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% HIGGS-FIELD GRAVITY (HI) 
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% Published in: 
%   International Journal of Theoretical Physics, 29(6):537-546, 1990

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\begin{center}
{\huge Higgs-Field Gravity.}
\vspace{2cm}

H. Dehnen and H. Frommert
\vspace{2cm}


Fakult\"at f\"ur Physik

Universit\"at Konstanz

7750 Konstanz 

Postfach 55 60


West Germany


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\section*{Summary}

It is shown that any excited Higgs-field mediates an attractive scalar
gravitational interaction of Yukawa-type between the elementary
particles, which become massive by the ground-state of the Higgs-field.

\section*{1. Introduction.}

Until now the origin of the mass of the elementary particles is unclear.
Usually mass is introduced by the interaction with the Higgs-field;
however in this way the mass is not explained, but only reduced to the
parameters of the Higgs-potential, whereby the physical meaning of the
Higgs-field and its potential remains non-understood. We want to give
here a contribution for its interpretation.

There exists an old idea of Einstein, the so called "principle of
relativity of inertia" according to which mass should be produced by the
interaction with the gravitational field [1]. Einstein argued that the
inertial mass is only a measure for the resistance of a particle against
the {\underline{relative}} acceleration with respect to other particles;
therefore within a consequent theory of relativity the mass of a
particle should be originated by the interaction with all other
particles of the Universe (Mach's principle), whereby this interaction
should be the gravitational one which couples to all particles, i.e. to
their masses or energies. He postulated even that the value of the mass
of a particle should go to zero, if one puts the particle in an infinite
distance of all other ones. 

This fascinating idea was not very successful in Einstein's theory of
gravity, i.e. general relativity, although it has caused, that Einstein
introduced the cosmological constant in order to construct a
cosmological model with finite space, and that Brans and Dicke developed
their scalar-tensor-theory [2]. But an explanation of the mass does not
follow from it until now.

In this paper we will show, that the successful Higgs-field mechanism 
lies in the direction of Einstein's idea of producing mass by
gravitational interaction; we find, that the Higgs-field as source of
the inertial mass of the elementary particles has to do something with
gravity [3], i.e. it mediates a {\underline{scalar}}
{\underline{gravitational}} interaction between {\underline{massive}}
particles, however of Yukawa type. This results from the fact, that the
Higgs-field itself becomes massive after symmetry breaking. On the other
hand an estimation of the coupling constants shows that it may be
unprobable that this Higgs-field gravity can be identified with any
experimental evidence. Perhaps its applicability lies beyond the scope
of the present experimental experiences.

\section*{2. Gravitational force and potential equation.}

We perform our calculations in full generality with the use of an U(N)
model and start from the Lagrange density of fermionic fields coupled
with the Higgs-field both belonging to the localized group U(N) $(c = 1,
\eta _{\mu \nu } = diag(1,-1,-1,-1))$:  
 $$
L = \frac{\hbar}{2} i \overline{\psi} \gamma ^\mu D_ \mu \psi  + h.c. - 
\frac{\hbar}{16 \pi } F^a _{\  \   \lambda \mu } F_a ^{\   \lambda \mu }+
\leqno (1)
$$ 
 $$
+ \frac{1}{2}(D_ \mu \phi)^{\dagger} D^\mu \phi - \frac{\mu ^2}{2
}\phi^{\dagger} \phi - \frac{\lambda }{4!}(\phi^{\dagger} \phi)^2 - k
\overline{\psi }\phi^{\dagger}\hat x\psi + h.c.
\leqno 
$$ 
($\mu^2 ,\lambda , k$ are real parameters of the Higgs-potential).
Herein $D_ \mu $ represents the covariant derivative with respect to the
localized group U(N)
$$D_ \mu = \partial_ \mu + igA_ \mu 
\leqno (1a)
$$  
($g$ gauge coupling constant, $A_ \mu  = A_ \mu ^{ \   a} \tau  _a$
gauge potentials, $\tau _a $ generators of the group U(N)) and the gauge
field strength $F_ {\mu \nu} $ is determined by its commutator $(F_{\mu
\nu } = (1/ig)\left [ D_ \mu , D_ \nu \right ] = F^a_ {\    \mu \nu
}\tau _a);$ furthermore $\hat x$ is the Yukawa coupling-matrix. For the
case of applying the Lagrange density (1) to a special model, as e.g.
the Glashow-Salam-Weinberg model or even the GUT-model, the wave
function $\psi $, the generators $\tau _a$, the Higgs-field $\phi$ and
the coupling matrix $\hat x$ must be specified explicitly [4]. 

From (1) we get immediately the field equations for the spinorial matter
fields ($\psi $-fields):
 $$
i \gamma ^ \mu D_ \mu  \psi  
- \frac{k}{\hbar}(\phi^{\dagger} \hat x + \hat {x}^{\dagger} \phi) \psi 
= 0,
\leqno (2)
$$ 
the Higgs-field $\phi$
 $$
D^\mu D_ \mu  \phi + \mu ^2 \phi + \frac{\lambda
}{3!}(\phi^{\dagger}\phi)\phi
= - 2k \overline{\psi }\hat x \psi 
\leqno (3)
$$  
and the gauge-fields $F^{a \mu \lambda }$
 $$
\partial_ \mu F^{a \mu \lambda}  + 
igf^a_{\    bc} A^{b \mu } F^{c \lambda} _ \mu   = 4 \pi j^{a \lambda }
\leqno (4)
$$ 
with the  gauge-current density
 $$
j^{a \lambda } = g ( \overline{\psi} \gamma ^ \lambda  \tau ^a \psi  +
\frac{i}{2\hbar}\left[\phi^{\dagger}\tau ^aD^\lambda \phi \right.
-\left. (D^\lambda \phi)^{\dagger} \tau ^a \phi\right]).
\leqno (4a)
$$  
Herein $f^a _{\    bc}$ are the totally skew symmetric structure
constants of the group U(N). The gauge invariant canonical energy
momentum tensor reads with the use of (2)
$$
T_ \lambda ^{\    \mu } = \frac{i\hbar}{2}
\left[ \overline{\psi }\gamma ^ \mu  D _\lambda \psi 
-(\overline{D_ \lambda \psi })\gamma ^ \mu  \psi \right] -
\leqno (5)
$$  
 $$
- \frac{\hbar}{4 \pi }
\left [F^\alpha _{\   \lambda \nu }
F^{\    \mu \nu }_a - 
\frac{1}{4}\delta _ \lambda ^{\   \mu } F^a _{\   \alpha \beta} 
F^{\     \alpha \beta }_a \right ]  + 
$$ 
 $$
+ \frac{1}{2} \left [
(D_ \lambda \phi)^{\dagger} D^ \mu  \phi +
 (D^ \mu \phi)^{\dagger} D_ \lambda \phi - \right. $$ $$ \left.  
  - \delta _\lambda ^{\    \mu } \{
(D_ \alpha \phi)^{\dagger} D^\alpha \phi - \mu ^2 \phi^{\dagger} \phi  - 
\frac{2 \lambda }{4!} (\phi^{\dagger} 
\phi)^2 \}  \right ] 
\leqno 
$$ 
and fulfils the conservation law
$$
\partial _ \mu  T_ \lambda ^{\   \mu} = 0.
\leqno (6)
$$  
Obviously, the current-density (4a) has a gauge-covariant matter-field
and Higgs-field part, i.e. $j^{a \lambda }(\psi )$ and $j^{a \lambda
}(\phi)$ respectively, whereas the energy-momentum tensor (5) consists
of a sum of three gauge-invariant parts:
$$
T^{\   \mu }_ \lambda  = T^{\   \mu }_ \lambda(\psi ) + 
T^{\   \mu }_ \lambda (F) + T^{\   \mu }_ \lambda(\phi),
\leqno (7)
$$ 
represented by the brackets on the right hand side of equ. (5).

In view of analyzing the interaction caused by the Higgs-field we
investigate at first the equation of motion for the expectation value of
the 4-momentum of the matter fields and the gauge-fields. From (6) and
(7) one finds under neglection of surface-integrals in the space-like
infinity:
$$
\partial _0 \int \left[ T_ \lambda ^{\   0}(\psi ) + 
T_ \lambda ^{\   0} (F) \right] d^3x =  - 
\int \partial _ \mu T_ \lambda  ^{\    \mu }(\phi)d^3 x .
\leqno (8)
$$
Insertion of $T_ \lambda  ^{\    \mu }(\phi)$ according to (5) and
elimination of the second derivatives of the Higgs-field by the
field-equations (3) results in:
 $$
\frac{\partial}{\partial t} \int \left[   T_ \lambda ^{\    0} (\psi ) +
T_ \lambda ^{\    0} (F) \right]  d^3 x = $$ $$ = k \int \overline{\psi
} \left[ (D_ \lambda \phi)^{\dagger} \hat x + \hat x ^{\dagger} (D_
\lambda \phi) \right ] \psi  d^3 x + 
\leqno (9)
$$ 
$$
+ \frac{ig}{2}\int F^a _{\    \mu \lambda } \left [ \phi^ {\dagger} 
\tau _a D^\mu  \phi - (D^\mu \phi)^{\dagger} \tau _a \phi \right ] d^3 x .
\leqno 
$$ 
The right hand side represents the expectation value of the 4-force,
which causes the change of the 4-momentum of the $\psi$ -fields and the
F-fields with time. However, the last expression can be rewritten with
the use of the field-equations (4) as follows:
 $$
\partial _ \mu T_ \lambda ^{\   \mu } (F) = \hbar F^a_ {\   \mu  \lambda
} ( j_a^{\    \mu } (\psi ) + j_a ^{\    \mu }(\phi)).
\leqno (9a)
$$ 
Herewith one obtains instead of (9):
$$
\frac{\partial}{\partial t} \int T_ \lambda ^{\    0} (\psi )d^3 x = 
\int \hbar F^a _ {\    \lambda \mu } j^\mu _ {\    a} (\psi ) d^3 x + 
\leqno (10)
$$  
 $$
+ k \int \overline{\psi } \left [ (D_ \lambda \phi)^{\dagger} \hat x +
\hat x ^{\dagger} (D_ \lambda  \phi ) \right]\psi d^3 x,
\leqno 
$$ 
where on the right hand side we have the 4-force of the gauge-field and
the Higgs-field, both acting on the matter-field. Evidently, the
gauge-field strength couples to the gauge-currents, i.e. to the
gauge-coupling constant g according to (4a), whereas the Higgs-field
strength (gradient of the Higgs-field) couples to the fermionic
mass-parameter k (c.f. [5]). This fact points to a
{\underline{gravitational} action of the scalar Higgs-field. 

\subsection*{a) Gravitational interaction on the level of the
  field-equations.}

For demonstrating the gravitational interaction explicitly 
we perform at first the spontaneous symmetry breaking, 
because in the case of a scalar gravity only massive particles 
should interact. {\footnote {The only possible source of a scalar 
gravity is the trace of the energy momentum tensor.}} 
For this $\mu ^2 < 0$ must be valid, and according to (3) and (5) 
the ground state $\phi_0$ of the Higgs-field is defined by 
  $$
\phi_0 ^{\dagger} \phi_0 = v^2 = \frac{-6 \mu ^2}{\lambda },
\leqno (11)
$$
which we resolve as
$$
\phi_0 = vN 
\leqno (12)
$$
with
 $$
N^{\dagger} N = 1, \quad \partial _ \lambda N \equiv 0.
\leqno (12a)
$$ 
The general Higgs-field $\phi$ is different from (12) by a local unitary
transformation:
 $$
\phi = \rho UN, \quad U^{\dagger} U =1
\leqno (13)
$$ 
with 
$$
 \phi^{\dagger} \phi = \rho ^2, \quad \rho  = v^{\dagger} \eta , 
\leqno (13a)
$$ 
where $\eta $ represents the real valued excited Higgs-field.

Now we use the possibility of a unitary gauge transformation which is
inverse to (13):
 $$
\phi ' = U^{-1} \phi, \quad \psi  ' = U^{-1} \psi ,
\leqno 
$$  
 $$
F'_{\mu \nu } = U^{-1} F_{\mu \nu } U,
\leqno (14)
$$ 
so that 
 $$
\phi ' = \rho N, 
\leqno (14a)
$$ 
and perform in the following all calculations in the gauge (14)
(unitary gauge). For this we note, that in the case of the symmetry
breaking of the group $G$
 $$
G \rightarrow \tilde G , 
\leqno (15)
$$ 
where $\tilde G$ represents the rest-symmetry group, we decompose the
unitary transformation:
 $$
U = \hat U \cdot \tilde U, \quad 
\tilde U \in \tilde G, \quad 
\hat U \in G/ \tilde G
\leqno (15a)
$$ 
with the isotropy property ($\tau_{\, \tilde{a}}$ generators of the
unbroken symmetry):
 $$
\tilde U N = e^{i \lambda ^{\, \tilde a} \tau _{\, \tilde{a}}}N = N , 
\leqno (16)
$$ 
so that
$$
\tau _{\, \tilde{a}} N = 0
\leqno (17)
$$ 
is valid. For $\, \hat U$ we write $ \hat U = e^{i \lambda ^{\, \hat{a}}
\tau _{\, \hat{a}}}$, where $\tau _{\, \hat{a}}$ are the generators of
the broken symmetry. 

Using (12) up to (17) the field-equations (2) through (4) take the form,
avoiding the strokes introduced in (14):
 $$
i \gamma ^ \mu  D_ \mu  \psi  - \frac{\hat m}{\hbar}(1 + \varphi ) \psi  = 0,
\leqno  (18)
$$ 
 $$
\partial_ \mu  F_a ^{ \   \mu  \lambda } + ig f_{abc} A^{b \mu } F^{c \  
\lambda } _{\  \mu } + 
$$ 
$$
+ \frac{1}{\hbar^2}M^2 _{ab} ( 1 + \varphi  )^2 A^{b \lambda } = 
4 \pi j_a ^\lambda (\psi ),
\leqno (19)
$$ 


$$
\partial ^\mu \partial_ \mu \varphi  + \frac{M^2}{\hbar^2} \varphi +
\frac{1}{2}\frac{M^2}{\hbar^2}
(3 \varphi ^2 + \varphi ^3) = $$ $$ 
- \frac{1}{v^2} \left[\overline{ \psi} \hat m \psi  - 
\frac{1}{4 \pi \hbar} M^2 _ {ab} A^a_{ \    \lambda } A^{b \lambda }
(1 + \varphi ) \right ] ,
\leqno (20)
$$
wherein $\varphi = \eta /v$ represents the excited Higgs-field and 
$$
\hat m = kv (N^{\dagger} \hat x + \hat x ^{\dagger} N )
\leqno (18a)
$$ 
is the mass-matrix of the matter-field ($\psi $-field), 
$$
M^2 _{\   ab} = 
M^2 _{\, \hat{a} \hat{b}} = 
4 \pi \hbar g^2v^2N^{\dagger} \tau_{(\, \hat{a}}  \tau_{\, \hat{b})} N
\leqno (19a)
$$ 
the symmetric matrix of the mass-square of the gauge-fields $(A_ \mu
^{\, \hat{a}}$  -fields) and 
 $$
M^2 = -2 \mu ^2 \hbar^2, \quad (\mu ^2 < 0)
\leqno (20a)
$$ 
is the square of the mass of Higgs-field ($\varphi$-field). Obviously in
the field-equations (18) up to (20) the Higgs-field $\varphi $ plays the
role of an attractive scalar gravitational potential between the
{\underline{massive}} particles: According to equ. (20) the source of
$\varphi $ is the mass of the fermions and of the gauge-bosons,
{\footnote{The second term in the bracket on the right hand side of equ.
(20) is positive with respect to the signature of the metric.}} whereby
this equation linearized with respect to $\varphi $ is a potential
equation of Yukawa-type. Accordingly the potential $\varphi $ has a
finite range 
 $$
l = \hbar /M
\leqno (21)
$$ 
given by the mass of the Higgs-particle and $v^{-2} $ has the meaning of
the gravitational constant, so that 
 $$v^{-2} = 4 \pi G \gamma 
\leqno (22)
$$
is valid, where $G$ is the Newtonian gravitational constant and $\gamma$
a dimensionless factor, which compares the strength of the Newtonian
gravity with that of the Higgs-field and which can be determined only
experimentally, see sect. 3. On the other hand, the gravitational
potential $\varphi $ acts back on the mass of the fermions and of the
gauge-bosons according to the field equations (18) and (19).
Simultaneously the equivalence between inertial and passive as well as
active gravitational mass is guaranteed. This feature results from the
fact that by the symmetry breaking only {\underline{one}} type of mass
is introduced. 

\subsection*{b) Gravitational interaction on the level of the momentum
  law.}

% b) Gravitational.....
At first we consider the potential equation from a more classical
standpoint. With respect to the fact of a {\underline{scalar}}
gravitational interaction we rewrite equation (20) with the help of the
trace of the energy-momentum tensor, because this should be the only
source of a scalar gravitational potential within a Lorentz-covariant
theory. From (5) one finds after symmetry breaking in analogy to (7):
$$
T_ \lambda ^{\  \mu } (\psi ) = 
\frac{i \hbar}{2} \left[ \overline{\psi }\gamma ^\mu D_ \lambda \psi - 
(\overline{ D_ \lambda \psi }) \gamma ^\mu \psi \right], 
\leqno (23a)
$$ 


$$
T_ \lambda ^{\  \mu }(A) = -\frac{\hbar}{4 \pi} (F^a _{\    \lambda \nu }
F_a ^{\    \mu \nu } - \frac{1}{4}\delta _ \lambda ^{\    \mu } F^a _ {\   
\alpha \beta } F_a ^{\    \alpha \beta }) + 
\leqno (23b)
$$ 
 $$
+ \frac{1}{4 \pi \hbar} (1 + \varphi )^2 M^2 _ {\    ab} 
(A^a_ {\    \lambda } 
A^{b \mu } -  \frac{1}{2}\delta _ \lambda ^{\  \mu }A_ \nu ^a  A^{b \nu }),
\leqno 
$$ 
 $$
T^{\    \mu } _\lambda  (\varphi ) = v^2 \left[  \partial_ \lambda
\varphi \partial^\mu \varphi  - 
\frac{1}{2}\delta _ \lambda ^{\    \mu }\left \{  \partial_ \alpha 
\varphi \partial^\alpha  \varphi   
+ \right. \right. $$ $$
+ \left.  \left. \frac{M^2}{4 \hbar^2} ( 1 + \varphi )^2 
(1 - 2 \varphi  - \varphi ^2) \right \} \right] . 
\leqno (23c)
$$ 
From this it follows immediately using the field equation (18):
 $$
T = T_ \lambda ^{\    \lambda } = \overline{\psi }\hat m \psi 
(1 + \varphi ) - 
\frac{1}{4 \pi \hbar}M^2 _{\    ab}A^a _ \lambda A^{b \lambda }(1 +
\varphi )^2 + 
\leqno 
$$ 
$$
+ v^2 \left [ \frac{M^2}{2 \hbar ^2}(\varphi ^4 + 4 \varphi ^3 + 
4 \varphi ^2 - 1) 
- \partial_ \lambda  \varphi \partial^\lambda \varphi \right ].
\leqno (23d)
$$ 
The comparison with equ. (20) shows, that the source of the potential
$\varphi $ is given by the first two terms of (23d), i.e. by $T(\psi )$
and $T(A)$ as expected.In this way we obtain as potential equation using
(22): 
 $$
\partial^\mu \partial_ \mu \varphi + \frac{M^2}{\hbar^2}\varphi  + 
\frac{1}{2} \frac{M^2}{\hbar^2}(3 \varphi ^2 + \varphi ^3)  = $$ $$ 
=  -4 \pi G \gamma (1 + \varphi )^{-1} (T(\psi ) + T(A)).
\leqno (24)
$$  
In the linearized version (with respect to $\varphi ) $ equ. (24)
represents a potential equation for $\varphi $ of Yukawa-type with the
trace of the energy-momentum tensor of the massive fermions and the
massive gauge-bosons as source.

Finally we investigate the gravitational force caused by the Higgs-field
more in detail. Insertion of the symmetry breaking according to (12) up
to (17) into the first integral of the right-hand side of (9) yields:
 $$
K_ \lambda = k \overline{\psi }\left [ (D_ \lambda  \phi)^{\dagger} \hat
x + \hat x ^{\dagger} (D_ \lambda \phi) \right ] \psi  =
\leqno 
$$ 
$$
= \overline{\psi }\hat m \psi \partial_ \lambda \varphi + v(1 + \varphi )
\left [ 
(D_ \lambda N)^{\dagger} k \overline{\psi }\hat x \psi  + 
k \overline{\psi }\hat x ^{\dagger} \psi  D_ \lambda N \right ].
\leqno (25)
$$ 
Substitution of the conglomerate $k \overline{\psi }\hat x \psi $ by the
left hand side of the field-equation (3) results with the use of (13a)
and (14a) in:
$$
K_ \lambda = \left [ 
\overline{\psi }\hat m \psi  - \right. \left.  
\frac{1}{4 \pi \hbar} M^2_ {\    ab}A^a _ \mu A^{b \mu } (1 + \varphi ) 
\right ]
\partial _ \lambda \varphi - 
\leqno 
$$ 
 $$
- \frac{1}{4 \pi \hbar}\partial_ \mu 
\left [
(1 + \varphi )^2 M^2_ {\    ab}(A^a _ {\    \lambda } A^{b \mu }    -
\frac{1}{2} \delta _ \lambda ^{\  \mu  } A^a _\nu A^{b \nu }) \right ] +
\leqno 
$$ 
 $$
+ \frac{v^2}{2}ig (1 + \varphi )^2 F^a _ {\    \lambda \mu } 
 \left [
N^{\dagger} \tau _a D^ \mu N - (D^ \mu N)^{\dagger} \tau _a N
\right ].
\leqno (26)
$$ 
By insertion of (26) into the right hand side of (9) the last term of
(26) drops out against the last term of (9), whereas the second term of
(26) can be combined with $\partial_ \mu T_ \lambda ^ \mu (F)$ to
$\partial_ \mu  T_ \lambda ^\mu (A)$ according to (23b). In this way we
obtain neglecting surface integrals in the space-like infinity:
 $$
\frac{\partial}{\partial t}\int \left [ T_ \lambda ^ {\    0} (\psi ) + 
T_ \lambda ^ {\    0}(A)
\right ]d^3 x =  
\int \left [ \overline{\psi }\hat m \psi \right. - $$ $$ \left. -
\frac{1}{4 \pi \hbar} M^2_ {\  ab} A^a _ \mu A^{b \mu }(1 + \varphi ) 
\right] \partial_ \lambda \varphi d^3 x.
\leqno  (27)
$$ 
In total analogy to the procedure yielding the potential equation (24)
we substitute the bracket of the 4-force in (27) by the traces $T(\psi
)$ and $T(A)$ given by (23d):
$$
\frac{\partial}{\partial t} \int \left [ T_ \lambda ^{\  0} (\psi )
 + 
T_ \lambda ^{\  0}(A) 
\right ] d^3 x = $$ $$ = \int (1 + \varphi )^{-1} \left [ T(\psi ) + T(A)
\right ] \partial_ \lambda  \varphi  d^3 x.
\leqno (28)
$$ 

Considering the transition from equ. (9) to (10) we can express the
time-derivative of the 4-momentum of the gauge-fields by a 4-force
acting on the matter currents. Restricting this procedure to the
{\underline{massless}} gauge-fields we get from (28):
 $$
\frac{\partial}{\partial t} \int 
\left [ T_ \lambda ^{\  0}(\psi ) + 
T_ \lambda ^{\  0}(A_ \sigma ^{\    \hat a}) 
\right ] d^3 x = 
\leqno 
$$ 
 $$
= \int \hbar F^{\tilde a} _ {\    \lambda  \mu } j^ \mu _{\   \tilde a}
(\psi )d^3 x + 
\leqno 
$$ 
 $$
+ \int (1 + \varphi )^{-1} \left [T(\psi ) + T(A_ \sigma ^{\    \hat a})
\right ] \partial_ \lambda \varphi d^3 x.
\leqno (29)
$$ 
Herein the first term of the right hand side describes the 4-force of
the massless gauge-bosons acting on the matter-field coupled by the
gauge-coupling constant g, see (4a), whereas the second term (identical
with the right hand side of (28)) is the attractive gravitational force
of the Higgs-field $\varphi $ acting on the masses of the fermions and
of the gauge-bosons, which are simultaneously the source of the
Higgs-potential $\varphi $ according to (24). This behaviour is exactly
that of classical gravity, coupling to the mass ($\equiv$ energy) only
and not to any charge. However the qualitative difference with respect
to the Newtonian gravity consists besides the non-linear terms in (24)
in the finite range of $\varphi $ caused by the Yukawa term.

\section*{3. Final Remarks.}

In the end we want to point to some interesting features of our result.
First of all we note, that in view of the right hand side of (28) it is
appropriate to define
 $$
\ln (1 + \varphi ) = \chi 
\leqno (30)
$$ 
as new gravitational potential, so that the momentum law reads:
 $$
\frac{\partial}{\partial t} \int \left [ T_ \lambda  ^{\   0} 
(\psi ) + 
T_ \lambda  ^{\   0} (A) \right ] 
d^3 x = 
\int \left [ T(\psi ) + T(A) \right ]  \partial_ \lambda  \chi d^3 x . 
\leqno (31)
$$ 
Then the non-linear terms concerning $\varphi $ in (24) can be expressed
by $T(\varphi ) \equiv T(\chi)$ according to the third term of the right
hand side of (23d). In this way the field equation for the potential
$\chi $ (excited Higgs-field) takes the very impressive form:
 $$
\partial _ \mu \partial^\mu  e^{2 \chi } +
\frac{M^2}{\hbar^2 } e^{2 \chi } =  -8 \pi G \gamma  \left [T(\psi ) +
T(A) + T(\chi ) \right ] .
\leqno (32)
$$ 
Equations (31) and (32) are indeed those of scalar gravity with self
interaction in a natural manner. For the understanding of the
Higgs-field it may be of interest, that the structure of equation (32)
exists already before the symmetry breaking. Considering the trace $T$
of the energy momentum tensor (5) one finds with the use of the
field-equations (2) and (3): 
$$
\partial_ \mu  \partial^ \mu (\phi^{\dagger} \phi) + 
\frac{M^2}{\hbar^2} (\phi^{\dagger} \phi) = -2T
\leqno (33)
$$ 
with $ M^2 = -2 \mu ^2 \hbar ^2$. Accordingly, the Yukawa-like
self-interacting scalar gravity of the Higgs-field is present within the
theory from the very beginning. Equation (33) possesses an interesting
behaviour with respect to the symmetry breaking. Then from the second
term on the left hand side there results in view of (11) in the first
step a cosmological constant $M^2 v^2/\hbar^2$; but this is compensated
exactly by the trace of the energy momentum tensor of the ground state.
It is our opinion that this is the property of the cosmological constant
at all, also in general relativity.

Furthermore because in (21) the mass M is that of the Higgs-particle,
the range $l$ of the potential $\varphi $ should be very short, so that
until now no experimental evidence for the Higgs-gravity may exist, at
least in the macroscopic limit. For this reason it also appears
unprobable, that it has to do something with the so called fifth force
[6]. Finally the factor $\gamma $ in (22) 
can be estimated as follows: Taking into consideration the unified 
theory of electroweak interaction the value of $v$ (see (19a)) is 
correlated with the mass $M_W$ of the $W$-bosons according 
to $v^{-2} =  \pi g_2 ^{\  2} \hbar / M^2 _W$ ($g_2 = $ gauge-coupling
constant of the group SU(2)). Combination with (22) results in 
$$
\gamma = \frac{g_2 ^2}{2} (\frac{M_P}{M_W})^2 = 2 \times 10^{32}
\leqno (34)
$$ 
($M_P$ Planck-mass). Consequently the Higgs-gravity represents a
relatively strong scalar gravitational interaction between massive
elementary particles, however with extremely short range and with the
essential property of quantizability. If any Higgs-field exists in
nature, this gravity is present. 

The expression (34) shows that in the case of a symmetry breaking where
the bosonic mass is of the order of the Planck-mass, the Higgs-gravity
approaches the Newtonian gravity, if the mass of the Higgs-particle is
sufficiently small. In this connection the question arises, following
Einstein's idea of relativity of inertia, if it is possible to construct
a tensorial quantum theory of gravity with the use of the
Higgs-mechanism, leading at last to Einstein's gravitational theory in
the classical macroscopic limit.

\section*{References}

\begin{itemize}
\item[{[1]}] A. Einstein, Sitzungsber. Preu\ss. Akad. d. Wiss. Berlin, 
  p. 142 \S  \/ 2 (1917).
\item[{[2]}] C. Brans and R. Dicke, Phys. Rev. {\underline{124}}, 925
  (1961).
\item[{[3]}] H. Dehnen, F. Ghaboussi and J. Schr\"oder, Wiss. Zs. d.
  Friedrich-Schiller-Universit\"at Jena, in press.
\item[{[4]}] H. Dehnen and H. Frommert, to be published.
\item[{[5]}] P. Becher, M. B\"ohm and H. Joos, Eichtheorien der starken
  und elektroschwachen Wechselwirkung, Teubner-Verlag (Stuttgart), p. 
  350 (1981).
\item[{[6]}] D. H. Eckhardt et al., Phys. Rev. Lett. {\underline {60}},
  2567 (1988).
\end{itemize}

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