG would agree with MOND $v^{4}=GMa_{0}$ if $a_{0}=2\gamma
c^{2}$MAN97 , for mass-independent $\gamma$. CG $\gamma=6.35\times 10^{-28}/m$
implies MOND $a_{0}=1.14\times 10^{-10}m/s^{2}$.

##  VIII Higgs parameter $\lambda$

The conformal scalar field equation, including parametrized $\Delta{\cal
L}_{\Phi}$, is MAN06 ; NESM1 ; NESM5

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}\partial^{\mu}\Phi)=-(\frac{1}{6}R-w_{0}^{2}+2\lambda\Phi^{\dagger}\Phi)\Phi.$
(16)

For $k=0$,
$\frac{1}{6}R(t)=h^{2}(t)(1+\Omega_{q}(t))>h^{2}(t)\Omega_{\Lambda}=w(t)^{2}$,
where $h(t)={\dot{a}}/a$ in Hubble unitsNESM1 .

Ricci scalar $R$ introduces gravitational effects. Time-dependent
$R(t)=6(\xi_{0}(t)+\xi_{1}(t))$, where $\xi_{0}(t)=\frac{\ddot{a}}{a}$ and
$\xi_{1}(t)=\frac{{\dot{a}}^{2}}{a^{2}}+\frac{k}{a^{2}}$NESM1 . For
$h(t)={\dot{a}}/a$ and $k=0$,
$\frac{1}{6}R(t)=h^{2}(t)(2-\Omega_{\Lambda}(t)-\Omega_{m}(t))=h^{2}(t)(1+\Omega_{q}(t))>h^{2}(t)\Omega_{\Lambda}(t)))=w(t)^{2}$.
Hence $\zeta(t)=\frac{1}{6}R(t)-w(t)^{2}>0$.

Only real-valued solution $\phi(t)$ is relevant in uniform, isotropic
geometry. The field equation is

$\displaystyle\frac{{\ddot{\phi}}}{\phi}+3\frac{{\dot{a}}}{a}\frac{{\dot{\phi}}}{\phi}=-(\frac{1}{6}R(t)-w(t)^{2}+2\lambda\phi^{2}).$
(17)

Omitting $R$ and assuming constant $\lambda>0$, Higgs solution
$\phi_{0}^{2}=w_{0}^{2}/2\lambda$ HIG64  is exact. All time derivatives drop
out. In the conformal scalar field equation, cosmological time dependence of
Ricci scalar $R(t)$, determined by the CHM Friedmann cosmic evolution
equation, introduces nonvanishing time derivatives.
$w_{0}^{2}=\frac{{\dot{\phi}}^{2}}{\phi^{2}}$ from the Higgs field covariant
derivative is consistent with constant $\lambda$. Time-dependent terms in the
scalar field equation can be included in
$w^{2}(t)=\frac{{\dot{\phi}}^{2}}{\phi^{2}}-\frac{{\ddot{\phi}}}{\phi}-3h(t)\frac{{\dot{\phi}}}{\phi}$.
For $\zeta(t)=\frac{1}{6}R(t)-w^{2}(t)>0$, $\phi^{2}(t)=-\zeta(t)/2\lambda$ is
an exact solution of Eq.(17). $\zeta>0$ for computed $R(t)$NESM1  implies
$\lambda<0$.

$\hbar\phi(t_{0})=174GeV$AMS08 $=1.203\times 10^{44}\hbar H_{0}$. In Hubble
units, for $\Omega_{m}=0$, $\zeta(t_{0})=2\Omega_{q}(t_{0})=0.536$. For
empirical $\phi(t)$, $\lambda(t)=\zeta/(-2\phi^{2})$. Given $\zeta(t_{0})$ and
$\phi(t_{0})$, dimensionless $\lambda(t_{0})=-0.185\times 10^{-88}$ NESM5 .

##  IX $W_{2}$ particle and $Z_{2}$ resonance

Conformal theory obtains empirical parameters that preserve the standard
electroweak model, but preclude a massive Higgs particleNESM5 . The observed
LHC 125GeV resonanceDJV12 ; ATL12 ; CMS12  requires an alternative
explanation.

A model Hamiltonian matrix can be defined NESM4  in which indices 0,1 refer
respectively to bare neutral scalar states
$WW=g_{\mu\nu}W_{-}^{\mu}W_{+}^{\nu}$, $ZZ=g_{\mu\nu}Z^{\mu*}Z^{\nu}$. Assumed
diagonal elements are $H_{00}=2m_{W}=160GeV$, $H_{11}=2m_{Z}=182GeV$, for
empirical masses $m_{W}$ and $m_{Z}$. Intermediate quark and lepton states
define a large complementary matrix ${\tilde{H}}$ indexed by $i,j\neq 0,1$,
with eigenvalues $\epsilon_{i}$, and off-diagonal elements
${\tilde{A}}_{i0},{\tilde{A}}_{i1}$. ${\tilde{H}}$ determines energy-dependent
increments in a $2\times 2$ reduced matrix

$\displaystyle H_{ab}-\mu_{ab}=H_{ab}-\sum_{i\neq
0,1}{\tilde{A}}^{\dagger}_{ai}(\epsilon_{i}-\epsilon)^{-1}{\tilde{A}}_{ib}.$
(18)

$H_{01}-\mu_{01}=(WW|H_{red}|ZZ)$ corresponds to Feynman diagrams for quark
and lepton exchange. The most massive and presumably most strongly coupled
intermediate field that interacts directly would be tetraquark
$T=t{\bar{b}}b{\bar{t}}$, whose mass is estimated as $\epsilon_{T}=350$GeV.
The very strong interaction between bare fields ZZ and WW is assumed to be
dominated by tetraquark exchange.

A simplified estimate of $W_{2}$ energy is obtained by restricting
intermediate states to the three color-indexed tetraquark states
$T=t{\bar{b}}b{\bar{t}}$, and assuming elements
${\tilde{A}}_{T0},{\tilde{A}}_{T1}$ of equal magnitude $\alpha/\sqrt{3}$. For
the reduced $2\times 2$ matrix, matrix increments
$\mu_{ab}\simeq\mu(\epsilon)=\frac{\alpha^{2}}{\epsilon_{T}-\epsilon}$ are all
defined by a single parameter $\alpha^{2}$. Secular equation

$\displaystyle(2m_{W}-\mu(\epsilon)-\epsilon)(2m_{Z}-\mu(\epsilon)-\epsilon)=\mu^{2}(\epsilon)$
(19)

is to be solved for two eigenvalues $\epsilon=E_{0},E_{1}$.

It is found that identifying the model diboson $W_{2}$ with the recently
observed LHC 125GeV resonanceDJV12 ; ATL12 ; CMS12  confirms the empirical
value of Higgs parameter $\lambda$. In Higgs
$V(\Phi^{\dagger}\Phi)=-(w^{2}-\lambda\Phi^{\dagger}\Phi)\Phi^{\dagger}\Phi$,
coefficient $w^{2}$ results from dressing the bare massless Higgs scalar field
by neutral gauge field $Z_{\mu}$, while coefficient $\lambda$ results from
dressing by $W_{2}$ NESM4 ; NESM5 .

Setting $E_{0}=125GeV=0.8644\times 10^{44}\hbar H_{0}$ for the $W_{2}$ state,
dominated by the bare $WW$ field, determines parameters $\alpha^{2}=4878$
GeV2, $\mu(E_{0})=21.68$GeV and $\tan\theta_{x}=0.6138$. Using $\alpha^{2}$
determined by $E_{0}$, the present model predicts $E_{1}=173$GeV, with
$\mu(E_{1})=27.62$GeV. This higher eigenvalue is the energy of a resonance
$Z_{2}$ dominated by the bare $ZZ$ field. $Z_{2}$ decay into bare $WW$, two
free charged gauge bosons, is allowed by energy conservation, but not into
bare $ZZ$. Composite field $W_{2}$ cannot decay spontaneously into either $WW$
or $ZZ$.

Identifying $E_{0}$ with the observed 125GeV resonance, and using
$g_{w}=0.6312$ and $g_{z}=0.7165$ with computed $\tan\theta_{x}=0.6138$, the
implied value of Higgs parameter $\lambda=-\frac{1}{4}g_{w}^{2}g_{z}^{2}\sin
2\theta_{x}(\frac{{\dot{\phi}}_{0}}{\phi_{0}})^{2}\hbar^{2}/m^{2}_{W_{2}}c^{4}=-0.455\times
10^{-88}$ is consistent with its empirical value $\lambda\simeq-10^{-88}$
NESM4 ; NESM5 .

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