%Paper: hep-th/9505114
%From: Fernando Tadeu Brandt <fbrandt@snfma2.if.usp.br>
%Date: Thu, 18 May 1995 12:42:17 -0300 (EST)

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\begin{document}
\draft
\title{Nonlinear couplings and tree amplitudes in \\
      gauge theories}

\author{F. T. Brandt and J. Frenkel}
\address{Instituto de F\'\i sica, Universidade de S\~ao Paulo,
S\~ao Paulo, 05389-970 SP, Brasil}

\date{\today}

\maketitle

\vskip 1.0cm

\begin{abstract}
Following a remark advanced by Feynman,
we study the connection between the form
of the nonlinear vertices involving
gauge particles and the Abelian gauge invariance of physical tree amplitudes.
We show that this requirement, together with some natural assumptions,
fixes uniquely the structure of the Yang-Mills theory.
However, the constraints imposed by the above property are not
sufficient to single out the gauge theory of gravitation.
\end{abstract}

\pacs{}

\section{Introduction}

In the Yang-Mills theory, the source of the Yang-Mills fields is the
conserved color current. Since these fields carry color, these will
self-interact leading to a non-Abelian gauge theory \cite{yangmills}.
Similarly, the source of the gravitational fields is the energy-momentum
tensor, a quantity which is locally conserved. These fields carry energy
and momentum and hence must couple to themselves. The non-Abelian gauge
theory of gravitation, which is invariant under local gauge
transformations, is identical to Einstein's theory \cite{weinberg}.
There has been much fundamental work on basic aspects of the non-Abelian
gauge theories
\cite{feynman,dewitt,faddeev,mandelstam,fradkin,thooft,thooftveltman}.
Feynman \cite{feynman} has shown that in these theories, the tree amplitudes
involving free external gauge fields must be invariant under Abelian
gauge transformations of the external fields. He remarked that this
property may be used
in order to investigate, in an alternative way, the structure of the
nonlinear graviton interactions.

The purpose of this work is to study the question whether the above
property of physical tree amplitudes is sufficient to determine completely
the form of the nonlinear interactions between the gauge particles. We consider
this problem in section \ref{sec2}, first in the simpler context of the
Yang-Mills theory. We assume that the nonlinear interactions between the
gluons are local and involve only dimensionless coupling constants. We find
that in this case the answer to the above question is affirmative, basically
due to the absence of gluon vertices of higher degree than four. In section
\ref{sec3}, we work out the corresponding expressions for gravity, whose
algebraic complexity is much greater. We assume that the interactions
between the gravitons are local and involve only two derivatives of these
fields. This allows for the presence of graviton self-couplings to all
orders. In the gravity case, it is always possible to make a local
redefinition of the basic fields, such that the physical amplitudes will
be the same \cite{thooftveltman}. We argue that, even accounting for this
possibility, the Abelian gauge invariance of the tree amplitudes does not
yield enough constraints to fix the form of the nonlinear
graviton couplings.

We report for simplicity only the results for pure gauge theories, since
the problem we study is basically connected with the self-interaction of
gauge particles. We have verified that the introduction of matter fields
adds only a further algebraic complication, without modifying the above
conclusions. Finally, we mention that other interesting aspects of tree
amplitudes in gauge theories have been discussed recently in the literature
\cite{choi,gould,bern}.


\section{The Yang-Mills theory}\label{sec2}

We start with the Yang-Mills case, characterized by a gauge field
$A^a_\alpha$, where $a$ denotes the color index and $\alpha$ is a
Lorentz index.
The quadratic part of the Yang-Mills Lagrangian
\begin{equation}\label{eq2.1}
{\cal L}^2_{YM}\left(A\right)=
\frac{1}{4}\left(\partial_\beta A^a_\alpha -\partial_\alpha A^a_\beta \right)
           \left(\partial_\beta A^a_\alpha -\partial_\alpha A^a_\beta \right),
\end{equation}
is invariant under the Abelian gauge transformation
\begin{equation}\label{eq2.2}
A^a_\alpha\rightarrow A^a_\alpha + \partial_\alpha \omega^a.
\end{equation}
This leads in momentum space to the free equation of motion
\begin{equation}\label{eq2.3}
\left(\eta_{\alpha\beta}k^2-k_\alpha k_\beta\right)A_\beta^a\left(k\right)=0,
\end{equation}
which is invariant under the gauge transformation
\begin{equation}\label{eq2.4}
\delta A^a_\alpha\left(k\right)=\omega^a k_\alpha.
\end{equation}

We now consider the interactions between the gluons, which we assume to be
local and characterized by dimensionless coupling constants. This natural
assumption allows for vertices involving 3 gluons with one derivative term and
4 gluons with no derivatives, but precludes the presence of higher order
gluon self-couplings. In this case, using Bose symmetry and Lorentz
invariance and disregarding total derivatives terms, we can write the
interaction Lagrangian as follows
\begin{eqnarray}\label{eq2.5}
{\cal L}^I_{YM}\left(A\right)&=&
\left(g\,f_{abc} + e_0\,d_{abc}\right)\left(\partial_\nu A^a_\mu\right)
A^b_\mu A^c_\nu +\nonumber \\
& & \left(l_0\,f_{abe} f_{cde} + l_1\,d_{abe} d_{cde} +
    l_2\,\delta_{ab} \delta_{cd}\right) A^a_\mu  A^b_\nu A^c_\mu A^d_\nu +
\nonumber \\
& & \left(l_3\,d_{abe} d_{cde} + l_4\,\delta_{ab} \delta_{cd}\right)
A^a_\mu  A^b_\mu A^c_\nu A^d_\nu.
\end{eqnarray}
Here $f_{abc}$ denote the antisymmetric color structure constants of the gauge
group SU(N) and $d_{abc}$ are the symmetric color factors. The coupling
constant $g$ sets the scale of the gluon interactions and $e_0$, $l_i$ are
dimensionless couplings which must be determined.

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\caption[f1]{\label{f1}{Basic tree diagrams involving gauge particles.
All momenta are inwards with $\sum k_i=0$.}}
\end{figure}
\bigskip


We proceed by imposing the condition that the gluon tree amplitudes
should be invariant under the Abelian gauge transformation given by
(\ref{eq2.4}). This property \cite{feynman} follows in consequence of the fact
that the external lines satisfy the free equation of motion (\ref{eq2.3}).
We use this constraint on the 3-gluon vertex shown in Fig. 1a and perform
a gauge transformation on the field $A^a_\alpha\left(k_1\right)$. Since
the trilinear gluon coupling proportional to $g\, f_{abc}$ satisfies
identically the above constraint, when we make use of momentum conservation,
we find the condition that
\begin{equation}\label{eq2.6}
e_0\left({k_2}_\beta\ {k_3}_\gamma - k_2\cdot k_3 \eta_{\beta\gamma}\right)
\omega^a\,d_{abc} A^b_\beta(k_2) A^c_\gamma(k_3)=0.
\end{equation}
Because $k_2$ and $k_3$ are arbitrary and independent momenta, this equation
requires the vanishing of the coupling constant $e_0$
\begin{equation}\label{eq2.7}
e_0=0.
\end{equation}
Therefore, in this case the Abelian gauge invariance determines basically
the structure of the trilinear vertex. As we shall see, this special feature
does not occur in the gravity case, which is much more complicated
algebraically.

We now evaluate the contributions from the graph in Fig. 1b and its
permutations to the gluon-gluon scattering amplitude.
In order to perform these calculations, it is simpler to use the Feynman
propagator
$\eta_{\mu\nu}/q^2$. In view of the Abelian gauge invariance
of this amplitude, we must equate the negative of the
gauge variation of these contributions
to the corresponding variations associated with the 4-gluon vertex shown
in Fig 1c. Then, under a gauge transformation
of the gluon field $A^a_\alpha(k_1)$, we find that
\begin{eqnarray}\label{eq2.8}
\left[\delta\;tree\right]_{1c}=-g^2 & \left[f_{abe}f_{cde}\left(
{k_1}_\beta \eta_{\sigma\gamma}+
{k_1}_\sigma \eta_{\beta\gamma}-
2 {k_1}_\gamma \eta_{\beta\sigma}\right)+\right.\nonumber\\
& \left. f_{ace}f_{bde}\left(
{k_1}_\gamma \eta_{\sigma\beta}+
{k_1}_\sigma \eta_{\beta\gamma}-
2 {k_1}_\beta \eta_{\gamma\sigma}\right)\right]\nonumber \\
& \times\omega^a A^b_\beta(k_2) A^c_\gamma(k_3) A^d_\sigma(k_4),
\end{eqnarray}
where we have used the Jacobi identity
\begin{equation}\label{eq2.9}
f_{abe}f_{cde}+f_{ace}f_{dbe}+f_{ade}f_{bce}=0
\end{equation}
to eliminate contributions proportional to $f_{ade}f_{bce}$.

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\caption[f2]{\label{f2}{
Higher order tree amplitude containing nonlinear couplings of gauge particles
}}
\end{figure}
\bigskip

We can now express the gauge variation on the left hand side of
(\ref{eq2.8}) in terms of the parameters introduced in (\ref{eq2.5}).
Using relations like
\begin{equation}\label{eq2.10}
f_{abe}f_{cde}=\frac{2} {N}\left(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}
\right)+d_{ace}d_{dbe}-d_{ade}d_{bce},
\end{equation}
and identifying the coefficients of the independent structures appearing
in (\ref{eq2.8}), we obtain the following relations:
\begin{eqnarray}\label{eq2.11}
&l_1&=-l_3=l_0-\frac{g^2}{4},\nonumber \\
& &\\
&l_2&=-l_4=\frac{2}{N}\left(l_0-\frac{g^2}{4}\right). \nonumber
\end{eqnarray}
We thus see that the parameters $l_i$ have not been fully determined by the
gauge invariance property of the gluon-gluon scattering amplitude.
However, we can now apply this condition also to the 5-gluon tree amplitude
represented by diagrams like the one shown in figures 2a and 2b. Due to
the absence of direct 5-gluon couplings, and using the equations
(\ref{eq2.11}),
it is straightforward to show that this constraint yields a further relation:
\begin{equation}\label{eq2.12}
l_0=\frac{g^2}{4}.
\end{equation}
Together with (\ref{eq2.11}), this relation implies the vanishing of the
coupling constants $l_i\;(i=1,2,3,4)$. Substituting these results in
equation (\ref{eq2.5}), and using (\ref{eq2.1}) and (\ref{eq2.7}), we arrive
at the well known expression for the Yang-Mills Lagrangian
\begin{eqnarray}\label{eq2.13}
{\cal L}_{YM}(A)=&\frac{1}{4}&
\left(\partial_\beta A^a_\alpha-\partial_\alpha A^a_\beta +
g\,f_{abc} A^b_\alpha A^c_\beta \right)
\nonumber \\ & &
\left(\partial_\beta A^a_\alpha-\partial_\alpha A^a_\beta +
g\,f_{a{b'}{c'}} A^{b'}_\alpha A^{c'}_\beta \right).
\end{eqnarray}

\section{The gravitational field}\label{sec3}

In this case, it is convenient to introduce a symmetric tensor field
$h_{\mu\nu}$ representing the deviation of the metric tensor $g_{\mu\nu}$
from the flat space Minkowski metric $\eta_{\mu\nu}$:
\begin{equation}\label{eq3.0}
g_{\mu\nu}=\eta_{\mu\nu}+\kappa\; h_{\mu\nu},
\end{equation}
where $\kappa$ is the usual gravitational constant. Gauge symmetry and
Lorentz invariance enable us to get the linearized gravitational Lagrangian
\begin{eqnarray}\label{eq3.1}
{\cal L}^2(h)=&\frac{1}{2}& h_{\mu\nu,\alpha}h_{\mu\nu,\alpha}-
               \frac{1}{2}  h_{\mu\mu,\alpha}h_{\nu\nu,\alpha}+
\nonumber \\ & &
                            h_{\mu\mu,\alpha}h_{\alpha\nu,\nu}-
                            h_{\mu\nu,\nu}h_{\mu\alpha,\alpha}\;\;,
\end{eqnarray}
where the index after a comma indicates differentiation. Although
we are not making explicit the distinction between up and down indices,
the Minkowski metric tensor $\eta_{\mu\nu}$ is implicitly present in all
the contractions of pairs of identical indices
(e. g. $h_{\mu\mu}=\eta_{\mu\nu}h_{\mu\nu}$).

%\begin{equation}\label{eq3.1a}
%\bar h_{\mu\nu}=h_{\mu\nu}-\frac{1}{2} \eta_{\mu\nu} h^\lambda_\lambda.
%\end{equation}

It is easy to verify that the above Lagrangian is invariant under the Abelian
gauge transformation
\begin{equation}\label{eq3.2}
h_{\mu\nu}\rightarrow h_{\mu\nu} + \xi_{\mu,\nu}+ \xi_{\nu,\mu}.
\end{equation}
By varying this Lagrangian one obtains in momentum space the equation of motion
satisfied by a free graviton
\begin{equation}\label{eq3.3}
\left(
k^2 \eta_{\alpha\mu} \eta_{\beta\nu} - k_{\mu} k_{\alpha} \eta_{\beta\nu}
                                     - k_{\mu} k_{\beta} \eta_{\alpha\nu}
                                     + k_{\alpha} k_{\beta} \eta_{\mu\nu}
\right) h_{\mu\nu}(k)=0,
\end{equation}
which is invariant under the gauge transformation
\begin{equation}\label{eq3.4}
\delta h_{\mu\nu}(k)= k_\nu \xi_\mu + k_\mu \xi_\nu
\end{equation}
In order to proceed, we need to parametrize the general structure of the
graviton self-interactions, which we assume to involve products of fields
with two derivative indices. The algebraic complexity is now so great that
we have made use of computer algebra to do the calculations. We start
constructing the 3-graviton vertex ${\cal L}^3$ as a sum over all possible
independent trilinear products of fields with two derivative terms. When
we write all possible such products and use Lorentz invariance, we find
an expression involving 16 independent constants $a_i$
\begin{equation}\label{eq3.5}
\begin{array}{llclclc}
{\cal L}^3\left(h\right)=\kappa\left(\right.&
 a_{1}\, h_{{\mu\nu}}\, h_{{\alpha\beta ,\mu}}\, h_{{\nu\alpha ,\beta}} &+&
 a_{2}\, h_{{\mu\nu}}\, h_{{\alpha\alpha ,\mu}}\, h_{{\nu\beta ,\beta}} &+&
 a_{3}\, h_{{\mu\nu}}\, h_{{\mu\alpha ,\alpha}}\, h_{{\nu\beta ,\beta}} &+
\\
&a_{4}\, h_{{\mu\nu}}\, h_{{\mu\alpha ,\nu}}\, h_{{\alpha\beta ,\beta}} &+&
 a_{5}\, h_{{\mu\nu}}\, h_{{\alpha\beta ,\mu}}\, h_{{\alpha\beta ,\nu}} &+&
 a_{6}\, h_{{\mu\nu}}\, h_{{\mu\alpha ,\beta}}\, h_{{\nu\alpha ,\beta}} &+
\\
&a_{7}\, h_{{\mu\mu}}\, h_{{\nu\alpha ,\beta}}\, h_{{\nu\alpha ,\beta}} &+&
 a_{8}\, h_{{\mu\mu}}\, h_{{\nu\alpha ,\nu}}\, h_{{\alpha\beta ,\beta}} &+&
 a_{9}\, h_{{\mu\nu}}\, h_{{\mu\nu ,\alpha}}\, h_{{\alpha\beta ,\beta}} &+
\\
&a_{10}\, h_{{\mu\mu}}\, h_{{\nu\alpha ,\nu}}\, h_{{\beta\beta ,\alpha}} &+&
 a_{11}\, h_{{\mu\nu}}\, h_{{\alpha\alpha ,\beta}}\, h_{{\mu\nu ,\beta}} &+&
 a_{12}\, h_{{\mu\nu}}\, h_{{\alpha\alpha ,\mu}}\, h_{{\beta\beta ,\nu}} &+
\\
&a_{13}\, h_{{\mu\nu}}\, h_{{\mu\alpha ,\nu}}\, h_{{\beta\beta ,\alpha}} &+&
 a_{14}\, h_{{\mu\nu}}\, h_{{\nu\alpha ,\beta}}\, h_{{\mu\beta ,\alpha}} &+&
 a_{15}\, h_{{\mu\mu}}\, h_{{\nu\alpha ,\beta}}\, h_{{\nu\beta ,\alpha}} &+
\\
&a_{16}\, h_{{\mu\mu}}\, h_{{\nu\nu ,\alpha}}\, h_{{\beta\beta ,\alpha}}&
 \left.\right)&{}&{}&{}&.
\end{array}
\end{equation}

The next steps are done in correspondence with the ones in the Yang-Mills
theory. We attempt to determine these constants, using the requirement of
gauge invariance under the transformation (\ref{eq3.4}) of the 3-graviton
vertex associated with Fig. 1a. Using the equation of motion (\ref{eq3.3})
for the free gravitons and momentum conservation, this results in a set of
7 independent equations for the 16 parameters, which yield the following
relations:
\begin{equation}\label{eq3.6}
\begin{array}{lll}
a_{1}&=&a_{14}-6\,a_{8}-6\,a_{15}-8\,a_{16}-8\,a_{10}-4\,a_{11}-4\,a_{9}
\\
a_{2}&=&-6\,a_{8}-8\,a_{15}-8\,a_{16}-8\,a_{10}+a_{13}-
         2\,a_{11}-2\,a_{9}-4\,a_{12}
\\
a_{3}&=&14\,a_{15}+12\,a_{8}+16\,a_{16}+16\,a_{10}-
         2\,a_{13}+4\,a_{11}+4\,a_{9}+4\,a_{12}  - a_{14}
\\
a_{4}&=&-a_{14}+2\,a_{15}-2\,a_{13}
\\
a_{5}&=&3\,a_{8}+3\,a_{15}+4\,a_{16}+4\,a_{10}+2\,a_{11}+2\,a_{9}
\\
a_{6}&=&-2\,a_{15}+2\,a_{13}
\\
2\,a_{7}&=&-{\displaystyle{{3\,a_{8}}}}-
          {\displaystyle{{3\,a_{15}}}}-4\,a_{16}-4\,a_{10}.
\\
\end{array}
\end{equation}
We remark that after inserting (\ref{eq3.6}) into  (\ref{eq3.5}) the resulting
expression is such that the coefficient of $a_{14}$ is a total derivative.

In contrast to the situation in the Yang-Mills theory [see eq. (\ref{eq2.7})]
we see that in this case we do not have enough conditions to determine all the
parameters of the trilinear graviton couplings. All we can do is to express
${\cal L}^3$ as a function of the parameters which appear on
the right hand side of eq. (\ref{eq3.6}), which we denote collectively be
the set $\tilde a\equiv{a_8, \cdots , a_{16}}$.

It is appropriate to
comment here on the possibility of making a local transformation of the
fields so that
\begin{eqnarray}\label{eq3.7a}
h_{\mu\nu}^{\prime}=h_{\mu\nu}+\kappa\left(\right. &A_1& \eta_{\mu\nu}
\left(h_{\alpha\alpha}\right)^2+A_2\eta_{\mu\nu}h_{\alpha\beta}h_{\beta\alpha}+
\nonumber \\
&A_3& h_{\mu\alpha} h_{\nu\alpha}+
A_4 h_{\mu\nu} h_{\alpha\alpha} \left. \right)
+\cdots,
\end{eqnarray}
where $\cdots$ denote terms of higher order in $\kappa$.
Note that the Abelian gauge transformation (\ref{eq3.4}) is the same for
both fields $h_{\mu\nu}$ and $h_{\mu\nu}^\prime$. Since the terms of order
$\kappa$ in (\ref{eq3.7a}) involve 4 arbitrary parameters, it is possible to
make a redefinition of the fields such that the number of independent
parameters in (\ref{eq3.5}) may be reduced from 9 to 5. Even allowing
for this possibility, we see that in contrast to the Yang-Mills case,
there remains a basic indetermination of the trilinear graviton couplings.

Following the analysis done in the Yang-Mills case, we may evaluate the
contributions from the graph 1b and its permutations to the graviton-graviton
tree amplitude, in terms of the parameters present in the set  $\tilde a$.
Since the gauge invariance condition of the physical tree amplitude
should be valid for any gauge-fixing term added onto (\ref{eq3.1}), it will
be convenient to choose this so that the graviton propagator becomes
\cite{feynman}
\begin{equation}\label{eq3.7}
P_{\mu\nu\alpha\beta}(q)=\frac{
\eta_{\mu\alpha}\eta_{\nu\beta}+
\eta_{\mu\beta}\eta_{\nu\alpha}-
\eta_{\mu\nu}\eta_{\alpha\beta}}{2q^2}.
\end{equation}
(We have verified, in the case of the gravitational Compton scattering by
scalar particles, that no additional information is obtained by
considering a more general class of gauges.)
The result of this evaluation, involving quadratic functions of the parameters
$\tilde a_i$ which are excessively long to write down here, will be employed
subsequently.

Next we must parametrize the structure of the 4-graviton vertex ${\cal L}^4$
indicated in Fig 1c, in terms of all possible quadrilinear products of fields
with two derivatives indices. Proceeding in this way, we find
for ${\cal L}^4$ the following expression  involving 43 independent constants:
%We write here explicitly only two typical terms
%\input lagra_four_reduct.tex
\begin{equation}\label{eq3.8}
\begin{array}{rcrcrc}
{\cal L}^4\left(h\right)=&\kappa^2\left(\right.&
 \,b_{1}\, h_{{\mu\nu}}\, h_{{\mu\nu}}\,
h_{{\alpha\beta,\rho}}\, h_{{\alpha\rho,\beta}}&+&
 \,b_{2}\, h_{{\mu\mu}}\, h_{{\nu\nu}}\, h_{{\alpha\beta,\rho}}\,
h_{{\alpha\rho,\beta}} &+
\\
\,b_{3}\, h_{{\mu\nu}}\, h_{{\mu\nu}}\, h_{{\alpha\alpha,\beta}}\,
h_{{\rho\rho,\beta}} &+&
 \,b_{4}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\beta,\alpha}}\,
h_{{\nu\rho,\rho}} &+&
 \,b_{5}\, h_{{\mu\nu}}\, h_{{\mu\nu}}\, h_{{\alpha\beta,\alpha}}\,
h_{{\beta\rho,\rho}} &+
\\
 \,b_{6}\, h_{{\mu\mu}}\, h_{{\nu\nu}}\, h_{{\alpha\beta,\alpha}}\,
h_{{\beta\rho,\rho}} &+&
 \,b_{7}\, h_{{\mu\mu}}\, h_{{\nu\nu}}\, h_{{\alpha\beta,\rho}}\,
h_{{\alpha\beta,\rho}} &+&
 \,b_{8}\, h_{{\mu\mu}}\, h_{{\nu\nu}}\, h_{{\alpha\alpha,\beta}}\,
h_{{\rho\rho,\beta}} &+
\\
 \,b_{9}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\beta,\rho}}\,
h_{{\alpha\nu,\rho}} &+&
 \,b_{10}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\mu,\nu}}\,
h_{{\rho\rho,\beta}} &+&
 \,b_{11}\, h_{{\mu\nu}}\, h_{{\alpha\alpha}}\, h_{{\beta\mu,\beta}}\,
h_{{\nu\rho,\rho}} &+
\\
 \,b_{12}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\nu,\beta}}\,
h_{{\alpha\rho,\rho}} &+&
 \,b_{13}\, h_{{\mu\nu}}\, h_{{\alpha\alpha}}\, h_{{\beta\mu,\nu}}\,
h_{{\beta\rho,\rho}} &+&
 \,b_{14}\, h_{{\mu\nu}}\, h_{{\alpha\alpha}}\, h_{{\beta\mu,\nu}}\,
h_{{\rho\rho,\beta}} &+
\\
 \,b_{15}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\rho,\beta}}\,
h_{{\alpha\nu,\rho}} &+&
 \,b_{16}\, h_{{\mu\mu}}\, h_{{\alpha\nu}}\, h_{{\alpha\nu,\beta}}\,
h_{{\beta\rho,\rho}} &+&
 \,b_{17}\, h_{{\mu\mu}}\, h_{{\alpha\nu}}\, h_{{\alpha\beta,\rho}}\,
h_{{\nu\rho,\beta}} &+
\\
 \,b_{18}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\rho,\mu}}\,
h_{{\nu\rho,\beta}} &+&
 \,b_{19}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\rho,\nu}}\,
h_{{\beta\rho,\alpha}} &+&
 \,b_{20}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\rho,\alpha}}\,
h_{{\beta\nu,\rho}} &+
\\
 \,b_{21}\, h_{{\mu\nu}}\, h_{{\alpha\alpha}}\, h_{{\beta\rho,\mu}}\,
h_{{\beta\nu,\rho}} &+&
 \,b_{22}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\mu\rho,\nu}}\,
h_{{\alpha\rho,\beta}} &+&
 \,b_{23}\, h_{{\mu\nu}}\, h_{{\alpha\alpha}}\, h_{{\beta\rho,\mu}}\,
h_{{\beta\rho,\nu}} &+
\\
 \,b_{24}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\mu,\rho}}\,
h_{{\beta\nu,\rho}} &+&
 \,b_{25}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\nu,\rho}}\,
h_{{\alpha\beta,\rho}} &+&
 \,b_{26}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\nu,\alpha}}\,
h_{{\beta\rho,\rho}} &+
\\
 \,b_{27}\, h_{{\mu\nu}}\, h_{{\mu\nu}}\, h_{{\alpha\beta,\rho}}\,
h_{{\alpha\beta,\rho}} &+&
 \,b_{28}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\beta,\rho}}\,
h_{{\mu\nu,\rho}} &+&
 \,b_{29}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\beta,\mu}}\,
h_{{\nu\rho,\rho}} &+
\\
 \,b_{30}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\mu,\nu}}\,
h_{{\beta\rho,\rho}} &+&
 \,b_{31}\, h_{{\mu\nu}}\, h_{{\mu\nu}}\, h_{{\alpha\beta,\alpha}}\,
h_{{\rho\rho,\beta}} &+&
 \,b_{32}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\beta,\nu}}\,
h_{{\rho\rho,\alpha}} &+
\\
 \,b_{33}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\nu,\rho}}\,
h_{{\alpha\rho,\beta}} &+&
 \,b_{34}\, h_{{\mu\mu}}\, h_{{\alpha\nu}}\, h_{{\beta\beta,\nu}}\,
h_{{\rho\rho,\alpha}} &+&
 \,b_{35}\, h_{{\mu\mu}}\, h_{{\nu\nu}}\, h_{{\alpha\alpha,\beta}}\,
h_{{\beta\rho,\rho}} &+
\\
 \,b_{36}\, h_{{\mu\mu}}\, h_{{\alpha\nu}}\, h_{{\beta\beta,\nu}}\,
h_{{\alpha\rho,\rho}} &+&
 \,b_{37}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\rho,\mu}}\,
h_{{\beta\rho,\nu}} &+&
 \,b_{38}\, h_{{\mu\nu}}\, h_{{\alpha\alpha}}\, h_{{\beta\mu,\rho}}\,
h_{{\beta\nu,\rho}} &+
\\
 \,b_{39}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\mu\rho,\nu}}\,
h_{{\alpha\beta,\rho}} &+&
 \,b_{40}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\rho,\mu}}\,
h_{{\beta\nu,\rho}} &+&
 \,b_{41}\, h_{{\mu\mu}}\, h_{{\alpha\nu}}\, h_{{\alpha\nu,\beta}}\,
h_{{\rho\rho,\beta}} &+
\\
 \,b_{42}\, h_{{\mu\nu}}\, h_{{\alpha\beta}}\, h_{{\alpha\beta,\mu}}\,
h_{{\rho\rho,\nu}} &+&
 \,b_{43}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\nu,\alpha}}\,
h_{{\rho\rho,\beta}}&\left.\right)&{}&.
\end{array}
\end{equation}

%\begin{equation}\label{eq3.8}
%{\cal L}^4=\kappa^2\left(b_{1}\, h_{{\mu\nu}}\, h_{{\mu\nu}}\,
%                         h_{{\alpha\beta,\rho}}\, h_{{\alpha\rho,\beta}}
%+ \cdots +
%b_{43}\, h_{{\mu\nu}}\, h_{{\alpha\mu}}\, h_{{\beta\nu,\alpha}}\,
%            h_{{\rho\rho,\beta}}
%\right).
%\end{equation}
%The complete expression is given in equation (\ref{eqA.1}) of the
%Appendix.

{}From the gauge invariance condition, one expects that a change in the
gravitational field $\delta h_{\mu\nu}$ given by (\ref{eq3.4}), should
have no effect on the graviton-graviton tree amplitude. Imposing this
requirement and using the results mentioned in the previous equations,
we obtain a relation which can be written in correspondence with (\ref{eq2.8})
as
\begin{equation}\label{eq3.9}
\left[\delta\; tree(b_i)\right]_{1c}=-\left[\delta\; tree(\tilde a)
\right]_{1b},
\end{equation}
where $\left[\delta\; tree\right]_{1b}$ represents the gauge variation
associated with the diagram in Fig. 1b and its corresponding permutations.
It is expressed as a function of the independent coupling constants
$\tilde a_i$ left over from the analysis of the trilinear graviton vertices.
The left-hand side of eq. (\ref{eq3.9}), denotes the gauge variation
resulting from the contributions associated with the graph in Fig. 1c,
which is a function of the independent constants $b_1\cdots b_{43}$, which
parametrize the 4-graviton vertex in (\ref{eq3.8}).

We now gather together the terms with the same structure and set the
coefficients of all independent structures in (\ref{eq3.9}) equal to zero.
We then obtain a system which comprises 27 algebraically independent
equations, expressing certain linear combinations of the $b_i$ in terms of
quadratic functions of the parameters $\tilde a_i$. Clearly, this set of
independent equations cannot determine all the parameters $b_i$, nor can it
lead to any additional relations among the $\tilde a_i$.
The solution of the above system in given by a set of equations
where the 27 coefficients $b_i$ $(i=1,\, 2,\,\cdots\, ,26,\, 27)$
are expressed in terms of the remaining 16
coefficients $b_i$ $(i=28,\, 29,\,\cdots\, ,42,\, 43)$ and of the parameters
$\tilde a_i$. We write here explicitly only a few typical equations:
%\input sol_four_grav_reduct.tex
\begin{equation}\label{eq3.8a}
\begin{array}{lll}
8\,b_{{1}}&=-&4\,b_{{29}}-
8\,b_{{42}}+
12\,a_{{11}}a_{{8}}+
a_{{9}}a_{{13}}+
16\,a_{{16}}a_{{11}}+
6\,a_{{9}}a_{{8}}+
16\,a_{{10}}a_{{11}}+
    \\ & &
14\,a_{{9}}a_{{11}}+
5\,a_{{9}}a_{{15}}+
8\,a_{{9}}a_{{10}}+
8\,a_{{9}}a_{{16}}+
12\,a_{{15}}a_{{11}}+
6\,{a_{{9}}}^{2}+
8\,{a_{{11}}}^{2}
 \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 8\,b_{{2}}&=-&2\,b_{{28}}-
4\,b_{{30}}-
2\,b_{{32}}-
4\,b_{{33}}+
2\,b_{{38}}-
8\,b_{{39}}-
2\,b_{{43}}-
3\,a_{{9}}a_{{8}}-
  \\ & &
3\,a_{{11}}a_{{8}}-
4\,a_{{10}}a_{{11}}-
16\,{a_{{10}}}^{2}-
4\,a_{{9}}a_{{10}}-
24\,a_{{10}}a_{{8}}+
4\,a_{{13}}a_{{11}}+
  \\ & &
4\,a_{{9}}a_{{13}}+
8\,a_{{13}}a_{{8}}-
18\,{a_{{15}}}^{2}+
10\,a_{{15}}a_{{13}}-
35\,a_{{15}}a_{{10}}-
7\,a_{{9}}a_{{15}}-
   \\ & &
7\,a_{{15}}a_{{11}}-
26\,a_{{15}}a_{{8}}+
11\,a_{{10}}a_{{13}}-
{a_{{13}}}^{2}-
4\,a_{{16}}a_{{11}}+
12\,a_{{16}}a_{{13}}-
     \\ & &
36\,a_{{15}}a_{{16}}-
32\,a_{{16}}a_{{10}}-
4\,a_{{9}}a_{{16}}-
24\,a_{{16}}a_{{8}}-
9\,{a_{{8}}}^{2}-
16\,{a_{{16}}}^{2}
 \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vdots &=& \vdots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \\
2\,b_{{26}}&=&2\,b_{{28}}+
4\,b_{{33}}-
4\,b_{{38}}-
12\,a_{{9}}a_{{8}}-
4\,{a_{{9}}}^{2}-
4\,{a_{{11}}}^{2}-
12\,a_{{11}}a_{{8}}-
      \\ & &
16\,a_{{10}}a_{{11}}-
8\,a_{{9}}a_{{11}}-
16\,{a_{{10}}}^{2}-
16\,a_{{9}}a_{{10}}-
24\,a_{{10}}a_{{8}}-
8\,a_{{13}}a_{{11}}-
     \\ & &
8\,a_{{9}}a_{{13}}-
12\,a_{{13}}a_{{8}}+
4\,{a_{{15}}}^{2}-
14\,a_{{15}}a_{{13}}-
8\,a_{{15}}a_{{10}}-
4\,a_{{9}}a_{{15}}-
      \\ & &
4\,a_{{15}}a_{{11}}-
6\,a_{{15}}a_{{8}}+
16\,a_{{10}}a_{{13}}+
{a_{{13}}}^{2}-
16\,a_{{16}}a_{{11}}-
16\,a_{{16}}a_{{13}}-
       \\ & &
8\,a_{{15}}a_{{16}}-
32\,a_{{16}}a_{{10}}-
16\,a_{{9}}a_{{16}}-
24\,a_{{16}}a_{{8}}-
9\,{a_{{8}}}^{2}-
16\,{a_{{16}}}^{2}
 \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
16\,b_{{27}}&=&
8\,b_{{42}}-
4\,a_{{9}}a_{{16}}-
4\,{a_{{9}}}^{2}-
3\,a_{{9}}a_{{15}}-
4\,a_{{9}}a_{{10}}-
3\,a_{{9}}a_{{8}}-
12\,a_{{9}}a_{{11}}-
                 \\ & &
12\,a_{{11}}a_{{8}}-
16\,a_{{16}}a_{{11}}-
16\,a_{{10}}a_{{11}}-
12\,a_{{15}}a_{{11}}-
8\,{a_{{11}}}^{2}.
\end{array}
\end{equation}

Although much more complicated in detail, these relations are basically
similar to the one encountered in the Yang-Mills case
[see eq. (\ref{eq2.11})].
The crucial difference occurs when attempting, in
parallel to the procedure used in the Yang-Mills case, to apply the gauge
invariance condition to the 5-graviton tree amplitude. Now, there exists
a basic 5-graviton vertex, shown in Fig 2c, which must be parametrized
in terms of the most general sum of independent products involving five
graviton fields with two derivatives. This parametrization can be done in terms
of a very large number of new constants, which we denote by the set
$c_i$. Following closely the analysis after equation (\ref{eq3.9}), is is clear
that the Abelian gauge invariance condition will merely lead to some
relations expressing certain $c_i$ in terms of the remaining $c_i$ and of
the parameters $\tilde a_i$ and $b_i$ left over from the previous analysis.

It is evident that this behavior is quite general, in view of the fact that
the graviton self-couplings occur to all orders. We thus conclude that the
constraint of Abelian gauge invariance of the physical tree amplitudes does
not determine completely the from of the nonlinear graviton interactions.
It is only when we impose the condition that the theory should be invariant
under (infinitesimal) non-Abelian gauge transformations
\begin{equation}\label{eq3.10}
h_{\mu\nu}\rightarrow
h_{\mu\nu} + \xi_{\mu,\nu} + \xi_{\nu,\mu} +
\kappa \left(
\xi^{\sigma}_{,\mu} h_{\sigma\nu}+\xi^{\sigma}_{,\nu} h_{\sigma\mu}+
\xi^{\sigma} h_{\mu\nu,\sigma}
\right).
\end{equation}
that it becomes essentially determined.
For example, using the parametrization given by (\ref{eq3.5}), we find
in this case for the trilinear graviton vertex:
\begin{equation}\label{eq3.11}
\begin{array}{lllll}
a_1 = 1+a_{14} \; ,\;\; &  2\,a_2 =-{3}-2\,a_{15}\; ,\;\;&{}
%\\{}&{}&{}\\
a_3 =1-a_{14} \; ,\;\; &  a_4 =a_3 \; ,\;\;&{}\\
%{}&{}&{}&{}&{}\\
2\,a_5 =-{1}\; ,\;\; &  a_6 =-1  \; ,\;\;&{}
%\\{}&{}&{}\\
4\, a_7 = 1\; ,\;\; &
2\,a_8=-{1}-2\,a_{15} & {}\\
%{}&{}&{}&{}&{}\\
a_9=-1 \; ,\;\; &  2\, a_{10}={1} \; ,\;\;&{}
%\\{}&{}&{}\\
a_{11}=1 \; ,\;\; &  2\, a_{12}={1} \; ,\;\;&{}\\
%{}&{}&{}&{}&{}\\
2\, a_{13}=2\, a_{15}-{1} \; ,\;\; &
4\, a_{16}=-{1} \; \;\;&{}.
\end{array}
\end{equation}
We remark that the structures which multiply $a_{14}$ and $a_{15}$ add up to
total derivative terms.
Since total derivatives are not relevant for our purpose, this
result is equivalent to the one obtained from the Einstein's general
relativity. Then, the theory becomes consistent with the existence of a locally
conserved energy-momentum ``tensor'' of matter
{and~gravitation\cite{weinberg}}.


\acknowledgements{We would like to thank CNPq (Brasil) for a grant. J. F. is
grateful to Prof. J. C. Taylor for a very helpful correspondence} and for
reading the manuscript.

%\appendix
%\section{Graviton scattering}\label{}
%
%In this appendix we present the results obtained from the calculation
%of graphs in figures 1b and 1c. The basic inputs to this calculation are
%the trilinear vertex given by equations (\ref{eq3.5}) and (\ref{eq3.6}),
%the graviton propagator given by equation (\ref{eq3.7}) and the following
%parametrization for the 4-graviton vertex:
%\input lagra_four.tex
%
%After imposing the gauge invariance and identifying independent structures,
%we obtain the following system of 27 independent relations:
%\input sol_four_grav.tex
%
%
%
%\newpage
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\end{document}




