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% Higgs-Field and a New Scalar-Tensor Theory of Gravity
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%   International Journal of Theoretical Physics, 31(1):109-114, 1992

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{ \Large{Higgs-Field and a New Scalar-Tensor Theory of Gravity }}
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H. Dehnen, H. Frommert, and F. Ghaboussi

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Universit\"at Konstanz 

Postfach 55 60 

7750 Konstanz 

West Germany 
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{\Large{Abstract}}

The combination of Brans and Dicke's idea of a variable gravitational
constant with the Higgs-field mechanism results in a renormalizable
theory of gravity. Einstein's theory is realized after symmetry breaking
in the neighbourhood of the Higgs-field ground-state. 

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There exist today in the literature two fundamental scalar fields
connected with the mass problem. First of all Brans and Dicke [1]
introduced a scalar field with the intention following Mach's principle
[2], that the active as well as passive gravitational mass $m_0 \sqrt
G$, that means the gravitational "constant" G, is not a constant but a
function determined by the other particles of the Universe. In this way
also the problem of Dirac's large cosmological numbers should be solved.
Secondly, in elementary particle physics the inertial mass $m_0$ is
generated with respect to the gauge invariance by the interaction with
the scalar Higgs-field, the source of which is also given by the
particles in the Universe [3]. Because of the identity of gravitational
and inertial mass (equivalence principle) it seems to be meaningful, if
not even necessary to identify these two approaches. Then the
Lagrange-density has the unique form $(\hbar = 1, \: c = 1)$: 
$$
{\cal{L}} = \left[ \frac{1}{16 \pi } \alpha \phi ^{\dagger} \phi R + \frac{1}{2}\phi ^{\dagger} _ {|| \mu } \phi^{|| \mu } 
- V(\phi) \right] \sqrt {-g} + L_M \sqrt {-g}
\leqno (1)
$$
with the Higgs-potential ($\mu ^2, \lambda $ real valued constants)
$$
V(\phi) = \frac{\mu ^2}{2} \phi^ {\dagger} \phi + 
\frac{\lambda }{4 !} (\phi^{\dagger} \phi)^2 + 
\frac{3}{2} \frac{\mu  ^4}{\lambda } \: . 
\leqno (1a)
$$
Herein $\phi$ is an U(N) iso-vector, $|| \mu $ means its covariant
derivative, $R$ is the Ricci-scalar and $\alpha $ a dimensionless
 factor, whereas $L_M$ contains the fermionic and massless bosonic
fields belonging to the inner gauge-group $U(N)$. Obviously, the
positive Higgs-field quantity $\phi^{\dagger} \phi$ (c.f. eq. (9a))
plays the role of a variable reciprocal gravitational "constant".
Formula (1) is related to a generalization of Brans and Dicke's theory
proposed by Bergmann [4] and Wagoner [5] as well as by Zee [6]. We want
to point here to some interesting features of the ansatz (1), which
unifies gravity with the other interactions using a minimum of effort. 

Before symmetry breaking the theory following from (1) contains no
gravitational constant; the only dimensional free parameters are those
of the Higgs-potential. Such a theory of gravity may be   renormalizable
according to the criterion given by de Witt [7], although it is not
unitary. - Concerning symmetry breaking the ground-state of the
Higgs-field is given by $( \mu ^2 < 0) $
$$
\phi^ {\dagger} _ 0 \phi_ 0 = v^2 = 
\frac{-6 \mu ^2}{\lambda } \quad 
\leqno (2)
$$
with 
$$
V( \phi_0) = 0. 
\leqno (2a)
$$
By this ground state the quantity 
$$
G = (\alpha v^2) ^{-1}
\leqno (3)
$$
related to Newton's gravitational constant (see below), as well as the
mass of the gauge bosons
$$
M_W = \sqrt {\pi} \, g v
\leqno (4)
$$
are determined ($g$ coupling constant of the gauge group $U(N)$).
Accordingly the factor $\alpha $ means the ratio
$$
\alpha \simeq ( M_P / M_W ) ^2 >> 1, 
\leqno (5)
$$
where $M_P $ is the Planck mass. 

The field equations for gravity and Higgs-field following from
 (1) take the form:
$$
R_{\mu \nu } - \frac{1}{2}R g _ {\mu \nu } + 
\frac{8 \pi }{\alpha \phi^{\dagger} \phi} V( \phi) g_{\mu \nu } = 
$$
$$
= - \frac{8 \pi }{\alpha \phi^ {\dagger} \phi} T_ {\mu \nu } - 
\frac{4 \pi }{\alpha \phi^ {\dagger} \phi} 
\left[
\phi^ {\dagger} _ {||\mu } \phi_ {||\nu } + 
\phi^ {\dagger} _ {||\nu } \phi_ {||\mu } 
\right] + 
$$
$$
+ \frac{4 \pi }{\alpha \phi^ {\dagger} \phi} 
\phi^ {\dagger} _{||\lambda } \phi^{||\lambda } g_ {\mu \nu } - 
\frac{1}{\phi^{\dagger} \phi} 
\left[( \phi^{\dagger} \phi) _{||\mu ||\nu } - 
( \phi^{\dagger} \phi)^{||\beta } {}_{|| \beta } \, g_{\mu \nu } \right]
\leqno (6)
$$
and
$$
\phi^{||\mu }{}_{||\mu } - \frac{1}{8 \pi }\alpha \phi R + 
\mu ^2 \phi + \frac{\lambda }{6} (\phi^{\dagger}  \phi)\phi = 0
\leqno (7)
$$
as well as the adjoint equation of (7). Herein $T_{\mu \nu }$ is the
symmetric energy momentum tensor belonging to $L_M \sqrt {-g}$ in (1)
alone. The conservation laws of energy and momentum read
$$
T_{\mu  } {}^{\nu }{}_{|| \nu } = 0 \: .
\leqno (8)
$$

Now we perform the symmetry breaking and introduce the unitary gauge. If
with respect to (2)
$$
\phi_0 = v N; \quad N^{\dagger} N = 1; \quad N = \mbox{const.}
\leqno (9)
$$
represents the ground-state, the Higgs-field $\phi$ can be brought
 within the unitary gauge into the form:
$$
\phi = \rho N, \quad \rho^2 = \phi^{\dagger} \phi. 
\leqno (9a)
$$
For the following we use therefore instead of $\phi$ the real valued
field quantity
$$
\varphi = \rho /v
\leqno (10)
$$
($\varphi = 1 $ represents the ground-state). 
Restricting ourselves to the field equations for gravity i.e. the metric
$g_ {\mu \nu }$ and the Higgs-field $\varphi $ we find from (6) and (7) 
( $| \mu $ means the usual partial derivative):
$$
R_{\mu \nu } - \frac{1}{2} R g_{\mu \nu } + \frac{12 \pi }{\alpha v^2}
\frac{\mu ^4}{\lambda } \varphi ^{-2} (\varphi ^2 -1)^2 g_{\mu \nu } = 
$$
$$
= - \frac{8 \pi }{\alpha v^2} \varphi ^{-2} T_{\mu \nu } 
- \frac{8 \pi }{\alpha }( 1 + \frac{\alpha }{4 \pi }) 
\varphi ^{-2} \varphi _{|\mu }\varphi _{|\nu } + 
$$
$$
+ \frac{4 \pi }{\alpha } ( 1 + \frac{\alpha }{2 \pi }) 
\varphi ^{-2} \varphi ^{|\lambda }\varphi _{|\lambda } 
g_{\mu \nu }- 
$$
$$
- 2 \varphi ^{-1} \left[ \varphi _{|\mu || \nu } - \varphi ^{|\lambda }
{}_ {|| \lambda } \, g_{\mu \nu } \right]
\leqno (11)
$$
and 
$$
\frac{4 \pi }{\alpha } (1 + \frac{3 \alpha }{4 \pi } )
\varphi ^{2|\mu } {}_{||\mu } + \frac{48 \pi }{\alpha v^2} 
\frac{\mu ^4}{\lambda } (\varphi ^2 -1) = 
\frac{8 \pi }{\alpha v^2}T \: .
\leqno (12)
$$
With respect to (3) and (5) we obtain from (11) and (12) the final result:
$$
R_{\mu \nu } - \frac{1}{2}R g_{\mu \nu } + 
12 \pi G \frac{\mu ^4}{\lambda } \varphi ^{-2}
(\varphi ^2-1)^2 g_ {\mu \nu } = 
$$
$$
= - 8 \pi G \varphi ^{-2} T_{\mu \nu } - 
2 \varphi ^{-2} \varphi _{|\mu }\varphi _{|\nu } + 
$$
$$
+ 2 \varphi ^{-2} \varphi ^{|\lambda }
\varphi _{| \lambda } \, g_{\mu \nu } - 
2 \varphi ^{-1} \left[\varphi _{|\mu ||\nu } - 
\varphi ^{| \lambda } {}_{||\lambda } g_{\mu \nu } \right]
\leqno (13)
$$
and 
$$\varphi ^{2| \mu } {}_{|| \mu } + 16 \pi G 
\frac{\mu ^4}{\lambda } ( \varphi ^2 -1) = 
\frac{8 \pi G}{3} T \: .
\leqno (14)
$$

There are two very important differences with respect to Brans and
Dicke's scalar tensor theory. First, the scalar field $\varphi $
possesses a finite range $l = M_ \varphi ^{-1}$ corresponding to the
mass term in (14) according to which the excited Higgs-field has the
mass square:
$$
M_ \varphi ^2 = 16 \pi G \frac{\mu ^4}{\lambda } \: .
\leqno (15)
$$This is smaller than the usual value by the factor $\alpha ^{-1}$. In
this connection we note, that $G$ in (3), (13) and (14) represents
Newton's gravitational constant only  up to a factor of the order of
one. The exact connection between $G$ and the Newtonian value $G_N$ is
given by the Newtonian limit of (13) and (14) and this depends, as shown
below, on the value of $l$ for the range of the scalar field. In case of
a suitable choice of this range also no difficulties with respect to the
solar-relativistic effects or gravitational waves are to be expected;
however the possibility of a fifth force of Yukawa type is given. 

Secondly there exists according to the left hand side of (13) a
cosmological function (instead of a cosmological constant) 
$$
\lambda (\varphi ) =  12 \pi G \frac{\mu ^4}{\lambda } \varphi ^{-2} (
\varphi ^2 -1)^2,
\leqno (16)
$$
which is necessarily positive. This is very interesting because a
positive value of a cosmological function (constant) corresponds to a
positive mass density, so that this theory could solve the problem of
missing mass in cosmology automatically. 

For the ground-state of the Higgs-field ($\varphi \equiv 1)$ the
cosmological function $\lambda(\varphi )$ vanishes (see also (2a)) and
from (13) and (14) it follows:
$$
R_{\mu \nu } - \frac{1}{2}R g_{\mu \nu } = - 8 \pi G T _{\mu \nu },
\quad T = 0.
\leqno (17)
$$
This is Einstein's theory with light-like matter. Einstein's theory is
realized only after symmetry breaking in the neighbourhood of the
ground-state. Of course, in case of vanishing energy momentum tensor the
ground-state is realized by the Minkowski space-time and $ \varphi = 1$.


Finally we investigate the Newtonian limit. For this we set 
$$
g_{\mu \nu } = \eta_{\mu \nu } + h_ {\mu \nu } ; \quad \varphi = 1 + \zeta 
\leqno (18)
$$
and linearize with respect to 
$|h_ {\mu \nu } | << 1$ and $|\zeta | <<1$ ( $ \eta _ {\mu \nu }  =
\mbox{diag} (1,-1,-1,-1)$. In this way we obtain from (13) and (14)
using the de Donder gauge $h_ \mu {}^\nu {}_{|\nu } - \frac{1}{2}h_{|\mu
} = 0$: 
$$
h_ {\mu \nu } {}^{|\lambda } {}_{|\lambda } = 
- 16 \pi G ( T_ {\mu \nu } - 
\frac{1}{3} T \eta _ {\mu \nu } ) + 
$$
$$
+ 32 \pi G \frac{\mu ^4}{\lambda } \zeta 
\eta _ {\mu \nu } - 4 \zeta _ {|\mu| \nu } 
\leqno (19)
$$
and 
$$
\zeta ^ {|\lambda } {}_ {|\lambda } + 
16 \pi G \frac{\mu ^4}{\lambda } \zeta = 
\frac{4 \pi G}{3} T \: .
\leqno (20)
$$
Because of the geodesic equation of motion of a free point particle in
consequence of (8)
$$
h_ {00} = 2 \phi_N
\leqno (21)
$$
is valid, where $\phi_N$ is the Newtonian gravitational potential. 
Insertion of (21) into (19) yields:
$$
\phi_N {}^{|\lambda } {}_ {|\lambda } = 
- 8 \pi G(T_ {00} - \frac{1}{3} T) + 
$$
$$
+ 16 \pi G \frac{\mu ^4}{\lambda } \zeta - 
2\zeta _ {|0|0} \: .
\leqno (22)
$$
For a point particle of mass $M$ at rest in the origin   the solution of
(20) reads:
$$
\zeta = \frac{MG}{3r} e ^ {-r/l} \: , 
\quad l^2 = \frac{\lambda }{16 \pi G \mu ^4} \: .
\leqno (23)
$$
Herewith the solution of (22) for a point particle takes the form: 
$$
\phi_N = - \frac{MG}{r} ( 1 + 
\frac{1}{3} e^ {-r/l} ).
\leqno (24)
$$
Consequently $G = G_\infty$ is valid, where $G_\infty$ is the Newtonian
gravitational constant $G_N$ determined by a torsion-balance experiment
in the laboratory in the case $r >> l$. In case of $r<<l$ one finds 
$G = \frac{3}{4} G_0$ with $G_0 = G_N$. It is interesting that such
gravitational potentials, where the usual $r^{-1} $-potential is
supplemented by a Yukawa term, are discussed in connection with the
fifth force [8] and in view of the flat rotation curves of spiral
galaxies [9]. 

In the static linear Newtonian limit, the potential equations following
from (20) and (22) are
$$
\Delta \zeta   - \frac{1}{l^2} \zeta = - \frac{4 \pi G}{3} \rho \: , 
$$
$$
\Delta \phi_N + \frac{1}{l^2} \zeta = \frac{16 \pi G}{3} \rho 
\leqno (25)
$$
instead of the Poisson-equation. Herein $\varphi^{-2} = 1 - 2 \zeta $
represents the variability of the gravitational "constant" in first
order (cf. eq. (13)); it decreases in view of (23) with decreasing
distance from a mass. The cosmological function (16) is of second order
in $\zeta $ and therefore not yet contained in (25); however, its
absolute value increases with decreasing distance from a mass.  Finally
we note, that the scalar-field $\zeta $ acts in the potential equation
(25) for $\phi_N$ as a negative mass-density (anti-gravity), c.f. [9].
The applications of these ideas to modern astrophysical and cosmological
questions are in preparation. 

{ \Large{References}}

\begin{itemize}
\item[{[1]}] C. Brans and R. Dicke, Phys. Rev. \underline{124}, 925
  (1961). \newline
  See also P. Jordan, ``Schwerkraft und Weltall", F. Vieweg, 
  Braunschweig (1955).
\item[{[2]}] A. Einstein, Sitzungsber. Preu\ss. Akad. d. Wiss. Berlin, 
p. 142 \S 2 (1917). \newline
  A. Einstein, ``Grundz\"uge der Relativit\"atstheorie", p. 98, F. 
  Vieweg, Braunschweig (1973).
\item[{[3]}] H. Dehnen et al., Int. J. theor. Phys. \underline{29}, 537
  (1990).
\item[{[4]}] P. G. Bergmann, Int. J. theor. Phys. \underline{1}, 25
  (1968).
\item[{[5]}] R. V. Wagoner, Phys. Rev. D \underline{1}, 3209 (1970). 
\item[{[6]}] A. Zee, Phys. Rev. Lett. \underline{42}, 417 (1979);
  \newline
  A. Zee, Phys. Rev. Lett. \underline{44}, 703 (1980). See also the 
  papers cited there.
\item[{[7]}] B. S. de Witt, "The formal Structure of Quantum Gravity" in
  "Recent Development in Gravitation" M. Levy/S. Deser (eds.), p. 300,
  Carg\`{e}se 1978, Plenum Press New York/London (1979).
\item[{[8]}] E. Fischbach et al., Phys. Rev. Lett. \underline{56}, 3
  (1986). \newline
  D. H. Eckhardt et al. Phys. Rev. Lett. \underline{60}, 2567 (1988). 
\item[{[9]}]  R. H. Sanders, Astron. Astrophys. \underline{154}, 135
  (1986).
\end{itemize}

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