\section{Introduction}
\label{intro}

The phenomenon of nucleation drives first order phase transitions in many applications in physics, most notably in particle physics, condensed matter physics, quantum field theory and cosmology. The semiclassical analysis of the rate of such a nucleation process was pioneered by Langer \cite{langer}, who identified a semiclassical saddle point solution that gives the dominant exponential contribution to the rate, with a prefactor to the exponential given by the quantum fluctuations about this classical solution. The nucleation rate is given by the quantum mechanical rate of decay of a metastable "false" vacuum, $\phi_-$, into the "true" vacuum, $\phi_+$. Decay proceeds by the nucleation of expanding bubbles of true vacuum within the metastable false vacuum \cite{langer,kobzarev,stone,coleman,affleck,voloshin1,weinberg1,voloshin-erice}. This picture may also be extended to finite temperature field theory \cite{linde,affleck2}. Computing the semiclassical prefactor requires the computation of the determinant of the differential operator associated with quantum fluctuations about the classical solution. This is a technically difficult problem. Thus, it is not possible to compute the fluctuation prefactor analytically, and so various approximation techniques have been developed. The most widely-studied is the so-called "thin-wall" approximation, in which the bubble wall thickness is small compared to the bubble radius \cite{stone,coleman,voloshin1}. This provides an elegant physical picture of nucleation and in this limit certain parts of the decay rate computation may be done analytically. A limited amount is also known analytically concerning the expansion beyond the leading thin-wall limit, in two dimensions \cite{konoplich,voloshin1,kiselev},  three dimensions \cite{munster,voloshin2+1}, and four dimensions \cite{kr}. Other important approaches to metastable decay use direct lattice simulations \cite{alford},  the average effective action \cite{strumia}, and phase shift techniques \cite{moss}.

The point of this present paper is to reduce the calculation of the decay rate to a straightforward numerical computation, without relying on any such approximation or expansion. The starting point for our approach is an extremely elegant and simple method, due to Gelfand and Yaglom \cite{gy}, for computing the determinant of a one-dimensional differential operator. This technique is ideal for numerical implementation. However, its naive generalization to higher dimensions is divergent \cite{forman}. This is because in higher dimensions renormalization is important, and so we must regulate and renormalize the determinant. We present an analytic method of doing this using an angular momentum cutoff, and apply it to the problem of false vacuum decay. (This method has previously been used to compute the quark mass dependence of the fermion determinant for quarks in the presence of an instanton background \cite{dunne}.)
A related approach to false vacuum decay, also based on the Gelfand-Yaglom formula, has been developed recently by Baacke and Lavrelashvili \cite{baacke}, and we comment in Section \ref{ren} on the similarities and differences between our approaches.

Consider the Euclidean classical action
\begin{eqnarray}
S_{\rm cl}[\phi]=\int d^4 x\left( \frac{1}{2}(\partial_\mu \phi)^2+U(\phi)\right)\quad ,
\label{action}
\end{eqnarray}
where the potential $U(\phi)$ has two nondegenerate classical minima, $\phi_-  < \phi_+$, with $U(\phi_-) > U(\phi_+)$. For quantitative computations, and comparison with previous work, we consider the standard quartic potential \cite{kobzarev,coleman,konoplich,kiselev}, whose form is illustrated in Figure \ref{fig1}:
\begin{eqnarray}
U(\phi)=\frac{\lambda}{8}\left(\phi^2-a^2\right)^2-\frac{\epsilon}{2 a}\left(\phi-a\right)\quad .
\label{pot1}
\end{eqnarray}
The parameter $\epsilon$ represents a constant external source breaking the degeneracy of the double well potential.
\begin{figure}[h]
\includegraphics{fig1.eps}
\caption{Field potential $U(\phi)$ showing the true and false vacua, $\phi_+$ and $\phi_-$, respectively.}
\label{fig1}
\end{figure}
In four dimensional spacetime, the mass dimensions of the couplings are: $[\lambda]=0$, $[a]=1$, and $[\epsilon]=4$. We choose $a>0$, $\lambda>0$ and $\epsilon>0$, in which case the  two minima are $\phi_\pm=\pm a(1\pm \frac{\epsilon}{2\lambda a^4}+\dots ).$ Note that
\begin{eqnarray}
U\left(\phi_-\right)-U\left(\phi_+\right)=\epsilon \left[1+O\left( \frac{\epsilon}{\lambda a^4}\right)\right]
\quad ,
\label{eps}
\end{eqnarray}
so for $\epsilon\ll \lambda a^4$, we see that  $\epsilon$ has the physical interpretation as the potential energy difference between the two classical vacua. This small $\epsilon$ limit is known as the ``thin-wall'' limit \cite{coleman} because in this case the bubbles of true vacuum within the false vacuum have thin walls compared to their radius.

Expanding the field $\phi$ about the false vacuum $\phi_-$
\begin{eqnarray}
\phi=\phi_- +\varphi\quad ,
\label{expansion}
\end{eqnarray}
and keeping terms up to dimension four, we find the potential, which is often considered directly in the literature \cite{baacke}
\begin{eqnarray}
U(\varphi)=\frac{m^2}{2}\varphi^2-\eta \varphi^3 +\frac{\lambda}{8}\varphi^4\quad .
\label{pot2}
\end{eqnarray}
Here $m^2$ and $\eta$ are related to the original couplings by
\begin{eqnarray}
m^2=\frac{\lambda}{2}\left(3\phi_-^2-a^2\right) \quad ; \quad \eta=\frac{\lambda}{2}|\phi_- | \quad .
\label{meta}
\end{eqnarray}
In order to describe semiclassical tunneling, it is useful to rescale the field $\varphi$ and the spacetime coordinates as
\begin{eqnarray}
\bar{x}= m x\quad ;  \quad\varphi =\frac{m^2}{2\eta} \Phi
\label{rescale}
\end{eqnarray}
Then the classical action in terms of these dimensionless quantities is :
\begin{eqnarray}
S_{\rm cl}[\Phi]=\left(\frac{m^2}{4\eta^2}\right) \int d^4 \bar{x}\left[\frac{1}{2}(\bar{\partial}_\mu \Phi)^2+\frac{1}{2}\Phi^2-\frac{1}{2}\Phi^3+\frac{\alpha}{8}\Phi^4\right]\quad .
\label{action2}
\end{eqnarray}
The overall factor is determined by the dimensionless parameter
\begin{eqnarray}
\beta=\frac{m^2}{4\eta^2}=\frac{1}{\lambda}\left(1-\frac{\epsilon}{2\lambda a^4}+\dots\right)\quad .
\label{beta}
\end{eqnarray}
For our semiclassical tunneling analysis, we assume $\beta\gg 1$. The quartic coupling strength in (\ref{action2})  is determined by the dimensionless quantity
\begin{eqnarray}
\alpha=\frac{\lambda m^2}{4\eta^2}=1-\frac{\epsilon}{2\lambda a^4}+\dots   \quad ,
\label{alpha}
\end{eqnarray}
which tends to 1 in the "thin-wall" limit where $\epsilon\to 0$. The dimensionless parameter $\alpha$ determines the shape of the potential, and its deviation from $1$ is a measure of the vacuum energy difference relative to the barrier height. Figure \ref{fig2} shows some plots, for various values of $\alpha$, of the dimensionless potential
\begin{eqnarray}
U(\Phi)=\frac{1}{2}\Phi^2-\frac{1}{2}\Phi^3+\frac{\alpha}{8}\Phi^4\quad .
\label{dimpot}
\end{eqnarray}
\begin{figure}[h]
\includegraphics[scale=0.75]{fig2.eps}
\caption{Plots of the rescaled potential, $U(\Phi)=\frac{1}{2}\Phi^2-\frac{1}{2}\Phi^3+\frac{\alpha}{8}\Phi^4$, for  $\alpha=0.6, 0.7, 0.8 , 0.9, 0.99.$ As $\alpha$ approaches 1, the vacua become degenerate.}
\label{fig2}
\end{figure}

The false vacuum decay rate per unit volume and unit time is denoted $\gamma$, and its one-loop expression is \cite{langer,kobzarev,stone,coleman,affleck,voloshin1,weinberg1,voloshin-erice}
\begin{eqnarray}
\gamma= \left(\frac{S_{\rm cl}[\Phi_{\rm cl}]}{2\pi}\right)^2 \left|\frac{{\rm det}^\prime \left(-\Box + U^{\prime\prime}(\Phi_{\rm cl})\right)}{{\rm det}\left(-\Box +U^{\prime\prime}(\Phi_-)\right)}\right|^{-1/2}\, e^{-S_{\rm cl}[\Phi_{\rm cl}]-\delta_{\rm ct}S[\Phi_{\rm cl}]}\quad ,
\label{rate}
\end{eqnarray}
where the prime on the determinant means that the zero modes (corresponding to translational invariance) are removed. Here $\Phi_{\rm cl}$ is a classical solution known as the ``bounce'' solution \cite{coleman}, defined below, and the prefactor terms in (\ref{rate}) correspond to quantum fluctuations about this bounce solution.  The second term in the exponent, $\delta_{\rm ct}S[\Phi_{\rm cl}]$, denotes the counterterms needed for renormalization. The computational challenge is to evaluate the rate $\gamma$ given a particular form of the classical potential. For our particular quartic model this corresponds to computing the rate for various values of $\beta$ and $\alpha$, the dimensionless parameters defined above in (\ref{beta}) and (\ref{alpha}).

In the language of quantum field theory \cite{stone,coleman,weinberg1}, $\gamma$ is half the imaginary part of the generating functional of the connected Green's functions with the constant external source, $-\frac{\epsilon}{2a}$, divided by the spacetime volume factor, and so we are essentially computing the renormalized effective action for this system \cite{cw,salam}. It is not possible to compute this tunneling rate $\gamma$ analytically. Indeed, it is not even possible to find the classical bounce solution, $\Phi_{\rm cl}$, analytically, let alone the quantum fluctuations about this classical solution. We present here a simple numerical technique to compute $\gamma$ for general forms of $U(\phi)$. This technique involves a combination of an analytical computation and a numerical computation. The analytical part of the computation is related to the regularization and renormalization of the one loop effective action. The numerical part is elementary, and can be implemented straightforwardly and efficiently in Mathematica.

In Section \ref{bounce} we describe how to compute the classical bounce solution numerically with very high precision. In Section \ref{prefactor} we describe how to compute the determinant prefactor arising from quantum fluctuations about the classical bounce solution. In Section \ref{renormalization} we explain how to regularize and renormalize our answer using the angular momentum cutoff regularization and renormalization scheme developed previously in \cite{dunne}. Our approach is related to an elegant technique presented recently by Baacke and Lavrelashvili \cite{baacke}, and in Section \ref{renormalization} we also compare and contrast these two methods. In Section \ref{conclusions} we conclude with some general comments about possible further applications, and in the Appendix we present the derivation of a simple new formula for the contribution to the decay rate $\gamma$ coming from the zero modes.

\section{Computing the Classical Bounce Solution}
\label{bounce}

The first step in computing the false vacuum decay rate $\gamma$ is to find the classical bounce solution, $\Phi_{\rm cl}(r)$, which is an $0(4)$-symmetric stationary point of the classical Euclidean action, with $\Phi_{\rm cl}(r)$ interpolating between the false and true vacuum as $r$ goes from 0 to $\infty$ \cite{coleman,spherical}. The action evaluated on the classical bounce solution determines the leading exponential factor in (\ref{rate}), and the quantum fluctuations about the classical bounce solution lead to the determinant prefactors in (\ref{rate}).

The bounce $\Phi_{\rm cl}(r)$ solves the nonlinear ordinary differential equation
\begin{eqnarray}
-\Phi_{\rm cl}^{\prime\prime} -\frac{3}{r}\Phi_{\rm cl}^\prime +\Phi_{\rm cl}-\frac{3}{2}\Phi_{\rm cl}^2+\frac{\alpha}{2} \Phi_{\rm cl}^3=0
\label{bounceeq}
\end{eqnarray}
with boundary conditions
\begin{eqnarray}
\Phi_{\rm cl}^\prime(0)&=&0
\label{bca}\\
\Phi_{\rm cl}(r)&\to& \Phi_-\equiv 0 \quad,\quad {\rm as}\quad r\to\infty\quad .
\label{bcb}
\end{eqnarray}
It is not known how to find $\Phi_{\rm cl}(r)$  analytically in any nontrivial field theory. Much work has been done in the thin-wall approximation, which in practice means finding $\Phi_{\rm cl}$ as an expansion about the point $\alpha=1$, where the two vacua are degenerate. The point of this present paper is to reduce the calculation of the tunneling rate $\gamma$ to a straightforward numerical computation, without relying on any such approximation or expansion. Thus, we begin by  determining $\Phi_{\rm cl}(r)$ numerically.

To obtain extremely good precision we use a combination of both forward and backward shooting to compute $\Phi_{\rm cl}(r)$. In forward shooting we numerically integrate (\ref{bounceeq}) [using 4th order Runge-Kutta], starting at $r=0$, and we adjust the initial value $\Phi_0\equiv \Phi_{\rm cl}(0)$ until the second boundary condition (\ref{bcb}) is satisfied to a certain level of accuracy. Since one cannot start exactly at $r=0$, we begin at some very small $r=r_0$ and use a Taylor expansion to write
\begin{eqnarray}
\Phi_{\rm cl}(r_0)&=&\Phi_0+\frac{1}{16} r_0^2\left(2\Phi_0-3\Phi_0^2+\alpha \Phi_0^3\right)\nonumber\\
\Phi_{\rm cl}^\prime(r_0)&=&\frac{1}{8} r_0\left(2\Phi_0-3\Phi_0^2+\alpha \Phi_0^3\right)\quad .
\label{forwardstart}
\end{eqnarray}
\begin{figure}[h]
\includegraphics[scale=0.9]{fig3.eps}
\caption{Plots of the bounce solution $\Phi_{\rm cl}(r)$ for various values of $\alpha$: $ \alpha= 0.5$, $0.9$, $0.95$, $0.96$, $0.97$, $0.98$, $0.99$, with the plateau ending farther to the right for increasing $\alpha$. Observe that as $\alpha\to 1$, the sharp falloff in $\Phi_{\rm cl}$ occurs at $r\sim \frac{1}{1-\alpha}$, and $\Phi_{\rm cl}$ can be approximated by a step function.}
\label{fig3}
\end{figure}

We adjust the parameter $\Phi_0$ until the large $r$ boundary condition (\ref{bcb}) is satisfied. In backward shooting we begin the numerical integration [also using 4th order Runge-Kutta] at some very large $r=R$, with starting conditions
\begin{eqnarray}
\Phi_{\rm cl}(R)=\Phi_\infty\, \frac{K_1(R)}{R}\quad , \quad \Phi_{\rm cl}^\prime(R)=-\Phi_\infty\, \frac{K_2(R)}{R}\quad ,
\label{backwardstart}
\end{eqnarray}
and adjust the parameter $\Phi_\infty$ until the $r=0$ boundary condition (\ref{bca}) is satisfied.

We first use forward shooting, which produces a good estimate of $\Phi_0$. Then using this bounce solution we estimate $\Phi_\infty$, and use this as a starting point for shooting backwards, which further refines this value of $\Phi_\infty$ and also leads to a refined value of $\Phi_0$. It is a simple matter to obtain 20 - 30 decimal precision for each of $\Phi_\infty$ and $\Phi_0$ in this way.
Some bounce profiles for various values of the coupling parameter $\alpha$ are shown in Figure \ref{fig3}.

\begin{figure}[h]
\includegraphics[scale=0.9]{fig4.eps}
\caption{Plots of the fluctuation potential $U^{\prime\prime}(\Phi_{\rm cl}(r))$ for various values of $\alpha$ : $\alpha= 0.5$,  $0.9$, $0.95$, $0.96$, $0.97$, $0.98$, $0.99$, with the binding well of the potential appearing farther to the right for increasing $\alpha$. Observe that as $\alpha\to 1$, the potential $U^{\prime\prime}(\Phi_{\rm cl}(r))$ is localized at $r\sim \frac{1}{1-\alpha}$, and is approximated well by the analytic form in \protect{(\ref{thinwall})}.}
\label{fig4}
\end{figure}

Given the bounce solution, $\Phi_{\rm cl}(r)$, the corresponding radial fluctuation potential is
\begin{eqnarray}
U^{\prime\prime}(\Phi_{\rm cl}(r))
= 1-3\Phi_{\rm cl}(r)+\frac{3\alpha}{2}\Phi_{\rm cl}^2(r)\quad .
\label{flucpot}
\end{eqnarray}
Clearly, since $\Phi_{\rm cl}(r)$ is only known numerically, the fluctuation potential $U^{\prime\prime}(\Phi_{\rm cl}(r))$ is also only known numerically.
Figure \ref{fig4} shows some profiles of this fluctuation potential, corresponding to the various bounce profiles in Figure \ref{fig3}. Note that this fluctuation potential is highly localized, with the localization radius depending strongly on $\alpha$. As $\alpha\to 1$, the fluctuation potential binds states at radius $r\sim \frac{1}{1-\alpha}$, the radius of the bubble of false vacuum. Moreover, in this thin-wall limit, the minimum of the potential approaches $-1/2$, and outside the binding well it approaches 1. In fact,  as noted in Langer's original work \cite{langer}, in this limit $U^{\prime\prime}(\Phi_{\rm cl}(r))$ is approximated well by the analytic potential
\begin{eqnarray}
U^{\prime\prime}(\Phi_{\rm cl}(r))\sim 1-\frac{3}{2}\, {\rm sech}^2\left(\frac{1}{2}\left[r-\frac{1}{1-\alpha}\right]\right) \quad .
\label{thinwall}
\end{eqnarray}

\section{Computing the Determinant Prefactor}
\label{prefactor}

Since the bounce solution $\Phi_{\rm cl}(r)$ is a function of $r$, the fluctuation operator $[-\Box +U^{\prime\prime}(\Phi_{\rm cl})]$ can be decomposed into partial waves, with (dimensionless) radial operators
\begin{eqnarray}
{\mathcal M}_{(l)}=-\frac{d^2}{d r^2}-\frac{3}{r}\frac{d}{d r} +\frac{l(l+2)}{r^2}+1+V(r)\quad ,
\label{radial}
\end{eqnarray}
of degeneracy $(l+1)^2$, with $l=0, 1, 2, \dots$. Here the radial potential $V(r)$ is equal to the fluctuation potential (\ref{flucpot}) with its large radius asymptotic value, $1$, subtracted:
\begin{eqnarray}
V(r)= -3\Phi_{\rm cl}(r)+\frac{3\alpha}{2}\Phi_{\rm cl}^2(r)\quad .
\label{radialpot}
\end{eqnarray}
Likewise, the free fluctuation operator $[-\Box +U^{\prime\prime}(\Phi_-)]$ can be decomposed into partial waves, with radial operators
\begin{eqnarray}
{\mathcal M}_{(l)}^{\rm free}=-\frac{d^2}{d r^2}-\frac{3}{r}\frac{d}{d r} +\frac{l(l+2)}{r^2}+1\quad ,
\label{radialfree}
\end{eqnarray}
also of degeneracy $(l+1)^2$.

For each $l$, the ratio of the determinants of ${\mathcal M}_{(l)}$ and ${\mathcal M}_{(l)}^{\rm free}$ can be computed efficiently and precisely using the Gelfand-Yaglom method \cite{gy,levit,erice,forman,kirsten,kirstenbook,kleinertbook}. This result states that for radial operators of the form (\ref{radial},\ref{radialfree}),
\begin{eqnarray}
\frac{{\rm det}({\mathcal M}_{(l)})}{{\rm det}({\mathcal M}_{(l)}^{\rm free})}=\left(\lim_{R\to\infty} \left[\frac{\psi_{(l)}(R)}{\psi^{\rm free}_{(l)} (R)}\right]\right)^{(l+1)^2}\quad .
\label{theorem}
\end{eqnarray}
Here  $\psi_{(l)}$ and $\psi_{(l)}^{\rm free}$ are regular solutions of
\begin{eqnarray}
{\mathcal M}_{(l)}\, \psi_{(l)}&=&0\nonumber\\
{\mathcal M}_{(l)}^{\rm free}\, \psi_{(l)}^{\rm free}&=&0\quad ,
\label{ode}
\end{eqnarray}
with the {\it same} leading behavior at $r=0$. By regularity we can choose this small $r$ behavior to be
\begin{eqnarray}
\psi_{(l)}&\sim& r^l \quad , \quad r\to 0 \nonumber\\
\psi_{(l)}^{\rm free}&\sim& r^l \quad , \quad r\to 0\quad .
\label{r0bc}
\end{eqnarray}
This normalization choice fixes the free solution to be
\begin{eqnarray}
\psi_{(l)}^{\rm free}(r)= 2^{l+1} (l+1)! \frac{I_{l+1}(r)}{r}\quad
\label{freesol}
\end{eqnarray}
where $I_{l+1}(r)$ is the modified Bessel function.

In fact, a numerical improvement comes from realizing that
both $\psi_{(l)}(r)$ and $\psi^{\rm free}_{(l)} (r)$ grow exponentially fast at large $r$, so it is numerically better to integrate directly the ratio \cite{baacke}
\begin{eqnarray}
T_{(l)}(r)\equiv \frac{\psi_{(l)}(r)}{\psi_{(l)}^{\rm free}(r)}\quad ,
\label{ratio}
\end{eqnarray}
which satisfies the equation
\begin{eqnarray}
T_{(l)}^{\prime\prime}+\left(\frac{1}{r}+ 2\frac{I_{l+1}^\prime(r)}{I_{l+1}(r)}\right) T_{(l)}^\prime-V(r) T_{(l)}=0\quad ,
\label{tequation}
\end{eqnarray}
with simple {\it initial value boundary conditions}:
\begin{eqnarray}
T_{(l)}(0)=1\quad ; \quad T_{(l)}^\prime(0)=0 \quad .
\label{ivbc}
\end{eqnarray}
Then the result (\ref{theorem}) is recast as
\begin{eqnarray}
\frac{{\rm det}({\mathcal M}_{(l)})}{{\rm det}({\mathcal M}_{(l)}^{\rm free})}=\left(T_{(l)}(\infty)\right)^{(l+1)^2}\quad .
\label{ttheorem}
\end{eqnarray}

We stress that the result (\ref{ttheorem}) provides a remarkably simple technique for computing the determinant of a radial differential operator. It does not require any detailed knowledge of the spectrum of the operator whose determinant is being computed, nor does it require evaluating and numerically integrating the associated phase shift. Furthermore, the result (\ref{ttheorem}) is ideally suited to numerical evaluation, as initial value boundary conditions are straightforward to implement numerically.

There are three different types of eigenvalue of the fluctuation operator, each having a different role physically and mathematically.

\begin{itemize}

\item
\underline{Negative Eigenvalue Mode : $(l=0)$}

The lowest eigenvalue mode in the $l=0$ sector is a negative eigenvalue mode of the fluctuation operator, and is responsible for the instability leading to decay. It can be identified with homogeneous swelling and shrinking of the bubble of true vacuum. This mode contributes a factor to the decay rate $\gamma$ related to the {\it absolute value}  of the determinant of the $l=0$ fluctuation operator \cite{langer,coleman}. This determinant is computed by numerically integrating (\ref{tequation}) for $l=0$ :
\begin{eqnarray}
\left |\frac{{\rm det}{\mathcal M}_{(l=0)}}{{\rm det}{\mathcal M}_{(l=0)}^{\rm free}}\right |^{-1/2}=\left |T_{(0)}(\infty)\right |^{-1/2} \quad .
\label{l=0}\end{eqnarray}

\item
\underline{Zero Eigenvalue Modes : $(l=1)$}

In the $l=1$ sector there is a four-fold degenerate zero eigenvalue of the fluctuation operator. Physically, these four zero modes are the Goldstone modes associated with the breaking of translational invariance. Integrating over the corresponding collective coordinates \cite{gervais} produces the factors of $\frac{S_{\rm cl}}{2\pi}$ in (\ref{rate}). In computing the rate $\gamma$, we need the determinant of the fluctuation operator with the zero modes removed \cite{langer,coleman}. We have found the following simple new formula for the $l=1$ sector prefactor contribution (see Appendix A for a derivation):
\begin{eqnarray}
\left(\frac{S_{\rm cl}[\varphi_{\rm cl}]}{2\pi}\right)^2 \left(\frac{{\rm det}^\prime{\mathcal M}_{(l=1)}}{{\rm det}{\mathcal M}_{(l=1)}^{\rm free}}\right)^{-1/2} = \left[\frac{\pi}{2} \Phi_\infty \left(\Phi_0-\frac{3}{2}\Phi_0^2+\frac{\alpha}{2}\Phi_0^3\right)\right]^2\quad ,
\label{zmformula}
\end{eqnarray}
where $\Phi_0=\Phi_{\rm cl}(0)$ is the bounce solution evaluated at the origin, and $\Phi_\infty$ is the coefficient of $K_1(r)/r$ in the large $r$ behavior of $\Phi_{\rm cl}(r)$, as in (\ref{backwardstart}). The advantage of the result (\ref{zmformula}) is that it is expressed entirely in terms of the asymptotic behavior of the classical bounce solution $\Phi_{\rm cl}(r)$. This asymptotic information is already generated in the precise numerical determination of the bounce solution, so no further computation is needed to extract the zero mode contribution to the prefactor.

\begin{figure}[h]
\includegraphics[scale=0.9]{fig5.eps}
\caption{Plots of $T_{(l)}(r)$ for $l=0$, $l=1$, and $l=2$. These plots are for $\alpha=0.5$. Note that the asymptotic value, $T_{(l)}(\infty)$, is negative for $l=0$, zero for $l=1$, and positive for $l=2$, confirming the discussion in the text concerning the three different types of modes. }
\label{fig5}
\end{figure}

\item
\underline{Positive Eigenvalue Modes : $(l\geq 2)$}

For $l\geq 2$, the fluctuation operator has positive eigenvalues, each of degeneracy $(l+1)^2$. These modes correspond to deformations of the bubble shape and thickness. For each $l$, the associated radial determinant is computed simply by numerical integration of (\ref{tequation}) with the initial value boundary conditions (\ref{ivbc}):
\begin{eqnarray}
\left(\frac{{\rm det}{\mathcal M}_{(l)}}{{\rm det}{\mathcal M}_{(l)}^{\rm free}}\right)^{-1/2} = \left[T_{(l)}(\infty)\right ]^{-(l+1)^2/2} \quad .
\label{lgeq2}
\end{eqnarray}
\end{itemize}

Figure \ref{fig5} shows plots of $T_{(l)}(r)$ as a function of $r$, for $l=0$, $l=1$, and $l=2$. These plots are for $\alpha=0.5$. Observe that $T_{(l)}(r)$ approaches very rapidly its asymptotic value, $T_{(l)}(\infty)$, for $r$ outside the range of the binding well of the fluctuation potential (compare with Figure \ref{fig4} for $\alpha=0.5$). Also note that the asymptotic value, $T_{(l)}(\infty)$, is negative for $l=0$, zero for $l=1$, and positive for $l=2$, confirming the discussion above of the three different types of modes.

\begin{figure}[ht]
\includegraphics[scale=0.9]{fig6.eps}
\caption{The $l$ dependence of $\ln T_{(l)}(\infty)$ for three different values of $\alpha$: $\alpha=0.01$ (top curve), $\alpha=0.7$ (middle curve), and $\alpha=0.9$ (bottom curve). The dots show the values of $\ln T_{(l)}(\infty)$ evaluated numerically using (\ref{tequation}), while the solid lines show the WKB prediction (\ref{largel}) of the leading large $l$ behavior. Notice the excellent agreement for $l\geq 20$.}
\label{fig6}
\end{figure}
For each partial wave with $l\geq 2$, the radial determinant, $\left[T_{(l)}(\infty)\right ]^{(l+1)^2}$, is finite and simple to evaluate.  In the discussion of renormalization it proves convenient to consider the {\it logarithm} of the determinant factors appearing in the rate (\ref{rate}):
\begin{eqnarray}
-\frac{1}{2}\ln \left(\frac{{\rm det}{\mathcal M}_{(l)}}{{\rm det}{\mathcal M}_{(l)}^{\rm free}}\right) = -\frac{1}{2}\left(l+1\right)^2 \ln T_{(l)}(\infty)    \quad .
\label{lnform}
\end{eqnarray}
For large $l$ we can use the radial WKB analysis of \cite{wkb,dunne} to find the leading behavior:
\begin{eqnarray}
\ln T_{(l)}(\infty)
\sim \frac{1}{(l+1)}\left[\frac{1}{2}\int_0^\infty dr\, r \, V(r)\right]+O\left(\frac{1}{(l+1)^3}\right)\quad .
\label{largel}
\end{eqnarray}
The leading large $l$ behavior (\ref{largel}) is illustrated in Figure \ref{fig6} for several different values of the quartic coupling parameter $\alpha$. Notice the very good agreement between the numerical results  for $\ln T_{(l)}(\infty)$ [solid points] and the WKB estimate [curves from (\ref{largel})]. Further subleading corrections to (\ref{largel}) are computed below in the next Section -- see (\ref{wkbnlo}).

\section{Regularization and Renormalization}
\label{renormalization}

The large $l$ behavior (\ref{largel}) means that the formal sum of contributions (\ref{lnform}) to $-\ln \gamma$,
\begin{eqnarray}
\frac{1}{2}\sum_{l=2}^\infty (l+1)^2 \ln \left[T_{(l)}(\infty)\right]\quad ,
\label{rawsum}
\end{eqnarray}
is quadratically divergent. This divergence should not be too surprising, as we have neither regulated nor renormalized the determinant prefactor in the expression (\ref{rate}) for $\gamma$. In the language of quantum field theory, we need  to compute the {\it renormalized} one-loop effective action for this interacting scalar field theory \cite{stone,coleman,voloshin1,weinberg1}. Here we will apply the {\it angular momentum cutoff regularization and renormalization technique} developed in \cite{dunne} which was successfully applied to the computation of the quark mass dependence of the fermion determinant in an instanton background in QCD.

\subsection{Regularization}

The first step is to introduce a regulator mass $\mu$ using the proper-time representation of the logarithm :
\begin{eqnarray}
\left[\ln  \left(\frac{{\rm det}\,{\mathcal M}_{(l)}}{{\rm det}\, {\mathcal M}_{(l)}^{\rm free}}\right)\right]_{\rm reg}=-(l+1)^2 \int_0^\infty \frac{ds}{s}\, \left(\mu^2 s\right)^\varepsilon\, {\rm tr}\left[e^{-s {\mathcal M}_{(l)}}- e^{-s {\mathcal M}_{(l)}^{\rm free}}\right]\quad ,
\label{mu}
\end{eqnarray}
where the space-time dimension is extended to $4-2\varepsilon$, and we have explicitly extracted the degeneracy factor $(l+1)^2$ from the trace of the radial operators ${\mathcal M}_{(l)}$ and ${\mathcal M}_{(l)}^{\rm free}$ defined in (\ref{radial}) and (\ref{radialfree}). We then split the sum over $l$ of these regulated terms into two parts:
\begin{eqnarray}
\frac{1}{2}\sum_{l=2}^\infty \left[\ln \left(\frac{{\rm det}\,{\mathcal M}_{(l)}}{{\rm det}\,{\mathcal M}_{(l)}^{\rm free}}\right)\right]_{\rm reg}
&=& \frac{1}{2}\sum_{l=0}^L\left[\ln \left(\frac{{\rm det}\,{\mathcal M}_{(l)}}{{\rm det}\,{\mathcal M}_{(l)}^{\rm free}}\right)\right]_{\rm reg}
 +\frac{1}{2}\sum_{l=L+1}^\infty\left[\ln \left(\frac{{\rm det}\,{\mathcal M}_{(l)}}{{\rm det}\,{\mathcal M}_{(l)}^{\rm free}}\right)\right]_{\rm reg}\nonumber\\\nonumber\\
 &\equiv& \Sigma_1+\Sigma_2
\label{split}
\end{eqnarray}
Here  $L$ is a large but finite integer (in practice, we take $L$ to be of the order of 50 to 100, depending on the value of $\alpha$). Since $\Sigma_1$ is a finite
sum we can safely remove the regulator $\mu$ and write
\begin{eqnarray}
\Sigma_1= \frac{1}{2} \ln \left |T_{(0)}(\infty)\right | -2 \ln \left[\frac{\pi}{2} \Phi_\infty \left(\Phi_0-\frac{3}{2}\Phi_0^2+\frac{\alpha}{2}\Phi_0^3\right)\right] +
\frac{1}{2}\sum_{l=2}^L (l+1)^2 \ln T_{(l)}(\infty)
\label{sum1}
\end{eqnarray}
The first term in $\Sigma_1$ corresponds to the negative mode contribution (\ref{l=0}), the second term is from the zero mode contribution (\ref{zmformula}), and the final sum corresponds to the contributions in partial waves with $2\leq l\leq L$.
Each term $T_{(l)}(\infty)$ in this last sum is evaluated numerically by solving (\ref{tequation}). The second sum in (\ref{split})
\begin{eqnarray}
\Sigma_2\equiv -\frac{1}{2}\sum_{l=L+1}^\infty (l+1)^2 \int_0^\infty \frac{ds}{s}\, \left(\mu^2 s\right)^\varepsilon\, {\rm tr}\left[e^{-s {\mathcal M}_{(l)}}- e^{-s {\mathcal M}_{(l)}^{\rm free}}\right] \quad ,
\label{sum2}
\end{eqnarray}
cannot be computed numerically using (\ref{tequation}) because of the infinite sum and the presence of the regulator mass $\mu$. Instead, we use radial WKB, which is a good approximation for large $l$, to compute {\it analytically} the leading large $L$ behavior of  $\Sigma_2$. This is a straightforward computation using the results of \cite{wkb,dunne}. The result is:
\begin{eqnarray}
\Sigma_2 &=&\frac{1}{2}\left\{ -\frac{(L+1)(L+2)}{4}\int_0^\infty dr \, r\, V(r)+\frac{\ln L}{8}\int_0^\infty dr\, r^3\, V(V+2)\right.\nonumber\\
&&\left. -\frac{1}{16} \int_0^\infty dr\,\left(\frac{1}{\varepsilon}+2-\gamma_E+\ln \frac{\mu^2 r^2}{4}\right)\, r^3\, V(V+2)\right\}+0\left(\frac{1}{L}\right) \quad .
\label{wkbresult}
\end{eqnarray}
Here $\gamma_E$ is Euler's constant.
Note that this WKB computation reveals large $L$ divergences going like $L^2$, $L$ and $\ln L$, in addition to a term which is finite in the large $L$ limit. These last two terms are exactly of the form expected for renormalization, as is discussed in the next section.

The most important observation at this stage is that the large $L$ divergences of $\Sigma_2$ cancel {\it exactly} the large $L$ divergence found numerically in $\Sigma_1$, leaving a finite answer. Indeed, comparing (\ref{wkbresult}) with (\ref{largel}) we see immediately that the quadratic divergence cancels. Extending (\ref{largel}) to include 2nd order WKB contributions proves the cancellation of the sub-leading divergences also.

\subsection{Renormalization}
\label{ren}

In standard perturbative renormalization theory, the self-interacting scalar field theory described
by the action (\ref{action}) involves two renormalization parameters, $\lambda$ and
$a^2$, together with the wave function renormalization $Z$. The renormalization counter
terms can be found by replacing those parameters with $\lambda+ \delta \lambda$, and
$a^2 + \delta a^2$. The one-loop approximations can be found in quantum field theory text books (see, {\it e.g.} \cite{peskin}) :
\begin{eqnarray}
\delta \lambda&=&\frac{9 \lambda^2}{32 \pi^2} \left(\frac{1}{\varepsilon} -\gamma_E\right)  \\
\delta (\lambda a^2)&=&\frac{3 \lambda^2 a^2}{32 \pi^2} \left( \frac{1}{\varepsilon}-\gamma_E\right),
\end{eqnarray}
and $Z=1$, in the $\overline{{\rm MS}}$ renormalization scheme using dimensional regularization.
With this replacement we identify the renormalization counterterms as
\begin{eqnarray}
\delta_{ct} S&=&\int d^4 x\left[\frac{ \delta \lambda}{8} \left(\phi^2-a^2\right)^2
-\frac{\lambda \delta a^2}{4} \left(\phi^2-a^2\right) \right]   \label{ct1}\\
&=&\frac{1}{32} \int_0^\infty dr \left( \frac{1}{\epsilon}-\gamma_E \right) r^3 V(V+2).
\label{ct2}
\end{eqnarray}
In obtaining (\ref{ct2}) from (\ref{ct1}) we have used  (\ref{expansion}), (\ref{meta}), (\ref{rescale}), (\ref{alpha}), and (\ref{radialpot}) to translate from the original physical field $\phi$, and couplings $\lambda$ and $a^2$, to the field $\Phi$ and coupling $\alpha$ which correspond to expanding the field about the false vacuum $\phi_-$. Notice the appearance in (\ref{ct2}) of the combination $V(V+2)$ when the counterterm is expressed in tems of these fields.
Furthermore, note that the counterterm only involves the pole-term $\frac{1}{\varepsilon}$ and $\gamma_E$,
independent of any other parameters, for instance the mass parameter,
in the  $\overline{{\rm MS}}$ renormalization scheme.  We identify this counterterm (\ref{ct2}) within the WKB result (\ref{wkbresult}), with precisely the right coefficient and structure. We choose this particular  $\overline{{\rm MS}}$ renormalization prescription in order to compare with the work of Baacke and Lavrelashvili \cite{baacke}, who also use an $\overline{{\rm MS}}$ prescription.

Combining $\Sigma_1$ and $\Sigma_2$ with the above renormalization counter term $\delta_{ct}S$, we get the
one-loop effective action:
\begin{eqnarray}
\Gamma_{\overline{{\rm MS}}}&=& \frac{1}{2} \ln \left |T_{(0)}(\infty)\right |-2 \ln \left[\frac{\pi}{2} \Phi_\infty \left(\Phi_0-\frac{3}{2}\Phi_0^2+\frac{\alpha}{2}\Phi_0^3\right)\right]  \nonumber\\
&& +\frac{1}{2} \lim_{L\to \infty}\left\{\sum_{l=2}^L (l+1)^2 \ln \left(T_{(l)}(\infty)\right)-\frac{(L+1)(L+2)}{4}\int_0^\infty dr \, r\, V(r)\right. \nonumber\\
&&\left.    +\frac{\ln L}{8}\int_0^\infty dr\, r^3\, V(V+2)\right\}-\frac{1}{16} \int_0^\infty dr\,\left(1+\ln \frac{r}{2}\right)\, r^3\, V(V+2)
\label{dm-answer}
\end{eqnarray}
Here, the first two terms correspond to the $l=0$ contribution (for the negative mode), computed using (\ref{l=0}), and the $l=1$ contribution (for the zero modes), computed using (\ref{zmformula}).
The renormalized effective action $\Gamma_{\overline{{\rm MS}}}$ is finite, and we obtain excellent convergence (accelerated by third order Richardson extrapolation \cite{bender}) for $L$ of the order of 50 to 100, depending on the value of $\alpha$. This result for $\Gamma_{\overline{{\rm MS}}}$ is plotted in Figure \ref{fig7} as a function of $\alpha$.
\begin{figure}
\includegraphics{fig7.eps}
\caption{Plot of the $\overline{{\rm MS}}$ renormalized effective action (\ref{dm-answer}) as a function of $\alpha$.}
\label{fig7}
\end{figure}

We can alternatively express (\ref{dm-answer}) with the $L$ dependent WKB subtraction terms included as subtractions {\it inside} the $l$ sum :
\begin{eqnarray}
\Gamma_{\overline{{\rm MS}}}&=& \frac{1}{2} \ln \left |T_{(0)}(\infty)\right |-2 \ln \left[\frac{\pi}{2} \Phi_\infty \left(\Phi_0-\frac{3}{2}\Phi_0^2+\frac{\alpha}{2}\Phi_0^3\right)\right]  \nonumber\\
&& +\frac{1}{2} \sum_{l=2}^\infty (l+1)^2 \left\{ \ln \left(T_{(l)}(\infty)\right)-\frac{\frac{1}{2}\int_0^\infty dr \, r\, V(r)}{(l+1)}+\frac{\frac{1}{8}\int_0^\infty dr\, r^3\, V(V+2)}{(l+1)^3}\right\} \nonumber\\
&&  -\frac{3}{4}\int_0^\infty dr \, r\, V(r)  +\frac{1}{16} \int_0^\infty dr\, r^3\, V(V+2)\,\left(\frac{1}{2}-\gamma_E-\ln \frac{r}{2}\right)
\label{dm-answer2}
\end{eqnarray}
Here we have simply used
\begin{eqnarray}
\sum_{l=2}^L (l+1)=\frac{1}{2}(L+1)(L+2)-3 \quad ;\quad \sum_{l=2}^L \frac{1}{(l+1)}\sim \ln L-\gamma_E
\label{sums}
\end{eqnarray}
Thus, (\ref{dm-answer2}) shows that we can extend to next-to-leading order the large $l$ behavior  in (\ref{largel}) of $\ln \left(T_{(l)}(\infty)\right)$, the logarithm of the determinant of the radial operators in (\ref{radial}) :
\begin{eqnarray}
 \ln \left(T_{(l)}(\infty)\right)\sim \frac{\frac{1}{2}\int_0^\infty dr \, r\, V}{(l+1)}-\frac{\frac{1}{8}\int_0^\infty dr\, r^3\, V(V+2)}{(l+1)^3}+O\left(\frac{1}{(l+1)^5}\right) \quad .
 \label{wkbnlo}
 \end{eqnarray}
We have confirmed numerically that the remainder is indeed $O\left(\frac{1}{(l+1)^5}\right)$. Note that (\ref{wkbnlo}) provides a simple closed-form expression for the large $l$ behavior of $\ln \left(T_{(l)}(\infty)\right)$, and this expression is {\it local} in the fluctuation potential $V(r)$.

It is instructive to compare (\ref{dm-answer2}) with an expression obtained by Baacke and Lavreshavili \cite{baacke} for the same quantity (note that the collective coordinate term, $-2\ln \left(\frac{S_{\rm cl}}{2\pi}\right)$, is separated out in \cite{baacke}), also using the $\overline{{\rm MS}}$ renormalization scheme :
\begin{eqnarray}
\Gamma_{BL}&=& \frac{1}{2} \ln \left |T_{(0)}(\infty)\right | +4 \frac{1}{2} \ln T_{(1)}^\prime(\infty)+\frac{1}{2}\left(A^{(1)}_{\rm fin}-\frac{1}{2}A^{(2)}_{\rm fin}\right) \nonumber\\
&& +\frac{1}{2} \sum_{l=2}^\infty (l+1)^2 \left\{ \ln \left(T_{(l)}(\infty)\right)-h_l^{(1)}(\infty)-\left[h_l^{(2)}(\infty)-\frac{1}{2}\left(h_l^{(1)}(\infty)\right)^2\right]\right\}
\label{bl-answer}
\end{eqnarray}
The second term corresponds to the fourfold degenerate zero modes, and is computed in \cite{baacke} by evaluating the slope at zero with an additional parameter added to the operator in order to displace the determinant from zero. This agrees numerically with our expression in (\ref{zmformula}), [allowing for the collective coordinate term, $-2\ln \left(\frac{S_{\rm cl}}{2\pi}\right)$], but is more cumbersome to evaluate. The subtractions $h_l^{(1)}(\infty)$ and $h_l^{(2)}(\infty)$ in (\ref{bl-answer}) are associated with first and second order Feynman diagrams \cite{baacke}, and are given explicitly as the asymptotic values of $h_l^{(1)}(r)$ and $h_l^{(2)}(r)$, satisfying the differential equations
\begin{eqnarray}
h_l^{(1)\,\prime\prime}+\left(\frac{1}{r}+ 2\frac{I_{l+1}^\prime(r)}{I_{l+1}(r)}\right) h_l^{(1)\,\prime}-V(r)&=&0 \nonumber\\
h_l^{(2)\,\prime\prime}+\left(\frac{1}{r}+ 2\frac{I_{l+1}^\prime(r)}{I_{l+1}(r)}\right) h_l^{(2)\,\prime}-V(r) h_l^{(1)}&=&0 \quad ,
\label{hequation}
\end{eqnarray}
with {\it initial value boundary conditions}: $h_l(0)=0$, and $h_l^\prime(0)=0$. The remaining terms in (\ref{bl-answer}) are Born approximation terms
\begin{eqnarray}
A^{(1)}_{\rm fin}&=&-\frac{1}{8}\int_0^\infty dr\, r^3\, V(r)\nonumber\\
A^{(2)}_{\rm fin}&=&\frac{1}{128\pi^4}\int_0^\infty dq\, q^3\,\left| \tilde{V}(q)\right|^2 \left(2-\sqrt{q^2+4}\,\ln\left[\frac{\sqrt{q^2+4}+q}{\sqrt{q^2+4}-q}\right]\right)\quad ,
\label{bla}
\end{eqnarray}
where $\tilde{V}(q)$ is the Fourier transform of the fluctuation potential $V(r)$.

Note that Baacke and Lavrelashvili's expression (\ref{bl-answer}) also gives a {\it finite} answer for the renormalized effective action. This means that we have another, quite different, expression for the large $l$ behavior of $\ln \left(T_{(l)}(\infty)\right)$:
\begin{eqnarray}
\ln \left(T_{(l)}(\infty)\right)\sim h_l^{(1)}(\infty)+\left[h_l^{(2)}(\infty)-\frac{1}{2}\left(h_l^{(1)}(\infty)\right)^2\right]+\dots
\label{bl-largel}
\end{eqnarray}
Comparing with (\ref{wkbnlo}), we deduce that the difference between these large $l$ behaviors should agree to order $O\left(\frac{1}{(l+1)^4}\right)$. We have verified this numerically, and in fact we find that  there is a nonzero difference at $O\left(\frac{1}{(l+1)^5}\right)$ :
\begin{eqnarray}
&&\left\{h_l^{(1)}(\infty)+\left[h_l^{(2)}(\infty)-\frac{1}{2}\left(h_l^{(1)}(\infty)\right)^2\right]\right\}\nonumber\\
&&\qquad\qquad  -\left\{\frac{\frac{1}{2}\int_0^\infty dr \, r\, V(r)}{(l+1)}-\frac{\frac{1}{8}\int_0^\infty dr\, r^3\, V(V+2)}{(l+1)^3}\right\}\sim O\left(\frac{1}{(l+1)^5}\right)
\label{diff}
\end{eqnarray}

Similarly, comparing the terms {\it outside} the $l$ sums in (\ref{dm-answer2}) and (\ref{bl-answer}), we see that these terms in (\ref{dm-answer2}) are {\it local} in the fluctuation potential $V(r)$, while $A^{(2)}_{\rm fin}$ in (\ref{bl-answer}) is {\it nonlocal} in $V(r)$. Nevertheless, the total quantities $\Gamma_{\overline{MS}}$ and $\Gamma_{BL}$ are {\it equal} (allowing for the collective coordinate term, $-2\ln \left(\frac{S_{\rm cl}}{2\pi}\right)$, that is separated out in \cite{baacke}), and we have confirmed this equality numerically. This means that (\ref{dm-answer2}) and (\ref{bl-answer}) are different ways of regulating the $l$ sum, and since each is renormalized using $\overline{MS}$, they should indeed be equal. In other words, in (\ref{dm-answer2}) and (\ref{bl-answer}), the different subtractions inside the $l$ sums are compensated for by different finite pieces outside the $l$ sums, in such a way that the total agrees. This is very similar to behavior found in the computation of mass shifts in soliton theories (see Appendix B of \cite{jaffe}). It is also a highly nontrivial check on both expressions. But, although the expressions (\ref{dm-answer2}) and (\ref{bl-answer}) are equal, in purely pragmatic computational terms the local expression (\ref{dm-answer2}) is considerably simpler. First, the zero mode part is more easily and more accurately computed using (\ref{zmformula}), since it only requires the asymptotic properties of the (already computed) bounce solution $\Phi_{\rm cl}(r)$. Second, one does not need to solve the differential equations (\ref{hequation}) to find $h_l^{(1)}(\infty)$ and $h_l^{(2)}(\infty)$; instead, the large $l$ behavior of $\ln T_{(l)}(\infty)$ is given by local expressions in $V(r)$. Third, one does not need to compute (numerically) the Fourier transform of the fluctuation potential, nor integrate the result over all momentum, as is necessary to compute $A_2$ in (\ref{bla}). Instead, all subtraction terms in (\ref{dm-answer2}) are purely {\it local} in the fluctuation potential $V(r)$.

In a series of papers studying the thin-wall approximation, Konoplich and Rubin \cite{konoplich,kr} have computed this same quantity with a related $\overline{\rm MS}$ renormalization prescription, and obtained the following result in the $\alpha\to 1$  limit [we have translated their notation to match ours]:
\begin{eqnarray}
\Gamma_{\rm KR}\sim \frac{9}{32}\, \left[1-\frac{2\pi}{9\sqrt{3}}\right]\,\frac{1}{\left(1-\alpha\right)^3}  +O\left(\frac{1}{1-\alpha}\right) \quad .
\label{kr-answer}
\end{eqnarray}
\begin{figure}[ht]
\includegraphics{fig8.eps}
\caption{Plot of the $\overline{{\rm MS}}$ renormalized effective action, $\Gamma_{\overline{{\rm MS}}}$, in (\ref{dm-answer}), rescaled by  a factor of $(1-\alpha)^3$, as a function of $\alpha$ in the thin-wall approximation limit as $\alpha\to 1$. The solid dots denote our data points for $\alpha$ in steps of $0.01$ from $0.9$ to $0.99$, and the solid  curve is an interpolation through these points. The horizontal line is the leading term of the Konoplich-Rubin result (\ref{kr-answer}), also rescaled by a factor of $(1-\alpha)^3$. Note the excellent agreement at $\alpha=1$.}
\label{fig8}
\end{figure}
Their renormalization prescription is such that this leading term can be compared directly to ours. The subleading terms in (\ref{kr-answer}) are scheme dependent, except for a universal logarithmic term \cite{konoplich,kr}. The agreement with our result (\ref{dm-answer2}) is striking.
Because of the divergence as $\alpha\to 1$, for plotting purposes it is useful to extract the overall factor of $\frac{1}{(1-\alpha)^3}$. In Figure \ref{fig8} we plot our result (\ref{dm-answer2}), rescaled by a factor of $(1-\alpha)^3$, compared with the Konoplich-Rubin thin-wall approximation answer (\ref{kr-answer}), also rescaled by a factor of $(1-\alpha)^3$. The intercept at $\alpha=1$ is given by (\ref{kr-answer}) to be
\begin{eqnarray}
\frac{9}{32}\, \left[1-\frac{2\pi}{9\sqrt{3}}\right]\approx 0.16788888 \quad,
\label{intercept}
\end{eqnarray}
which matches perfectly the $\alpha\to 1$ limit of our result (\ref{dm-answer2}), as can be clearly seen in Figure \ref{fig8}.

The $\overline{{\rm MS}}$ prescription provides simple renormalization structures,
but we can also use a ``physical'' renormalization prescription. For example, we may impose the following normalization condition to the one-loop effective potential,
\begin{eqnarray}
\frac{d^2}{d \phi^2} \left(U_{\rm cl}+U_{\rm eff}\right)|_{\phi=\phi_{1}}= \frac{\lambda }{2}(3\phi_1^2-a^2),
\qquad  \frac{d^4}{d \phi^4} \left(U_{\rm cl}+U_{\rm eff}\right)|_{\phi=\phi_{1}}= 3\lambda.
\label{normal}
\end{eqnarray}
In (\ref{normal}), $ \phi_1\equiv\phi_{-}+\delta \phi_{-} $ is the solution of the one-loop modified field equation
\begin{eqnarray}
\frac{d}{d \phi} \left(U_{\rm cl}+U_{\rm eff}\right)|_{\phi=\phi_1}=0.
\end{eqnarray}
These normalization conditions can be satisfied when we change the renormalized parameters by finite
amounts as $ \lambda+\delta\lambda \to \lambda+\delta\lambda+\delta_{fin}\lambda$, and
$ \lambda a^2 +\delta(\lambda a^2) \to \lambda a^2 +\delta(\lambda a^2)+\delta_{fin}(\lambda a^2)$.

\section{Conclusions}
\label{conclusions}

In conclusion, we have presented a simple and practical technique for evaluating the prefactor determinant in the expression for the metastable decay rate in scalar field theories. The technique for computing the determinant is based on the Gelfand-Yaglom formula \cite{gy} for computing the determinant of a one-dimensional (here, radial) differential operator in terms of the asymptotic boundary value of an associated differential equation with initial value boundary conditions. This technique is extremely easy to implement for any given partial wave, but the naive sum over partial waves is divergent. Thus the direct application of the Gelfand-Yaglom formula to higher dimensions is not possible. However, this divergence can be regulated in various ways. Here we propose using the {\it angular momentum cutoff} regularization and renormalization scheme, which has been used previously to compute the explicit mass dependence of the fermion determinant in QCD for massive quarks in an instanton background \cite{dunne}. In this approach the contribution of the low partial waves is computed numerically using the Gelfand-Yaglom formula, while the contribution of the high partial waves is computed analytically using radial WKB, which is a good approximation for large $l$. The merging of these two parts involves renormalization and we have illustrated our technique using an  $\overline{{\rm MS}}$ scheme in order to compare with the work of Baacke and Lavrelashvili \cite{baacke}, who also use the Gelfand-Yaglom technique, also with an $\overline{{\rm MS}}$ prescription, but with a different regularization technique. We have shown that the two techniques agree, and have argued that the one presented here is somewhat easier to implement as it is purely {\it local} in the numerically determined fluctuation potential. The conversion to other renormalization shemes can be done using conventional field theory techniques, and corresponds to including finite polynomial terms in the effective potential. In an appendix we have derived a simple new formula for the $l=1$ determinant with the zero modes removed, solely in terms of the asymptotic values of the bounce solution.

The goal of this work has been to reduce the computation of field theoretic  one loop fluctuation determinants to a straightforward numerical exercise. As a by-product, we have found in (\ref{dm-answer2}) a simple extension of the Gelfand-Yaglom result to higher dimensional radially separable problems.
There are many possible further applications of this technique, as there are many semiclassical problems where the classical solution, about which one is computing the quantum fluctuations, has radial symmetry. Closely related possible applications include: (i) metastable decay in theories with more than one field \cite{kusenko}, where analytic and approximate approaches to the prefactor are quite difficult, but a direct numerical approach might be more useful; (ii) models in dimensions other than 4, which have been studied in and beyond the thin-wall approximation \cite{voloshin1,nicole,munster,voloshin2+1}; (iii) an extension to finite temperature, where the high temperature limit is essentially a dimensionally reduced 3d radial problem \cite{linde,affleck2}, but for intermediate temperatures the explicit summation over Matsubara modes is necessary.
This technique for computing precisely the fluctuation contribution may also be useful for a set of fascinating questions concerning the validity of Langer's homogeneous nucleation picture itself, as well as the semiclassical approximation \cite{strumia2}. More generally, our technique for extending the Gelfand-Yaglom formula to higher dimensions can also be applied to other symmetric semiclassical configurations such as vortices and monopoles (instantons were considered already in \cite{dunne}).

\vskip .5cm
{\bf Acknowledgments:}  We thank Holger Gies for helpful discussions. GD thanks the US DOE for support through the grant DE-FG02-92ER40716, and gratefully acknowledges the hospitality and support of Choonkyu Lee and the Center for Theoretical Physics at Seoul National University where part of this work was done.

\section{Appendix A : Zero Mode Contribution}
\label{zeromode}

In this Appendix we present a derivation of the expression (\ref{zmformula}) for the factor contributed by the $l=1$ zero modes. We adapt a method of McKane and Tarlie \cite{mckanetarlie} (see also \cite{kirsten,kirstenbook,kleinert}). First, add a small quantity, $k^2$, to the operator possessing the zero mode, and to the corresponding free operator (although this latter addition is not important in the end). Then for small $k^2$ the existence of the four zero modes for ${\mathcal M}$ [we suppress the $(l=1)$ subscript] implies that
\begin{eqnarray}
\frac{{\rm det}({\mathcal M}+k^2)}{{\rm det}({\mathcal M}^{\rm free}+k^2)}\sim (k^2)^4 \, \frac{{\rm det}^\prime{\mathcal M}}{{\rm det}\,{\mathcal M}^{\rm free}}\quad , \quad k^2\to 0 \quad .
\label{addk}
\end{eqnarray}
The result (\ref{theorem}) means we can evaluate the LHS of (\ref{addk}) for arbitrary but small $k^2$ as
\begin{eqnarray}
\frac{{\rm det}({\mathcal M}+k^2)}{{\rm det}({\mathcal M}^{\rm free}+k^2)}=\left(\lim_{R\to\infty} \left(\frac{\psi_{k^2}(R)}{\psi_{k^2}^{\rm free} (R)}\right)\right)^4\quad ,
\label{zmratio}
\end{eqnarray}
where $\psi_{k^2}$ satisfies
\begin{eqnarray}
-\psi_{k^2}^{\prime\prime}-\frac{3}{r}\psi_{k^2}^\prime +\left(\frac{3}{r^2}+1+k^2+V(r)\right)\psi_{k^2}&=&0\quad ,\nonumber\\
\psi_{k^2}(0)=0\quad ; \quad \psi^\prime_{k^2}(0)=1\, . &&
\label{psik}
\end{eqnarray}
Note that for $l=1$ we can express the $r\to 0$ boundary conditions (\ref{r0bc}) in this initial value form.
The function $\psi_{k^2}^{\rm free}$ satisfies the same equation and boundary conditions as in (\ref{psik}), but with the potential $V$ set to 0. So, one possible approach, as suggested in \cite{baacke}, to computing ${\rm det}^\prime$ is to compute the derivative of $\lim_{R\to\infty} \left(\frac{\psi_{k^2}(R)}{\psi_{k^2}^{\rm free} (R)}\right)$  as $k^2\to 0$. But a more direct and accurate method is as follows.

Define the function $\psi_0(r)$ to be the solution of (\ref{psik}) with $k^2=0$. Then by elementary integration by parts it follows that
\begin{eqnarray}
0&=&\int_0^R dr\, r^3 \psi_0 \left\{-\frac{1}{r^3}\left(r^3 \psi_{k^2}^\prime\right)^\prime +\left(\frac{3}{r^2}+1+k^2+V(r)\right)\psi_{k^2}\right\}\nonumber\\
&=&\left[r^3\left(\psi_0^\prime \psi_{k^2}-\psi_0 \psi_{k^2}^\prime\right)\right]^R_0 +k^2 \int_0^R dr\, r^3 \psi_0 \psi_{k^2} \quad .
\label{byparts}
\end{eqnarray}
Applying the boundary conditions at $r=0$ we obtain
\begin{eqnarray}
\psi_{k^2}(R)\left(1-\left[\frac{\psi_0(R)}{\psi_0^\prime(R)}\right] \left[\frac{\psi_{k^2}^\prime(R)}{\psi_{k^2}(R)}\right]\right)=k^2\frac{ \int_0^R dr\, r^3 \psi_0 \psi_{k^2}}{(-R^3 \psi_0^\prime(R))} \quad .
\label{magic}
\end{eqnarray}
The important observation now is that $\psi_0(r)$ is actually the normalizable zero mode of ${\mathcal M}$, and decreases exponentially  as $e^{-r}$ at large $r$. On the other hand, for arbitrarily small but nonzero $k^2$, the solution $\psi_{k^2}(r)$ {\it increases} exponentially as $e^{r\sqrt{1+k^2}}$ at large $r$. Thus at large $R$ and arbitrarily small but nonzero $k^2$, the identity (\ref{magic}) implies that the leading $k^2$ dependence at small $k^2$ is :
\begin{eqnarray}
\psi_{k^2}(R)\sim \frac{1}{2}\,k^2\frac{\int_0^R dr\, r^3 \psi_0^2 }{(- R^3 \psi_0^\prime(R))}\quad , \quad k^2\to 0 \quad .
\label{magic2}
\end{eqnarray}
Then (\ref{zmratio}) leads to the following expression for the determinant with zero mode removed:
\begin{eqnarray}
\frac{{\rm det}^\prime{\mathcal M}}{{\rm det}\,{\mathcal M}^{\rm free}}=\left(\frac{\int_0^\infty dr\, r^3 \psi_0^2}{\lim_{R\to\infty} \left[-2 R^3 \psi_0^\prime(R) \psi_0^{\rm free}(R)\right]}\right)^4\, .
\label{zeroremoved}
\end{eqnarray}

To simplify this general expression further, recall that the zero mode $\psi_0$ can be expressed in terms of the classical bounce solution :
\begin{eqnarray}
\psi_0(r)=\frac{\Phi_{\rm cl}^\prime(r)}{\Phi_{\rm cl}^{\prime\prime}(0)} \quad .
\label{bouncezm}
\end{eqnarray}
This has two important implications. First, the numerator on the RHS of (\ref{zeroremoved}) can be expressed in terms of the classical bounce action :
\begin{eqnarray}
\int_0^\infty dr\, r^3 \psi_0^2&=&\frac{1}{2\pi^2 \left(\Phi_{\rm cl}^{\prime\prime}(0)\right)^2} \int d^4 x \left(\phib^\prime\right)^2\nonumber\\
&=& \frac{1}{2\pi^2 \left(\Phi_{\rm cl}^{\prime\prime}(0)\right)^2}\, 4\, S_{\rm cl}[\Phi_{\rm cl}]
\label{innerproduct}
\end{eqnarray}
Since $\frac{{\rm det}^\prime\,{\mathcal M}}{{\rm det}\,{\mathcal M}^{\rm free}}$ is raised to the power $-1/2$ on the LHS of (\ref{zmformula}), we see that the $S_{\rm cl}[\Phi_{\rm cl}]$ factors cancel.
The second important implication of (\ref{bouncezm}) is that the leading large $R$ behavior of $\psi_0^\prime(R)$ is determined by the coefficient $\Phi_\infty$ in (\ref{backwardstart}). Since $\psi_0^{\rm free}(r)=8 I_2(r)/r$, the denominator in (\ref{zeroremoved}) is
\begin{eqnarray}
\left[-2 R^3 \psi_0^\prime(R) \psi_0^{\rm free}(R)\right] &\sim& -2 R^3 \frac{\Phi_\infty}{\Phi_{\rm cl}^{\prime\prime}(0)} \left(\frac{K_1(r)}{r}\right)^{\prime\prime} \bigg|_{r=R} \left(\frac{8 I_2(R)}{R}\right)\nonumber\\
&\to & -8 \frac{\Phi_\infty}{\Phi_{\rm cl}^{\prime\prime}(0)}\quad , \quad R\to\infty\quad .
\label{zmr}
\end{eqnarray}
Thus we obtain the simple formula
\begin{eqnarray}
\left(\frac{S_{\rm cl}[\Phi_{\rm cl}]}{2\pi}\right)^2 \left(\frac{{\rm det}^\prime{\mathcal M}_{(l=1)}}{{\rm det}\,{\mathcal M}_{(l=1)}^{\rm free}}\right)^{-1/2} = \left[2 \pi\,\Phi_\infty \Phi_{\rm cl}^{\prime\prime}(0)\right]^2 \, .
\label{zmf}
\end{eqnarray}
The final result (\ref{zmformula}) follows by noting that $\Phi_{\rm cl}^{\prime\prime}(0)$ may be expressed in terms of $\Phi_{\rm cl}(0)$ using the bounce differential equation (\ref{bounceeq}) and the boundary condition (\ref{bca}).

We stress that the final expression (\ref{zmformula}) only requires knowing the two asymptotic constants, $\Phi_\infty$ and $\Phi_{\rm cl}(0)$, each of which is determined to great precision in the course of finding the classical bounce solution numerically, as described in Section \ref{prefactor}.


\section{Introduction}

The Spin Foam model of the BF theory \cite{BF} for the gauge group $SO(4,C)$
is discussed. The Barrett-Crane model \cite{BCReimmanion} of the
$SO(4,C)\ $general relativity is systematically derived. The $SO(4,C)\ $%
Barrett-Crane model has been used to develop the concept of reality conditions
for the Barrett-Crane models \cite{RealitySpinFoam}.

The asymptotic limit of the $SO(4,C)$ general relativity is discussed. The
asymptotic limit \cite{PonzanoReggeModel}, \cite{AngMomClassLimit} is
controlled by the $SO(4,C)$ Regge calculus which unifies the Regge calculus
\cite{ReggeCalc} theories for all the real general relativity cases for the
four dimensional signatures.

\section{Spin foam of the $SO(4,C)$ BF model}

Consider a four dimensional submanifold $M$. Let $A$ be a $SO(4,C)$ connection
1-form and $B^{ij}$ a complex bivector valued $2$-form on $M$. Let $F$ be the
curvature 2-form of the connection $A$. Then I define a real continuum
BF\ theory action \cite{RealitySpinFoam},%
\begin{equation}
S_{BF}(A,B_{ij},\bar{A},\bar{B}_{ij})=\operatorname{Re}\int_{M}B\wedge F,
\label{BFaction}%
\end{equation}
where $A,B_{ij}$ and their complex conjugates are considered as independent
free variables.

The Spin foam model for the $SO(4,C)$ BF\ theory action can be derived
from\ the discretized BF action by using the path integral quantization as
illustrated in Ref:\cite{ooguriBFderv} for compact groups. Let $\Delta$ be a
simplicial manifold obtained by a triangulation of $M$. Let $g_{e}\in SO(4,C)$
be the parallel propagators associated with the edges (three-simplices)
representing the discretized connection. Let $H_{b}=%
{\textstyle\prod_{e\supset b}}
g_{e}$ be the holonomies around the bones (two-simplices) in the four
dimensional matrix representation of $SO(4,C)$ representing the curvature. Let
$B_{b}$ be the $4\times4$ antisymmetric complex matrices corresponding to the
dual Lie algebra of $SO(4,C)$ corresponding to the discrete analog of the $B$
field. Then the discrete BF action is
\[
S_{d}=\operatorname{Re}\sum_{b\in M}tr(B_{b}\ln H_{b}),
\]
which is considered as a function of the $B_{b}$'s and $g_{e}$'s. Here $B_{b}$
the discrete analog of the $B$ field are $4\times4$ antisymmetric complex
matrices corresponding to dual Lie algebra of $SO(4,C)$. The $\ln$ maps from
the group space to the Lie algebra space. The trace is taken over the Lie
algebra indices. Then the\ quantum partition function can be calculated using
the path integral formulation as,%

\[
Z_{BF}(\Delta)=\int\prod_{b}dB_{b}d\bar{B}_{b}\exp(iS_{d})\prod_{e}dg_{e}%
\]%
\begin{equation}
=\int\prod_{b}\delta(H_{b})\prod_{e}dg_{e}, \label{eq.del}%
\end{equation}
where $dg_{e}$ is the invariant measure on the group $SO(4,C)$. The invariant
measure can be defined as the product of the bi-invariant measures on the left
and the right $SL(2,C)$ matrix components. Please see appendix A and B\ for
more details. Similar to the integral measure on the $B$'s an explicit
expression for the $dg_{e}$ involves product of conjugate measures of complex coordinates.

Now consider the identity
\begin{equation}
\delta(g)=\frac{1}{64\pi^{8}}\int d_{\omega}tr(T_{\omega}(g))d\omega,
\label{eq.del.exp}%
\end{equation}
where the $T_{\omega}(g)$ is a unitary representation of $SO(4,C),$ where
$\omega=$ $\left(  \chi_{L},\chi_{R}\right)  $ such that $n_{L}+n_{R}$ is
even, $d_{\omega}=\left\vert \chi_{L}\chi_{R}\right\vert ^{2}$. The details of
the representation theory are discussed in appendix B. The integration with
respect to $d\omega$ in the above equation is interpreted as the summation
over the discrete $n$'s and the integration over the continuous $\rho$'s.

By substituting the harmonic expansion for $\delta(g)$ into the equation
(\ref{eq.del}) we can derive the spin foam partition of the $SO(4,C)\ BF$
theory as explained in Ref:\cite{JCBintro} or Ref:\cite{ooguriBFderv}. The
partition function is defined using the $SO(4,C)$ intertwiners and the
$\{15\omega\}$ symbols.

The relevant intertwiner for the $BF\ $spin foam is%
\[
i_{e}=%
\raisebox{-0.3814in}{\includegraphics[
height=0.7628in,
width=0.518in
]%
{4dintw.eps}%
}%
.
\]
The nodes where the three links meet are the Clebsch-Gordan coefficients of
$SO(4,C)$. The Clebsch-Gordan coefficients of $SO(4,C)$ are just the product
of the Clebsch-Gordan coefficients of the left and the right handed $SL(2,C)$
components. The Clebsch-Gordan coefficients of $SL(2,C)$ are discussed in the
references \cite{IMG} and \cite{NaimarckClebsch}.

The quantum amplitude associated with each simplex $s$ is given below and can
be referred to as the $\{15\omega\}$ symbol,%

\[
\{15\omega\}=%
\raisebox{-0.8026in}{\includegraphics[
height=1.548in,
width=1.6016in
]%
{15w.eps}%
}%
.
\]
The final partition function is
\begin{equation}
Z_{BF}(\Delta)=\int\limits_{\{\omega_{b,}\omega_{e}\}}\prod_{b}\frac
{d_{\omega_{b}}}{64\pi^{8}}\prod_{s}Z_{BF}(s)\prod_{b}d\omega_{b}\prod
_{e}d\omega_{e}, \label{eq.6}%
\end{equation}
where the $Z_{BF}(s)=\{15\omega\}$ is the amplitude for a four-simplex $s$.
The $d_{\omega_{b}}=\left\vert \chi_{L}\chi_{R}\right\vert ^{2}$ term is the
quantum amplitude associated with the bone $b$.\ Here $\omega_{e}$ is the
internal representation used to define the intertwiners. Usually $\omega_{e}$
is replaced by $i_{e}$ to indicate the intertwiner. The set $\left\{
\boldsymbol{\omega}_{b,}\boldsymbol{\omega}_{e}\right\}  $ of all $\omega_{b}%
$'s and $\omega_{e}$'s is usually called a \textbf{coloring} of the bones and
the edges. This partition function may not be finite in general.

It is well known that the$\ BF$ theories are topological field theories. A
priori one cannot expect this to be true for the case of the BF spin foam
models because of the discretization of the BF action. For the spin foam
models of the BF\ theories for compact groups, it has been shown that the
partition functions are triangulation independent up to a factor
\cite{SpinFoamDiag}. This analysis is purely based on spin foam diagrammatics
and is independent of the group used as long the BF spin foam is defined
formally by equation (\ref{eq.del}) and the harmonic expansion in equation
(\ref{eq.del.exp}) is formally valid. So one can apply the spin foam
diagrammatics analysis directly to the $SO(4,C)$ BF spin foam and write down
the triangulation independent partition function as%

\[
Z_{BF}^{^{\prime}}(\Delta)=\tau^{n_{4}-n_{3}}Z_{BF}(\Delta)
\]
using the result from \cite{SpinFoamDiag}. In the above equation $n_{4},n_{3}$
is number of four bubbles and three bubbles in the triangulation $\Delta$ and
\begin{align*}
\tau &  =\delta_{SO(4,C)}(I)\\
&  =\frac{1}{64\pi^{8}}\int d_{\omega}^{2}d\omega.
\end{align*}
The above integral is divergent and so the partition functions need not be
finite. The normalized partition function is to be considered as the proper
partition function because the BF\ theory is supposed to be topological and so
triangulation independent.

\section{The $SO(4,C)$ Barrett-Crane Model}

\subsection{Classical $SO(4,C)$ General Relativity}

Consider a four dimensional manifold $M$. Let $A$ be a $SO(4,C)$ connection
1-form and $B^{ij}$ be a complex bivector valued $2$-form on $M$. I\ would
like to restrict myself to the non-degenerate general relativity in this
section by assuming $b=\frac{1}{4}\epsilon^{abcd}B_{ab}\wedge B_{cd}\neq0$.
The Plebanski action for the $SO(4,C)$ general relativity is obtained by
adding a Lagrange multiplier term to impose the Plebanski constraint to the BF
theory action given in equation :(\ref{BFaction}). A\ simple way of writing
the action \cite{MPR1} is%
\begin{equation}
S_{C}(A,B_{ij},\bar{A},\bar{B}_{ij},\phi)=\operatorname{Re}\left[  \int
_{M}tr(B\wedge F)+\frac{b}{2}\phi^{abcd}B_{ab}\wedge B_{cd}\right]  \text{,}
\label{GRactionComplex}%
\end{equation}
where $\phi$ is a complex tensor with the symmetries of the Riemann curvature
tensor such that $\phi^{abcd}\epsilon_{abcd}=0$. The physics corresponding to
the extrema of the above action has been discussed by me in Ref:
\cite{RealitySpinFoam}. Two important results are

\begin{itemize}
\item The Plebanski constraint imposes the condition $B_{ab}^{ij}=$
$\theta_{a}^{[i}\theta_{b}^{j]}$ where $\theta_{a}^{i}$ is a complex tetrad
field \cite{Plebanski}, \cite{ClassicalAreaReal}.

\item The field equations correspond to the $SO(4,C)$ general relativity on
the manifold $M$ \cite{ClassicalAreaReal}.
\end{itemize}

\subsubsection{Relation to Complex Geometry}

Let $M$ be a real analytic manifold. Let $M_{c}$ be the complex analytic
manifold which is obtained by analytically continuing the real coordinates on
$M$. The analytical continuation of the field equations and their solutions on
$M$ to complex $M_{c}$ can be used to define complex general relativity.
Conversely, the field equations of complex general relativity or their
solutions on $M_{c}$ when restricted to $M$ defines the $SO(4,C)$ general
relativity. Because of these properties the action $S$ can also be considered
as an action for complex general relativity.

Now consider the relation between the complex general relativity on $M_{c}$
and\ the $SO(4,C)$ general relativity on $M$. This relation critically depends
on $M$ being a real analytic manifold. It also depends on the fields on it
being analytic on some region may be except for some singularities. If the
fields and the field equations are discretized we lose the relation to complex
general relativity. Thus it is also not meaningful to relate a $SO(4,C)$
Barrett-Crane Model to complex general relativity. If the $SO(4,C)$
Barrett-Crane model has a semiclassical continuum general relativity limit
then a relation to complex general relativity may be recovered.

\subsection{The $SO(4,C)$ Barrett-Crane Constraints}

My goal here is to systematically construct the Barrett-Crane model of the
$SO(4,C)$ general relativity. In the previous section I\ discussed the
$SO(4,C)$ BF spin foam model. The basic elements of the BF spin foams are spin
networks built on graphs dual to the triangulations of the four simplices with
arbitrary intertwiners and the principal unitary representations of $SO(4,C)$
discussed in appendix B. These closed spin networks can be considered as
quantum states of four simplices in the BF\ theory and the essence of these
spin networks is mainly gauge invariance. To construct a spin foam model of
general relativity these spin networks need to be modified to include the
Plebanski Constraints in the discrete form.

A\ quantization of a four-simplex for the Riemannian general relativity was
proposed by Barrett and Crane \cite{BCReimmanion}. The bivectors $B_{i}$
associated with the ten triangles of a four-simplex in a flat Riemannian space
satisfy the following properties called the Barrett-Crane
constraints\footnote{I\ would like to refer the readers to the original paper
\cite{BCReimmanion} for more details.}:

\begin{enumerate}
\item The bivector changes sign if the orientation of the triangle is changed.

\item Each bivector is simple.

\item If two triangles share a common edge, then the sum of the bivectors is
also simple.

\item The sum of the bivectors corresponding to the edges of any tetrahedron
is zero. This sum is calculated taking into account the orientations of the
bivectors with respect to the tetrahedron.

\item The six bivectors of a four-simplex sharing the same vertex are linearly independent.

\item The volume of a tetrahedron calculated from the bivectors is real and non-zero.
\end{enumerate}

The items two and three can be summarized as follows:
\[
B_{i}\wedge B_{j}=0~\forall i,j,
\]
where $A\wedge B=\varepsilon_{IJKL}A^{IJ}B^{KL}$ and the $i,j$ represents the
triangles of a tetrahedron. If $i=j$, it is referred to as the simplicity
constraint. If $i\neq j$ it is referred as the cross-simplicity constraints.

Barrett and Crane have shown that these constraints are sufficient to restrict
a general set of ten bivectors $E_{b}$ so that they correspond to the
triangles of a geometric four-simplex up to translations and rotations in a
four dimensional flat Riemannian space.

The Barrett-Crane constraints theory can be trivially extended to the
$SO(4,C)$ general relativity. In this case the bivectors are complex and so
the volume calculated for the sixth constraint is complex. So we need to relax
the condition of the reality of the volume.

A quantum four-simplex for Riemannian general relativity is defined by
quantizing the Barrett-Crane constraints \cite{BCReimmanion}. The bivectors
$B_{i}$ are promoted to the Lie operators $\hat{B}_{i}$ on the representation
space of the relevant group and the Barrett-Crane constraints are imposed at
the quantum level. A\ four-simplex has been quantized and studied in the case
of the Riemannian general relativity before \cite{BCReimmanion}. All the first
four constraints have been rigorously implemented in this case. The last two
constraints are inequalities and they are difficult to impose. This could be
related to the fact that the Riemannian Barrett-Crane model reveal the
presence of degenerate sectors \cite{BaezEtalAsym}, \cite{JWBCS} in the
asymptotic limit \cite{JWBRW} of the model. For these reasons here after
I\ would like to refer to a spin foam model that satisfies only the first four
constraints as an \textit{essential Barrett-Crane model}, While a spin foam
model that satisfies all the six constraints as a \textit{rigorous
Barrett-Crane model}.

Here I\ would like to derive the essential $SO(4,C)$ Barrett-Crane model. For
this one must deal with complex bivectors instead of real bivectors. The
procedure that I\ would like to use to solve the constraints can be carried
over directly to the Riemannian Barrett-Crane\ model. This derivation
essentially makes the derivation of the Barrett-Crane intertwiners for the
real and the complex Riemannian general relativity more rigorous.

\subsubsection{The Simplicity Constraint}

The group $SO(4,C)$ is locally isomorphic to $\frac{SL(2,C)\times
SL(2,C)}{Z_{2}}$. An element $B$ of the Lie algebra space of $SO(4,C)$ can be
split into the left and the right handed $SL(2,C)$ components,%
\begin{equation}
B=B_{L}+B_{R}.
\end{equation}
There are two Casimir operators for $SO(4,C)$ which are $\varepsilon
_{IJKL}B^{IJ}B^{KL}$ and $\eta_{IK}\eta_{JL}B^{IJ}B^{KL}$, where $\eta_{IK}$
is the flat Euclidean metric. In terms of the left and right handed split I
can expand the Casimir operators as%
\[
\varepsilon_{IJKL}B^{IJ}B^{KL}=B_{L}\cdot B_{L}-B_{R}\cdot B_{R}~~\text{and}%
\]%
\[
\eta_{IK}\eta_{JL}B^{IJ}B^{KL}=B_{L}\cdot B_{L}+B_{R}\cdot B_{R},
\]
where the dot products are the trace in the $SL(2,C)$ Lie algebra coordinates.

The bivectors are to be quantized by promoting the Lie algebra vectors to Lie
operators on the unitary representation space of $SO(4,C)\approx$
$\frac{SL(2,C)\times SL(2,C)}{Z_{2}}$. The relevant unitary representations of
$SO(4,C)\simeq$ $SL(2,C)\otimes SL(2,C)/Z_{2}$ are labeled by a pair
($\chi_{L},$ $\chi_{R}$) such that $n_{L}+n_{R}$ is even (appendix B). The
elements of the representation space $D_{\chi_{L}}\otimes$ $D_{\chi_{R}}$ are
the eigen states of the Casimirs and on them the operators reduce to the
following:
\begin{equation}
\varepsilon_{IJKL}\hat{B}^{IJ}\hat{B}^{KL}=\frac{\chi_{L}^{2}-\chi_{R}^{2}}%
{2}\hat{I}~~\text{and} \label{eq.1}%
\end{equation}%
\begin{equation}
\eta_{IK}\eta_{JL}\hat{B}^{IJ}\hat{B}^{KL}=\frac{\chi_{L}^{2}+\chi_{R}^{2}%
-2}{2}\hat{I}. \label{eq.2}%
\end{equation}
The equation (\ref{eq.1}) implies that on $D_{\chi_{L}}\otimes$ $D_{\chi_{R}}$
the simplicity constraint $B\wedge B=0$ is equivalent to the condition
$\chi_{L}=\pm\chi_{R}$. I would like to find a representation space on which
the representations of $SO(4,C)$ are restricted precisely by $\chi_{L}$ $=$
$\pm\chi_{R}$. Since a $\chi$ representation is equivalent to $-\chi$
representations \cite{IMG}, $\chi_{L}=+\chi_{R}$ case is equivalent to
$\chi_{L}=-\chi_{R}$ \cite{IMG}.

Consider a square integrable function $f$ $(x)$ on the complex sphere $CS^{3}$
defined by%

\[
x\cdot x=1,\forall x\in\boldsymbol{C}^{4}.
\]
It can be Fourier expanded in the representation matrices of $SL(2,C)$ using
the isomorphism $CS^{3}\simeq$ $SL(2,C)$,%
\begin{equation}
f(x)=\frac{1}{8\pi^{4}}\int Tr(F(\chi)T_{\chi}(\mathfrak{g}(x)^{-1})\chi
\bar{\chi}d\chi, \label{Cs3Exp}%
\end{equation}
where the isomorphism $\mathfrak{g}\mathrm{:}CS^{3}$ $\longrightarrow SL(2,C)$
is defined in the appendix A. The group action of $g=(g_{L},g_{R})\in SO(4,C)$
on $x$ $\in CS^{3}$ is given by
\begin{equation}
\mathfrak{g}(gx)=g_{L}^{-1}\mathfrak{g}(x)g_{R}. \label{Cs3Action}%
\end{equation}
Using equation (\ref{Cs3Exp}) I can consider the $T_{\chi}(\mathfrak{g}%
(x))(z_{1},z_{2})$ as the basis functions of $L^{2}$ functions on $CS^{3}.$
The matrix elements of the action of $g$ on $CS^{3}$ is given by (appendix B)%
\[
\int\bar{T}_{\acute{\chi}}(\mathfrak{g}(x))(\acute{z}_{1},\acute{z}%
_{2})T_{\chi}(\mathfrak{g}(gx))(z_{1},z_{2})dx=T_{-\acute{\chi}}(g_{L}%
)(\acute{z}_{1},z_{1})T_{\chi}(g_{R})(\acute{z}_{2},z_{2})\delta(\acute{\chi
}-\chi).
\]
I see that the representation matrices are precisely those of $SO(4,C)$ only
restricted by the constraint $\chi_{L}=-\chi_{R}\approx\chi_{R}$. So the
simplicity constraint effectively reduces the Hilbert space $H$ to the space
of $L^{2}$ functions on $CS^{3}$. In Ref:\cite{BFfoamHighD} the analogous
result has been shown for $SO(N,R)$ where the Hilbert space is reduced to the
space of the $L^{2}$ functions on $S^{N-1}$.

\subsubsection{The Cross-simplicity Constraints}

Next let me quantize the cross-simplicity constraint part of the Barrett-Crane
constraint. Consider the quantum state space associated with a pair of
triangles $1$ and $2$ of a tetrahedron. A general quantum state that just
satisfies the simplicity constraints $B_{1}\wedge B_{1}=0$ and $B_{2}\wedge
B_{2}=0$ is of the form $f(x_{1},x_{2})$ $\in L^{2}(CS^{3}\ast CS^{3})$,
$x_{1},x_{2}\in CS^{3}$.

On the elements of $L^{2}(CS^{3}\ast CS^{3})$ the action $B_{1}\wedge B_{2}$
is equivalent to the action of $\left(  B_{1}+B_{2}\right)  \wedge\left(
B_{1}+B_{2}\right)  $\footnote{Please notice that
\[
\left(  \hat{B}_{1}+\hat{B}_{2}\right)  \wedge\left(  \hat{B}_{1}+\hat{B}%
_{2}\right)  =\hat{B}_{1}\wedge\hat{B}_{1}+\hat{B}_{2}\wedge\hat{B}_{2}%
+2B_{1}\wedge\hat{B}_{2}.
\]
But since $\hat{B}_{1}\wedge\hat{B}_{1}=\hat{B}_{2}\wedge\hat{B}_{2}=0$ on
$f(x_{1},x_{2})$ we have
\[
\left(  \hat{B}_{1}+\hat{B}_{2}\right)  \wedge\left(  \hat{B}_{1}+\hat{B}%
_{2}\right)  f(x_{1},x_{2})=\hat{B}_{1}\wedge\hat{B}_{2}f(x_{1},x_{2}).
\]
}. This implies that the cross-simplicity constraint $B_{1}\wedge B_{2}=0$
requires the simultaneous rotation of $x_{1},x_{2}$ involve only the $\chi
_{L}$ $=$ $\pm\chi_{R}$ representations. The simultaneous action of
$g=(g_{L},g_{R})$ on the arguments of $f(x_{1},x_{2})$ is%
\begin{equation}
gf(x_{1},x_{2})=f(g_{L}^{-1}x_{1}g_{R},g_{L}^{-1}x_{2}g_{R}). \label{action}%
\end{equation}
The harmonic expansion of $f(x_{1},x_{2})$ in terms of the basis function
$T_{\chi}(\mathfrak{g}(x))(z_{1},z_{2})$ is%
\[
f(x_{1},x_{2})=F_{z_{1}z_{2}\chi_{1}\chi_{2}}^{\acute{z}_{1}\acute{z}_{2}%
}T_{\acute{z}_{1}\chi_{1}}^{z_{1}}(\mathfrak{g}(x_{1}))T_{\acute{z}_{2}%
\chi_{2}}^{z_{2}}(\mathfrak{g}(x_{2})),
\]
where I have assumed all the repeated indices are either integrated or summed
over for equation only. The rest of the calculations can be understood
graphically. The last equation can be graphically written as follows:%

\[
f(x_{1},x_{2})=\iint\limits_{\chi_{1}\chi_{2}}%
\raisebox{-0.5613in}{\includegraphics[
height=0.9193in,
width=1.3318in
]%
{Xsimplicity1.eps}%
}%
d\chi_{1}d\chi_{2},
\]
where the box $F$ represents the tensor $F_{z_{1}z_{2}\chi_{1}\chi_{2}%
}^{\acute{z}_{1}\acute{z}_{2}}$. The action of $g\in SO(4,C)$ on $f$ is%

\begin{equation}
gf(x_{1},x_{2})=\iint\limits_{\chi_{1}\chi_{2}}%
\raisebox{-0.5518in}{\includegraphics[
trim=0.014528in 0.000000in -0.014528in 0.000000in,
height=1.0032in,
width=2.2935in
]%
{Xsimplicity2.eps}%
}%
d\chi_{1}d\chi_{2}. \label{gfresult}%
\end{equation}
Now for any $h\in SL(2,C),$%
\[
T_{a_{1}\chi_{1}}^{b_{1}}(h)T_{a_{2}\chi_{1}}^{b_{2}}(h)=C_{\chi_{1}\chi
_{2}b_{3}}^{b_{1}b_{2}\chi_{3}}\bar{C}_{a_{1}a_{2}\chi_{3}}^{\chi_{1}\chi
_{2}a_{3}}T_{a_{3}\chi_{3}}^{b_{3}}(h),
\]
where $C$'s are the Clebsch-Gordan coefficients of $SL(2,C)$ \cite{IMG},
\cite{NaimarckClebsch}\footnote{I\ derived this equation explicitly in the
appendix of Ref:\cite{MyRigorousSpinFoam}.}. I have assumed all the repeated
indices are either integrated or summed over for the previous and the next two
equations. Using this I can rewrite the $g_{L}$ and $g_{R}$ parts of the
result (\ref{gfresult}) as follows:%
\begin{equation}
T_{a_{1}\chi_{1}}^{z_{1}}(g_{L}^{-1})T_{a_{2}\chi_{2}}^{z_{2}}(g_{L}%
^{-1})=C_{\chi_{1}\chi_{2}z_{3}}^{z_{1}z_{2}\chi_{L}}\bar{C}_{a_{1}a_{2}%
\chi_{L}}^{\chi_{1}\chi_{2}a_{3}}T_{a_{3}\chi_{L}}^{z_{3}}(g_{L}^{-1})
\label{exp1}%
\end{equation}
and%
\begin{equation}
T_{\acute{z}_{1}\chi_{1}}^{b_{1}}(g_{R})T_{\acute{z}_{2}\chi_{2}}^{b_{2}%
}(g_{R})=C_{\chi_{1}\chi_{2}b_{3}}^{b_{1}b_{2}\chi_{R}}\bar{C}_{\acute{z}%
_{1}\acute{z}_{2}\chi_{R}}^{\chi_{1}\chi_{2}\acute{z}_{3}}T_{\acute{z}_{3}%
\chi_{R}}^{b_{3}}(g_{R}). \label{exp2}%
\end{equation}
Now we have
\[
gf(x_{1},x_{2})=\idotsint\limits_{\chi_{1}\chi_{2}\chi_{L}\chi_{R}}%
\raisebox{-0.5708in}{\includegraphics[
height=1.1588in,
width=2.2701in
]%
{Xsimplicity3.eps}%
}%
d\chi_{1}d\chi_{2}.
\]

To satisfy the cross-simplicity constraint the expansion of $gf(x_{1},x_{2})$
must have contribution only from the terms with $\chi_{L}=\pm\chi_{R}$. In the
expansion in equation (\ref{exp1}) and equation (\ref{exp2}) in the right hand
side the terms are defined only up to a sign of $\chi_{L}$ and $\chi_{R}%
$\footnote{Please see appendix A for the explanation.}. Let me remove all the
terms which does not satisfy $\chi_{L}=\pm\chi_{R}$ (say = $\pm\chi$). Also
let me set $g=I.$ Now we can deduce that the functions denoted by $\tilde
{f}(x_{1},x_{2})$ obtained by reducing $f(x_{1},x_{2})$ using the
cross-simplicity constraints must have the expansion \footnote{The factor of 2
has been introduced to include the terms with $\chi_{L}=-\chi_{R}.$},%

\[
f(x_{1},x_{2})=2\iiint\limits_{\chi_{1}\chi_{2}\chi}c_{\chi}%
\raisebox{-0.5016in}{\includegraphics[
height=1.0006in,
width=2.2035in
]%
{Xsimplicity35.eps}%
}%
d\chi_{1}d\chi_{2}d\chi,
\]
where the coefficients $c_{\chi}$ are arbitrary. Now the Clebsch-Gordan
coefficient terms in the expansion can be re-expressed using the following
equation :%
\begin{equation}
C_{\chi_{1}\chi_{2}z_{3}}^{z_{1}z_{2}\chi}\bar{C}_{\acute{z}_{1}\acute{z}%
_{2}\chi}^{\chi_{1}\chi_{2}\acute{z}_{3}}=\frac{8\pi^{4}}{\chi\bar{\chi}}%
\int_{SL(2,C)}T_{\acute{z}_{1}\chi_{1}}^{z_{1}}(h)T_{\acute{z}_{2}\chi_{2}%
}^{z_{2}}(h)\bar{T}_{z_{3}\chi}^{\acute{z}_{3}}(h)dh, \label{clebsmpl}%
\end{equation}
where $h$, $\tilde{h}$ $\in$ $SL(2,C)$ and $dh$ the bi-invariant measure on
$SL(2,C)$. Using this in the middle two Clebsch-Gordan coefficients of
$\tilde{f}(x_{1},x_{2})$ we get
\[
\tilde{f}(x_{1},x_{2})=2\iiint\limits_{\chi_{1}\chi_{2}\chi}\int
_{SL(2,C)}\frac{8\pi^{4}c_{\chi}}{\chi\bar{\chi}}%
\raisebox{-0.6028in}{\includegraphics[
height=1.0499in,
width=1.8983in
]%
{Xsimplicity4.eps}%
}%
dhd\chi_{1}d\chi_{2}d\chi.
\]
This result can be rewritten for clarity as%
\[
\tilde{f}(x_{1},x_{2})=2\iiint\limits_{\chi_{1}\chi_{2}\chi}\int
_{SL(2,C)}\frac{8\pi^{4}c_{\chi}}{\chi\bar{\chi}}%
\raisebox{-0.5016in}{\includegraphics[
height=1.0715in,
width=1.3214in
]%
{Xsimplicity45.eps}%
}%
dhd\chi_{1}d\chi_{2}d\chi.
\]
Once again applying equation (\ref{clebsmpl}) to the remaining two
Clebsch-Gordan coefficients we get,%

\[
\tilde{f}(x_{1},x_{2})=2\iiint\limits_{\chi_{1}\chi_{2}\chi}c_{\chi}%
\iint_{SL(2,C)\times SL(2,C)}\left(  \frac{8\pi^{4}}{\chi\bar{\chi}}\right)
^{2}%
\raisebox{-0.5016in}{\includegraphics[
height=1.0404in,
width=1.203in
]%
{Xsimplicity5.eps}%
}%
dhd\tilde{h}d\chi_{1}d\chi_{2}d\chi.
\]

By rewriting the above expression, I deduce that a general function $\tilde
{f}(x_{1},x_{2})$ that satisfies the cross-simplicity constraint must be of
the form,
\begin{subequations}
\label{xsimpfunc}%
\begin{align*}
\tilde{f}(x_{1},x_{2})  &  =\iint\limits_{\chi_{1}\chi_{2}}c_{\chi}%
\int_{SL(2,C)}F_{\chi_{1}\chi_{2}}(h)%
\raisebox{-0.3684in}{\includegraphics[
height=0.7178in,
width=1.5368in
]%
{Xsimplicity55.eps}%
}%
dhd\chi_{1}d\chi_{2},\\
&  =\iint\limits_{\chi_{1}\chi_{2}}c_{\chi}\int_{SL(2,C)}F_{\chi_{1}\chi_{2}%
}(h)tr(T_{\chi_{1}}(\mathfrak{g}(x_{1})h)tr(T_{\chi_{2}}(\mathfrak{g}%
(x_{2})h)dhd\chi_{1}d\chi_{2},
\end{align*}
where $F_{\chi_{1}\chi_{2}}(h)$ is arbitrary. Then if $\Psi(x_{1},x_{2}%
,x_{3},x_{4})$ is the quantum state of a tetrahedron that satisfies all of the
simplicity constraints and the cross-simplicity constraints, it must be of the
form,
\end{subequations}
\begin{align*}
&  \Psi(x_{1},x_{2},x_{3},x_{4})\\
&  =\int F_{\chi_{1}\chi_{2}\chi_{3}\chi_{4}}(h)tr(T_{\chi_{1}}(\mathfrak{g}%
(x_{1})h)tr(T_{\chi_{2}}(\mathfrak{g}(x_{2})h)\\
&  tr(T_{\chi_{3}}(\mathfrak{g}(x_{3})h)tr(T_{\chi_{4}}(\mathfrak{g}%
(x_{4})h)dh\prod\limits_{i}d\chi_{i}.
\end{align*}
This general form is deduced by requiring that for every pair of variables
with the other two fixed, the function must be the form of the right hand side
of equation (\ref{xsimpfunc}).

\subsubsection{The $SO(4,C)$ Barrett-Crane Intertwiner}

Now the quantization of the fourth Barrett-Crane constraint demands that
$\Psi$ is invariant under the simultaneous complex rotation of its variables.
This is achieved if $F_{\chi_{1}\chi_{2}\chi_{3}\chi_{4}}(h)$ is constant
function of $h$. Therefore the quantum state of a tetrahedron is spanned by%
\begin{equation}
\Psi(x_{1},x_{2},x_{3},x_{4})=\int_{n\in CS^{3}}%
{\displaystyle\prod\limits_{i}}
T_{\chi_{i}}(\mathfrak{g}(x_{i})\mathfrak{g}(n))dn, \label{BCintertwiner}%
\end{equation}
where the measure $dn$ on $CS^{3}$ is derived from the bi-invariant measure on
$SL(2,C)$. I\ would like to refer to the functions $T_{\chi_{i}}%
(\mathfrak{g}(x_{i})$ as the $T-$\textbf{functions} here after.

\paragraph{Alternative forms}

The quantum state can be diagrammatically represented as follows:%
\[
\Psi(x_{1},x_{2},x_{3},x_{4})=\int%
\raisebox{-0.6599in}{\includegraphics[
height=1.3059in,
width=1.3059in
]%
{BCIntGroup.eps}%
}%
dn.
\]
A unitary representation $T_{\chi}$ of $SL(2,C)$ can be considered as an
element of $D_{\chi}\otimes D_{\chi}^{\ast}$ where $D_{\chi}^{\ast}$ is the
dual representation of $D_{\chi}$. So using this the Barrett-Crane intertwiner
can be written as an element $\left\vert \Psi\right\rangle \in\bigotimes
\limits_{i}D_{\chi_{i}}\otimes D_{\chi_{i}}^{\ast}$ as follows:%
\[
\left\vert \Psi\right\rangle =\int\limits_{CS^{3}}%
\raisebox{-0.6529in}{\includegraphics[
height=1.2955in,
width=1.3474in
]%
{BCIntRep.eps}%
}%
dn.
\]
Since $SL(2,C)\approx CS^{3},$ using the following graphical identity:%

\[
\int_{SL(2,C)}%
\raisebox{-0.5708in}{\includegraphics[
height=1.1338in,
width=0.9824in
]%
{4intertwine.eps}%
}%
dg=\int%
\raisebox{-0.5613in}{\includegraphics[
height=1.1122in,
width=2.0877in
]%
{edge4int3.eps}%
}%
\frac{8\pi^{4}}{\chi\bar{\chi}}d\chi,
\]
the Barrett-Crane solution can be rewritten as%

\[
\left\vert \Psi\right\rangle =\int%
\raisebox{-0.4817in}{\includegraphics[
height=0.9617in,
width=2.1932in
]%
{4XXBC.eps}%
}%
\frac{8\pi^{4}}{\chi\bar{\chi}}d\chi,
\]
which emerges as an intertwiner in the familiar form in which Barrett and
Crane proposed it for the Riemannian general relativity. It can be clearly
seen that the simple representations for $SO(4,R)$ ($J_{L}=$ $J_{R}$) has been
replaced by the simple representation of $SO(4,C)$ ($\chi_{L}=$ $\pm\chi_{R}$).

\paragraph{Relation to the Riemannian Barrett-Crane model:}

All the analysis done until for the $SO(4,C)\ $Barrett-Crane theory can be
directly applied to the Riemannian$\ $Barrett-Crane theory. The
correspondences between the two models are listed in the following
table\footnote{BC stands for Barrett-Crane. For $\chi_{L}$ and $\chi_{R}$ we
have $n_{L}+n_{R}=even$.
\par
{}}:

\begin{center}%
\begin{tabular}
[c]{lll}%
\textbf{Property} & $SO(4,R)$ BC model & $SO(4,C)$ BC model\\
Gauge group & $SO(4,R)\approx\frac{SL(2,C)\otimes SL(2,C)}{Z_{2}}$ &
$SO(4,C)\approx\frac{SU(2)\otimes SU(2)}{Z_{2}}$\\
Representations & $J_{L},J_{R}$ & $\chi_{L},\chi_{R}$\\
Simple representations & $J_{L}=J_{R}$ & $\chi_{L}=\pm\chi_{R}$\\
Homogenous space & $S^{3}\approx SU(2)$ & $CS^{3}\approx SL(2,C)$%
\end{tabular}

\end{center}

\subsubsection{The Spin Foam Model for the $SO(4,C)$ General Relativity.}

The $SO(4,C)$ Barrett-Crane intertwiner derived in the previous section can be
used to define a $SO(4,C)$ Barrett-Crane spin foam model. The amplitude
$Z_{BC}(s)$ of a four-simplex $s$ is given by the $\{10\chi\}_{SO(4,C)}$
symbol given below:%

\begin{equation}
\{10\chi\}_{SO(4,C)}=%
\raisebox{-0.7022in}{\includegraphics[
height=1.4131in,
width=1.5134in
]%
{10XBC.eps}%
}%
\text{,} \label{10w}%
\end{equation}
where the circles are the Barrett-Crane intertwiners. The integers represent
the tetrahedra and the pairs of integers represent triangles. The intertwiners
use the four $\chi$'s associated with the links that emerge from it for its
definition in equation (\ref{10w}). In the next subsection, the propagators of
this theory are defined and the $\{10\chi\}$ symbol is expressed in terms of
the propagators in the subsubsection that follows it.

The $SO(4,C)$ Barrett-Crane partition function of the spin foam associated
with the four dimensional simplicial manifold with a triangulation $\Delta$
is
\begin{equation}
Z(\Delta)=\sum_{\left\{  \chi_{b}\right\}  }\left(  \prod_{b}\frac{d_{\chi
_{b}}^{2}}{64\pi^{8}}\right)  \prod_{s}Z(s), \label{PartitionFunction}%
\end{equation}
where $Z(s)$ is the quantum amplitude associated with the $4$-simplex $s$ and
the $d_{\chi_{b}}$ adopted from the spin foam model of the $BF\ $theory can be
interpreted as the quantum amplitude associated with the bone $b$.

\subsubsection{The Features of the $SO(4,C)$ Spin Foam}

\begin{itemize}
\item Areas: The squares of the areas of the triangles (bones) of the
triangulation are given by $\eta_{IK}\eta_{JL}B^{IJ}B^{KL}$. The eigen values
of the squares of the areas in the $SO(4,C)$ Barrett-Crane model from equation
(\ref{eq.2}) are given by
\begin{align*}
\eta_{IK}\eta_{JL}\hat{B}_{b}^{IJ}\hat{B}_{b}^{KL}  &  =\left(  \chi
^{2}-1\right)  \hat{I}\\
&  =\left(  \frac{n^{2}}{2}-\rho^{2}-1+i\rho n\right)  \hat{I}.
\end{align*}
One can clearly see that the area eigen values are complex. The $SO(4,C)$
Barrett-Crane model relates to the $SO(4,C)$ general relativity. Since in the
$SO(4,C)$ general relativity the bivectors associated with any two dimensional
flat object are complex, it is natural to expect that the areas defined in
such a theory are complex too. This is a generalization of the concept of the
space-like and the time-like areas for the real general relativity models:
Area is imaginary if it is time-like and real if it is space-like.

\item Propagators: Laurent and Freidel have investigated the idea of
expressing simple spin networks as Feynman diagrams \cite{SimpleSpinNetworks}.
Here we will apply this idea to the $SO(4,C)$ simple spin networks. Let
$\Sigma$ be a triangulated three surface. Let $n_{i}\in CS^{3}$ be a vector
associated with the $i^{th}$ tetrahedron of the $\Sigma$. The propagator of
the $SO(4,C)$ Barrett-Crane model associated with the triangle $ij$ is given
by%
\begin{align*}
G_{\chi_{ij}}(n_{i},n_{j})  &  =Tr(T_{\chi_{ij}}(g(n_{i}))T_{\chi_{ij}}^{\dag
}(g(n_{j})))\\
&  =Tr(T_{\chi_{ij}}(g(n_{i})g^{-1}(n_{j}))),
\end{align*}
where $\chi_{ij}$ is a representation associated with the triangle common to
the $i^{th}$ and the $j^{th}$ tetrahedron of $\Sigma$. If $X$ and $Y$ belong
to $CS^{3}$ then
\[
tr\left(  \mathfrak{g}(X)\mathfrak{g}(Y)^{-1}\right)  =2X.Y,
\]
where $X.Y$ is the Euclidean dot product and $tr$ is the matrix trace. If
$\lambda=e^{t}$ and $\frac{1}{\lambda}$ are the eigen values of $g(X)g(Y)^{-1}%
$ then,%
\begin{align*}
\lambda+\lambda^{-1}  &  =2X.Y\\
X.Y  &  =\cosh(t).
\end{align*}
From the expression for the trace of the $SL(2,C)$ unitary representations,
(appendix A, \cite{IMG}) I have the propagator for the $SO(4,C)$ Barrett-Crane
model calculated as%
\[
G_{\chi_{ij}}(n_{i},n_{j})=\dfrac{\cos(\rho_{ij}\eta_{ij}+n_{ij}\theta_{ij}%
)}{\left\vert \sinh(\eta_{ij}+i\theta_{ij})\right\vert ^{2}},
\]
where $\eta_{ij}+i\theta_{ij}$ is defined by $n_{i}.n_{j}=\cosh(\eta
_{ij}+i\theta_{ij})$. Two important properties of the propagators are listed below.

\begin{enumerate}
\item Using the expansion for the delta on $SL(2,C)$ I have
\begin{align*}
\delta_{CS^{3}}(X,Y)  &  =\delta_{SL(2,C)}(g(X)g^{-1}(Y))\\
&  =\frac{1}{8\pi^{4}}\int\bar{\chi}\chi Tr(T_{\chi}(g(X)g^{-1}(Y))d\chi,
\end{align*}
where the suffix on the deltas indicate the space in which it is defined.
Therefore%
\[
\int\bar{\chi}\chi G_{\chi}(X,Y))=8\pi^{4}\delta_{CS^{3}}(X,Y).
\]

\item Consider the orthonormality property of the principal unitary
representations of $SL(2,C)$ given by%
\begin{align*}
&  \int_{CS^{3}}T_{\acute{z}_{1}\chi_{1}}^{z_{1}}(g(X))T_{\acute{z}_{2}%
\chi_{2}}^{\dag z_{2}}(g(X))dX\\
&  =\frac{8\pi^{4}}{\chi_{1}\bar{\chi}_{1}}\delta(\chi_{1}-\chi_{2}%
)\delta(z_{1}-\acute{z}_{1})\delta(z_{2}-\acute{z}_{2}),
\end{align*}
where the delta on the $\chi$'s is defined up to a sign of them$.$ From this I
have%
\[
\int_{CS^{3}}G_{\chi_{1}}(X,Y)G_{\chi_{2}}(Y,Z)dY=\frac{8\pi^{4}}{\chi_{1}%
\bar{\chi}_{1}}\delta(\chi_{1}-\chi_{2})G_{\chi_{1}}(X,Z).
\]

\end{enumerate}

\item The $\{10\chi\}$ symbol can be defined using the propagators on the
complex three sphere as follows:%
\begin{align*}
Z(s)  &  =\int_{x_{k}\in CS^{3}}%
{\displaystyle\prod\limits_{i<j}}
T_{\chi_{ij}}(\mathfrak{g}(x_{i})\mathfrak{g}(x_{j}))%
{\displaystyle\prod\limits_{k}}
dx_{k},\\
&  =\int_{\forall x_{k}\in CS^{3}}%
{\displaystyle\prod\limits_{i<j}}
G_{\chi_{ij}}(x_{i}\mathfrak{,}x_{j})%
{\displaystyle\prod\limits_{k}}
dx_{k},
\end{align*}
where $i$ denotes a tetrahedron of the four-simplex. For each tetrahedron
$k,~$a free variable $x_{k}\in CS^{3}$ is associated. For each triangle $ij$
which is the intersection of the $i$'th and the $j$'th tetrahedron, a
representation of $SL(2,C)$ denoted by $\chi_{ij}$ is associated.

\item Discretization Dependence and Local Excitations: It is well known that
the BF theory is discretization independent and is topological. The$\ $spin
foam for the $SO(4,C)$ general relativity is got by imposing the Barrett-Crane
constraints on the $BF\ $Spin foam. After the imposition of the Barrett-Crane
constraints the theory loses the discretization independence and the
topological nature. This can be seen in many ways.

\begin{itemize}
\item The simplest reason is that the $SO(4,C)\ $Barrett-Crane model
corresponds to the quantization of the discrete $SO(4,C)$ general relativity
which has local degrees of freedom.

\item After the restriction of the representations involved in BF spin foams
to the simple representations and the intertwiners to the Barrett-Crane
intertwiners, various important identities used in the spin foam diagrammatics
and proof of the discretization independence of the BF\ theory spin foams in
Ref:\cite{SpinFoamDiag} are no longer available.

\item The BF partition function is simply gauge invariant measure of the
volume of space of flat connections. Consider the following harmonic expansion
of the delta function which was used in the derivation of the $SO(4,C)$ BF
theory:%
\[
\delta(g)=\frac{1}{8\pi^{4}}\int d_{\omega}tr(T_{\omega}(g))d\omega.
\]
Imposition of the Barrett-Crane constraints on the BF theory spin foam,
suppresses the terms corresponding to the non-simple representations. If only
the simple representations are allowed in the right hand side, it is no longer
peaked at the identity. This means that the partition function for
the\ $SO(4,C)$\ Barrett-Crane model involves contributions only from the
non-flat connections which has local information.

\item In the asymptotic limit study of the $SO(4,C)$ spin foams in section
four the discrete version of the $SO(4,C)$ general relativity (Regge calculus)
is obtained. The Regge calculus action is clearly discretization dependent and non-topological.
\end{itemize}
\end{itemize}

\section{The Asymptotic Limit of the $SO(4,C)$ Barrett-Crane models.}

The asymptotic limit of the real Barrett-Crane models has been studied before
\cite{JWBCS}, \cite{JWBRW}, \cite{PonzanoReggeModel}, \cite{BaezEtalAsym} to a
certain degree. Here I will discuss the asymptotic limit of the $SO(4,C)$
Barrett-Crane model. For the first time I show here that we can extract
bivectors which satisfy the essential Barrett-Crane constraints from the
asymptotic limit. Consider the amplitude of a four-simplex given by Eq.
(\ref{10w}) with a real scale parameter $\lambda$,%
\begin{align*}
Z_{\lambda}  &  =\int_{n_{k}\in CS^{3}}%
{\displaystyle\prod\limits_{i<j}}
G_{\lambda\chi_{ij}}(n_{i}\mathfrak{,}n_{j})%
{\displaystyle\prod\limits_{k}}
dn_{k},\\
&  =\int_{n_{k}\in CS^{3}}%
{\displaystyle\prod\limits_{i<j}}
\dfrac{\cos(\lambda\rho_{ij}\eta_{ij}+\lambda n_{ij}\theta_{ij})}{\left\vert
\sinh(\lambda\eta_{ij}+i\lambda\theta_{ij})\right\vert ^{2}}%
{\displaystyle\prod\limits_{k}}
dx_{k},\\
&  =\int_{n_{k}\in CS^{3}}%
{\displaystyle\prod\limits_{i<j}}
\sum\limits_{\varepsilon_{ij}=\pm1}\dfrac{\exp(i\varepsilon_{ij}\lambda
(\rho_{ij}\eta_{ij}+n_{ij}\theta_{ij}))}{2\left\vert \sinh(\lambda\eta
_{ij}+i\lambda\theta_{ij})\right\vert ^{2}}%
{\displaystyle\prod\limits_{k}}
dx_{k},
\end{align*}
where the $\eta_{ij}+i\theta_{ij}$ is defined by $n_{i}.n_{j}=\cosh(\eta
_{ij}+i\theta_{ij})$. Here the $\zeta_{ij}=$ $\eta_{ij}+i\theta_{ij}$ is the
complex angle between $n_{i}$ and $n_{j}$. The asymptotic limit of
$Z_{\lambda}(s)\ $under $\lambda\longrightarrow\infty$ is controlled by%

\begin{align*}
&  S(\{n_{i},\bar{n}_{i}\},\{\chi_{ij},\bar{\chi}_{ij}\})\\
&  =\sum_{i<j}\varepsilon_{ij}(\rho_{ij}\eta_{ij}+n_{ij}\theta_{ij}%
)+\operatorname{Re}\left(  \sum_{i}q_{i}(n_{i}.n_{i}-1)\right) \\
&  =\operatorname{Re}\left(  \sum_{i<j}\varepsilon_{ij}\bar{\chi}_{ij}%
\zeta_{ij}+\sum_{i}q_{i}(n_{i}.n_{i}-1)\right)  ,
\end{align*}
where the $q_{i}$ are the Lagrange multipliers to impose $n_{i}.n_{i}%
=1,\forall i$. My goal now is to find stationary points for this action. The
stationary points are determined by
\begin{subequations}
\begin{equation}
\sum_{~~~~~i\neq j}\varepsilon_{ij}\bar{\chi}_{ij}\frac{\partial\zeta_{ij}%
}{\partial n_{i}}+q_{j}n_{j}=0,\forall j, \label{ext}%
\end{equation}
and $n_{j}.n_{j}=1,\forall j$ where the $j$ is a constant in the summation.%

\end{subequations}
\begin{equation}
\frac{\partial\zeta_{ij}}{\partial n_{i}}=\frac{n_{j}}{\sinh(\zeta_{ij})}.
\label{eq.dif.as}%
\end{equation}

Using equation (\ref{eq.dif.as}) in equation :(\ref{ext}) and taking the wedge
product of the equation with $n_{j}$ we have,
\[
\sum_{~~~~~i\neq j}\varepsilon_{ij}\bar{\chi}_{ij}\frac{n_{i}\wedge n_{j}%
}{\sinh(\zeta_{ij})}=0,\forall j.
\]

If
\[
\bar{E}_{ij}=i\varepsilon_{ij}\bar{\chi}_{ij}\frac{n_{i}\wedge n_{j}}%
{\sinh(\zeta_{ij})},
\]
then the last equation can be simplified to%

\begin{equation}
\sum_{~~~~~i\neq j}E_{ij}=0,\forall j. \label{sumzero}%
\end{equation}

We now consider the properties of $E_{ij}:$

\begin{itemize}
\item Each $i$ represents a tetrahedron. There are ten $E_{ij}$'s, each one of
them is associated with one triangle of the four-simplex.

\item The square of $E_{ij}$:%
\begin{align*}
\bar{E}_{ij}\cdot\bar{E}_{ij}  &  =\frac{-\bar{\chi}_{ij}^{2}}{\sinh^{2}%
(\zeta_{ij})}(n_{j}^{2}n_{i}^{2}-\left(  n_{i}\cdot n_{j}\right)  ^{2})\\
&  =\frac{-\bar{\chi}_{ij}^{2}}{\sinh^{2}(\zeta_{ij})}(1-\left(  \cosh
(\zeta_{ij}\right)  ^{2})\\
&  =\bar{\chi}_{ij}^{2}.
\end{align*}

\item The wedge product of any two $E_{ij}$ is zero if they are equal to each
other or if their corresponding triangles belong to the same tetrahedron.

\item Sum of all the $E_{ij}$ belonging to the same tetrahedron are zero
according to equation (\ref{sumzero}).
\end{itemize}

It is clear that these properties contain the first four Barrett-Crane
constraints. So we have successfully extracted the bivectors corresponding to
the triangles of a general flat four-simplex in $SO(4,C)$ general relativity
and the $n_{i}$ are the normal vectors of the tetrahedra. The $\chi_{ij}$ are
the complex areas of the triangle as one would expect. Since we did not impose
any non-degeneracy conditions, it is not guaranteed that the tetrahedra or the
four-simplex have non-zero volumes.

The asymptotic limit of the partition function of the entire simplicial
manifold with triangulation $\Delta$ is%
\[
S(\Delta,\{n_{is}\in CS^{3},\chi_{ij},\bar{\chi}_{ij},\varepsilon
_{ijs}\})=\operatorname{Re}\sum_{i<j,s}\varepsilon_{ijs}\bar{\chi}_{ij}%
\zeta_{ijs},
\]
where I\ have assumed variable $s$ represents the four simplices of $\Delta$
and $i,$ $j$ represents the tetrahedra. The $\varepsilon_{ijs}$ can be
interpreted as the orientation of the triangles. Each triangle has a
corresponding $\chi_{ij}$. The $n_{is}$ denote the unit complex vector
associated with the side of the tetrahedron $i$ facing the inside of a simplex
$s$. Now there is one bivector $E_{sij}$ associated with each side facing
inside of a simplex $s$ of a triangle $ij$ defined by%

\[
\bar{E}_{ijs}=i\varepsilon_{ijs}\bar{\chi}_{ij}\frac{n_{i}\wedge n_{js}}%
{\sinh(\zeta_{ijs})}.
\]
If the $n_{is}$ are chosen such that they satisfy stationary conditions%

\[
\sum_{~~~~~i\neq j}E_{ijs}=0,\forall j,s,
\]
and if
\[
\theta_{ij}=\left(  \sum_{s}\varepsilon_{ijs}\zeta_{ijs}\right)  ,
\]

then%

\begin{align*}
S(\Delta,\{\chi_{ij},\bar{\chi}_{ij},\varepsilon_{ijs}\}) &
=\operatorname{Re}\sum_{i<j,s}\varepsilon_{ijs}\bar{\chi}_{ij}\zeta_{ijs},\\
&  =\operatorname{Re}\sum_{i<j}\bar{\chi}_{ij}\theta_{ij}%
\end{align*}
can be considered to describe the Regge calculus for the $SO(4,C)$ general
relativity. The angle $\theta_{ij}$ are the deficit angles associated with the
triangles and the $n_{is}$ are the complex vector normals associated with the
tetrahedra. From the analysis that has been done in this section, it is easy
see that the $SO(4,C)$ Regge calculus contains the Regge calculus theories for
all the signatures. The Regge calculus for each signature can be obtained by
restricting the $n_{is}$ and the $\chi_{ij}$ to the corresponding homogenous
space and representations \cite{RealitySpinFoam}. Also by the properly
restricting the $n_{is}$ and the $\chi_{ij}$ we can derive the Regge calculus
corresponding to the mixed Lorentzian and multi-signature Barrett-Crane models
described in the previous subsections. The details of the relation between the
$SO(4,C)$ Regge Calculus and the real Regge Calculus for different signature
will be studied elsewhere.