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% SCALAR GRAVITY AND HIGGS POTENTIAL (H. FROMMERT) FR 
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% Published in: 
%   International Journal of Theoretical Physics, 29(4):361-370, 1990 

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\begin{center}
{\huge Scalar Gravity and Higgs Potential}
\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*{Abstract}

A general Lorentz-invariant scalar gravitational interaction theory with
self-interaction is presented. It is shown that this theory leads to the
recently discovered Higgs-field gravity and thereby provides a new
approach to the Higgs-potential. 

\section*{1. Introduction.}

Scalar theories of elementary particles and their interactions are of
interest due to their importance as Higgs fields in the theory of
spontaneous symmetry breaking. In addition, scalar theories of
gravitation have a long history; the classical example is the Newtonian
theory of gravity, but also more modern theories, for example the
Dicke-Brans-Jordan theory (see [1]) and the examples in ref. [2], deal
with scalar interactions. 

Recently we have pointed out (ref. [3]-[4]) that the scalar interaction
mediated by the Higgs field in theories with spontaneous symmetry
breaking is of gravitational type, i.e., it is coupled to the masses of
the elementary particles and not to any other charges: Mass and not some
currents is the source of the scalar Higgs field, and the Higgs-field
acts back by its gradient on the mass in the momentum law. Moreover, the
spontaneous symmetry breaking generates exactly the mass of the
elementary particles which then serves as active and passive
gravitational mass. Thereby, Einstein's "principle of relativity of
inertia" (Mach's principle, see ref. [5]) is fulfilled: Mass is
generated by the same mechanism as the gravitational interaction. In
this sense, the inertial mass as  a measure for the resistance of a
particle against the relative accelleration with respect to other
particles has its origin in the gravitational interaction with all other
particles in the Universe. 

Here we go the opposite way and construct a very general scalar
gravitational theory between elementary particles on the level of
special relativity. It is imposed to contain self-interaction, and to
obey a Yukawa-type field equation, i.e. we consider a massive scalar
field. This is even necessary, because the only source of
Lorentz-invariant  scalar gravity is the trace of the
energy-momentum-tensor, which vanishes in the massless case. We find
that the scalar field equation is exactly that of the Higgs field with
the correct Higgs potential. 

\section*{2. Structure of the Lagrange-density.}

The most general Lagrange density of a pure scalar field $\varphi $
containing the derivatives of $\varphi $ at most quadratically is given
by 
$$
L_0 = L_0 ( \partial_ \lambda \varphi , \varphi ) = 
\frac{1}{2}( \partial_ \lambda \varphi) \partial^\lambda \varphi - 
V(\varphi ), 
\leqno (2.1)
$$
where $V(\varphi )$ is some arbitrary functional of the field $\varphi$,
usually called the potential term of $L_0$. 

To construct a theory of scalar {\underline{interaction}}, a "matter"
term $L_M$ and an "interaction" part $L_{int}$ must be added to (2.1) in
order to obtain the complete Lagrange density:
$$
L = L_0 + L_M + L_{int}, 
\leqno (2.2)
$$
where $L_M$ is the Lorentz-invariant Lagrange density of the pure matter
fields $\psi^A$ (A represents some set of inner, spinor or tensor
indices which are not specified here) and $L_{int}$ is the interaction
part depending on $\varphi $ and $\psi ^A$ only and not on their
derivatives:
$$
L_M = L_M ( \partial_ \lambda  \psi ^A, \psi ^A),
\leqno (2.3)
$$
$$
L_{int} = L_{int} (\varphi , \psi ^A).
\leqno (2.4)
$$
The field equations for $\varphi $ and $\psi ^A$ obtained from (2.2) by
the variational principle are :
$$ 
\partial_ \lambda \partial^\lambda \varphi + 
\frac{\partial V}{\partial \varphi } = 
\frac{\partial L_ {int}}{\partial \varphi } =: - \eta (\varphi , \psi ^A),
\leqno (2.5)
$$
$$
\partial_ \lambda p_A ^{\  \lambda}  - 
(\frac{\partial L_M}{\partial \psi ^A} + 
\frac{\partial L_{int}}{\partial \psi ^A}) = 0
\leqno (2.6)
$$
with the {\underline{source}} $ \eta (\varphi,  \psi ^A)$ of the scalar
field $\varphi $ and the canonical momentum of the matter field 
$$
p_A ^{\  \lambda} := \frac{\partial L_M}{\partial (\partial_ \lambda
\psi ^A)}.
\leqno (2.7)
$$


The canonical energy-momentum tensor is given by
$$
T_ \lambda ^{\   \mu } = T_ \lambda ^{\   \mu }(\varphi ) + 
T_ \lambda ^{\   \mu }(\psi ),
\leqno (2.8)
$$
where
$$
T_ \lambda ^{\   \mu }(\varphi ) = (\partial_ \lambda \varphi )
\partial^\mu \varphi -
\delta_ \lambda ^\mu \left[ \frac{1}{2}(\partial_v \varphi )
\partial^v \varphi - 
V(\varphi )\right]
\leqno (2.8a)
$$
and
$$
T_ \lambda ^{\   \mu }(\psi ) = 
p_A ^{\  \mu } \partial _ \lambda \psi ^A - \delta ^\mu _\lambda 
(L_M + L_{int})
\leqno (2.8b)
$$
are the parts of $T_ \lambda ^{\   \mu } $ resulting from the pure
scalar field $\varphi $ and the matter fields $\psi ^A$, respectively.
It obeys with respect to the field equations (2.5) and (2.6) the
equation of continuity: 
$$
\partial_ \mu T_ \lambda ^{\   \mu } = 0
\leqno (2.9)
$$
and has the trace 
$$
T = T_ \lambda ^\lambda = T(\varphi ) + T(\psi ) = 
\left[ - (\partial_ \lambda \varphi ) \partial^\lambda  \varphi   + 4
V(\varphi ) \right]
$$
$$
+ \left[ p_A ^{\   \lambda}  \partial_\lambda \psi ^A - 4(L_M + L_{int})
\right],
\leqno (2.10)
$$
which represents the rest mass-energy densities of the scalar field and
the matter field, respectively.

Splitting $T_ \lambda ^{\   \mu }$ according to (2.8), the equation of
continuity (2.9) yields 
$$
0 = \partial_ \mu T_ \lambda ^{\   \mu }(\varphi ) + 
\partial_ \mu T_ \lambda ^{\   \mu }(\psi ) = 
- (\partial_ \lambda \varphi ) \eta + 
\partial_ \mu T_ \lambda ^{\   \mu }(\psi ),
\leqno (2.11)
$$
where equation (2.8a) and the field equation of the scalar field (2.5)
are inserted. Obviously, equ. (2.11) can be rewritten as 
$$
\partial_ \mu  T_ \lambda ^{\   \mu }(\psi ) = 
(\partial_ \lambda \varphi ) \cdot  \eta .
\leqno (2.11a)
$$
By intergration over a spacelike hypersurface one obtains, neglecting
boundary terms on the left-hand side of (2.11a) 
$$
\frac{d}{dt} P_ \lambda:= 
\frac{\partial}{\partial t} \int d^3 x T^0 _ \lambda (\psi ) 
=  \int d^3 x (\partial_ \lambda \varphi ) \eta  =: K_ \lambda ,
\leqno (2.12)
$$
which is the {\underline{momentum law}} for the matter field: The
4-momentum $P_ \lambda $ of the matter fields on the left-hand side of
(2.12) is changed with time by the 4-force $K_ \lambda $ caused by the
4-gradient of the scalar field $\varphi $ acting on the matter field
described by $\eta $, which is simultaneously the source of the scalar
field $\varphi $ according to (2.5). As it must be, particles that do
not participate in the interaction are not influenced by the scalar
force, due to $\eta = 0$ in this case.  

Evidently equs. (2.5) and (2.12) describe a self-consistent
{\underline{gravitational}} interaction only if $\eta $ is proportional
to the trace $T(\psi )$ of the energy-momentum tensor of the matter
field according to (2.10). 

\section*{3. Determination of the potential.}

Now, in a {\underline{physical}} theory, the rest mass-energy must have
a lower bound in order to avoid infinite negative energies, and,
therefore, according to (2.10) the potential term $V(\varphi )$ should
have a minimum, say, at $\varphi = v$. Expanding the potential in the
neighbourhood of this minimum, one obtains 

$$
V(\varphi ) = V_0 + \frac{1}{2} (\frac{M}{\hbar})^2 (\varphi -v)^2 + 
{\cal{O}} \left[ (\varphi - v )^3 \right], 
\leqno (3.1)
$$
with $V_0 = V(v)$ and $M = const.$. At this stage we assume $v \neq 0$
{\footnote{It should be noted that in the case of $v = 0$ no meaningful
scalar gravity can be constructed.}}; furthermore it is convenient to
introduce the {\underline{excited scalar field}} $\chi $ according to 
$$
\varphi = v(1 + \chi ).
\leqno (3.2)
$$
With the {\underline{new source}} 
$$
\hat \eta = - \frac{\partial L_{int}}{\partial \chi } = 
v \eta 
\leqno (3.3)
$$
and the {\underline{new potential}} 
$$
\hat V = \hat V (\chi )= 
\frac{V (\chi )}{v^2} = 
\frac{V_0}{v^2}  
+ \frac{1}{2} (\frac{M}{\hbar})^2 \chi ^2 + {\cal{O}} (\chi ^3)
\leqno (3.4)
$$
one obtains from (2.5) the field equations for the excited scalar field
$\chi $ 
$$
\partial_ \lambda \partial^\lambda  \chi + 
\frac{\partial \hat V }{\partial \chi } = 
- \frac{1}{v^2} \hat \eta 
\leqno (3.5)
$$
with 
$$
\frac{\partial \hat V}{\partial \chi } = 
(\frac{M}{\hbar})^2 \chi  + {\cal{O}} (\chi ^2) ,
\leqno (3.6)
$$
according to which $M$ is the mass of the excited scalar field $\chi $.
Analogously, the new equation of continuity, see (2.11a), reads 
$$
\partial_ \mu  T_ \lambda ^{\   \mu }( \psi ) = (\partial_ \lambda  \chi
) \cdot \hat \eta , 
\leqno (3.7)
$$
and the momentum law (2.12) takes the new form:
$$
\frac{d}{dt} P_ \lambda  = 
\frac{\partial}{\partial t} \int d^3 x T_ \lambda ^{\   0}(\psi ) 
= \int d^3 x (\partial_ \lambda \chi )
\hat \eta  = K_ \lambda. 
\leqno (3.8)
$$
Comparison with Newtonian gravity shows that $\frac{1}{v^2}$ 
plays the role of the gravitational constant. 

For the establishment of the {\underline{gravitational}} character of
the scalar interaction in detail, it remains (see the last remark of
sect. 2) to postulate that the source $\hat \eta $ in (3.5) is
proportional to the trace $T(\psi )$ of the energy momentum tensor of
the matter fields. This means
$$
\hat \eta = F(\chi ) T(\psi ) 
\leqno (3.9)
$$
with some functional $F(\chi )$. To realize the gravitational
{\underline{self-interaction}} we add $F(\chi ) T(\chi)/v^2$ on both
sides of (3.5) where 
$$ 
T(\chi ) = T(\varphi ). 
\leqno (3.10)
$$ 
In this way we get from (3.5) the field equation for the scalar field in
the form:
$$
\partial_ \lambda \partial^\lambda \chi  + 
\frac{\partial \hat V}{\partial \chi } - 
 \frac{F(\chi )}{v^2} T(\chi) = 
- \frac{F(\chi )}{v^2} (T(\psi  ) + T(\chi )) .
\leqno (3.11)
$$
Simultaneously the momentum law (3.8) reads:
$$
\frac{d}{dt} P_ \lambda = \int d^3 x (\partial_ \lambda \chi ) F(\chi )
T(\psi ).
\leqno (3.12)
$$
Dividing (3.11) by $F(\chi )$ we obtain: 
$$
\frac{1}{F(\chi) } \partial_ \lambda  \partial^ \lambda \chi  + 
\frac{1}{F(\chi) } \frac{\partial \hat V}{\partial \chi } - 
\frac{1}{v^2} T(\chi ) = - \frac{1}{v^2} (T(\psi ) + T(\chi )).
\leqno (3.13)
$$

In case of a self-interacting scalar gravity, a functional $u(\chi )$
should exist in such a way, that equ. (3.13) takes the form ($a$ =
const.):
$$
\partial_ \lambda \partial^\lambda  u + a^2 u = 
- \frac{1}{v^2} (T(\psi ) + T(u )).
\leqno (3.14)
$$
This is a Yukawa-equation with the mass-term $a^2u$ and self-interaction
described by the term $T(u)$; for $T(u )$ one obtains from (2.10) and
(3.10) using (3.2):
$$
T(u) = T(\chi ) = - v^2 \left[ (\partial_ \lambda \chi )
\partial^\lambda \chi  - 4 \hat V (\chi )\right] .
\leqno (3.15)
$$
Inserting the identity 
$$
\partial_ \lambda \partial^ \lambda  u(\chi ) = 
\frac{\partial u}{\partial \chi } \partial_ \lambda \partial^ \lambda 
\chi + 
(\frac{\partial^2 u}{\partial \chi ^2}) ( \partial_ \lambda \chi )
\partial^ \lambda \chi 
\leqno (3.16)
$$
into (3.14) and subtracting of (3.13) after insertion of (3.15) we find 
$$
( \frac{\partial u }{\partial \chi  } - \frac{1}{F}) 
\partial_ \lambda \partial^\lambda \chi  + 
( \frac{\partial^2 u }{d \chi ^2} - 1) (\partial_ \lambda \chi )
\partial^\lambda \chi  
- (\frac{1}{F} \frac{\partial {\hat V}}{\partial \chi } - 4 {\hat V} -
a^2 u) = 0 .
\leqno (3.17)
$$
This equation must hold for arbitrary $\partial_ \lambda \chi , \partial
_ \lambda  \partial^ \lambda \chi $; this requires that each of the
three terms in (3.17) vanishes independently, resulting in 
$$
\frac{\partial^2 u}{\partial \chi ^2} = 1, 
\leqno (3.18a)
$$
$$
\frac{1}{F} \frac{\partial \hat V}{\partial \chi } - 
4 \hat V (\chi ) - a^2 u(\chi ) = 0,
\leqno (3.18b) 
$$
$$
F(\chi ) = \frac{1}{\partial u / \partial \chi }. 
\leqno (3.18c)
$$
Integration of (3.18a) determines $u$ up to two constants of integration
A and B:
$$
u = \frac{1}{2} (\chi ^2 + 2A \chi + B) = 
\frac{1}{2}\left[ (\chi + A)^2 + (B - A^2) \right].
\leqno (3.19)
$$
From this one has 
$$
\frac{\partial u}{\partial \chi } = \chi  + A , 
\leqno (3.19a)
$$
and (3.18c) gives the functional $F(\chi ) $:
$$
F(\chi ) = \frac{1}{\chi + A},
\leqno (3.20)
$$
by which (3.18b) yields after integration:
$$
\hat V (\chi ) = \hat V_0 + 
\frac{C}{2}a^2 \left[(\chi + A)^2 - \frac{1}{4C} \right]^2
\leqno (3.21)
$$
with 
$$
\hat V_0 = \frac{1}{8}a^2 \left[ A^2 - B - \frac{1}{4C}\right] ,
\leqno (3.21a)
$$
where C is the new integration constant. Herewith, the demanded
structure of (3.14) is achieved. 

Now, in order to specify the potential $\hat V$ or $V$ explicitely, the
constants $A,B,C$ and $a$ must be determined. Remembering that $\hat V$
has a minimum at $\chi = 0$ results in the relation 
$$
C = \frac{1}{4A^2}.
\leqno (3.22)
$$
Herewith $\hat V_0 $ is simplified to 
$$
\hat V_0 = - \frac{B}{8}a^2
\leqno (3.22a)
$$
and $\hat V (\chi )$ takes the form:
$$
\hat V (\chi ) = \hat V_0 + \frac{1}{2} (\frac{Aa}{2})^2
\left[ (1 + \frac{\chi }{A})^2 - 1 \right]^2 . 
\leqno (3.23)
$$
Comparison of (3.23) with (3.4) gives 
$$
a = \frac{M}{\hbar}, 
\leqno (3.24)
$$
so that $M$ represents also the mass of the field $u$ (cf. (3.14)).

The integration constant $A$ lacks a deeper physical meaning because it
can be eliminated by the substitution of the scalar field $\varphi $ by
a {\underline{new}} one $\phi$ differing from $\varphi $ by a constant
only:
$$
\varphi \rightarrow \phi = \varphi  + (A - 1)v .
\leqno (3.25)
$$
$\phi$ obeys the same field equation (2.5) as $\varphi $ because of $
\frac{\partial V}{\partial \phi} = 
\frac{\partial V}{ \partial \varphi }$: 
$$
\partial_ \lambda \partial^\lambda  \phi+ 
\frac{\partial V}{\partial \phi} = - \eta .
\leqno (3.26)
$$
The use of $\phi$ instead of $\varphi $ leads to a minimum of $V(\phi) $
at 
$$
\phi = v' = \phi(\varphi  = v) = Av. 
\leqno (3.27)
$$
The new excited scalar field is 
$$
\chi ' = \frac{\phi - v'}{v'} = \frac{ \chi  }{A},
\leqno (3.28)
$$
which obeys a field equation of the form of (3.5):
$$
\partial_ \lambda  \partial^\lambda  \chi ' + 
\frac{\partial \hat V'}{\partial \chi ' } = - \frac{1 }{v'^2}
\hat \eta ' , 
\leqno (3.29)
$$
where 
$$
\hat V'(\chi ') = \frac{V}{v'^2 }= 
\frac{\hat V}{A^2}
\leqno (3.29a)
$$
and 
$$
\hat \eta ' = v' \eta =   A \hat \eta = 
- \frac{\partial L_{int}}{\partial \chi '} .
\leqno (3.29b)
$$
With the use of the relations (3.24), (3.28) and (3.29a)the following 
expression for $\hat V'$ follows from equation (3.23): 
$$
\hat V' (\chi ') = \hat V'_0 + \frac{1}{8} 
(\frac{M}{\hbar})^2 \left[ (1 + \chi ')^2 -1 \right] ^2 ,
\leqno (3.30)
$$
where
$$
\hat V'_0 = \frac{\hat V_0}{A^2} = - \frac{1}{8}
\frac{B}{A^2} ( \frac{M}{ \hbar })^2 . 
\leqno (3.30a)
$$
Obviously only the quotient $B/A^2$ appears in the potential (3.30). 

Now we are able to write down the potential $V(\phi)$ in the field
equation (3.26) explicitely; insertion of (3.30) into (3.29a) and
resolution with respect to $V$ give with the use of (3.28):
$$
V(\phi) = \frac{v'^2}{8} \left[ 1 - \frac{B}{A^2}\right] 
(\frac{M}{\hbar})^2 - \frac{1}{4} ( \frac{M}{\hbar})^2 \phi^2 +
\frac{1}{8} ( \frac{M}{\hbar})^2 \frac{1}{v'^2} \phi^4 ; 
\leqno (3.31)
$$
the minimum lies at $\phi = v'$ and this minimum value is 
$V(v') = - \frac{B}{A^2} ( \frac{M}{\hbar})^2 \frac{v'^2}{8}$. 
The quantity $B/A^2$  has no deeper meaning because it is  contained
only in the additive constant of the potential (3.31) or in its minimum
value. For simplicity, we choose without restriction of generality the
additive constant of the potential $V(\phi)$ to be equal to zero; this
means:
$$
\frac{B}{A^2} = 1.
\leqno (3.31a)
$$
Evidently, equation (3.31) represents the Higgs potential, which is here
derived by postulating a scalar self-interacting massive gravitational
field. Moreover the field equation (3.26) for the scalar field $\phi$ is
exactly the Higgs-field equation.

\section*{4. Scalar-Gravity with self-interaction. }

Finally we give an explicite representation of the self-interacting
massive scalar gravity introduced in sect. 3. In view of the
ground-state $v'$ for the scalar field $\phi$ according to (3.27) 
and the excited scalar field $\chi ' $ given by (3.28) we rewrite 
the momentum law (3.12) and the Yukawa-field-equation (3.14) 
with the use of (3.24) as follows:
$$
\frac{d}{dt}P_ \lambda = \int d^3 x (\partial_ \lambda \chi ') F'(\chi
') T( \psi )
\leqno (4.1)
$$
and 
$$
\partial_ \lambda \partial ^\lambda u' + (\frac{M}{\hbar})^2 u' = 
- \frac{1}{v'^2} (T(\psi ) + T(u '))
\leqno (4.2)
$$
($T(u') = T(\chi ' )$) with 
$$
u' = \frac{v^2}{v'^2} u = \frac{1}{A^2 }u = 
\frac{1}{2}(1 + \chi ')^2
\leqno (4.3a)
$$
and 
$$
F'(\chi ') = A F(\chi  ') = \frac{1}{1 + \chi '}
\leqno (4.3b)
$$
using (3.19), (3.20) and (3.31a). With respect to the 4-force on the
right hand side of (4.1) it is convenient to choose instead of $\chi '$ 
$$
\zeta  = \ln (1 + \chi ')
\leqno (4.4)
$$
as new excited scalar field. Herewith it follows $u'= \frac{1}{2}e^{2
\zeta }$ and the equations (4.1) and (4.2) take the form:
$$
\frac{d}{dt} P_ \lambda  = \int d^3 x 
(\partial _ \lambda \zeta  ) T(\psi ), 
\leqno (4.5)
$$
$$
\partial_ \lambda  \partial ^\lambda e^{2 \zeta  } + 
( \frac{M}{\hbar})^2 e^{2 \zeta  } = - \frac{2}{v'^2}
(T(\psi ) + T( \zeta ))
\leqno (4.6)
$$
where $T(\zeta ) = T(\chi ')$. Evidently, these equations describe a
massive scalar gravitational interaction with self-interaction, where
with respect to the Newtonian limit (linearization in $\zeta $) 
$$
\frac{1}{v'^2} = 4 \pi  G \gamma 
\leqno (4.6a)
$$
has the meaning of the gravitational constant (G Newtonian gravitational
constant) and $\gamma $ is a numerical factor comparing the strength of
the scalar gravity in question with the Newtonian one. Furthermore in
equ. (4.6) M is the mass of the excited scalar field $\zeta $;
simultaneously there appears a cosmological constant $ \frac{1}{2}(M /
\hbar)^2$, which however drops out against the trace of the
energy-momentum tensor of the ground-state $T(\zeta = 0$). 

It may be of interest that a special-relativistic version of the
Newtonian gravity is included in the general theory for the special case

$$
v'^{-2} = 4 \pi  G , \quad ( \gamma  = 1) 
\leqno  (4.7a)
$$
and 
$$
M < \frac{\hbar}{c \cdot (10^4 ly)} \approx 10^{-26} ( \frac{eV}{c^2}), 
\leqno (4.7b)
$$
so that the range of this scalar gravity is at least $10^4$ light-years
because there is experimental evidence that Newton's law of gravitation
is valid at least up to this distance. 

\section*{References}

\begin{itemize}
\item[{[1]}] C. Brans and  R. Dicke, Phys. Rev. {\underline{124}}, 925
  (1961). 
\item[{[2]}] C. W. Misner, K. S. Thorne  and  J. A. Wheeler,
  {\underline{Gravitation}}, W. H. Freeman, San Francisco, 1973 (p. 178,
  exercise 7.1, and p. 1070).
\item[{[3]}] H. Dehnen, F. Ghaboussi and J. Schr\"oder, \newline
  {\em Gravitational Interaction by the Higgs-Field\/} will appear in 
  Wiss. Zeitschr. d. F.-Schiller-Univ. Jena (1990).
\item[{[4]}] H. Dehnen  and  H. Frommert, {\em On the Higgs-Field 
  Gravity within the Standard Model\/}, to be published. 
\item[{[5]}] A. Einstein, Sitzungsber. Preu\ss. Akad. d. Wiss. Berlin,
  1917, 2, p. 142.
\end{itemize}

\end{document}