spacetime. In (256), $G^{(1)}_{\rm con}(x_{1},x_{2},x_{3},x_{4})$ is the
coefficient of $g^{2}$ in the expansion of the Greens function ${G}_{\rm
con}(x_{1},x_{2},x_{3},x_{4})$ in a power series expansion in $g^{2}$.

###  A.3 The solution at first subleading order

The solution of the second of (243) (subject to the boundary conditions listed
in the second of (242)) is given by

$\phi_{1}(x)=\int d^{D}x_{4}G(x_{4},x)\phi^{3}_{0}(x_{4})$ (254)

Once again, in order to avoid writing lengthy equations we have not
substituted for $\phi_{0}$ and displayed our answer explicitly in terms of
$\beta$; the answer is given by three bulk to boundary propagators, sewn
together at the bulk point $x_{4}$ which itself then connects to a bulk to
bulk propagator ending at $x$. The bulk point $x_{4}$ is integrated over, as
are the boundary points (weighted with boundary values of the fields).

As in the previous subsection, the knowledge of $\phi_{1}$ (even without
knowing $\phi_{2}$) is sufficient to evaluate the action to second order. At
this order, $\phi_{2}$ could only have appeared in the quadratic part of the
action in the schematic combination $\phi_{0}\phi_{2}$. Once again the
coefficient of this contribution vanishes because $\phi_{0}$ is a solution to
the equation of motion that follows from the quadratic part of the action. The
action at this order is simply given by

$\begin{split}S_{2}&=\int d^{D}x_{5}~{}\phi_{1}(x_{5})\phi^{3}_{0}(x_{5})\\\
&=\int d^{D}x_{5}\int d^{D}x_{4}\int
dA_{1}dA_{2}dA_{3}~{}G(x_{5},x_{4})G_{\partial B}(x_{4},y_{1})G_{\partial
B}(x_{4},y_{2})G_{\partial
B}(x_{4},y_{3})~{}\beta(y_{1})\beta(y_{2})\beta(y_{3})\\\ &\int
dA_{4}dA_{5}dA_{6}~{}G_{\partial B}(x_{5},y_{4})G_{\partial
B}(x_{5},y_{5})G_{\partial
B}(x_{5},y_{6})~{}\beta(y_{4})\beta(y_{5})\beta(y_{6})\end{split}$ (255)

Note that

$\begin{split}\frac{\delta^{6}S_{2}}{\delta\beta(y_{1})\cdots\delta\beta(y_{6})}&=6!\int
d^{D}x_{5}\int d^{D}x_{4}\int dA_{1}\cdots dA_{6}~{}G(x_{5},x_{4})G_{\partial
B}(x_{4},y_{1})G_{\partial B}(x_{4},y_{2})G_{\partial B}(x_{4},y_{3})\\\
&~{}G_{\partial B}(x_{5},y_{4})G_{\partial B}(x_{5},y_{5})G_{\partial
B}(x_{5},y_{6})\\\
&=n_{1}.\nabla_{1}n_{2}.\nabla_{2}n_{3}.\nabla_{3}n_{4}.\nabla_{4}n_{5}.\nabla_{5}n_{6}.\nabla_{6}G^{(1)}_{\rm
con}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})\big{|}_{x_{i}=y_{i}}\end{split}$
(256)

We obtain a sum of 10 such exchange graphs (three boundary points leading to a
bulk point connected to another bulk point and going back to three other
boundary points) with both bulk points integrated over all possible values,
and all six boundary points smeared against $\beta$ and then integrated over
all values.

###  A.4 Generalization to arbitrary order and to the quantum theory

The generalization of the low order computations presented above, both to
higher orders and also to the quantum theory is clear. Let us define

$Z=\int{\cal
D}\phi~{}e^{-S}=e^{\sum\limits_{n}\frac{1}{n!}\int\left(\prod_{i=1}^{n}dy_{i}\beta(y_{i})\right)G_{\rm
bdry,con}(y_{1}\cdots
y_{n})}=\sum_{n}\int\prod_{i=1}^{n}dy_{i}\beta(y_{i})\frac{G_{\rm
bdry}(y_{1}\cdots y_{n})}{n!}$ (257)

Note the boundary Greens function $G_{\rm bdry}$ appeared in (22), which is
the full Greens function including the disconnected diagrams. Then

$G_{\rm bdry,con}(y_{1}\cdots
y_{n})=\prod_{i=1}^{n}{\hat{n}}_{i}.\nabla_{i}{G}_{\rm
con}(x_{1},x_{2}\ldots,x_{n})$ (258)

where the function ${G}_{\rm con}(x_{1},x_{2}\ldots,x_{n})$ was defined in
(253). The explicit computations presented above may be viewed as an explicit
verification of (23).

###  A.5 Propagators in the limit $mT\gg 1$

The analysis presented so far in this Appendix has been exact; in particular,
it applies to every choice of cut-off surface. Given a general choice of cut-
off surface, however, the two point Greens functions (in terms of which we
have presented our analysis) would, in general, be very difficult to compute.

Of course, this problem is ameliorated in especially symmetrical situations.
For instance, if our spacetime is the ‘slab’ defined by $-T\leq t\leq T$ (here
$t$ is Euclidean time) then it is possible to use the method of images to find
reasonably explicit formulae for all Greens functions (see Appendix B below).
However even in this situation, the explicit results are given in terms of
infinite sums, and so are relatively complicated.

There is one situation, in which, however, all Greens functions can be well
approximated in a relatively simple manner even for arbitrary smooth cut-off
surfaces. This situation arises when spacetime bounded by the cut-off is
convex, the cut-off surface of our spacetime only varies on the length scale
of the IR cut-off $T$, and finally when $mT\gg 1$. Let us consider two bulk
points which are separated from all boundary points by a distance that is held
fixed in units of $T$ as $T$ is taken to infinity. As we explain in more
detail in subsection B.3 (see the paragraph above (284)) the bulk to bulk
propagator in such a situation is well approximated by the Euclidean Greens
function on $R^{D}$. Moreover, the bulk to boundary propagator in the same
situation (again assuming that the bulk point in question is separated from
all boundary points by a distance that is held fixed in units of $T$) is well
approximated by evaluating $n.\nabla G$ after accounting for the contribution
of a single ‘image’ charge: the reflection of the ‘source’ that approaches the
boundary. In order to evaluate $n.\nabla G$ we only need to reflect a source
that is infinitesimally separated from the boundary. In such a situation the
effective image charge is obtained by reflecting in the boundary pretending it
is flat. The error generated by this process - stemming from the fact that the
boundary is curved rather than flat is of order $e^{-\alpha mT}$ where
$\alpha$ is a positive number. Implementing this procedure, it follows that in
this situation the bulk to boundary Greens function is well approximated by

$G_{\rm
bdry}(x,x^{\prime})=-\frac{d}{da}\left({\widetilde{G}}(x,x^{\prime}-a{\hat{n}^{\prime}})-{\widetilde{G}}(x,x^{\prime}+a{\hat{n}^{\prime}})\right)=2{\hat{n}}^{\prime}.\nabla^{\prime}{\widetilde{G}}(x,x^{\prime})$
(259)

where ${\widetilde{G}}$ the Euclidean propagator.

In summary, both the bulk to bulk and the bulk to boundary propagators have a
simple explicit form in the limit $mT\gg 1$.

###  A.6 Propagators in Momentum space

In this section, we give momentum space expressions for the bulk-to-boundary
and boundary-to-boundary propagators discussed previously in the case of a
Euclidean ‘slab’ $-T<\tau<T$. It is convenient to parametrize time by shifting
it by $\frac{T}{2}$, so we will take $0\leq\tau\leq 2T$. The equation of
motion is given by

$\delta^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi-m^{2}\phi=0,$ (260)

which we can write in momentum space along the spatial directions as

$\phi(\tau,\vec{x})=\int\frac{d^{d}\vec{k}}{(2\pi)^{d}}e^{i\vec{k}\cdot\vec{x}}\phi_{\vec{k}}(\tau),$
(261)
$\partial_{\tau}^{2}\phi_{\vec{k}}-\omega_{\vec{k}}^{2}\phi_{\vec{k}}=0,\;\;\;\cdots\;\;\;(\omega^{2}_{\vec{k}}=\vec{k}^{2}+m^{2}).$
(262)

The solutions are linear combinations of the form:

$\phi_{\vec{k}}(\tau)=a_{\vec{k}}\sinh(\omega_{\vec{k}}\tau)+b_{\vec{k}}\cosh(\omega_{\vec{k}}\tau).$
(263)

#### Bulk to boundary propagator

In order to get the bulk to boundary propagator, we need to impose boundary
conditions. Let us first consider the bulk to boundary propagator
$K(X|\vec{y})$, where $X=(\tau,\vec{x})$ is the bulk point and $\vec{y}$ is a
boundary point at either the future or past boundary where the propagator
approaches a delta function. More explicitly, if the source point is on the
past boundary, the bulk to boundary propagator satisfies the boundary
condition:

$K_{E}^{(-)}(\tau=0,\vec{x}|\vec{y})=\delta^{d}(\vec{x}-\vec{y}),\;\;K_{E}^{(-)}(\tau=2T,\vec{x}|\vec{y})=0,$
(264)

while if the source point is on the future boundary, the propagator satisfies

$K_{E}^{(+)}(\tau=0,\vec{x}|\vec{y})=0,\;\;K_{E}^{(+)}(\tau=2T,\vec{x}|\vec{y})=\delta^{d}(\vec{x}-\vec{y}).$
(265)

Thus, we get

$K_{E}^{(+)}(\tau,\vec{x}|\vec{y})=\int\frac{d^{d}\vec{k}}{(2\pi)^{d}}e^{i\vec{k}\cdot(\vec{x}-\vec{y})}\frac{\sinh(\omega_{\vec{k}}\tau)}{\sinh(2\omega_{\vec{k}}T)},$
(266)
$K_{E}^{(-)}(\tau,\vec{x}|\vec{y})=\int\frac{d^{d}\vec{k}}{(2\pi)^{d}}e^{i\vec{k}\cdot(\vec{x}-\vec{y})}\frac{\sinh(\omega_{\vec{k}}(2T-\tau))}{\sinh(2\omega_{\vec{k}}T)}.$
(267)

### Bulk to bulk propagator

In order to get the bulk to bulk propagator, we first solve the eigenvalue
equation:

$(\partial_{\tau}^{2}-\omega_{\vec{k}}^{2})\psi_{E,\vec{k}}(\tau)=-E^{2}\psi_{E,\vec{k}}(\tau),$
(268)

with Dirichlet boundary conditions at $\tau=0$ and $\tau=2T$. For
$E^{2}>\omega_{\vec{k}}^{2}$, we get

$\psi_{E,\vec{k}}(\tau)=\alpha_{E,\vec{k}}\sin(\sqrt{E^{2}-\omega^{2}_{\vec{k}}}\tau)+\beta_{E,\vec{k}}\cos(\sqrt{E^{2}-\omega^{2}_{\vec{k}}}\tau).$
(269)

This can only satisfy Dirichlet boundary conditions for

$\sqrt{E^{2}_{n}-\omega_{\vec{k}}^{2}}=\frac{\pi
n}{2T},\;\;\;\cdots\;\;\;(n\in\mathbb{Z}).$ (270)

Thus, we get a complete set of (normalized) eigenfunctions for $n>0$:

$\psi_{n,\vec{k}}(\tau)=\frac{1}{\sqrt{T}}\sin\left(\frac{\pi
n\tau}{2T}\right),$ (271)

There are no non-trivial solutions for $E^{2}\leq\omega_{\vec{k}}^{2}$. Thus,
the bulk to bulk propagator is given by

$G_{E}(\tau_{1},\vec{x}_{1}|\tau_{1},\vec{x}_{2})=-\frac{1}{T}\sum_{n=1}^{\infty}\int\frac{d^{d}\vec{k}}{(2\pi)^{d}}e^{i\vec{k}\cdot(\vec{x}_{1}-\vec{x}_{2})}\frac{\sin(\frac{\pi
n\tau_{1}}{2T})\sin(\frac{\pi
n\tau_{2}}{2T})}{\omega_{\vec{k}}^{2}+\frac{\pi^{2}n^{2}}{4T^{2}}}.$ (272)

Note that in the large $T$ limit, the sum becomes an integral over the
intermediate energy.

#### Example: Tree level contact diagram

As an example of diagrams one can compute using these propagators, consider
the boundary correlation function where we take two derivatives with respect
to the source on the future boundary (labeled by the points $\vec{y}_{1,2}$
below) and two derivatives on the past boundary (labeled by the points
$\vec{y}_{3,4}$ below). In $\phi^{4}$ theory, this boundary correlation
function has a tree-level contact diagram contribution, which we can compute
as

$G_{\text{bdry}}\Big{|}_{\text{contact}}=\lambda_{4}\int_{0}^{2T}d\tau\int
d^{d}\vec{x}K^{(+)}(\tau,\vec{x}|\vec{y}_{1})K^{(+)}(\tau,\vec{x}|\vec{y}_{2})K^{(-)}(\tau,\vec{x}|\vec{y}_{3})K^{(-)}(\tau,\vec{x}|\vec{y}_{4}).$
(273)

The integral over spatial coordinates gives a momentum-conserving delta
function. The integral over $\tau$ is of the form:

$\int_{-T}^{T}d\tau\,\frac{\sinh(\omega_{1}(T+\tau))}{\sinh(2\omega_{1}T)}\frac{\sinh(\omega_{2}(T+\tau))}{\sinh(2\omega_{2}T)}\frac{\sinh(\omega_{3}(T-\tau))}{\sinh(2\omega_{3}T)}\frac{\sinh(\omega_{4}(T-\tau))}{\sinh(2\omega_{4}T)},$
(274)

where $\omega_{1}=\omega_{\vec{k}_{1}}$ and so on, and we have shifted the
integration variable. In the large $T$ limit, the denominators give a
suppression factor of $e^{-2T\sum_{i}\omega_{i}}$. The only term that survives
in the integral in this limit is the exponentially growing part of the
$\sinh$, which gives

$e^{-T\sum_{i}\omega_{i}}\int_{-T}^{T}d\tau\,e^{\tau(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})}+\cdots=e^{-T\sum_{i}\omega_{i}}\frac{2\sinh\left((\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})T\right)}{(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})}+\cdots,$
(275)

where $\cdots$ denotes terms that are exponentially subleading in $T$.
Continuing analytically in $T$ to the imaginary axis, we get666666The terms we
dropped were exponentially smaller in $T$, and even after analytic
continuation, they remain so. This is because under Wick rotation we send
$T\to T(i+\epsilon)$, and we take $T\to\infty$ along this contour before
sending $\epsilon\to 0$.

$e^{-iT\sum_{i}\omega_{i}}2\pi\frac{\sin((\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})T)}{\pi(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})},$
(276)

which in the $T\to\infty$ limit gives

$e^{-iT\sum_{i}\omega_{i}}2\pi\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}).$
(277)

The overall oscillating factor is precisely canceled by the wavefunctions in
equation (27), and thus, this boundary correlation function when inserted in
(27) precisely reproduces the expected result for the S-matrix.

##  Appendix B Euclidean Two Point Functions in position space: massive
particles

In this Appendix, we first review the result for the exact Euclidean massive
two point function on $R^{D}$, and then present a series expansion for the two
point function on the ‘slab’ of $R^{D}$ defined by $-T<t<T$ where $t$ is one
of the Cartesian coordinates in $R^{D}$ and the two point function is computed
after imposing Dirichlet boundary conditions.

###  B.1 Euclidean Greens Function on $R^{d}$ equals the bulk to bulk
propagator

The Euclidean Greens function on $R^{D}$, ${\widetilde{G}}$, is the solution
of the equation

$\left(\nabla^{2}-m^{2}\right)\widetilde{G}(\vec{r}_{1},\vec{r}_{2})=\delta^{D}(\vec{r}_{1}-\vec{r}_{2})$

that vanishes at infinity. Translational and rotational invariance tell us
that $\widetilde{G}=\widetilde{G}(|\vec{r}_{1}-\vec{r}_{2}|)$. Let
$y=|\vec{r}_{1}-\vec{r}_{2}|$. It follows that at $y\neq 0$

$\frac{1}{y^{D-1}}\partial_{y}y^{D-1}\partial_{y}{\widetilde{G}}(y)-m^{2}{\widetilde{G}}=0$
(278)

The general solution to (278) is,

${\widetilde{G}}(y)=a(my)^{\frac{2-D}{2}}I_{\frac{D-2}{2}}(my)+b(my)^{\frac{2-D}{2}}K_{\frac{D-2}{2}}(my)$
(279)

At small and large values of the arguments

$\begin{split}&I_{\frac{D-2}{2}}(z)\approx\frac{1}{\Gamma(\frac{D-2}{2}+1)}\left(\frac{z}{2}\right)^{\frac{D-2}{2}},~{}~{}~{}{\rm
small}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\approx\frac{e^{z}}{\sqrt{2\pi
z}}~{}~{}~{}~{}~{}{\rm large}\\\
&K_{\frac{D-2}{2}}(z)\approx\frac{\Gamma(\frac{D-2}{2})}{2}\left(\frac{2}{z}\right)^{\frac{D-2}{2}},~{}~{}~{}{\rm
small}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\approx\pi\frac{e^{-z}}{\sqrt{2\pi
z}}~{}~{}~{}~{}~{}{\rm large}\\\ \end{split}$ (280)

Since we are interested in the decaying solution, conclude that $a=0$ in
(278). The value of $b$ is fixed by the requirement that the action of
$\nabla^{2}-m^{2}$ produces a delta function at zero on the right hand side.
Using the fact that the volume of the unit $D-1$ sphere is
$\frac{\pi^{\frac{D}{2}}}{\Gamma(\frac{D}{2}+1)}$ we conclude that

${\widetilde{G}}(y)=b_{D}(my)^{\frac{2-D}{2}}K_{\frac{D-2}{2}}(my)$ (281)

where
$y(t,t^{\prime},\vec{r},\vec{r^{\prime}})=\sqrt{(t-t^{\prime})^{2}+|\vec{r}-\vec{r^{\prime}}|^{2}}$
and $b_{D}=\frac{m^{D}}{(2\pi)^{\frac{D}{2}}}$.

In the large $y$ limit

${\widetilde{G}}(y)\approx\left(\frac{\pi}{(2\pi)^{\frac{D+1}{2}}}m^{\frac{D+1}{2}}\right)~{}e^{-my}y^{\frac{1-D}{2}}$
(282)

###  B.2 Greens function for the ‘slab’

We now work in a space in which the Euclidean time direction has a cutoff at
$t=\pm T$ and the Greens function satisfies $G(y)|_{t=\pm T}=0$. The Greens
function in this space is easily obtained using the method of images. The
total number of images is infinite (as the image in one boundary must also be
reflected in the other boundary, and so on). The final answer for the Greens
function is given by:

$\begin{split}G(t,t^{\prime})&=\widetilde{G}(|t-t^{\prime}|)+\\\
&-\sum_{n=0}^{\infty}\left[\widetilde{G}\left(|t+t^{\prime}+(4n+2)T|\right)+\widetilde{G}\left(|t+t^{\prime}-(4n+2)T|\right)\right]\\\
&+\sum_{n=1}^{\infty}\left[\widetilde{G}\left(|t-t^{\prime}-4nT|\right)+\widetilde{G}\left(|t-t^{\prime}+4nT|\right)\right]\\\
\end{split}$ (283)

where we have suppressed the dependence of the Greens functions on ${\vec{r}}$
and ${\vec{r}}^{\prime}$ (these dependence just go along for the ride).

###  B.3 Approximate Greens Functions when $mT\gg 1$

In the limit $mT\gg 1$, and for all values of $t^{\prime}$ and $t$ lying
within the slab, we obtain a good approximation to (283) by keeping the terms
in (283) with $n\leq 2$.

In fact when $t$ and $t^{\prime}$ are separated from both boundaries by a
finite fraction of $T$, we get an excellent approximation 676767For the
purposes of this Appendix, we define an approximation to the Greens function
to be good if the fractional error in the Greens function is of order
$e^{-\alpha mT}$ where $\alpha$ is any positive number that is held fixed as
$T$ is taken to infinity. to the Greens function by retaining only those terms
in (283) with $n=0$. That this is the case is easily seen from the exact
formula (283), but may also be understood physically in the following terms.
The Greens function $G_{\text{full}}(t,t^{\prime})$ may be thought of as the
‘potential’ created at $t$ by a ‘charge’ placed at $t^{\prime}$. This
potential has two contributions. The first comes directly from the charge. The
second comes from the ‘rearrangement of surface charges’ that are needed in
order to enforce the Dirichlet boundary conditions of our problem. As the
theory we are dealing with is massive, the direct effect is of order $e^{-md}$
where $d$ is the distance between the source and the measuring point. On the
other hand, the second effect is proportional to $e^{-m(d_{1}+d_{2})}$ where
$d_{1}$ is the distance between the source and any point on the boundary,
while $d_{2}$ is the distance between the same boundary point and the
measuring point. If the source and measuring point are both separated by the
boundary by a finite fraction of $T$, it is obvious that $d_{1}+d_{2}-d$ is
positive and is also a finite fraction of $d$. It follows that the ‘direct’
contribution (first line of the RHS of (283)) is a good approximation to the
full result. In other words, in this case

$G\approx\widetilde{G}(t,t^{\prime})$ (284)

When $t^{\prime}$ approaches one of the boundaries but $t$ is in the bulk (and
is separated from both boundaries by a distance that is a finite fraction of
$T$), the argument presented above would suggest that, in addition to the
‘direct’ term on the first line of the RHS of (283), we need to include one of
the two $n=0$ terms on the second line of the RHS of (283) 686868This term can
be thought of as capturing the reflection of the source at $t^{\prime}$ in the
boundary that it is tending towards.; all other terms in this expansion may be
neglected. It is easy to verify directly from (283) that this approximation is
indeed accurate. In this situation, in other words, if either $t$ or
$t^{\prime}$ or bout approaches the boundary at time $T$

$G\approx\widetilde{G}(t,t^{\prime})-\widetilde{G}\left(t,2T-t^{\prime}\right)$
(285)

while if $t$ or $t^{\prime}$ or both approach the boundary at time $-T$

$G\approx\widetilde{G}(t,t^{\prime})-\widetilde{G}\left(t,-2T-t^{\prime}\right)$
(286)

Finally, if $t$ approaches the boundary at time $T$ while if $t^{\prime}$
approaches the boundary at time $-T$ then

$G\approx\widetilde{G}(t,t^{\prime})-\widetilde{G}\left(t,2T-t^{\prime}\right)-\widetilde{G}\left(t,-2T-t^{\prime}\right)+\widetilde{G}\left(|t-t^{\prime}-4T|\right)$
(287)

On the other hand of $t$ approaches the boundary at time $-T$ while
$t^{\prime}$ approaches the boundary at time $T$,

$G\approx\widetilde{G}(t,t^{\prime})-\widetilde{G}\left(t,2T-t^{\prime}\right)-\widetilde{G}\left(t,-2T-t^{\prime}\right)+\widetilde{G}\left(|t-t^{\prime}+4T|\right)$
(288)

Assuming that all bulk points are separated from both boundaries by a finite
fraction of $T$, therefore, it follows that the bulk to bulk propagator is
well approximated by (284). On the other hand, the bulk to boundary propagator
is well approximated by

$G_{\rm
bdry}=\partial_{t^{\prime}}G(t,t^{\prime})|_{t^{\prime}=t}=2\partial_{t^{\prime}}{\widetilde{G}}(t,t^{\prime})|_{t^{\prime}=t}=2{\hat{n}}.\nabla^{\prime}\widetilde{G}(x,x^{\prime})$
(289)

In Lorentzian signature, this relationship is given by

$G_{\rm bdry}=2i{\hat{n}}.\nabla^{\prime}\widetilde{G}(x,x^{\prime})$ (290)

This is because in the usual definition of Lorentzian Greens function,
$-i\delta(x-x^{\prime})$ appears on the RHS whereas in the Euclidean case
discussed in the previous Appendix, we defined Greens function with just the
delta function on RHS. Tracking the $i$’s in our Euclidean derivation, it is
easy to verify (290).

Using the explicit expression for ${\widetilde{G}}$ presented in the previous
Appendix, it is easy to explicitly evaluate the RHS of (289). Depending on
whether the boundary is at time $T$ or time $-T$ we find

$\begin{split}G_{\partial
B}(r,t,r^{\prime},T)&=\partial_{t^{\prime}}G(y)|_{t^{\prime}=T}=2b_{D}m^{2}(t-T)\frac{K_{\frac{D}{2}}\left(my_{T-t}\right)}{\left(my_{T-t}\right)^{\frac{D}{2}}}\\\
G_{\partial
B}(r,t,r^{\prime},-T)&=-\partial_{t^{\prime}}G(y)|_{t^{\prime}=-T}=-2b_{D}m^{2}(t+T)\frac{K_{\frac{D}{2}}(my_{T+t})}{(my_{T+t})^{D/2}}\end{split}$
(291)

where, $y_{t}=\sqrt{|\vec{r}-\vec{r^{\prime}}|^{2}+t^{2}}$. In the limit that
$my_{T}$ is large we obtain:

$\lim_{T\to\infty}G_{\partial
B}(r,t,r^{\prime},T)=\lim_{T\to\infty}G_{\partial
B}(r,t,r^{\prime},-T)=b_{D}\sqrt{2\pi
m^{3}T}{e^{-my_{T}}\over\left(my_{T}\right)^{D\over 2}}$ (292)

Finally, the boundary to boundary propagators, (249), are also easily
evaluated. We have two types of boundary to boundary propagators, one in which
both points are at the same boundary (we denote them by
$G_{\partial\partial}^{++}$ and $G_{\partial\partial}^{--}$) and the other one
in which one point is at boundary $T$ and other point is at $-T$ (we denote
them by $G_{\partial\partial}^{+-}$).

$\begin{split}G_{\partial\partial}^{++}(r,T,r^{\prime},T)&=2b_{D}m^{2}\frac{K_{\frac{D}{2}}(my_{0})}{(my_{0})^{\frac{D}{2}}}\\\
G_{\partial\partial}^{+-}(r,T,r^{\prime},-T)&={4b_{D}m^{2}\over\left(my_{2T}\right)^{D+2\over
2}}\left(4(mT)^{2}K_{\frac{D+2}{2}}\left(my_{2T}\right)-my_{2T}K_{\frac{D}{2}}\left(my_{2T}\right)\right)\end{split}$
(293)

In the large $T$ limit these expressions simplify to

$\begin{split}G_{\partial\partial}^{++}(r,T,r^{\prime},T)&=2b_{D}m^{2}\frac{K_{\frac{D}{2}}(my_{0})}{(my_{0})^{\frac{D}{2}}}\\\
G_{\partial\partial}^{+-}(r,T,r^{\prime},-T)&=8b_{D}m^{2}(mT)^{2}\sqrt{2\pi}{e^{-my_{2T}}\over\left(my_{2T}\right)^{\frac{D+3}{2}}}\end{split}$
(294)

##  Appendix C Toy Model: Harmonic oscillator

In this Appendix, we study the coherent state path integral for the
‘anharmonic’ harmonic oscillator defined by the Hamiltonian

$\mathcal{H}=\frac{P^{2}}{2}+\frac{\omega^{2}X^{2}}{2}+{\tilde{V}}(X)$ (295)

where $X$ and $P$ are the usual position and momentum operators. We also
recall the usual position space path integral for the same system, and comment
on the relationship between these two path integrals.

###  C.1 Coherent State Quantization for a Harmonic Oscillator

Recall that coherent states for the Harmonic oscillator are defined so that

$a|z\rangle=z|z\rangle\qquad\text{where}\qquad|z\rangle=e^{za^{\dagger}}|0\rangle=\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n!}}|n\rangle$
(296)

where $|n\rangle$ is the n-particle state. The inner product is given by:

$\langle z^{\prime}|z\rangle=e^{z^{\prime}z}$ (297)

and obey the (over) completeness relation

$\int\frac{d^{2}z}{\pi}e^{-z{\bar{z}}}|z\rangle\langle z|=I$ (298)

Recall that the operator $X(t)$ can be expanded in terms of $a$ and
$a^{\dagger}$

$X=\frac{1}{\sqrt{2\omega}}\left(ae^{-i\omega t}+a^{\dagger}e^{i\omega
t}\right)$ (299)

It follows that $z$, the eigenvalue of the operator $a$, is ‘positive energy
data’, while ${\bar{z}}$ is ‘negative energy data’.

Let us define the matrix elements (between coherent states) of the time
evolution operator $e^{-iHt}$ ($H$ is the Hamiltonian operator) as

$U(\bar{z}_{f},t_{f};z_{i},t_{i})=\langle
z_{f}|e^{-iH(t_{f}-t_{i})}|z_{i}\rangle$ (300)

Using the usual Feynman time slicing method (and inserting (298) into each
time step), it is not difficult to demonstrate that
$U(\bar{z}_{f},t_{f};z_{i},t_{i})$ is given by the Lorentzian path integral
Itzykson:1980rh

$U(\bar{z}_{f},t_{f};z_{i},t_{i})=\int\mathcal{D}(z,\bar{z})\,e^{i\mathcal{S}}$
(301)

where

$\mathcal{S}=-i\frac{\bar{z}_{f}z_{f}+\bar{z}_{i}z_{i}}{2}-i\int_{t_{i}}^{t_{f}}dt\left[\frac{\dot{\bar{z}}z-\dot{z}\bar{z}}{2}-i\mathcal{H}(z,\bar{z})\right]$
(302) 696969The action (302) can be rewritten as
$\begin{split}\mathcal{S}&=-i\frac{\bar{z}_{f}z_{f}+\bar{z}_{i}z_{i}}{2}-i\int_{t_{i}}^{t_{f}}dt\left[\frac{({\dot{z}}+{\dot{\bar{z}}})(z-{\bar{z}})-z{\dot{z}}+{\bar{z}}{\dot{\bar{z}}}}{2}-i\mathcal{H}(z,\bar{z})\right]\\\
&=i\frac{z_{f}^{2}-z_{i}^{2}-{\bar{z}}_{f}^{2}+{\bar{z}}_{i}^{2}-2\bar{z}_{f}z_{f}-2\bar{z}_{i}z_{i}}{4}-i\int_{t_{i}}^{t_{f}}dt\left[\frac{({\dot{z}}+{\dot{\bar{z}}})(z-{\bar{z}})}{2}-i\mathcal{H}(z,\bar{z})\right]\\\
\end{split}$ (303) The point of this rewriting is that the integral that
appears in the last of (303) is simply
$\int_{t_{i}}^{t_{f}}dt\left[p{\dot{x}}-\mathcal{H}(x,p)\right]$ (304) (where
we have used $z=x+ip$ and ${\bar{z}}=x-ip$). (304) is (upto normal ordering
issues, see below) the ‘usual’ form of the action - i.e. the action
appropriate for the computation of the action (or the path integral) as a
function of $x$ at initial and final times.

The function ${\cal H}$ given by The Hamiltonian is given by

$\mathcal{H}=\omega z{\bar{z}}+V\left(\frac{z+\bar{z}}{\sqrt{2\omega}}\right)$
(305) 707070Roughly speaking, $z$ and $\bar{z}$ are related to the variables
$x$ and $p$ in the phase space path integral via  $\omega
x+ip=\sqrt{2\omega}z,\qquad\qquad\omega x-ip=\sqrt{2\omega}{\bar{z}}$ (306)

The function $V(x)$ is related to the potential ${\tilde{V}}(x)$ that appears
in (295) as

${\tilde{V}}(x)=e^{-\frac{\partial_{x}^{2}}{4\omega}}V(x),~{}~{}~{}{V}(x)=e^{+\frac{\partial_{x}^{2}}{4\omega}}{\tilde{V}}(x)$
(307) 717171The nontrivial relationship between $V(x)$ and ${\tilde{V}}(x)$ is
a manifestation of operator ordering issues. Consider a Harmonic oscillator
with potential ${\tilde{V}}(X)$ (here $X$ is the position operator). This
potential may be rewritten in terms of the operators $a$ and $a^{\dagger}$,
but the expression one obtains is not normal ordered. Wicks theorem tells us
that
${\tilde{V}}(X)=~{}:e^{+\frac{\partial_{X}^{2}}{4\omega}}{\tilde{V}}(X):$
(308) where the symbol $::$ denotes normal ordering (recall that $\langle
XX\rangle=\frac{1}{2\omega}$). Once we have normal ordered the potential, the
repeated insertion of (298) into (300) converts the potential as a function of
$a$ and $a^{\dagger}$ into a function of ${\bar{z}}$ and $z$. It is crucial
that this simple replacement rule applies only once we have normal ordered.
These facts, taken together, explain (307).

The path integral in (301) is evaluated subject to the boundary conditions
that $z(t_{i})=z_{i}$ and ${\bar{z}}(t_{f})={\bar{z}}_{f}$. In the above
expression, $\mathcal{H}(z,\bar{z})$ is the Hamiltonian of the system. Note
that the boundary conditions fix positive energy data at the initial time
$t_{i}$ but negative energy data at the final time $t_{f}$, exactly as in our
discussion of the S matrix (this is the reason the discussion of this
subsection is relevant to the S matrix).

The action in (302) (that comes out of time slicing the evolution operator in
the Feynman manner) is the sum of two terms; the explicit counterterm
$-i\frac{\bar{z}_{f}z_{f}+\bar{z}_{i}z_{i}}{2}$ plus an integral of the
‘Lagrangian’ over time. We emphasize that it is the sum of these two terms
that gives rise to an action that leads to a good variational principle (whose
variation implies the usual Harmonic Oscillator equations of motion). In
particular, the ‘counterterm’ is needed to cancel boundary variations, as we
now explain. The bulk action in $z$, ${\bar{z}}$ variables is given by

$\ \mathcal{S}_{\rm bulk}=-i\int
dt\left[\frac{\dot{\bar{z}}z-\dot{z}\bar{z}}{2}-i\mathcal{H}\right]$ (309)

We are interested in the path integral computed by specifying $z=z_{i}$ at
initial time $t_{i}$ and $\bar{z}=\bar{z}_{f}$ at final time $t_{f}$. As
mentioned above, these boundary conditions are equivalent to specifying
positive energy data in the past and negative energy data in the future. As we
have mentioned above, the measure for this path integral is $e^{iS}$ where $S$
is the time integral of the Lagrangian plus a counter-term. The form of this
counter-term (which comes out independently from the Feynman time slice
procedure) may independently have been guessed from the following
considerations. The variation of ${\mathcal{S}}_{\rm bulk}$ (see (309)) with
respect to $z$ and $\bar{z}$ is given by

$\displaystyle\delta{\mathcal{S}}_{\rm bulk}$ $\displaystyle=$
$\displaystyle\int dt\left[-i\frac{\dot{\bar{z}}\delta z-\bar{z}\dot{\delta
z}}{2}-\omega\bar{z}\delta
z-V^{\prime}\left(\frac{z+\bar{z}}{\sqrt{2\omega}}\right)\delta
z-i\frac{\dot{\delta\bar{z}}z-\delta\bar{z}\dot{z}}{2}-\omega
z\delta\bar{z}-V^{\prime}\left(\frac{z+\bar{z}}{\sqrt{2\omega}}\right)\delta\bar{z}\right]$
$\displaystyle=$ $\displaystyle\int
dt\left[\left(-i\dot{\bar{z}}-\omega\bar{z}-V^{\prime}\left(\frac{z+\bar{z}}{\sqrt{2\omega}}\right)\right)\delta
z+\left(i\dot{z}-\omega
z-V^{\prime}\left(\frac{z+\bar{z}}{\sqrt{2\omega}}\right)\right)\delta\bar{z}+\frac{d}{dt}\left(i\frac{\bar{z}\delta
z-\delta\bar{z}z}{2}\right)\right]$ $\displaystyle=$ $\displaystyle\int
dt\left[\left(\text{EOM of}\,\bar{z}\right)\delta z+\left(\text{EOM
of}\,z\right)\delta\bar{z}\right]+\frac{i}{2}\left(\bar{z}_{f}\delta
z_{f}+z_{i}\delta\bar{z}_{i}\right)$

where we have used the fact that $z(t_{i})=z_{i}$ hence $\delta z_{i}=0$,
similarly $\bar{z}(t_{f})=\bar{z}_{f}$ hence $\delta\bar{z}_{f}=0$. We have
also used the fact that the potential $V$ is a function of combination
$z+\bar{z}$, hence it’s derivative w.r.t. $z$ and $\bar{z}$ is same, which we
denote by $V^{\prime}\left(\frac{z+\bar{z}}{\sqrt{2\omega}}\right)$. Now we
see the need to add counter-terms in the action. The variation of action
should give us just the equation of motion but here we are getting some extra
boundary contributions to that variation. So we need to add appropriate
counter-terms to the action that cancels these boundary terms. Hence the full
action, with a well-defined variational principle is given by:

$\begin{split}&\mathcal{S}=\mathcal{S}_{\rm bdry}+\mathcal{S}_{\rm bulk}\\\
&{\rm where}\\\ &\mathcal{S}_{\rm
bdry}=-i\frac{\bar{z}_{f}z_{f}+\bar{z}_{i}z_{i}}{2},\qquad\qquad\mathcal{S}_{\rm
bulk}=-i\int_{t_{i}}^{t_{f}}dt\left[\frac{\dot{\bar{z}}z-\dot{z}\bar{z}}{2}-i\omega
z\bar{z}-iV\left(\frac{z+\bar{z}}{\sqrt{2\omega}}\right)\right]\end{split}$
(310)

which matches (302). Therefore the boundary term in (302) is required to make
the variational principle well-defined.

###  C.2 Diagrammatic Computation of the coherent state path integral

In the free theory, $z$ and $\bar{z}$ obey the following equations

$\dot{{z}}+i\omega z=0,\qquad\qquad-\dot{\bar{z}}+i\omega\bar{z}=0$ (311)

Let us define the coherent state ‘bulk to bulk’ propagators as

$\begin{split}\langle
z(t)\bar{z}(t^{\prime})\rangle&:=G_{F}^{z\bar{z}}(t,t^{\prime})=e^{-i\omega(t-t^{\prime})}\Theta(t-t^{\prime})\\\
\end{split}$ (312)

These propagators obey the following equations

$\begin{split}\frac{d}{dt}G_{F}^{z\bar{z}}(t,t^{\prime})+i\omega
G_{F}^{z\bar{z}}(t,t^{\prime})&=\delta(t-t^{\prime})\\\
-\frac{d}{dt^{\prime}}G_{F}^{z\bar{z}}(t,t^{\prime})+i\omega
G_{F}^{z\bar{z}}(t,t^{\prime})&=\delta(t-t^{\prime})\end{split}$ (313)

and satisfy the following boundary conditions

$G_{F}^{z\bar{z}}(0,t^{\prime})=0,\qquad\qquad G_{F}^{z\bar{z}}(t,T)=0$ (314)

Note that the bulk to boundary propagator (312) is discontinuous at
$t=t^{\prime}$. The value of this propagator at precisely $t=t^{\prime}$ is
relevant only for ‘tadpole’ contractions in Feynman diagrams. Since the our
potential $V$ is normal ordered, all such tadpoles must vanish. In order to
ensure this we set

$G_{F}^{z\bar{z}}(t,t)=0$ (315)

Similarly, the coherent state ‘bulk to boundary’ propagators are given by

$K_{F}^{(+)}(t)=e^{-i\omega t},\qquad\qquad K_{F}^{(-)}(t)=e^{i\omega(t-T)}$
(316)

These propagators obey the first and second of (313) respectively but with the
following boundary conditions

$K_{F}^{(+)}(0)=1,\qquad\qquad K_{F}^{(-)}(T)=1$ (317)

It is easy to check that

$\begin{split}&K_{F}^{(+)}(t)=G_{F}^{{z\bar{z}}}(t,0)\\\
&K_{F}^{(-)}(t)=G_{F}^{z{\bar{z}}}(T,t)\end{split}$ (318)

as could have been anticipated on general grounds. 727272The equations (318)
can be abstractly argued for as follows. Let $z(t)$ and ${\bar{z}}(t)$ be
solutions of the free harmonic equations (311), and consider the two integrals
$\int_{t=0}^{T}\partial_{t}\left(z(t)G^{z\bar{z}}_{F}(t^{\prime},t)\right),~{}~{}~{}\int_{t=0}^{T}\partial_{t}\left({\bar{z}}(t)G_{F}^{z\bar{z}}(t^{\prime},t)\right)$
We can perform each of these integrals in two different ways: first by
reducing them to boundary terms, and second by opening out the derivatives and
using the (313) and (311). This procedure gives a formula for $z$ in terms of
$z_{i}$ and for ${\bar{z}}(t)$ in terms of ${\bar{z}}_{f}$ (i.e. for the bulk
to boundary propagators) in terms of the bulk to bulk propagators,
establishing (318).

As a short diversion, it is useful to analytically study the analytic
continuation of the bulk to bulk propagator (312) to Euclidean space.
Performing the continuation $t\rightarrow-i\tau$, we obtain the Euclidean bulk
to bulk propagators

$\begin{split}\langle
z(\tau)\bar{z}(\tau^{\prime})\rangle&:=G_{E}^{z\bar{z}}(\tau,\tau^{\prime})=e^{-\omega(\tau-\tau^{\prime})}\Theta(\tau-\tau^{\prime})\\\
\end{split}$ (319)

It follows that

$\begin{split}\langle\frac{z(\tau)+\bar{z}(\tau)}{\sqrt{2\omega}}~{}~{}\frac{z(\tau^{\prime})+\bar{z}(\tau^{\prime})}{\sqrt{2\omega}}\rangle&=G_{E}(\tau,\tau^{\prime})=\frac{1}{2\omega}e^{-\omega|\tau-\tau^{\prime}|}\\\
\end{split}$ (320)

But $G_{E}(\tau,\tau^{\prime})$ is simply the well known Euclidean propagator,
$\langle x(\tau)x(\tau^{\prime})\rangle$ of the Harmonic oscillator (i.e. the
propagator that is required to vanish when $|t|\to\infty$).

The relationship between the bulk to boundary propagators (318) and the
Euclidean propagator is

$\begin{split}&K_{F}^{(+)}(t)=\theta(t)(-2i\partial_{t})G_{F}(t,0)=\theta(t)(2i{\hat{n}}.\partial)G_{F}(t,t^{\prime})\bigg{|}_{t^{\prime}=0}\\\
&K_{F}^{(-)}(t)=\theta(T-t)(2i\partial_{t})G_{F}(T,t)=\theta(T-t)(2i{\hat{n}}.\partial)G_{F}(t,t^{\prime})\bigg{|}_{t=T}\end{split}$
(321)

where $G_{F}(t,t^{\prime})$ is the Feynman propagator obtained by analytic
continuation of the Euclidean propagator given in (320) and $\hat{n}$ is the
outward pointing normal vector to the boundary.

It follows that the in out path integral as a functional of boundary
conditions, has the following simple relation to purely Euclidean Greens
functions. If we denote the logarithm of the path integral as a functional of
$z_{i}$ and ${\bar{z}}_{f}$ as $Z(z_{i},{\bar{z}}_{f})$, then

$Z_{E}(z_{i},{\bar{z}}_{f})\approx\sum_{m,n}\frac{(\sqrt{2\omega}z_{i})^{m}(\sqrt{2\omega}{\bar{z}}_{f})^{n}}{m!n!}G_{E}^{m,n}(0,T)$
(322)

where the Euclidean propagator $G_{E}^{m,n}$ is defined by

$G_{E}^{m,n}(\tau,\tau^{\prime})=\langle
X^{m}(\tau)X^{n}(\tau^{\prime})\rangle$

Defining $\beta=\sqrt{2\omega}z_{i}$ and
${\bar{\beta}}=\sqrt{2\omega}{\bar{z}}_{f}$ (322) may be rewritten as

$Z_{E}(\beta,{\bar{\beta}})\approx\sum_{m,n}\frac{(2\omega\beta)^{m}(2\omega{\bar{\beta}})^{n}}{m!n!}G_{E}^{m,n}(0,T)\approx\sum_{m,n}\frac{(2\partial_{t}\beta)^{m}(2\partial_{t^{\prime}}{\bar{\beta}})^{n}}{m!n!}G_{E}^{m,n}(t,t^{\prime})\Bigg{|}_{t=0,t^{\prime}=T}$
(323)

The approximation signs in (322) and (323) have the following meaning. The
Euclidean Greens functions that appear on the RHS of these equations can, in
perturbation theory, be computed in the usual manner using the Feynman
expansion. This expansion involves an integral over the location of all
interaction vertices. In evaluating the Euclidean Greens functions on the RHS
of (322) and (323), we are instructed to integrate the integration location of
all interaction vertices over all of spacetime. We get the LHS of these
equations, however, only if we restrict the location of these interaction
vertices to the slab region, $t\in(0,T)$. The LHS is an approximation to the
RHS in the following sense. The extra region of integration space does give
contributions that scale like $T$ (i.e. an energy conserving $\delta$
function) and so do not contribute to the S matrix (see around (42)).

While the discussion of the last few paragraphs has focussed on the case of
quantum mechanics, the generalization to quantum field theory (regarded as a
theory of an infinite number of harmonic oscillators) is straightforward.

The propagators listed above can be used for a number of purposes. At the
classical level, one can sum tree diagrams built using these propagators to
(perturbatively) find the $z(t)$ and ${\bar{z}}(t)$ subject to the boundary
conditions described above. At leading order, for instance, one finds

$z_{0}(t)=z_{i}K_{F}^{(+)}(t),\qquad\qquad\bar{z}_{0}(t)={\bar{z}}_{f}K_{F}^{(-)}(t)$
(324) 737373A similar procedure, but without the restriction to tree diagrams,
yields expressions for time ordered correlators of $a(t)$ and $a^{\dagger}(t)$
in the quantum case.

One can also use the propagators listed above to evaluate the logarithm of the
Harmonic Oscillator path integral as a functional of $z_{i}$ and
${\bar{z}}_{f}$. In order to find the coefficient of
$z_{i}^{n}{\bar{z}}_{f}^{m}$, one simply evaluates the sum of all connected
Feynman diagrams with $n$ legs on the boundary at $t=0$ and $m$ legs on the
boundary at $t=T$ 747474The restriction to tree diagrams evaluates yields the
classical action as a function of this data..

As a simple check of this formalism (and especially of the fact that all
boundary terms are working out right) we have computed the classical action
resulting from the potential $V(x)=gx^{3}$, at order $g^{2}$, in two different
ways: first by summing (tree exchange) diagrams, and second by performing the
computation using the formalism of Faddeev:1980be ; Kraus:2002cb . We briefly
describe this second procedure.

The classical solutions of the nonlinear equations of motion

$\begin{split}\dot{\bar{z}}-i\omega\bar{z}-iV^{\prime}(z+\bar{z})&=0\\\
-\dot{{z}}-i\omega z-iV^{\prime}(z+\bar{z})&=0\end{split}$ (325)

that obey the boundary conditions $z(0)=z_{i}$ and
${\bar{z}}(0)={\bar{z}}_{f}$, are formally given by

$\begin{split}z(t)&=z_{0}(t)+\tilde{z}(t)=\left(z_{i}-i\int_{0}^{T}V^{\prime}(z+\bar{z})\Theta(t-t^{\prime})e^{i\omega
t^{\prime}}dt^{\prime}\right)e^{-i\omega t}\\\
&=z_{i}K^{(+)}_{F}(t)-i\int_{0}^{T}V^{\prime}(z+\bar{z})G^{z\bar{z}}_{F}(t,t^{\prime})dt^{\prime}\\\
\bar{z}(t)&=\bar{z}_{0}(t)+{\tilde{\bar{z}}}(t)=\left(\bar{z}_{f}e^{-i\omega
T}-i\int_{0}^{T}V^{\prime}(z+\bar{z})\Theta(t^{\prime}-t)e^{-i\omega
t^{\prime}}dt^{\prime}\right)e^{i\omega t}\\\
&=\bar{z}_{f}K^{(-)}_{F}(t)-i\int_{0}^{T}V^{\prime}(z+\bar{z})G^{z\bar{z}}_{F}(t^{\prime},t)dt^{\prime}\end{split}$
(326)

Here $z_{0}(t)$ and $\bar{z}_{0}(t)$ satisfy the free equation of motion and
are given in (324) and ${\tilde{z}}(t)$, ${\tilde{\bar{z}}}(t)$ are the
corrections to the free solutions and obey trivial boundary conditions i.e.
${\tilde{z}}(0)=0$ and ${\tilde{\bar{z}}}(T)=0$. (326) are implicit solutions
to the equations of motion, as the RHS of the equations in (326) depends on
the solutions $z$ and ${\bar{z}}$. This equation can be made more explicit by
iterating (326). It is easy to check that the solution obtained in this manner
agrees with the solution obtained by expanding in Feynman diagrams (following
the procedure described above).

Inserting (326) into the action (310), we find

$\begin{split}\mathcal{S}_{\rm
bdry}&=-i\frac{\bar{z}_{f}z_{f}+\bar{z}_{i}z_{i}}{2}\\\
&=-iz_{i}{\bar{z}}_{f}e^{-i\omega
T}-\frac{1}{2}\int_{0}^{T}\left(z_{i}e^{-i\omega
t}+{\bar{z}}_{f}e^{i\omega(t-T)}\right)V^{\prime}(z,\bar{z})dt\\\
&=-iz_{i}{\bar{z}}_{f}e^{-i\omega
T}-\frac{1}{2}\int_{0}^{T}\left(z_{0}(t)+{\bar{z}}_{0}(t)\right)V^{\prime}(z,\bar{z})dt\\\
\mathcal{S}_{\rm
bulk}&=\int_{0}^{T}\left(\frac{1}{2}\left(z(t)+{\bar{z}}(t)\right)V^{\prime}(z,\bar{z})-V(z+\bar{z})\right)dt\end{split}$
(327)

Using the above equations, we find that the full on-shell action, ${\cal
S}=\mathcal{S}_{\rm bdry}+\mathcal{S}_{\rm bulk}$

$\mathcal{S}=-iz_{i}{\bar{z}}_{f}e^{-i\omega
T}+\int_{0}^{T}\left(\frac{1}{2}\left(\tilde{z}(t)+\tilde{\bar{z}}(t)\right)V^{\prime}(z,\bar{z})-V(z+\bar{z})\right)dt$
(328)

As in the discussion around (326), the expression (328) is implicit, as it
depends on $z(t)$ and ${\bar{z}}(t)$. The expression can be made explicit by
inserting the iterated solutions to (326) into (328). It is not difficult to
check that this procedure reproduces the action as evaluated by summing over
all tree diagrams.

Note that, on-shell, the boundary terms and the linear piece in the free bulk
action cancel each other.

###  C.3 Diagrammatic Computation of the Dirichlet path integral

Let us now study the on-shell action with Dirichlet boundary conditions for
$x$ i.e.

$x(0)=x_{i}\qquad\qquad\qquad x(T)=x_{f}.$ (329)

In this case the variational principle is well-defined without the addition of
any boundary terms. Hence the full action (in the variable $x$) is just given
by

$\mathcal{S}_{\rm
bulk}=\int_{0}^{T}dt~{}\left(\frac{{\dot{x}}^{2}}{2}-\frac{\omega^{2}x^{2}}{2}-\tilde{V}(x)\right)$
(330)

and the equation of motion is given by

$\ddot{x}+\omega^{2}x+\tilde{V}^{\prime}(x)=0$ (331)

(in the equation above ${\tilde{V}}^{\prime}$ denotes the derivative of
$\tilde{V}$ w.r.t the variable $x$). The ‘bulk to bulk’ propagator

$G_{D}(t,t^{\prime})=\frac{i}{\omega\sin\omega T}\left[\sin\omega
t\sin\omega(T-t^{\prime})\Theta(t^{\prime}-t)+\sin\omega
t^{\prime}\sin\omega(T-t)\Theta(t-t^{\prime})\right]$ (332)

satisfies the equation of motion

$\frac{d^{2}}{dt^{2}}G_{D}(t,t^{\prime})+\omega^{2}G_{D}(t,t^{\prime})=-i\delta(t-t^{\prime})$
(333)

and obeys the boundary conditions

$G_{D}(0,t^{\prime})=0,\qquad\qquad G_{D}(T,t^{\prime})=0.$ (334)

Note that the function $G_{D}(t,t^{\prime})$ is continuous at $t=t^{\prime}$
(even though its first derivative is discontinuous at this point). In
particular, we have

$G_{D}(t,t)=i\frac{\sin\omega t\sin\omega(T-t)}{\omega\sin\omega T}$ (335)

Similarly, the ‘bulk to boundary’ propagators are given by

$K_{D}^{(-)}(t)=\frac{\sin(\omega t)}{\sin(\omega T)},\qquad\qquad
K_{D}^{(+)}(t)=\frac{\sin(\omega(T-t))}{\sin(\omega T)}.$ (336)

They satisfy (333) with the following boundary conditions

$K_{D}^{(-)}(T)=1,\qquad\qquad K_{D}^{(+)}(0)=1$ (337)

Note that

$\begin{split}&K_{D}^{+}(t)=-i\partial_{t^{\prime}}G_{D}(t,t^{\prime})\left|{}_{t^{\prime}=0}\right.\\\
&K_{D}^{-}(t)=+i\partial_{t^{\prime}}G_{D}(t,t^{\prime})\left|{}_{t^{\prime}=T}\right.\\\
\end{split}$ (338)

in agreement with (246)757575In Lorentzian signature, the bulk to boundary
propagator is related to bulk to bulk propagator via
$K_{D}(t)=in.\nabla^{\prime}G_{D}(t,t^{\prime})\bigg{|}_{\rm bdry}$ . As in
the previous subsection, the propagators listed above can be used to evaluate
the logarithm of the harmonic oscillator path integral as a functional of
$x_{i}$ and $x_{f}$.

An implicit formula for the Dirichlet on-shell action, similar to (328) is
given by

$\mathcal{S}_{\rm bulk}=\frac{\omega}{2\sin(\omega
T)}\left(\left(x_{f}^{2}+x_{i}^{2}\right)\cos(\omega
T)-2x_{f}x_{i}\right)+\int_{0}^{T}dt~{}\left(\frac{1}{2}{\tilde{x}}\tilde{V}^{\prime}(x)-\tilde{V}(x)\right)$
(339)

###  C.4 Relationship between the Dirichlet and Positive/Negative energy path
integrals

In the previous two subsections, we have argued that

  * •

The coefficient of $z_{i}^{m}\bar{z}_{f}^{n}$ in the logarithm of the ‘In Out’
path integral is given by summing all Feynman diagrams with $m$ legs on the
past boundary and $n$ legs on the future boundary.

  * •

The coefficient of $x_{i}^{m}x_{f}^{n}$ in the logarithm of the Dirichlet path
integral is given by summing all Feynman diagrams with $m$ legs on the past
boundary and $n$ legs on the future boundary.

It follows that the ‘in-out’ partition function as a function of $z_{i}$ and
${\bar{z}}_{f}$, is very similar to the Dirichlet partition function as a
function of $x_{i}$ and $x_{f}$. The differences between these two functions
are due to

  * 1

The ‘in-out’ propagators differ from the Dirichlet counterparts.

  * 2

The ‘in-out’ partition function is given by a path integral with the potential
$V(x)$, while the Dirichlet path integral uses the potential ${\tilde{V}}(x)$

  * 3

Tadpole graphs are absent in the ‘in-out’ path integral, while they are
present in the Dirichlet path integral (see (315) and (335)).

The differences above tell us that the two path integrals are, indeed,
distinct functions of their arguments. Nonetheless, we will now argue that an
important part of the two answers is identical.

Recall that the contribution of any Feynman diagram (to the path integral) is
given by multiplying propagators, and then performing an integral over the
location of all interaction vertices. We will now explain that the two
integrands (i.e. the one for the in/out problem and the one for the Dirichlet
problem) generically disagree with each other when one or more of the
interaction vertices are located near one of the two boundary surfaces ($t=0$
or $t=T$) but agree with each other (upto corrections that are exponentially
suppressed in $\omega T$) when all interaction vertices are located far from
the boundary surfaces. The implication of this observation is the following.
As we have seen in (45), the S matrix is encoded in the part of the path
integral that develops an energy conserving delta function (see (47)). This
delta function is a consequence of contributions that scale like $T$ when
energies are conserved. This scaling comes about because the integrand becomes
independent of the time of the insertion of the vertex operator precisely when
energy is conserved. Of course terms that scale like $T$ do so precisely
because they receive their dominant contribution from interaction vertices
that are located in an ‘elevator’ that is far from each of the boundaries.

It follows that the demonstration (which we will turn to immediately below)
that the integrand for the in-out and Dirichlet Feynman diagrams agree when
all insertion times are far from boundaries, guarantees that the terms in the
path integral that are proportional to an energy conserving delta function
(i.e. the terms in the action that contribute to scattering) are, in fact,
identical between these two path integrals.

We now explain why the two integrands agree when interaction vertices lie far
from the boundaries. To see this, let us give a small Euclidean time to time
coordinates appearing in the Dirichlet propagators i.e. $t\rightarrow
t(1-i\epsilon)$. When the interaction point $t$ lies deep in the bulk i.e.
$0\ll t\ll T$, we can use

$\sin[\omega t(1-i\epsilon)]\approx\frac{e^{i\omega t}}{2i}$ (340)

where we have dropped the exponentially decaying piece. Using (332) and (336),
we find that in the limit, the ‘bulk to bulk’ and ‘bulk to boundary’
propagators in the Dirichlet case simplify to

$\begin{split}&G_{D}(t,t^{\prime})\approx\frac{1}{2\omega}\left[e^{-i\omega(t^{\prime}-t)}\Theta(t^{\prime}-t)+e^{-i\omega(t-t^{\prime})}\Theta(t-t^{\prime})\right]\\\
&K_{D}^{(+)}(t)\approx e^{i\omega(t-T)},\qquad\qquad K_{D}^{(-)}(t)\approx
e^{-i\omega t}\end{split}$ (341)

Comparing the above equations with the ‘in-out’ propagators given in (312) and
(316), we see that in this limit

$G_{D}(t,t^{\prime})=\langle\frac{z(t)+{\bar{z}}(t)}{\sqrt{2\omega}}\frac{z(t^{\prime})+{\bar{z}}(t^{\prime})}{\sqrt{2\omega}}\rangle$
(342)

and that

$K_{D}^{(+)}(t)=K_{F}^{(+)}(t),~{}~{}~{}K_{D}^{(-)}(t)=K_{F}^{(-)}.$ (343)

In other words, in the above limit, all propagators in the Dirichlet and ‘in-
out’ problems become identical to each other.

We now turn to a discussion of the vertex factors. The second apparent
difference between the Dirichlet answer and the ‘in-out’ answer lies in the
potentials: recall that potentials $V(x)$ and $\tilde{V}(x)$ are not identical
but are related to each other by normal ordering (308). However, this
difference is precisely compensated for by the different treatment of self
contractions (tadpoles) of two $x(t)s$ from the same interaction vertex.
Recall from (315) that such self contraction tadpoles are absent in the in out
path integral (this is a consequence of the fact that the potential, in this
formalism, was already normal ordered). On the other hand, the analogous
tadpoles are present in the Dirichlet problem, and their contributions are
given by (335). In the limit that $t$ and $T-t$ are both large, it is easily
checked that the RHS of (335) reduces to $\frac{1}{2\omega}$. In the free
harmonic oscillator $<0|x^{2}|0>=\frac{1}{2\omega}$. It follows that (in the
relevant time window) the net effect of the tadpoles is simply to affect the
replacement

$V\rightarrow V_{\rm eff}=e^{\frac{\partial_{x}^{2}}{4\omega}}{V},$

and then to work with the effective potential $V_{\rm eff}$ with no tadpole
contractions. From (308), that $V_{\rm eff}$ equals ${\tilde{V}}$. In other
words the effective potential for the Dirichlet problem (once we have
accounted for Tadpoles) is identical to that for the in-out problem.

In summary, we have demonstrated that propagators and (effective) vertex
factors for the Dirichlet and in-out problems both agree exactly with each
other when all vertices are inserted at times $t$ such that $\omega t\gg 1$
and $\omega(T-t)\gg 1$, as we set out to show.

###  C.5 Explicit computation of the On-shell Action

In this subsection, we explicitly compute the on-shell action as a function of
boundary values both for the ‘in-out’ boundary conditions and for Dirichlet
boundary conditions. We consider an anharmonic oscillator with quartic
potential and compute the on-shell action to the leading order in the quartic
coupling.
In-out action
Let us first consider the on-shell action (310) as a function of positive
energy data $z_{i}$ in the past and negative energy data $\bar{z}_{f}$ in the
future for a quartic coupling i.e.

$V(z+\bar{z})=\frac{\lambda}{16\omega^{2}}(z+\bar{z})^{4}.$ (344)

Using (328), we find that at leading order in $\lambda$, the on shell action
is given by

$\begin{split}i\mathcal{S}&={\bar{z}}_{f}z_{i}e^{-i\omega T}-\frac{i\lambda
e^{-2i\omega
T}}{32\omega^{3}}\left(\left({\bar{z}}_{f}^{4}+z_{i}^{4}\right)\sin 2\omega
T+12\omega
T{\bar{z}}_{f}^{2}z_{i}^{2}+8{\bar{z}}_{f}z_{i}\left({\bar{z}}_{f}^{2}+z_{i}^{2}\right)\sin\omega
T\right)+O\left(\lambda^{2}\right)\end{split}$ (345)

In the large $T$ limit, we obtain

$\begin{split}i\mathcal{S}&={\bar{z}}_{f}z_{i}e^{-i\omega
T}-\frac{\lambda}{64\omega^{3}}\left({\bar{z}}_{f}^{4}+z_{i}^{4}\right)-i\frac{3\lambda
T}{8\omega^{2}}{\bar{z}}_{f}^{2}z_{i}^{2}e^{-2i\omega
T}-\frac{\lambda}{8\omega^{3}}{\bar{z}}_{f}z_{i}\left({\bar{z}}_{f}^{2}+z_{i}^{2}\right)e^{-i\omega
T}+O\left(\lambda^{2}\right)\end{split}$ (346)

As we are working only to leading order in $\lambda$, the action receives
contributions from no loop diagrams (recall that tadpoles vanish in the in-out
path integral). Exchange diagrams also do not contribute at this order.

Dirichlet action
Using (307), we find that the potential $\tilde{V}(x)$ corresponding the the
potential (344) is given by

$\tilde{V}(x)=\lambda\left(\frac{x^{4}}{4}-\frac{3x^{2}}{4\omega}+\frac{3}{16\omega^{2}}\right)$
(347)

Finally, in the large $T$ limit767676To take the large $T$ limit, we give
small Euclidean time to all the terms in the action, the on-shell action (330)
with the above potential contains three pieces

$i\mathcal{S}_{\rm bulk}=i\mathcal{S}_{f}+i\mathcal{S}_{\rm
contact}+i\mathcal{S}_{\rm tadpole}$ (348)

where

$\begin{split}i\mathcal{S}_{\rm
f}&=-\frac{i\omega}{2}\left(x_{f}^{2}+x_{i}^{2}\right)+2i\omega e^{-i\omega
T}x_{f}x_{i}\\\ i\mathcal{S}_{\rm
contact}&=-\frac{\lambda}{8\omega}\left(-\frac{3}{\omega}\left(x_{f}^{2}+x_{i}^{2}\right)+\frac{1}{2}\left(x_{f}^{4}+x_{i}^{4}\right)\right)-\frac{\lambda
e^{-iT\omega}}{4\omega}\left(x_{f}^{3}x_{i}+x_{f}x_{i}^{3}\right)\\\
&-i\frac{3}{2}\lambda Te^{-2iT\omega}~{}x_{f}^{2}x_{i}^{2}+\frac{3i\lambda
T}{2\omega}e^{-iT\omega}x_{f}x_{i}-i\frac{3\lambda T}{16\omega^{2}}\\\
i\mathcal{S}_{\rm
tadpole}&=-\frac{3\lambda}{16\omega}\left(x_{f}^{2}+x_{i}^{2}\right)-\frac{3i\lambda
T}{2\omega}e^{-iT\omega}x_{f}x_{i}+i\frac{3\lambda
T}{16\omega^{2}}\end{split}$ (349)

Therefore the full action is given by

$\begin{split}i\mathcal{S}_{\rm
bulk}&=-\frac{\omega}{2}\left(x_{f}^{2}+x_{i}^{2}\right)+2\omega e^{-i\omega
T}x_{f}x_{i}-\frac{\lambda}{16\omega}\left(-\frac{3}{\omega}\left(x_{f}^{2}+x_{i}^{2}\right)+\left(x_{f}^{4}+x_{i}^{4}\right)\right)\\\
&-\frac{\lambda
e^{-iT\omega}}{4\omega}\left(x_{f}^{3}x_{i}+x_{f}x_{i}^{3}\right)-i\frac{3}{2}\lambda
Te^{-2iT\omega}~{}x_{f}^{2}x_{i}^{2}\end{split}$ (350)

Comparing S-matrix
By comparing the ‘in-out’ action given in (346) and the Dirichlet action given
in (350), it is easy to check that the S-matrix piece i.e.
$\bar{z}_{f}^{2}z_{i}^{2}$ piece in (346) and $x_{i}^{2}x_{f}^{2}$ piece in
(350), match (after taking into account the factors of
$\frac{1}{\sqrt{2\omega}}$ that appear in the relationship between the
variables $z$ and $x$.).

Comparing Vacuum Wavefunction
Using the onshell action (346), we find that the vacuum wavefunctional of the
harmonic oscillator receives corrections at order $\lambda$. At the leading
order it is given by

$\Psi(\bar{z})_{c}=1-\frac{\lambda}{64\omega^{3}}\bar{z}^{4}$ (351)

For the Dirichlet problem, the vacuum wavefunctional can be computed by
setting $x_{i}$ to zero in (350). We find that, at leading order in $\lambda$,
the vacuum wavefunctional is given by

$\Psi(x)_{D}=\left(1-\lambda\left(\frac{1}{16\omega}x_{f}^{4}+\frac{3}{16\omega^{2}}x_{f}^{2}\right)\right)e^{-\frac{\omega}{2}x_{f}^{2}}$
(352)

Both the answers exactly match the corrections to the wavefunction computed
using Hamiltonian perturbation theory as shown below.

Using time-independent perturbation theory, at linear order, the corrections
to the vacuum state are given by

$|\psi\rangle=|0\rangle+\sum_{m\neq
0}^{\infty}\frac{V_{m0}}{E_{0}^{(0)}-E_{m}^{(0)}}|m\rangle$ (353)

where $V_{m0}=\langle m|V(x)|0\rangle$ and $V(x)$ is the potential given in
(344). Since the potential is quartic and is normal ordered, only one term in
the above sum contributes and we find

$|\psi\rangle=|0\rangle+\frac{V_{40}}{E_{0}^{(0)}-E_{4}^{(0)}}|4\rangle$ (354)

Using the above formula, we can compute the coherent state wavefunction
($\Psi(\bar{z})_{c}$) as well as the Dirichlet wavefunction
($\Psi(x_{f})_{D}$), by taking the inner product with $\langle z|$ and
$\langle x|$ respectively. We find the following wavefunctions

$\begin{split}\Psi(\bar{z})_{c}&:=\langle
z|\psi\rangle=1-\frac{\lambda}{64\omega^{3}}\bar{z}^{4}\\\
\Psi(x_{f})_{D}&:=\langle
x_{f}|\psi\rangle=1-\frac{\lambda}{256\omega^{3}}H_{4}(\sqrt{\omega}x_{f})e^{-\frac{\omega}{2}x_{f}^{2}}\end{split}$
(355)

where $H_{4}(x)$ is the fourth Hermite polynomial. The above expressions
exactly match (351) and (352) respectively 777777Hermite polynomials have a
constant piece which is absent in (352), but from the perspective of path
integral, the constant piece just shifts the zero point energy..

The map between the coherent state wave function (351) and the position space
wave function (352) may be understood by noting that

$e^{-\frac{1}{4\omega}\partial^{2}_{x}}x^{n}=H_{n}(x)$ (356)

Therefore both wavefunctions are related to each other via the following
relation

$e^{-\frac{1}{4\omega}\partial^{2}_{x}}\Psi(x)_{c}=\Psi(x)_{D}~{}e^{-\frac{\omega}{2}x^{2}}$
(357)

Note that the above relation is exactly the same as the relationship between
the Dirichlet potential $\tilde{V}(x)$ and the ‘in-out’ potential $V(x)$ given
in (307).

####  C.5.1 Four different oscillators

In this subsubsection, we consider four different oscillators interacting via
a quartic coupling and compute the ‘in-out’ onshell action for this case. This
Lagrangian can be thought of as a toy version of the field theory Lagrangian
with $\phi^{4}$ coupling. This is because the scalar field $\phi$ can be
written as an infinite sum over harmonic oscillators labeled via $\vec{k}$ and
the quartic coupling couples four such harmonic oscillators. The Hamiltonian
for this system is given by

$\begin{split}\mathcal{H}&=\sum_{k=1}^{4}\omega_{k}z_{k}{\bar{z}}_{k}+\lambda\prod_{k=1}^{4}\frac{1}{\sqrt{2\omega_{k}}}(z_{k}+\bar{z}_{k})\end{split}$
(358)

Using the ‘bulk to boundary’ propagators given in (316), we find that, at the
leading order, the on-shell action takes the following form

$\mathcal{S}=\mathcal{S}_{f}+\mathcal{S}_{\lambda}$ (359)

where $\mathcal{S}_{f}$ is the free part i.e. the $\lambda$ independent
contribution to the action whereas $\mathcal{S}_{\lambda}$ is the first order
correction to the on-shell action. Both these pieces are given by

$\begin{split}i\mathcal{S}_{f}=&\sum_{k=1}^{4}z_{i_{k}}{\bar{z}}_{f_{k}}e^{-i\omega_{k}T}\\\
i\mathcal{S}_{\lambda}=&-\frac{i\lambda{\bar{z}}_{f_{1}}{\bar{z}}_{f_{2}}{\bar{z}}_{f_{3}}{\bar{z}}_{f_{4}}\left(-1+e^{-iT\left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right)}\right)}{4\sqrt{\omega_{1}\omega_{2}\omega_{3}\omega_{4}}\left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right)}+\frac{i\lambda
z_{i_{1}}z_{i_{2}}z_{i_{3}}z_{i_{4}}\left(1-e^{-iT\left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right)}\right)}{4\sqrt{\omega_{1}\omega_{2}\omega_{3}\omega_{4}}\left(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4}\right)}\\\
&+\frac{i\lambda{\bar{z}}_{f_{3}}{\bar{z}}_{f_{4}}z_{i_{1}}z_{i_{2}}\left(e^{-iT\left(\omega_{3}+\omega_{4}\right)}-e^{-iT\left(\omega_{1}+\omega_{2}\right)}\right)}{4\left(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}\right)\sqrt{\omega_{1}\omega_{2}\omega_{3}\omega_{4}}}-\frac{i\lambda{\bar{z}}_{f_{1}}{\bar{z}}_{f_{2}}z_{i_{3}}z_{i_{4}}\left(e^{-iT\left(\omega_{1}+\omega_{2}\right)}-e^{-iT\left(\omega_{3}+\omega_{4}\right)}\right)}{4\left(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}\right)\sqrt{\omega_{1}\omega_{2}\omega_{3}\omega_{4}}}\end{split}$
(360)

In the main text, we work with a ‘slab’ boundary which consists of two
spacelike slices at initial time $-T$ and final time $T$. In this Appendix, we
work with the boundaries at time $t_{in}=0$ and $t_{f}=T$. The results derived
above can be easily translated to the case where boundaries are at $-T$ and
$T$. This changes the ‘bulk to boundary’ propagator $K_{F}^{(+)}(t)$ to

$K_{F}^{(+)}(t)=e^{-i\omega(t+T)}$

so that $K_{F}^{(+)}(-T)=1$. This amounts to replacing $T$ by $2T$ in the on-
shell action (360).

The results of this computation can be easily generalized to field theory,
where we have an infinite number of different harmonic oscillators labeled by
the momentum $\vec{k}$. We find,

$i\mathcal{S}_{\lambda}\propto-\int\prod_{i}d^{d}k_{i}~{}\frac{i\lambda\left(e^{-2iTE_{in}}-e^{-2iTE_{out}}\right)}{(E_{in}-E_{out})}(2\pi)^{d}\delta^{d}\left(\sum_{i}\vec{k}_{i}^{\rm
in}-\sum_{j}\vec{k}_{j}^{\rm
out}\right)\prod_{j}\bar{\beta}_{k_{j}}\prod_{i}\beta_{k_{i}}$ (361)

where we used the fact that $z_{i}=(2\pi)^{d}\sqrt{2\omega}\beta_{k_{i}}$ and
similarly for $\bar{z}_{i}$.

##  Appendix D Exchange Saddles: Lorentzian and Euclidean

In subsection 5.4 we have studied correlators with ‘scattering kinematics’,
i.e. correlators whose insertions were located in such a manner that the
equations (105) admit real solutions. This is not always the case. In this
section, we study a particular one parameter class of insertion
configurations, and study the existence (and nature) of the saddle point
solutions as a function of this parameter.

Consider a configuration in which our four insertions are located a the
vertices of a rectangle, whose sides are parallel to the $x$ and $y$ axes in
spacetime. Let us label the insertions, respectively, as $1,2,3,4$, moving
anticlockwise starting from ‘southwest’. Our insertions have coordinates
$(x,y)$ (all other coordinates are zero)

$1:(-a,-b),~{}~{}~{}2:(a,-b),~{}~{}~{}3:(a,b),~{}~{}~{}4:(-a,b)$ (362)

We assume that the external insertions are all of a field that creates a
particle of mass $m$, and search for exchange saddles in which particles 1 and
2 come together and fuse into an intermediate particle, of mass $M$, at the
location $(0,-(b-y))$. The intermediate particle then propagates to the point
$(0,b-y)$ where it, once again, bifurcates into two particles each of mass
$m$, which propagates upto insertions 3 and 4.

As we have explained above, in this section we focus on exchange saddles of
the form $(1,2)\rightarrow$ intermediate $\rightarrow(3,4)$. There, of course
also exist other crossed saddles (e.g. $(1,4)\rightarrow$ intermediate
$\rightarrow((2,3)$. The analysis of these ‘crossed’ saddles is very similar
to that of the ‘direct’ saddles studied in this section and is left as an
exercise to the interested reader.

When $b$ is real the $y$ direction is spacelike. When $b$ is imaginary the $y$
direction is timelike. We will consider both possibilities.

In the situation presented above, it is not difficult to analyze when saddle
points of the sort described above exist, and how many such solutions exist.
In this Appendix, we summarize the results of this investigation. The results
turn out to depend sensitively on whether $M>2m$ or $M<2m$, so we consider the
two cases separately.

###  D.1 $M<2m$

In this case a single real solution exists provided that $b$ is real and

$\frac{b}{a}>\frac{M}{\sqrt{4m^{2}-M^{2}}}$ (363)

The solution exists at a value of $y$ given by

$y=\frac{aM}{\sqrt{4m^{2}-M^{2}}}$ (364)

No solutions (real or complex) exist when $b$ is real and (363) is violated.
No solutions exist when $b$ is imaginary

###  D.2 $M>2m$

In this case, we have a pair of complex solutions for every real value of $b$.
The solutions are given by the analytic continuation of (365), i.e. by

$y=\pm i\frac{aM}{\sqrt{M^{2}-4m^{2}}}$ (365)

On this saddle point, $y^{2}>a^{2}$ so the external and intermediate particles
both propagate in a complex manner (the spacetime interval for propagation, in
both cases, is complex). The solution presented above, however - atleast
formally - solves the saddle point equations.

When $b$ is imaginary, on the other hand, solutions to the saddle point
equation exist only when

$\left|\frac{b}{a}\right|>\frac{M}{\sqrt{M^{2}-4m^{2}}}$ (366)

When this condition is met, the solution is once again given by (365), except
that the sign in the solution is chosen so that $\frac{y}{b}>0$. This is the
scattering exchange saddle point studied in §5.4.

###  D.3 Analytic Continuation

As is clear from the results of this subsection, the saddle point contribution
to correlators is not an analytic function of either the masses or of the
insertion locations. Imagine starting with imaginary $b$ and in a
configuration that obeys (366). If we now lower $M$ (keeping $b$ and $a$
fixed), the saddle described around (366) continues to exist down to a
critical value of $M$. At values of $M$ smaller than this critical value, the
saddle point has simply disappeared - it does not exist anymore. It follows,
in other words, that the scattering saddle point exists only in a certain
domain of parameters. Similar statements apply to the Euclidean saddle.

##  Appendix E Geometry of intersection of the lightcones

In this Appendix, we discuss the geometry of the intersection of lightcones of
$m$ bulk points (with coordinates $x_{i}$, $i=1\ldots m$, in $(D-1,1)$ flat
spacetime dimensions.

###  E.1 $m\leq D+1$

Let us first focus on the case of $c=1$ i.e. on the case when the number of
insertions $m\leq D+1$. As in the main text, we define the scattering manifold
to be the subset of $R^{D-1,1}$ spanned by vectors $x_{i}-x_{j}$ (for
$i,j=1\ldots m$), i.e. the vectors that run between two insertion points. This
scattering manifold is generically $m-1$ dimensional. Generically, this $m-1$
dimensional subspace is either an $R^{m-2,1}$ or an $R^{m-1,0}$ (exceptional
situations, in which the subspace is null or less than $m-1$ dimensional are
also possible, but we will not consider these cases here).

We wish to find the manifold of points, $y$, that are null separated from each
$x_{i}$, i.e. the manifold of points $y$ that obey the equation

$(x_{i}-y)^{2}=0,~{}~{}~{}~{}i=1\ldots m$ (367)

(367) is a set of $m$ nonlinear equations. By taking linear combinations,
however, we can regroup these into a set of $m-1$ linear equations and one
nonlinear equations. The $m-1$ linear equations are

$\begin{split}&(x_{i}-y)^{2}=(x_{j}-y)^{2}~{}~{}~{}~{}~{}~{}\forall~{}~{}i,j=1\ldots
m\\\ &2(x_{j}-x_{i}).y=x_{j}^{2}-x_{i}^{2}\end{split}$ (368)

The first line of (368) asserts that $y$ is equidistant from each of the
insertion points. The (algebraically equivalent) second line of (368) tells us
that this condition is satisfied when the dot products of the vector $y$ with
a basis of vectors in the scattering plane, each take a certain value. This
set of linear equations completely determines $y_{\parallel}$ the projection
of $y$ in the scattering plane. However, these equations leave the part of $y$
that is orthogonal to the scattering plane, $y_{\perp}$ completely unfixed.

Let us first consider the case of maximal physical interest to us, namely the
case in which the scattering manifold is $R^{m-2,1}$ (this is the case that is
relevant to scattering from our slab boundary). In this case the space
orthogonal to $R^{m-2,1}$ is the Euclidean space $R^{D-m+1}$. Depending on the
details of the insertions, we have three possibilities.

  * •

The insertions are all spacelike separated from $y_{\parallel}$ i.e.
$(x_{i}-y_{\parallel})^{2}=r^{2}>0$ It follows that

$(x_{i}-y)^{2}=(x_{i}-y_{\parallel})^{2}+y_{\perp}^{2}\geq r^{2}>0.$

In this case, the lightcones emanating out of $\\{x-i\\}$ do not intersect.

  * •

The insertions are all timelike separated from $y$ i.e.
$(x_{i}-y)^{2}=-r^{2}<0$ In this case

$(x_{i}-y)^{2}=-r^{2}+y_{\perp}^{2}\geq r^{2}$

and so the condition $(x_{i}-y)^{2}=0$ is met on an $S^{D-m}$ of radius $r$ in
$R^{D-m+1}$. We see that, in this case, the lightcones intersect on a compact
spacelike surface, namely an $D-m$ dimensional sphere. As explained above, the
radius of this sphere is $r$. This radius shrinks to zero when
$r^{2}=(x_{i}-y)^{2}\to 0$. This is the case we turn to next.

  * •

All the insertions are null separated from $y$ i.e.
$(x_{i}-y_{\parallel})^{2}=0$. In this case lightcones intersect only when
$y_{\perp}^{2}=0$, i.e. when $y_{\perp}=0$. In this case the $y=y_{\parallel}$
and the common intersection of the lightcones is a single point - one that
lies on the scattering plane. This is the case that gives rise to a scattering
singularity in the main text, as, in this case, it is always possible to
satisfy momentum conservation. (momentum cannot be conserved in the previous
case because the perpendicular component of the vector
$\sum_{i}(\omega_{i}(x_{i}-y)$ equals $(\sum_{i}\omega_{i})y_{\perp}$, which
cannot vanish if all $\omega_{i}$ are non negative, and at least one of them
is nonzero).

Although this will not be of great interest to the current paper, let us also
briefly consider the case that the scattering plane is spacelike, i.e. that it
is an $R^{m-1,0}$. In this case, the orthogonal plane is $R^{D-m,1}$. In this
situation, it is always the case that $(x_{i}-y_{\parallel})^{2}=r^{2}>0$. The
simultaneous intersection of lightcones occurs when

$y_{\perp}^{2}=-r^{2}$ (369)

i.e. on a hyperboliod of ‘radius’ $r$ in $R^{D-m,1}$. Note that the lightcone
intersection manifold is noncompact in this case. Of course, this case is
incompatible with momentum conservation as above.

###  E.2 $c>1$

We have argued above that in the $c=1$ case, whenever the lightcones
intersect, there is a unique bulk point for which momentum conservation can be
satisfied.

Next, we consider the $c>1$ case i.e. $m>D+1$. In this case, the scattering
manifold generically covers the full D dimensional spacetime: we consider this
generic case.

Let us again try to find a bulk point $y$ which is equidistant from all the
boundary insertions. We need to find solutions to equation (368). For $c>1$,
the number of equations ($m-1$) is greater than the number of variables in $y$
($D$), generically there are no solutions. But we can perform $m-D-1$ tunings
of the boundary points such that (368) can be solved. Generically, the
solution of this linear equation is unique (further tuning of $\\{x_{i}\\}$can
lead to non unique solutions: we do not consider this possibility here).

Once (368) is solved, one additional tuning is required to make
$(x_{i}-y)^{2}$ zero. The total number of tunings is, thus $m-D$, explaining
the codimension of the singular manifold in the case $c$. Notice that the
common intersection of lightcones - when it exists - is generically unique.

##  Appendix F Zero eigenvalues of $N$ implies existence of bulk point

In this Appendix, we show that if the boundary insertions are chosen such that
(125) is satisfied, then it implies that there exists a bulk point $y$ which
is null separated from all the boundary insertions.

Let us assume that the boundary distance matrix $N_{ij}$ has a zero eigenvalue
and all entries of the corresponding eigenvector have positive entries i.e.

$\begin{split}&\sum_{i}\omega_{i}N_{ij}=\sum_{i}\omega_{i}(x_{i}-x_{j})^{2}=0\\\
&\sum_{i}\omega_{i}x^{2}_{i}+\sum_{i}\omega_{i}x_{j}^{2}-2x_{j}.\sum_{i}x_{i}\omega_{i}=0\end{split}$
(370)

Completing the square, we obtain

$\left(\sum_{i}\omega_{i}\right)\left(x_{j}-\frac{\sum_{i}\omega_{i}x_{i}}{\sum_{i}\omega_{i}}\right)^{2}+\sum_{i}\omega_{i}x^{2}_{i}-\frac{(\sum_{i}\omega_{i}x_{i})^{2}}{\sum_{i}\omega_{i}}=0$
(371)

Note that, in $m>D$ case, for the generic zero eigenvectors satisfying (130),
the above manipulations cannot be performed because $\sum_{i}\omega_{i}$
vanishes for those eigenvectors. Therefore, division by that quantity is not
allowed. But for the insertions on the singular manifold, all the entries of
the zero eigenvector $\omega$ are positive and, hence, $\sum_{i}\omega_{i}\neq
0$.

Let us next analyse the equation obtained by multiplying the eigenvalue
equation by $\omega_{j}$ and summing over $j$

$\begin{split}&\sum_{i,j}\omega_{i}\omega_{j}(x_{i}-x_{j})^{2}=0\\\
&2\sum_{i}\omega_{i}\sum_{j}\omega_{j}x_{j}^{2}-2\left(\sum_{i}\omega_{i}x_{i}\right)^{2}=0\end{split}$
(372)

Using (372) in (372), we obtain:

$\left(\sum_{i}\omega_{i}\right)\left(x_{j}-\frac{\sum_{i}\omega_{i}x_{i}}{\sum_{i}\omega_{i}}\right)^{2}=0$
(373)

Since $\sum_{i}\omega_{i}\neq 0$, we obtain

$\left(x_{j}-\frac{\sum_{i}\omega_{i}x_{i}}{\sum_{i}\omega_{i}}\right)^{2}=0$
(374)

Above equation implies that all boundary points $x_{j}$’s are null separated
from a point given by $y=\frac{\sum_{i}\omega_{i}x_{i}}{\sum_{i}\omega_{i}}$.

Hence, as promised, we proved that if the distance matrix has a zero
eigenvalue and the corresponding eigenvector has all positive entries, there
exists a point $y$, which is null separated from all the insertions.

##  Appendix G Conformal Invariance of the submanifold $detN=0$

We shall now show that the vanishing of $detN$ is a conformally invariant
condition on the matrix elements of $N$. It suffices to show that the matrix
elements of $N$ transforms homogeneously under a special conformal
transformation ${x^{\mu}}\rightarrow x^{\prime\mu}$. A special conformal
transformation, being decomposable as an inversion followed by a translation
by an arbitrary vector $b^{\mu}$ and another inversion, is given by

$\frac{x^{\prime\mu}}{x^{\prime 2}}=\frac{x^{\mu}}{x^{2}}+b^{\mu}$ (375)

It immediately follows that under such a transformation, each insertion point
$x_{i}\rightarrow x^{\prime}_{i}$ such that

$\begin{split}x_{i}^{\prime 2}=\frac{x_{i}^{2}}{1+2b\cdot
x_{i}+b^{2}x_{i}^{2}}\equiv\frac{x_{i}^{2}}{\Omega(x_{i})}\end{split}$ (376)

and also that

$\begin{split}x_{i}^{\prime}\cdot
x_{j}^{\prime}&=x^{{}^{\prime}2}_{i}x^{{}^{\prime}2}_{j}\left(\frac{x_{i}^{\mu}}{x_{j}^{2}}+b^{\mu}\right)\left(\frac{x_{j}^{\mu}}{x_{j}^{2}}+b^{\mu}\right)\\\
&=\frac{1}{\Omega(x_{i})\Omega(x_{j})}\left(x_{i}.x_{j}+x_{j}^{2}(b\cdot
x_{i})+x_{i}^{2}(b\cdot x_{j})+b^{2}x_{i}^{2}x_{j}^{2}\right)\end{split}$
(377)

With this, we see that under a special conformal transformation, the
individual matrix elements of $N$ transform as
$(x_{i}-x_{j})^{2}\rightarrow(x^{\prime}_{i}-x^{\prime}_{j})^{2}$ given by,

$\begin{split}(x^{\prime}_{i}-x^{\prime}_{j})^{2}&=\frac{1}{\Omega(x_{i})\Omega(x_{j})}\Big{(}x_{i}^{2}\left(\Omega(x_{j})-2(b\cdot
x_{j})-b^{2}x_{j}^{2}\right)+x_{j}^{2}(\Omega(x_{i})-2(b\cdot
x_{i})-b^{2}x_{i}^{2})-2x_{i}\cdot x_{j}\Big{)}\\\
&=\frac{1}{\Omega(x_{i})\Omega(x_{j})}\left(x_{i}-x_{j}\right)^{2}\end{split}$
(378)

This means under the special conformal transformation, the $i^{th}$ row and
the $i^{th}$ coloumn of the matrix $N$ each pick up a factor of
$\frac{1}{\Omega(x_{i})}$ so the determinant of the matrix $N$ trasnforms into

$\rm{Det}N^{\prime}=\frac{1}{\Omega(x_{1})^{2}\Omega(x_{2})^{2}\ldots\Omega(x_{m})^{2}}{\rm
Det}N$ (379)

This proves that the condition ${\rm Det}N=0$ is left invariant by special
conformal transformations. In fact, as we have already noted, (379) also
represents the transformation of the determinant of $N$ under every other
conformal transformation (with a constant conformal factor $\Omega(x_{i})$,
the constant being equal to unity for Poincare transformations). Totally then,
the condition on the coordinates of the insertions obtained by setting ${\rm
Det}N=0$ does not change under any conformal transformation. This proves that
the vanishing of the determinant of $N$ is a condition that only involves the
cross ratios of the insertions.

We now illustrate this assertion for the special case of 4 and 5
points787878For the case of 2 and 3 points, it is clear that $detN$ vanishes
only when some of the insertions are coincident.. For $m$ points, the number
of cross ratios is $\frac{m(m-3)}{2}$797979When $m<D$ these cross ratios are
not all independent of each other.. In the case of four points, we define the
two independent cross ratios as follows

$u=\frac{N_{12}N_{34}}{N_{13}N_{24}},\qquad\qquad
v=\frac{N_{14}N_{23}}{N_{13}N_{24}}.$

where $N_{ij}$ are the Lorentzian distances between the insertion points $i$
and $j$ (the $ij^{\rm th}$ element of the matrix $N$). We find that the
determinant of $N$ can be written in the following form

${\rm Det}N=(N_{13})^{2}(N_{24})^{2}(1+u^{2}+v^{2}-2u-2v-2uv).$ (380)

Further making the redefinitions $u=Z\bar{Z}$ and $v=(1-Z)(1-\bar{Z})$, we
find

${\rm Det}N=(N_{13})^{2}(N_{24})^{2}(Z-\bar{Z})^{2}$ (381)

Hence, as long as no two insertions coincide, the singularity condition ${\rm
Det}N=0$ is met precisely when the cross ratios satisfy $Z=\bar{Z}$.

In the case of five points, there are 5 cross ratios, and it turns out that
they can be formed as follows. First, we pick two sets of four points each,
say $\\{x_{1},x_{2},x_{3},x_{5}\\}$ and $\\{x_{1},x_{2},x_{4},x_{5}\\}$, and
form four-point cross ratios involving these two sets of points. There are two
of these ratios corresponding to each set, for a total of four ratios. We
denote the four-point cross ratios involving the set of points
$\\{x_{1},x_{2},x_{3},x_{5}\\}$ as

$\begin{split}&u_{1}=\frac{N_{13}N_{25}}{N_{12}N_{35}},\qquad
v_{1}=\frac{N_{15}N_{23}}{N_{12}N_{35}},\end{split}$ (382)

and the four-point cross ratios involving the set of points
$\\{x_{1},x_{2},x_{4},x_{5}\\}$ as

$\begin{split}&u_{2}=\frac{N_{14}N_{25}}{N_{12}N_{45}},\qquad
v_{2}=\frac{N_{15}N_{24}}{N_{12}N_{45}}.\end{split}$ (383)

The fifth cross ratio is formed from all five points and can be written as

$\begin{split}s_{5}=\frac{N_{15}N_{25}N_{34}}{N_{12}N_{35}N_{45}}.\end{split}$
(384)

Now, we can express the determinant of $N$ as

${\rm
det}N=-2\frac{N^{3}_{12}N^{2}_{35}N^{2}_{45}}{N_{15}N_{25}}\left(s_{5}-(Y-Z)(\bar{Y}-\bar{Z}\right)\left(s_{5}-(\bar{Y}-Z)(Y-\bar{Z}\right)$
(385)

where we used $u_{1}=Z\bar{Z},v_{1}=(1-Z)(1-\bar{Z})$ and
$u_{2}=Y\bar{Y},v_{2}=(1-Y)(1-\bar{Y})$. Once again, we see that the vanishing
of $detN$ is a condition on the conformal cross ratios only, namely it
vanishes when either $s_{5}-(Y-Z)(\bar{Y}-\bar{Z})=0$ or
$s_{5}-(\bar{Y}-Z)(Y-\bar{Z})=0$808080It might seem that there are other
potential zeros of $detN$, namely when the denominator of the prefactor in
(385) becomes infinite i.e $N_{15}$ or $N_{25}$ becomes infinite. However, it
is easy to see that these are not zeros of $detN$ either by explicitly
evaluating the rest of the expression at these points, or by noting that the
elements of the matrix $N_{ij}$ are distances, so its determinant, being just
a sum of products of distances, can’t possibly vanish when any of the
distances becomes infinite..

##  Appendix H $\mathcal{D}^{\flat}$ functions

In this section, we will compute exact answers for the contact contributions
of Euclidean $n$-point massless correlators, given by (179).

As shown in Sec. 6, we can write this in the form818181As in Sec. 6, we have
dropped an overall numerical factor given by
$\mathcal{A}=\frac{ig}{4^{4}\pi^{3D/2}}$, and we shall restore this in the
final answers.

$\begin{split}{\cal G}(x_{1},\ldots,x_{n})&=\Gamma\left(\Delta-{D\over
2}\right)\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\omega_{i}^{\Delta_{i}-1}\delta(1-\sum\limits_{i=1}^{n}\lambda_{i}\omega_{i}){(\sum\limits_{i=1}^{n}\omega_{i})^{\Delta-D}\over\left(\sum\omega_{i}\omega_{j}N_{ij}\right)^{\Delta-D/2}}\end{split}$
(386)

3-point functions

Four point conformal correlators are expressed in terms of the so-called
$\mathcal{D}$-functions (DHoker:1999kzh ) of the conformal cross ratios,
invariants of the conformal symmetry. We will encounter the analogues of these
functions, which we call $\mathcal{D}^{\flat}$ functions, already at the 3
point stage. It is simple to see that the 3-point answer is a function of two
invariants. Given three points, we use translations to put one to the origin,
then rotations about the origin to put a second point on a fixed coordinate
axis and then choose its distance from the origin by using overall scaling
invariance. Then, we use rotations about the axis defined by the two points to
put the third point in the same plane. All the information is then in
specifying the third point which can be done using two numbers.

We evaluate the expression Eq.(386) for the case of 3 boundary points to find

$\begin{split}{\cal G}(x_{1},\ldots,x_{n})&=\Gamma\left(\Delta-{D\over
2}\right)\prod_{i=1}^{3}\int_{0}^{\infty}d\omega_{i}\omega_{i}^{\Delta_{i}-1}\delta(1-\sum\limits_{i=1}^{3}\lambda_{i}\omega_{i}){(\sum\limits_{i=1}^{3}\omega_{i})^{\Delta-D}\over\left(\sum\omega_{i}\omega_{j}N_{ij}\right)^{\Delta-D/2}}\end{split}$
(387)

We use the Dirac delta function to choose $\omega_{3}=1$, and write the
remaining integral in the form

$\begin{split}\int dpp^{\Delta-D/2-1}\int
d\omega_{1}\omega_{1}^{\Delta_{1}-1}\int
d\omega_{2}\omega_{2}^{\Delta_{2}-1}(1+\omega_{1}+\omega_{2})^{\Delta-D}e^{-p\left(\omega_{1}s_{13}+\omega_{1}\omega_{2}s_{12}+\omega_{2}s_{23}\right)}\end{split}$
(388)

Now, we can make use of the identity

$e^{-x}=\int_{c-i\infty}^{c+i\infty}{ds\over{2\pi i}}\Gamma(-s)x^{s}$ (389)

where $c$ is a positive number smaller than unity, and the contour is closed
to the right so that the integral picks up the residues of all the poles of
the Gamma function at positive integer values of $s$, and reproduces the
series expansion of the exponential on the left. We use this identity to
replace two of the exponentials in the integrand, $e^{-p\omega_{1}s_{13}}$ and
$e^{-p\omega_{1}\omega_{2}s_{12}}$ with

$\begin{split}e^{-p\omega_{1}\omega_{2}s_{12}}&=\int_{c-i\infty}^{{c+i\infty}}{dq\over{2\pi
i}}\Gamma(-q)(p\omega_{1}\omega_{2}s_{12})^{q}\\\
e^{-p\omega_{1}s_{13}}&=\int_{c-i\infty}^{{c+i\infty}}{dr\over{2\pi
i}}\Gamma(-r)(p\omega_{1}s_{13})^{r}\end{split}$ (390)

Henceforth, we will keep the integration limits implicit for the above
integrals. The $\omega_{1}$ integral is then elementary so that we can easily
evaluate it and find

$\begin{split}&\int{dq\over{2\pi i}}\int{dr\over{2\pi
i}}s_{12}^{q}s_{13}^{r}\Gamma(-q)\Gamma(-r)\frac{\Gamma(\Delta_{1}+q+r)\Gamma(D-\Delta-\Delta_{1}-q-r)}{\Gamma(D-\Delta)}\\\
&\times\int dpp^{\Delta+q+r-D/2-1}\int
d\omega_{2}\omega_{2}^{\Delta_{2}+q-1}(1+\omega_{2})^{\Delta+\Delta_{1}+q+r-D}e^{-p\,\omega_{2}s_{23}}\end{split}$
(391)

We note that the $\omega_{1}$ integral is not convergent when the sum of the
scaling dimensions $\Delta=D$. This is precisely the conformally invariant
case which can be treated separately to find the usual three-point function
for scalar fields in a conformal theory. We are, of course, interested in the
case $\Delta\neq D$. Now, we can integrate over $p$ first and then over
$\omega_{2}$, and find that

$\begin{split}{\cal
G}(x_{1},x_{2},x_{3})=&\frac{1}{s_{23}^{\Delta-D/2}\Gamma(D-\Delta)}\int{dq\over{2\pi
i}}\int{dr\over{2\pi i}}\,s^{q}\,t^{r}\,\Gamma(-q)\Gamma(-r)\\\
&\times\Gamma(\Delta_{1}+q+r)\Gamma(\Delta-D/2+q+r)\Gamma(D/2-\Delta_{1}-\Delta_{3}-q)\Gamma(D/2-\Delta_{1}-\Delta_{2}-r)\\\
\end{split}$ (392)

where we have defined the two cross ratios $s=\frac{s_{12}}{s_{23}}$ and
$t=\frac{s_{13}}{s_{23}}$. We are now in a position to introduce our first
$\mathcal{D}^{\flat}_{n,D}$ as a function of these two cross ratios,

$\begin{split}\mathcal{D}^{\flat}_{3,D}(s,t)=\int{dq\over{2\pi
i}}\int{dr\over{2\pi
i}}\,s^{q}\,t^{r}\,&\Gamma(-q)\Gamma(-r)\Gamma(\Delta_{1}+q+r)\Gamma(\Delta-{D\over
2}+q+r)\\\ &\times\Gamma({D\over 2}-\Delta_{1}-\Delta_{3}-q)\Gamma({D\over
2}-\Delta_{1}-\Delta_{2}-r)\end{split}$ (393)

so that,

$\begin{split}{\cal
G}(x_{1},x_{2},x_{3})=&\frac{ig}{4^{4}\pi^{3D/2}}{1\over\Gamma(D-\Delta)}\frac{1}{s_{23}^{\Delta-D/2}}\,\mathcal{D}^{\flat}_{3,D}(s,t),\\\
\end{split}$ (394)

where we have restored the overall numerical factor.

We can find an explicit form for $\mathcal{D}^{\flat}_{3,D}(s,t)$ as follows.
We perform the $q$ and $r$ integrals by closing the contours to the right, i.e
while keeping $\text{Re}(q)>0$ and $\text{Re}(r)>0$, and the points ${D\over
2}-\Delta_{1}-\Delta_{3}$ and ${D\over 2}-\Delta_{1}-\Delta_{2}$ to the left
of the contours respectively. We then pick up residues of the poles of Gamma
functions lying along the positive real axis i.e. when the arguments become
negative integers. Doing so lands us at a closed form answer given in terms of
one of the Appell Hypergeometrics of two variables, $F_{4}$,

$F_{4}(a,b,c,d;x,y)=\sum_{m,n=0}^{\infty}\frac{(a)_{m+n}(b)_{m+n}}{c_{(m)}d_{(n)}}\frac{x^{m}y^{n}}{m!n!}$
(395)

in terms of which we find

$\begin{split}\mathcal{D}^{\flat}_{3,D}(s,t)=&\Gamma(\Delta_{1})\Gamma(\Delta-D/2)\Gamma(D/2-\Delta_{13})\Gamma(D/2-\Delta_{12})\\\
&\times
F_{4}(\Delta_{1},\Delta-D/2,1-D/2-\Delta_{13},1-D/2-\Delta_{12};s,t)\\\
+&t^{D/2-\Delta_{12}}\Gamma(\Delta_{3})\Gamma(D/2-\Delta_{2})\Gamma(D/2-\Delta_{13})\Gamma(\Delta_{12}-D/2)\\\
&\times
F_{4}(\Delta_{3},D/2-\Delta_{2},1-D/2-\Delta_{13},1+D/2+\Delta_{12};s,t)\\\
+&s^{D/2-\Delta_{13}}\,\Gamma(\Delta_{2})\Gamma(D/2-\Delta_{3})\Gamma(D/2-\Delta_{12})\Gamma(\Delta_{13}-D/2)\\\
&\times
F_{4}(\Delta_{2},D/2-\Delta_{3},1+D/2+\Delta_{13},1-D/2-\Delta_{12};s,t)\\\
+&s^{D/2-\Delta_{13}}t^{D/2-\Delta_{12}}\,\Gamma(D-\Delta)\Gamma(D/2-\Delta_{1})\Gamma(\Delta_{12}-D/2)\Gamma(\Delta_{13}-D/2)\\\
\times&F_{4}(D-\Delta,D/2-\Delta_{1},1+D/2+\Delta_{13},1+D/2+\Delta_{12};s,t)\end{split}$
(396)

where we have defined $\Delta_{ij}=\Delta_{i}+\Delta_{j}$.

4-point functions
The 4-point $\mathcal{D}^{\flat}$ functions are functions of five invariants.
We needed two numbers to specify three points. Rotations in axes perpendicular
to the plane containing these three points can be used to fix a 3-plane
containing all 4 points. Specifying the fourth point then requires three more
numbers taking the total to 5.

Using similar reasoning as for the 3-point function, we find that

$\begin{split}{\cal
G}(x_{1},\ldots,x_{4})&=\frac{ig}{4^{4}\pi^{3D/2}}\frac{1}{\Gamma(D-\Delta)}{1\over{s_{14}^{\Delta-D/2}}}\mathcal{D}^{\flat}_{4,D}(s_{1},s_{2},s_{3},s_{4},s_{5})\end{split}$
(397)

where $\mathcal{D}^{\flat}_{4,D}$ is a function of the simple ratios

$s_{1}={s_{34}\over s_{14}},\ s_{2}={s_{23}\over s_{14}},\ s_{3}={s_{13}\over
s_{14}},\ s_{4}={s_{24}\over s_{14}},\ s_{5}={s_{12}\over s_{14}}$

given by

$\begin{split}\mathcal{D}^{\flat}_{4,D}(\\{s_{i}\\})=\prod_{i=1}^{5}\int
dt_{i}\,s_{i}^{t_{i}}\,\Gamma(-t_{i})\,&\Gamma(\Delta_{3}+t_{123})\Gamma(\Delta_{2}+t_{245})\\\
&\times\Gamma(D/2-\Delta_{234}-t_{124})\Gamma(D/2-\Delta_{123}-t_{235})\Gamma(\Delta-D/2+\sum_{i=1}^{5}t_{i})\end{split}$
(398)

where $t_{ijk}=t_{i}+t_{j}+t_{k}$ and
$\Delta_{ijk}=\Delta_{i}+\Delta_{j}+\Delta_{k}$. Performing the $s_{i}$
integrals lands us at a closed form answer which has 48 terms each involving a
hypergeometric of five variables. One of these terms, for example, is given by

$\begin{split}&\Gamma(\Delta_{2})\Gamma(\Delta_{3})\Gamma(\Delta-D/2)\Gamma(D/2-\Delta_{234})\Gamma(D/2-\Delta_{123})\\\
&\times
D_{5}(\Delta_{3},\Delta_{2},\Delta-D/2,1-D/2+\Delta_{234},1-D/2+\Delta_{123},\\{s_{i}\\})\end{split}$
(399)

where we have introduced the following hypergeometric function of five
variables,

$D_{5}(a_{1},a_{2},b,c,d;\\{s_{i}\\})=\sum_{i_{1},\ldots
i_{5}=0}^{\infty}\frac{(a_{1})_{i_{123}}(a_{2})_{i_{245}}(b)_{I}}{(c)_{i_{124}}(d)_{i_{235}}}\prod_{j=1}^{5}\frac{s_{j}^{i_{j}}}{i_{j}!},$
(400)

with the definitions $i_{jkl}=i_{j}+i_{k}+i_{l}$ and $I=\sum_{j=1}^{5}i_{j}$.

In one special case, the answer is much simpler, namely when the conformality
condition of $\Delta=D$ is satisfied. This happens in $D=4$, and can be
thought of as the consequence of the $\phi^{4}$ operator being marginal in
four dimensions. We have worked out this special case in §6.7, leading to the
correlator in (192).

###  H.1 Massive fields

In the case of massive fields, we are interested in computing

$\mathcal{I}(x_{1},\ldots,x_{n})={\mathcal{A}\over\pi^{D/2}}\int
d^{D}y\prod_{i=1}^{n}G(x_{i}-y)$ (401)

where the bulk to bulk propagator can be expressed in terms of the modified
Bessel function of the second kind as

$G(x_{i}-y)=\left({m_{i}^{2}\over
4}\right)^{\Delta_{i}}\frac{K_{\Delta_{i}}(\sqrt{s_{i}})}{s_{i}^{\Delta_{i}/2}}$
(402)

with $\Delta_{i}={D-2\over 2}$, $s_{i}=m_{i}^{2}(x_{i}-y)^{2}$, and the
overall normalization has been chosen to reproduce the massless result
exactly, in the limit of all masses going to zero. As in the massless case, we
drop the overall numerical factor $\mathcal{A}$ in the following.

In order to evaluate the bulk integral, we will find it convenient to use the
following representation of the Bessel function, which will supply to us, a
natural analog of the Schwinger parametrization employed in the massless case,

$K_{\nu}(z)=\frac{1}{2}\left({z\over
2}\right)^{\nu}\int_{0}^{\infty}\frac{e^{-t-{z^{2}\over 4t}}}{t^{\nu+1}}dt$
(403)

With this, we get,

$\mathcal{I}(x_{1},\ldots,x_{n})={1\over\pi^{D/2}}\frac{1}{2}\prod_{i=1}^{n}{1\over
2^{\Delta_{i}}}\left({m_{i}^{2}\over
4}\right)^{\Delta_{i}}\int_{0}^{\infty}dt_{i}t_{i}^{-\Delta_{i}-1}e^{-t_{i}}\int
d^{D}y\,e^{-{1\over 4}\sum_{i=1}^{n}{m_{i}^{2}\over t_{i}}(x_{i}-y)^{2}},$
(404)

so that the bulk integral is in the same form as in the massless case.
Redefining ${m_{i}^{2}\over 4t_{i}}\rightarrow t_{i}$, introducing
$T\equiv\sum\limits_{i=1}^{n}{t_{i}}$, and performing the bulk integral lands
us at

$\begin{split}\mathcal{I}(x_{1},\ldots,x_{n})&=\prod_{i}^{n}\int_{0}^{\infty}dt_{i}t_{i}^{\Delta_{i}-1}e^{-{m_{i}^{2}\over
4t_{i}}}{1\over T^{D/2}}\exp(-{1\over
T}\sum\limits_{\begin{subarray}{c}i,j=1\\\ i\leq
j\end{subarray}}^{n}t_{i}t_{j}s_{ij}),\end{split}$ (405)

Comparing with the massless expression in Eq. 180, we see that the only
difference is the additional factor of $e^{-{m_{i}^{2}\over 4t_{i}}}$ for each
$i$, so that the massive answer smoothly goes over to the massless answer as
all the masses are taken to zero. Further, we can find correlators with both
massive and massless fields by taking some of the masses to zero in the above
expression. We can also express our answer as an integral over the
corresponding massless $n$-point function by using

$e^{-{m_{i}^{2}\over 4t_{i}}}=\int_{c-i\infty}^{c+i\infty}{dp_{i}\over{2\pi
i}}\Gamma(-p_{i})\left({m_{i}^{2}\over 4t_{i}}\right)^{p_{i}}$ (406)

so that

$\begin{split}\mathcal{I}(x_{1},\ldots,x_{n})&=\prod_{i}^{n}\int_{0}^{\infty}{dp_{i}\over
2\pi i}\Gamma(-p_{i})\left({m_{i}^{2}\over
4}\right)^{p_{i}}\int_{0}^{\infty}dt_{i}t_{i}^{\Delta_{i}-p_{i}-1}{1\over
T^{D/2}}\exp(-{1\over T}\sum\limits_{\begin{subarray}{c}i,j=1\\\ i\leq
j\end{subarray}}^{n}t_{i}t_{j}s_{ij})\\\
&=\prod_{i}^{n}\int_{0}^{\infty}{dp_{i}\over 2\pi
i}\Gamma(-p_{i})\left({m_{i}^{2}\over 4}\right)^{p_{i}}\tilde{{\cal
G}}(x_{1},\ldots,x_{n})\end{split}$ (407)

where $\tilde{{\cal G}}(x_{1},\ldots,x_{n})$ is the $n$-point function of
massless fields with scaling dimensions $\Delta_{i}-p_{i}$, given by Eq. 386
with the replacement $\Delta_{i}\rightarrow\Delta_{i}-p_{i}$.

###  H.2 Singularity structure of exchanges and loops

As a first example, consider the exchange contribution to the 4-point
correlator coming from two cubic vertices,

${{\cal G}}_{e}=\int d^{D}y_{1}\int
d^{D}y_{2}\frac{1}{((x_{1}-y_{1})^{2})^{\Delta_{1}}}\frac{1}{((x_{2}-y_{1})^{2})^{\Delta_{2}}}\frac{1}{((y_{1}-y_{2})^{2})^{\Delta_{0}}}\frac{1}{((x_{3}-y_{2})^{2})^{\Delta_{3}}}\frac{1}{((x_{4}-y_{2})^{2})^{\Delta_{4}}}\\\
$ (408)

We make a change of spacetime coordinates to work in the frame of the ”center
of mass” and the ”difference” coordinates,

$y_{+}=\frac{y_{1}+y_{2}}{2},y_{-}=\frac{y_{1}-y_{2}}{2}$ (409)

so that the exchange contribution (408) becomes

${\cal G}_{e}=\int d^{D}y_{-}\frac{1}{(4y_{-}^{2})^{\Delta_{0}}}\int
d^{D}y_{+}\frac{1}{((x^{\prime}_{1}-y_{+})^{2})^{\Delta_{1}}}\frac{1}{((x^{\prime}_{2}-y_{+})^{2})^{\Delta_{2}}}\frac{1}{((x^{\prime}_{3}-y_{+})^{2})^{\Delta_{3}}}\frac{1}{((x^{\prime}_{4}-y_{+})^{2})^{\Delta_{4}}}\\\
$ (410)

where

$x^{\prime}_{i}=\begin{cases}x_{i}+y_{-}&\ i=1,2\\\ x_{i}-y_{-}&\
i=3,4\end{cases}$ (411)

The integral over the center of mass $y_{+}$ gives rise to a 4-point contact
correlator with the insertion points shifted by the difference coordinate, so
that we can express the exchange contribution as a spacetime integral over the
difference coordinate of a $4$-point contact correlator weighted by a power of
the difference coordinate thus

${\cal G}_{e}=\int d^{d}y_{-}\frac{1}{4(y_{-})^{2(\Delta_{0})}}\,{\cal
G}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4},}(x^{\prime}_{1},\ldots,x^{\prime}_{4}),\\\
$ (412)

We note that the $y_{-}$ integral in (412) is singular at $y_{-}=0$ and the
residue at this singularity is the contact contribution to the four point
correlator with insertions at $\\{x_{i}\\}$. This is an endpoint singularity
in $y_{-}$. At finite $y_{-}$, the integral is singular whenever ${\cal
G}_{\Delta_{1},\Delta_{4},\Delta_{3},\Delta_{2}}(x^{\prime}_{1},\ldots,x^{\prime}_{4})$
is singular. This singularity is again of the contact type with the insertions
now at $\\{x^{\prime}_{i}\\}$.

Let us now consider loop contributions to the four-point function. The one-
loop contribution with two quartic vertices is given by essentially the same
integral as in (412). The one-loop contribution with four cubic vertices is
given by

$\begin{split}{\cal G}_{\ell}=\prod_{i=1}^{4}\int
d^{D}y_{i}&\frac{1}{((x_{1}-y_{1})^{2})^{\Delta_{1}}}\frac{1}{((x_{2}-y_{2})^{2})^{\Delta_{2}}}\frac{1}{((x_{3}-y_{3})^{2})^{\Delta_{3}}}\frac{1}{((x_{4}-y_{4})^{2})^{\Delta_{4}}}\\\
&\frac{1}{((y_{1}-y_{2})^{2})^{\Delta_{-1}}}\frac{1}{((y_{2}-y_{3})^{2})^{\Delta_{-2}}}\frac{1}{((y_{3}-y_{4})^{2})^{\Delta_{-3}}}\frac{1}{((y_{4}-y_{1})^{2})^{\Delta_{-4}}}\\\
&\\\ \end{split}$ (413)

Only three of the differences in $y$ coordinates that appear are linearly
independent, we choose to replace
$(y_{4}-y_{1})=-(y_{1}-y_{2})-(y_{2}-y_{3})-(y_{3}-y_{4})$ and move to the
”center of mass” and ”difference” frame defined by

$y_{+}=\frac{\sum_{j=1}^{4}y_{j}}{4};y_{-i}=\frac{y_{i}-y_{i+1}}{4},\ \
i=1,2,3$ (414)

In these coordinates, we find that the one-loop contribution is given by

$\begin{split}{\cal G}_{\ell}=&\prod_{i=1}^{3}\int
d^{D}y_{-i}\frac{1}{(16y_{-i}^{2})^{\Delta_{-i}}}\frac{1}{((y_{-1}+y_{-2}+y_{-3})^{2})^{\Delta_{-4}}}\\\
&\int
d^{D}y_{+}\frac{1}{((x^{\prime}_{1}-y_{+})^{2})^{\Delta_{1}}}\frac{1}{((x^{\prime}_{2}-y_{+})^{2})^{\Delta_{2}}}\frac{1}{((x^{\prime}_{3}-y_{+})^{2})^{\Delta_{3}}}\frac{1}{((x^{\prime}_{4}-y_{+})^{2})^{\Delta_{4}}}\\\
\end{split}$ (415)

where $x^{\prime}_{i}=x_{i}+\sum_{j=1}^{3}A_{ij}y_{-j}$ with a matrix of
numbers $A_{ij}$ .

As before, we recognize that the center of mass integral is the same as for a
contact contribution and so, the 1-loop answer becomes

$\begin{split}{\cal G}^{\flat}_{\ell}=&\prod_{i=1}^{3}\int
d^{D}y_{-i}\frac{1}{(16y_{-i}^{2})^{\Delta_{-i}}}\frac{1}{((y_{-1}+y_{-2}+y_{-3})^{2})^{\Delta_{-4}}}{\cal
G}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(x^{\prime}_{1},\ldots,x^{\prime}_{4})\\\
\end{split}$ (416)

The integrand is singular when any one of the difference coordinates vanishes.
This happens when one or more of the four sides of the loop is of zero length.
When one side of the loop is contracted, we get a different one-loop diagram
with three vertices - two cubic vertices and one quartic vertex. This diagram,
in turn, has singularities when or more of the internal lines are contracted.
If one internal line is contracted, we end up with a single exchange diagram
and when both two internal lines are contracted, we find a (six-point) contact
diagram. In totality then, all the singularities of the one-loop diagram with
four cubic vertices arise from singularities of ”sub-diagrams”, i.e ones with
lesser number of internal lines and same or lesser number of vertices.

In general then, any contribution to the 4-point function can be expressed as
spacetime integrals over the difference coordinates with the integrand being
the product of a 4-point $\mathcal{D}^{\flat}$ function with a product of
polynomials of difference coordinates. The integrand becomes singular when one
or more of the internal lines collapse to give a ”sub diagram” of the original
diagram. The integrand is most singular when all the differences are zero i.e.
all internal lines are collapsed, in which case one gets the contact
contribution. Thus, the most singular part of any contribution to the
$4$-point function is the one coming from the contact diagram, and so it
suffices to study contact diagrams in order to extract the most singular part
of correlators. We display this general form for a diagram with $\ell$ loops,
$I$ internal lines

$\begin{split}{\cal G}_{\ell}=&\prod_{i=1}^{I-\ell}\int
d^{D}y_{-i}\frac{1}{(y_{-i}^{2})^{\Delta_{-i}}}\prod_{j=1}^{\ell}\frac{1}{((\sum_{i=1}^{\ell_{j}}y_{i})^{2})^{\Delta_{-j}}}{\cal
G}_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}(x^{\prime}_{1},\ldots,x^{\prime}_{4})\\\
\end{split}$ (417)

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