%Paper: hep-th/9506173
%From: ahmed@ovrut.hep.upenn.edu (Ahmed Hindawi)
%Date: Mon, 26 Jun 1995 18:20:55 -0400
%Date (revised): Mon, 26 Jun 1995 21:05:44 -0400
%Date (revised): Sun, 03 Sep 95 00:37:00
%Date (revised): Tue, 26 Sep 95 19:57:28

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\begin{document}

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\scriptsize{HEP-TH/9506173, UPR-672T}
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\title[Higher-Derivative Gravitation]{Higher-Derivative Gravitation in
Bosonic \\
and Superstring Theories}
\author{Ahmed Hindawi, Burt A. Ovrut, and Daniel Waldram}
\thanks{Talk presented at SUSY'95, Ecole Polytechnique in Palaiseau,
Palaiseau, France, May 15-19, 1995. Work supported in part by DOE
Contract DOE-AC02-76-ERO-3071 and NATO Grant CRG. 940784.}
\maketitle
\vspace*{-0.3in}
\begin{center}
\small{\textit{Department of Physics, University of Pennsylvania}} \\
\small{\textit{Philadelphia, PA 19104-6396, USA}}
\end{center}

\begin{abstract}

A discussion of the number of degrees of freedom, and their dynamical
properties, in higher-derivative gravitational theories is presented.
The complete non-linear sigma model for these degrees of freedom is
exhibited using the method of auxiliary fields. As a by-product we
present a consistent non-linear coupling of a spin-2 tensor to
gravitation. It is shown that non-vanishing
$(C_{\mu\nu\alpha\beta})^{2}$ terms arise in $N=1$, $D=4$ superstring
Lagrangians due to one-loop radiative corrections with light field
internal lines.

\end{abstract}

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\section{Bosonic Gravitation}

The usual Einstein theory of gravitation involves a symmetric tensor
$g_{\mu\nu}$ whose dynamics is determined by the Lagrangian
\begin{equation}
\mathcal{L}=\frac{1}{2\kappa^2}\mathcal{R}.
\end{equation}
The diffeomorphic gauge group reduces the number of degrees of freedom
from ten down to six. Einstein's equations further reduce the degrees of
freedom to two, which correspond to a physical spin-2 massless graviton.
Now let us consider an extension of Einstein's theory by including terms
in the action which are quadratic in the curvature tensors. This
extended Lagrangian is given by
\begin{equation}
\mathcal{L} = \frac{1}{2\kappa^2}\mathcal{R} + \alpha\mathcal{R}^2 +
\beta(C_{\mu\nu\alpha\beta})^2 + \gamma(\mathcal{R}_{\mu\nu})^2,
\end{equation}
where $\mathcal{R}^2$, $(C_{\mu\nu\alpha\beta})^2$, and
$(\mathcal{R}_{\mu\nu})^2$ are a complete set of CP-even quadratic
curvature terms. The topological Gauss-Bonnet term is given by
\begin{equation}
\label{GB}
\GB=(C_{\mu\nu\alpha\beta})^2 - 2(\mathcal{R}_{\mu\nu})^2 +
\frac{2}{3}\mathcal{R}^2.
\end{equation}
Therefore, we can write
\begin{equation}
\mathcal{L} = \frac{1}{2\kappa^2} \mathcal{R} + a \mathcal{R}^2 - b
(C_{\mu\nu\alpha\beta})^2 + c\, \GB.
\end{equation}
In this case, it can be shown \cite{A} that there is still a physical
spin-2 massless graviton in the spectrum. However, the addition of the
$\mathcal{R}^2$ term introduces a new physical spin-0 scalar, $\phi$,
with mass $m=(12a\kappa^2)^{-1/2}$. Similarly, the
$(C_{\mu\nu\alpha\beta})^2$ term introduces a spin-2 symmetric tensor,
$\phi_{\mu\nu}$, with mass $m=(4b\kappa^2)^{-1/2}$ but this field,
having wrong sign kinetic energy, is ghost-like. The $\GB$ term, being a
total divergence, is purely topological and does not lead to any new
degrees of freedom. The scalar $\phi$ is perfectly physical and can lead
to very interesting new physics \cite{B}. The new tensor
$\phi_{\mu\nu}$, however, appears to be problematical. There have been a
number of attempts to show that the ghost-like behavior of
$\phi_{\mu\nu}$ is illusory, being an artifact of linearization
\cite{C}. Other authors have pointed out that since the mass of
$\phi_{\mu\nu}$ is near the Planck scale, other Planck scale physics may
come in to correct the situation \cite{D}. In all these attempts, the
gravitational theories being discussed were not necessarily consistent
and well defined. However, in recent years, superstring theories have
emerged as finite, unitary theories of gravitation. Superstrings,
therefore, are an ideal laboratory for exploring the issue of the ghost-
like behavior of $\phi_{\mu\nu}$, as well as for asking whether the
scalar $\phi$ occurs in the superstring Lagrangian. Hence, we want to
explore the question ``\textit{Do quadratic gravitation terms appear in
the $N=1$, $D=4$ superstring Lagrangian?}''

Before doing this, however, we would like to present further details
about the emergence of the new degrees of freedom in quadratic
gravitation. We begin by adding to Einstein gravitation, quadratic terms
associated with the scalar curvature only. That is, we consider the
action
\begin{equation}
\label{R2}
\mathcal{S} = \int d^4x\sqrt{-g}\left(\mathcal{R} + \frac{1}{6} m^{-2}
\mathcal{R}^2 \right).
\end{equation}
The equations of motion derived from this action are of fourth order and
their physical meaning is somewhat obscure. These equations can be
reduced to second order, and their physical content illuminated, by
introducing an auxiliary field $\phi$. The action then becomes
%
\begin{align}
\label{R2phi}
\mathcal{S} &= \int{d^4x \sqrt{-g} \left(\mathcal{R} +
\frac{1}{6} m^{-2} \mathcal{R}^2 - \frac{1}{6} m^{-2} \left[\mathcal{R}
- 3m^2 \left\{e^\phi - 1 \right\}\right]^2\right)} \\
\label{Rphi}
&= \int{d^{4}x\sqrt{-g}\left(e^{\phi}\mathcal{R} -
\frac{3}{2}m^{2}\left[e^{\phi}-1 \right]^{2}\right)}.
\end{align}
%
Note that the $\phi$ equation of motion sets the square bracket in
equation \eqref{R2phi} to zero. Hence, action \eqref{Rphi} with the
auxiliary field $\phi$ is equivalent to the original action \eqref{R2}.
Now, let us perform a Weyl rescaling of the metric
\begin{equation}
g_{\mu\nu}=e^{-\phi} \overline{g}_{\mu\nu}.
\end{equation}
It follows that
%
\begin{equation}
\begin{split}
\sqrt{-g} & = e^{-2\phi}\sqrt{-\overline{g}}\, , \\
\mathcal{R} & = e^{\phi}\left(\overline{\mathcal{R}} +
3\overline{\nabla}^{2} \phi - \frac{3}{2}
\left[\overline{\nabla}\phi\right]^{2}\right),
\end{split}
\end{equation}
%
where $\overline{\nabla}_{\lambda}\overline{g}_{\mu\nu}=0$. Therefore,
%
\begin{equation}
\begin{split}
\sqrt{-g}e^{\phi}\mathcal{R} &= \sqrt{- \overline{g}} \left(
\overline{R}+3\overline{\nabla}^2 \phi - \frac{3}{2}
\left[\overline{\nabla}\phi\right]^2\right), \\
-\frac{3}{2}m^{2}\sqrt{-g}\left(e^{\phi}-1\right)^{2} &=
-\frac{3}{2}m^{2} \sqrt{-\overline{g}} \left(1-e^{-\phi}\right)^{2},
\end{split}
\end{equation}
%
and the action becomes
%
\begin{equation}
\mathcal{S} = \int{d^{4}x \sqrt{-
\overline{g}}\left(\overline{\mathcal{R}}-\frac{3}{2}
\left[\overline{\nabla}\phi\right]^2-\frac{3}{2}m^{2}\left[1-e^{-\phi}
\right]^2 \right)},
\end{equation}
%
where we have dropped a total divergence term. It follows that the
higher-derivative pure gravity theory described by action \eqref{R2} is
equivalent to a theory of normal Einstein gravity coupled to a real
scalar field $\phi$. It is important to note that, with respect to the
metric signature $(-,+,+,+)$ we are using, the kinetic energy term for
$\phi$ has the correct sign and, hence, that $\phi$ is not ghost like.
Also, note that a unique potential energy function
%
\begin{equation}
V(\phi)=\frac{3}{2}m^2 \left(1-e^{-\phi}\right)^2
\end{equation}
%
emerges which has a stable minimum at $\phi=0$. We conclude that
$\mathcal{R}+\mathcal{R}^2$ gravitation with metric $g_{\mu\nu}$ is
equivalent to $\overline{\mathcal{R}}$ gravitation with metric
$\overline{g}_{\mu\nu}$ plus a non-ghost real scalar field $\phi$ with a
fixed potential energy and a stable vacuum state. The property that
$\phi$ is non-ghost like is sufficiently important that we will present
yet another proof of this fact. This proof was first presented in
\cite{B}. If we expand the metric tensor as
%
\begin{equation}
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},
\end{equation}
%
then the part of action \eqref{R2} quadratic in $h_{\mu\nu}$ is given by
%
\begin{equation}
\mathcal{S}=\int{d^{4}x\left[\frac{1}{4}h^{\mu\nu}\left(\nabla^{2}
\left\{ P_{\mu\nu\rho\sigma}^{(2)} - 2P_{\mu\nu\rho\sigma}^{(0)}
\right\} + 2m^{-2}\left(\nabla^{2} \right)^2
P_{\mu\nu\rho\sigma}^{(0)}\right)h^{\rho\sigma}\right]},
\end{equation}
%
where $P_{\mu\nu\rho\sigma}^{(2)}$ and $P_{\mu\nu\rho\sigma}^{(0)}$ are
transverse projection operators for $h_{\mu\nu}$. Inverting the kernel
yields the propagator
%
\begin{align}
\Delta_{\mu\nu\rho\sigma}^{-1} &=
\left(\nabla^{2}P_{\mu\nu\rho\sigma}^{(2)}
+ 2m^{-2}\nabla^{2}\left[ \nabla^{2}-m^{2}\right]P_{\mu\nu\rho\sigma}
\right)^{-1}
\nonumber \\
&= \frac{1}{\nabla^{2}}\left(P_{\mu\nu\rho\sigma}^{(2)}-
\frac{1}{2}P_{\mu\nu\rho\sigma}^{(0)}\right)+\frac{1}{2(\nabla^{2}-
m^{2})}P_{\mu\nu\rho\sigma}^{(0)}.
\end{align}
%
The term proportional to $(\nabla^2)^{-1}$ corresponds to the usual two
helicity massless graviton. However, the term proportional to
$(\nabla^2-m^2)^{-1}$ represents the propagation of a real scalar field
with positive energy and, hence, not a ghost. This corresponds to the
results obtained using the auxiliary field above. We would like to point
out that there may be very interesting physics associated with the
scalar field $\phi$. For example, as emphasized in \cite{s1}, $\phi$ may
act as a natural inflaton in cosmology of the early universe.

Now let us consider Einstein gravity modified by quadratic terms
involving the Weyl tensor only. That is, consider the action
%
\begin{equation}
\mathcal{S} = \int{d^{4}x\sqrt{-g}\left(\mathcal{R}-\frac{1}{2}
m^{-2} C_{\mu\nu\alpha\beta} C^{\mu\nu\alpha\beta}\right)}.
\end{equation}
%
Using the identity
%
\begin{equation}
C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}= \GB +
2\left(\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}-
\frac{1}{3}\mathcal{R}^2\right),
\end{equation}
%
where $\GB$ is the topological Gauss-Bonnet combination defined in
\eqref{GB}, the action becomes
%
\begin{equation}
\mathcal{S}=\int{d^{4}x\sqrt{-g}\left(\mathcal{R}
-m^{-2}\left[\mathcal{R}_{\mu\nu} \mathcal{R}^{\mu\nu}-
\frac{1}{3}\mathcal{R}^2\right]\right)},
\label{Rmunu2}
\end{equation}
%
where we have dropped a total divergence. The fourth order equations of
motion can be reduced to second order equations by introducing an
auxiliary symmetric tensor field $\phi_{\mu\nu}$. Using this field, the
action can be written as
%
\begin{equation}
{\mathcal{S}}=\int{d^{4}x\sqrt{-g}\left({\mathcal{R}}-
G_{\mu\nu}\phi^{\mu\nu} + \frac{m^{2}}{4}
\left[\phi_{\mu\nu}\phi^{\mu\nu}-\phi^{2}\right]\right)},
\label{Rphimunu}
\end{equation}
%
where $\phi=\phi_{\mu\nu}g^{\mu\nu}$ and
$G_{\mu\nu}=\mathcal{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathcal{R}$ is
the Einstein tensor. Note that the $\phi_{\mu\nu}$ equation of motion is
%
\begin{equation}
\phi_{\mu\nu}=2m^{-2}\left(\mathcal{R}_{\mu\nu}-
\frac{1}{6}g_{\mu\nu}\mathcal{R}\right).
\end{equation}
%
Substituting this into \eqref{Rphimunu} gives back the original action
\eqref{Rmunu2}. As it stands, action \eqref{Rphimunu} is somewhat
obscure since the $G_{\mu\nu}\phi^{\mu\nu}$ term mixes $g_{\mu\nu}$ and
$\phi_{\mu\nu}$ at the quadratic level. They can, however, be decoupled
by a field redefinition. First write the above action as
%
\begin{equation}
\mathcal{S}=\int{d^{4}x \sqrt{-g}
\left(\left[\left\{1+\frac{1}{2}\phi\right\}g^{\mu\nu}
-\phi^{\mu\nu} \right]
\mathcal{R}_{\mu\nu}+\frac{m^{2}}{4}\left[\phi_{\mu\nu}
\phi^{\mu\nu}-\phi^{2}\right]\right)}.
\end{equation}
%
Now transform the metric as
\begin{equation}
\sqrt{-\overline{g}}\overline{g}^{\mu\nu}=\sqrt{-g}
\left(\left[1+\frac{1}{2}\phi \right]g^{\mu\nu} -
{\phi^{\mu}}_{\alpha}g^{\alpha\nu}\right),
\end{equation}
or, equivalently,
%
\begin{equation}
\begin{split}
g_{\mu\nu} &= \left(\det A\right)^{-1/2} {A_\mu}^\alpha
\overline{g}_{\alpha\nu}, \\
{A_\mu}^\alpha &= \left(1+\frac{1}{2}\phi\right){\delta_\mu}^{\alpha}
-{\phi_\mu}^{\alpha}.
\end{split}
\end{equation}
%
Under this transformation
%
\begin{equation}
\mathcal{R}_{\mu\nu}=\overline{\mathcal{R}}_{\mu\nu} -
\overline{\nabla}_{\mu} {C^{\alpha}}_{\alpha\nu} +
\overline{\nabla}_{\alpha} {C^{\alpha}}_{\mu\nu}+{C^{\alpha}}_{\mu\nu}
{C^{\beta}}_{\alpha\beta} -
{C^{\alpha}}_{\mu\beta}{C^{\beta}}_{\nu\alpha},
\end{equation}
%
where $\overline{\nabla}_{\lambda}\overline{g}_{\mu\nu}=0$ and
%
\begin{equation}
\begin{split}
{C^{\alpha}}_{\mu\nu} &= \frac{1}{2}\left(X^{-
1}\right)^{\alpha\beta}\left(\overline{\nabla}_{\mu}
X_{\nu\beta}+\overline{\nabla}_{\nu}X_{\mu\beta} -
\overline{\nabla}_\beta X_{\mu\nu}\right), \\
X_{\mu\nu} &= g_{\mu\nu}=\left(\det A\right)^{-1/2}{A_\mu}^\alpha
\overline{g}_{\alpha\nu}.
\end{split}
\end{equation}
%
Inserting these transformations into the above and dropping a total
divergence, the action becomes \cite{s2}
%
\begin{equation}
\mathcal{S}=\int{d^{4}x\sqrt{-
\overline{g}}\left[\overline{\mathcal{R}}+\overline{g}^
{\mu\nu}\left({C^{\alpha}}_{\mu\nu}{C^{\beta}}_{\beta\alpha}-
{C^{\alpha}}_{\mu\beta}
{C^{\beta}}_{\nu\alpha}\right)-\frac{m^{2}}{4}\left(\det A\right)^{-
1/2}\left(\phi_{\mu\nu}\phi^{\mu\nu}-\phi^{2}\right)\right]}.
\label{Rbarphimunu}
\end{equation}
%
Note that the action for $\phi_{\mu\nu}$ is a complicated non-linear
sigma model since $C=C(X)$ and $X=X(\phi)$. It is useful to consider the
kinetic energy part of the action expanded to quadratic order in
$\phi_{\mu\nu}$ only. It is found to be
%
\begin{equation}
\mathcal{S}^{\text{quad}}_\phi = \int d^4 x \sqrt{-\overline{g}} \left(
\frac{1}{4} \overline{\nabla}^\alpha \phi^{\mu\nu}
\overline{\nabla}_\alpha \phi_{\mu\nu}
-\frac{1}{2} \overline{\nabla}^\alpha \phi^{\mu\nu}
\overline{\nabla}_\mu \phi_{\nu\alpha} + \frac{1}{2}
\overline{\nabla}_\mu \phi^{\mu\nu} \overline{\nabla}_\nu \phi
-\frac{1}{4} \overline{\nabla}^\alpha \phi \overline{\nabla}_\alpha \phi
\right).
\end{equation}
%
This action is clearly the curved space generalization of the
Pauli-Fierz action for a spin-2 field except that every term has the
wrong sign! This implies, of course, that $\phi_{\mu\nu}$ propagates as
a ghost. It is interesting to note that the action \eqref{Rbarphimunu}
is invariant under the gauge transformation
%
\begin{equation}
\begin{split}
\phi'_{\mu\nu} &= \phi_{\mu\nu}+\overline{\nabla}_{(\mu}\xi_{\nu)}
-C^{\alpha}_{\mu\nu}\xi_{\alpha}, \\
 \overline{g}'_{\mu\nu} &= \left(1+\frac{1}{2}\phi'\right)^{-1}
\left[\frac {\det^{1/2}A'}{\det^{1/2}A} \left(1+\frac{1}{2}\phi\right)
\overline{g}_{\mu\nu} + \left(\phi'_{\mu\nu}-\phi_{\mu\nu}\right)
\right]. \\
\end{split}
\end{equation}
%
This insures that the above action describes a consistent coupling of a
spin-2 symmetric tensor field $\phi_{\mu\nu}$ to Einstein gravitation at
the full non-linear level \cite{s3}. We conclude, therefore, that
$\mathcal{R}+C^{2}$ gravitation with metric $g_{\mu\nu}$ is equivalent
to $\overline{\mathcal{R}}$ gravity with metric $\overline{g}_{\mu\nu}$
plus a ghost-like symmetric tensor field $\phi_{\mu\nu}$ with a
consistent non-linear coupling to gravity and a fixed potential energy.
The physics in the field $\phi_{\mu\nu}$ is obscured by its ghost-like
nature. However, its ghost nature can be altered by yet
higher-derivative terms, such as those one would expect to find
generated in superstring theories. Therefore, at long last, we turn to
our discussion of quadratic supergravitation in superstring theory.

\section{Superspace Formalism}

In the K{\"a}hler (Einstein frame) superspace formalism, the most
general Lagrangian for Einstein gravity, matter and gauge fields is
\begin{equation}
\mathcal{L}_E=-
\frac{3}{2\kappa^{2}}\int{d^{4}{\theta}E[K]}+\frac{1}{8}\int{d^{4}
\theta\frac{E}{R}f(\Phi_{i})_{ab}W^{{\alpha}a} {W_\alpha}^b} +
\text{h.c.},
\label{eq:fifth}
\end{equation}
where we have ignored the superpotential term which is irrelevant for
this discussion. The fundamental supergravity superfields are $R$ and
$W_{\alpha\beta\gamma}$, which are chiral, and $G_{\alpha{\dot\alpha}}$,
which is Hermitian. The bosonic $\mathcal{R}^2$, $(C_{mnpq})^{2}$ and
$(\mathcal{R}_{mn})^{2}$ terms are contained in the highest components
of the superfields $\bar{R}R$, $(W_{\alpha\beta\gamma})^{2}$ and
$(G_{\alpha\dot{\alpha}})^{2}$ respectively. One can also define the
superGauss-Bonnet combination
\begin{equation}
\SGB = 8(W_{\alpha\beta\gamma})^{2}+(\bar{\mathcal{D}}^2-
8R)(G_{\alpha\dot{\alpha}}^2-4\bar{R}R).
\end{equation}
The bosonic Gauss-Bonnet term is contained in the highest chiral
component of $\SGB$. It follows that the most general quadratic
supergravity Lagrangian is given by
\begin{equation}
\mathcal{L}_Q = \int d^4 \theta E \left[
\Sigma(\bar{\Phi}_i,\Phi_i) \bar{R} R +
\frac{1}{R} g(\Phi_i) (W_{\alpha\beta\gamma})^2
+ \Delta(\bar{\Phi}_i,\Phi_i) (G_{\alpha\dot{\alpha}})^2 + \text{h.c.}
\right].
\end{equation}

\section{(2,2) Symmetric $Z_{N}$ Orbifolds}

Although our discussion is perfectly general, we will limit ourselves to
orbifolds, such as $Z_{4}$, which have $(1,1)$ moduli only. The relevant
superfields are the dilaton, $S$, the diagonal moduli $T^{II}$, which
we'll denote as $T^{I}$, and all other moduli and matter superfields,
which we denote collectively as $\phi^{i}$. The associated K{\"a}hler
potential is
%
\begin{equation}
\begin{split}
K &= K_{0} + Z_{ij}\bar{\phi}^{i}\phi^{j} +
{\mathcal{O}}((\bar{\phi}\phi)^{2}), \\
\kappa^{2}K_{0} &= -\ln(S+\bar{S})-\sum{(T^{I}+\bar{T}^{I})}, \\
Z_{ij} &= \delta_{ij} \prod{(T^{I}+\bar{T}^{I})^{q^{i}_I}}.
\end{split}
\end{equation}
%
The tree level coupling functions $f_{ab}$ and $g$ can be computed
uniquely from amplitude computations and are given by
\begin{equation}
f_{ab} = \delta_{ab}k_{a}S, \qquad g = S.
\end{equation}
There is some ambiguity in the values of $\Delta$ and $\Sigma$ due to
the ambiguity in the definition of the linear supermultiplet. We will
take the conventional choice
\begin{equation}
\Delta=-S, \qquad \Sigma=4S.
\end{equation}
It follows that, at tree level, the complete $Z_{N}$ orbifold Lagrangian
is given by $\mathcal{L}=\mathcal{L}_E+\mathcal{L}_Q$ where
\begin{equation}
\mathcal{L}_Q = \frac{1}{4}\int{d^4\theta\frac{E}{R}S\, \SGB}+
\text{h.c.}
\end{equation}
Using this Lagrangian, we now compute the one-loop
moduli-gravity-gravity anomalous threshold correction \cite{E}. This
must actually be carried out in the conventional (string frame)
superspace formalism and then transformed to K{\"a}hler superspace
\cite{F}. We also compute the relevant superGreen-Schwarz graphs. Here
we will simply present the result. We find that
\begin{align}
\mathcal{L}_{\mathrm{massless}}^{\text{1-loop}} = &
\frac{1}{24(4\pi)^{2}} \sum \left[h^{I}\int{d^{4}\theta
(\bar{\mathcal{D}}^{2} - 8
R)\bar{R}R\frac{1}{\partial^{2}}D^{2}\ln(T^{I}+\bar{T}^{I})} \right.
\nonumber \\
&+ (b^{I}-8p^{I}) \int{d^{4}\theta(W_{\alpha\beta\gamma})^{2}
\frac{1}{\partial^{2}} D^{2}\ln(T^{I}+\bar{T}^{I})}
\nonumber \\
&\left. +p^{I}\int{d^{4}\theta(8(W_{\alpha\beta\gamma})^{2}+
(\bar{\mathcal{D}}^2 - 8R)((G_{\alpha\dot{\alpha}})^{2}-
4\bar{R}R))\frac{1}{\partial^{2}}
D^{2}\ln(T^{I}+\bar{T}^{I})}+ \text{h.c.} \right],
\end{align}
where
%
\begin{equation}
\begin{split}
h^I &= \frac{1}{12}(3\gamma {T}+3\vartheta_{T}q^{I}+\varphi), \\
b^I &= 21+1+n_{M}^{I}-\dim G + \sum (1+2q_{I}^{i})-24\delta_{GS}^{I}, \\
p^{I} &= -\frac{3}{8}\dim G-\frac{1}{8}-\frac{1}{24}\sum 1+\xi-3
\delta_{GS}^{I}.
\end{split}
\end{equation}
%
The coefficients $\gamma_{T}$ and $\vartheta_{T}$, which arise from
moduli loops, and $\varphi$ and $\xi$, which arise from gravity and
dilaton loops, are unknown. However, as we shall see, it is not
necessary to know their values to accomplish our goal. Now note that if
$h^{I}\neq0$ then there are non-vanishing $\mathcal{R}^2$ terms in the
superstring Lagrangian. If $b^{I}-8p^{I}\neq0$ then the Lagrangian has
$C^{2}$ terms. Coefficient $p^{I}\neq0$ merely produces a Gauss-Bonnet
term. With four unknown parameters what can we learn? The answer is, a
great deal! Let us take the specific example of the $Z_{4}$ orbifold. In
this case, the Green-Schwarz coefficients are known \cite{G}
\begin{equation}
\delta_{GS}^{1,2}=-30, \qquad \delta_{GS}^{3}=0,
\end{equation}
which gives the result
\begin{equation}
b^{1,2}=0, \qquad b^{3}=11\times24.
\end{equation}
Now, let us try to set the coefficients of the
$(C_{\mu\nu\alpha\beta})^{2}$ terms to zero simultaneously. This implies
that
\begin{equation}
b^{I}=8p^{I}
\end{equation}
for $I=1,2,3$ and therefore that
\begin{equation}
p^{1,2}=0, \qquad p^{3}=33.
\end{equation}
{}From this one obtains two separate equations for the parameter $\xi$
given by
\begin{equation}
\xi=\frac{3}{8}\dim G+\frac{1}{8}+\frac{1}{24}\sum 1-90,
\end{equation}
for $I=1,2$ and
\begin{equation}
\xi=\frac{3}{8}\dim G+\frac{1}{8}+\frac{1}{24}\sum 1,
\end{equation}
for $I=3$. Clearly these two equations are incompatible and, hence, it
is impossible to have all vanishing $(C_{\mu\nu\alpha\beta})^{2}$ terms
in the 1-loop corrected Lagrangian of $Z_{4}$ orbifolds. We find that
the same results hold in other orbifolds as well.

\section{Conclusion}

We conclude that non-vanishing $(C_{\mu\nu\alpha\beta})^{2}$ terms are
generated by light field loops in the $N=1$, $D=4$ Lagrangian of $Z_{N}$
orbifold superstrings. It is conceivable that loops containing the heavy
tower of states might cancel these terms, but we find no reason, be it
duality or any other symmetry, for this to be the case \cite{H}. This is
presently being checked by a complete genus-one string calculation
\cite{I}. We conjecture that cancellation will not occur. Unfortunately,
since $\gamma_{T}$, $\vartheta_{T}$ and $\varphi$ are unknown, we can
say nothing concrete about the existence of $\mathcal{R}^2$ terms in the
Lagrangian. This issue will also be finally resolved in the complete
superstring calculation.

\begin{thebibliography}{99}

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\end{document}

