2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},r_{2})\equiv$
$\displaystyle\int_{-r_{2}}^{0}{\mathrm{d}}r\displaystyle\frac{\,{(-r)^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(-r)}\,}{r(r-r_{1})},$ (98) $\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},r_{2})\equiv$
$\displaystyle\int_{-1}^{-r_{2}}{\mathrm{d}}r\displaystyle\frac{\,{(-r)^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(-r)}\,}{r(r-r_{1})}.$ (99)

Unlike the 3-point case where the integrals extend from $-\infty$ to 0, the
integrals here are defined on finite intervals $(-r_{2},0)$ and $(-1,-r_{2})$,
making the calculation more involved. Still, we can get their analytical
results by using the PMB representation, although the actual computation is
quite lengthy. We collect the main steps and the final results for these two
integrals in App. C.

Once $\mathbf{I}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}$ and $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$ are obtained,
we get the full expression of the seed integral $\mathcal{Y}(r_{1},r_{2})$,
which can be further simplified and separated into the signal and the
background, namely,
$\mathcal{Y}(r_{1},r_{2})=\mathcal{Y}_{\text{S}}(r_{1},r_{2})+\mathcal{Y}_{\text{BG}}(r_{1},r_{2})$.
The simplification is spelled out in App. C. Here, we only show the final
result for the background $\mathcal{Y}_{\text{BG}}(r_{1},r_{2})$, since the
signal $\mathcal{Y}_{\text{S}}(r_{1},r_{2})$ has been given in (91):

$\displaystyle\mathcal{Y}_{\text{BG}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{Y}_{\text{BG},>}(r_{1},r_{2})\theta(r_{1}-r_{2})+\mathcal{Y}_{\text{BG},>}(r_{2},r_{1})\theta(r_{2}-r_{1});$
$\displaystyle\mathcal{Y}_{\mathrm{BG},>}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{\mathrm{i}2^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{\sqrt{2\pi}}\sum_{n=0}^{\infty}\displaystyle\frac{\,{(1+n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n-1/2}}\,}{n!}{\,}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}$
$\displaystyle\times{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{3}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}r_{2}^{2}\biggr{]}r_{1}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{2n}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (100)

This expression appears different from the known results in the literature
[39], but a direct numerical check shows that they agree with each other.
Therefore, we have successfully bootstrapped the 4-point correlation functions
with single massive exchange by dispersion integrals.

As we can see, for this particular 4-point example, performing the dispersion
integral is by no means simpler than performing the nest time integral
directly [39]. Rather, our calculation here serves as a proof of principle,
and shows that the dispersion relations really work for correlators with more
complicated kinematics than 3-point single-exchange diagram. On the other
hand, we can anticipate that the use of dispersion relation can bring
significant simplification to the 4-point correlators at 1-loop level. We will
pursue this 1-loop calculation in a future work.

## 4 Analytical Structure on the Complex Line-Energy Plane

In the previous two sections, we considered the analytical properties and
dispersion relations of inflation correlators in the complex plane of a vertex
energy. Starting from this section, we are going to study the analytical
properties of inflation correlators from a different perspective, by treating
a line energy as a complex variable. In general, inflation correlators with
massive exchanges also develop branch cuts on the complex plane of line
energies. Therefore, it is possible to develop a different type of dispersion
relations on the line energy plane, which we call _line dispersion relations_.
As we shall show, branch cuts on the complex plane of a line energy can all be
connected to the _nonlocal signal_ of the inflation correlator. Therefore, a
line dispersion relation allows us to compute the entire inflation correlator
from its nonlocal signal alone.

At the first sight, it may appear trivial that the branch cuts on a line
energy plane can be entirely attributed to the nonlocal signal. Indeed, recall
that the nonlocal signal with respect to a line energy $K_{i}$ refers to the
part of the correlator which develops complex powers in $K_{i}$ in the soft
$K_{i}$ limit:

$\lim_{K_{i}\rightarrow 0}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{G}\mkern-2.0mu}\mkern
2.0mu\Big{(}\\{E_{\ell}\\},\\{K_{j}\\}\Big{)}\sim
f\Big{(}\\{E_{\ell}\\},\\{K_{j}\\}\Big{)}K_{i}^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+g\Big{(}\\{E_{\ell}\\},\\{K_{j}\\}\Big{)},$ (101)

where both $f$ and $g$ are analytic at $K_{i}=0$, i.e., they have ordinary
Taylor expansions at $K_{i}=0$. Therefore, the nonlocal signal, by definition,
is associated with the branch point at $K=0$ generated by the complex-power
term $f(\\{E_{\ell}\\},\\{K_{j}\\})K_{i}^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$. However, things
are less trivial than they appear: The functions $f$ and $g$ are analytic in
$K_{i}$ only within a finite domain around $K_{i}=0$ where their Taylor
expansions converge. Outside the convergence domain, these two functions could
well develop new nonanalytic behaviors, including branch cuts, on the $K_{i}$
plane. These new nonanalyticities, in particular the ones in $g$, are not
obviously related to the nonlocal signal. Therefore, it is quite remarkable
that all branch cuts on the $K_{i}$ plane, including those not generated by
nonlocal signals, can actually be connected to the nonlocal signal alone. In
this section, we will spell out the details of reducing the entire correlator
to its nonlocal signal. In this sense, we may say that the nonlocal signal by
itself knows all about the whole correlator.

Recall from the previous two sections that a vertex dispersion integral
relates an inflation correlator with its _signal_ , both local and nonlocal.
On the other hand, the line dispersion enables the recovery of full correlator
from the nonlocal signal alone. Therefore, we see that the line dispersion is
more “economic” than vertex dispersion in that it can generate the full
correlator from a smaller set of data. This may have a practical advantage for
bootstrapping inflation correlators, since the nonlocal signal appears easier
to identify and to compute than the local signal, especially at the loop level
[37, 49, 50]. Therefore, we may expect that the line dispersion relation may
be a useful tool to bootstrap some complicated loop correlators whose full
analytical results remain out of reach with currently known methods.

#### Defining the nonlocal signal

Clearly, the nonlocal signal plays a central role in the line dispersion
relation. By definition, the nonlocal signal is a term in the correlator that
develops complex powers $K^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ in the soft line
energy limit $K\rightarrow 0$, namely the
$f(E_{\ell},K_{j})K_{i}^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ term in (101). Now
let us identify this piece in the four-point seed integral
$\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ in (11) without really
computing it.

When we fix the two vertex energies $k_{12}$ and $k_{34}$ in their physical
domain and let $k_{s}\rightarrow 0$, the seed integral
$\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ is well convergent in the
early time limit. Thus, its analytical behavior at $k_{s}=0$ is fully
determined by the analytical property of the integrand in $k_{s}$, which in
turn is determined by the bulk propagator
$D_{\mathsf{a}\mathsf{b}}(k_{s};\tau_{1},\tau_{2})$. Clearly, all four bulk
propagators listed in (4)-(6) are constructed from a pair of Hankel functions
$\mathrm{H}_{+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$ and
$\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$. Thus, we can
regroup these Hankel functions to separate all bulk propagators into a piece
analytic at $k_{s}=0$ and a piece that contains complex powers in $k_{s}$. In
practice, this can be neatly done by rewriting each Hankel function as a
linear combination of Bessel function of the first kind
$\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$; See (159). Then,
the Hankel product in the propagator can be rewritten as:

$\displaystyle\mathrm{H}$ ${}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k_{s}\tau_{2})$
$\displaystyle=\ $ $\displaystyle\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)^{2}\mathrm{J}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{J}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})+\bigl{(}1+\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}^{2}\mathrm{J}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{J}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k_{s}\tau_{2})$
$\displaystyle-\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigl{(}1+\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}\Big{[}\mathrm{J}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{J}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})+\mathrm{J}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{1})\mathrm{J}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k_{s}\tau_{2})\Big{]}.$ (102)

In this expression, we have two types of terms: One involves a product of two
$\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ with opposite
orders, namely $\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{J}_{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, as listed in the
first line on the right hand side of (4); The other type involves a product of
two $\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ with the same
order, namely $\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, listed in the last
line of (4). By expanding these Bessel J functions in the $k_{s}\rightarrow 0$
limit, it is straightforward to see that the opposite-order terms
$\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{J}_{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ are analytic as
$k_{s}\rightarrow 0$, while the same-order terms
$\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ behaves like
$k_{s}^{\pm 2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ as
$k_{s}\rightarrow 0$. Thus, the same-order terms in the propagators precisely
give rise to the nonlocal-signal part of the seed integral, while the
opposite-order terms contain no nonlocal signal. Either by more careful
inspection of the integral or by direct calculation, one can confirm that the
opposite-order terms correspond to the local signal and the background, but we
will not need this detailed separation between the local signal and the
background in this section. Incidentally, from the boundary viewpoint, the
same-order part can be viewed as the two-point correlator of a given conformal
block with dimension $\Delta=\mbox{$\frac{\,{3}\,}{2}$}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$, while the opposite-
order part the correlator between a conformal block and its shadow.

Based on the above observation, we now separate all four bulk propagators
$D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ according to their analytic
property at $k\rightarrow 0$ in the following way:

$\displaystyle
D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})=\Sigma(k;\tau_{1},\tau_{2})+\Omega_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2}).$
(103)

Here the _same-order propagators_ $\Sigma(k;\tau_{1},\tau_{2})$ involve terms
with same-order Bessel-J products, and thus are nonanalytic at $k=0$:

$\displaystyle\Sigma(k;\tau_{1},\tau_{2})\equiv$
$\displaystyle-\displaystyle\frac{\,{\pi(\tau_{1}\tau_{2})^{3/2}}\,}{4\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Big{[}\mathrm{J}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau_{1})\mathrm{J}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau_{2})+\mathrm{J}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau_{1})\mathrm{J}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k\tau_{2})\Big{]},$
(104)

while the four _opposite-order propagators_
$\Omega_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ involve terms with
opposite-order Bessel-J products, and thus are analytic at $k=0$:

$\displaystyle\Omega_{\pm\mp}(k;\tau_{1},\tau_{2})\equiv$
$\displaystyle~{}\displaystyle\frac{\,{\pi(\tau_{1}\tau_{2})^{3/2}}\,}{4\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Big{[}\big{(}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)-1\big{)}\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-k\tau_{1})\mathrm{J}_{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-k\tau_{2})-(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\Big{]},$
$\displaystyle\Omega_{\pm\pm}(k;\tau_{1},\tau_{2})\equiv$
$\displaystyle~{}\Omega_{\mp\pm}(k;\tau_{1},\tau_{2})\theta(\tau_{1}-\tau_{2})+\Omega_{\mp\pm}(k;\tau_{1},\tau_{2})\theta(\tau_{2}-\tau_{1}).$
(105)

We have deliberately removed the SK indices in the same-order propagators
$\Sigma(k;\tau_{1},\tau_{2})$, to highlight the fact that this propagator is
actually independent of the SK contours: All four choices of the SK labels
$\mathsf{a},\mathsf{b}=\pm$ yield the same expression
$\Sigma(k;\tau_{1},\tau_{2})$. This is closely tied to the fact that the
nonanalytic part of the propagator is real and symmetric in the two time
variables $\tau_{1}$ and $\tau_{2}$. In particular, the symmetry under
$\tau_{1}\leftrightarrow\tau_{2}$ renders the time-ordering $\theta$ functions
ineffective in the same-sign propagators.

However, let us immediately clarify that the same-order propagator
$\Sigma(k;\tau_{1},\tau_{2})$ is _not_ the symmetrization of the original bulk
propagator $D_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ with respect to
$\tau_{1}\leftrightarrow r_{2}$. As one can directly check, the opposite-order
propagator $\Omega_{\pm\pm}(k;\tau_{1},\tau_{2})$ also contains a piece that
is symmetric with respect to $\tau_{1}\leftrightarrow r_{2}$ but is
nevertheless analytic at $k_{s}=0$. In fact, this additional piece corresponds
to a part of the local signal that is symmetric in $k_{12}\leftrightarrow
k_{34}$.

Now, we can put the above separated bulk propagator back into the seed
integral, and separate the seed integral accordingly:

$\displaystyle\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=$
$\displaystyle~{}\mathcal{P}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})+\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s}),$
(106)

where $\mathcal{P}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$
and $\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ are
respectively nonanalytic and analytic at $k_{s}\rightarrow 0$ when $k_{12}$
and $k_{34}$ staying in the interior of their physical domain, whose
definitions are:

$\displaystyle\mathcal{P}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\equiv$
$\displaystyle-\mathsf{ab}\
k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}\Sigma(k_{s};\tau_{1},\tau_{2}),$
(107)
$\displaystyle\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\equiv$
$\displaystyle-\mathsf{ab}\
k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}\Omega_{\mathsf{a}\mathsf{b}}(k_{s};\tau_{1},\tau_{2}).$
(108)

We note that, although the same-order propagator itself
$\Sigma(k;\tau_{1},\tau_{2})$ is independent of SK indices, the nonanalytic
integrals
$\mathcal{P}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ still
have nontrivial dependences on $\mathsf{a},\mathsf{b}=\pm$ through the
exponential factors
$e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}$.

To complete our list of new definitions, we can also define the integrals with
SK indices summed:

$\displaystyle\mathcal{P}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\equiv\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{P}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s});$
(109)
$\displaystyle\mathcal{Q}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\equiv\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s}).$
(110)

From the above discussion, we see that $\mathcal{P}^{p_{1}p_{2}}$ is nothing
but the nonlocal signal, while $\mathcal{Q}^{p_{1}p_{2}}$ is the sum of the
local signal and the background:

$\displaystyle\mathcal{P}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=$
$\displaystyle~{}\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s}),$
(111) $\displaystyle\mathcal{Q}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=$
$\displaystyle~{}\mathcal{I}_{\text{LS}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})+\mathcal{I}_{\text{BG}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s}).$
(112)

#### Same-order integral

Now let us briefly look at the two integrals defined in (109) and (110).
First, consider the same-order integral
$\mathcal{P}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$. Combining (104), (107), and
(109), we see that the nonlocal signal can be directly expressed as a sum of
factorized time integrals:

$\displaystyle\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=\displaystyle\frac{\,{\pi}\,}{4\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\sum_{\mathsf{a},\mathsf{b},\mathsf{c}=\pm}\mathsf{a}\mathsf{b}\mathcal{V}_{\mathsf{c}}^{p_{1}}(\mathsf{a}k_{12},k_{s})\mathcal{V}_{\mathsf{c}}^{p_{2}}(\mathsf{b}k_{34},k_{s}),$
(113)

where we have introduced two single-layer integrals
$\mathcal{V}_{\pm}^{p}(E,K)$, defined by:

$\displaystyle\mathcal{V}_{\pm}^{p}(E,K)\equiv$
$\displaystyle~{}K^{5/2+p}\int_{-\infty}^{0}{\mathrm{d}}\tau\,(-\tau)^{3/2+p}e^{+\mathrm{i}E\tau}\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(-K\tau).$ (114)

This integral can be directly done and the result is expressed in terms of the
(dressed) Gauss’s hypergeometric function: (See App. B for our definition of
the dressed hypergeometric functions.)

$\displaystyle\mathcal{V}_{\pm}^{p}(E,K)=$
$\displaystyle~{}\displaystyle\frac{\,{2^{3/2+p}}\,}{\sqrt{\pi}}\Big{(}\displaystyle\frac{\,{K}\,}{\mathrm{i}E}\Big{)}^{5/2+p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{}_{2}\mathcal{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}(\mbox{$\frac{\,{5}\,}{2}$}+p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu),\mbox{$\frac{\,{1}\,}{2}$}(\mbox{$\frac{\,{7}\,}{2}$}+p\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\\\
1\pm\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\displaystyle\frac{\,{K^{2}}\,}{E^{2}}\right].$ (115)

Thus, the computation of the nonlocal signal involves only single-layer
integrals, which is a direct consequence of the nonlocal-signal cutting rule
studied in the literature [33, 49, 50].

#### Parity of the opposite-order integral

Next, let us turn to the opposite-order integrals
$\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}$. Unlike the nonlocal signal,
these integrals involve genuine time orderings that cannot be removed,
resulting in final expressions of higher “transcendental weight” [67], and
thus are more difficult to compute. We are going to compute them using
dispersion relations below. Here, without computing them directly, we point
out that the integral $\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}$ has a
very useful property: It possesses a fixed parity under the parity
transformation of the line energy: $k_{s}\rightarrow-k_{s}$.

To see this point, we make use of a property of Bessel J function, given in
App. B, which shows that $\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(e^{\mathrm{i}\pi}z)\mathrm{J}_{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(e^{\mathrm{i}\pi}w)=\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z)\mathrm{J}_{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(w)$. As a result,
the opposite-order propagator
$\Omega_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2})$ is invariant under the
sign flip of its energy:

$\Omega_{\mathsf{a}\mathsf{b}}(-k;\tau_{1},\tau_{2})=\Omega_{\mathsf{a}\mathsf{b}}(k;\tau_{1},\tau_{2}).$
(116)

With this property and the definition of the opposite-order integral in (108),
it is straightforward to see that
$\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}$ has a fixed parity
$(-1)^{1+p_{12}}$ under the $k_{s}$-parity transformation $k_{s}\rightarrow-
k_{s}$:

$\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},-k_{s})=(-1)^{1+p_{12}}\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s}),$
(117)

where $(-1)^{1+p_{12}}$ comes entirely from the prefactor $k_{s}^{5+p_{12}}$
in our definition of $\mathcal{Q}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}$. This
property will be very useful for our following derivation of the line
dispersion relation.

#### Analyticity along the positive real axis

After a brief analysis of same-order and opposite-order integrals, now let us
come back to the main goal of this section, namely, to diagnose the
nonanalyticity of the seed integral
$\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ on the complex $k_{s}$ plane.

The strategy is similar to what we adopted in Sec. 3, namely, to use the
contour-deformation method. With this method, we will show that the seed
integral $\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ is analytic
everywhere on the complex $k_{s}$ plane, expect for a possible branch cut
lying on the whole negative real axis. In the next part, we shall relate the
discontinuity of this branch cut to the one in the nonlocal signal
$\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}$.

Similar to the behavior in the vertex energy plane, the seed integral is
obviously analytic in $k_{s}$ for $\text{Im}\,k_{s}\neq 0$, a direct
consequence of contour deformation argument. More nontrivial is the following
fact:

$\displaystyle\theta(k_{s})\mathop{\text{Disc}}_{k_{s}}\mathcal{I}^{p_{1}p_{2}}_{\mathsf{a}\mathsf{b}}(k_{12},k_{34},k_{s})=0.$
(118)

That is, the seed integral is analytic in $k_{s}$ for all $k_{s}>0$. This is
quite remarkable because the region $k_{s}>0$ is not entirely physical:
Physically allowed $k_{s}$ satisfies $0\leq
k_{s}\leq\text{min}\\{k_{12},k_{34}\\}$. Thus, the statement (118) in
particular implies that seed integrals
$\mathcal{I}^{p_{1}p_{2}}_{\mathsf{a}\mathsf{b}}$ are regular on the
boundaries of the physical region when $k_{s}=k_{12}$ and $k_{s}=k_{34}$. As
is well known, the absence of singularities at these _folded_ configurations
is a consequence of choosing the Bunch-Davies initial state for all
fluctuating modes in the bulk. Now let us prove (118) rigorously using our
contour-deformation method.

Once again, the analytical behavior of seed integrals on the complex $k_{s}$
plane is governed by the UV behavior of the integrands, namely the convergence
of the integral as $\tau_{1,2}\rightarrow-\infty$. So let us look at these UV
regions for the opposite-sign integrals $\mathcal{I}_{\pm\mp}^{p_{1}p_{2}}$
and same-sign integrals $\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}$, respectively.

First, we consider the opposite-sign integral:

$\displaystyle\mathcal{I}_{+-}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=$
$\displaystyle~{}\displaystyle\frac{\,{\pi e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{4}k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}$
$\displaystyle~{}\times
e^{\mathrm{i}k_{12}\tau_{1}-\mathrm{i}k_{34}\tau_{2}}\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(-k_{s}\tau_{1})\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k_{s}\tau_{2}).$ (119)

Clearly, the integrand is well defined in the entire integration region for
all $k_{s}>0$. Furthermore, we only consider IR finite processes so that the
integral is convergent as $\tau_{1,2}\rightarrow 0$. Thus, any potential
singularity of the integral on the complex $k_{s}$ plane must come from the UV
divergences when $\tau_{1,2}\rightarrow-\infty$. However, it is easy to see
this never happens for $k_{s}>0$. In fact, using the asymptotic behavior of
the Hankel functions (31), we see that the integrand behaves like:

$e^{+\mathrm{i}(k_{12}+k_{s})\tau_{1}}e^{-\mathrm{i}(k_{34}+k_{s})\tau_{2}},$
(120)

up to irrelevant power factors of $\tau_{1}$ and $\tau_{2}$. We see that, for
physical values of $k_{12}$ and $k_{34}$, and for any $k_{s}>0$, the phases of
these factors never change sign or hit zero. Thus, we conclude that, with the
original choices of the integration contour (with proper $\mathrm{i}\epsilon$
prescriptions), the opposite-sign integral
$\mathcal{I}_{+-}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ is regular for any
$k_{s}>0$. With completely the same argument, we can also show that the other
opposite-sign integral $\mathcal{I}_{-+}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ is
regular for any $k_{s}>0$ as well.

The two same-sign seed integrals can be analyzed similarly. Let us consider
the all-plus integral:

$\displaystyle\mathcal{I}_{++}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=-\displaystyle\frac{\,{\pi
e^{-\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{4}k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}e^{\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}}$
$\displaystyle~{}\times\Big{[}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k_{s}\tau_{1})\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(-k_{s}\tau_{2})\theta(\tau_{1}-\tau_{2})+\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(-k_{s}\tau_{1})\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k_{s}\tau_{2})\theta(\tau_{2}-\tau_{1})\Big{]}.$ (121)

Parallel to the previous argument, the integral can develop singular behaviors
on the complex $k_{s}$ plane only through UV divergences. They occur when
either the earlier time variable or both the time variables go to $-\infty$.
Taking the $\theta(\tau_{1}-\tau_{2})$ part of (4) as an example:

$\displaystyle-\displaystyle\frac{\,{\pi e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{4}k_{s}^{5+p_{12}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{2}\int_{\tau_{2}}^{0}{\mathrm{d}}\tau_{1}\,(-\tau_{1})^{3/2+p_{1}}(-\tau_{2})^{3/2+p_{2}}e^{\mathrm{i}k_{12}\tau_{1}+\mathrm{i}k_{34}\tau_{2}}\mathrm{H}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(-k_{s}\tau_{1})\mathrm{H}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(-k_{s}\tau_{2}).$ (122)

In the UV limit $\tau_{1},\tau_{2}\rightarrow-\infty$, the integrand behaves
like
$e^{+\mathrm{i}(k_{12}-k_{s})\tau_{1}}e^{+\mathrm{i}(k_{34}+k_{s})\tau_{2}}$
up to unimportant power factors. Then, after finishing the $\tau_{1}$
integral, we get two terms, each behaves in the $\tau_{2}\rightarrow-\infty$
limit as $e^{+\mathrm{i}(k_{12}+k_{34})\tau_{2}}$ and
$e^{+\mathrm{i}(k_{34}+k_{s})\tau_{2}}$, respectively. Clearly, both phases
stay positive for all $k_{s}>0$, so that the integral is well convergent with
its original contour. One can similarly analyze the
$\theta(\tau_{2}-\tau_{1})$ term in (4) and gets the same result. The analysis
for $\mathcal{I}_{--}^{p_{1}p_{2}}$ is also the same. Thus we conclude that
the same-sign seed integrals are also regular for all $k_{s}>0$. This
completes the proof of (118).

#### Discontinuity in the line energy

Figure 9: The analytical structure of the seed integral
$\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})$ in (11) on the complex $k_{s}$
plane, with $k_{12}$ and $k_{34}$ staying in their physical regions
$k_{12},k_{34}>0$. The physical region of $k_{s}$ ($0\leq
k_{s}\leq\text{min}\\{k_{12},k_{34}\\}$) is marked in green. The seed integral
is analytic in $k_{s}$ everywhere except when $k_{s}\in(-\infty,0]$. In this
interval, we have three branch points: two partial-energy (PE) poles at
$k_{s}=-k_{12}$ and $k_{s}=-k_{34}$, and the signal branch point at
$k_{12}=0$. The total-energy pole $k_{1234}=0$ is never met for finite $k_{s}$
when $k_{12}\neq-k_{34}$, and can be thought of as sitting at $k_{s}=-\infty$
due to the scale invariance of the seed integral. The branch cut can be chosen
to be lying in the entire interval $k_{12}\in(-\infty,-k_{s}]$, with the
discontinuity itself being discontinuous at the two partial-energy poles.

From (118) we see that any possible branch cuts of the seed integrals
$\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}$ must lie in the negative
real axis. In other words, we have:

$\mathop{\text{Disc}}_{k_{s}}\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=\theta(-k_{s})\mathop{\text{Disc}}_{k_{s}}\Big{[}\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})+\mathcal{Q}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\Big{]}.$
(123)

Here we have used (106) with all SK indices summed, as well as (111), namely,
the same-order integral $\mathcal{P}^{p_{1}p_{2}}$ with all SK indices summed
is nothing but the nonlocal signal $\mathcal{I}^{p_{1}p_{2}}_{\text{NS}}$.

Now we are ready to relates this discontinuity with that of the nonlocal
signal. To this end, we make use of the above result (118), written in terms
of $\mathcal{I}^{p_{1}p_{2}}_{\text{NS}}$ and $\mathcal{Q}^{p_{1}p_{2}}$:

$\theta(k_{s})\mathop{\text{Disc}}_{k_{s}}\Big{[}\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})+\mathcal{Q}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})\Big{]}=0.$
(124)

Then, using the parity of the opposite-order integral $\mathcal{Q}$ in (117),
we get:

$\theta(k_{s})\mathop{\text{Disc}}_{k_{s}}\Big{[}\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})-(-1)^{1+p_{12}}\mathcal{Q}^{p_{1}p_{2}}(k_{12},k_{34},-k_{s})\Big{]}=0.$
(125)

This can be equivalently written as:

$\theta(-k_{s})\mathop{\text{Disc}}_{k_{s}}\mathcal{Q}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=(-1)^{1+p_{12}}\theta(-k_{s})\mathop{\text{Disc}}_{k_{s}}\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}(k_{12},k_{34},-k_{s}).$
(126)

Substituting this relation back to (123), we finally get:

$\displaystyle\mathop{\text{Disc}}_{k_{s}}\mathcal{I}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})=$
$\displaystyle~{}\mathop{\text{Disc}}_{k_{s}}\Big{[}\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}(k_{12},k_{34},k_{s})-(-1)^{p_{12}}\mathcal{I}_{\text{NS}}^{p_{1}p_{2}}(k_{12},k_{34},-k_{s})\Big{]}\theta(-k_{s}).$
(127)

That is, the discontinuity of the full seed integral can be completely related
to that of the nonlocal signal alone. This is the central result of the
current section, and forms the basis for the line energy dispersion relation,
to be discussed below.

## 5 Bootstrapping Correlators with Line Dispersion Relation

From the analytical structure of the seed integral on the complex $k_{s}$
plane, it is straightforward to construct dispersion integrals, which relate
the whole seed integral with its nonlocal signal alone. For clarity, let us
still specialize to the case of $p_{1}=p_{2}=-2$. Once again, we use the fact
that the dimensionless seed integral depends only on two independent momentum
ratios, and we have freedom to choose them. A convenient choice is
$r_{1}=k_{s}/k_{12}$ and $x\equiv k_{34}/k_{12}$, so that the analytical
structure of the seed integral in the line energy $k_{s}$ is manifest on the
complex $r_{1}$ plane. Again, to avoid potential confusions, we introduce a
new variable for the seed integral with this particular choice of arguments:

$\mathcal{Z}(r_{1}=k_{s}/k_{12},x=k_{34}/k_{12})\equiv\mathcal{I}^{-2,-2}(k_{12},k_{34},k_{s}).$
(128)

The integral for the nonlocal signal $\mathcal{Z}_{\text{NS}}(r_{1},x)$ is
likewise defined. Then, with this new notation, the discontinuity of the seed
integral (127) can be rewritten as:

$\displaystyle\mathop{\text{Disc}}_{r_{1}}\mathcal{Z}(r_{1},x)=\mathop{\text{Disc}}_{r_{1}}\Big{[}\mathcal{Z}_{\text{NS}}(r_{1},x)-\mathcal{Z}_{\text{NS}}(-r_{1},x)\Big{]}\theta(-r_{1}),$
(129)

where $x$ remains in the physical region $x>0$. We show this result in Fig.
10.

From this result we learn a lesson: The singularity structure of the seed
integral as a function of one momentum ratio, say $r_{1}$, is crucially
dependent on how we choose and fix other ratios. This is made clear by
comparing Fig. 8, where we fix $r_{2}$, and Fig. 10, where we fix
$x=r_{1}/r_{2}$. There is nothing mysterious here: In the most general
situation, the seed integral is to be treated as a function of multiple
complex variables, whose singularity structure on a multidimensional complex
space can be quite complicated. The dispersion relations considered in this
work, on the other hand, are always formulated on a fixed complex dimension-1
submanifold, where we only see the projections of higher dimensional
singularities. By fixing different ratios, we are working on different complex
dimension-1 submanifolds, and it is not surprising that the projections of
singularities on these submanifolds are different.

With the discontinuity given in (129), we can directly write down a dispersion
integral for the seed integral $\mathcal{Z}(r_{1},x)$:

$\displaystyle\mathcal{Z}(r_{1},x)=\displaystyle\frac{\,{r_{1}}\,}{2\pi\mathrm{i}}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}r}\,}{r(r-r_{1})}\mathop{\text{Disc}}_{r_{1}}\Big{[}\mathcal{Z}_{\text{NS}}(r_{1},x)-\mathcal{Z}_{\text{NS}}(-r_{1},x)\Big{]}.$
(130)

Here we have introduced a first-order subtraction. This choice follows from
the asymptotic behavior of the seed integral $\mathcal{Z}(r_{1},x)$ in the
limit $|r_{1}|\rightarrow\infty$. Note that $r_{1}=\infty$ is the total energy
limit where $\mathcal{Z}(r_{1},x)$ diverges at most logarithmically by power
counting of time in the time integral. So, a first-order subtraction is
sufficient to make the dispersion integral well defined.

Figure 10: The analytical structure of the 4-point seed integral
$\mathcal{Z}(r_{1}^{\prime},x)$ on the complex $r_{1}^{\prime}$ plane, with
$x\in(0,\infty)$ fixed in the interior of its physical region. There are three
branch points at $r_{1}^{\prime}=0$ (signal branch point), $r_{1}^{\prime}=-1$
(partial-energy pole $k_{s}=-k_{12}$), and $r_{1}^{\prime}=-x$ (partial-energy
pole $k_{s}=-k_{34}$), with a branch cut connecting them on the whole negative
real axis. In this plot, we also show the insertion of a pole at
$r_{1}^{\prime}=r_{1}$ at which we compute the seed integral via the line
dispersion integral (130), whose contour is shown by the blue curve
surrounding the branch cut.

The nonlocal signal of the seed integral has been presented in (113), and here
we rewrite it as a function of $r_{1}$ and $x$:

$\displaystyle\mathcal{Z}_{\text{NS}}(r_{1}=k_{s}/k_{12},x=k_{34}/k_{12})=\displaystyle\frac{\,{\pi}\,}{4\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\sum_{\mathsf{a},\mathsf{b},\mathsf{c}=\pm}\mathsf{a}\mathsf{b}\mathcal{V}_{\mathsf{c}}^{-2}(\mathsf{a}k_{12},k_{s})\mathcal{V}_{\mathsf{c}}^{-2}(\mathsf{b}k_{34},k_{s}),$
(131)

Using the result for $\mathcal{V}^{-2}_{\mathsf{c}}$ in (115), we get an
explicit expression for $\mathcal{Z}_{\mathrm{NS}}$ as:

$\displaystyle\mathcal{Z}_{\mathrm{NS}}(r_{1},x)=\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi}x^{-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}r_{1}^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1}/x)\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu),$ (132)

where $\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p}$ is defined in (63).

#### Discontinuity of the nonlocal signal

To evaluate the dispersion integral (130), we need the discontinuity of the
nonlocal signal $\mathcal{Z}_{\mathrm{NS}}$. It is possible to get this
discontinuity by analyzing the integral expression for $\mathcal{V}_{\pm}^{p}$
without really evaluating it, like we did for $\mathcal{U}_{\pm}^{p}$ before.
However, here we choose to present the discontinuity directly by known
analytical properties of power functions and Gauss’s hypergeometric functions.
Notice that the power function $r_{1}^{-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ has a branch cut in
the negative real axis, and that the Gauss’s hypergeometric function in
$\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}$ factors has a branch cut when its argument $z\in(1,\infty)$.
Thus, all three $r_{1}$-dependent factors in (132) make contributions to the
discontinuity of the nonlocal signal. More explicitly:

  * •

$r_{1}^{\pm 2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ contributes a
branch cut for $r_{1}\in(-\infty,0)$.

  * •

$\mathbf{F}_{\pm\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1}/x)$ contributes a branch cut for
$r_{1}\in(-\infty,-x)\cup(x,\infty)$.

  * •

$\mathbf{F}_{\pm\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1})$ contributes a branch cut for
$r_{1}\in(-\infty,-1)\cup(1,\infty)$.

Then, it is straightforward to see that the nonlocal signal
$\mathcal{Z}_{\text{NS}}(r_{1},x)$ in (132) has a branch cut for any real
value of $r_{1}$ except when $0<r_{1}<\text{min}\\{x,1\\}$, in which
$\mathcal{Z}_{\text{NS}}(r_{1},x)$ is real. In addition, the discontinuity
across the branch cut is itself discontinuous at $r_{1}=\pm x$ and $r_{1}=\pm
1$. We show these branch cuts in Fig. 11, where we make manifest the
contributions from different factors in (132).

Figure 11: The analytical structure of the nonlocal signal
$\mathcal{Z}_{\text{NS}}(r_{1},x)$ on the complex $r_{1}$ plane, with
$x\in(0,\infty)$ fixed in the interior of its physical region. The nonlocal
signal is discontinuous for all real $r_{1}$ except in the physical interval
$0<x<\text{min}\\{x,1\\}$ in which $\mathcal{Z}_{\text{NS}}(r_{1},x)$ is real.
The gray, green, and cyan wiggly lines show the branch cut arising from
$r^{\pm 2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}$, $\mathbf{F}_{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(\pm r_{1}/x)$,
and $\mathbf{F}_{\pm\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(\pm r_{1})$, respectively. All branch cuts are on the real axis.
However, we offset the green and cyan cuts to highlight our description for
computing the discontinuity of a product of functions, and also for clearer
illustration.

Incidentally, when we compute a quantity like
$\mathop{\text{Disc}}\limits_{z}[f(z)g(z)]$ where both $f$ and $g$ have
discontinuities, there are multiple equivalent ways to express it in terms of
the discontinuity of individual factor. For instance, when $z>0$, we have

$\displaystyle\mathop{\text{Disc}}_{z}\Big{[}f(z)g(z)\Big{]}=f(z^{+})\mathop{\text{Disc}}_{z}\Big{[}g(z)\Big{]}+\mathop{\text{Disc}}_{z}\Big{[}f(z)\Big{]}g(z^{-})$
$\displaystyle=f(z^{-})\mathop{\text{Disc}}_{z}\Big{[}g(z)\Big{]}+\mathop{\text{Disc}}_{z}\Big{[}f(z)\Big{]}g(z^{+}).$
(133)

Here and below, we introduce the shorthand notation $z^{\pm}\equiv
ze^{\pm\mathrm{i}\epsilon}$ with $\epsilon$ an infinitesimal positive real.
Thus, when computing the discontinuity of products of functions, one can make
various choices. To fix our choice, we infinitesimally displace some branch
cuts into complex plane, as shown in Fig. 11. According to this prescription,
for instance, when we compute the discontinuity across the cyan branch cut
(from $\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1})$ factor) in the negative real axis, we should evaluate the
other two factors on the lower edges of the gray and green branch cut, namely,
we take $(r_{1}^{+})^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ and
$\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1}^{+}/x)$.111111Note that, for negative $r_{1}$, $r_{1}^{+}$
corresponds to the lower edge of the branch cut and $r^{-}$ to the upper edge.

The structure of the branch cut suggests that we should break the line
dispersion integral (130) into three pieces, each corresponding to the branch
cut from a given factor, and also to a wiggly line of a given color in Fig.
11. Below we work out the discontinuity across each of these branch cuts.

First, the gray branch cut in Fig. 11 is contributed by the power factor
$r_{1}^{1-2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}$ in (130). We define the discontinuity from this branch cut into the
following function:

$\displaystyle\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},x)\equiv$
$\displaystyle~{}\mathop{\text{Disc}}_{r_{1}}\Big{[}r_{1}^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{]}\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}^{-}/x)\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}^{-})$
$\displaystyle=$ $\displaystyle-2\sinh(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-r_{1})^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}^{-}/x)\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}^{-}).$ (134)

Here the two hypergeometric factors are taking values from the upper edges of
their branch cuts on $r_{1}$ plane, consistent with our displacement of the
branch cuts in Fig. 11.

Second, the green branch cut in Fig. 11 is contributed by the hypergeometric
factor $\mathbf{F}_{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(\pm r_{1}/x)$
in (130). For negative $r_{1}$, the discontinuity across this branch cut is
given by the following function:

$\displaystyle\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},x)\equiv$ $\displaystyle~{}(r_{1}^{+})^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathop{\text{Disc}}_{r_{1}}\Big{[}\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}/x)\Big{]}\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}^{-})$
$\displaystyle=$ $\displaystyle-e^{-2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-r_{1})^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}/x)\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1}^{-}),$
(135)

where we have defined the discontinuity of $\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(z)$ along its
branch cut to be $\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$. Using the known
property of the Gauss’s hypergeometric function in (156), we can find an
explicit expression for $\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$:

$\displaystyle\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z)\equiv\mathop{\text{Disc}}_{z}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(z)=-\sqrt{2\pi^{3}}\,\mathrm{csch}\,(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\,{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}\frac{1}{4}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{3}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1\end{matrix}\bigg{|}1-z^{2}\biggr{]}.~{}~{}~{}~{}(z<-1)$ (136)

Third, the cyan branch cut in Fig. 11 is contributed by the hypergeometric
factor $\mathbf{F}_{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(\pm r_{1})$ in
(130). For negative $r_{1}$, we define the discontinuity across this branch
cut into the following function:

$\displaystyle\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(3)}(r_{1},x)\equiv$ $\displaystyle~{}(r_{1}^{+})^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}^{+}/x)\mathop{\text{Disc}}_{r_{1}}\Big{[}\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1})\Big{]}$
$\displaystyle=$ $\displaystyle-e^{-2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-r_{1})^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{[}\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}^{-}/x)-\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}/x)\Big{]}\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}),$ (137)

where we have used the relation $\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}^{+}/x)=\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}^{-}/x)+\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}/x)$ to rewrite
a hypergeometric factor in terms of its value across the branch cut.

In summary, for negative values of $r_{1}$, the discontinuity across the
branch cuts of the nonlocal signal $\mathcal{Z}_{\text{NS}}(r_{1},x)$ is given
by:

$\displaystyle\mathop{\text{Disc}}_{r_{1}}\mathcal{Z}_{\text{NS}}(r_{1}<0,x)=\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},x)\theta(-r_{1})+\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},x)\theta(-r_{1}-x)+\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(3)}(r_{1},x)\theta(-r_{1}-1).$ (138)

On the other hand, as shown in Fig. 11, the green and cyan branch cuts also
extend to positive real values of $r_{1}$. However, the discontinuities for
these “positive” branch cuts are not independent, since the two “positive”
branch cuts can be related to the corresponding two “negative” branch cuts by
a $180^{\circ}$ rotation around the origin $r_{1}=0$ via the _lower_ plane (in
order not to cross the gray branch cut). Now, using the facts that the
function $\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1})$ is even in
$r_{1}$, and that $(e^{-\mathrm{i}\pi}r_{1})^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}=-e^{-2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}r_{1}^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ for $r_{1}>0$, we
have:

$\displaystyle\mathop{\text{Disc}}_{r_{1}}\mathcal{Z}_{\text{NS}}(r_{1}>0,x)=e^{2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},x)\theta(r_{1}-x)+e^{2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(3)}(r_{1},x)\theta(r_{1}-1).$ (139)

Thus we have found the explicit expressions for all five branch cuts shown in
Fig. 11.

#### Line dispersion integral and the result

Based on the previous analysis of the branch cut of the nonlocal signal, we
are now ready to find the explicit expression for the line dispersion
integral. Combining (138) and (139), we see that the dispersion integral (130)
now boils down to three integrals $\mathbf{J}_{i}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ $(i=1,2,3)$:

$\displaystyle{\mathcal{Z}}(r_{1},x)=-\displaystyle\frac{\,{\mathrm{i}+\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{4\pi^{2}}x^{-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{r_{1}}\Bigl{[}\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},x)+\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},x)+\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(3)}(r_{1},x)\Bigr{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu),$ (140)

where the three terms $(\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)},\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)},\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(3)})$ correspond
to integrals around the gray, green, and cyan branch cuts in Fig. 11,
respectively:

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},x)\equiv$
$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}r\displaystyle\frac{\,{\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r,x)}\,}{r(r-r_{1})},$ (141) $\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},x)\equiv$
$\displaystyle\int_{-\infty}^{-x}{\mathrm{d}}r\displaystyle\frac{\,{(1-e^{2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu})\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r,x)}\,}{r(r-r_{1})},$ (142) $\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(3)}(r_{1},x)\equiv$
$\displaystyle\int_{-\infty}^{-1}{\mathrm{d}}r\displaystyle\frac{\,{(1-e^{2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu})\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(3)}(r,x)}\,}{r(r-r_{1})}.$ (143)

These three integrals can be computed analytically via PMB representation,
although the details are quite lengthy. We collect them in App. D, and present
the results below.

As mentioned many times before, the 4-point seed integral
$\mathcal{Z}(r_{1},x)$ can be written as a sum of nonlocal signal (NS), the
local signal (LS), and the background (BG):

$\mathcal{Z}(r_{1},x)=\mathcal{Z}_{\text{NS}}(r_{1},x)+\mathcal{Z}_{\text{LS}}(r_{1},x)+\mathcal{Z}_{\text{BG}}(r_{1},x).$
(144)

It turns out that the nonlocal signal $\mathcal{Z}_{\text{NS}}$ is contributed
only by the integral around the gray branch cut, namely $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$. The result
is simply identical to our input (132), which we collect here for
completeness:

$\displaystyle\mathcal{Z}_{\mathrm{NS}}(r_{1},x)=\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi}x^{-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}r_{1}^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1}/x)\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu),$ (145)

The local signal $\mathcal{Z}_{\text{LS}}$ receives contributions from the
integrals around the gray and the green branch cuts, namely
$\mathbf{J}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}$ and $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$,
respectively. The result is:

$\displaystyle\mathcal{Z}_{\mathrm{LS}}(r_{1},x)=$
$\displaystyle~{}\mathcal{Z}_{\mathrm{LS},>}(r_{1},x)\theta\big{(}1-|x|\big{)}+\mathcal{Z}_{\mathrm{LS},>}(r_{1}/x,1/x)\theta\big{(}|x|-1\big{)},$
(146) $\displaystyle\mathcal{Z}_{\mathrm{LS},>}(r_{1},x)=$
$\displaystyle~{}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi}x^{-1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}r_{1}\mathbf{F}^{-2}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}/x)\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (147)

Finally, the background $\mathcal{Z}_{\text{BG}}$ receives contributions from
the integrals around all three branch cuts, namely $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(i)}$ $(i=1,2,3)$,
whose result can be simplified into the following form:

$\displaystyle\mathcal{Z}_{\mathrm{BG}}(r_{1},x)=$
$\displaystyle~{}\mathcal{Z}_{\mathrm{BG},>}(r_{1},x)\theta\big{(}1-|x|\big{)}+\mathcal{Z}_{\mathrm{BG},>}(r_{1}/x,1/x)\theta\big{(}|x|-1\big{)},$
(148) $\displaystyle\mathcal{Z}_{\mathrm{BG},>}(r_{1},x)=$
$\displaystyle\sum_{n=0}^{\infty}\frac{8(-x)^{n}r_{1}}{(1+2n)^{2}+4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}\,{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}1,\frac{1}{2}+\frac{n}{2},1+\frac{n}{2}\\\
\frac{5}{4}+\frac{n}{2}-\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},\frac{5}{4}+\frac{n}{2}+\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2}\end{matrix}\bigg{|}r_{1}^{2}\biggr{]}.$ (149)

This expression for $\mathcal{Z}_{\mathrm{BG}}$ has a different look from
known results in [39], but is identical to the latter. In fact, the background
part is a two-variable hypergeometric function known as Kampé de Fériet
function and allows for many different series representations [7, 39, 42].

## 6 Conclusions and Outlooks

As the dS counterparts of flat-space scattering amplitudes, inflation
correlators possess distinct analytical structure. For general massive-
exchange processes, branch cuts usually appear in the complex plane of
appropriate kinematic variables, which connect physics in the UV and IR
regions. In the IR regions, such branch cuts are closely related to
logarithmic oscillations of the correlators in the physical regions, known as
CC signals in the context of Cosmological Collider physics. The CC signals
have received many studies in recent years. In particular, it has been shown
that, although the computation of general inflation correlators is difficult,
we can apply the cutting rule and the factorization theorem to extract CC
signals in the squeezed limit [33, 49, 50]. In comparison, the corresponding
branch cuts on the complex domain beyond the physical region are less
understood and deserve more studies.

In this work, we explore the analytical properties of massive inflation
correlators as functions of two types of kinematic variables: the vertex
energies and the line energies. For both types of energies of a tree-level
correlator, we identified the total-energy and partial-energy poles, the
signal branch point, together with the branch cuts connecting them. Based on
this structure, we developed two distinct dispersion relations: a vertex
dispersion relation which relates the correlator to its full signal, and a
line dispersion relation which relates the correlator to the nonlocal signal
alone. With these dispersion relations, we have successfully bootstrapped a
few tree-level and 1-loop massive inflation correlators. At 1-loop level, our
method is manifestly UV finite and free from any regularization procedure.
This allows us to neatly single out the renormalization-independent part of
the correlator, which is unambiguously determined by analyticity.

Although there have been scattered studies on analytical properties of
inflation correlators (and the related wavefunction coefficients), to our best
knowledge, the dispersion relations have not been used to bootstrap the full
massive inflation correlators. Our work filled this gap by providing a few
proof-of-principle calculations. While the computation itself can often become
lengthy compared to other existing methods for simple examples, it
nevertheless shows the potential power of the dispersion techniques in
bootstrapping more complicated diagrams. Thus, we consider this work a first
step in carrying out a more extensive program of dispersive bootstrap.
Naturally, many directions are open to further explorations, and we conclude
this work by mentioning some of them.

A natural first task is to chart all nonanalyticities of a given tree diagram,
beyond the 4-point single exchange. This includes not only the locations of
poles and branch cuts, but also the discontinuities across all branch cuts.
With these data, we can imagine to recursively bootstrap more complicated
diagrams from simple sub-diagrams, either analytically or numerically.

Next, it would be very interesting to explore the potential of dispersive
bootstrap for loop diagrams. We have seen that dispersion relations could be
advantageous in bootstrapping one-loop diagrams, including the absence of UV
divergences, the simplified expressions, and the separation of renormalization
dependent and independent parts. They encourage us to consider more
complicated loop processes. As a concrete first step, we may try to combine
the dispersive and spectral methods and bootstrap 1-loop bubble processes with
spinning exchanges and derivative couplings, and this will be explored in a
follow-up work. Beyond the bubble topology, it is not immediately clear that
techniques like spectral decomposition are still available. Nevertheless, it
looks promising to us to numerically implement the dispersive techniques for
loop processes. We plan to investigate this route in a future work.

As we pointed out many times, the dispersive bootstrap at its core is an idea
to reconstruct the whole diagram from a knowledge of sub-diagrams. In this
regard, what we have considered in this work is the most straightforward
realization, namely, exploiting the complex energy planes. It is also
interesting to search for “dispersion relations” with not only complex
energies, but also other complex parameters. In flat space, it has been very
fruitful to consider scattering amplitudes on complex planes of mass, angular
momentum, and even spacetime dimensions. We can imagine that the analytical
structures in these complex parameters could also bring us new insights and
new methods for inflation correlators. Also, it has been shown recently that
the parity-odd part of a cosmological correlator (or a wavefunction
coefficient) automatically factorized under rather general conditions. [104,
40, 55, 71] Thus, it would be very interesting to develop dispersion
techniques for parity-violating theories.

Last but not least, the dispersion relations in flat spacetime or in CFT are
usually tied to nonperturbative properties of amplitudes, and are used to make
nonperturbative statements about the unitarity and positivity of the theory.
On the other hand, in this work, we only apply the dispersion techniques at
the diagrammatic level. How are the two approaches related? From pure
diagrammatic analysis, is it possible to gain insights applicable to all
orders in perturbation theory? Similar to flat-space situations, we believe
that, at least for simple kinematics with full dS isometries, it is possible
to make progress along these directions. We leave all these interesting topics
for future studies.

#### Acknowledgments

We thank Xingang Chen, Enrico Pajer, Carlos Duaso Pueyo, Sébastien Renaux-
Petel, Xi Tong, Lian-Tao Wang, Yi Wang, Denis Werth, Jiayi Wu, Hongyu Zhang,
and Yuhang Zhu for useful discussions. This work is supported by NSFC under
Grants No. 12275146 and No. 12247103, the National Key R&D Program of China
(2021YFC2203100), and the Dushi Program of Tsinghua University.

##  Appendix A Notations

In this appendix, for readers’ convenience, we collect some frequently used
variables in Table 1, together with the numbers of equations where they are
defined or first appear.

Table 1: List of selected symbolic notations  Notation | Description | Equation
---|---|---
$E_{i}$ | External energy | Above (7)
$K_{i}$ | Internal energy | Above (7)
$u$ | Momentum ratio $u\equiv 2k_{3}/k_{123}$ | Above (59)
$r_{1},r_{2}$ | Momentum ratios $r_{1}\equiv k_{s}/k_{12}$ and $r_{2}\equiv k_{s}/k_{34}$ | (89)
$x$ | Momentum ratio $x\equiv k_{34}/k_{12}$ | Above (128)
$C_{\mathsf{a}}(k;\tau)$ | External propagator of conformal scalar $\phi_{c}$ | (3)
$D_{\mathsf{ab}}(k;\tau_{1},\tau_{2})$ | Internal propagator of massive scalar $\sigma$ | (4), (5), and (6)
$\Sigma(k;\tau_{1},\tau_{2})$ | Same-order part of bulk propagator | (104)
$\Omega_{\mathsf{ab}}(k;\tau_{1},\tau_{2})$ | Opposite-order part of bulk propagator | (4)
$\mathbf{Q}_{\mathsf{ab}}(k;\tau_{1},\tau_{2})$ | 1-loop bubble function of $\sigma$ | (81)
$\mathsf{a},\mathsf{b},\cdots$ | SK or non-SK indices taking value from $\pm 1$ | (2)
$\mathcal{U}^{p}_{\pm}(E_{i},K_{i})$ | Subgraph integral with one Hankel H | (33)
$\mathcal{V}^{p}_{\pm}(E_{i},K_{i})$ | Subgraph integral with one Bessel J | (114)
$\mathcal{G}$ | A general tree graph for $\phi_{c}$ correlators | (2)
$\mathcal{I}_{\mathsf{ab}}(k_{12},k_{34},k_{s})$ | 4-point single-exchange seed integral | (10)
$\mathcal{P}_{\mathsf{ab}}$ | Same-order part of $\mathcal{I}_{\mathsf{a}\mathsf{b}}$ | (107)
$\mathcal{Q}_{\mathsf{ab}}$ | Opposite-order part of $\mathcal{I}_{\mathsf{a}\mathsf{b}}$ | (108)
$\mathcal{X}(u)$ | 3-point tree integral in $u$ | (59)
$\mathcal{J}(u)$ | 3-point 1-loop bubble integral in $u$ | (80)
$\mathcal{Y}(r_{1},r_{2})$ | 4-point tree integral in $r_{1}$ and $r_{2}$ | (90)
$\mathcal{Z}(r_{1},x)$ | 4-point tree integral in $r_{1}$ and $x$ | (128)
$\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p}(z)$ | A rescaled Gauss’s hypergeometric funtion | (63)
$\mathbf{G}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z)$ | Discontinuity of $\mathbf{F}^{-2}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z)$ when $z<-1$ | (136)

##  Appendix B Useful Functions and Properties

In this appendix, we collect a few special functions and their properties used
in the main text. These are standard material, and we quote them from [170,
171].

#### Euler $\mathbf{Gamma}$ products and fractions

In this work we use the following shorthand notation for the productions and
fractions of Euler $\Gamma$ functions:

$\Gamma\big{[}a_{1},\cdots,a_{n}\big{]}\equiv\Gamma(a_{1})\cdots\Gamma(a_{n});$
(150) $\displaystyle\Gamma\biggl{[}\begin{matrix}a_{1},a_{2},\cdots,a_{m}\\\
b_{1},b_{2},\cdots,b_{n}\end{matrix}\biggr{]}\equiv\displaystyle\frac{\,{\Gamma(a_{1})\Gamma(a_{2})\cdots\Gamma(a_{m})}\,}{\Gamma(b_{1})\Gamma(b_{2})\cdots\Gamma(b_{n})}.$
(151)

With this notation, the Pochhammer symbol $(a)_{n}$ is defined as

$\displaystyle(a)_{n}\equiv\Gamma\biggl{[}\begin{matrix}a+n\\\
a\end{matrix}\biggr{]}.$ (152)

#### Hypergeometric functions

The (generalized) hypergeometric function is used in this work, whose standard
form is defined by the following series when convergent, and by analytical
continuation otherwise:

${}_{p}\mathrm{F}_{q}\biggl{[}\begin{matrix}a_{1},a_{2},\cdots,a_{p}\\\
b_{1},b_{2},\cdots,b_{q}\end{matrix}\bigg{|}z\biggr{]}\equiv\sum_{n=0}^{\infty}\displaystyle\frac{\,{(a_{1})_{n}(a_{2})_{n}\cdots(a_{p})_{n}}\,}{(b_{1})_{n}(b_{2})_{n}\cdots(b_{q})_{n}}\displaystyle\frac{\,{z^{n}}\,}{n!}.$
(153)

In particular, ${}_{2}\text{F}_{1}$ is known as the Gauss’s or ordinary
hypergeometric function.

There are a few useful variations whose definitions are different from the
standard form only in prefactors. First, the _regularized_ hypergeometric
function ${}_{p}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern 2.0mu_{q}$ is
defined by:

$\displaystyle{}_{p}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{q}\biggl{[}\begin{matrix}a_{1},a_{2},\cdots,a_{p}\\\
b_{1},b_{2},\cdots,b_{q}\end{matrix}\bigg{|}z\biggr{]}\equiv\displaystyle\frac{\,{1}\,}{\Gamma[b_{1},b_{2},\cdots,b_{q}]}{}_{p}\mathrm{F}_{q}\biggl{[}\begin{matrix}a_{1},a_{2},\cdots,a_{p}\\\
b_{1},b_{2},\cdots,b_{q}\end{matrix}\bigg{|}z\biggr{]}.$ (154)

It is called regularized, because, when the argument $z$ is not at the
singular points, the regularized hypergeometric function is an entire function
of all the parameters $(a_{1},\cdots,a_{p},b_{1},\cdots,b_{q})$.

Second, we frequently use the “dressed” hypergeometric function
${}_{p}{\mathcal{F}}_{q}$ in the main text, because it simplifies a lot of
expressions:

$\displaystyle{}_{p}{\mathcal{F}}_{q}\biggl{[}\begin{matrix}a_{1},a_{2},\cdots,a_{p}\\\
b_{1},b_{2},\cdots,b_{q}\end{matrix}\bigg{|}z\biggr{]}\equiv\displaystyle\frac{\,{\Gamma[a_{1},a_{2},\cdots,a_{p}]}\,}{\Gamma[b_{1},b_{2},\cdots,b_{q}]}{}_{p}\mathrm{F}_{q}\biggl{[}\begin{matrix}a_{1},a_{2},\cdots,a_{p}\\\
b_{1},b_{2},\cdots,b_{q}\end{matrix}\bigg{|}z\biggr{]}.$ (155)

It is useful to note that the Gauss’s hypergeometric function
${}_{2}\text{F}_{1}[\cdots|z]$ in general has two branch points at $z=1$ and
$z=\infty$. It is our convention to choose the branch cut connecting these two
points to lie in the interval $z\in(1,\infty)$ on the real axis. We define the
value of ${}_{2}\text{F}_{1}[\cdots|z]$ when $z>1$ by its value on the _lower_
edge of the branch cut. Then, the value on the upper edge is determined by the
discontinuity across the branch cut. More explicitly:

$\left\\{\begin{aligned} &{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,b\\\
c\end{matrix}\bigg{|}z^{+}\biggr{]}\\\ =&\
2\pi\mathrm{i}e^{\pi\mathrm{i}(a+b-c)}\Gamma\biggl{[}\begin{matrix}c\\\
a+b-c+1,c-a,c-b\end{matrix}\biggr{]}{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,b\\\
a+b-c+1\end{matrix}\bigg{|}1-z\biggr{]}\\\
&+e^{2\pi\mathrm{i}(a+b-c)}{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,b\\\
c\end{matrix}\bigg{|}z\biggr{]},\\\
&{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,b\\\
c\end{matrix}\bigg{|}z^{-}\biggr{]}={}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,b\\\
c\end{matrix}\bigg{|}z\biggr{]}.\end{aligned}\right.$ (156)

For power functions with non-integer powers, we can get a branch cut along the
negative real axis by restricting the argument of variable in $(-\pi,\pi]$.

$\displaystyle\begin{aligned} &(ze^{\mathrm{i}\pi})^{p}=e^{\mathrm{i}\pi
p}z^{p},\\\ &(ze^{-\mathrm{i}\pi})^{p}=e^{-\mathrm{i}\pi
p}z^{p}.~{}~{}~{}~{}(z>0)\end{aligned}$ (157)

#### Bessel functions

In this work, we used the standard Bessel J function, especially its
analytical property. Generically, a Bessel J function $J_{\nu}(z)$ has a
branch cut on the negative real axis, connecting $z=0$ and $z=-\infty$. The
discontinuity across this branch cut is conveniently captured by the following
identity:

$\displaystyle\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(e^{m\pi\mathrm{i}}z)=e^{\mp m\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\pi}\mathrm{J}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z).~{}~{}~{}~{}(z>0)$ (158)

More frequently appeared in the main text are Hankel functions
$\mathrm{H}_{\nu}^{(1)}$ and $\mathrm{H}_{\nu}^{(2)}$, which can be expressed
in terms of Bessel J function as:

$\displaystyle\begin{aligned} &\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z)=\bigl{(}1+\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}\mathrm{J}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z)-\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathrm{J}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z),\\\
&\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z)=-\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathrm{J}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z)+\bigl{(}1+\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}\mathrm{J}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z).\end{aligned}$
(159)

Consequently, the Hankel functions $\mathrm{H}_{\nu}^{(j)}(z)$ $(j=1,2)$
possess branch cuts on the negative real axis of $z$, whose discontinuity can
be found from the following identities:

$\displaystyle\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(ze^{\mathrm{i}\pi})=-e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{H}^{(2)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z),$ (160)
$\displaystyle\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(ze^{-\mathrm{i}\pi})=2\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathrm{H}^{(1)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z)+e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{H}^{(2)}_{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z),$
$\displaystyle\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(ze^{\mathrm{i}\pi})=e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{H}^{(1)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z)+2\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(z),$
$\displaystyle\mathrm{H}^{(2)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(ze^{-\mathrm{i}\pi})=-e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{H}^{(1)}_{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(z).~{}~{}~{}~{}(z>0)$

##  Appendix C Vertex Dispersion Integral with PMB Representation

In this section we collect some details of computing the 4-point single-
exchange correlator from the vertex dispersion integral (3.3). As shown in
Sec. 3.3, the vertex dispersion integral for the 4-point tree seed integral
can be reduced to (3.3), which in turns amount to the computation of two
integrals $\mathbf{I}^{(j)}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ ($j=1,2$), which we
collect here again:

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},r_{2})\equiv\int_{-r_{2}}^{0}{\mathrm{d}}r\displaystyle\frac{\,{(-r)^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(-r)}\,}{r(r-r_{1})},$ (161) $\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},r_{2})\equiv\int_{-1}^{-r_{2}}{\mathrm{d}}r\displaystyle\frac{\,{(-r)^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(-r)}\,}{r(r-r_{1})}.$ (162)

#### Computing $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$ and
$\mathbf{I}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}$

Now we compute the two integrals above with PMB representation. For
$\mathbf{I}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}$, we take the MB representation of $\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(-r)$, which is
given by:

$\displaystyle\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(-r)=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}(-r)^{-s}\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s),$ (163)

where

$\displaystyle\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s)\equiv-\displaystyle\frac{\,{\mathrm{i}\pi 2^{-1+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}e^{-\mathrm{i}\pi
s/2}}\,}{\sinh(\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{s}\,}{2}$},\mbox{$\frac{\,{1}\,}{2}$}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1-\mbox{$\frac{\,{s}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}.$ (164)

then the original integral $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$ becomes:

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}=\int_{-r_{2}}^{0}{\mathrm{d}}r\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(-r)^{1/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{r(r-r_{1})}\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s).$ (165)

The integral over $r$ can then be finished directly, which gives:

$\displaystyle\int_{-r_{2}}^{0}{\mathrm{d}}r\displaystyle\frac{\,{(-r)^{1/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{r(r-r_{1})}=\displaystyle\frac{\,{r_{2}^{1/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{r_{1}}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,1\\\
\mbox{$\frac{\,{3}\,}{2}$}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\biggr{]}.$
(166)

Now, the original integral $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$ has been
recasted into an integral over a Mellin variable $s$:

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{r_{2}^{1/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{r_{1}}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,1\\\
\mbox{$\frac{\,{3}\,}{2}$}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\biggr{]}\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s).$ (167)

Again, we use residue theorem to compute the integral over $s$. For
$r_{2}\in(0,1)$, we need to close the contour from the left side, and get a
set of poles coming from $\Gamma[s/2]$ in $\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s)$:

$\displaystyle s=-2n.~{}~{}~{}~{}(n=0,1,2,\cdots)$ (168)

Summing up all residues we get:121212Here we introduce the notation
$\mathrm{Res}(\mathbf{I},s)$ to represent the residue of integral $\mathbf{I}$
of the pole at $s$, multiplying an extra factor $2\pi\mathrm{i}$ for
simplicity. For example, if for integral $\mathbf{I}$ there is only one pole
$s$ inside the contour, then the final result is simply
$\mathbf{I}=\pm\mathrm{Res}(\mathbf{I},s)$, where the plus/minus sign depends
on the direction of the contour.

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}=\sum_{n=0}^{\infty}\mathrm{Res}(\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)},-2n),$ (169)

where

$\displaystyle\mathrm{Res}(\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)},-2n)=-\displaystyle\frac{\,{\mathrm{i}\pi 2^{-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}r_{2}^{{1}/{2}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\biggr{]}.$
(170)

This completes the computation of $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$.

Next we consider $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$ in (162).
Again we use the MB representation of $\mathbf{F}^{-2}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ (164) and get:

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}=\int_{-1}^{-r_{2}}{\mathrm{d}}r\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(-r)^{1/2-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{r(r-r_{1})}\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s).$ (171)

So the integral over $r$ can be done:

$\displaystyle\int_{-1}^{-r_{2}}{\mathrm{d}}r\
\displaystyle\frac{\,{(-r)^{\frac{1}{2}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{r(r-r_{1})}=\Gamma\Bigl{[}\mbox{$\frac{\,{1}\,}{2}$}+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Bigr{]}$
$\displaystyle\times\biggl{(}{r_{2}^{-\frac{1}{2}-s-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}{}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}\biggl{[}\begin{matrix}1,\frac{1}{2}+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\frac{3}{2}+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}-{}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}\biggl{[}\begin{matrix}1,\frac{1}{2}+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\frac{3}{2}+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-r_{1}\biggr{]}\biggr{)},$ (172)

where ${}_{2}\mkern 2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}[\cdots]$ is the regularized Gauss’s hypergeometric function whose
definition is collected in App. B. Again the integral over $s$ can be finished
via residue theorem. Closing the contour from the left side, there are two
sets of poles: one from $\Gamma[s/2]$ in $\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s)$, another from
$\Gamma[1/2+s+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$ contributed by the
integral over $r$ (C):

$\displaystyle\left\\{\begin{aligned} &s=-2n,\\\
&s=-\mbox{$\frac{\,{1}\,}{2}$}-n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu.~{}~{}~{}~{}(n=0,1,2,\cdots)\end{aligned}\right.$ (173)

Then we get

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}=\sum_{n=0}^{\infty}\mathrm{Res}(\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)},-2n)+\sum_{n=0}^{\infty}\mathrm{Res}(\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)},-\mbox{$\frac{\,{1}\,}{2}$}-n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu),$ (174)

where

$\displaystyle\mathrm{Res}(\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)},-2n)=-\displaystyle\frac{\,{\mathrm{i}\pi 2^{-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}$
$\displaystyle\times\biggl{(}r_{2}^{-1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}-{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-r_{1}\biggr{]}\biggr{)},$ (175)

and

$\displaystyle\mathrm{Res}(\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)},-\mbox{$\frac{\,{1}\,}{2}$}-n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)=$ $\displaystyle\
\displaystyle\frac{\,{\pi 2^{-3/2-n}e^{\mathrm{i}\pi(-1-2n+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)/4}}\,}{\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}-\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{n}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
\mbox{$\frac{\,{5}\,}{4}$}+\mbox{$\frac{\,{n}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}\end{matrix}\biggr{]}$
$\displaystyle\times\biggl{(}r_{2}^{n}{}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}\biggl{[}\begin{matrix}1,-n\\\
1-n\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}-{}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}\biggl{[}\begin{matrix}1,-n\\\
1-n\end{matrix}\bigg{|}-r_{1}\biggr{]}\biggr{)}.$ (176)

Note that

$\displaystyle{}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}\biggl{[}\begin{matrix}1,-n\\\
1-n\end{matrix}\bigg{|}x\biggr{]}=\Gamma[n](-x)^{n},~{}~{}~{}~{}(n=0,1,2,\cdots)$
(177)

so residues from the second set of poles actually vanish:

$\displaystyle\mathrm{Res}(\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)},-\mbox{$\frac{\,{1}\,}{2}$}-n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)=0.$ (178)

This completes the computation of $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$. Let us
collect the explicit results for both integrals $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(j)}$ $(j=1,2)$
here for future reference:

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}(r_{1},r_{2})=$
$\displaystyle\sum_{n=0}^{\infty}-\displaystyle\frac{\,{\mathrm{i}\pi
2^{-2n+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}r_{2}^{{1}/{2}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\biggr{]},$
(179) $\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}(r_{1},r_{2})=$
$\displaystyle\ \sum_{n=0}^{\infty}-\displaystyle\frac{\,{\mathrm{i}\pi
2^{-2n+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{\sinh(\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}$
$\displaystyle\times\biggl{(}r_{2}^{-1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}-{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-r_{1}\biggr{]}\biggr{)}.$ (180)

#### Simplifying the result

In principle, we can just substitute the above results for $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(j)}$ $(j=1,2)$
into (3.3) to get an analytical expression for the tree seed integral, but
this expression is obviously to be simplified. Now we describe how to massage
this expression, using various functional identities, to get a reasonably
simplified result.

For definiteness, we consider the case $0<r_{1}<r_{2}<1$ without loss of
generality. Given this relation, the result can be separated into the signal
and the background without introducing $\theta$ factors.

Then, for $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$ shown in
(179), we use the following relation:

$\displaystyle{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,b\\\
b+1\end{matrix}\bigg{|}-x\biggr{]}=x^{-b}\Gamma\biggl{[}\begin{matrix}a-b,1+b\\\
a\end{matrix}\biggr{]}-\displaystyle\frac{\,{b\times
x^{-a}}\,}{a-b}{}_{2}\mathrm{F}_{1}\biggl{[}\begin{matrix}a,a-b\\\
1+a-b\end{matrix}\bigg{|}-\displaystyle\frac{\,{1}\,}{x}\biggr{]},$ (181)

through which we get

$\displaystyle{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\biggr{]}=\pi~{}\mathrm{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Bigl{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Bigr{)}^{1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}.$
(182)

Then

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}(r_{1},r_{2})=$
$\displaystyle-\sum_{n=0}^{\infty}\displaystyle\frac{\,{\mathrm{i}\pi
2^{-2n+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{\sinh(\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}$
$\displaystyle\times\biggl{(}\pi~{}\mathrm{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu){r_{1}}^{-1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}-{r_{2}}^{-1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}\biggr{)}$
$\displaystyle=$ $\displaystyle~{}\mathbf{\Lambda}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})-\mathbf{\Lambda}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2}),$ (183)

where

$\displaystyle\mathbf{\Lambda}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2})\equiv$
$\displaystyle\sum_{n=0}^{\infty}-\displaystyle\frac{\,{\mathrm{i}\pi^{2}2^{-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{r_{1}}^{-1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]},$ (184) $\displaystyle\mathbf{\Lambda}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2})\equiv$
$\displaystyle\sum_{n=0}^{\infty}-\displaystyle\frac{\,{\mathrm{i}\pi
2^{-2n+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{r_{2}}^{-1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}.$
(185)

Also, if we define

$\displaystyle\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})\equiv\sum_{n=0}^{\infty}-\displaystyle\frac{\,{\mathrm{i}\pi
2^{-2n+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{\sinh(\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-r_{1}\biggr{]},$ (186)

then the second integral $\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$ can be
expressed as:

$\displaystyle\mathbf{I}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},r_{2})=\mathbf{\Lambda}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})-\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2}).$ (187)

Now, substituting (C) and (187) into (3.3), we get

$\displaystyle\mathcal{Y}(r_{1},r_{2})=$ $\displaystyle\
\biggl{[}\Bigl{(}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi^{2}}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu){r_{1}}\Bigl{[}\mathbf{\Lambda}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})-\mathbf{\Lambda}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})\Bigr{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Bigr{)}r_{2}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{2})$
$\displaystyle+\Bigl{(}{r_{1}}\Bigl{[}\mathbf{\Lambda}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})-\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})\Bigr{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Bigr{)}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi^{2}}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{2}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{2})\biggr{]}$ $\displaystyle+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (188)

#### Signal

One can show that the terms associated with $\mathbf{\Lambda}_{1}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ give the signal
part. In fact, the summation in (184) can be done:

$\displaystyle\mathbf{\Lambda}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2})$
$\displaystyle=\sum_{n=0}^{\infty}-\displaystyle\frac{\,{\mathrm{i}\pi^{2}2^{-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{r_{1}}^{-1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\,}{\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}$ $\displaystyle=\pi~{}\mathrm{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}^{-1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1}),$ (189)

As a result, the terms involving (184) in (C) gives:

$\displaystyle\mathcal{Y}_{\mathrm{S},>}(r_{1},r_{2})=\Bigl{(}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi}r_{1}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Bigr{)}\Bigl{(}r_{2}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{2})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\Bigr{)}.$ (190)

This is exactly the signal part of the 4-point tree seed integral.

#### Background

It follows that all other terms besides the $\mathbf{\Lambda}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ terms give rise to
the background:

$\displaystyle\mathcal{Y}$ ${}_{\mathrm{BG},>}(r_{1},r_{2})$ $\displaystyle=$
$\displaystyle\
\biggl{[}\Bigl{(}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi^{2}}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu){r_{1}}\Bigl{[}-\mathbf{\Lambda}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})\Bigr{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Bigr{)}r_{2}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{2})$
$\displaystyle+\Bigl{(}{r_{1}}\Bigl{[}\mathbf{\Lambda}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})-\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})\Bigr{]}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Bigr{)}\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi^{2}}\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{2}^{1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{2})\biggr{]}$ $\displaystyle+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (191)

This expression can be further simplified thanks to several cancellations.
First, it is easy to see that all terms including
$\mathbf{\Lambda}_{2}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})\mathbf{F}^{-2}_{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{2})$ cancel out.
Second, it can be shown that all terms involving
$\mathbf{\Lambda}_{3}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2})$ cancel
out. As a result, the background of 4-point seed integral can be simplified
into:

$\displaystyle\mathcal{Y}_{\mathrm{BG},>}(r_{1},r_{2})=$ $\displaystyle\
\displaystyle\frac{\,{\mathrm{i}}\,}{\pi^{2}}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\times{r_{1}}\times
r_{2}^{1/2+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{\Lambda}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})\mathbf{F}^{-2}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{2})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ $\displaystyle=$
$\displaystyle\
\sum_{n=0}^{\infty}\biggl{(}\displaystyle\frac{\,{\mathrm{i}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{\pi^{1/2}2^{1/2+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}}\times r_{1}\times
r_{2}^{2n}\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}$
$\displaystyle\times{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\biggr{]}{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{3}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}r_{2}^{2}\biggr{]}\biggr{)}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (192)

This is still not the expression found in previous works, but we have checked
numerically that it agrees with known results, as mentioned in the main text.
There are a large number of functional identities and resummation tricks with
which one may prove the agreement analytically, but we shall not pursue this
pure mathematical exercise in this work.

Instead, in the rest of this appendix, we prove the cancellation of
$\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ terms. More
precisely, we shall prove $\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})+\mathbf{\Lambda}_{3}^{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2})=0$.

To this end, we use the standard series representation for the dressed
hypergeometric function in $\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$:

$\displaystyle{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}a,b\\\
c\end{matrix}\bigg{|}x\biggr{]}=\sum_{m=0}^{\infty}\Gamma\biggl{[}\begin{matrix}a+m,b+m\\\
c+m,1+m\end{matrix}\biggr{]}x^{m}.$ (193)

Then, the expression (186) for $\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ can be rewritten
as:

$\displaystyle\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}-\displaystyle\frac{\,{\mathrm{i}\pi
2^{-2n+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{\sinh(\pi\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n,1+n-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}\displaystyle\frac{\,{(-r_{1})^{m}}\,}{\mbox{$\frac{\,{1}\,}{2}$}-2n+m+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}.$ (194)

In this expression, the sum over $n$ can be directly finished:

$\displaystyle\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})=\sum_{m=0}^{\infty}-\displaystyle\frac{\,{\mathrm{i}\pi
2^{1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(-r_{1})^{m}}\,}{(1+2m+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{3}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},-\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{m}\,}{2}$}-\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2}\\\
\frac{3}{4}-\mbox{$\frac{\,{m}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}1\biggr{]}.$ (195)

Then, we use the following identity of ${}_{3}\mathrm{F}_{2}$ (Eq. (4.3.4) of
[172]):

$\displaystyle{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}a,b,c\\\
d,e\end{matrix}\bigg{|}1\biggr{]}=$ $\displaystyle\
\Gamma\biggl{[}\begin{matrix}1-a,d,e,c-b\\\
e-b,d-b,1+b-a,c\end{matrix}\biggr{]}{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}b,1+b-d,1+b-e\\\
1+b-a,1+b-c\end{matrix}\bigg{|}1\biggr{]}$
$\displaystyle+\Gamma\biggl{[}\begin{matrix}1-a,d,e,b-c\\\
e-c,d-c,1+c-a,b\end{matrix}\biggr{]}{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}c,1+c-d,1+c-e\\\
1+c-a,1+c-b\end{matrix}\bigg{|}1\biggr{]}.$ (196)

Then $\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ can be rewritten
as:

$\displaystyle\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2})=$
$\displaystyle\
\sum_{m=0}^{\infty}\displaystyle\frac{\,{\mathrm{i}(-r_{1})^{m}}\,}{\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\bigg{\\{}2^{-7/2-m}\Bigl{(}(-1)^{m+1}-1\Bigr{)}\Gamma\Bigl{[}1+m,-\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{m}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},-\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{m}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\Bigr{]}$
$\displaystyle+\displaystyle\frac{\,{2^{3/2}\pi^{2}}\,}{(2+m)\cosh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma\bigl{[}\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}\bigr{]}}{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}1+\mbox{$\frac{\,{m}\,}{2}$},\mbox{$\frac{\,{3}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{3}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
\mbox{$\frac{\,{3}\,}{2}$},2+\mbox{$\frac{\,{m}\,}{2}$}\end{matrix}\bigg{|}1\biggr{]}\bigg{\\}}.$
(197)

From this expression, it is easy to see $\mathbf{\Lambda}_{3}^{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},r_{2})=-\mathbf{\Lambda}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},r_{2})$.

##  Appendix D Line Dispersion Integral with PMB Representation

In this appendix, we spell out the details of computing the three integrals
(141)-(143) arising from the line dispersion relation for the 4-point seed
integral (130). The strategy is again the PMB representation.

#### Computing $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$

For the first integral $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$, we take the
MB representation for $\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$ appeared in
the integrand, whose expression is given in (5).131313A fine point is that the
arguments of the two hypergeometric factors in (5) are taken values from the
lower edge of the branch cut, where our MB representation for
$\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}$ in (164) is valid. On the contrary, if we want to evaluate the
$\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}$ on the upper edge of the branch cut, such as $\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(z^{+})$ when
$z>1$, we need to begin with $\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(z^{-})$, and
add back the discontinuity across the branch cut using (156). Then we get:

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},x)={-2\sinh(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\int_{-\infty}^{0}{\mathrm{d}}r\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(-r)^{1-s_{1}-s_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}x^{s_{2}}}\,}{r(r-r_{1})}\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{1})\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s_{2}),$ (198)

where $\mathbb{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s)$ is given in (164). Then, the $r$ integral is again directly done,
giving:

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},x)=-2\pi\sinh(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{r_{1}^{-s_{1}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1}/x)^{-s_{2}}}\,}{\sin\bigl{(}\pi(s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{)}}\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{1})\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s_{2}).$ (199)

When applying the residue theorem to compute this integral, we meet a subtlety
here due to the “mixed poles” such as those from
$1/\sin[\pi(s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)]$, and we need to
deal with these poles carefully. Below we spell out some details.

First, consider the $s_{1}$-integral. Since we want to obtain a result for
physical $r_{1}\in(0,1)$, the factor $r_{1}^{-s_{1}}$ says that we should
close the contour from the left side on the $s_{1}$ plane. There are two sets
of left poles in $s_{1}$, respectively from $\Gamma(s_{1}/2)$ and
$1/\sin[\pi(s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)]$:

$\begin{cases}s_{1}=-2n_{1},\\\ s_{1}=-s_{2}-n_{1}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu.~{}~{}~{}~{}(n_{1}=0,1,2,\cdots)\end{cases}$ (200)

After evaluating the Mellin integrand in (199) on these two sets of poles
respectively, we are left with an $s_{2}$ integral. The analysis of the
$s_{2}$ integral depends on which sets of $s_{1}$-poles we take.

Suppose we take the first set of poles $s_{1}=-2n_{1}$. Then, we are left with
a factor of $(r_{1}/x)^{-s_{2}}=r_{2}^{-s_{2}}$. For physical $r_{2}\in(0,1)$,
we should pick up left $s_{2}$-poles of the integrand. Examining the integrand
in (199), we see that there are two sets of left poles:141414One may naively
think that the second left poles $s_{2}=2n_{1}-n_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ with fixed $n_{1}$
and $n_{2}\in\mathbb{N}$ are not all “left,” in the sense that some of these
poles have positive real part when $2n_{1}-n_{2}>0$. However, we emphasize
that the criterion for a pole being left or right is not the sign of its real
part, but rather the sign in front of the natural number $n_{2}$ that
parameterize the set of poles. Therefore, in this case, all poles with a
$-n_{2}$ term should be counted as left poles.

$\begin{cases}s_{2}=-2n_{2},\\\ s_{2}=2n_{1}-n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu.~{}~{}~{}~{}(n_{2}=0,1,2,\cdots)\end{cases}$ (201)

However, some of “poles” from the second set (e.g., the second line in (201))
coincide with zeros from the factor $1/\Gamma(1-s_{2}/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ in
$\mathbb{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{2})$, which locate at $s_{2}=2n+2-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ with
$n=0,1,2,\cdots$. Thus, the poles in the second line of (201) clash with these
zeros if $2n_{1}-n_{2}$ happens to be a positive even integer. As a result,
among all poles in (201), only the following ones make nonzero contributions
to the final results:

$\begin{cases}s_{2}=-2n_{2},~{}~{}~{}~{}&(n_{2}=0,1,2,\cdots)\\\
s_{2}=2n_{1}-(2n_{2}+1)-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,&(n_{2}=0,1,2,\cdots)\\\ s_{2}=2n_{1}-2n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu.&(n_{2}=n_{1},n_{1}+1,n_{1}+2,\cdots)\end{cases}$ (202)

So much for the poles involving $s_{1}=-2n_{1}$. Now let us return to (200)
and consider the second set of $s_{1}$-poles, namely
$s_{1}=-s_{2}-n_{1}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$. After evaluating
the Mellin integrand (199) at these poles, we get a factor $x^{s_{2}}$. Since
we are considering the region $k_{12}>k_{34}$, namely $x=k_{34}/k_{12}<1$, the
factor $x^{s_{2}}$ suggests that we should take the right $s_{2}$-poles. There
is only one set of right poles from the factor
$\Gamma[1/2-s_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$ in
$\mathbb{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{2})$:

$\displaystyle s_{2}=\mbox{$\frac{\,{1}\,}{2}$}+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu.~{}~{}~{}~{}(n_{1,2}=0,1,2,\cdots)$ (203)

Naively, one may expect that there is another set of right poles coming from
the factor $\Gamma(s_{1}/2)=\Gamma[(-s_{2}-n_{1}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)/2]$ in
$\mathbb{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{1})$, since we are now evaluating $s_{1}$ at
$s_{1}=-s_{2}-n_{1}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$. However, this is
overcounting since the pole from the $\Gamma(s_{1}/2)$ factor has already been
included in (200). So, we should not include them again here.

In summary, to compute $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}(r_{1},x)$ we
need to pick up the following four sets of poles from the Mellin integrand in
(199):

$\displaystyle\left\\{\begin{aligned}
&s_{1}=-2n_{1},s_{2}=-2n_{2},~{}~{}~{}~{}&&(n_{1},n_{2}\in\mathbb{N})\\\
&s_{1}=-2n_{1},s_{2}=2n_{1}-(2n_{2}+1)-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,&&(n_{1},n_{2}\in\mathbb{N})\\\
&s_{1}=-2n_{1},s_{2}=2n_{1}-2n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,&&(n_{1}\in\mathbb{N};n_{2}-n_{1}\in\mathbb{N})\\\
&s_{1}=-s_{2}-n_{1}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}=\mbox{$\frac{\,{1}\,}{2}$}+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu.&&(n_{1},n_{2}\in\mathbb{N})\end{aligned}\right.$ (204)

The contributions from these poles in order are:

$\displaystyle\mathbf{\Upsilon}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}-\mathrm{i}\times
2^{1-2n_{1}-2n_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\pi^{3}\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)^{2}r_{1}^{2n_{1}+2n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}x^{-2n_{2}}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n_{1},1+n_{2},1+n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,1+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]},$ (205) $\displaystyle\mathbf{\Upsilon}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}(-1)^{3/2+n_{1}+n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}2^{-2n_{2}}\pi^{2}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}^{1+2n_{2}}x^{-1+2n_{1}-2n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-\mbox{$\frac{\,{1}\,}{2}$}+n_{1}-n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{3}\,}{2}$}-2n_{1}+2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n_{1},\mbox{$\frac{\,{3}\,}{2}$}-n_{1}+n_{2},1+n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]},$ (206) $\displaystyle\mathbf{\Upsilon}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=n_{1}}^{\infty}(-1)^{-n_{1}+n_{2}}2^{1-2n_{2}}\pi^{2}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}^{2n_{2}}x^{2n_{1}-2n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,n_{1}-n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-2n_{1}+2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n_{1},1-n_{1}+n_{2},1+n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]},$ (207) $\displaystyle\mathbf{\Upsilon}_{4}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\mathrm{i}^{2n_{2}-n_{1}}2^{-n_{1}}\pi^{2}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}^{n_{1}}x^{1/2+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}1+n_{1}+n_{2},-\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{n_{1}}\,}{2}$}-\mbox{$\frac{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1+n_{2},\mbox{$\frac{\,{3}\,}{4}$}-\mbox{$\frac{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{5}\,}{4}$}+\mbox{$\frac{\,{n_{1}}\,}{2}$}+\mbox{$\frac{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}\end{matrix}\biggr{]}.$ (208)

Then we get the full result for the integral $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(1)}$ as:

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(1)}(r_{1},x)=\mathbf{\Upsilon}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{4}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x).$ (209)

#### Computing $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$

For the second integral $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$ in (142), we
again take the MB representation of its numerator $\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$ in (5). Then,
the integral $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}$ becomes:

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}(r_{1},x)=$
$\displaystyle\int_{-\infty}^{-x}{\mathrm{d}}r\displaystyle\frac{\,{(1-e^{-2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu})(-r)^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{r(r-r_{1})}\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r/x)\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r)$
$\displaystyle=$ $\displaystyle~{}(1-e^{-2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu})\int_{-\infty}^{-x}{\mathrm{d}}r\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(-r)^{1-s_{1}-s_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}x^{s_{1}}}\,}{r(r-r_{1})}\mathbb{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{1})\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s_{2}),$ (210)

where $\mathbb{G}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{1})$ is the MB representation of $\mathbf{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r/x)$:

$\displaystyle\mathbb{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{1})=-2^{2s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma\Bigl{[}s_{1},s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-2s_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Bigr{]}.$ (211)

After finishing the integral over $r$, we get:

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(2)}(r_{1},x)=$
$\displaystyle\ (1-e^{-2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu})\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}x^{-s_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma[s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$
$\displaystyle\times{}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}\biggl{[}\begin{matrix}1,s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{x}\biggr{]}\mathbb{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{1})\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s_{2}).$ (212)

The analysis of poles are very similar to the previous case and here we only
list the result. That is, there are two sets of poles contributing to the
integral when all momentum ratios taking values from their physical region,
together with the condition $k_{12}>k_{34}$:

$\displaystyle\left\\{\begin{aligned}
&s_{1}=\mbox{$\frac{\,{1}\,}{2}$}+n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,s_{2}=-2n_{2},\\\
&s_{1}=\mbox{$\frac{\,{1}\,}{2}$}+n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}=-\mbox{$\frac{\,{1}\,}{2}$}-n_{1}-n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu.~{}~{}~{}~{}(n_{1,2}=0,1,2,\cdots)\end{aligned}\right.$ (213)

The contribution from the first set of poles is:

$\displaystyle\mathbf{\Upsilon}_{5}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$