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% Higgs-Field Gravity within the Standard Model
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% Published in: 
%   International Journal of Theoretical Physics, 30(7):985-998, 1991

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\begin{titlepage}
\begin{center}
{\huge Higgs-Field Gravity within the Standard Model}
\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}

Within the frame-work of the Glashow-Salam-Weinberg model it is shown
that the 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 after symmetry breaking.

\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.

On the other hand 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 relative acceleration with respect to other
particles; therefore, within a consequent theory of relativity, the mass
of a particle should be originated by interaction with all other
particles of the Universe, 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 the other ones.

This fascinating idea was not very successful within 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 has to do something with gravity [3], i.e. it mediates
a scalar gravitational interaction between the 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 Action of the Higgs-Field.}

In a previous publication [3] we have shown approximatively 
the gravitational interaction of the Higgs-field between 
massive fermions. In the present paper we extend our investigation in an
exact manner on fermions and bosons. Due to this reason we perform our
calculations within the well established Glashow-Salam-Weinberg model of
electro-weak interaction based on the localized group $ SU(2) \times
U(1)$, taking into account all families of elementary particles. 
For this we start with the following definitions: {\footnote{Spinor- and 
 isospin- indices are suppressed.}}
$$
\psi ^i = \psi ^{m_i} = 
{{\psi ^{l_i}} \choose {\psi ^{q_i}}} , \quad
m = l,q 
\leqno (2.1)
$$
represents the spinorial wave-functions of the $i$-th familiy 
$(i = 1, ...., N_f)$, wherein 
$$
\psi ^{l_i} = \psi ^{l_i} _L +  \psi ^{l_i} _R
\leqno (2.2)
$$
is the leptonic part with 
$$
\psi ^{l_i} _L = {{\nu ^i _L} \choose {e^i_L}} , \quad \psi ^{l_i} _R =
e^i _R ,
\leqno (2.2a)
$$
and 
$$
\psi ^{q_i} = \psi ^{q_i} _L + \psi ^{q_i} _R
\leqno (2.3)
$$
means the part of the quarks with
$$
\psi ^{q_i} _L = {{u^i _ L} \choose {d^{\  'i}_L}}, \qquad
\psi ^{q_i} _R = {{u^i _ R} \choose {d^{\  'i}_R}}
\leqno (2.3a)
$$
and 
$$
d^{\  'i} = U^i_{(c)j} d^j
\leqno (2.3b)
$$
as the Cabibbo transformed quark wave-functions. Here the left-handed
fermions $\psi ^{l_i} _L$ and $\psi ^{q_i} _L$ are doublets with respect
to the localized group $SU(2)$, whereas the right-handed ones $\psi
^{l_i} _R$ and $\psi ^{q_i} _R$ are singlets. Correspondingly the
covariant derivatives take the form:
$$
D_ \lambda \psi ^{l_i} _L = 
(\partial_ \lambda  + ig_2 W^a _ \lambda \tau _a - 
\frac{1}{2}ig_1 B_ \lambda )\psi ^{l_i} _L ,
$$
$$
D_ \lambda \psi ^{q_i} _L = 
(\partial_ \lambda  + ig_2 W^a _ \lambda \tau _a + 
\frac{1}{6}ig_1 B_ \lambda )\psi ^{q_i} _L ,
$$
$$
D_ \lambda \psi ^{l_i} _R = (\partial_ \lambda  - 
ig_1 B_ \lambda )\psi ^{l_i} _R ,
$$
$$
D_ \lambda \psi ^{q_i} _R = D_ \lambda {{u^i _ R} 
\choose {d^{\  'i}_R}} = 
\left \{ 
 {( \partial_ \lambda + \frac{2}{3}ig_1 B_ 
\lambda )u^i _R ,}
\atop  { ( \partial_ \lambda -  \frac{1}{3}ig_1 B_ \lambda )
d^{\  'i} _R.} 
\right. 
\leqno (2.4)
$$
Herein $\tau ^a$ are the generators of the group $SU(2), \/   W^{\   a}
_ \lambda $ represent the corresponding gauge-potentials and $B_ \lambda
$ is the $U(1)$ gauge-potential with $g_1$ and $g_2$ as gauge-coupling
constants. The covariant gauge-field strengths are given by the
commutators
$$
{\cal F}_{(2) \mu \nu } = F^a _{(2) \mu \nu } \tau _a = 
\frac{1}{ig_2} 
\left[  
D_ \mu ^{(2)}, D_ \nu ^{(2)} 
\right] ,   
\leqno 
$$
$$
{\cal F}_{(1) \mu \nu } = F_{(1) \mu \nu } Y = 
\frac{1}{ig_1} 
\left[  
D_ \mu ^{(1)}, D_ \nu ^{(1)} 
\right]   
\leqno (2.5)
$$
((1) and (2) refer to the group $U(1)$ and$ SU(2)$ respectively). Here
$Y$ is the $U(1)$-generator of the weak hypercharge in the different
representations according to (2.4), where we follow the notation of ref.
[5] and not of [4]. Finally we introduce a scalar Higgs-field $\phi$
belonging to the fundamental representation of 
$ SU(2)$; its covariant derivative reads
$$
D_ \lambda  \phi = ( \partial _\lambda  + 
ig_2 W_ \lambda ^{\    a} \tau _a + 
\frac{1}{2} ig_1 B_ \lambda ) \phi.
\leqno (2.6)
$$

Herewith we construct the gauge invariant minimally coupled
Lagrange-density:
$$
L = L(\psi ) + L(F) + L(\phi) ,
\leqno (2.7)
$$
where 
$$
L(\psi ) = i \frac{\hbar}{2} 
\left[   
\overline{\psi }_{Lm_i} \gamma ^ \lambda 
D_ \lambda \psi _L ^{m_i} + 
\overline{\psi }_{Rm_i} \gamma ^ \lambda 
D_ \lambda \psi _R ^{m_i} 
\right]   + h.c.,
\leqno (2.7a)
$$
$$
L(F) =  - \frac{\hbar}{16 \pi } ( F^{\    a}_{(2) \lambda \mu }
                            F^{\    \lambda \mu }_{(2)a} + 
                            F_{(1) \lambda \mu }
                            F^{\    \lambda \mu }_{(1)} )
\leqno (2.7b)
$$
and
$$
L(\phi) = \frac{1}{2}(D_ \lambda \phi)^\dagger 
D^\lambda \phi - 
\frac{\mu ^2}{2} \phi^\dagger \phi - 
\frac{\lambda }{4!}(\phi^\dagger \phi)^2 - 
k \phi^\dagger \overline{\psi }_{Rm_i} 
\hat x ^{m_i}_{\ \ n_j} \psi _L ^{\  n_j} + h.c. .
\leqno (2.7c)
$$
($\mu ^2, \lambda ,k $ 
real parameters of the Higgs-potential 
and $\hat x ^{m_i}_{\  \  n_j} $ 
Yukawa coupling matrix). The field equations 
following from Hamilton's action principle result in the 
wave equations for the left and right handed fermions 
$$
i \gamma ^ \mu  D_ \mu \psi _L ^{\    m_i} - 
\frac{k}{\hbar} 
\hat x ^{\dagger m_i} _{\   \   n_j} 
\phi \psi _R ^{\    n_j} = 0 ,
\leqno (2.8a)
$$
$$ 
i \gamma ^ \mu  D_ \mu \psi _R ^{\    m_i} - 
\frac{k}{\hbar} \phi^{\dagger}
\hat x ^{ m_i} _{\  \   n_j}  \psi _L ^{\    n_j} = 0 ,
\leqno (2.8b)
$$
in the Yang-Mills equations 
$$
D_ \mu F_{(2) a} ^{\    \mu \lambda } \equiv 
\partial _ \mu F_{(2)a }^{\  \mu \lambda} - 
g_2 \epsilon _{abc} W^b _ \mu  
F_{(2)} ^{c \mu  \lambda } = 
4 \pi j_{(2)a} ^{\    \lambda } ,
\leqno (2.9a)
$$
$$
\partial _ \mu F_{(1)} ^{\  \mu \lambda } = 
4 \pi j_{(1)} ^{\    \lambda }
\leqno (2.9b)
$$
$(\epsilon _{abc}$ Levi-Civita symbol) with the current densities
$$
j_{(2)a} ^{\    \lambda } = 
g_2 \overline{\psi }_{L m_i} 
\gamma  ^ \lambda \tau _a \psi _L ^{\    m_i} + 
i \frac{g_2}{2 \hbar} 
\left[  
\phi^\dagger \tau _a D^\lambda  \phi - 
(D^\lambda  \phi)^\dagger \tau  _a \phi 
\right]   , 
\leqno (2.10a)
$$
$$
j_{(1)} ^{\    \lambda } = 
g_1 
\left[  
Y \overline{\psi }_{L m_i}
\gamma ^\lambda \psi _L ^{\    m_i} + 
Y \overline{\psi }_{R m_i} 
\gamma ^\lambda \psi _R ^{\    m_i}
\right]   + 
i \frac{g_1}{4 \hbar} 
\left[  
\phi^\dagger D^\lambda  \phi - 
(D^\lambda  \phi)^\dagger \phi 
\right]   
\leqno (2.10b)
$$
and in the Higgs-field equation
$$
D_ \mu D^\mu  \phi + \mu ^2 \phi +
\frac{\lambda }{6}(\phi^\dagger \phi) \phi = 
-2k \overline{\psi }_{R m_i} 
\hat x ^{m_i} _{\    n_j} 
\psi ^{\    n_j} _L .
\leqno (2.11)
$$
Obviously the current densities separate into two gauge-covariant 
parts $j_{(2)a} ^\lambda (\psi )$ and $j_{(2)a} ^\lambda (\phi )$ 
as well as $j_{(1)} ^\lambda (\psi )$ and $j_{(1)} ^\lambda (\phi )$. 
In a similar way the gauge-invariant canonical energy-momentum 
tensor consists of three gauge-invariant parts:
$$
T_ \lambda ^{\    \mu } = 
T_ \lambda ^{\    \mu } (\psi ) + 
T_ \lambda ^{\    \mu } (F ) + 
T_ \lambda ^{\    \mu } (\phi ) 
\leqno (2.12)
$$
with 
$$
T_ \lambda ^{\    \mu } (\psi ) = 
i \frac{\hbar}{2} 
\left[ \rule{0mm}{5mm}
\overline{\psi}_{Lm_i} 
\gamma ^ \mu 
D_ \lambda \psi _ L ^{\    m_i} + 
 \overline{\psi}_{Rm_i} 
\gamma ^ \mu D_ \lambda \psi _ R ^{\    m_i} 
\right]   
+ h.c.,
\leqno (2.12a)
$$
$$
T_ \lambda ^ {\    \mu } (F) = 
- \frac{\hbar}{4 \pi} 
\left[   \rule{0mm}{5mm}
( F^a _{(2) \lambda \nu} F^{\    \mu \nu } _{(2)a} - 
\frac{1}{4}
 \delta _ \lambda ^{\    \mu } 
F^a _{(2) \alpha \beta }
F ^{\    \alpha \beta }_ {(2) a} ) + \right. 
 $$
 $$
\left. + ( F_{(1) \lambda \nu} 
\rule{0mm}{5mm}
F^{\    \mu \nu } _{(1)} - 
\frac{1}{4}
\delta _ \lambda ^{\    \mu } F_{(1) \alpha  \beta } 
F^{\    \alpha \beta }_{(1)} ) 
 \right]   
\leqno (2.12b)
$$
and 
$$
T_ \lambda ^{\    \mu } (\phi) = \frac{1}{2} 
\left[   \rule{0mm}{5mm}
( D_ \lambda \phi)^\dagger D^ \mu \phi + 
(D^ \mu \phi)^\dagger D_ \lambda \phi - \right.
$$
$$
\left. - \delta _ \lambda ^{\    \mu } 
\left \{ 
(D_\alpha \phi )^\dagger D^\alpha \phi - 
\mu ^2 \phi^\dagger \phi - 
\frac{\lambda }{12} ( \phi^\dagger \phi)^2 
 \right \} 
 \right]    .
\leqno (2.12c)
$$
With respect to the field-equations the conservation laws for 
energy and momentum of the whole system of fields are valid:
$$
\partial_ \mu T^{\    \mu }_ \lambda = 0 .
\leqno (2.13)
$$

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 fermionic matter fields ($\psi $-fields) and the 
gauge-fields ($F$-fields). From (2.12) and (2.13) one finds immediately 
under neglection of surface-integrals in the space-like infinity:
$$
\partial_0 \int 
\left[ \rule{0mm}{5mm}
T^{\  0}_ \lambda (\psi ) +  T^{\  0}_\lambda (F) 
\right]  
d^3 x = 
 - \int \partial_ \mu 
T_ \lambda ^{\  \mu } (\phi)
d^3 x . 
\leqno (2.14)
$$
Insertion of $T_ \lambda ^{\  \mu }(\phi)$ according to (2.12c) and 
elimination of the second derivatives of the Higgs-field by the 
field-equation (2.11) results 
with the use of the definitions of the field-strengths $ F_{(1) \mu \nu }$ 
and $ F^a _{(2) \mu \nu }$  in:
$$
\frac{\partial}{\partial t} \int 
\left[ \rule{0mm}{5mm}
T^{\  0}_ \lambda (\psi ) + 
T^{\  0}_ \lambda (F) 
\right]  
d^3 x = 
\leqno 
$$
$$
= k \int 
\left[   
(D_ \lambda \phi)^\dagger \overline{\psi } _{Rm_i} 
\hat x ^{m_i} _{\  n_j} \psi _L ^{\  n_j} + 
\overline{\psi } _{Lm_i} \hat x ^{\dagger m_i} _{\  \  n_j} 
\psi _R ^{\  n_j}D_ \lambda \phi \right]   d^3 x  +  
\leqno 
$$
$$
+ \frac{i}{2}
\int 
\left[   \rule{0mm}{5mm}
g_2 F^a _{(2) \mu \lambda } 
\left \{ 
\phi^\dagger \tau _a D^\mu \phi - 
(D^\mu \phi) ^\dagger \tau _a \phi 
 \right \} + \right.
\leqno 
$$
$$
\left. \rule{0mm}{5mm} + \frac{1}{2}g_1 F_{(1) \mu \lambda } 
 \left \{ 
\phi^\dagger D^\mu \phi - (D^\mu \phi) ^\dagger \phi 
 \right \} 
 \right]   d^3 x. 
\leqno (2.15)
$$
The right hand side represents the expectation value of the 4-force, which 
changes the 4-momentum of the $\psi $-fields and of the $F$-fields with 
time. However, the latter expression can be rewritten with the use of 
the field-equations (2.9a) and (2.9b) as follows:
$$
\partial_ \mu T_ \lambda ^{\    \mu } (F) = 
\hbar \left[ 
F_{(2) \mu \lambda }^a
\left \{ 
j_{(2)a}^ {\    \mu } (\psi ) + 
j_{(2)a}^ {\    \mu } (\phi) 
\right \} + \right. 
\leqno 
$$
$$
+ \left. 
F_{(1) \mu \lambda } 
\left \{ 
j_{(1)}^ {\    \mu }(\psi ) + j_{(1)}^ {\    \mu }(\phi ) 
\right \} \
\right]  . 
\leqno (2.16)
$$
Herewith one obtains instead of (2.15):
$$
\frac{\partial}{\partial t} 
\int T^{\    0}_ \lambda  (\psi )d^3 x = 
\int \hbar
 \left[ 
F_{(2) \mu \lambda }^a 
j_{(2)a}^ {\    \mu }(\psi ) + 
F_{(1) \mu \lambda }
j_{(1)} ^\mu (\psi ) 
\right]  
d^3 x + 
\leqno 
$$
$$
+ k \int 
\left[
(D_ \lambda \phi)^\dagger 
\overline{\psi }_{Rm_i} 
\hat x ^{m_i} _{\    n_j} \psi ^{n_j}_L + 
\overline{\psi }_{Lm_i} 
\hat x ^{\dagger m_i} _{\   \  n_j}\psi ^{n_j} _R D_ \lambda \phi
\right]  
d^3 x,
\leqno (2.17)
$$
where on the right hand side we have the 4-force of the gauge-fields and 
the Higgs-field, both acting on the matter field and changing its
4-momentum. Evidently, the gauge-field strengths couple to the
gauge-currents $j_{(2)a} ^{\    \mu }(\psi ) $ and 
$j_{(1)}^ {\    \mu }(\psi )$, i.e. to the gauge-coupling 
constants $g_1$ and $g_2$ according to (2.10a) and (2.10b), whereas the
Higgs-field strength (gradient of the Higgs-field) couples to the
fermionic mass-parameter $k$ only (c.f. [4]). 
This fact points to a {\underline{gravitational}} action of the scalar 
Higgs-field.

\section*{3. Field-Equations of Higgs-Gravity.}

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 classical scalar gravity is the trace of the
energy-momentum tensor.}} For this $\mu^2 < 0$ must be valid, and
according to (2.11) and (2.12c) the ground-state $\phi _0$ of the
Higgs-field is defined by 
$$
\phi _0^ {\  \dagger} \phi _0 = v^2 = \frac{-6 \mu ^2}{\lambda }, 
\leqno (3.1)
$$
which we resolve as 
$$
\phi_0 = vN
\leqno (3.2)
$$
with 
$$
N^\dagger N = 1,  \quad \partial_ \lambda N 
= 0 .
\leqno (3.2a)
$$
The general Higgs-field $\phi$ is different from (3.2) by a local
unitary transformation:
$$
\phi = \rho UN, \quad U^\dagger U = 1
\leqno (3.3)
$$
with 
$$
\phi^\dagger \phi = \rho ^2 , \quad \rho = v (1 + \varphi ) ,
\leqno (3.3a)
$$
where $\varphi $ represents the real valued excited Higgs-field. Now we
use the possibility of a unitary gauge transformation which is inverse
to (3.3):
$$
\phi ' = U^{-1} \phi, \quad \psi ' = U^{-1} \psi , \quad 
{\cal F}' _{\mu \nu } = U^{-1} {\cal F}_{\mu \nu }U ,
\leqno (3.4)
$$
so that 
$$
\phi ' = \rho N
\leqno (3.4a)
$$
is valid, and perform in the following all calculations in the gauge
(3.4) (unitary gauge). 

Using (3.2) up to (3.4a) the field equations (2.8a) through (2.11) take
the form, avoiding the strokes introduced in (3.4):
$$
i \gamma ^ \mu D_ \mu  \psi ^{m_i} _L 
- \frac{1}{\hbar}( 1 + \varphi ) \hat m ^{m_i} _{\    n_j}
\psi ^{\    n_j} _R = 0,
\leqno (3.5a)
$$
$$
i \gamma ^ \mu D_ \mu  \psi ^{m_i} _R 
- \frac{1}{\hbar}( 1 + \varphi ) \hat m ^{m_i} _{\    n_j}
\psi ^{\    n_j} _L = 0,
\leqno (3.5b)
$$
$$
D_ \mu F_{(2)a}^{\    \mu \lambda } + 
\frac{1}{\hbar^2}(1 + \varphi)^2  \left[ 
M^2 _{(2)ab} W^{b \lambda } + 
M_{(1,2)a}^2 B^ \lambda \right]  = 4 \pi  j_{(2)a}^ {\    \lambda }
(\psi ), 
\leqno (3.6a)
$$
$$
\partial_ \mu  F^ {\    \mu \lambda } _{(1)} + 
\frac{1}{\hbar^2}(1 + \varphi )^2 
\left[  M^2 _{(1,2)a} W^{a \lambda } + M^2 _{(1)} 
B^\lambda  \right]  = 
4 \pi j^{\    \lambda }_ {(1)} (\psi ), 
\leqno (3.6b)
$$
$$
\partial _\mu \partial ^\mu \varphi + 
\frac{M^2}{\hbar^2 }\varphi + 
\frac{1}{2} \frac{M^2}{\hbar^2 } (3 \varphi ^2 +\varphi ^3)
 = 
\leqno $$
$$
= -\frac{1}{v^2} \left[   \rule{0mm}{5mm}
\overline{\psi }_{Lm_i}\hat m ^{m_i} _{\    n_j} \psi ^{n_j} _R + 
\overline{\psi }_{Rm_i}\hat m ^{m_i} _{\    n_j} \psi ^{n_j} _L - \right. 
\leqno 
$$
$$
\left. \rule{0mm}{5mm} - \frac{1}{4 \pi \hbar} \left \{
M^2 _{\   (2)ab} W^a _ {\    \lambda } W^{b \lambda } + 
2 M^2 _{(1,2)a} W^a _ {\    \lambda } B^\lambda + 
M^2 _{(1)} B_ \lambda B^\lambda \right \} (1 + \varphi ) \right]  , 
\leqno (3.7)
$$
wherein 
$$
M^2 = -2 \mu ^2 \hbar^2, \quad (\mu ^2 < 0)
\leqno (3.7a)
$$
is the square of the mass of the Higgs-field ($\varphi $-field) and 
$$
\hat m ^{m_i} _{\    n_j} = kv( N^\dagger \hat x ^{m_i} _{\    n_j} + 
\hat x ^{\dagger m_i} _{\  \    n_j}N )
\leqno (3.8)
$$
is the mass-matrix of the fermionic $\psi $-fields, which must be
adjusted to the observed mass-values of the fermions. The matrices of
the mass-squares of the gauge fields are defined by 
$$
M^2 _{(2)ab} = 4 \pi \hbar v^2 g^2_2 N^\dagger \tau _{(a} \tau _{b)} N =  
M^2 _W \delta _{ab},
\leqno (3.9a)
$$
$$
M^2 _{(1,2)a} = 4 \pi \hbar v^2 g_1 g_2 \frac{1}{2}N^\dagger \tau _a N = 
- M^2_W \frac{g_1}{g_2} \delta^{\  3} _a ,
\leqno (3.9b)
$$
$$
M^2 _{(1)} = \pi \hbar v^2 g_1^2 = M^2_W ( \frac{g_1}{g_2})^2 ,
\leqno (3.9c)
$$
where $N = {{0} \choose {1}} $ is chosen and 
$$
M_W = \sqrt {\pi  \hbar} vg_2. 
\leqno (3.10)
$$
Diagonalization of (3.9a) up to (3.9c) yields the four eigenvalues:
$$
M^2 _W ; \quad M^2 _W; \quad M^2 _Z = 
\pi \hbar v^2 (g^2_1 + g^2_2 ); \quad 0
\leqno (3.11)
$$
with the corresponding eigenvectors:
$$
W^1 _{\   \lambda } ; \quad W^2 _{\   \lambda } ; \quad 
Z_ \lambda = c_W W^3 _{\   \lambda } - s_W B_ \lambda ; \quad 
A_ \lambda =  s_W W^3 _{\   \lambda } + c_W B_ \lambda , 
\leqno (3.11a)
$$
wherein $c_W = \cos \theta _W$ and $ s_W = \sin \theta _W$ ($\theta _W$
Weinberg-angle). The field-strengths belonging to (3.11a) are given by:
$$
F_{(W^1)}^{\    \mu \lambda } = F_{(2)}^{ 1 \mu \lambda } ; \quad 
F_{(W^2)}^{\    \mu \lambda } = F_{(2)}^{ 2 \mu \lambda } ; 
\leqno 
$$
$$
F_{(Z)}^{\    \mu \lambda } = c_W F_{(2)}^{ 3 \mu \lambda } -  
s_W F_{(1)}^{\  \mu \lambda };
\leqno 
$$
$$
F_{(A)}^{\    \mu \lambda } = s_W F_{(2)}^{ 3 \mu \lambda } + 
c_WF_{(1)}^{\  \mu \lambda } . 
\leqno (3.12)
$$
Herewith we obtain from (3.6a) and (3.6b) in view of (3.9a) through
(3.11) the gauge-field equations: {\footnote{The covariant derivative in
(3.13a, b, c) is defined by the covariant derivative of the right hand
side of (3.12) according to (2.9a).}}
$$
D_ \mu F^{\    \mu \lambda }_{(W^{1,2})} + 
(1 + \varphi )^2 \left ( \frac{M_W}{\hbar}\right ) ^2 W^{1,2 \lambda } = 
4 \pi j^{1,2 \lambda }_{(2)} (\psi ) , 
\leqno (3.13a)
$$
$$
D_ \mu F^{\    \mu \lambda }_{(Z)} + 
(1 + \varphi )^2 \left ( \frac{M_Z}{\hbar}\right ) ^2 Z^ \lambda  = 
4 \pi j^{\   \lambda }_{(Z)} (\psi ) , 
\leqno (3.13b)
$$
$$
D_ \mu F^{\    \mu \lambda }_{(A)} = 4 \pi j^{\   \lambda }_{(A)} (\psi ) 
\leqno (3.13c)
$$
with the matter current densities corresponding to (3.12):
$$
j_{(Z)}^{\    \lambda }(\psi ) = 
c_W j_{(2)}^{\  3  \lambda }(\psi ) -  
s_W j_{(1)}^{\    \lambda }(\psi ) ,
\leqno (3.14a)
$$
$$
j_{(A)}^{\    \lambda }(\psi ) = s_W j_{(2)}^{\  3  \lambda }(\psi ) + 
c_W j_{(1)}^{\    \lambda }(\psi ). 
\leqno (3.14b)
$$
In the same way we find from (3.7) for the Higgs-field $\varphi $:
$$
\partial_ \mu \partial^\mu  \varphi + \frac{M^2}{\hbar^2} \varphi + 
\frac{1}{2} \frac{M^2}{\hbar^2} (3 \varphi ^2 + \varphi ^3) = 
\leqno 
$$
$$
= - \frac{1}{v^2} \left[   \rule{0mm}{5mm}
\overline{\psi }_{Lm_i} \hat m ^{m_i}_{\    n_j}\psi ^{\  n_j}_R + 
\overline{\psi }_{Rm_i} \hat m ^{m_i}_{\    n_j}\psi ^{\  n_j}_L - \right. 
\leqno 
$$
$$
\left. \rule{0mm}{5mm} - \frac{1}{4 \pi  \hbar} \left \{ 
M^{\  2} _W (W^1_ \lambda  W^{1 \lambda } + 
        W^2_ \lambda  W^{2 \lambda }) + 
M^2 _Z Z_ \lambda Z^\lambda  \right \} (1 + \varphi )   \right]  . 
\leqno (3.15)
$$

Obviously, in the field-equations (3.5a), (3.5b), (3.13a) through
(3.13c) and (3.15) the Higgs-field $\varphi $ plays the role of an
(attractive) scalar gravitational potential between the
{\underline{massive}} particles: According to equ. (3.15) the source of
$\varphi $ is the mass of the fermions and of the gauge bosons $W^{1,2}$
and $Z$, {\footnote{The second term on the right hand side of equ.
(3.15) 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
$$
l = \hbar / M
\leqno (3.16)
$$
given by the mass of the Higgs-particle, and $v^{-2}$ has the meaning of
the gravitational constant, so that 
$$
v^{-2} = 4 \pi G \gamma 
\leqno (3.17)
$$
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. 5. 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 (3.5a), (3.5b) and (3.13)
through (3.13c). 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. Evidently, the neutrinos
$\nu ^i_L$ and the photon $A$ do not participate in this gravitational
interaction. 

\section*{4. Gravitational Force and Potential Equation.}

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 (3.15) 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 (2.12) and (2.12a) through (2.12c) one finds after symmetry
breaking:
$$
T_ \lambda ^{\    \mu } = 
T_ \lambda ^{\   \mu } (\psi ) + 
T_ \lambda ^{\    \mu } (W,Z,A) + 
T_ \lambda ^{\    \mu } (\varphi )
\leqno (4.1)
$$
with $T_ \lambda ^{\    \mu  } (\psi )$ given by (2.12a) and
$$
T_ \lambda ^{\    \mu  } (W,Z,A) = 
T_ \lambda ^{\    \mu } (F) + 
\frac{1}{4 \pi \hbar} 
\left[  \rule{0mm}{5mm}
  M_W ^2 
\left \{  
( W^1_ \lambda W^{1 \mu } + 
           W^2_ \lambda W^{2 \mu } ) - \right. \right. 
\leqno 
$$
$$
\left.- \rule{0mm}{5mm} \frac{1}{2} \delta ^{\    \mu }_ \lambda  
( W^1_ \alpha  W^{1 \alpha  } + 
           W^2_ \alpha  W^{2 \alpha  } ) 
  \right \} + 
M^2 _Z \left \{ \rule{0mm}{5mm} Z_ \lambda Z^\mu  - 
\frac{1}{2} \delta ^{\    \mu }_ \lambda Z_ \alpha Z^\alpha 
\right \} \rule{0mm}{5mm}  \left.  \rule{0mm}{5mm} \right] 
\leqno (4.1a)
$$
$(T_ \lambda ^{\    \mu }(F)$ according to (2.12b)) as well as 
$$
T_ \lambda ^{\    \mu }(\varphi ) = 
v^2 \left[  
\partial_ \lambda \varphi \partial^\mu \varphi - 
\frac{1}{2} \delta ^{\    \mu }_ \lambda 
\left \{ 
\partial_\alpha \varphi  \partial^ \alpha  \varphi  + 
\frac{M^2}{4 \hbar^2}(1 + \varphi )^2 (1 - 2 \varphi  - \varphi ^2)
\right \} \right]  . 
\leqno (4.1b)
$$
From this it follows immediately using the field equations (3.5a) and
(3.5b):
$$
T = T_ \lambda ^{\    \lambda } = 
T( \psi ) + T(W,Z,A) + T(\varphi ) 
\leqno (4.2)
$$
with 
$$
T(\psi ) = \left[  \overline{\psi }_{Lm_i} 
\hat m ^{m_i} _{\    n_j} \psi ^{\    n_j}_R + 
\overline{\psi }_{Rm_i} \hat m ^{m_i} _{\    n_j} 
\psi ^{\    n_j}_L \right]  (1 + \varphi ) , 
\leqno (4.2a)
$$
$$
T(W,Z,A) = T(W,Z) = - \frac{1}{4 \pi \hbar} \left[  
M^2 _W ( W^1_ \lambda W^{1 \lambda } + \right. 
$$
$$
 \left.
 + W^2 _ \lambda W^{2 \lambda } ) + 
M^2 _Z Z_ \lambda Z^\lambda \right] (1 + \varphi )^2 
\leqno (4.2b)
$$
and 
$$
T(\varphi ) = v^2 \left[  \frac{M^2}{2 \hbar^2} ( \varphi ^4 + 4 \varphi
^3 + 4 \varphi ^2 -1) - \partial_ \lambda \varphi  \partial  ^\lambda
\varphi \right]  . 
\leqno (4.2c) 
$$
In the appendix it is shown that $T(\psi )$ separates in total analogy
to $T(W,Z)$ into the masses of the single fermions:
$$
T(\psi ) = \sum_{i}  ( m_{e^i} \overline{e}_i e^i + 
m_{u^i} \overline{u}_i u^i + 
m_{d^i} \overline{d}_i d^i) (1 + \varphi ) . 
\leqno (4.2a')
$$
Comparing (4.2a) and (4.2b) with the right hand side of the Higgs-field
equation (3.15) one finds that the source of the potential $\varphi$  is
given by the first two terms of the trace (4.2). In this way we find
using (3.17):
$$
\partial_ \mu \partial ^\mu  \varphi  + 
\frac{M^2}{\hbar ^2} \varphi + 
\frac{1}{2} \frac{M^2}{\hbar ^2} (3 \varphi ^2 + \varphi ^3) = 
\leqno 
$$
$$
= - 4 \pi G \gamma (1 + \varphi )^{-1} (T(\psi ) + T (W,Z)) . 
\leqno (4.3)
$$
In the linearized version (with respect to $\varphi $) equ. (4.3)
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 $W^{1,2} $ and $Z$ as source.

Finally we investigate the gravitational force caused by the Higgs-field
more in detail. Insertion of the symmetry breaking according to (3.1) up
to (3.4a) into the first integral of the right hand side of (2.15)
yields:
$$
K_ \lambda = k \left[  (D_ \lambda  \phi) ^\dagger \overline{\psi }_{Rm_i}
\hat x ^{m_i} _{\    n_j} \psi _L ^{\    n_j} + 
\overline{\psi }_{Lm_i} \hat x ^{\dagger m_i} _{\   \  n_j} 
\psi ^{\    n_j}_R D_ \lambda \phi \right]   = 
\leqno 
$$
$$
 = (\overline{\psi }_{Rm_i} \hat m ^{m_i} _{\    n_j} 
\psi ^{\    n_j}_L + 
\overline{\psi }_{Lm_i} \hat m ^{m_i} _{\    n_j} 
\psi ^{\    n_j}_R ) \partial _ \lambda \varphi + 
\leqno 
$$
$$
+ v(1 + \varphi ) \left[ (D_ \lambda N)^\dagger k 
\overline{\psi }_{Rm_i} \hat x ^{m_i} _{\    \   n_j}
\psi ^{\    n_j}_L + 
k \overline{\psi }_{Lm_i}\hat x ^{\dagger  m_i} _{\   \    n_j}
\psi ^{\    n_j}_R D_ \lambda N \right]  . 
\leqno (4.4)
$$
Substitution of the conglomerate $ k \overline{\psi }_{Rm_i} \hat x
^{m_i} _{\    n_j}\psi ^{\    n_j}_L$ by the left hand side of the field
equation (2.11) results with the use of (3.3a) and (3.4a) in:
$$
K_ \lambda = \left[   \rule{0mm}{5mm} \overline{\psi }_{Lm_i}
\hat m ^{m_i} _{\    n_j} \psi ^{\    n_j}_R + 
\overline{\psi }_{Rm_i}\hat m ^{m_i} _{\    n_j} 
\psi ^{\    n_j}_L - \right. 
\leqno 
$$
$$
\left. - \frac{1}{4 \pi  \hbar} \left\{  
M^2 _W ( W^1_ {\    \alpha } W^{1 \alpha } + 
W^2 _\alpha  W^{2 \alpha }) + M^2 _Z Z_ \alpha Z^ \alpha \right\} (1 +
\varphi )
 \right]  \partial_ \lambda \varphi - 
\leqno 
$$
$$
- \frac{1}{4 \pi  \hbar} \partial_ \mu \left[  \rule{0mm}{5mm} (1 +
\varphi )^2 
\left \{   
M^2 _W 
( W^1_ \lambda  W^{1 \mu } + \right.
 W^2_ \lambda  W^{2 \mu } - \right.
\leqno 
$$
$$
- \frac{1}{2}\delta ^{\    \mu }_ \lambda \left[  \rule{0mm}{5mm}
W^1 _ \alpha   W^{1 \alpha  } + 
 W^2 _ \alpha   W^{2 \alpha } \right] ) + 
\left. \rule{0mm}{5mm} M^2 _Z ( Z_ \lambda Z^\mu  - \frac{1}{2} \delta
^{\    \mu }_ \lambda
Z_ \alpha Z^\alpha ) \left. \right \} \right] + 
\leqno 
$$
$$
+ i \frac{v^2}{2}(1 + \varphi )^2 \left[  \rule{0mm}{5mm} g_2 F^a _{(2)
\mu \lambda } \left \{ 
N^\dagger \tau _a D^\mu N -  
(D^\mu N)^\dagger \tau _a N \right \} + \right. 
$$
$$
+ \left. \rule{0mm}{5mm} g_1 F_{(1) \mu \lambda } 
\left \{ N^\dagger D^ \mu N  
- (D^\mu N)^\dagger N \right \} \right] .
\leqno (4.5)
$$
By insertion of (4.5) into the right hand side of 
(2.15) the last brackets of (4.5) and (2.15) cancel out, 
whereas the second bracket of (4.5) can be combined 
with $ \partial_ \mu T_ \lambda  ^{\    \mu } (F)$ 
to $\partial_ \mu T_ \lambda  ^{\    \mu } (W,Z,A)$ according to 
(4.2b). In this way we obtain neglecting surface integrals in the 
space-like infinity:
$$
\frac{\partial}{\partial t}
\int \left[  \rule{0mm}{5mm} T_ \lambda ^{\    0} (\psi ) + 
T_ \lambda ^{\    0}(W,Z,A) \right] d^3 x = 
\leqno 
$$
$$
= \int \left[ \rule{0mm}{5mm} \overline{\psi }_{Lm_i} \hat m ^{m_i} _{\   
n_j}
\psi ^{\    n_j}_R  + 
\overline{\psi }_{Rm_i}
\hat m ^{m_i} _{\    n_j}\psi ^{\    n_j}_L - \right. 
\leqno 
$$
$$
\left. \rule{0mm}{5mm}
- \frac{1}{4 \pi \hbar} 
\left \{ 
M^2 _W 
( W^1 _ \alpha  W^{1 \alpha } + 
W^2 _ \alpha  W^{2 \alpha }) + 
M^2 _Z Z_ \alpha Z^ \alpha 
\right \} (1 + \varphi ) 
\right] 
\partial _ \lambda \varphi d^3 x . 
\leqno (4.6)
$$
In total analogy to the procedure yielding the potential 
equation (4.3) we substitute the bracket of the 4-force in (4.6) by the 
traces $T(\psi) $ and $T(W,Z) $ given by (4.2a) and (4.2b) 
respectively; so we find:
$$
\frac{\partial}{\partial t} \int \left[  \rule{0mm}{5mm} 
T_ \lambda ^{\    0} (\psi ) + 
T_ \lambda ^{\    0}(W,Z,A) \right] d^3 x = 
\leqno 
$$
$$
= \int (1 + \varphi ) ^{-1} \left[ \rule{0mm}{5mm} T(\psi ) + 
T(W,Z) \right] \partial_ \lambda \varphi d^3 x. 
\leqno (4.7)
$$


Considering the transition from equ. (2.15) to (2.17) we can 
express the time derivative of the 4-momentum of the 
gauge-fields by a 4-force acting on the fermionic matter currents. 
Restricting this procedure to the {\underline{massless}} gauge-field 
$A^ \lambda $ (photon) we get from (4.7): 
$$
\frac{\partial}{\partial t} \int 
\left[  T_ \lambda ^{\    0} (\psi ) + 
T_ \lambda ^{\    0}(W,Z) \right] d^3 x  
= \int \hbar F_{(A) \lambda \mu } j^ \mu _{(A)} (\psi ) d^3 x +
\leqno 
$$
$$
+ \int (1 + \varphi ) ^{-1} \left[  T(\psi ) + T( W,Z) \right] \partial_
\lambda \varphi d^3 x.
\leqno (4.8)
$$
Herein the first term of the right hand side describes the 4-force of
the massless gauge-boson acting on the matter fields, i.e. the
electromagnetic Lorentz-force coupled by the electric charge, see
(3.14b):
$$
e = s_W g_2 = c_W g_1.
\leqno (4.8a)
$$
The second term (identical with the right hand side of (4.7)) is the
attractive gravitational force on the masses of the fermions and of the
gauge-bosons $W^{1,2}$ and $Z$, which are simultaneously the source of
the Higgs-potential $\varphi $ according to (4.3). 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 (4.3) in the finite range of $\varphi $ caused by the
Yukawa-term.

\section*{5. Final Remarks.}

In the end we point to some interesting features of our result. 
First of all we note, that in view of the right hand side of (4.7) 
it is appropriate to define
$$
\ln (1+ \varphi ) = \chi 
\leqno (5.1)
$$
as new gravitational potential, so that the momentum law reads:
$$
\frac{\partial}{\partial t} \int \left[ 
T_ \lambda ^ {\    0} (\psi ) + 
T_ \lambda ^ {\    0} (W,Z,A) \right] d^3 x = 
\leqno 
$$
$$
= \int \left[ T(\psi ) + T( W,Z) \right] \partial _ \lambda \chi d^3 x . 
\leqno (5.2) 
$$
Then the non-linear terms concerning $\varphi $ in (4.3) can be 
expressed by $T(\varphi ) \equiv T( \chi )$ according to (4.2c). In this
way the field equation for the potential $\chi $ (excited Higgs-field)
takes the very impressive form:
$$
 \partial_ \mu \partial^\mu  e^{2 \chi } + \frac{M^2}{\hbar^2} 
e^{2 \chi } = 
- 8 \pi G \gamma \left[ T(\psi ) + 
T(W,Z) + T(\chi ) \right] .
\leqno (5.3)
$$
Equations (5.2) and (5.3) 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 (5.3)
exists already before the symmetry breaking. Considering the trace $T$
of the energy-momentum tensor (2.12) one finds with the use of the
field-equations (2.8a), (2.8b) and (2.11): 
$$
\partial_ \mu \partial^\mu (\phi^ \dagger \phi) + 
( \frac{M}{\hbar})^2 (\phi^ \dagger \phi) = -2T
\leqno (5.4)
$$
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 (5.4) possesses an interesting
behaviour with respect to the symmetry breaking. From the second term on
the left hand side there results in view of (3.1) 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. In our
opinion this is the property of the cosmological constant at all, also
in general relativity. 

Furthermore we emphasize that the gravitational action of the
Higgs-field is not restricted to the Glashow-Salam-Weinberg model, but
it is valid in all cases of mass producing by symmetry breaking via the
Higgs-mechanism [6], e.g. also in the GUT-model. However, because in
(3.16) 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 non-Newtonian gravity currently discussed
as so called fifth force [7]. 

Finally, the factor $\gamma $ in (3.17) can be calculated from (3.10) by
the use of the mass of the W-bosons and the value of the gauge-coupling
constant $g_2$; one finds:
$$
\gamma = \hbar g^2 _2 / 4G M^2 _W = \frac{1}{2}g^2 _2 (\frac{M_P}{M_W})^2 = 
2 \times 10^{32}
\leqno (5.5)
$$
($M_P$ Planck mass). Consequently, the Higgs-gravity represents a
relatively strong scalar gravitational interaction between the massive
elementary particles, however with extremely short range and with the
essential property of quantizability. If any Higgs-field exists in
nature, this type of gravity is present. 

The expression (5.5) 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. \newpage
%
%
%
%
{\underline{\large{Appendix}}}: In order to show the separation of
$T(\psi )$ into the single fermionic masses it is necessary to specify
the fermionic mass-matrix as follows (without suppression of the $SU(2)$
indices $I,J, ...)$: {\footnote{In equ. (A1) the sum-convention does not
hold.}}
$$
\hat m ^{m_i} _{\    n_j} = \hat m ^{Im_i} _{\    J n_j}= 
\sum _k 
m_{Im_k} \delta ^I_ {\    J} \delta ^m _{\    n}
(U^{\   I_m} _{(c)})^i _k 
(U^{\   I_m} _{(c)})^{-1k} _{\  \    j}
\leqno (A1)
$$
where
$$
(U^{I_m} _{(c)})^i _k = 
\left \{ 
\begin{array}{cc}
U_{(c)k}^{\  i} \quad \mbox{if} & (I,m) = (d',q), \\
\delta ^i_ {\    k} & \mbox{otherwise}
\end{array}
\right. 
\leqno (A2)
$$
with the Cabibbo-matrix $U_{(c)  k}^{\  i} $ according to (2.3b).
Insertion of (A1) into the right hand side of (4.2a) yields:
$$
\overline{\psi } _{Lm_i} \hat m ^{m_i} _{\    n_j}\psi ^{\    n_j}_R + 
\overline{\psi }_{Rm_i}\hat m ^{m_i} _{\    n_j}\psi ^{\    n_j}_L = 
\leqno 
$$
$$
= \sum _i \left[ m_{e^i} ( \overline e _{R_i} e^{\    i}_L + 
                           \overline e_{L_i} e^{\    i}_R) +
m_{u^i} ( \overline u_{R_i} u^{\    i}_L + 
                           \overline u_{L_i} u^{\    i}_R) + 
m_{d^i} ( \overline d_{R_i} d^{\    i}_L + 
                           \overline d_{L_i} d^{\    i}_R)  
\right], 
\leqno (A3)
$$
which immediately goes over into the expression (4.2a').

\section*{References.}

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  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. Zeitschr. 
  d. Friedrich-Schiller- Univ. Jena \underline{39}, 41 (1990).
\item[{[4]}] P. Becher, M. B\"ohm and H. Joos, Eichtheorien der starken
  und elektroschwachen Wechselwirkung, Teubner-Verlag (Stuttgart),
  p. 350 (1981).
\item[{[5]}] H. Saller, Vereinheitlichte Feldtheorien der
  Elementarteilchen, Springer-Verlag, p. 41 (1985).
\item[{[6]}] H. Dehnen, H. Frommert and F. Ghaboussi, Int. J. theor.
  Phys. in press.
\item[{[7]}] D. H. Eckhardt et al. Phys. Rev. Lett. {\underline{60}},
  2567 (1988).
\end{itemize}

\end{document}