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\begin{document}
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\begin{titlepage}
\title{\bf On the power counting of loop diagrams in general relativity}
\author{\hspace{3ex} John F. Donoghue\protect\thanks{e--mail:
donoghue@phast.umass.edu}\hfill{\it and}\hfill
Tibor Torma\protect\thanks{e--mail:
kakukk@phast.umass.edu}\hspace{5ex}
\protect\\
\protect\\{\it University of Massachusetts, Department of Physics,}
\protect\\{\it Amherst MA 01003}\protect\\}
\date{\today}
\maketitle
\begin{center}{hep-ph/9602121}\end{center}

\begin{center}{UMHEP-426}\end{center}
\vspace{.4in}
\begin{abstract}
A class of loop diagrams in general relativity appears to have a
behavior which would upset the utility of the energy expansion for
quantum effects. We show through the study of specific diagrams that
cancellations occur which restore the expected behaviour of the energy
expansion. By considering the power counting in  a physical gauge we
show that the apparent bad behavior is a gauge artifact, and that the
quantum loops enter with a well behaved energy expansion.
\end{abstract}
\renewcommand{\thepage}{\mbox{}}
\end{titlepage}

\pagenumbering{arabic}
\renewcommand{\thesection}{\Roman{section}}
\Section{\bf Introduction}\llabel{sec:intro}

Loop calculations in general relativity are readily interpreted using the
techniques of effective field theory~\ccite{1,2}. As in all effective field
theories, the utility of such calculations is tied to an expansion in
powers of the energy or inverse distance. In chiral theories,
Weinberg~\ccite{3}
has provided an important theorem which states that diagrams with
increasing numbers of loops contribute to an amplitude with
increasing powers of the energy, with each extra loop adding an extra
factor of $E^2$. For example, if one is working to order $E^4$
accuracy one needs to include only one loop diagrams. While pure gravity
behaves exactly in the same way, if we try a
simple extension of this same argument to gravity interacting with
matter, we will see in Sec.~\rref{sec:powcov} that the desired behavior
is not obtained. There is a class of diagrams which appears to have
$Gm^2$ as the expansion parameter. This would upset the utility of the
energy expansion. The purpose of this paper is to explore this problem
and to see if it obstructs the energy expansion.

The desired expansion parameters for quantum corrections in an effective
theory of gravity is $Gq^2\sim\frac{G}{r^2}$,
such that at low energies/long
distances the higher order loop effects are suppressed with respect to
tree diagrams and low order loops. Thus we can obtain predictions to a
given order with a finite amount of calculation. General relativity
also contains the classical expansion parameter $Gmq\sim\frac{Gm}{r}$
which represents the nonlinearities of the classical theory. This can
be found in the loop expansion from the nonanalytic terms of the form
$Gq^2\sqrt{\frac{m^2}{-q^2}}$. However, $Gm^2$ as an expansion parameter
is a major problem. In the first place, the mass can be extremely
large in units of the Planck mass (e.~g.~$m=M_{\mbox{Sun}}$) so that
$Gm^2$ can be a number very
much larger than unity. In addition if we restore factors of $\hbar$,
this dimensionless combination goes like $\frac{Gm^2}{\hbar}$. The
classical limit $\hbar\to0$ would be upset by corrections of this
form.

We will see that the apparent difficulty with the loop expansion appears
to be a gauge artifact. When calculating in harmonic gauge, where the
power counting is first discussed in Sec.~\rref{sec:powcov}, there occur
cancellations between individual diagrams, cancelling the bad behavior.
We detail the calculation for the box and crossed box diagrams. Part
of the problem is due to the occurrence of both classical and quantum
effects in the same Feynman diagram, when treated in covariant gauges.
This suggests that separating the classical physics from the physical
quantum (transverse and traceless) degrees of freedom will improve the
power counting. For the interaction of two nearly static masses, we show
that this is in fact the case.

The organization of this paper is as follows. In Sec.~\rref{sec:powcov} we
make a na\"\i ve generalization of the Weinberg power counting theorem
and isolate those diagrams which appear to give a problem. In order to
explore this without all the tensor indices of gravitons, we introduce a
scalar toy model with the same behavior in Sect.~\rref{sec:toy} in order
to see how cancellations occur. Sec.~\rref{sec:harm} applies the lessons
so learned to the gravitational interaction. Sec.~\rref{sec:fizg} was
devoted to development of the power counting scheme in a physical gauge,
and to the interpretation of the apparent problem as a gauge artifact. We
end with some concluding comments. 

\Section{Power counting in covariant gauges}\llabel{sec:powcov}

\begin{figure}[hb]
%
\begin{picture}(390,85)(0,0)
\Photon(90,65)(110,85)2 4
\Photon(90,65)(70,85)2 4
\Photon(90,45)(90,65)2 3
\Vertex(90,65)2
\put(70,20){$\sim\kappa q^2$}
\put(90,0){\it (a)}
\Photon(220,45)(260,85)3 5
\Photon(260,45)(220,85)3 5
\Vertex(240,65)2
\put(220,20){$\sim\kappa^2q^2$}
\put(233,0){\it (b)}
\end{picture}
%
\caption{\protect\llabel{fig:p1}Three and four graviton couplings.}
\end{figure}

We are interested in treating powers of energies and masses in vertices
and propagators in order to determine the overall energy dependence of a
given multiloop diagram. The mass of the matter field is not a small
parameter, but we can treat the external three-momenta as small if we are
working at low enough energies. Let us review the Feynman rules, and
extract the essential dependence of the vertices. Starting from the action
\eqn{p1}{S=\int d^4x\,\sqrt{g}\cdot\frac{2}{\kappa^2}\cdot R}
with $\kappa^2=32\pi G$, we expand this metric
\eqn{p2}{g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}}
where $h_{\mu\nu}$ is the fluctuating field. Expanding
$\frac{2}{\kappa^2}R$
in powers of $h_{\mu\nu}$ we see that a term which involves $n$ graviton
fields, i.~e. $h^n$, carries a coupling constant $\kappa^{n-2}$. Since
the curvature is second order in derivatives, all terms emerging from the
Einstein action will be of order $q^2$. Thus the triple graviton coupling
of Fig.~\rref{fig:p1} is of order $\kappa q^2$, while the four graviton
vertex is of order $\kappa^2q^2$ etc. The matter fields couple to
gravitons through $T_{\mu\nu}$, which for a scalar field has matrix
elements
\eqn{p3}{<p^{\prime}\mid T_{\mu\nu}\mid p>=p_\mu p^\prime_\nu+
p_\nu p^{\prime}_\mu-\frac{1}{2}g_{\mu\nu}
\left(p\cdot p^\prime-m^2\right)}
with $p_\mu=\left(\sqrt{m^2+\vec{p}^2}\mid\vec{p}\right)_\mu$. Treating
the mass as a large parameter leads to a one graviton
vertex (see Fig.~\rref{fig:p2}) which behaves as $\kappa m^2$ while the
two graviton diagram is of order $\kappa^2m^2$ etc.
\begin{figure}[bh]
%
\begin{picture}(390,70)(0,0)
\ArrowLine(100,65)(120,65)
\ArrowLine(80,65)(100,65)
\Photon(100,45)(100,65)2 3
\Vertex(100,65)2
\put(90,20){$\kappa m^2$}
\put(90,0){\it (a)}
\Photon(220,45)(240,65)3 5
\Photon(260,45)(240,65)3 5
\ArrowLine(240,65)(260,65)
\ArrowLine(220,65)(240,65)
\Vertex(240,65)2
\put(230,20){$\kappa^2m^2$}
\put(233,0){\it (b)}
\end{picture}
%
\caption{Matter-graviton couplings from $T_{\mu\nu}$.
\protect\llabel{fig:p2}}
\end{figure}

The graviton propagator, like all massless boson propagators, scales as
$\frac{1}{q^2}$. The matter field propagator requires a bit more
explanation.
Because we are dealing with an effective theory at low energies, we
need not consider loops of heavy matter fields. These loops have
already been integrated out in order to define the low energy effective
theory. However, we do need to consider matter fields which appear as
external states and which propagate through a given diagram
interacting with each other and with gravitons.
The explicit form of the propagator is
\begin{equation}\llabel{eq:p4}
D(p+q)=\frac{i}{(p+q)^2-m^2}=\frac{i}{2p\cdot q+q^2+(p^2-m^2)}
\end{equation}
where $p$ is the momentum that the matter field has as an external
particle, and $q$ is the momentum which has been added to it
through interactions with gravitons (internal or external). The external
momentum is on shell ($p^2-m^2=0$) so that the matter propagator is
counted as a factor of $\frac{1}{mq}$. Note that if we had chosen a
different normalization for our matter fields
$\left(\vstrut{1em}\right.$e.~g. a nonrelativistic normalization such
that $T_{00}\sim m\ \mbox{and}\ D(q)\sim\frac{1}{q}\left.\vstrut{1em}
\right)$ both the vertices and propagators change in a way that
compensates each other, leading to the same counting rules as in our
normalization.

Before giving the general power counting theorem, let us illustrate the
idea with two specific examples, one of which illustrates the "good"
behavior and one which shows the problem. First consider
graviton-graviton scattering, whose overall matrix element is
dimensionless. At lowest order we have a $\frac{1}{\kappa^2}$ factor from
the coupling in the Einstein action, one of $\kappa^4$ from the four
graviton fields and one $q^2$ because the Einstein action involves two
derivatives. This leads to an overall matrix element
\eqn{p5}{{\cal M}_{tree}\sim\kappa^2q^2.}
\begin{figure}[thb]
%
\begin{picture}(390,85)(0,0)
\Photon(70,45)(110,85)3 5
\Photon(110,45)(70,85)3 5
\Vertex(90,65)2
\put(73,20){$\sim\kappa^2q^2$}
\put(83,0){\it (a)}
\Photon(260,65)(280,85)3 5
\Photon(260,65)(280,45)3 5
\Photon(200,85)(220,65)3 5
\Photon(200,45)(220,65)3 5
\Vertex(220,65)2
\PhotonArc(240,45)(28.28,45,135)1 6
\PhotonArc(240,85)(28.28,225,315)1 6
\Vertex(260,65)2
\put(223,20){$\sim\kappa^4q^4$}
\put(233,0){\it (b)}
\end{picture}
%
\caption{Sample four-graviton interaction diagrams to one loop,
illustrating the expected behavior in the energy expansion.
\protect\llabel{fig:p3}}
\end{figure}

If we try to iterate this vertex to produce the one loop diagram of
Fig.~\rref{fig:p3}b we obtain schematically
\eqn{p6}{{\cal M}_{loop}\sim\kappa^4\int\frac{d^4l}{(2\pi)^4}
\frac{(l-p_1)^2(l-p_2^2)^2}{l^2(l-q)^2}}
where $p_1,p_2,q$ are various combination of external momenta. If this
loop integral is regularized dimensionally, which does not introduce
powers
of any new scale, the integral will be represented in terms of the
exchanged momentum to the appropriate power. Thus we have
\eqn{p7}{{\cal M}_{loop}\sim\kappa^4q^4}
where again $q$ represents some combination of external momenta. [There
may also be logarithms of $\frac{q^2}{\mu^2}$ where $\mu$ is the usual
scale introduced in dimensional regularization.] In this case adding a
loop has generated an effect which is higher order in the energy
expansion. The expansion is in terms of powers of $\kappa^2q^2$.
\begin{figure}[hbt]
%
\begin{picture}(390,100)(0,0)
\ArrowLine(0,100)(180,100)
\Vertex(90,100)2
\Photon(90,25)(90,100)3 5
\Vertex(90,25)2
\ArrowLine(0,25)(180,25)
\put(81,0){\it (a)}
\ArrowLine(210,100)(270,100)
\ArrowLine(270,100)(330,100)
\ArrowLine(330,100)(390,100)
\Vertex(270,100)2
\Vertex(330,100)2
\Photon(270,25)(270,100)3 5
\Photon(330,25)(330,100)3 5
\Vertex(270,25)2
\Vertex(330,25)2
\ArrowLine(210,25)(270,25)
\ArrowLine(270,25)(330,25)
\ArrowLine(330,25)(390,25)
\put(297,0){\it (b)}
\end{picture}
%
\caption{\protect\llabel{fig:new}Sample interactions of two massive
particles.}
\end{figure}

A different behavior is shown by the interactions of two massive
particles, such as \mbox{in~Fig.~\rref{fig:new}a,b}. The tree level result
in our normalization is
\eqn{p8}{{\cal M}_{tree}=\kappa^2\cdot\frac{m_1^2m_2^2}{q^2}}
which again is dimensionless. Iterating this to form a loop gives us
\eqn{p9}{{\cal M}_{loop}\sim\kappa^4m_1^4m_2^4\cdot\int{d^4l\cdot
\frac{1}{m_1(l+p)}\cdot\frac{1}{m_2(l+p^\prime)}\cdot
\frac{1}{(l+q^\prime)^2}\cdot\frac{1}{(l+q)^2}}}
which by the same reasoning is
\eqn{p10}{{\cal M}_{loop}\sim\kappa^4\cdot\frac{m_1^3m_2^3}{q^2}\sim
\kappa^2\cdot\frac{m_1^2m_2^2}{q^2}\cdot\kappa^2m_1m_2.}

Here the expansion parameter appears as $\kappa^2m^2$. An explicit
calculation of this diagram later in this paper confirms that this is the
correct result for the diagram by itself. This expansion parameter
$\kappa^2m^2$ would cause the problem described in the introduction.

Now let us turn to the general power counting result. Our goal is to
obtain the power of $q$ (with $q$ being a typical external momentum)
that a general diagram would yield. This will tell us what order in
the energy expansion that diagram will contribute to. The problematic
class will be manifest by having diagrams with increasing number of
loops which yield the same power of $q$, so that to calculate to this
order in the energy expansion one would apparently needs to sum all the
diagrams in this class. For a general result we need to allow for
vertices not just from the lowest order gravitational action, but also
from ones which contain more derivatives. Let us write this
schematically as
\eqn{hpp1}{S_g=\int d^4x\,\sqrt{g}\cdot\frac{2}{\kappa^2}\cdot\left[
R+\kappa_0^2R^2+\kappa_0^4R^3+\dots\right]}
such that the coefficients of a gravitational Lagrangian with $n$
derivatives will be $\frac{\kappa_0^{n-2}}{\kappa^2}$. Note that
$\kappa_0\sim\frac{1}{\mbox{\tiny energy}}$. In a pure gravitational
theory one would expect $\kappa_0\sim\kappa$, but there is no need to
impose such a restriction here. Likewise the matter Lagrangian can
involve extra derivatives on the light fields. We let the
coefficients of the higher derivative terms involve a scale
$\overline{\kappa_0}$, i.~e.
\begin{eqnarray}\llabel{eq:hpp2}
S_m&=&\int d^4x\,\sqrt{g}\cdot\\
&&\mbox{}\cdot\left[
\frac{1}{2}\left(\partial_\mu\Phi\partial^\mu\Phi-m^2\Phi^2\right)+
\overline{\kappa_0}^2R\partial_\mu\Phi\partial^\mu\Phi
+\kappa_0^4R^2\partial_\mu\Phi\partial^\mu\Phi+\dots\right]\nonumber
\end{eqnarray}
so that the coefficient of a Lagrangian with $l$ derivatives on the
gravitational field is $m^2\overline{\kappa_0}^l$.
[Again, $\overline{\kappa_0}\sim\frac{1}{\mbox{\tiny energy}}$ and
$\overline{\kappa_0}$ can be kept distinct from $\kappa_0$ and $\kappa$
if desired.]

Our procedure is to count powers of $\kappa,\kappa_0,\overline{\kappa_0}$
and $m^2$ in a general diagram. The remaining energy factor of the
diagram, needed to give the proper overall dimension, will come from
factors of the external momenta. Consider a diagram
with $N^m_E$ external matter legs and $N^g_E$ external graviton legs,
with a series of interactions between these particles. Correspondingly
let $N^m_I$ and $N^g_I$ be the number of internal matter and graviton
propagators respectively. There are $N^g_V$ vertices involving only
gravitons, and $N^m_V$ vertices which involve matter fields plus any
number of gravitons, and a total of $N_L$ loops. However, these vertices
need to be categorized by the number of derivatives that are involved.
For example, let $N^g_V[n]$ be the number of graviton vertices which come
from a Lagrangian with $n$ derivatives.
Clearly, $N^g_V=\sum_nN^g_V[n]$.
Likewise the number of matter vertices with $l$ derivatives on the light
fields will be called $N^m_V[l]$ with $N^m_V=\sum_l{N^m_V[l]}$.
We illustrate this with a sample diagram in Fig.~\rref{fig:p4}. All matter
lines propagate all the way through a diagram without terminating.
\begin{figure}[thb]
\begin{picture}(390,140)(0,0)
\Vertex(95,100)2
\Vertex(120,100)2
\Vertex(145,100)2
\Vertex(200,100)2
\Vertex(120,0)2
\Vertex(200,0)2
\Vertex(120,40)2
\Vertex(200,50)2
\Photon(200,0)(200,50)3 5
\Photon(200,50)(200,100)3 5
\Photon(120,40)(200,50)3 5
\Photon(120,40)(95,100)3 5
\Photon(120,40)(145,100)3 5
\Photon(120,100)(60,140)3 5
\Photon(120,0)(120,40)3 5
\ArrowLine(200,100)(245,100)
\ArrowLine(145,100)(200,100)
\ArrowLine(120,100)(145,100)
\ArrowLine(95,100)(120,100)
\ArrowLine(45,100)(95,100)
\ArrowLine(200,0)(245,0)
\ArrowLine(120,0)(200,0)
\ArrowLine(45,0)(120,0)
\end{picture}
%
\caption{\protect\llabel{fig:p4}
Sample diagram with $N_E^g=1,\ N_E^m=4,\ N_I^g=6,\ N_I^m=4,
\protect\\ N_V^g=2,\ N_V^m=6,\ N_L=3$.}
\end{figure}

With these definitions the coupling constants contribute the dimensionful
factors
\eqn{p11}{
\left(\kappa^2\vstrut{2ex}\right)^{-N^g_V}
\cdot\left(\kappa_0\vstrut{2ex}\right)^{\sum_n{(n-2)N^g_V[n]}}\cdot
\left(\vstrut{2ex}m\right)^{2N^m_V}\cdot
\left(\vstrut{2ex}\overline{\kappa_0}\right)^{\sum_l{l\cdot N^m_V[l]}}.}
In addition, because each internal graviton line is formed using two
vertices, the graviton fields will contribute a power of
\eqn{p12}{\left(\vstrut{2ex}\kappa\right)^{2N^g_I+N^g_E}} 
from the normalization of the metric in~(\rref{eq:p2}). On a matter line,
there will be $(v-1)$ propagators if there are $v$ vertices. Thus the
number of matter propagators $N^m_I$ satisfies
\eqn{p13}{N^m_I=N^m_V-\frac{1}{2}N^m_E.}

Since each propagator counts as a power of $\frac{1}{m}$, this
contributes mass factors
\eqn{p14}{\left(\vstrut{2ex}\frac{1}{m}\right)^{N^m_V-\frac{1}{2}N^m_E}.}

These constitute all of the general dimensionful parameters except the
external momenta and the loop momenta. When the loop integrals are
regularized dimensionally there will not be any powers of a regulator
mass and the remaining dimensions after integration will be carried by
the external momenta. Let us generically call these momenta $q$, and
describe the power of the momenta by a factor $q^D$. It is the dimension
$D$ which we are seeking in this exercise.

Overall, this matrix element carries a dimension
\eqn{p15}{{\cal A}\sim\left(\vstrut{2ex}\mbox{Energy}
\right)^{4-N^m_E-N^g_E}.}
From our identification above, this is decomposed as
\begin{eqnarray}\llabel{eq:p16}
{\cal A}&\sim&\left(\vstrut{2ex}\mbox{Energy}\right)^{4-N^m_E-N^g_E}\\
&=&
\left(\vstrut{2ex}\kappa_0\right)^{\sum_n{(n-2)\cdot N^g_V[n]}}\cdot
\left(\vstrut{2ex}m\right)^{ 2N^m_V}\cdot
\left(\vstrut{2ex}\overline{\kappa_0}\right)^{\sum_l{l\cdot N^m_V[l]}}\cdot
\left(\vstrut{2ex}\kappa\right)^{ 2N^g_I+N^g_E-2N^g_V}\cdot\nonumber\\
&&\mbox{}\cdot\left(\vstrut{2ex}\frac{1}{m}\right)^{N^m_V-
\frac{1}{2}N^m_E}\cdot q^D\nonumber.
\end{eqnarray}
There are however some relations among all the variables. For example, the
total number of internal lines can be expressed in terms of the total
number of vertices and the number of loops. The relation is
\begin{eqnarray}\llabel{eq:p17}
N^m_I+N^g_I&=&N_L+(N^m_V+N^g_V)-1\\
&=&N_L+\sum_lN^m_V[l]+\sum_nN^g_V[n]-1
\nonumber.
\end{eqnarray}
We can use this to eliminate $N^g_I$, using also 
$N^m_I=N^m_V-\frac{1}{2}N^m_E$, to find
\begin{eqnarray}\llabel{eq:p18}
N^g_I&=&\left(N^g_I+N^m_I\right)-\left(N^m_V-\frac{1}{2}N^m_E\right)\\
&=&N_L+\frac{1}{2}N^m_E+\sum_nN^g_V[n]-1
\nonumber.
\end{eqnarray}
Plugging this into the general formula Eq.~(\rref{eq:p16}), using
$\sum_nN^g_V[n]=N^g_V$ and recalling that $\kappa,\kappa_0,
\overline{\kappa_0}$ all go as $\frac{1}{\mbox{\tiny Energy}}$ allow us to
solve for the parameter $D$, resulting in
\eqn{p19}{D=2-\frac{N^m_E}{2}+2N_L-N^m_V+\sum_n{(n-2)N^g_V[n]}+
\sum_l{l\cdot N^m_V[l]}.}

This is our general power counting result for the momentum dependence of a
general diagram. If we disregard the matter vertices, 
$N^m_E=N^m_V[l]=N^m_V=0$, it is identical to Weinberg's theorem for
chiral theories. The momentum dimension of a diagram is higher if we
increase the number of loops or if we use a gravitational Lagrangian
with more than two derivatives. This shows that the power counting of loop
diagrams in pure gravity involves the parameter $\kappa^2q^2$ (or
$\kappa_0^2q^2$ if $\kappa_0\neq\kappa$.)

In the presence of matter, the last term also behaves as expected: using
the Lagrangian with extra derivatives on light fields ($l>0$) only
increases the
power of the momentum. However, the problem arises because of the minus
sign in front of $N^m_N$. Increasing the number of matter vertices in
diagrams does not increase the order in the energy expansion. This
cannot actually make the momentum power
$D$ decrease by increasing the number of matter vertices, since to add
matter vertices to a given process we also have to change the number of
loops. However, there are diagrams where one can increase the number of
loops by one while increasing the number of matter vertices by two.
Fig.~\rref{fig:new} is one such example. This leaves $D$ unchanged. Thus
higher loop processes contribute at the same level to the energy
expansion as tree processes. This gives a loop expansion of $\kappa^2m^2$
instead of $\kappa^2q^2.$

In summary, we have computed the momentum power or a given
process~(\rref{eq:p11}), and found a class of dangerous diagrams where the
addition of two matter vertices adds only one loop to the process,
leading to no net increase in the momentum power $D$.

\Section{Analysis of a related model}\llabel{sec:toy}

Because the couplings of general relativity are complicated by the
tensor indices, it is easier to analyse a simpler model
first. Although it is clear that the model is not identical with
general relativity, it will nevertheless exhibit several interesting
features which we will be able to generalize to the case of gravity in
the next section.

Consider a massless scalar field $h$ coupled to one or more massive
scalars $\Phi$ with a trilinear coupling which carries the same
strength as our counting rule given in~(\rref{eq:p11}). That is, we
identify the Lagrangian
\eqn{t1}{{\cal
L}=\frac{1}{2}\partial_\mu\Phi\partial^\mu\Phi+m^2\Phi^2(1+\kappa
h)+\frac{1}{2}\partial_\mu h\partial^\mu h.}
\begin{figure}[thb]
\begin{picture}(320,100)(0,0)
\ArrowLine(60,100)(160,100)
\Vertex(160,100)2
\ArrowLine(160,100)(260,100)
\Photon(160,0)(160,100)3 5
\Vertex(160,0)2
\ArrowLine(60,0)(160,0)
\ArrowLine(160,0)(260,0)
\put(150,50){$\uparrow$}
\put(165,50){$q^{\mu}$}
\put(60,10){$p_3^{\mu}(m_2)$}
\put(200,10){$p_4^{\mu}(m_2)$}
\put(60,80){$p_1^{\mu}(m_1)$}
\put(200,80){$p_2^{\mu}(m_1)$}
\end{picture}
%
\caption{\protect\llabel{fig:toy1}
Tree level graph for heavy scalar scattering.}\end{figure}

The coupling $\kappa m^2\Phi^2h$ enters into the power counting
derivation in the same way as the lowest order graviton coupling and
in this theory we obtain the same momentum power as in~(\rref{eq:p11})
with $N^g_V[n]=0\ \mbox{for}\ n>2$ and $N^m_V[l]=0\ \mbox{for}\ l>0$. The
dangerous class of diagrams identified in the previous section also
are equally problematic for this model.

Let us verify the result of the counting theorem by considerations of
the gravitational interaction of two heavy masses, as
\mbox{in~Fig.~\rref{fig:toy1},~\ref{fig:toy2}a,b}. The single "graviton"
exchange vertex, Fig.~\rref{fig:toy1}, has magnitude
\eqn{t2}{{\cal M}=\kappa^2\frac{m_1^2m_2^2}{q^2}}
as expected. Note that in forming a nonrelativistic static potential one
divides by
$2m_1\cdot2m_2$ to account for our normalization of the states,
obtaining 
\begin{eqnarray}\llabel{eq:t3}
V(r)&=&\int\frac{d^3q}{(2\pi)^3}\cdot\frac{1}{2m_1\cdot2m_2}\cdot
\kappa^2\frac{m_1^2m_2^2}{q^2}\cdot
e^{i\vec{q}\cdot\vec{r}}\nonumber\\
&=&-\kappa^2\cdot\frac{m_1m_2}{32\pi\cdot r}.
\end{eqnarray}
\begin{figure}[hbt]
%
\begin{picture}(390,100)(0,0)
\ArrowLine(0,100)(60,100)
\ArrowLine(60,100)(120,100)
\ArrowLine(120,100)(180,100)
\Vertex(60,100)2
\Vertex(120,100)2
\Photon(60,25)(60,100)3 5
\Photon(120,25)(120,100)3 5
\Vertex(60,25)2
\Vertex(120,25)2
\ArrowLine(0,25)(60,25)
\ArrowLine(60,25)(120,25)
\ArrowLine(120,25)(180,25)
\put(50,55){$\uparrow$}
\put(65,55){$k^\mu$}
\put(110,55){$\uparrow$}
\put(125,55){$q^\mu\!\!\!-\!\!k^\mu$}
\put(5,35){$p_3^{\mu}(m_2)$}
\put(140,35){$p_4^{\mu}(m_2)$}
\put(5,86){$p_1^{\mu}(m_1)$}
\put(140,86){$p_2^{\mu}(m_1)$}
\put(90,0){\it (a)}
\ArrowLine(210,100)(270,100)
\ArrowLine(270,100)(330,100)
\ArrowLine(330,100)(390,100)
\Vertex(270,100)2
\Vertex(330,100)2
\Photon(270,25)(330,100)3 5
\Photon(330,25)(270,100)3 5
\Vertex(270,25)2
\Vertex(330,25)2
\ArrowLine(210,25)(270,25)
\ArrowLine(270,25)(330,25)
\ArrowLine(330,25)(390,25)
\ArrowLine(284,40)(291,46)
\put(270,45){$k^\mu$}
\ArrowLine(312,40)(305,46)
\put(320,45){$q^\mu\!\!\!-\!\!k^\mu$}
\put(215,35){$p_3^{\mu}(m_2)$}
\put(350,35){$p_4^{\mu}(m_2)$}
\put(215,86){$p_1^{\mu}(m_1)$}
\put(350,86){$p_2^{\mu}(m_1)$}
\put(300,0){\it (b)}
\end{picture}
%
\caption{\protect\llabel{fig:toy2}
The (a) box and (b) crossed box graphs which have the wrong na\"\i ve
power counting behavior.}\end{figure}

Now consider the box diagram. Fig.~\rref{fig:toy2}a:
\begin{eqnarray}\llabel{eq:t4}
{\cal M}_{box}=
i\cdot(\kappa m_1^2)^2&\cdot&(\kappa m_2^2)^2\cdot
\int\frac{d^4k}{(2\pi)^4}\,
\frac{1}{[k^2+i\epsilon][(q-k)^2+i\epsilon]}\cdot\nonumber\\
&&\mbox{}\cdot\frac{1}{(p_1+k)^2-m_1^2}\cdot\frac{1}{(p_2-k)^2-m_2^2}.
\end{eqnarray}

We combine the graviton propagators using the usual identity
\begin{eqnarray}\llabel{eq:t5}
\frac{1}{k^2}\frac{1}{(k+q)^2}
&=& \int^1_0\frac{dx}{\left[k^2-2xk\cdot q+x^2q^2\right]^2}
\nonumber\\
&=& \int^1_0\frac{dx}{\left[(k-xq)^2-\overline{Q}^2\right]}
\end{eqnarray}
with $\overline{Q}^2=-x(1-x)q^2$, and do likewise for the
matter propagator. The diagram is ultraviolet finite but has an
infrared divergence which we regulate dimensionally. Integrating in
$d$ dimensions we obtain
\begin{equation}\llabel{eq:t6}
{\cal M}_{box}=-(\kappa m_1^2)^2\cdot(\kappa m_2^2)^2\cdot\mu^{4-d}
\cdot\frac{\Gamma\left(4-\frac{d}{2}\right)}{(4\pi)^{\frac{d}{2}}}
\cdot\int^1_0dx\int^1_0dy\int^1_0dz\cdot\frac{y(1-y)}{D^{4-\frac{d}{2}}}
\end{equation}
with
\begin{eqnarray}\llabel{eq:t7}
D&=&M_z^2y^2\overline{Q}^2(1-y)^2\\
M_z^2&=&z^2m_1^2+(1-z)^2m_2^2-2z(1-z)p_1\cdot p_2.\nonumber
\end{eqnarray}
For small $\frac{\overline{Q}}{m}$, the $y$ integration can be done
near $d=4$ dimensions,
\eqn{t8}{\int_0^1\cdot\frac{y(1-y)}{D^{4-\frac{d}{2}}}=
\frac{1}{2M^2}\cdot\frac{1}{\overline{Q}^{6-d}}\cdot\left[1+
(d-4)\left(\frac{1}{2}-\frac{\overline{Q}}{m}\right)\right].}
The infrared divergence as $d\to4$ is in the $x$ integration.
In the $z$ integration there is a pole when
the intermediate matter state is on shell and this is made well
defined by the usual $i\epsilon$ factors. However, both of these
integrals can be done straightforwardly. If we define $p_1\cdot
p_2=m_1m_2+w$, we end up with
\begin{eqnarray}\llabel{eq:t9}
{\cal M}_{box}&=&\frac{\kappa^2m_1^2m_2^2}{q^2}\cdot\frac{\kappa^2m_1m_2}
{16\pi^2}\cdot\left[-1+
\frac{w}{3m_1m_2}+i\pi\frac{m_1m_2}{p(m_1+m_2)}\right]\cdot\\
&&\mbox{}\cdot\left\{\frac{2}{4-d}-\ln\left(-\frac{q^2}{m^2}\right)
+\mbox{constant}+{\cal O}\left(\frac{q}{M}\right)\right\}
\nonumber
\end{eqnarray}
with $p=\mid\vec{p}\mid$ in the center of mass. If we defer comment
on the imaginary part of this amplitude to below, we see that this
does obey the expectation of the power counting theorem, with a
correction of order $\kappa^2m_1m_2$ compared to the tree level
amplitude.

However, we must also consider the crossed box diagram in
Fig.~\rref{fig:toy2}b. This can be handled in a similar fashion, and is
slightly easier because it does not have an imaginary part. Defining
$p_1\cdot p_4=m_1m_2+w^\prime$, we obtain
\begin{eqnarray}\llabel{eq:t10}
{\cal M}_{crossed}&=&\frac{\kappa^2m_1^2m_2^2}{q^2}\cdot
\frac{\kappa^2m_1m_2}
{16\pi^2}\cdot\left[+1-
\frac{w^\prime}{3m_1m_2}\right]\cdot\\
&&\mbox{}\cdot\left\{\frac{2}{4-d}-\ln{\left(-\frac{q^2}{m^2}\right)}
+\mbox{constant}+{\cal O}\left(\frac{q}{M}\right)\right\}
\nonumber
\end{eqnarray}
with the {\it same} constant as in~(\rref{eq:t9}). We see that the most
dangerous terms cancel between the diagrams. Using
$w-w^\prime=-\frac{q^2}{2}$ we get the final result
\begin{eqnarray}\llabel{eq:t11}
{\cal M}_{total}&=&{\cal M}_{tree}+{\cal M}_{box}+{\cal M}_{crossed}\\
&=&\frac{\kappa^2m_1^2m_2^2}{q^2}\cdot
\left\{\vstrut{1.5em}\right.1+\frac{1}{16\pi}\left[-\frac{1}{6}
\kappa^2q^2+i\pi\frac{\kappa^2m_1m_2}{p(m_1+m_2)}\right]\cdot\nonumber\\
&&\hspace{15ex}
\cdot\left[\frac{2}{4-d}-
\ln\left(-\frac{q^2}{m^2}\right)+\mbox{constant}
\right]\left.\vstrut{1.5em}\right\}.\nonumber
\end{eqnarray}

In the real part of the amplitude, the expansion parameter has become
$\kappa^2q^2$, which is well behaved. The imaginary part of the
amplitude is just a phase and does not contribute to observables at
this order when the matrix element is squared. It is simply the
analogue of the well known "Coulomb phase" and is generated from the
rescattering of the on shell intermediate state of matter
particles. Like the Coulomb phase, it has been shown by
Weinberg~\ccite{4} to exponentiate to all orders in general
relativity (the proof extends to this simpler theory as well), so that
this term does not cause any trouble. The same paper by Weinberg also
proves that infrared divergencies cancel in general relativity and by
extension in this theory, by the considerations of virtual corrections of
these diagrams plus the bremsstrahlung radiation of real
particles. These can be regulated dimensionally also~\ccite{5}, and
will yield as finite effects residual corrections of order
$\kappa^2q^2$
and $\kappa^2q^2\ln{q^2}$. We are not here directly interested in the
exact answer; for us the important result was the cancellation of
$\kappa^2m^2$ effects in~(\rref{eq:t11}).
\begin{figure}[htb]
%
\begin{picture}(390,70)(0,0)
\put(242,35){$\Longrightarrow$}
\put(117,35){$+$}
%
\ArrowLine(270,60)(295,60)
\put(280,67){$p$}
\ArrowLine(295,59)(345,59)
\ArrowLine(295,61)(345,61)
\put(350,67){$p\!\!+\!q$}
\ArrowLine(345,60)(370,60)
\Vertex(295,60)2
\Vertex(345,60)2
\Photon(295,20)(295,60)3 5
\put(286,34){$\uparrow$}
\put(300,32){$k_1$}
\Photon(345,20)(345,60)3 5
\put(336,34){$\uparrow$}
\put(350,32){$k_2$}
\put(292,16){$\times$}
\put(342,16){$\times$}
\put(312,0){(c)}
%
\ArrowLine(130,60)(155,60)
\put(140,67){$p$}
\ArrowLine(205,60)(230,60)
\ArrowLine(155,60)(205,60)
\put(210,67){$p\!\!+\!q$}
\Vertex(155,60)2
\Vertex(205,60)2
\Photon(205,20)(155,60)3 5
\ArrowLine(166,23)(171,29)
\put(155,27){$k_1$}
\ArrowLine(190,23)(185,29)
\put(200,27){$k_2$}
\Photon(155,20)(205,60)3 5
\put(152,16){$\times$}
\put(202,16){$\times$}
\put(172,0){(b)}
%
\ArrowLine(8,60)(33,60)
\put(18,67){$p$}
\ArrowLine(83,60)(108,60)
\put(88,67){$p\!\!+\!q$}
\ArrowLine(33,60)(83,60)
\Vertex(33,60)2
\Vertex(83,60)2
\Photon(33,20)(33,60)3 5
\put(23,34){$\uparrow$}
\put(36,32){$k_1$}
\Photon(83,20)(83,60)3 5
\put(73,34){$\uparrow$}
\put(86,32){$k_2$}
\put(30,16){$\times$}
\put(80,16){$\times$}
\put(50,0){(a)}
\end{picture}
%
\caption{\protect\llabel{fig:toy4}
The definition of the 'Bose-symmetrized' propagators.}
\end{figure}

We can show that this cancellation is not peculiar to the box
diagram, but rather is a general feature of this theory. This can be
seen by considering the basic unit of a single line with two interactions
with off-shell gravitons, as in Fig.~\rref{fig:toy4}. Any time a given
ordering is possible, the crossed order is also possible,
cf.~Fig.~\rref{fig:toy4}a and~\rref{fig:toy4}b. If we add these two
diagrams and allow the external legs to be on-shell, we find that the
sum of propagators behaves as $\frac{1}{m^2}$ whereas each individual
propagator was of order $\frac{1}{m}$:
\begin{eqnarray}\llabel{eq:t12}
{\cal V}&=&\kappa m^2\cdot\left[\frac{1}{(p+k)^2-m^2}+
\frac{1}{(p^\prime-k)^2-m^2}\right]\cdot\kappa m^2\nonumber\\
&=&(\kappa m^2)^2\cdot\left[\frac{1}{2p\cdot k+k^2}+
\frac{1}{-2p^\prime\cdot k+k^2}\right]\cdot\\
&=&(\kappa m^2)^2\cdot\left[\frac{2(p^\prime-p)\cdot k-2k^2}{[2p\cdot
k+k^2]\cdot[2p^\prime\cdot k-k^2]}\right].\nonumber
\end{eqnarray}

However, since $p^\prime-p=q$, there is no factor of the large mass $m$ in
the numerator,
\begin{equation}\llabel{eq:t13}
{\cal V}=(\kappa m^2)^2\cdot\left[\frac{q^2-k^2-(k-q)^2}{[2p\cdot
k+k^2]\cdot[2p^\prime\cdot k-k^2]}\right].
\end{equation}

Because there are two factors of $p\cdot k\sim m$ in the denominator,
this double vertex counts as
\eqn{t14}{{\cal V}\sim\frac{\kappa^2m^4}{m^2}\sim\kappa^2m^2}
rather than the ${\cal V}\sim\kappa^2m^3$ that the na\"\i ve counting
would imply. \mbox{For~Fig.~\rref{fig:toy4}a~or~b} individually would give
the extra two factors of $\frac{1}{m}$ which converts the undesirable
expression to $\kappa^2q^2$, thereby explaining the result found above.

The only exceptions to this power counting occurs for what can be
termed "exceptional momenta." This refers to momenta where the
propagator is not of order $\frac{1}{m}$, and can occur when the
intermediate line goes on shell, $(p+k)^2-m^2=0=2p\cdot k+k^2$ so that
$p\cdot k\sim k^2$. In
this case we do not gain a power the power of $\frac{1}{m}$ from the
propagator. This is exactly what was found above in the explicit
calculation of the box diagram. The on-shell intermediate states
generate the imaginary part of the diagram which has a different
dependence on the masses than does the remainder of the diagram.
This leads to the Coulomb phase in the box diagram.
\begin{figure}[htb]
%
\begin{picture}(390,50)(0,0)
%
\ArrowLine(95,41)(195,41)
\ArrowLine(95,39)(195,39)
\put(68,47){$p$}
\ArrowLine(195,41)(295,41)
\ArrowLine(195,39)(295,39)
\ArrowLine(45,40)(95,40)
\put(312,47){$p\!\!+\!q$}
\ArrowLine(295,40)(345,40)
\Vertex(95,40)2
\Vertex(195,40)2
\Vertex(295,40)2
\Photon(95,0)(95,40)3 5
\put(86,14){$\uparrow$}
\put(100,12){$l_1\!\!+\!\frac{q}{3}$}
\Photon(195,0)(195,40)3 5
\put(186,14){$\uparrow$}
\put(200,12){$l_2\!\!+\!\frac{q}{3}$}
\Photon(295,0)(295,40)3 5
\put(286,14){$\uparrow$}
\put(300,12){$l_3\!\!+\!\frac{q}{3}$}
\put(92,-4){$\times$}
\put(192,-4){$\times$}
\put(292,-4){$\times$}
%
\end{picture}
%
\caption{\protect\llabel{fig:toy6}
The two-loop 'Bose-symmetrized' propagator, defined
for $\sum_1^3l_i\equiv0$.}
\end{figure}

We have been able to extend the demonstration of cancellations to
three and four vertices (using computer algebra). The difficulty here
appears because only the external lines are on shell. The desired
cancellation does not occur for just the symmetrized sum of any two of the
permutations, e.~g.~when we permute $l_1$ and $l_2$
in~Fig.~\rref{fig:toy6}, but requires the sum of all six permutations.

We take the 'Bose-symmetrized propagator' denoted by a double line
in~Fig.~\rref{fig:toy6}:
\eqn{toy12}{
D^{(3)}_m(p,l_1,l_2,l_3)=\frac{1}{3!}\cdot\sum_{\mbox{perm}}
\frac{i}{(p+l_1+\frac{q}{3})^2-m^2}\cdot
\frac{i}{(p+l_1+l_2+2\frac{q}{3})^2-m^2}}
where the summation goes for all permutations of the $l_i$, subject to
$\sum_i{l_i}\equiv0$.

Na\"\i ve power counting would say 
$D^{(3)}\sim O\left(\frac{1}{m^2q^2}\right)$ and for the total
disappearance of the $\kappa m$ expansion parameter we would need
$D^{(3)}\sim O\left(\frac{1}{m^4}\right)$ because the expected behavior
of the amplitude is

in power counting:
\eqn{toy13}{
\frac{(\kappa m_1^2)(\kappa m_2^2)}{q^2}\cdot
\left\{1+(\kappa m)^2_{\mbox{one-loop}}+(\kappa m)^2_{\mbox{two-loop}}+
\dots\right\}
}

and in the effective theory
\eqn{toy14}{\frac{(\kappa m_1^2)(\kappa m_2^2)}{q^2}\cdot
\left\{1+(\kappa q)^2_{\mbox{one-loop}}+(\kappa q)^2_{\mbox{two-loop}}+
\dots\right\}.}

A tedious but straightforward algebra shows indeed that, disregarding
special points in the integration region where some of the denominators
are less than $O(m_i)$, cancellations occur with the result
\eqn{toy15}{D^{(3)}_m(p,l_1,l_2,l_3)=O\left(\frac{1}{m^4}\right).}

Although we do not have an inductive proof valid for all orders we did
the same calculation for $D^{(4)}_m(p,l_1,l_2,l_3,l_4)$, figuring in
ladder-type three-loop diagrams, which indeed showed a similar behavior
\eqn{toy16}{D^{(4)}_m(p,l_1,l_2,l_3,l_4)=O\left(\frac{1}{m^6}\right)}
in accordance with the requirement of the validity of the energy
expansion. This calculation was done with a Mathematica program and
attempts to calculate $D^{(6)}_m$ did not succeed because of the
prohibitively large amount of CPU time required. We have unfortunately
not found an inductive proof that lets us extend these results to all
orders. These calculations show that in the toy model, which is free
of gauge complications, the power counting theorem works
correspondingly to Weinberg's power counting theorem in chiral
effective QCD due to unexpected cancellations.

\Section{Lessons for general relativity}\llabel{sec:harm}

Many of the features uncovered in the last section are also applicable for
general relativity. Although general relativity contains additional
multigraviton vertices, the matter vertex considered in the previous
section is the most dangerous. We have not been able to reformulate
the general power counting formula (in harmonic gauge) such that the
cancellation is explicit in all diagrams. However, we can display how
the desired behavior is restored in examples which are physically
relevant.
\begin{figure}[tbh]
%
\begin{picture}(390,70)(0,0)
\ArrowLine(100,65)(135,65)
\ArrowLine(65,65)(100,65)
\Photon(100,30)(100,65)2 3
\Vertex(100,65)2
\put(90,0){\it (a)}
\Photon(240,30)(240,55)3 5
\ArrowLine(260,55)(280,55)
\ArrowLine(240,55)(260,55)
\ArrowLine(220,55)(240,55)
\ArrowLine(200,55)(220,55)
\Vertex(220,55)2
\Vertex(240,55)2
\Vertex(260,55)2
\PhotonArc(240,55)(20,0,180)2 9
\put(233,0){\it (b)}
\end{picture}
%
\caption{\protect\llabel{fig:hv}
Graviton vertex and one of the loop corrections.}
\end{figure}

Before we turn to the box diagram once again, let us consider the
vertex correction, Fig.~\rref{fig:hv}. Of the several contributions
to the vertex at one loop only Fig.~\rref{fig:hv}b and the
self-energy diagrams are of the dangerous category, with two extra
matter vertices and one loop.

Because the vertex coupling is the energy momentum tensor, and the
energy momentum tensor is conserved, there is a nonrenormalization
theorem for the matrix element at $q^2=0$. The general form for
the vertex, consistent with the conservation $\partial^\mu
T_{\mu\nu}=0$ is
\begin{equation}\llabel{eq:hformf}
<p^\prime\mid T_{\mu\nu}\mid p>=F_1(q^2)\cdot\left[p_\mu p^\prime_\nu+
p_\nu p^\prime_\mu-\eta_{\mu\nu}\right]+
F_2(q^2)\cdot\left[q_\mu q_\nu-\eta_{\mu\nu} q^2\right]
\end{equation}
and at tree level $F_1=1$, $F_2=0$.

The dangerous diagrams na\"\i vely give a correction to $F_1$ of order
$\kappa^2m^2$, and individually will do so. [Because powers of $q^2$
multiply $F_2$ it automatically is not a problem -- a one loop
contribution to $F_2(q^2)$ of order $\kappa^2m^2$ is allowable in a
well behaved energy expansion.] However, since $T_{\mu\nu}$ measures
the physical energy and momentum we have the constraint
$F_1(0)=1$. Thus all contributions to $F_1(q^2)$ which are independent
of $q^2$, in particular all corrections of order $\kappa^2m^2$, must
cancel when expressed in terms of the physical mass and momenta. This
occurs by the cancellation of the vertex~\ref{fig:hv}b with the
renormalization due to the self energy. This is entirely analogous to
the nonrenormalization of the charge form factor in QED at $q^2=0$.

One can also repeat the exercise to show that the sum of the two
diagrams \mbox{in~Fig.~\ref{fig:toy2}a,b}, but with two gravitons instead
of scalars, behaves as $\kappa^2m^2$ instead of the $\kappa^2m^3$ behavior
as given by the na\"\i ve power counting. The gravitational vertex is
more complicated than the scalar one, for example involving $\kappa
p_\mu(p+k)_\nu$ at a given vertex instead of $\kappa m^2$. However, in
the counting of powers of mass $p_\mu k_\nu$ is already one factor
fewer power of the mass than is $p_\mu p_\nu$. Therefore in showing
that the $\kappa^2m^3$ behavior is not present in the sum of
diagrams, we need only consider the $\kappa p_\mu p_\nu$ portion of
the vertex, which is common to all vertices and which will not upset
the cancellation of the two diagrams. The proof then goes through
exactly as in the last section.

Finally, we seek to demonstrate the cancellation in the box plus
crossed box diagrams.

In order to show that the situation, including the necessary
cancellations, is analogous to the one in the toy model, we calculate the
contribution from the box and crossed box graphs
\mbox{on~Fig.~\rref{fig:toy2}a,b} and we will see that the cancellation
of the $\geq O\left(\frac{1}{q^2}\right)$ terms {\it does} occur in the
{\it real} part of the amplitude and that in the nonrelativistic limit,
i.~e. for small values of spatial momenta $p\ll\sqrt{s}$, the toy model
gives the correct coefficients of the leading terms. This could be
expected as we will explain later. 
\begin{figure}[htb]
\begin{picture}(390,100)(0,0)
%
\put(33,90){$\mu\nu$}
\put(180,90){$\Longrightarrow$}
\Photon(50,93)(120,93)3 5
\Vertex(50,93)2
\Vertex(120,93)2
\put(125,90){$\alpha\beta$}
\put(250,90){$\frac{i}{q^2+i\epsilon}\cdot P_{\mu\nu,\alpha\beta}$}
%
\put(180,57){$\Longrightarrow$}
\ArrowLine(50,60)(120,60)
\Vertex(50,60)2
\Vertex(120,60)2
\put(250,57){$\frac{i}{p^2+m_i^2}$}
%
\put(180,13){$\Longrightarrow$}
\ArrowLine(50,30)(85,30)\put(65,33){$p$}
\ArrowLine(85,30)(120,30)\put(100,33){$p^\prime$}
\Vertex(85,30)2
\Photon(85,5)(85,30)3 5
\put(77,-2){$\mu\nu$}
\put(250,13){$-\frac{i}{2}\kappa\cdot V_{\mu\nu}^{(m)}(p,p\prime)$}
%
\end{picture}
\caption{\protect\llabel{fig:h1}Propagators and the relevant vertex in
harmonic gauge.}
\end{figure}

In harmonic gauge the graviton propagator is
\eqn{h1}{\frac{i}{q^2-i\epsilon}\cdot P_{\mu\nu,\alpha\beta}
\mbox{\hspace{4ex}with\hspace{4ex}}P_{\mu\nu,\alpha\beta}=
\eta_{(\mu}^\alpha\eta_{\nu)}^\beta-
\frac{1}{2}\eta_{\mu\nu}\eta^{\alpha\beta},}
the heavy mass propagators are
\eqn{h2}{\frac{i}{p^2-m^2}}
and the graviton-scalar-scalar vertex is
\eqn{h3}{-i\kappa\cdot V_{\mu\nu}^{(m)}(p,p^{\prime})
\mbox{\hspace{4ex}with\hspace{4ex}}V_{\mu\nu}^{(m)}(p,p^{\prime})=
p_{(\mu}p_{\nu)}^\prime-\frac{1}{2}
\eta_{\mu\nu}\cdot(p\cdot p^\prime-m^2),}
see also Fig.~\rref{fig:h1}.
Using these rules and the usual Feynman parameterization in the loop
integral we arrive at
\begin{eqnarray}\llabel{eq:h4}
{\cal M}_{(c)box}&=&-\frac{\kappa^4}{64}\cdot\frac{1}{(4\pi)^{2+\frac{
\epsilon}{2}}}\cdot\int_{-1}^1\frac{dx}{2}
\int_0^1dy\cdot y(1-y)\cdot\int_0^1dz\cdot\nonumber\\
&\cdot&\left\{\vstrut{2em}\right.4\cdot
\Gamma\left(2-\frac{\epsilon}{2}\right)\cdot
\frac{F_{(c)box}}{[M_{(c)box}^2-i\epsilon]^{2-\frac{\epsilon}{2}}}-\\
&-&2\cdot\Gamma\left(1-\frac{\epsilon}{2}\right)\cdot
\frac{g_{\mu\nu}F^{\mu\nu}_{(c)box}}{[M_{(c)box}^2-
i\epsilon]^{1-\frac{\epsilon}{2}}}+\nonumber\\
&+&\Gamma\left(-\frac{\epsilon}{2}\right)\cdot
\frac{(g_{\mu\nu}g_{\alpha\beta}+g_{\mu\alpha}g_{\nu\beta}+
g_{\mu\beta}g_{\nu\alpha})\cdot F^{\mu\nu\alpha\beta}_{(c)box}}
{[M_{(c)box}^2-i\epsilon]^{-\frac{\epsilon}{2}}}
\left.\vstrut{2em}\right\}\nonumber,
\end{eqnarray}
where the IR singularities (at $\epsilon\to0$) come from the
first term and the UV singularities come from the last term. The functions
$F$ are determined as
\begin{eqnarray}\llabel{eq:h5}
F_{box}&=&\left[V_{\mu\nu}^{m_1}(p_1,p_1+k)\cdot
P^{\mu\nu,\alpha\beta}\cdot V_{\alpha\beta}^{m_2}(p_2,p_2-k)\right]\cdot
\nonumber\\
&&\mbox{}\cdot\left[V_{\lambda\rho}^{m_1}(p_1+k,p_1+q)\cdot
P^{\lambda\rho,\kappa\delta}\cdot
V_{\kappa\delta}^{m_2}(p_2-k,p_2-q)\right]
\nonumber\\
F_{cbox}&=&\left[V_{\mu\nu}^{m_1}(p_1,p_1+k)\cdot
P^{\mu\nu,\alpha\beta}\cdot V_{\alpha\beta}^{m_2}(p_2-q+k,p_2-q)\right]
\cdot\\
&&\mbox{}\cdot\left[V_{\lambda\rho}^{m_1}(p_1+k,p_1+q)\cdot
P^{\lambda\rho,\kappa\delta}\cdot
V_{\kappa\delta}^{m_2}(p_2,p_2-q+k)\right]\nonumber
\end{eqnarray}
and
\eqn{h6a}{[M_{(c)box}]^2=(1-x^2)(1-y)^2\cdot\frac{q^2}{4}+y^2\cdot[\ ]_z}
with
\begin{eqnarray}\llabel{eq:h6b}
{[\ ]}_{z}^{box~}&=&(1-z)^2\cdot m_1^2+z^2m_2^2-z(1-z)(s-m_1^2-m_2^2)
\nonumber\\
{[\ ]}_{z}^{cbox}&=&(1-z)^2\cdot m_1^2+z^2m_2^2-z(1-z)(u-m_1^2-m_2^2),\\
u&\equiv&2(m_1^2+m_2^2)-s-q^2\equiv u_0-q^2.\nonumber
\end{eqnarray}
\begin{figure}[hbt]
\begin{picture}(400,180)(-50,0)
\epsfbox{powDTfig.eps}
\end{picture}
\caption{\protect\llabel{fig:h2}
(a) The complex path of $z$ integration. (b) What this path becomes
in the complex $y$ plane. Singularities are shown in red, the path
of integration in green.}
\end{figure}

The quantity $[\ ]_z^{cbox}$ is always positive, corresponding to the fact
that the crossed box diagram has no imaginary part (which would emerge from
the point where $M_{(c)box}^2\rightarrow0$). The corresponding
$[\ ]_z^{box}$ does have zeroes which can be avoided by a complex
continuation in $z$ as shown in Fig.~\rref{fig:h2}.

After some a tedious algebra we managed to separate out the IR divergent
part which only stems form the $x=\pm 1, y=0$ corners of the integration
region. The coefficients were extracted with a REDUCE program and cast in
the form
\eqn{h7}{i{\cal M}_{(c)box}=-\frac{\kappa^4}{8(4\pi)^2}\int_0^1dz
\int_0^{\pi/2}d\beta\cdot\cos\beta\sin\beta\cdot \left(\sum_{k=1}^4
h_k(\gamma,z)\psi_k(b)\right)}
with
\eqn{h8}{\gamma=\frac{\cos\beta-\sin\beta}{\cos\beta+\sin\beta}}
and
\eqn{h9}{b=\frac{q^2}{4}\cdot\cos^2{\beta}+[\ ]_z\cdot\sin^2{\beta}
-i\epsilon,}
and
\eqn{h10}{\psi_1(b)=\frac{1}{b},
\ \ \psi_2(b)=\frac{1}{b^2},
\ \ \psi_3(b)=\ln b,
\ \ \psi_4(b)=\frac{\ln b}{b^2},}
and the functions $h_k(\gamma,z)$ turn out to be polynomials in $q^2$;
they contain about 932 terms that we do not display here. Their algebraic
structure can be understood though and we find that the only nonpolynomial
dependences in {\it any} of the variables come from seven special
functions of $\gamma$ and their singularities are avoided by the above
prescription to use a complex path of integration for $z$.

The calculation at this point contains {\it no} approximation at all.
However, the integrals cannot be done by quadrature. For our purposes it
is only necessary to separate the terms which are divergent in the limit
$q\to0$, and due to the fact that the integration contours defined
above avoid the singularities we can bring the $q\to0$ limit into
the integrands. For lack of space we do not describe all the details of
the thorough analysis of the divergent terms. What we do is we first
observe that cutting the $\beta$ integration in two parts, 
$\int_0^{\rho(s,m_1^2,m_2^2)}+\int_{\rho}^{\pi/2}$, the integrand in the
latter is uniformly bounded and so provides no divergent terms. Using
a Taylor-formula for the former part, the remainder is also seen to be
uniformly bounded (this was proven for each $k$ using Weierstrass's
theorem) and what remains is
\begin{eqnarray}\llabel{eq:h11}
{\cal M}_{(c)box}&=&-\frac{\kappa^4}{8(4\pi)^2}
\cdot\left\{\vstrut{2em}\right.
\frac{4}{q^2}\ln{\frac{q}{2}}\cdot\tilde h^{^{(0)}}_4\cdot
\int_0^1\frac{dz}{[\ ]^{^{(0)}}_z}\nonumber\\
&+&\frac{2}{q^2}\cdot\int_0^1\frac{dz}{[\ ]^{^{(0)}}_z}\cdot
\left(\tilde h^{^{(0)}}_4+\tilde h^{^{(0)}}_2(0,z)\right)\nonumber\\
&-&\frac{i\pi}{2q}\cdot\int_0^1\frac{dz
\cdot\left({\tilde h^{^{(0)}}_2}\right)^{\prime}(0,z)
}{[\ ]^{^{(0)}}_z\cdot
\sqrt{i\epsilon-[\ ]^{^{(0)}}_z}}\\
&-&\ln{\frac{q}{2}}\cdot
\left[\vstrut{1.8em}\right.
\int_0^1\frac{dz}{[\ ]^{^{(0)}}_z}
\left(\tilde h^{^{(0)}}_1(0,z)-4\tilde h^{^{(1)}}_4 \right)+\nonumber\\
&&\ \ \ \ \ \ \ \ \ +\ \int_0^1\frac{dz}{\left([\ ]^{^{(0)}}_z\right)^2}
\cdot\left(\left(\tilde h^{^{(0)}}_2\right)^{\prime\prime}(0,z)-
\tilde h^{^{(0)}}_4\cdot\phi(z)\right)
\left.\vstrut{1.8em}\right]\left.\vstrut{2em}\right\},\nonumber
\end{eqnarray}
where $[\ ]^{^{(0)}}_z\equiv\left([\ ]_z\right)_{q\rightarrow0}$
and $\tilde h^{^{(0)}}(x,z)\equiv
h(x\!\rightarrow\!\tan{\beta},z)_{q\rightarrow0}$.

Also,
$\tilde h^{^{(1)}}_4\equiv \lim_{q\to0}\frac{h_4-h^{^{(0)}}_4}
{q^2}$ and the prime always refers to the derivative in $x$ and we used
the notation
\[\phi(z)=\left\{\begin{array}{ll}
                       1+4z(1-z) & \mbox{for {\it cbox}}\\
                       1         & \mbox{for {\it box.}}
                      \end{array}
          \right.\]

Here, $h^{^{(0)}}_4\!=\!8\left[s^2-2s(m_1^2+m_2^2)+
(m_1^4+m_2^4)\right]^2$ is
independent of $\beta,z$. This term generates the leading divergence,
and as we will see this term is the only one present in the toy model of
Sec.~\rref{sec:toy}.

The most divergent terms are multiplied by the integrals
$\int_0^1\frac{dz}{[\ ]^{^{(0)}}_z}$. A substitution of variables
$z\!\rightarrow\!y$ with $\frac{1}{z}+\frac{1}{y}=2$, has the interesting
property \[[]^{box}_z=\frac{[\ ]^{^{(0,cbox)}}_y}{(2y-1)^2},\] and
tracing the path of the complex $y$ integration as shown in
Fig.~\rref{fig:h2} we find a residue term in
\eqn{h12}{\int_0^1\frac{dz}{[\ ]^{box}_z}=
-\int_0^1\frac{dz}{[\ ]^{^{(0,cbox)}}_z}+\frac{i\pi}{p\sqrt{s}},}
i.e. the sum of the two integrals figuring in~(\rref{eq:h11}) is a pure
residue. We find that the leading divergence is IR and UV finite:
\eqn{h13}{\sim-\frac{i}{16\pi}\cdot\frac{\ln q}{q^2}\cdot
\frac{\kappa^4}{p\sqrt{s}}\cdot\left[s^2-2s(m_1^2+m_2^2)+(m_1^4+m_2^4)
\right]^2}
is pure imaginary and as such can be included in an unobservable phase.

The next-to-leading, $\frac{1}{q^2}$ divergence contains the same factor
and also results in a pure imaginary contribution
\eqn{h14}{\sim-\frac{i}{16\pi}\cdot\frac{1}{q^2}\cdot
\frac{\kappa^4}{p\sqrt{s}}\cdot\left[s^2-2s(m_1^2+m_2^2)+(m_1^4+m_2^4)
\right]^2\cdot\left(\frac{1}{\epsilon}+\frac{C-1-\ln{4\pi}}{2}\right)}
where the $\epsilon\to0$ singularity is of IR origin.

The ambiguity in~(\rref{eq:h13}) in defining
$\ln q\equiv\ln{\frac{q}{\mu}}+\ln{\mu}$ is a reflection of the ambiguity
in the IR regulator to remove IR divergencies. Total removal is achieved
only after taking into account soft graviton processes as in~\ccite{4}
and different choices of $\mu$ show up as different constants
in~(\rref{eq:h14}):
\eqn{hh1}{\frac{C-1-\ln{4\pi}}{2}\longrightarrow\frac{C-1-\ln{4\pi}}{2}+
{\ln{\frac{\mu^\prime}{\mu}}.}}

The $O\left(\frac{1}{q}\right)$ divergencies are much harder to calculate.
We found, by a similar calculation, that the $O\left(\frac{1}{q}\right)$
part has {\it no} imaginary part, contains {\it no} IR or UV singular
terms. Its coefficient is a complicated function of $s$, $m_1^2$ and
$m_2^2$ which we do not display for lack of space but whose
$p\to0$ limit is
\eqn{hh3}{\sim\frac{\kappa^4m_1^2m_2^2}{128q}\cdot\left[m_1+m_2+
O\left(\frac{p}{m}\right)\right].}

This term, which still contains {\it one} power of $\kappa m$ too
much, corresponds to a contribution to the classical potential
\eqn{label:hh5}{V(r)\longrightarrow-G\frac{m_1m_2}{r}\cdot\left[
\dots-\frac{G(m_1+m_2)}{4r}+\dots\right]}
and forms part of the {\it classical} general relativity corrections
which are encoded in the one-loop diagrams of~Fig.~\rref{fig:toy2}
when the harmonic gauge is used. We note that no similar real term
arises in the toy model of Eq.~(\rref{eq:t11}).

The $p\!\to\!0$ limit of the coefficients of $q\!\to\!0$ divergent
terms turn out to be smooth except for the imaginary part which has a
$\frac{\kappa^4m^7}{q^4\cdot p}$ term in~(\rref{eq:h14}). The limit we
use in this calculation,
\eqn{eq:hh6}{\frac{p}{m}\gg\frac{q}{p}\to0}
exactly corresponds to the requirement that no bound states are formed
\begin{eqnarray}
Gm_1m_2\cdot q&\sim&\frac{p^2}{2m_1}+\frac{p^2}{2m_2}\nonumber\\
\mbox{i. e.}\hfill&&\\
2Gm_1m_2&\ll&\frac{p^2}{q\cdot m_{red}}\to\infty.\nonumber
\end{eqnarray}

We finally comment on why out toy model gave the correct leading
divergencies. At leading order in $\frac{q}{m}$, the heavy scalar vertex
becomes
\eqn{hu1}{-\frac{i}{2}\kappa\cdot[
p_\mu\cdot p_\nu^\prime+p_\nu\cdot p_\mu^\prime-\eta_{\mu\nu}\cdot(
p\cdot p^\prime-m^2)]\longrightarrow-i\kappa m^2v_\mu v_\nu.}
This vertex is attached to a graviton propagator and is multiplied by
$P^{\mu\nu}_{\alpha\beta}$ and so becomes
\eqn{hu2}{-i\kappa m^2\cdot\left(v_\alpha v_\beta-\frac{1}{2}
\eta_{\alpha\beta}\right),}
so when we multiply this by the heavy scalar on the other end of the
graviton propagator, the full theory, as compared to the toy model,
provides and extra factor
\eqn{hu3}{(v_1\cdot v_2)^2-\frac{1}{2}\equiv
\frac{(2p_1\cdot p_2)^2-2m_1^2m_2^2}{4m_1^2m_2^2}
\equiv\frac{[s^2-2s(m_1^2+m_2^2)+(m_1^4+m_2^4)]}{4m_1^2m_2^2}.}
Comparing now~(\rref{eq:h13}) with the appropriate,
$O\left(\frac{\ln{q}}{q^2}\right)$ part of~(\rref{eq:t11}) we see that the
square of~(\rref{eq:hu3}) shows up, corresponding to the presence of
{\it two} graviton lines. A similar comparison between~(\rref{eq:hh3})
and~(\rref{eq:t11}) shows that in the toy model the $\frac{1}{q}$
divergence is absent. It is nonzero in~(\rref{eq:hh3}) only due to
subleading contributions to~(\rref{eq:hu1}) and~(\rref{eq:hu2}).

\Section{Power counting in a physical gauge}\llabel{sec:fizg}

In covariant gauges, both classical and quantum effects are included
in the same Feynman diagram. The simplest example is the one graviton
exchange diagram, which includes the classical Newton potential, but
loop diagrams also exhibit this property~\ccite{2}. However, in
physical gauges, classical and quantum effects are separated.The
physical quantum degrees of freedom are transverse and
traceless~\ccite{6}, corresponding to massless spin-two quanta. In a
multipole expansion the monopole term, which generates the Newton
potential, is classical, while the spin-two degrees of freedom
couple to the quadrupole term. This suggests that the dominant $\kappa
m^2$ coupling that caused us do much trouble in Sec.~\rref{sec:powcov} is
to be associated with classical physics while the quantum degrees of
freedom have a milder behavior. We will show that this is the case for
the interaction of two heavy masses. This allows a quantum power counting
which is well behaved.

Covariant gauges, in particular the harmonic gauge treated
covariantly, are preferred for practical calculations~\ccite{7}. When
combined with the background field technique, they can explicitly
retain the invariances of general relativity. In contrast the
construction of the physical gauge quantum theory severely disturbs the
underlying symmetry of general coordinate invariance~\ccite{8}. One
picks a preferred frame for the quantization, and the split of quantum
and classical degrees of freedom depends on that frame. Nevertheless a
physical gauge is attractive conceptually because only the physical
radiation degrees of freedom are quantized. This is analogous to the
Coulomb gauge quantization of QED.

Although the general metric tensor $g_{\mu\nu}$ has ten real
components, the radiation field has only two independent degrees of
freedom, corresponding to helicity $\pm2$~\ccite{6}. For a wave
propagating in the $z$ direction, a harmonic gauge constraint plus
residual gauge freedom can be used to reduce the ten original components
of a polarization tensor $\epsilon_{\mu\nu}$ such that only
$\epsilon_{11}=-\epsilon_{22}$ and $\epsilon_{12}=\epsilon_{21}$ are
nonvanishing (other choices are also possible.) The helicity states
with $\lambda=\pm2$ can be found from these via
\eqn{f1}{\epsilon_{\pm}=\epsilon_{11}\mp i\epsilon_{12}=
-\epsilon_{22}\mp i\epsilon_{12}.}
When quantized, these become the graviton degrees of freedom.

Let us consider the interaction of two heavy masses, which have a
small relative momentum. The momentum of one of the masses can be
written as
\begin{eqnarray}\llabel{f2} 
p_\mu&=&mv_\mu+\overline{p}_\mu\\
v_\mu&=&(1,0,0,0)_\mu.\nonumber
\end{eqnarray}

\begin{figure}[thb]
%
\begin{picture}(390,70)(0,0)
\ArrowLine(60,65)(100,65)
\ArrowLine(100,65)(140,65)
\Photon(100,65)(130,30)3 5
\Photon(100,65)(70,30)3 5
\Vertex(100,65)2
\put(90,0){\it (a)}
\ArrowLine(200,65)(240,65)
\ArrowLine(240,65)(280,65)
\Vertex(240,65)2
\DashLine(240,30)(240,65)3
\Photon(240,65)(270,30)3 5
\put(233,0){\it (b)}
\end{picture}
%
\caption{\protect\llabel{fig:f1}Suppressed
matter--gravity interaction vertices.}
\end{figure}
\noindent Since
\eqn{f3}{q_\mu=\overline{p}_\mu-\overline{p}_\mu^\prime,}
we will treat the residual $\overline{p}$ as of order $q$ in the
energy expansion. If we quantize in the frame where $v_\mu$ defines
the timelike direction, then the physical graviton degrees of freedom
will only couple to the spacelike traceless components of the
energy-momentum tensor $T_{ij}$. Since the large mass does not
contribute to these, we know
\eqn{f4}{T_{ij}=\overline{p}_i\overline{p}_j^\prime+\overline{p}_j^\prime
\overline{p}_i}
and we count $T_{ij}$ as order $q^2$, whereas $T_{00}$ is of order
$m^2$. The same suppression is present for the two graviton
interaction of Fig.~\rref{fig:f1}a, and the coupling of one classical
Newtonian field and one physical quantum field as shown in
Fig.~\rref{fig:f1}b. From the general vertex
\begin{eqnarray}\llabel{eq:f5}
\tau_{\lambda\eta,\rho\sigma}=i\frac{\kappa^2}{2}\cdot
&\left\{\vstrut{2em}\right.&
I_{\eta\lambda,\alpha\delta}\cdot I^\delta_{\ \beta,\rho\sigma}\cdot\left(
p^\alpha p^{\prime\beta}p^{\prime\alpha}p^\beta\right)\nonumber\\
&&\vstrut{2em}-\frac{1}{2}\cdot\left(
\eta_{\eta\lambda}\cdot
I_{\beta\sigma,\alpha\beta}+\eta_{\rho\sigma}\cdot I_{\eta\lambda,
\alpha\beta}\right)\cdot p^{\prime\alpha}p^\beta\\
&&\mbox{}-\frac{1}{2}\left(I_{\eta\lambda,\rho\sigma}-\frac{1}{2}
\eta_{\eta\lambda}\eta{\rho\sigma}\right)\cdot[p\cdotp^\prime-m^2]
\left.\vstrut{2em}\right\}.\nonumber
\end{eqnarray}
we have
\begin{eqnarray}\llabel{f6}
\tau_{00,00}&\sim&\kappa^2m^2\nonumber\\
\tau_{00,ij}&\sim&\kappa^2q^2\\
\tau_{ij,kl}&\sim&\kappa^2q^2\nonumber.
\end{eqnarray}
What we see from these couplings is that the physical quantum degrees
of freedom have a reduced matter coupling with two fewer powers of $m$
compared to the harmonic gauge counting rules.

\begin{figure}[htb]
%
\begin{picture}(390,150)(0,0)
%
\ArrowLine(270,140)(295,140)
\ArrowLine(295,140)(370,140)
\ArrowLine(270,96)(295,96)
\ArrowLine(295,96)(370,96)
\Vertex(320,140)2
\Vertex(320,96)2
\Vertex(320,107)2
\Vertex(320,129)2
\DashLine(320,96)(320,107)2
\DashLine(320,129)(320,140)2
\PhotonArc(320,118)(11,270,90)2 5
\PhotonArc(320,118)(11,90,270)2 5
\put(312,80){(c)}
%
\ArrowLine(140,140)(185,140)
\ArrowLine(185,140)(240,140)
\ArrowLine(140,96)(185,96)
\ArrowLine(185,96)(240,96)
\Vertex(185,96)2
\Vertex(185,140)2
\Vertex(185,107)2
\Vertex(185,129)2
\DashLine(185,96)(185,107)2
\DashLine(185,107)(185,129)2
\DashLine(185,129)(185,140)2
\PhotonArc(185,118)(11,270,90)2 5
\put(182,80){(b)}
%
\ArrowLine(195,60)(220,60)
\ArrowLine(220,60)(295,60)
\ArrowLine(195,16)(220,16)
\ArrowLine(220,16)(270,16)
\ArrowLine(270,16)(300,16)
\Vertex(220,60)2
\Vertex(270,16)2
\Vertex(220,16)2
\DashLine(220,16)(220,60)3
\Photon(270,16)(220,60)3 5
\put(237,0){(e)}
%
\ArrowLine(70,60)(95,60)
\ArrowLine(140,60)(170,60)
\ArrowLine(95,60)(140,60)
\ArrowLine(70,16)(95,16)
\ArrowLine(140,16)(170,16)
\ArrowLine(95,16)(140,16)
\Vertex(95,16)2
\Vertex(140,16)2
\Vertex(95,60)2
\Vertex(140,60)2
\DashLine(140,16)(140,60)3
\Photon(95,16)(95,60)3 5
\put(107,0){(d)}
%
\ArrowLine(8,140)(58,140)
\ArrowLine(58,140)(108,140)
\ArrowLine(8,96)(58,96)
\ArrowLine(58,96)(108,96)
\Vertex(58,96)2
\Vertex(58,140)2
\DashLine(58,96)(58,140)3
\put(50,80){(a)}
\end{picture}
%
\caption{\protect\llabel{fig:ff}
Interaction of two masses in a physical gauge.}
\end{figure}

Various other diagrams describing the interaction of two masses in
physical gauges are shown in Fig.~\ref{fig:ff}. The dashed line
represents the classical Newtonian potential while the wavy line
describes the quantum degrees of freedom. In the normalization used
throughout this work, the classical interaction is again of order
$\frac{\kappa^2m^4}{q^2}$. Corrections due to vacuum polarization
\mbox{in~Fig.~\ref{fig:ff}b,c} have the same powers of $m$ but two
extra powers of $\kappa^2$, leading to a result of order
\eqn{f7}{\frac{\kappa^2m^4}{q^2}\cdot\kappa^2q^2\cdot(a+b\log{q^2})}
so that these diagrams have the expected expansion parameter $\kappa^2q^2$
as desired. An important feature of the physical gauge is that
diagrams with extra matter couplings are suppressed. For example, the
mixed box diagram of Fig.~\ref{fig:ff}d has two factors
of~$\frac{1}{m}$ from the propagators, but no compensating factors of the
mass in the vertex coupling or the physical gravitons. This means that
this diagram is suppressed by~$\frac{1}{m^2}$ compared to the vacuum
polarization corrections. All of the diagrams with graviton--matter
vertices are suppressed by powers of~$\frac{1}{m}$. This leaves the
vacuum polarization diagram as the leading quantum correction is this
gauge, with a well behaved expansion parameter.

The above results may be seen more easily in different normalization
for the fields, typical for the nonrelativistic limit. Here, we divide
all matter vertices by a factor of $2m$, so that $T_{00}=m$ for a
particle at rest. In this normalization (used also in Heavy Quark
Effective Theory~\ccite{9}) propagators have no factor
of~~$\frac{1}{m}$.
However, the coupling of the transverse traceless degrees of freedom
to matter fields are proportional to
\eqn{f8}{T_{ij}=\frac{\overline{p}_i\overline{p}_j^\prime+
\overline{p}_i^\prime\overline{p}_j}{2m}}
and vanish as $m\to0$. Then all diagrams with matter coupling of graviton
simply drop out, leaving only the vacuum polarization diagram in this
gauge.

To close this section, we note that the use of a physical gauge seems
to be required if we are to be able to introduce the idea of a purely
classical source. In the harmonic gauge, the inclusion of the matter
propagator was required to properly identify the classical and
quantum corrections. Both vertex and vacuum polarization diagrams are
important. One could not at the start of the calculation take the
mass to infinity and treat this resulting field in the classical
limit. The reason is that the vertex coupling strength also goes to
infinity in this limit. However, in the physical gauge, the diagrams
with matter propagators and couplings to transverse quantum fields are
unimportant in the limit $m\to\infty$. By taking this limit in this gauge,
one obtains a classical source, with an interaction which receives
quantum corrections.

\Section{Summary}\llabel{sec:sum}

We were motivated for this study by the observation of a class of
diagrams which in harmonic gauge would apparently upset the utility of
the energy expansion, and indeed also spoil the classical limit of the
theory. Part of the problem is due to the fact that the graviton
propagator in harmonic gauge includes both classical and quantum effects.
By consideration of several diagrams we were able to demonstrate the
nature of the cancellations in harmonic gauge which removed the
undesirable expansion parameter and led to a well defined energy
expansion. The logic for this behavior is clearer in a physical gauge,
analogous to Coulomb gauge in QED, even if explicit calculations are
much more painful in such a gauge. The physical transverse traceless
quantum degrees of freedom only couple with a reduced strength in
problems with nearly static matter fields. In the limit that $m$
becomes very large, the effect of matter couplings become negligible,
and the modification to the gravitational self-interactions (i.~e.~vacuum
polarization) become the most important quantum corrections. These
self-interactions of the sector satisfy the Weinberg power counting
theorem without any problems. This indicates that the quantum energy
expansion is well behaved in physical gauges, and hence by extension
in all gauges as long as one is calculating gauge invariant quantities.

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\end{thebibliography}

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