$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}(-1)^{1/2+n_{1}}2^{{1}/{2}+n_{1}-2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{\pi}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{n_{1}}\,}{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_{1}}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1+n_{1},1+n_{2},1+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\end{matrix}\biggr{]}$
$\displaystyle\times{}_{2}\mathcal{F}_{1}\biggl{[}\begin{matrix}1,\frac{1}{2}+n_{1}-2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}+n_{1}-2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-\displaystyle\frac{\,{r_{1}}\,}{x}\biggr{]}x^{2n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu},$ (214)

and the contribution from the second set of poles is:

$\displaystyle\mathbf{\Upsilon}_{6}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}{\mathrm{i}^{{3}/{2}+3n_{2}+3n_{1}}}2^{-1-n_{1}}{\pi}e^{-\mbox{$\frac{\,{3}\,}{2}$}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}^{n_{1}}x^{1/2+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}1+n_{1}+n_{2},\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{n_{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}$},-\mbox{$\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}$}\\\
1+n_{2},\mbox{$\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}$}\end{matrix}\biggr{]}.$ (215)

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

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(2)}(r_{1},x)=\mathbf{\Upsilon}_{5}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{6}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x).$ (216)

#### Computing $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(3)}$

Finally, we consider the third integral $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(3)}$ in (143). We
again take the MB representation of the numerator $\mathbf{D}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(3)}$ in (5) and
finish the $r$ integral, which gives:

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(3)}(r_{1},x)=\ $
$\displaystyle(1-e^{-2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu})\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}x^{s_{1}}\Gamma[s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]$
$\displaystyle\times{}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{1}\biggl{[}\begin{matrix}1,s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+s_{1}+s_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-r_{1}\biggr{]}\mathbb{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{2})\Bigl{(}\mathbb{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(s_{1})-\mathbb{G}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(s_{1})\Bigr{)}.$
(217)

After closing the contour from the right plane of $s_{1}$ then from the left
plane of $s_{2}$, we get three sets of poles contributing residues:

$\displaystyle\left\\{\begin{aligned}
&s_{1}=\mbox{$\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}$}+n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}=-2n_{2}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,\\\
&s_{1}=\mbox{$\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}$}-n_{1}-n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu.~{}~{}~{}~{}(n_{1,2}=0,1,2,\cdots)\end{aligned}\right.$ (218)

The contributions from these poles are given respectively by:

$\displaystyle\mathbf{\Upsilon}_{7}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}(-1)^{n_{1}}2^{-1/2-2n_{1}+n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigl{(}\mathrm{i}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+(-1)^{1+n_{2}}\bigr{)}$ $\displaystyle\
\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\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}$},\mbox{$\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}$}\\\
1+n_{1},1+n_{2}\end{matrix}\biggr{]}$ $\displaystyle\
\times{}_{2}\mathcal{{F}}_{1}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}-2n_{1}+n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-2n_{1}+n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-r_{1}\biggr{]}x^{1/2+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu},$ (219)
$\displaystyle\mathbf{\Upsilon}_{8}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}(-1)^{n_{1}}2^{-1/2-2n_{1}+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigl{(}\mathrm{i}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+(-1)^{1+n_{2}}\bigr{)}$ $\displaystyle\
\times\Gamma\biggl{[}\begin{matrix}\frac{1}{2}+2n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\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}$},\mbox{$\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}$}\\\
1+n_{1},1+n_{2}\end{matrix}\biggr{]}$ $\displaystyle\
\times{}_{2}\mathcal{{F}}_{1}\biggl{[}\begin{matrix}1,\frac{1}{2}-2n_{1}+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
\frac{3}{2}-2n_{1}+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}-r_{1}\biggr{]}x^{1/2+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu},$ (220)
$\displaystyle\mathbf{\Upsilon}_{9}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}2^{-2-n_{1}}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigl{(}\mathrm{i}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+(-1)^{1+n_{2}}\bigr{)}(-r_{1})^{n_{1}}x^{1/2+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ $\displaystyle\
\times\Gamma\biggl{[}\begin{matrix}1+n_{1}+n_{2},-\mbox{$\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_{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}$}\\\
1+n_{2}\end{matrix}\biggr{]}$ $\displaystyle\
\times\Gamma\Bigl{[}\mbox{$\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}$},\mbox{$\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}$}\Bigr{]}.$
(221)

Then, the result for the integral $\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{(3)}$ is:

$\displaystyle\mathbf{J}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(3)}(r_{1},x)=\mathbf{\Upsilon}_{7}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{8}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{9}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x).$ (222)

#### Simplifying the result

Now we have finished the computation of the three integrals
$\mathbf{J}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{(j)}$ $(j=1,2,3)$ in (141)-(143). According to (140), the final result
of the seed integral is the sum of nine series
$\mathbf{\Upsilon}_{\ell}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ with
$\ell=1,\cdots,9$. By looking at the dependence on various momentum ratios, it
is straightforward to observe the following patterns:

$\displaystyle\\{\mathbf{\Upsilon}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\\}\subset\text{nonlocal signal};$ (223)
$\displaystyle\\{\mathbf{\Upsilon}_{\ell}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu};\ell=2,3,5\\}\subset\text{local signal};$ (224)
$\displaystyle\\{\mathbf{\Upsilon}_{\ell}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu};\ell=4,6,7,8,9\\}\subset\text{background}.$ (225)

Below, we will simply these 9 series according the above grouping.

#### Nonlocal signal

The nonlocal signal, which only comes from $\mathbf{\Upsilon}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}_{1}$, can be
directly obtained by finishing the double sum in (D), and the result is:

$\displaystyle\mathbf{\Upsilon}_{1}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)=2\mathrm{i}\pi\ r_{1}^{-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\ \mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1})\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1}/x),$
(226)

This gives exactly the nonlocal signal which is the starting point of the line
dispersion integral:

$\displaystyle\mathcal{Z}_{\mathrm{NS}}(r_{1},x)=\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi}r_{1}^{1-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}x^{-1/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1})\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1}/x)+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (227)

#### Local signal

Local signal comes from $\mathbf{\Upsilon}_{2}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$,
$\mathbf{\Upsilon}_{3}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, and
$\mathbf{\Upsilon}_{5}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, whose explicit
results are respectively (D), (D), and (D). For $\mathbf{\Upsilon}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, its double sum can
be directly cpmputed, and the result is

$\displaystyle\mathbf{\Upsilon}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)=\displaystyle\frac{\,{2\mathrm{i}\pi}\,}{1+\tanh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}x^{-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1})\mathbf{F}^{-2}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}/x).$ (228)

The simplification of $\mathbf{\Upsilon}_{5}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ is more
complicated. We first expand the ${}_{2}\mathcal{F}_{1}$ factor in (D) and get

$\displaystyle\mathbf{\Upsilon}_{5}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\displaystyle\frac{\,{(-1)^{1/2+n_{1}+n_{3}}2^{{1}/{2}+n_{1}-2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\pi}}\,}{\mbox{$\frac{\,{1}\,}{2}$}+n_{1}-2n_{2}+n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}^{n_{3}}x^{2n_{2}-n_{3}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{n_{1}}\,}{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_{1}}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1+n_{1},1+n_{2},1+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}.$ (229)

Then the sum over $n_{1}$ can be finished:

$\displaystyle\mathbf{\Upsilon}_{5}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\ \mathrm{i}2^{-1/2-2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\pi e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-r_{1})^{n_{3}}x^{2n_{2}-n_{3}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n_{2},1+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\end{matrix}\biggr{]}$
$\displaystyle\times\bigg{\\{}{}_{3}\mathcal{F}_{2}\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{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}$}-n_{2}+\mbox{$\frac{\,{n_{3}}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
\mbox{$\frac{\,{1}\,}{2}$},\mbox{$\frac{\,{5}\,}{4}$}-n_{2}+\mbox{$\frac{\,{n_{3}}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}\end{matrix}\bigg{|}1\biggr{]}$
$\displaystyle-2\times{}_{3}\mathcal{F}_{2}\biggl{[}\begin{matrix}\mbox{$\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}\,}{4}$}-n_{2}+\mbox{$\frac{\,{n_{3}}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
\mbox{$\frac{\,{3}\,}{2}$},\mbox{$\frac{\,{7}\,}{4}$}-n_{2}+\mbox{$\frac{\,{n_{3}}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}\end{matrix}\bigg{|}1\biggr{]}\bigg{\\}}.$ (230)

Thereafter, we use the formula (C) again and get

$\displaystyle\mathbf{\Upsilon}_{5}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\mathrm{i}\pi^{3}2^{1-n_{3}}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-r_{1})^{n_{3}}x^{2n_{2}-n_{3}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-2n_{2}+n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n_{2},1+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,1-n_{2}+\mbox{$\frac{\,{n_{3}}\,}{2}$},1-n_{2}+\mbox{$\frac{\,{n_{3}}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}.$ (231)

The next key step is to devide this result into two parts, based on the parity
of the summation index $n_{3}$. When $n_{3}$ is odd, we replace $n_{3}$ by
$2n_{3}+1$ and get

$\displaystyle\mathbf{\Upsilon}_{5,\mathrm{odd}}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}-\mathrm{i}\pi^{3}4^{-n_{3}}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}^{1+2n_{3}}x^{-1+2n_{2}-2n_{3}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{3}\,}{2}$}-2n_{2}+2n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n_{2},1+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{3}\,}{2}$}-n_{2}+n_{3},\mbox{$\frac{\,{3}\,}{2}$}-n_{2}+n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}.$ (232)

When $n_{3}$ is even, we replace $n_{3}$ by $2n_{3}$ and get

$\displaystyle\mathbf{\Upsilon}_{5,\mathrm{even}}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=n_{2}}^{\infty}\mathrm{i}\pi^{3}2^{1-2n_{3}}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)r_{1}^{2n_{3}}x^{2n_{2}-2n_{3}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-2n_{2}+2n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n_{2},1+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,1-n_{2}+n_{3},1-n_{2}+n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\biggr{]}.$ (233)

Note that the sum over $n_{3}$ is not from $0$ to $\infty$ as some terms
vanish because of the factor $\Gamma[1-n_{2}+n_{3}]$ in the denominator. This
explains our split of $\mathbf{\Upsilon}_{5}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ into two parts:
Comparing (D) with (D), we find

$\displaystyle\mathbf{\Upsilon}_{2}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{5,\mathrm{odd}}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=0.$ (234)

Then we only need to compute $\mathbf{\Upsilon}_{5,\mathrm{even}}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, which can be
directly done:

$\displaystyle\mathbf{\Upsilon}_{5,\mathrm{even}}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)=\displaystyle\frac{\,{2\mathrm{i}\pi}\,}{1+\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}x^{-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1})\mathbf{F}^{-2}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}/x).$ (235)

Then we get the whole local signal:

$\displaystyle\mathbf{\Upsilon}_{3}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{5,\mathrm{even}}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)=2\mathrm{i}\pi\ x^{-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\ \mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1})\mathbf{F}^{-2}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}/x).$ (236)

Consequently,

$\displaystyle\mathcal{Z}_{\mathrm{LS}}(r_{1},x)=\displaystyle\frac{\,{1-\mathrm{i}\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\pi}r_{1}x^{-1/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{-2}(r_{1})\mathbf{F}^{-2}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}/x)+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (237)

#### Background

The background comes from 5 terms: $\mathbf{\Upsilon}_{4}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$,
$\mathbf{\Upsilon}_{6}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$,
$\mathbf{\Upsilon}_{7}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$,
$\mathbf{\Upsilon}_{8}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, and
$\mathbf{\Upsilon}_{9}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$. First, for
$\mathbf{\Upsilon}_{4}^{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, one can directly
finish the sum over $n_{1}$ in (D) and get:

$\displaystyle\mathbf{\Upsilon}_{4}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\ \sum_{n=0}^{\infty}(-2)^{n}\pi^{5/2}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)x^{1/2+n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma\biggl{[}\begin{matrix}\mbox{$\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}$}\\\
1+n,\mbox{$\frac{\,{3}\,}{4}$}-\mbox{$\frac{\,{n}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}\end{matrix}\biggr{]}$
$\displaystyle\times\bigg{\\{}\mathrm{i}r_{1}\sec\bigl{[}\mbox{$\frac{\,{\pi}\,}{4}$}(1+2n+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{]}{}_{3}\mathcal{F}_{2}\biggl{[}\begin{matrix}1,1+\mbox{$\frac{\,{n}\,}{2}$},\mbox{$\frac{\,{3}\,}{2}$}+\mbox{$\frac{\,{n}\,}{2}$}\\\
\mbox{$\frac{\,{7}\,}{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{\,{7}\,}{4}$}+\mbox{$\frac{\,{n}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}\end{matrix}\bigg{|}r_{1}^{2}\biggr{]}$
$\displaystyle-\csc\bigl{[}\mbox{$\frac{\,{\pi}\,}{4}$}(1+2n+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigr{]}{}_{3}\mathcal{F}_{2}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}+\mbox{$\frac{\,{n}\,}{2}$},1+\mbox{$\frac{\,{n}\,}{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}$},\mbox{$\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}$}\end{matrix}\bigg{|}r_{1}^{2}\biggr{]}\bigg{\\}}.$ (238)

We can also deal with $\mathbf{\Upsilon}_{6}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ and
$\mathbf{\Upsilon}_{9}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ in a similar way.
Adding these three terms together, we get

$\displaystyle\mathbf{\Upsilon}_{4}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{6}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{9}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)$
$\displaystyle=\displaystyle\frac{\,{(-1)^{1+n}16\pi^{2}\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{(1+2n)^{2}+4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}x^{1/2+n-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}1,\mbox{$\frac{\,{1}\,}{2}$}+\mbox{$\frac{\,{n}\,}{2}$},1+\mbox{$\frac{\,{n}\,}{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}$},\mbox{$\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}$}\end{matrix}\bigg{|}r_{1}^{2}\biggr{]}.$ (239)

For the rest two terms $\mathbf{\Upsilon}_{7}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ and
$\mathbf{\Upsilon}_{8}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$, we apply the same
procedure used when simplifying $\mathbf{\Upsilon}_{5}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$. Taking
$\mathbf{\Upsilon}_{7}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ as an example, we
expand the ${}_{2}\mathcal{F}_{1}$ factor in (D) and get

$\displaystyle\mathbf{\Upsilon}_{7}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=$
$\displaystyle\
\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\displaystyle\frac{\,{(-1)^{1+n_{1}+n_{3}}2^{-1/2-2n_{1}+n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}}\,}{\mbox{$\frac{\,{1}\,}{2}$}-2n_{1}+n_{2}+n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\bigl{(}(-1)^{n_{2}}-\mathrm{i}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\bigr{)}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$
$\displaystyle\times r_{1}^{n_{3}}x^{1/2+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma\biggl{[}\begin{matrix}-n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+2n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\mbox{$\frac{\,{1}\,}{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{\,{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}$}\\\
1+n_{1},1+n_{2}\end{matrix}\biggr{]}.$ (240)

Then the sum over $n_{1}$ can be finished:

$\displaystyle\mathbf{\Upsilon}_{7}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)=\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}2^{-2+n_{2}}\sqrt{\pi}\bigl{(}\mathrm{i}(-1)^{1+n_{2}}-e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\bigr{)}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-r_{1})^{n_{3}}x^{1/2+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\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}$},\mbox{$\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}$}\\\
1+n_{2}\end{matrix}\biggr{]}{}_{3}\mathcal{F}_{2}\biggl{[}\begin{matrix}\mbox{$\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{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{n_{3}}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
\mbox{$\frac{\,{3}\,}{4}$}-\mbox{$\frac{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{n_{3}}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}1\biggr{]}.$ (241)

One can simplify $\mathbf{\Upsilon}_{8}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ in the same way and
get:

$\displaystyle\mathbf{\Upsilon}_{8}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)=\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}2^{-2+n_{2}}\sqrt{\pi}\bigl{(}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+\mathrm{i}(-1)^{n_{2}}\bigr{)}\coth(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathrm{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(-r_{1})^{n_{3}}x^{1/2+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\biggl{[}\begin{matrix}\mbox{$\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}$},\mbox{$\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}$}\\\
1+n_{2}\end{matrix}\biggr{]}{}_{3}\mathcal{F}_{2}\biggl{[}\begin{matrix}\mbox{$\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{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{n_{3}}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
\mbox{$\frac{\,{3}\,}{4}$}-\mbox{$\frac{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{n_{3}}\,}{2}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}1\biggr{]}.$ (242)

After that, we use the transformation of ${}_{3}\mathrm{F}_{2}$ in (C) again,
and find:

$\displaystyle{}_{3}\mathcal{F}_{2}\biggl{[}\begin{matrix}\mbox{$\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{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{n_{3}}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
\mbox{$\frac{\,{3}\,}{4}$}-\mbox{$\frac{\,{n_{2}}\,}{2}$}-\mbox{$\frac{\,{n_{3}}\,}{2}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\bigg{|}1\biggr{]}-(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)=0,$ (243)

which leads to a nontrivial result:

$\displaystyle\mathbf{\Upsilon}_{7}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1},x)+\mathbf{\Upsilon}_{8}^{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1},x)=0.$ (244)

Therefore, the background is totally from the three series in (D), and
therefore we get:

$\displaystyle\mathcal{Z}_{\mathrm{BG},>}(r_{1},x)=\sum_{n=0}^{\infty}\frac{8(-x)^{n}r_{1}}{(1+2n)^{2}+4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}{}_{3}\mathrm{F}_{2}\biggl{[}\begin{matrix}1,\frac{1}{2}+\frac{n}{2},1+\frac{n}{2}\\\
\frac{5}{4}+\frac{n}{2}-\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},\frac{5}{4}+\frac{n}{2}+\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2}\end{matrix}\bigg{|}r_{1}^{2}\biggr{]}.$ (245)

This completes our computation of line dispersion integral for the 4-point
tree seed integral.

##  Appendix E Dispersion Integral for a Minkowski One-Loop Correlator

One feature of the dispersive bootstrap is that the UV divergence in the
ordinary computation of 1-loop correlators is totally absent. This may be
unfamiliar to some readers, so we use a simple example to connect our
dispersion method with the more familiar traditional calculation by
dimensional regularization.

Figure 12: The 4-point correlator of four massive scalars
$\phi_{1},\cdots,\phi_{4}$ with 1-loop bubble exchange of a pair of massive
scalar $\sigma$ in the $s$-channel.

Our example will be a 4-point 1-loop equal-time correlator of four scalar
particles $\phi_{i}$ with masses $m_{i}$ ($i=1,2,3,4$), mediated by a pair of
massive scalar $\sigma$ with mass $m$ running in a bubble loop, shown in Fig.
12. We take the two vertices to be $\phi_{1}\phi_{2}\sigma^{2}$ and
$\phi_{3}\phi_{4}\sigma^{2}$. Then, the diagram in Fig. 12 is computed by the
following integral:

$\displaystyle\mathcal{G}=$
$\displaystyle-\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int_{-\infty}^{0}{\mathrm{d}}t_{1}{\mathrm{d}}t_{2}\,D_{\mathsf{a}}^{(m_{1})}(k_{1};\tau_{1})D_{\mathsf{a}}^{(m_{2})}(k_{2};\tau_{1})D_{\mathsf{b}}^{(m_{3})}(k_{3};\tau_{2})D_{\mathsf{b}}^{(m_{4})}(k_{4};\tau_{2})$
$\displaystyle\times\int\displaystyle\frac{\,{{\mathrm{d}}^{d}\bm{q}}\,}{(2\pi)^{d}}D^{(m)}_{\mathsf{a}\mathsf{b}}(q;\tau_{1},\tau_{2})D^{(m)}_{\mathsf{a}\mathsf{b}}\Big{(}\big{|}\bm{k}_{s}-\bm{q}\big{|};\tau_{1},\tau_{2}\Big{)},$
(246)

where $D_{\mathsf{a}\mathsf{b}}^{(m)}(k;\tau_{1},\tau_{2})$ is the bulk scalar
propagator with mass $m$, and is given by:

$\displaystyle
D_{\pm\mp}^{(m)}(k;t_{1},t_{2})=\displaystyle\frac{\,{e^{\pm\mathrm{i}E(t_{1}-t_{2})}}\,}{2E},$
(247) $\displaystyle
D_{\pm\pm}^{(m)}(k;t_{1},t_{2})=D_{\mp\pm}^{(m)}(k;t_{1},t_{2})\theta(\tau_{1}-\tau_{2})+D_{\pm\mp}^{(m)}(k;t_{1},t_{2})\theta(\tau_{2}-\tau_{1}),$
(248)

where $E\equiv\sqrt{k^{2}+m^{2}}$.

A computation of this diagram with dimensional regularization has been done in
App. F of [42]. Here we directly quote the result. By setting $d=3-\epsilon$
and let $\epsilon\rightarrow 0$, we have:

$\displaystyle\mathcal{G}_{\text{DR}}=\displaystyle\frac{\,{1}\,}{256\pi^{2}E_{1}E_{2}E_{3}E_{4}E_{1234}}\bigg{[}\displaystyle\frac{\,{2}\,}{\epsilon}-\gamma_{\text{E}}+\log
4\pi+2$
$\displaystyle+\displaystyle\frac{\,{2}\,}{E_{12}-E_{34}}\int_{0}^{1}{\mathrm{d}}\xi\,\bigg{(}E_{34}\log\displaystyle\frac{\,{E_{12}+E_{\text{min}}}\,}{\mu_{R}}-E_{12}\log\displaystyle\frac{\,{E_{34}+E_{\text{min}}}\,}{\mu_{R}}\bigg{)}\bigg{]}+\mathcal{O}({\epsilon}),$
(249)

where $\xi$ is a Feynman parameter,
$E_{\text{min}}\equiv\sqrt{k_{s}^{2}+m^{2}/[\xi(1-\xi)]}$,
$E_{i}=\sqrt{k_{i}^{2}+m_{i}^{2}}$, and $\mu_{R}$ is the renormalization
scale. Note that the divergent term $\mathcal{O}({1/\epsilon})$ is
proportional to $1/(E_{1}E_{2}E_{3}E_{4}E_{1234})$, and is what we would get
by computing a contact diagram with $\phi_{1}\phi_{2}\phi_{3}\phi_{4}$
interaction. This is nothing but the local counterterm we should separate from
the bare Lagrangian. The finite part of the counterterm is determined by a
renormalization condition, and here we can choose the standard $\mkern
2.0mu\overline{\mkern-2.0mu\text{MS}\mkern-2.0mu}\mkern 2.0mu$ scheme, and
remove the term proportional to $2/\epsilon-\gamma_{\text{E}}+\log 4\pi$ in
(E) altogether. Then, we get:

$\displaystyle\mathcal{G}_{\mkern
2.0mu\overline{\mkern-2.0mu\text{MS}\mkern-2.0mu}\mkern 2.0mu}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{256\pi^{2}E_{1}E_{2}E_{3}E_{4}E_{1234}}\mathcal{I}(E_{12},E_{34},k_{s}),$
$\displaystyle\mathcal{I}_{m}(E_{12},E_{34},k_{s})\equiv$
$\displaystyle~{}2+\displaystyle\frac{\,{2}\,}{E_{12}-E_{34}}\int_{0}^{1}{\mathrm{d}}\xi\,\bigg{(}E_{34}\log\displaystyle\frac{\,{E_{12}+E_{\text{min}}}\,}{\mu_{R}}-E_{12}\log\displaystyle\frac{\,{E_{34}+E_{\text{min}}}\,}{\mu_{R}}\bigg{)}.$
(250)

To make things even simpler, we set the loop mass $m=0$, and so that the
integral over $\xi$ can be done, leading to the following expression:

$\displaystyle\mathcal{I}_{0}(E_{12},E_{34},k_{s})=2+\displaystyle\frac{\,{2}\,}{E_{12}-E_{34}}\bigg{(}E_{34}\log\displaystyle\frac{\,{E_{12}+k_{s}}\,}{\mu_{R}}-E_{12}\log\displaystyle\frac{\,{E_{34}+k_{s}}\,}{\mu_{R}}\bigg{)}.$
(251)

Now, let us consider the dispersion integral for
$\mathcal{I}_{0}(E_{12},E_{34},k_{s})$ on the complex-$E_{12}$ plane, with
$E_{34}$ and $k_{s}$ fixed in the physical region. Clearly the only
discontinuity comes from the first logarithmic factor:

$\mathop{\text{Disc}}_{E_{12}}\mathcal{I}_{0}(E_{12})=\displaystyle\frac{\,{4\pi\mathrm{i}E_{34}}\,}{E_{12}-E_{34}},~{}~{}~{}~{}~{}E_{12}\in(-\infty,-k_{s}).$
(252)

Then, we can use this discontinuity to form a dispersion integral. To
determine the subtraction order, we note that the full correlator
$\mathcal{I}_{0}$ approaches to a constant as $E_{12}\rightarrow\infty$:

$\lim_{E_{12}\rightarrow\infty}\mathcal{I}_{0}(E_{12})=2-2\log\displaystyle\frac{\,{E_{34}+k_{s}}\,}{\mu_{R}}.$
(253)

Therefore our dispersion integral should have a first-order subtraction:

$\displaystyle\mathcal{I}_{0}(E_{12})=\mathcal{I}_{0}(0)+\displaystyle\frac{\,{E_{12}}\,}{2\pi\mathrm{i}}\int_{-\infty}^{-k_{s}}{\mathrm{d}}E\,\displaystyle\frac{\,{\mathop{\text{Disc}}_{E}\mathcal{I}_{0}(E)}\,}{E(E-E_{12})}.$
(254)

This equality can be directly verified by finishing the integral. The lesson
to be learned here is that the dispersion integral itself is independent of
$\mu_{R}$ and is convergent. Similarly, had we started from the discontinuity
of the regularized version (E) to do the dispersion integral, we will not get
any term $\propto 1/\epsilon$. So, the dispersion method here is free of UV
regularization procedure; On the other hand, the renormalization-scale
dependence cannot be removed. In the dispersion calculation, this dependence
is introduced by the subtraction point $\mathcal{I}_{0}(0)$. Clearly, our “
further modified” $\mkern
2.0mu\overline{\mkern-2.0mu\text{MS}\mkern-2.0mu}\mkern 2.0mu$ subtraction
corresponds to choosing $\mathcal{I}_{0}(0)=2-2\log(k_{s}/\mu_{R})$.

## References

  * [1] D. Baumann, D. Green, A. Joyce, E. Pajer, G. L. Pimentel, C. Sleight, and M. Taronna, “Snowmass White Paper: The Cosmological Bootstrap,” in 2022 Snowmass Summer Study.  3, 2022.  arXiv:2203.08121 [hep-th].
  * [2] J. M. Maldacena and G. L. Pimentel, “On graviton non-Gaussianities during inflation,” JHEP 09 (2011) 045, arXiv:1104.2846 [hep-th].
  * [3] V. Assassi, D. Baumann, and D. Green, “On Soft Limits of Inflationary Correlation Functions,” JCAP 11 (2012) 047, arXiv:1204.4207 [hep-th].
  * [4] N. Arkani-Hamed, P. Benincasa, and A. Postnikov, “Cosmological Polytopes and the Wavefunction of the Universe,” arXiv:1709.02813 [hep-th].
  * [5] D. Baumann, G. Goon, H. Lee, and G. L. Pimentel, “Partially Massless Fields During Inflation,” JHEP 04 (2018) 140, arXiv:1712.06624 [hep-th].
  * [6] N. Arkani-Hamed and P. Benincasa, “On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes,” arXiv:1811.01125 [hep-th].
  * [7] N. Arkani-Hamed, D. Baumann, H. Lee, and G. L. Pimentel, “The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities,” JHEP 04 (2020) 105, arXiv:1811.00024 [hep-th].
  * [8] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “The cosmological bootstrap: weight-shifting operators and scalar seeds,” JHEP 12 (2020) 204, arXiv:1910.14051 [hep-th].
  * [9] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization,” SciPost Phys. 11 (2021) 071, arXiv:2005.04234 [hep-th].
  * [10] C. Sleight, “A Mellin Space Approach to Cosmological Correlators,” JHEP 01 (2020) 090, arXiv:1906.12302 [hep-th].
  * [11] C. Sleight and M. Taronna, “Bootstrapping Inflationary Correlators in Mellin Space,” JHEP 02 (2020) 098, arXiv:1907.01143 [hep-th].
  * [12] C. Sleight and M. Taronna, “From AdS to dS exchanges: Spectral representation, Mellin amplitudes, and crossing,” Phys. Rev. D 104 no. 8, (2021) L081902, arXiv:2007.09993 [hep-th].
  * [13] C. Sleight and M. Taronna, “On the consistency of (partially-)massless matter couplings in de Sitter space,” JHEP 10 (2021) 156, arXiv:2106.00366 [hep-th].
  * [14] C. Sleight and M. Taronna, “From dS to AdS and back,” JHEP 12 (2021) 074, arXiv:2109.02725 [hep-th].
  * [15] A. Hillman, “Symbol Recursion for the dS Wave Function,” arXiv:1912.09450 [hep-th].
  * [16] E. Pajer, D. Stefanyszyn, and J. Supeł, “The Boostless Bootstrap: Amplitudes without Lorentz boosts,” JHEP 12 (2020) 198, arXiv:2007.00027 [hep-th]. [Erratum: JHEP 04, 023 (2022)].
  * [17] E. Pajer, “Building a Boostless Bootstrap for the Bispectrum,” JCAP 01 (2021) 023, arXiv:2010.12818 [hep-th].
  * [18] H. Goodhew, S. Jazayeri, and E. Pajer, “The Cosmological Optical Theorem,” JCAP 04 (2021) 021, arXiv:2009.02898 [hep-th].
  * [19] S. Jazayeri, E. Pajer, and D. Stefanyszyn, “From locality and unitarity to cosmological correlators,” JHEP 10 (2021) 065, arXiv:2103.08649 [hep-th].
  * [20] S. Melville and E. Pajer, “Cosmological Cutting Rules,” JHEP 05 (2021) 249, arXiv:2103.09832 [hep-th].
  * [21] H. Goodhew, S. Jazayeri, M. H. Gordon Lee, and E. Pajer, “Cutting cosmological correlators,” JCAP 08 (2021) 003, arXiv:2104.06587 [hep-th].
  * [22] D. Baumann, W.-M. Chen, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “Linking the singularities of cosmological correlators,” JHEP 09 (2022) 010, arXiv:2106.05294 [hep-th].
  * [23] H. Gomez, R. L. Jusinskas, and A. Lipstein, “Cosmological Scattering Equations,” Phys. Rev. Lett. 127 no. 25, (2021) 251604, arXiv:2106.11903 [hep-th].
  * [24] H. Gomez, R. Lipinski Jusinskas, and A. Lipstein, “Cosmological scattering equations at tree-level and one-loop,” JHEP 07 (2022) 004, arXiv:2112.12695 [hep-th].
  * [25] J. Bonifacio, E. Pajer, and D.-G. Wang, “From amplitudes to contact cosmological correlators,” JHEP 10 (2021) 001, arXiv:2106.15468 [hep-th].
  * [26] D. Meltzer, “The inflationary wavefunction from analyticity and factorization,” JCAP 12 no. 12, (2021) 018, arXiv:2107.10266 [hep-th].
  * [27] M. Hogervorst, J. a. Penedones, and K. S. Vaziri, “Towards the non-perturbative cosmological bootstrap,” JHEP 02 (2023) 162, arXiv:2107.13871 [hep-th].
  * [28] L. Di Pietro, V. Gorbenko, and S. Komatsu, “Analyticity and unitarity for cosmological correlators,” JHEP 03 (2022) 023, arXiv:2108.01695 [hep-th].
  * [29] G. Cabass, E. Pajer, D. Stefanyszyn, and J. Supeł, “Bootstrapping large graviton non-Gaussianities,” JHEP 05 (2022) 077, arXiv:2109.10189 [hep-th].
  * [30] L.-T. Wang, Z.-Z. Xianyu, and Y.-M. Zhong, “Precision calculation of inflation correlators at one loop,” JHEP 02 (2022) 085, arXiv:2109.14635 [hep-ph].
  * [31] A. Premkumar, “Regulating loops in de Sitter spacetime,” Phys. Rev. D 109 no. 4, (2024) 045003, arXiv:2110.12504 [hep-th].
  * [32] A. Hillman and E. Pajer, “A differential representation of cosmological wavefunctions,” JHEP 04 (2022) 012, arXiv:2112.01619 [hep-th].
  * [33] X. Tong, Y. Wang, and Y. Zhu, “Cutting rule for cosmological collider signals: a bulk evolution perspective,” JHEP 03 (2022) 181, arXiv:2112.03448 [hep-th].
  * [34] T. Heckelbacher, I. Sachs, E. Skvortsov, and P. Vanhove, “Analytical evaluation of cosmological correlation functions,” JHEP 08 (2022) 139, arXiv:2204.07217 [hep-th].
  * [35] G. L. Pimentel and D.-G. Wang, “Boostless cosmological collider bootstrap,” JHEP 10 (2022) 177, arXiv:2205.00013 [hep-th].
  * [36] D.-G. Wang, G. L. Pimentel, and A. Achúcarro, “Bootstrapping multi-field inflation: non-Gaussianities from light scalars revisited,” JCAP 05 (2023) 043, arXiv:2212.14035 [astro-ph.CO].
  * [37] Z. Qin and Z.-Z. Xianyu, “Phase information in cosmological collider signals,” JHEP 10 (2022) 192, arXiv:2205.01692 [hep-th].
  * [38] S. Jazayeri and S. Renaux-Petel, “Cosmological bootstrap in slow motion,” JHEP 12 (2022) 137, arXiv:2205.10340 [hep-th].
  * [39] Z. Qin and Z.-Z. Xianyu, “Helical inflation correlators: partial Mellin-Barnes and bootstrap equations,” JHEP 04 (2023) 059, arXiv:2208.13790 [hep-th].
  * [40] G. Cabass, S. Jazayeri, E. Pajer, and D. Stefanyszyn, “Parity violation in the scalar trispectrum: no-go theorems and yes-go examples,” JHEP 02 (2023) 021, arXiv:2210.02907 [hep-th].
  * [41] G. Cabass, M. M. Ivanov, and O. H. E. Philcox, “Colliders and ghosts: Constraining inflation with the parity-odd galaxy four-point function,” Phys. Rev. D 107 no. 2, (2023) 023523, arXiv:2210.16320 [astro-ph.CO].
  * [42] Z.-Z. Xianyu and H. Zhang, “Bootstrapping one-loop inflation correlators with the spectral decomposition,” JHEP 04 (2023) 103, arXiv:2211.03810 [hep-th].
  * [43] J. Bonifacio, H. Goodhew, A. Joyce, E. Pajer, and D. Stefanyszyn, “The graviton four-point function in de Sitter space,” JHEP 06 (2023) 212, arXiv:2212.07370 [hep-th].
  * [44] S. A. Salcedo, M. H. G. Lee, S. Melville, and E. Pajer, “The Analytic Wavefunction,” JHEP 06 (2023) 020, arXiv:2212.08009 [hep-th].
  * [45] H. Lee and X. Wang, “Cosmological double-copy relations,” Phys. Rev. D 108 no. 6, (2023) L061702, arXiv:2212.11282 [hep-th].
  * [46] Z. Qin and Z.-Z. Xianyu, “Closed-form formulae for inflation correlators,” JHEP 07 (2023) 001, arXiv:2301.07047 [hep-th].
  * [47] D. Werth, L. Pinol, and S. Renaux-Petel, “Cosmological Flow of Primordial Correlators,” arXiv:2302.00655 [hep-th].
  * [48] L. Pinol, S. Renaux-Petel, and D. Werth, “The Cosmological Flow: A Systematic Approach to Primordial Correlators,” arXiv:2312.06559 [astro-ph.CO].
  * [49] Z. Qin and Z.-Z. Xianyu, “Inflation correlators at the one-loop order: nonanalyticity, factorization, cutting rule, and OPE,” JHEP 09 (2023) 116, arXiv:2304.13295 [hep-th].
  * [50] Z. Qin and Z.-Z. Xianyu, “Nonanalyticity and on-shell factorization of inflation correlators at all loop orders,” JHEP 01 (2024) 168, arXiv:2308.14802 [hep-th].
  * [51] M. H. G. Lee, C. McCulloch, and E. Pajer, “Leading loops in cosmological correlators,” JHEP 11 (2023) 038, arXiv:2305.11228 [hep-th].
  * [52] M. Loparco, J. Penedones, K. Salehi Vaziri, and Z. Sun, “The Källén-Lehmann representation in de Sitter spacetime,” JHEP 12 (2023) 159, arXiv:2306.00090 [hep-th].
  * [53] S. Agui Salcedo and S. Melville, “The cosmological tree theorem,” JHEP 12 (2023) 076, arXiv:2308.00680 [hep-th].
  * [54] S. De and A. Pokraka, “Cosmology meets cohomology,” JHEP 03 (2024) 156, arXiv:2308.03753 [hep-th].
  * [55] D. Stefanyszyn, X. Tong, and Y. Zhu, “Cosmological Correlators Through the Looking Glass: Reality, Parity, and Factorisation,” arXiv:2309.07769 [hep-th].
  * [56] Z.-Z. Xianyu and J. Zang, “Inflation correlators with multiple massive exchanges,” JHEP 03 (2024) 070, arXiv:2309.10849 [hep-th].
  * [57] D. Green, Y. Huang, C.-H. Shen, and D. Baumann, “Positivity from Cosmological Correlators,” JHEP 04 (2024) 034, arXiv:2310.02490 [hep-th].
  * [58] C. Duaso Pueyo and E. Pajer, “A cosmological bootstrap for resonant non-Gaussianity,” JHEP 03 (2024) 098, arXiv:2311.01395 [hep-th].
  * [59] N. Arkani-Hamed, D. Baumann, A. Hillman, A. Joyce, H. Lee, and G. L. Pimentel, “Kinematic Flow and the Emergence of Time,” arXiv:2312.05300 [hep-th].
  * [60] N. Arkani-Hamed, D. Baumann, A. Hillman, A. Joyce, H. Lee, and G. L. Pimentel, “Differential Equations for Cosmological Correlators,” arXiv:2312.05303 [hep-th].
  * [61] J. Chen and B. Feng, “Towards Systematic Evaluation of de Sitter Correlators via Generalized Integration-By-Parts Relations,” arXiv:2401.00129 [hep-th].
  * [62] P. Benincasa and G. Dian, “The Geometry of Cosmological Correlators,” arXiv:2401.05207 [hep-th].
  * [63] P. Benincasa and F. Vazão, “The Asymptotic Structure of Cosmological Integrals,” arXiv:2402.06558 [hep-th].
  * [64] D. Werth, L. Pinol, and S. Renaux-Petel, “CosmoFlow: Python Package for Cosmological Correlators,” arXiv:2402.03693 [astro-ph.CO].
  * [65] Y. Donath and E. Pajer, “The In-Out Formalism for In-In Correlators,” arXiv:2402.05999 [hep-th].
  * [66] Z.-Z. Du and D. Stefanyszyn, “Soft theorems for boostless amplitudes,” JHEP 07 (2024) 011, arXiv:2403.05459 [hep-th].
  * [67] B. Fan and Z.-Z. Xianyu, “Cosmological Amplitudes in Power-Law FRW Universe,” arXiv:2403.07050 [hep-th].
  * [68] T. W. Grimm, A. Hoefnagels, and M. van Vliet, “Structure and Complexity of Cosmological Correlators,” arXiv:2404.03716 [hep-th].
  * [69] S. Melville and G. L. Pimentel, “A de Sitter S-matrix from amputated cosmological correlators,” arXiv:2404.05712 [hep-th].
  * [70] T. Cohen, D. Green, and Y. Huang, “Operator Origin of Anomalous Dimensions in de Sitter Space,” arXiv:2407.08581 [hep-th].
  * [71] D. Stefanyszyn, X. Tong, and Y. Zhu, “There and Back Again: Mapping and Factorising Cosmological Observables,” arXiv:2406.00099 [hep-th].
  * [72] X. Chen and Y. Wang, “Large non-Gaussianities with Intermediate Shapes from Quasi-Single Field Inflation,” Phys. Rev. D 81 (2010) 063511, arXiv:0909.0496 [astro-ph.CO].
  * [73] X. Chen and Y. Wang, “Quasi-Single Field Inflation and Non-Gaussianities,” JCAP 1004 (2010) 027, arXiv:0911.3380 [hep-th].
  * [74] D. Baumann and D. Green, “Signatures of Supersymmetry from the Early Universe,” Phys. Rev. D85 (2012) 103520, arXiv:1109.0292 [hep-th].
  * [75] X. Chen and Y. Wang, “Quasi-Single Field Inflation with Large Mass,” JCAP 1209 (2012) 021, arXiv:1205.0160 [hep-th].
  * [76] S. Pi and M. Sasaki, “Curvature Perturbation Spectrum in Two-field Inflation with a Turning Trajectory,” JCAP 10 (2012) 051, arXiv:1205.0161 [hep-th].
  * [77] T. Noumi, M. Yamaguchi, and D. Yokoyama, “Effective field theory approach to quasi-single field inflation and effects of heavy fields,” JHEP 06 (2013) 051, arXiv:1211.1624 [hep-th].
  * [78] J.-O. Gong, S. Pi, and M. Sasaki, “Equilateral non-Gaussianity from heavy fields,” JCAP 11 (2013) 043, arXiv:1306.3691 [hep-th].
  * [79] N. Arkani-Hamed and J. Maldacena, “Cosmological Collider Physics,” arXiv:1503.08043 [hep-th].
  * [80] X. Chen, M. H. Namjoo, and Y. Wang, “Quantum Primordial Standard Clocks,” JCAP 1602 no. 02, (2016) 013, arXiv:1509.03930 [astro-ph.CO].
  * [81] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Loop Corrections to Standard Model Fields in Inflation,” JHEP 08 (2016) 051, arXiv:1604.07841 [hep-th].
  * [82] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Standard Model Background of the Cosmological Collider,” Phys. Rev. Lett. 118 no. 26, (2017) 261302, arXiv:1610.06597 [hep-th].
  * [83] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Standard Model Mass Spectrum in Inflationary Universe,” JHEP 04 (2017) 058, arXiv:1612.08122 [hep-th].
  * [84] H. Lee, D. Baumann, and G. L. Pimentel, “Non-Gaussianity as a Particle Detector,” JHEP 12 (2016) 040, arXiv:1607.03735 [hep-th].
  * [85] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Schwinger-Keldysh Diagrammatics for Primordial Perturbations,” JCAP 1712 no. 12, (2017) 006, arXiv:1703.10166 [hep-th].
  * [86] H. An, M. McAneny, A. K. Ridgway, and M. B. Wise, “Quasi Single Field Inflation in the non-perturbative regime,” JHEP 06 (2018) 105, arXiv:1706.09971 [hep-ph].
  * [87] H. An, M. McAneny, A. K. Ridgway, and M. B. Wise, “Non-Gaussian Enhancements of Galactic Halo Correlations in Quasi-Single Field Inflation,” Phys. Rev. D 97 no. 12, (2018) 123528, arXiv:1711.02667 [hep-ph].
  * [88] A. V. Iyer, S. Pi, Y. Wang, Z. Wang, and S. Zhou, “Strongly Coupled Quasi-Single Field Inflation,” JCAP 1801 no. 01, (2018) 041, arXiv:1710.03054 [hep-th].
  * [89] S. Kumar and R. Sundrum, “Heavy-Lifting of Gauge Theories By Cosmic Inflation,” JHEP 05 (2018) 011, arXiv:1711.03988 [hep-ph].
  * [90] X. Tong, Y. Wang, and S. Zhou, “Unsuppressed primordial standard clocks in warm quasi-single field inflation,” JCAP 1806 no. 06, (2018) 013, arXiv:1801.05688 [hep-th].
  * [91] X. Chen, W. Z. Chua, Y. Guo, Y. Wang, Z.-Z. Xianyu, and T. Xie, “Quantum Standard Clocks in the Primordial Trispectrum,” JCAP 1805 no. 05, (2018) 049, arXiv:1803.04412 [hep-th].
  * [92] R. Saito and T. Kubota, “Heavy Particle Signatures in Cosmological Correlation Functions with Tensor Modes,” JCAP 06 (2018) 009, arXiv:1804.06974 [hep-th].
  * [93] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Neutrino Signatures in Primordial Non-Gaussianities,” JHEP 09 (2018) 022, arXiv:1805.02656 [hep-ph].
  * [94] X. Chen, A. Loeb, and Z.-Z. Xianyu, “Unique Fingerprints of Alternatives to Inflation in the Primordial Power Spectrum,” Phys. Rev. Lett. 122 no. 12, (2019) 121301, arXiv:1809.02603 [astro-ph.CO].
  * [95] W. Z. Chua, Q. Ding, Y. Wang, and S. Zhou, “Imprints of Schwinger Effect on Primordial Spectra,” JHEP 04 (2019) 066, arXiv:1810.09815 [hep-th].
  * [96] S. Kumar and R. Sundrum, “Seeing Higher-Dimensional Grand Unification In Primordial Non-Gaussianities,” JHEP 04 (2019) 120, arXiv:1811.11200 [hep-ph].
  * [97] Y.-P. Wu, “Higgs as heavy-lifted physics during inflation,” JHEP 04 (2019) 125, arXiv:1812.10654 [hep-ph].
  * [98] L. Li, T. Nakama, C. M. Sou, Y. Wang, and S. Zhou, “Gravitational Production of Superheavy Dark Matter and Associated Cosmological Signatures,” JHEP 07 (2019) 067, arXiv:1903.08842 [astro-ph.CO].
  * [99] S. Alexander, S. J. Gates, L. Jenks, K. Koutrolikos, and E. McDonough, “Higher Spin Supersymmetry at the Cosmological Collider: Sculpting SUSY Rilles in the CMB,” JHEP 10 (2019) 156, arXiv:1907.05829 [hep-th].
  * [100] S. Lu, Y. Wang, and Z.-Z. Xianyu, “A Cosmological Higgs Collider,” JHEP 02 (2020) 011, arXiv:1907.07390 [hep-th].
  * [101] A. Hook, J. Huang, and D. Racco, “Searches for other vacua. Part II. A new Higgstory at the cosmological collider,” JHEP 01 (2020) 105, arXiv:1907.10624 [hep-ph].
  * [102] A. Hook, J. Huang, and D. Racco, “Minimal signatures of the Standard Model in non-Gaussianities,” Phys. Rev. D 101 no. 2, (2020) 023519, arXiv:1908.00019 [hep-ph].
  * [103] S. Kumar and R. Sundrum, “Cosmological Collider Physics and the Curvaton,” JHEP 04 (2020) 077, arXiv:1908.11378 [hep-ph].
  * [104] T. Liu, X. Tong, Y. Wang, and Z.-Z. Xianyu, “Probing P and CP Violations on the Cosmological Collider,” JHEP 04 (2020) 189, arXiv:1909.01819 [hep-ph].
  * [105] L.-T. Wang and Z.-Z. Xianyu, “In Search of Large Signals at the Cosmological Collider,” JHEP 02 (2020) 044, arXiv:1910.12876 [hep-ph].
  * [106] D.-G. Wang, “On the inflationary massive field with a curved field manifold,” JCAP 01 (2020) 046, arXiv:1911.04459 [astro-ph.CO].
  * [107] Y. Wang and Y. Zhu, “Cosmological Collider Signatures of Massive Vectors from Non-Gaussian Gravitational Waves,” JCAP 04 (2020) 049, arXiv:2001.03879 [astro-ph.CO].
  * [108] L. Li, S. Lu, Y. Wang, and S. Zhou, “Cosmological Signatures of Superheavy Dark Matter,” JHEP 07 (2020) 231, arXiv:2002.01131 [hep-ph].
  * [109] L.-T. Wang and Z.-Z. Xianyu, “Gauge Boson Signals at the Cosmological Collider,” JHEP 11 (2020) 082, arXiv:2004.02887 [hep-ph].
  * [110] J. Fan and Z.-Z. Xianyu, “A Cosmic Microscope for the Preheating Era,” JHEP 01 (2021) 021, arXiv:2005.12278 [hep-ph].
  * [111] A. Bodas, S. Kumar, and R. Sundrum, “The Scalar Chemical Potential in Cosmological Collider Physics,” JHEP 02 (2021) 079, arXiv:2010.04727 [hep-ph].
  * [112] S. Aoki and M. Yamaguchi, “Disentangling mass spectra of multiple fields in cosmological collider,” JHEP 04 (2021) 127, arXiv:2012.13667 [hep-th].
  * [113] N. Maru and A. Okawa, “Non-Gaussianity from $X,Y$ gauge bosons in Cosmological Collider Physics,” arXiv:2101.10634 [hep-ph].
  * [114] S. Lu, “Axion isocurvature collider,” JHEP 04 (2022) 157, arXiv:2103.05958 [hep-th].
  * [115] C. M. Sou, X. Tong, and Y. Wang, “Chemical-potential-assisted particle production in FRW spacetimes,” JHEP 06 (2021) 129, arXiv:2104.08772 [hep-th].
  * [116] Q. Lu, M. Reece, and Z.-Z. Xianyu, “Missing scalars at the cosmological collider,” JHEP 12 (2021) 098, arXiv:2108.11385 [hep-ph].
  * [117] L. Pinol, S. Aoki, S. Renaux-Petel, and M. Yamaguchi, “Inflationary flavor oscillations and the cosmic spectroscopy,” Phys. Rev. D 107 no. 2, (2023) L021301, arXiv:2112.05710 [hep-th].
  * [118] Y. Cui and Z.-Z. Xianyu, “Probing Leptogenesis with the Cosmological Collider,” Phys. Rev. Lett. 129 no. 11, (2022) 111301, arXiv:2112.10793 [hep-ph].
  * [119] X. Tong and Z.-Z. Xianyu, “Large spin-2 signals at the cosmological collider,” JHEP 10 (2022) 194, arXiv:2203.06349 [hep-ph].
  * [120] M. Reece, L.-T. Wang, and Z.-Z. Xianyu, “Large-field inflation and the cosmological collider,” Phys. Rev. D 107 no. 10, (2023) L101304, arXiv:2204.11869 [hep-ph].
  * [121] X. Chen, R. Ebadi, and S. Kumar, “Classical cosmological collider physics and primordial features,” JCAP 08 (2022) 083, arXiv:2205.01107 [hep-ph].
  * [122] X. Niu, M. H. Rahat, K. Srinivasan, and W. Xue, “Gravitational wave probes of massive gauge bosons at the cosmological collider,” JCAP 02 (2023) 013, arXiv:2211.14331 [hep-ph].
  * [123] X. Niu, M. H. Rahat, K. Srinivasan, and W. Xue, “Parity-odd and even trispectrum from axion inflation,” JCAP 05 (2023) 018, arXiv:2211.14324 [hep-ph].
  * [124] S. Aoki, “Continuous spectrum on cosmological collider,” JCAP 04 (2023) 002, arXiv:2301.07920 [hep-th].
  * [125] X. Chen, J. Fan, and L. Li, “New inflationary probes of axion dark matter,” JHEP 12 (2023) 197, arXiv:2303.03406 [hep-ph].
  * [126] X. Tong, Y. Wang, C. Zhang, and Y. Zhu, “BCS in the sky: signatures of inflationary fermion condensation,” JCAP 04 (2024) 022, arXiv:2304.09428 [hep-th].
  * [127] S. Jazayeri, S. Renaux-Petel, and D. Werth, “Shapes of the cosmological low-speed collider,” JCAP 12 (2023) 035, arXiv:2307.01751 [hep-th].
  * [128] S. Jazayeri, S. Renaux-Petel, X. Tong, D. Werth, and Y. Zhu, “Parity violation from emergent nonlocality during inflation,” Phys. Rev. D 108 no. 12, (2023) 123523, arXiv:2308.11315 [hep-th].
  * [129] P. Chakraborty and J. Stout, “Light scalars at the cosmological collider,” JHEP 02 (2024) 021, arXiv:2310.01494 [hep-th].
  * [130] P. Chakraborty and J. Stout, “Compact scalars at the cosmological collider,” JHEP 03 (2024) 149, arXiv:2311.09219 [hep-th].
  * [131] S. Aoki, T. Noumi, F. Sano, and M. Yamaguchi, “Analytic formulae for inflationary correlators with dynamical mass,” JHEP 03 (2024) 073, arXiv:2312.09642 [hep-th].
  * [132] N. Craig, S. Kumar, and A. McCune, “An Effective Cosmological Collider,” arXiv:2401.10976 [hep-ph].
  * [133] C. McCulloch, E. Pajer, and X. Tong, “A cosmological tachyon collider: enhancing the long-short scale coupling,” JHEP 05 (2024) 262, arXiv:2401.11009 [hep-th].
  * [134] Y.-P. Wu, “The cosmological collider in $R^{2}$ inflation,” arXiv:2404.05031 [astro-ph.CO].
  * [135] S. Aoki, L. Pinol, F. Sano, M. Yamaguchi, and Y. Zhu, “Cosmological Correlators with Double Massive Exchanges: Bootstrap Equation and Phenomenology,” arXiv:2404.09547 [hep-th].
  * [136] X. Chen and M. H. Namjoo, “Standard Clock in Primordial Density Perturbations and Cosmic Microwave Background,” Phys. Lett. B 739 (2014) 285–292, arXiv:1404.1536 [astro-ph.CO].
  * [137] J. Quintin, X. Chen, and R. Ebadi, “Fingerprints of a Non-Inflationary Universe from Massive Fields,” arXiv:2405.11016 [astro-ph.CO].
  * [138] P. D. Meerburg, M. Münchmeyer, J. B. Muñoz, and X. Chen, “Prospects for Cosmological Collider Physics,” JCAP 1703 no. 03, (2017) 050, arXiv:1610.06559 [astro-ph.CO].
  * [139] A. Moradinezhad Dizgah and C. Dvorkin, “Scale-Dependent Galaxy Bias from Massive Particles with Spin during Inflation,” JCAP 01 (2018) 010, arXiv:1708.06473 [astro-ph.CO].
  * [140] A. Moradinezhad Dizgah, H. Lee, J. B. Muñoz, and C. Dvorkin, “Galaxy Bispectrum from Massive Spinning Particles,” JCAP 1805 no. 05, (2018) 013, arXiv:1801.07265 [astro-ph.CO].
  * [141] K. Kogai, K. Akitsu, F. Schmidt, and Y. Urakawa, “Galaxy imaging surveys as spin-sensitive detector for cosmological colliders,” JCAP 03 (2021) 060, arXiv:2009.05517 [astro-ph.CO].
  * [142] D. Green, Y. Guo, J. Han, and B. Wallisch, “Light fields during inflation from BOSS and future galaxy surveys,” JCAP 05 (2024) 090, arXiv:2311.04882 [astro-ph.CO].
  * [143] G. Cabass, O. H. E. Philcox, M. M. Ivanov, K. Akitsu, S.-F. Chen, M. Simonović, and M. Zaldarriaga, “BOSS Constraints on Massive Particles during Inflation: The Cosmological Collider in Action,” arXiv:2404.01894 [astro-ph.CO].
  * [144] W. Sohn, D.-G. Wang, J. R. Fergusson, and E. P. S. Shellard, “Searching for Cosmological Collider in the Planck CMB Data,” arXiv:2404.07203 [astro-ph.CO].
  * [145] O. H. E. Philcox, S. Kumar, and J. C. Hill, “Too Hot to Handle: Searching for Inflationary Particle Production in Planck Data,” arXiv:2405.03738 [astro-ph.CO].
  * [146] S. Goldstein, O. H. E. Philcox, J. C. Hill, and L. Hui, “Massive-ish Particles from Small-ish Scales: Non-Perturbative Techniques for Cosmological Collider Physics from Large-Scale Structure Surveys,” arXiv:2407.08731 [astro-ph.CO].
  * [147] R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The analytic S-matrix.  Cambridge Univ. Press, Cambridge, 1966.
  * [148] S. Mandelstam, “Determination of the pion - nucleon scattering amplitude from dispersion relations and unitarity. General theory,” Phys. Rev. 112 (1958) 1344–1360.
  * [149] J. S. Toll, “Causality and the Dispersion Relation: Logical Foundations,” Phys. Rev. 104 (1956) 1760–1770.
  * [150] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix bootstrap II: two dimensional amplitudes,” JHEP 11 (2017) 143, arXiv:1607.06110 [hep-th].
  * [151] I. V. Danilkin, C. Fernández-Ramírez, P. Guo, V. Mathieu, D. Schott, M. Shi, and A. P. Szczepaniak, “Dispersive analysis of $\omega$/$\phi$→3$\pi$,$\pi$$\gamma$*,” Phys. Rev. D 91 no. 9, (2015) 094029, arXiv:1409.7708 [hep-ph].
  * [152] M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, “Dispersion relation for hadronic light-by-light scattering: pion pole,” JHEP 10 (2018) 141, arXiv:1808.04823 [hep-ph].
  * [153] R. Garcia-Martin, R. Kaminski, J. R. Pelaez, and J. Ruiz de Elvira, “Precise determination of the f0(600) and f0(980) pole parameters from a dispersive data analysis,” Phys. Rev. Lett. 107 (2011) 072001, arXiv:1107.1635 [hep-ph].
  * [154] J. D. Bjorken and S. D. Drell, “Relativistic quantum fields,”.
  * [155] M. L. Goldberger, H. Miyazawa, and R. Oehme, “Application of Dispersion Relations to Pion-Nucleon Scattering,” Phys. Rev. 99 (1955) 986–988.
  * [156] M. L. Goldberger, “Causality Conditions and Dispersion Relations. 1. Boson Fields,” Phys. Rev. 99 (1955) 979–985.
  * [157] S. Caron-Huot, “Analyticity in Spin in Conformal Theories,” JHEP 09 (2017) 078, arXiv:1703.00278 [hep-th].
  * [158] L. F. Alday and S. Caron-Huot, “Gravitational S-matrix from CFT dispersion relations,” JHEP 12 (2018) 017, arXiv:1711.02031 [hep-th].
  * [159] D. Carmi and S. Caron-Huot, “A Conformal Dispersion Relation: Correlations from Absorption,” JHEP 09 (2020) 009, arXiv:1910.12123 [hep-th].
  * [160] J. Penedones, J. A. Silva, and A. Zhiboedov, “Nonperturbative Mellin Amplitudes: Existence, Properties, Applications,” JHEP 08 (2020) 031, arXiv:1912.11100 [hep-th].
  * [161] A. Bissi, A. Sinha, and X. Zhou, “Selected topics in analytic conformal bootstrap: A guided journey,” Phys. Rept. 991 (2022) 1–89, arXiv:2202.08475 [hep-th].
  * [162] S. Céspedes, A.-C. Davis, and S. Melville, “On the time evolution of cosmological correlators,” JHEP 02 (2021) 012, arXiv:2009.07874 [hep-th].
  * [163] H. A. Kramers, “La diffusion de la lumiere par les atomes,” in Atti Cong. Intern. Fisica (Transactions of Volta Centenary Congress) Como, vol. 2, pp. 545–557.  1927\.
  * [164] R. de L. Kronig, “On the theory of dispersion of x-rays,” J. Opt. Soc. Am. 12 no. 6, (Jun, 1926) 547–557. https://opg.optica.org/abstract.cfm?URI=josa-12-6-547.
  * [165] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,” JHEP 10 (2006) 014, arXiv:hep-th/0602178.
  * [166] A. Nicolis, R. Rattazzi, and E. Trincherini, “Energy’s and amplitudes’ positivity,” JHEP 05 (2010) 095, arXiv:0912.4258 [hep-th]. [Erratum: JHEP 11, 128 (2011)].
  * [167] C. de Rham, S. Melville, A. J. Tolley, and S.-Y. Zhou, “Positivity bounds for scalar field theories,” Phys. Rev. D 96 no. 8, (2017) 081702, arXiv:1702.06134 [hep-th].
  * [168] C. de Rham, S. Melville, A. J. Tolley, and S.-Y. Zhou, “UV complete me: Positivity Bounds for Particles with Spin,” JHEP 03 (2018) 011, arXiv:1706.02712 [hep-th].
  * [169] D. Marolf and I. A. Morrison, “The IR stability of de Sitter: Loop corrections to scalar propagators,” Phys. Rev. D 82 (2010) 105032, arXiv:1006.0035 [gr-qc].
  * [170] “NIST Digital Library of Mathematical Functions.”  https://dlmf.nist.gov.
  * [171] “The Mathematical Functions Site.”  https://functions.wolfram.com.
  * [172] L. J. Slater, Generalized Hypergeometric Functions.  Cambridge University Press, 1966.