derivative of parameters of ${}_{7}\mathrm{F}_{6}$ function thrice, which is
analytically nontrivial. Fortunately, it is easy to do a numerical fit from
the explicit expression of $\mkern
2.0mu\widehat{\mkern-2.0mu\rho\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{\text{dS}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime})$ in (B.3).
For large $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$
(and not too large $\nu^{\prime}$), we find the following expression a perfect
fit:

$\lim_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\rightarrow\infty}\mkern
2.0mu\widehat{\mkern-2.0mu\rho\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{\text{dS}}(-\mathrm{i}{\nu^{\prime}})\sim\frac{1}{16\pi^{2}}\bigg{[}\log(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2})+\Big{(}\displaystyle\frac{\,{2\nu^{\prime}}\,}{5\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\Big{)}^{2}\bigg{]}.$
(189)

In Fig. 6, we show the full spectral function with the $\overline{\text{MS}}$
scheme against several analytical approximations. We see good asymptotic
agreements between the full spectral function $\mkern
2.0mu\widehat{\mkern-2.0mu\rho\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{\text{dS}}(-\mathrm{i}{\nu^{\prime}})$, evaluated with the explicit
formula (B.3), and the analytical approximations given in (189).

Figure 6: The renormalized spectral function $\mkern
2.0mu\widehat{\mkern-2.0mu\rho\mkern-2.0mu}\mkern 2.0mu_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{\text{dS}}(-\mathrm{i}{\nu^{\prime}})$ in (B.3) and its approximations
in the large $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$
limit. The left panel compares the leading (logarithmic) order behavior for a
range of $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$,
the middle panel compares the subleading (inverse quadratic) order behavior,
and the right panel shows the dependence on $\nu^{\prime}$ for a fixed value
of $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu=30$.

###  C.2 Large $\bm{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}}$ limit

The large $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}$ limit of the spectral function $\rho_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{\text{dS}}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime})$ can be
worked out similarly. We shall be brief. First, the large $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime}$ limit of
the ${}_{7}\mathrm{F}_{6}$ function can be worked out as:

$\displaystyle~{}{}_{7}\mathrm{F}_{6}\left[\begin{matrix}1-\tfrac{d}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\tfrac{3-d/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},1-\tfrac{d}{2},1-\tfrac{d}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,1-\tfrac{d}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime},\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2}{2},\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime}+d/2}{2}\\\
\tfrac{1-d/2+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime},1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\tfrac{4+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-3d/2}{2},\tfrac{4+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu-3d/2}{2}\end{matrix}\middle|1\right]$ $\displaystyle\sim$
$\displaystyle~{}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}1-\tfrac{d}{2},1-\tfrac{d}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\ 1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|1\right]+\mathcal{O}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime-1}).$ (190)

Next, the pre-factors of the ${}_{7}\mathrm{F}_{6}$ function in (3.1) have the
following asymptotic behavior for large $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime}$:

$\displaystyle~{}\displaystyle\frac{\,{1}\,}{8\pi^{d/2}}\displaystyle\frac{\,{\cos[\frac{\pi}{2}(-d+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\sin(\pi\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\Bigg{[}\begin{matrix}2-d,\tfrac{d}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,1-\tfrac{d}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime},2-\tfrac{d}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2}{2},\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime}+d/2}{2}\\\
1-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime},1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\tfrac{4+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu-3d/2}{2},\tfrac{4+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}-3d/2}{2}\end{matrix}\Bigg{]}$ $\displaystyle\sim$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2^{2d-1}\pi^{d/2}}\displaystyle\frac{\,{\cos[\frac{\pi}{2}(-d+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\sin(\pi\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\Gamma\Bigg{[}\begin{matrix}2-d,\tfrac{d}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\ 1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\Bigg{]}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime})^{d-3}+\mathcal{O}(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime d-4}).$ (191)

Combining the above two results, we conclude that the spectral density
$\rho_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{\text{dS}}(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime})\sim(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime})^{d-3}$ when $|\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{\prime}|\gg 1$.

##  Appendix D Tree Seed Integral in Arbitrary Dimensions

The 4-point function with tree-level massive exchange is one of the
ingredients for our bootstrap program of 1-loop correlators. To implement the
dimensional regularization, we need the result for the tree-level correlator
in general $d$ spatial dimensions. In this appendix, we compute the tree seed
integral defined in (25) for arbitrary $(p_{1},p_{2},d)$.

We shall use the partial Mellin-Barnes representation to compute the integral
and we will compute the four terms in the summation in (25) one by one. The
computation is largely in parallel with the one given in [72], in which the
readers can find more details about the method. Here we shall be brief.

The four terms in the summation in (25) are given by: (We omit the label for
the mass $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$.)

$\mathcal{I}_{\mathsf{ab}}^{p_{1}p_{2}}(r_{1},r_{2})\equiv-\mathsf{ab}\,k_{s}^{d+2+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\left(-\tau_{1}\right)^{p_{1}}\left(-\tau_{2}\right)^{p_{2}}\text{e}^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{a}\mathsf{b}}\left(k_{s};\tau_{1},\tau_{2}\right).$
(192)

In $d$ spatial dimensions, the Mellin-Barnes representation for the two
homogeneous propagators in (5) are:

$\displaystyle D_{\gtrless}\left(k_{s};\tau_{1},\tau_{2}\right)=$
$\displaystyle~{}\frac{1}{4\pi}\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\frac{{\mathrm{d}}s_{1}}{2\pi\mathrm{i}}\frac{{\mathrm{d}}s_{2}}{2\pi\mathrm{i}}\text{e}^{\pm\mathrm{i}\pi\left(s_{1}-s_{2}\right)}\Big{(}\frac{k_{s}}{2}\Big{)}^{-2s_{12}}\left(-\tau_{1}\right)^{-2s_{1}+d/2}\left(-\tau_{2}\right)^{-2s_{2}+d/2}$
$\displaystyle~{}\times\Gamma\bigg{[}s_{1}-\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},s_{1}+\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},s_{2}-\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},s_{2}+\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2}\bigg{]}.$ (193)

Together with (3) and (4), we can finish the time integrals in the seed
integral. For the two opposite-sign integrals
$\mathcal{I}_{\mp\pm}^{p_{1}p_{2}}$, we have

$\displaystyle\mathcal{I}_{\mp\pm}^{p_{1}p_{2}}(r_{1},r_{2})=\frac{1}{4\pi}\text{e}^{\pm\mathrm{i}\pi\left(p_{1}-p_{2}\right)/2}r_{1}^{d/2+1+p_{1}}r_{2}^{d/2+1+p_{2}}\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\frac{{\mathrm{d}}s_{1}}{2\pi\mathrm{i}}\frac{{\mathrm{d}}s_{2}}{2\pi\mathrm{i}}\left(\frac{r_{1}}{2}\right)^{-2s_{1}}\left(\frac{r_{2}}{2}\right)^{-2s_{2}}$
$\displaystyle\times\Gamma\Big{[}s_{1}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{1}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},-2s_{1}+\mbox{$\frac{\,{d}\,}{2}$}+1+p_{1},-2s_{2}+\mbox{$\frac{\,{d}\,}{2}$}+1+p_{2}\Big{]}.$
(194)

The two same-sign integrals $\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}$ can be
separated into two parts, one called the “factorized” (F) and the other called
the “time-ordered” (TO): (See [72] for the details.)

$\mathcal{I}_{\pm\pm}^{p_{1}p_{2}}(r_{1},r_{2})=\mathcal{I}_{\pm\pm,\text{F},>}^{p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\pm\pm,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2}).~{}~{}~{}~{}(r_{1}<r_{2})$
(195)

More explicitly,

$\displaystyle\mathcal{I}_{\pm\pm,\text{F},>}^{p_{1}p_{2}}(r_{1},r_{2})=\frac{\text{e}^{-\mathrm{i}\pi(p_{12}+d)/2}}{4\pi}r_{1}^{d/2+1+p_{1}}r_{2}^{d/2+1+p_{2}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{{\mathrm{d}}s_{1}}{2\pi\mathrm{i}}\frac{{\mathrm{d}}s_{2}}{2\pi\mathrm{i}}\text{e}^{2\mathrm{i}\pi
s_{1}}\Big{(}\frac{r_{1}}{2}\Big{)}^{-2s_{1}}\Big{(}\frac{r_{2}}{2}\Big{)}^{-2s_{2}}$
$\displaystyle\times\Gamma\Big{[}s_{1}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{1}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},-2s_{1}+\mbox{$\frac{\,{d}\,}{2}$}+1+p_{1},-2s_{2}+\mbox{$\frac{\,{d}\,}{2}$}+1+p_{2}\Big{]},$
(196)

and,

$\displaystyle\mathcal{I}_{\pm\pm,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2})=-\frac{\text{e}^{\mp\mathrm{i}\pi(p_{12}+d)/2}}{4\pi}r_{1}^{d+2+p_{12}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{{\mathrm{d}}s_{1}}{2\pi\mathrm{i}}\frac{{\mathrm{d}}s_{2}}{2\pi\mathrm{i}}\Gamma\Big{[}s_{1}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{1}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},s_{2}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\Big{]}$
$\displaystyle\times(\text{e}^{\pm 2\mathrm{i}\pi s_{1}}-\text{e}^{\pm
2\mathrm{i}\pi
s_{2}})\Big{(}\frac{r_{1}}{2}\Big{)}^{-2s_{12}}{}_{2}\mathcal{F}_{1}\left[\begin{matrix}-2s_{2}+\mbox{$\frac{\,{d}\,}{2}$}+p_{2}+1,-2s_{12}+d+p_{12}+2\\\
-2s_{2}+\mbox{$\frac{\,{d}\,}{2}$}+p_{2}+2\end{matrix}\middle|-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\right].$
(197)

To finish the integration over the Mellin variables $s_{1}$ and $s_{2}$, we
close the contours from the left, since we are assuming $r_{1}<r_{2}<1$. Then,
the enclosed poles can be classified into two groups:

$\displaystyle s_{1}=-n_{1}\mp\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2},\ $
$\displaystyle s_{2}=-n_{2}\pm\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2},$ (198)
$\displaystyle s_{1}=-n_{1}\mp\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2},\ $
$\displaystyle s_{2}=-n_{2}\mp\frac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2}.$ (199)

Using the residue theorem for all these poles, and also summing over the four
integrals in (192), we can find the final result for the tree seed integral.
As shown in (28), the tree seed integral can be broken into three parts:

$\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}_{\mathsf{ab}}^{p_{1}p_{2}}(r_{1},r_{2})=\mathcal{I}_{\text{NL},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\text{L},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\text{BG},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}(r_{1},r_{2}).$ (200)

The three pieces are respectively given by:

$\displaystyle\mathcal{I}_{\text{NL},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}=$
$\displaystyle~{}\frac{1}{4\pi}r_{1}^{d/2+1+p_{1}}r_{2}^{d/2+1+p_{2}}\sum_{n_{1},n_{2}=0}^{\infty}\frac{(-1)^{n_{12}}}{n_{1}!n_{2}!}\Big{[}\text{e}^{\mathrm{i}\pi\overline{p}_{12}/2}+\text{e}^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu-\mathrm{i}\pi(p_{12}+d)/2}\Big{]}\Big{(}\frac{r_{1}}{2}\Big{)}^{2n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}\frac{r_{2}}{2}\Big{)}^{2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\Big{[}2n_{1}+\mbox{$\frac{\,{d}\,}{2}$}+1+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,2n_{2}+\mbox{$\frac{\,{d}\,}{2}$}+1+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}+\text{c.c.}$
$\displaystyle=$
$\displaystyle~{}\frac{1}{2\pi}\Big{[}\cos\mbox{$\frac{\,{\pi\overline{p}_{12}}\,}{2}$}+\cos\mbox{$\frac{\,{\pi(2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+p_{12}+d)}\,}{2}$}\Big{]}r_{1}^{d/2+1+p_{1}}r_{2}^{d/2+1+p_{2}}\Big{(}\frac{r_{1}r_{2}}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\Big{[}\mbox{$\frac{\,{d}\,}{2}$}+1+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{d}\,}{2}$}+1+p_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}$
$\displaystyle\times{}_{2}F_{1}\left[\begin{matrix}\mbox{$\frac{\,{d+2p_{1}+2}\,}{4}$}+\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},\mbox{$\frac{\,{d+2p_{1}+4}\,}{4}$}+\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|r_{1}^{2}\right]{}_{2}F_{1}\left[\begin{matrix}\mbox{$\frac{\,{d+2p_{2}+2}\,}{4}$}+\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},\mbox{$\frac{\,{d+2p_{2}+4}\,}{4}$}+\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|r_{2}^{2}\right]+\text{c.c.};$ (201)
$\displaystyle\mathcal{I}_{\text{L},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}=$
$\displaystyle~{}\frac{1}{4\pi}r_{1}^{d/2+1+p_{1}}r_{2}^{d/2+1+p_{2}}\sum_{n_{1},n_{2}=0}^{\infty}\frac{(-1)^{n_{12}}}{n_{1}!n_{2}!}\Big{[}\text{e}^{\mathrm{i}\pi\overline{p}_{12}/2}+\text{e}^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu-\mathrm{i}\pi(p_{12}+d)/2}\Big{]}\Big{(}\frac{r_{1}}{2}\Big{)}^{2n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}\frac{r_{2}}{2}\Big{)}^{2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times\Gamma\Big{[}2n_{1}+\mbox{$\frac{\,{d}\,}{2}$}+1+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,2n_{2}+\mbox{$\frac{\,{d}\,}{2}$}+1+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,-n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}+\text{c.c.}$
$\displaystyle=$
$\displaystyle~{}\frac{1}{2\pi}\Big{[}\cos\mbox{$\frac{\,{\pi\overline{p}_{12}}\,}{2}$}+\cos\mbox{$\frac{\,{\pi(2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+p_{12}+d)}\,}{2}$}\Big{]}r_{1}^{d/2+1+p_{1}}r_{2}^{d/2+1+p_{2}}\Big{(}\frac{r_{1}}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle\times$
$\displaystyle\Gamma\Big{[}\mbox{$\frac{\,{d}\,}{2}$}+1+p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{d}\,}{2}$}+1+p_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}$
$\displaystyle\times{}_{2}F_{1}\left[\begin{matrix}\mbox{$\frac{\,{d+2p_{1}+2}\,}{4}$}+\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},\mbox{$\frac{\,{d+2p_{1}+4}\,}{4}$}+\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|r_{1}^{2}\right]{}_{2}F_{1}\left[\begin{matrix}\mbox{$\frac{\,{d+2p_{2}+2}\,}{4}$}-\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{2},\mbox{$\frac{\,{d+2p_{2}+4}\,}{4}$}-\tfrac{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}{2}\\\
1-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|r_{2}^{2}\right]+\text{c.c.};$ (202)
$\displaystyle\mathcal{I}_{\text{BG},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}=$
$\displaystyle~{}\frac{-1}{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\sin\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{12}+d)\big{]}r_{1}^{d+p_{12}+2}\sum_{n_{1},n_{2}=0}^{\infty}\bigg{\\{}\frac{(-1)^{n_{12}}}{n_{1}!n_{2}!}\Big{(}\frac{r_{1}}{2}\Big{)}^{2n_{1}+2n_{2}}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{1}}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times{}_{2}\mathcal{F}_{1}\left[\begin{matrix}2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+1,2n_{12}+p_{12}+d+2\\\ 2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+2\end{matrix}\middle|-\frac{r_{1}}{r_{2}}\right]\bigg{\\}}+\text{c.c.}.$
(203)

The nonlocal and local signal pieces, namely $\mathcal{I}_{\text{NL},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}$ and
$\mathcal{I}_{\text{L},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}$, can
be written in more compact forms, as shown in (29) and (30). On the other
hand, the result for the background piece $\mathcal{I}_{\text{BG},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}$ in (D)
has resummed all $r_{1}/r_{2}$ dependence into a hypergeometric function,
which is typically what we would get from a partial Mellin-Barnes
representation [72]. On the other hand, the result shown in (31) was expressed
as a double series in $r_{1}$ and $r_{1}/r_{2}$, which is typically what we
would find from recursively solving the bootstrap equation [58, 72]. In [72],
it was checked numerically that the partially resummed result (D) agrees with
the bootstrapped double series (31). We shall prove analytically the
equivalence of these two results in the next appendix.

##  Appendix E From Partial Mellin-Barnes to Bootstrapped Series

As commented at the end of the previous appendix, the background piece of the
tree seed integral can be computed in two different ways, namely the partial
Mellin-Barnes representation and solving the bootstrap equations. They yield
two results of different looks, given respectively by (D) and (31). In this
appendix, we shall show the equivalence between the two by deriving (31) from
(D).

First, we use the definition of the (dressed) hypergeometric function (104)
and (106) to rewrite ${}_{2}\mathcal{F}_{1}$ in (D) as a series:

$\displaystyle\mathcal{I}_{\text{BG},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}=$
$\displaystyle~{}\frac{-1}{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\sin\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{12}+d)\big{]}r_{1}^{d+p_{12}+2}\sum_{n_{1},n_{2},\ell=0}^{\infty}\frac{(-1)^{n_{12}+\ell}}{n_{1}!n_{2}!\ell!}\Big{(}\frac{r_{1}}{2}\Big{)}^{2n_{12}}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\ell}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{1}}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times\displaystyle\frac{\,{\Gamma(\ell+2n_{12}+p_{12}+d+2)}\,}{\ell+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+1}+\text{c.c.}.$ (204)

Using the new summation variable $m=n_{1}+n_{2}$, we can rewrite the summation
as

$\displaystyle\mathcal{I}_{\text{BG},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}=$
$\displaystyle~{}\frac{-1}{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\sin\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{12}+d)\big{]}r_{1}^{d+p_{12}+2}\sum_{m,\ell=0}^{\infty}\sum_{n_{2}=0}^{m}\binom{m}{n_{2}}\frac{(-1)^{m+\ell}}{m!\ell!}\Big{(}\frac{r_{1}}{2}\Big{)}^{2m}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\ell}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n_{2}-m}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times\displaystyle\frac{\,{\Gamma(\ell+2m+p_{12}+d+2)}\,}{\ell+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+1}$ $\displaystyle~{}+\frac{1}{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\sin\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{12}+d)\big{]}r_{1}^{d+p_{12}+2}\sum_{m,\ell=0}^{\infty}\sum_{n_{2}=0}^{m}\binom{m}{n_{2}}\frac{(-1)^{m+\ell}}{m!\ell!}\Big{(}\frac{r_{1}}{2}\Big{)}^{2m}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\ell}(+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n_{2}-m}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times\displaystyle\frac{\,{\Gamma(\ell+2m+p_{12}+d+2)}\,}{\ell+2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+1}$ (205)

There are two summations in this expression. For the second summation, we
change the summation variable $n_{2}\rightarrow m-n_{2}$:

$\displaystyle\sum_{m,\ell=0}^{\infty}\sum_{n_{2}=0}^{m}\binom{m}{n_{2}}\frac{(-1)^{m+\ell}}{m!\ell!}\Big{(}\frac{r_{1}}{2}\Big{)}^{2m}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\ell}(+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n_{2}-m}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{2}}\displaystyle\frac{\,{\Gamma(\ell+2m+p_{12}+d+2)}\,}{\ell+2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+1}$ $\displaystyle=$
$\displaystyle\sum_{m,\ell=0}^{\infty}\sum_{n_{2}=0}^{m}\binom{m}{n_{2}}\frac{(-1)^{m+\ell}}{m!\ell!}\Big{(}\frac{r_{1}}{2}\Big{)}^{2m}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\ell}(+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{2}}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n_{2}-m}\displaystyle\frac{\,{\Gamma(\ell+2m+p_{12}+d+2)}\,}{\ell+2(m-n_{2})+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+1}.$ (206)

Then, combining this result with the first summation in (E), we get:

$\displaystyle\mathcal{I}_{\text{BG},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}=$
$\displaystyle~{}\frac{-1}{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\sin\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{12}+d)\big{]}r_{1}^{d+p_{12}+2}\sum_{m,\ell=0}^{\infty}\sum_{n_{2}=0}^{m}\binom{m}{n_{2}}\frac{(-1)^{m+\ell}}{m!\ell!}\Big{(}\frac{r_{1}}{2}\Big{)}^{2m}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\ell}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n_{2}-m}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times\displaystyle\frac{\,{2(m-2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma(\ell+2m+p_{12}+d+2)}\,}{(\ell+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+1)[\ell+2(m-n_{2})+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\tfrac{d}{2}+p_{2}+1]}.$ (207)

Next, we finish the summation over $n_{2}$. Isolating the terms dependent on
$n_{2}$, we have:

$\displaystyle\sum_{n_{2}=0}^{m}\binom{m}{n_{2}}\displaystyle\frac{\,{(n_{2}-\mbox{$\frac{\,{m+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$})(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{n_{2}-m}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{2}}}\,}{(n_{2}+\mbox{$\frac{\,{\ell-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2+p_{2}+1}\,}{2}$})(n_{2}-m-\mbox{$\frac{\,{\ell+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2+p_{2}+1}\,}{2}$})}$ $\displaystyle=$
$\displaystyle\sum_{n_{2}=0}^{m}\displaystyle\frac{\,{1}\,}{n_{2}!}\displaystyle\frac{\,{\Gamma(-m+n_{2})}\,}{\Gamma(-m)}\displaystyle\frac{\,{(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)(n_{2}-\mbox{$\frac{\,{m+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$})}\,}{(n_{2}+\mbox{$\frac{\,{\ell-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2+p_{2}+1}\,}{2}$})(n_{2}-m-\mbox{$\frac{\,{\ell+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2+p_{2}+1}\,}{2}$})}\Gamma\begin{bmatrix}-m+n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+n_{2}-\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{bmatrix}$ $\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{\Gamma(-m)}{}_{5}\mathcal{F}_{4}\left[\begin{matrix}-m,-\mbox{$\frac{\,{m+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$}+1,\mbox{$\frac{\,{\ell-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2+p_{2}+1}\,}{2}$},-m-\mbox{$\frac{\,{\ell+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2+p_{2}+1}\,}{2}$},-m-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
-\mbox{$\frac{\,{m+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{\ell-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2+p_{2}+3}\,}{2}$},-m-\mbox{$\frac{\,{\ell+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+d/2+p_{2}-1}\,}{2}$},1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|1\right].$ (208)

The dressed hypergeometric function ${}_{5}\mathcal{F}_{4}$ in the last line
can be simplified by using the Rogers–Dougall very well-poised sum for
${}_{5}\mathrm{F}_{4}$ [93, 94]:

$\displaystyle{}_{5}F_{4}\left[\begin{matrix}a,\tfrac{a}{2}+1,b,c,e\\\
\tfrac{a}{2},a-b+1,a-c+1,a-e+1\end{matrix}\middle|1\right]$
$\displaystyle=\Gamma\left[\begin{matrix}a-b+1,a-c+1,a-e+1,a-b-c-e+1\\\
a+1,a-b-c+1,a-b-e+1,a-c-e+1\end{matrix}\right],$ (209)

when $\text{Re}(b+c+d-a)<1$, or when the series terminates with
$e=-n$($n=0,1,2,\cdots$). Here we can choose

$\displaystyle a=-m-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu,$ $\displaystyle
b=\frac{\ell-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+p_{2}+1}{2}+\frac{d}{4},$ $\displaystyle
c=-m-\frac{\ell+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+p_{2}+1}{2}-\frac{d}{4},$ $\displaystyle e=-m.$ (210)

Then, we get

$\displaystyle\mathcal{I}_{\text{BG},\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p_{1}p_{2}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2}\sum_{m,\ell=0}^{\infty}\frac{\left(-1\right)^{m+\ell}\sin[\mbox{$\frac{\,{\pi}\,}{2}$}(p_{12}+d)]}{\ell!}\Big{(}\frac{r_{1}}{2}\Big{)}^{2m}\Big{(}\frac{r_{1}}{r_{2}}\Big{)}^{\ell}r_{1}^{d+p_{12}+2}$
$\displaystyle\times\Gamma\left[\begin{matrix}\tfrac{\ell-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}+p_{2}+1}{2}+\tfrac{d}{4},-m-\tfrac{\ell+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}+p_{2}+1}{2}-\tfrac{d}{4},2m+\ell+d+p_{12}+2\\\
-\tfrac{\ell+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}+p_{2}-1}{2}-\tfrac{d}{4},m+\tfrac{\ell-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}+p_{2}+3}{2}+\tfrac{d}{4}\end{matrix}\right]$ $\displaystyle=$
$\displaystyle\sum_{m,\ell=0}^{\infty}\frac{(-1)^{\ell+1}\sin[\mbox{$\frac{\,{\pi}\,}{2}$}(p_{12}+d)](\ell+1)_{2m+d+p_{12}+1}}{2^{2m+1}\big{(}\tfrac{\ell-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}+p_{2}+1}{2}+\tfrac{d}{4}\big{)}_{m+1}\big{(}\tfrac{\ell+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{\prime}+p_{2}+1}{2}+\tfrac{d}{4}\big{)}_{m+1}}r_{1}^{2m+d+p_{12}+2}\Big{(}\frac{r_{1}}{r_{2}}\Big{)}^{\ell},$
(211)

which is nothing but the double series (31).

##  Appendix F Bootstrapping Loop Correlators in Minkowski Spacetime

In this section, we compute the 4-point correlation with 1-loop massive scalar
exchange in the $s$-channel in Minkowski spacetime. The diagram is again given
by the left-hand side of Fig. 1. It turns out a direct computation of SK time
integrals is already complicated enough in $(d+1)$-dimensional Minkowski
space. Therefore, we also take the spectral decomposition approach here.
Below, we first compute the corresponding tree-level correlator, namely the
right-hand side of Fig. 1, and then derive a spectral function in flat space.
Finally, we compute the 1-loop correlator by finishing the spectral integral
over the tree-level correlator.

###  F.1 Tree-level correlator

We again apply the SK diagrammatic method [25] to compute the correlator. In
flat space, algebra is much simplified. For instance, one of the homogeneous
propagators for a scalar of mass $m$ is given by

$D_{m,>}(\mathbf{k};t_{1},t_{2})=\displaystyle\frac{\,{e^{-\mathrm{i}\sqrt{\mathbf{k}^{2}+m^{2}}(t_{1}-t_{2})}}\,}{2\sqrt{\mathbf{k}^{2}+m^{2}}},$
(212)

and $D_{m,<}$ is the complex conjugation of $D_{m,>}$. The four SK propagators
$D_{\mathsf{a}\mathsf{b}}$ can be built as in (3) and (4). Then, the
$s$-channel diagram of the 4-point function can be computed as

$\displaystyle\mathcal{T}^{\text{Mink}}_{m,\mathsf{a}\mathsf{b}}=$
$\displaystyle-\mathsf{a}\mathsf{b}\int_{-\infty}^{0}{\mathrm{d}}t_{1}{\mathrm{d}}t_{2}\,G_{\mathsf{a}}(\mathbf{k}_{1};\tau_{1})G_{\mathsf{a}}(\mathbf{k}_{2};\tau_{1})G_{\mathsf{b}}(\mathbf{k}_{3};\tau_{2})G_{\mathsf{b}}(\mathbf{k}_{4};\tau_{2})D_{m,\mathsf{a}\mathsf{b}}(\mathbf{k}_{s};\tau_{1},\tau_{2}).$
(213)

Here $G_{\mathsf{a}}(\mathbf{k}_{i},\tau)$ are boundary propagators for the
four external legs, and $D_{m,\mathsf{a}\mathsf{b}}$ is the bulk propagator
for the intermediate states. We take all these lines to be massive scalars.
The simple algebra of Minkowski space allows us to introduce (different)
masses to all these legs without any pain. So, we associate $m_{i}$
$(i=1,2,3,4)$ to the four external legs and $m$ to the intermediate state.
Then, we have $E_{i}\equiv\sqrt{\mathbf{k}_{i}^{2}+m_{i}^{2}}$ and
$E_{s}\equiv\sqrt{\mathbf{k}_{s}^{2}+m^{2}}$. With these quantities, the time
integrals can be done directly. First, the two same-sign integrals read:

$\displaystyle\mathcal{T}^{\text{Mink}}_{m,\pm\pm}=$
$\displaystyle-\displaystyle\frac{\,{1}\,}{32E_{1}E_{2}E_{3}E_{4}E_{s}}\bigg{[}\int_{-\infty}^{0}{\mathrm{d}}t_{1}\int_{-\infty}^{t_{1}}{\mathrm{d}}t_{2}\,e^{\pm\mathrm{i}(E_{12}-E_{s})t_{1}\pm\mathrm{i}(E_{34}+E_{s})t_{2}}$
$\displaystyle+\int_{-\infty}^{0}{\mathrm{d}}t_{1}\int_{t_{1}}^{0}{\mathrm{d}}t_{2}\,e^{\pm\mathrm{i}(E_{12}+E_{s})t_{1}\pm\mathrm{i}(E_{34}-E_{s})t_{2}}\bigg{]}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{32E_{1}E_{2}E_{3}E_{4}E_{s}(E_{12}+E_{s})(E_{34}+E_{s})}\bigg{(}1+\displaystyle\frac{\,{2E_{s}}\,}{E_{1234}}\bigg{)}.$
(214)

Then, the two opposite-sign integrals read:

$\displaystyle\mathcal{T}^{\text{Mink}}_{m,s,\pm\mp}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{32E_{1}E_{2}E_{3}E_{4}E_{s}}\int_{-\infty}^{0}{\mathrm{d}}t_{1}{\mathrm{d}}t_{2}\,e^{\pm\mathrm{i}(E_{12}+E_{s})t_{1}\mp\mathrm{i}(E_{34}+E_{s})t_{2}}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{32E_{1}E_{2}E_{3}E_{4}E_{s}(E_{12}+E_{s})(E_{34}+E_{s})}.$
(215)

Summing up the four integrals, we get the final answer for the tree-level
4-point diagram with $s$-channel massive scalar exchange:

$\displaystyle\mathcal{T}_{m}^{\text{Mink}}\equiv\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{T}_{m,\mathsf{a}\mathsf{b}}^{\text{Mink}}=\displaystyle\frac{\,{1}\,}{8E_{1}E_{2}E_{3}E_{4}E_{s}(E_{12}+E_{s})(E_{34}+E_{s})}\bigg{(}1+\displaystyle\frac{\,{E_{s}}\,}{E_{1234}}\bigg{)}.$
(216)

This will be the resource for us to bootstrap the 1-loop correlator.

###  F.2 Spectral function

Now we turn to the spectral decomposition in flat space. The goal is to find a
spectral function $\rho^{\text{Mink}}_{m^{2}}(m^{\prime\,2})$ in Minkowski
spacetime such that

$\Big{[}D_{m}(x,y)\Big{]}^{2}=\int_{0}^{\infty}{\mathrm{d}}m^{\prime}\displaystyle\frac{\,{m^{\prime}}\,}{\pi\mathrm{i}}\rho_{m^{2}}^{\text{Mink}}(m^{\prime\,2})D_{m^{\prime}}(x,y),$
(217)

or, in spatial-momentum space,

$\int\displaystyle\frac{\,{{\mathrm{d}}^{d}\mathbf{q}}\,}{(2\pi)^{d}}D_{m,\mathsf{a}\mathsf{b}}\Big{(}|\mathbf{q}|;t_{1},t_{2}\Big{)}D_{m,\mathsf{a}\mathsf{b}}\Big{(}|\mathbf{q}-\mathbf{k}|;t_{1},t_{2}\Big{)}=\int_{0}^{\infty}{\mathrm{d}}m^{\prime}\displaystyle\frac{\,{m^{\prime}}\,}{\pi\mathrm{i}}\rho_{m^{2}}^{\text{Mink}}(m^{\prime\,2})D_{m^{\prime},\mathsf{a}\mathsf{b}}\Big{(}|\mathbf{k}|;t_{1},t_{2}\Big{)}.$
(218)

It turns out convenient to start from the Euclidean version. Let us consider
the product of a pair of scalar propagators $[D_{m}^{\text{Euc}}(x,y)]^{2}$ in
Euclidean space, and take the Fourier transform:

$\Big{[}D_{m}^{\text{Eucl}}(x,y)\Big{]}^{2}=\int\displaystyle\frac{\,{{\mathrm{d}}^{d+1}k}\,}{(2\pi)^{d+1}}\,B_{m^{2}}^{\text{Eucl}}(k^{2})\text{e}^{\mathrm{i}k_{\mu}(x_{1}-x_{2})^{\mu}},$
(219)

where $B_{m^{2}}^{\text{Eucl}}(k^{2})$ is the 1-loop bubble function in
Euclidean space. By the symmetry of the theory, the bubble function
$B_{m^{2}}^{\text{Eucl}}(k^{2})$ depends only on the magnitude of the
(Euclidean) momentum $k^{2}=k_{\mu}k^{\mu}$.

Now, suppose that we know the bubble function
$B_{m^{2}}^{\text{Eucl}}(k^{2})$. Then, it is straightforward to express the
spectral function $\rho^{\text{Mink}}_{m^{2}}(m^{\prime\,2})$ in terms of the
bubble function. More explicitly, we have:

$\displaystyle\int\frac{{\mathrm{d}}^{d+1}k}{(2\pi)^{d+1}}B_{m^{2}}^{\text{Eucl}}(k^{2})\text{e}^{\mathrm{i}k_{\mu}(x-y)^{\mu}}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2^{d}\pi^{d/2+1}\Gamma(\frac{d}{2})}\int_{0}^{\infty}{\mathrm{d}}k\,k^{d}\int_{0}^{\pi}{\mathrm{d}}\theta\,\sin^{d-1}\theta\,\text{e}^{\mathrm{i}kr\cos\theta}B_{m^{2}}^{\text{Eucl}}(k^{2})$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{{\mathrm{d}}k}{(2\pi)^{(d+1)/2}}\frac{k^{(d+1)/2}}{r^{(d-1)/2}}\mathrm{J}_{(d-1)/2}(kr)B_{m^{2}}^{\text{Eucl}}(k^{2})$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{{\mathrm{d}}m^{\prime}}{(2\pi)^{(d+1)/2}}\frac{\mathrm{i}}{\pi}\frac{m^{\prime\,(d+1)/2}}{r^{(d-1)/2}}\mathrm{K}_{(d-1)/2}(m^{\prime}r)\Big{[}B_{m^{2}}^{\text{Eucl}}(m^{\prime\,2}\text{e}^{\mathrm{i}\pi})-B_{m^{2}}^{\text{Eucl}}(m^{\prime\,2}\text{e}^{-\mathrm{i}\pi})\Big{]}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\mathrm{d}}m^{\prime}\,\frac{m^{\prime}}{\pi\mathrm{i}}D_{m^{\prime}}^{\text{Eucl}}(x,y)\Big{[}B_{m^{2}}^{\text{Eucl}}(m^{\prime\,2}\text{e}^{-\mathrm{i}\pi})-B_{m^{2}}^{\text{Eucl}}(m^{\prime\,2}\text{e}^{\mathrm{i}\pi})\Big{]},$
(220)

where $\mathrm{J}_{\nu}(z)$ is the Bessel function and $\mathrm{K}_{\nu}(z)$
is a modified Bessel function. Here we have used the fact that the Euclidean
propagator for a scalar of mass $m$ is given by

$\displaystyle
D_{m}^{\text{Eucl}}(x,y)=\displaystyle\frac{\,{1}\,}{(2\pi)^{(d+1)/2}}\Big{(}\displaystyle\frac{\,{m}\,}{r}\Big{)}^{(d-1)/2}\mathrm{K}_{(d-1)/2}(mr).$
(221)

Now we can Wick-rotate back to the Minkowski spacetime. Then, comparing (219),
(F.2), and (217), we find

$\rho_{m^{2}}^{\text{Mink}}(m^{\prime\,2})=B_{m^{2}}^{\text{Eucl}}(m^{\prime\,2}\text{e}^{-\mathrm{i}\pi})-B_{m^{2}}^{\text{Eucl}}(m^{\prime\,2}\text{e}^{\mathrm{i}\pi}).$
(222)

It remains to find the bubble function $B_{m^{2}}^{\text{Eucl}}(k^{2})$. To
this end, we Fourier transform each of the two propagators
$D_{m}^{\text{Eucl}}(x,y)$ separately:

$\displaystyle\Big{[}D_{m}^{\text{Eucl}}(x,y)\Big{]}^{2}=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}q_{1}^{d+1}}\,}{(2\pi)^{d+1}}\displaystyle\frac{\,{{\mathrm{d}}q_{2}^{d+1}}\,}{(2\pi)^{d+1}}\displaystyle\frac{\,{e^{\mathrm{i}(q_{1}+q_{2})_{\mu}(x-y)^{\mu}}}\,}{(q_{1}^{2}+m^{2})(q_{2}^{2}+m^{2})}$
$\displaystyle=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}p^{d+1}}\,}{(2\pi)^{d+1}}\displaystyle\frac{\,{e^{\mathrm{i}p_{\mu}(x-y)^{\mu}}}\,}{[(p-q)^{2}+m^{2}](q^{2}+m^{2})},$
(223)

where $p=q_{1}+q_{2}$ and $q=q_{2}$. Comparing (F.2) with (219), we find:

$\displaystyle B_{m^{2}}^{\text{Eucl}}(k^{2})=$
$\displaystyle\int\displaystyle\frac{\,{{\mathrm{d}}^{d+1}q}\,}{(2\pi)^{d+1}}\displaystyle\frac{\,{1}\,}{((k-q)^{2}+m^{2})(q^{2}+m^{2})}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{\Gamma(\frac{3-d}{2})}\,}{(4\pi)^{(d+1)/2}}\int_{0}^{1}\displaystyle\frac{\,{{\mathrm{d}}\xi}\,}{[m^{2}+\xi(1-\xi)k^{2}]^{(3-d)/2}},$
(224)

where we use the standard Feynman parametrization in the last step. Finally,
using (222), we get an expression for the 1-loop spectral function in
Minkowski spacetime:

$\displaystyle\rho_{m^{2}}^{\text{Mink}}(m^{\prime\,2})=$
$\displaystyle~{}\displaystyle\frac{\,{\Gamma(\frac{3-d}{2})}\,}{(4\pi)^{(d+1)/2}}\int_{0}^{1}{\mathrm{d}}\xi\bigg{\\{}\displaystyle\frac{\,{1}\,}{[m^{2}+\xi(1-\xi)m^{\prime\,2}e^{-\mathrm{i}\pi}]^{(3-d)/2}}-\displaystyle\frac{\,{1}\,}{[m^{2}+\xi(1-\xi)m^{\prime\,2}e^{\mathrm{i}\pi}]^{(3-d)/2}}\bigg{\\}}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{\Gamma(\frac{3-d}{2})}\,}{(4\pi)^{(d+1)/2}}\int_{0}^{1}{\mathrm{d}}\xi\,2\mathrm{i}\,\text{Im}\,\displaystyle\frac{\,{1}\,}{[m^{2}-\xi(1-\xi)m^{\prime\,2}-\mathrm{i}\epsilon]^{(3-d)/2}}.$
(225)

###  F.3 1-loop correlator

Now we are ready to bootstrap the 1-loop correlator. Using the Feynman rule in
SK formalism, the 1-loop correlator has the following form:

$\displaystyle\mathcal{L}^{\text{Mink}}_{m}\equiv\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{L}^{\text{Mink}}_{m,\mathsf{a}\mathsf{b}}=$
$\displaystyle-\displaystyle\frac{\,{1}\,}{2}\sum_{\mathsf{a},\mathsf{b}=\pm}\mathsf{a}\mathsf{b}\int_{-\infty}^{0}{\mathrm{d}}t_{1}{\mathrm{d}}t_{2}\,G_{\mathsf{a}}(\mathbf{k}_{1};\tau_{1})G_{\mathsf{a}}(\mathbf{k}_{2};\tau_{1})G_{\mathsf{b}}(\mathbf{k}_{3};\tau_{2})G_{\mathsf{b}}(\mathbf{k}_{4};\tau_{2})$
$\displaystyle\times\int\displaystyle\frac{\,{{\mathrm{d}}^{d}\mathbf{q}}\,}{(2\pi)^{d}}D_{m,\mathsf{a}\mathsf{b}}\Big{(}|\mathbf{q}|;t_{1},t_{2}\Big{)}D_{m,\mathsf{a}\mathsf{b}}\Big{(}|\mathbf{q}-\mathbf{k}_{s}|;t_{1},t_{2}\Big{)}.$
(226)

Comparing this expression with the tree amplitude (213) and using the spectral
function (217), we have

$\displaystyle\mathcal{L}^{\text{Mink}}_{m}=\int_{0}^{\infty}{\mathrm{d}}m^{\prime}\displaystyle\frac{\,{m^{\prime}}\,}{2\pi\mathrm{i}}\rho_{m^{2}}^{\text{Mink}}(m^{\prime\,2})\mathcal{T}^{\text{Mink}}_{m^{\prime}}.$
(227)

Now, with the explicit expressions for the tree-level correlator
$\mathcal{T}^{\text{Mink}}_{m}$ in (216) and the spectral function
$\rho_{m^{2}}^{\text{Mink}}(m^{\prime\,2})$ in (F.2), we can compute the
1-loop correlator as

$\displaystyle\mathcal{L}_{m}^{\text{Mink}}=\int_{0}^{\infty}{\mathrm{d}}m^{\prime}\displaystyle\frac{\,{m^{\prime}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(1+E_{s}/E_{1234})}\,}{8E_{1}E_{2}E_{3}E_{4}E_{s}(E_{12}+E_{s})(E_{34}+E_{s})}\rho_{m^{2}}^{\text{Mink}}(m^{\prime\,2}).$
(228)

Change the integration variable from $m^{\prime}$ to
$E_{s}=\sqrt{\mathbf{k}_{s}^{2}+m^{\prime\,2}}$, we have
${\mathrm{d}}m^{\prime}(m^{\prime}/E_{s})={\mathrm{d}}E_{s}$. Then,

$\displaystyle\mathcal{L}_{m}^{\text{Mink}}=$
$\displaystyle\int_{0}^{1}{\mathrm{d}}\xi\int_{E_{\text{min}}}^{\infty}\displaystyle\frac{\,{{\mathrm{d}}E_{s}}\,}{\pi}\displaystyle\frac{\,{(1+E_{s}/E_{1234})}\,}{8E_{1}E_{2}E_{3}E_{4}(E_{12}+E_{s})(E_{34}+E_{s})}$
$\displaystyle~{}\times\displaystyle\frac{\,{\Gamma(\frac{3-d}{2})}\,}{(4\pi)^{(d+1)/2}}\displaystyle\frac{\,{-\cos(\pi
d/2)}\,}{[\xi(1-\xi)(E_{s}^{2}-E_{\text{min}}^{2})]^{(3-d)/2}}.$ (229)

Here $E_{\text{min}}\equiv\sqrt{\mathbf{k}_{s}^{2}+m^{2}/[\xi(1-\xi)]}$. The
$E_{s}$ integral can be finished directly. In the $d\rightarrow 3$ limit, the
result is

$\displaystyle\mathcal{L}_{m}^{\text{Mink}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{256\pi^{2}E_{1}E_{2}E_{3}E_{4}E_{1234}}\bigg{[}\displaystyle\frac{\,{2}\,}{3-d}-\gamma_{E}+\log
4\pi+2$
$\displaystyle~{}+\displaystyle\frac{\,{2}\,}{E_{12}-E_{34}}\int_{0}^{1}{\mathrm{d}}\xi\,\bigg{(}E_{34}\log\displaystyle\frac{\,{E_{12}+E_{\text{min}}}\,}{\mu_{R}}-E_{12}\log\displaystyle\frac{\,{E_{34}+E_{\text{min}}}\,}{\mu_{R}}\bigg{)}\bigg{]}.$
(230)

This is the final result for the 1-loop correlator in Minkowski spacetime. We
can now directly use $\overline{\text{MS}}$ scheme to subtract the divergent
piece $\propto 2/(3-d)-\gamma_{E}+\log 4\pi$. It is also of interest to have
an expression for the renormalized loop correlator in the large mass limit
$m\rightarrow\infty$:

$\displaystyle\Big{[}\mathcal{L}_{m}^{\text{Mink}}\Big{]}_{\overline{\text{MS}}}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{256\pi^{2}E_{1}E_{2}E_{3}E_{4}E_{1234}}\bigg{[}\log\displaystyle\frac{\,{\mu_{R}^{2}}\,}{m^{2}}-\displaystyle\frac{\,{E_{12}E_{34}+\mathbf{k}_{s}^{2}}\,}{6m^{2}}+\mathcal{O}\Big{(}\displaystyle\frac{\,{1}\,}{m^{3}}\Big{)}\bigg{]}.$
(231)

## References

  * [1] Planck Collaboration, Y. Akrami et al., “Planck 2018 results. X. Constraints on inflation,” Astron. Astrophys. 641 (2020) A10, arXiv:1807.06211 [astro-ph.CO].
  * [2] Planck Collaboration, Y. Akrami et al., “Planck 2018 results. IX. Constraints on primordial non-Gaussianity,” Astron. Astrophys. 641 (2020) A9, arXiv:1905.05697 [astro-ph.CO].
  * [3] S. Ferraro, N. Sailer, A. Slosar, and M. White, “Snowmass2021 Cosmic Frontier White Paper: Cosmology and Fundamental Physics from the three-dimensional Large Scale Structure,” arXiv:2203.07506 [astro-ph.CO].
  * [4] J. B. Muñoz, Y. Ali-Haïmoud, and M. Kamionkowski, “Primordial non-gaussianity from the bispectrum of 21-cm fluctuations in the dark ages,” Phys. Rev. D92 no. 8, (2015) 083508, arXiv:1506.04152 [astro-ph.CO].
  * [5] Snowmass 2021 Cosmic Frontier 5 Topical Group Collaboration, A. Liu, L. Newburgh, B. Saliwanchik, and A. Slosar, “Snowmass2021 Cosmic Frontier White Paper: 21cm Radiation as a Probe of Physics Across Cosmic Ages,” arXiv:2203.07864 [astro-ph.CO].
  * [6] A. Achúcarro et al., “Inflation: Theory and Observations,” arXiv:2203.08128 [astro-ph.CO].
  * [7] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP 05 (2003) 013, arXiv:astro-ph/0210603.
  * [8] 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].
  * [9] X. Chen and Y. Wang, “Quasi-Single Field Inflation and Non-Gaussianities,” JCAP 1004 (2010) 027, arXiv:0911.3380 [hep-th].
  * [10] D. Baumann and D. Green, “Signatures of Supersymmetry from the Early Universe,” Phys. Rev. D85 (2012) 103520, arXiv:1109.0292 [hep-th].
  * [11] X. Chen and Y. Wang, “Quasi-Single Field Inflation with Large Mass,” JCAP 1209 (2012) 021, arXiv:1205.0160 [hep-th].
  * [12] 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].
  * [13] 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].
  * [14] J.-O. Gong, S. Pi, and M. Sasaki, “Equilateral non-Gaussianity from heavy fields,” JCAP 11 (2013) 043, arXiv:1306.3691 [hep-th].
  * [15] N. Arkani-Hamed and J. Maldacena, “Cosmological Collider Physics,” arXiv:1503.08043 [hep-th].
  * [16] X. Chen, M. H. Namjoo, and Y. Wang, “Quantum Primordial Standard Clocks,” JCAP 1602 no. 02, (2016) 013, arXiv:1509.03930 [astro-ph.CO].
  * [17] 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].
  * [18] 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].
  * [19] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Standard Model Mass Spectrum in Inflationary Universe,” JHEP 04 (2017) 058, arXiv:1612.08122 [hep-th].
  * [20] H. Lee, D. Baumann, and G. L. Pimentel, “Non-Gaussianity as a Particle Detector,” JHEP 12 (2016) 040, arXiv:1607.03735 [hep-th].
  * [21] 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].
  * [22] 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].
  * [23] 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].
  * [24] S. Kumar and R. Sundrum, “Heavy-Lifting of Gauge Theories By Cosmic Inflation,” JHEP 05 (2018) 011, arXiv:1711.03988 [hep-ph].
  * [25] 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].
  * [26] 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].
  * [27] 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].
  * [28] X. Chen, Y. Wang, and Z.-Z. Xianyu, “Neutrino Signatures in Primordial Non-Gaussianities,” JHEP 09 (2018) 022, arXiv:1805.02656 [hep-ph].
  * [29] 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].
  * [30] 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].
  * [31] Y.-P. Wu, “Higgs as heavy-lifted physics during inflation,” JHEP 04 (2019) 125, arXiv:1812.10654 [hep-ph].
  * [32] R. Saito and T. Kubota, “Heavy Particle Signatures in Cosmological Correlation Functions with Tensor Modes,” JCAP 06 (2018) 009, arXiv:1804.06974 [hep-th].
  * [33] 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].
  * [34] S. Lu, Y. Wang, and Z.-Z. Xianyu, “A Cosmological Higgs Collider,” JHEP 02 (2020) 011, arXiv:1907.07390 [hep-th].
  * [35] 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].
  * [36] 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].
  * [37] 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].
  * [38] S. Kumar and R. Sundrum, “Seeing Higher-Dimensional Grand Unification In Primordial Non-Gaussianities,” JHEP 04 (2019) 120, arXiv:1811.11200 [hep-ph].
  * [39] S. Kumar and R. Sundrum, “Cosmological Collider Physics and the Curvaton,” JHEP 04 (2020) 077, arXiv:1908.11378 [hep-ph].
  * [40] 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].
  * [41] 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].
  * [42] D.-G. Wang, “On the inflationary massive field with a curved field manifold,” JCAP 01 (2020) 046, arXiv:1911.04459 [astro-ph.CO].
  * [43] 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].
  * [44] L. Li, S. Lu, Y. Wang, and S. Zhou, “Cosmological Signatures of Superheavy Dark Matter,” JHEP 07 (2020) 231, arXiv:2002.01131 [hep-ph].
  * [45] L.-T. Wang and Z.-Z. Xianyu, “Gauge Boson Signals at the Cosmological Collider,” JHEP 11 (2020) 082, arXiv:2004.02887 [hep-ph].
  * [46] J. Fan and Z.-Z. Xianyu, “A Cosmic Microscope for the Preheating Era,” JHEP 01 (2021) 021, arXiv:2005.12278 [hep-ph].
  * [47] S. Aoki and M. Yamaguchi, “Disentangling mass spectra of multiple fields in cosmological collider,” JHEP 04 (2021) 127, arXiv:2012.13667 [hep-th].
  * [48] A. Bodas, S. Kumar, and R. Sundrum, “The Scalar Chemical Potential in Cosmological Collider Physics,” JHEP 02 (2021) 079, arXiv:2010.04727 [hep-ph].
  * [49] N. Maru and A. Okawa, “Non-Gaussianity from $X,Y$ gauge bosons in Cosmological Collider Physics,” arXiv:2101.10634 [hep-ph].
  * [50] S. Lu, “Axion isocurvature collider,” JHEP 04 (2022) 157, arXiv:2103.05958 [hep-th].
  * [51] 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].
  * [52] Q. Lu, M. Reece, and Z.-Z. Xianyu, “Missing scalars at the cosmological collider,” JHEP 12 (2021) 098, arXiv:2108.11385 [hep-ph].
  * [53] L. Pinol, S. Aoki, S. Renaux-Petel, and M. Yamaguchi, “Inflationary flavor oscillations and the cosmic spectroscopy,” arXiv:2112.05710 [hep-th].
  * [54] 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].
  * [55] X. Tong and Z.-Z. Xianyu, “Large Spin-2 Signals at the Cosmological Collider,” arXiv:2203.06349 [hep-ph].
  * [56] M. Reece, L.-T. Wang, and Z.-Z. Xianyu, “Large-Field Inflation and the Cosmological Collider,” arXiv:2204.11869 [hep-ph].
  * [57] X. Chen, R. Ebadi, and S. Kumar, “Classical Cosmological Collider Physics and Primordial Features,” arXiv:2205.01107 [hep-ph].
  * [58] 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].
  * [59] 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].
  * [60] 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].
  * [61] C. Sleight, “A Mellin Space Approach to Cosmological Correlators,” JHEP 01 (2020) 090, arXiv:1906.12302 [hep-th].
  * [62] C. Sleight and M. Taronna, “Bootstrapping Inflationary Correlators in Mellin Space,” JHEP 02 (2020) 098, arXiv:1907.01143 [hep-th].
  * [63] 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].
  * [64] 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].
  * [65] C. Sleight and M. Taronna, “From dS to AdS and back,” JHEP 12 (2021) 074, arXiv:2109.02725 [hep-th].
  * [66] E. Pajer, D. Stefanyszyn, and J. Supeł, “The Boostless Bootstrap: Amplitudes without Lorentz boosts,” JHEP 12 (2020) 198, arXiv:2007.00027 [hep-th].
  * [67] E. Pajer, “Building a Boostless Bootstrap for the Bispectrum,” JCAP 01 (2021) 023, arXiv:2010.12818 [hep-th].
  * [68] G. Cabass, E. Pajer, D. Stefanyszyn, and J. Supeł, “Bootstrapping Large Graviton non-Gaussianities,” arXiv:2109.10189 [hep-th].
  * [69] G. L. Pimentel and D.-G. Wang, “Boostless Cosmological Collider Bootstrap,” arXiv:2205.00013 [hep-th].
  * [70] S. Jazayeri and S. Renaux-Petel, “Cosmological Bootstrap in Slow Motion,” arXiv:2205.10340 [hep-th].
  * [71] Z. Qin and Z.-Z. Xianyu, “Phase Information in Cosmological Collider Signals,” arXiv:2205.01692 [hep-th].
  * [72] Z. Qin and Z.-Z. Xianyu, “Helical Inflation Correlators: Partial Mellin-Barnes and Bootstrap Equations,” arXiv:2208.13790 [hep-th].
  * [73] 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].
  * [74] H. Goodhew, S. Jazayeri, and E. Pajer, “The Cosmological Optical Theorem,” JCAP 04 (2021) 021, arXiv:2009.02898 [hep-th].
  * [75] S. Melville and E. Pajer, “Cosmological Cutting Rules,” JHEP 05 (2021) 249, arXiv:2103.09832 [hep-th].
  * [76] H. Goodhew, S. Jazayeri, M. H. Gordon Lee, and E. Pajer, “Cutting cosmological correlators,” JCAP 08 (2021) 003, arXiv:2104.06587 [hep-th].
  * [77] L. Di Pietro, V. Gorbenko, and S. Komatsu, “Analyticity and unitarity for cosmological correlators,” JHEP 03 (2022) 023, arXiv:2108.01695 [hep-th].
  * [78] 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].
  * [79] J. Bonifacio, E. Pajer, and D.-G. Wang, “From amplitudes to contact cosmological correlators,” JHEP 10 (2021) 001, arXiv:2106.15468 [hep-th].
  * [80] M. Hogervorst, J. a. Penedones, and K. S. Vaziri, “Towards the non-perturbative cosmological bootstrap,” arXiv:2107.13871 [hep-th].
  * [81] D. Meltzer, “The inflationary wavefunction from analyticity and factorization,” JCAP 12 no. 12, (2021) 018, arXiv:2107.10266 [hep-th].
  * [82] T. Heckelbacher, I. Sachs, E. Skvortsov, and P. Vanhove, “Analytical evaluation of cosmological correlation functions,” arXiv:2204.07217 [hep-th].
  * [83] H. Gomez, R. L. Jusinskas, and A. Lipstein, “Cosmological Scattering Equations,” Phys. Rev. Lett. 127 no. 25, (2021) 251604, arXiv:2106.11903 [hep-th].
  * [84] 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].
  * [85] D. Baumann, W.-M. Chen, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, “Linking the Singularities of Cosmological Correlators,” arXiv:2106.05294 [hep-th].
  * [86] 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].
  * [87] 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].
  * [88] E. Joung, J. Mourad, and R. Parentani, “Group theoretical approach to quantum fields in de Sitter space. I. The Principle series,” JHEP 08 (2006) 082, arXiv:hep-th/0606119.
  * [89] E. Joung, J. Mourad, and R. Parentani, “Group theoretical approach to quantum fields in de Sitter space. II. The complementary and discrete series,” JHEP 09 (2007) 030, arXiv:0707.2907 [hep-th].
  * [90] J. Penedones, “Writing CFT correlation functions as AdS scattering amplitudes,” JHEP 03 (2011) 025, arXiv:1011.1485 [hep-th].
  * [91] A. L. Fitzpatrick and J. Kaplan, “Analyticity and the Holographic S-Matrix,” JHEP 10 (2012) 127, arXiv:1111.6972 [hep-th].
  * [92] L. U. Ancarani and G. Gasaneo, “Derivatives of any order of the hypergeometric function ${}_{p}{F}_{q}(a_{1},\cdots,a_{p};b_{1},\cdots,b_{q};z)$ with respect to the parameters $a_{i}$ and $b_{i}$,” Journal of Physics A: Mathematical and Theoretical 43 no. 8, (Feb, 2010) 085210. https://doi.org/10.1088/1751-8113/43/8/085210.
  * [93] L. J. Slater, Generalized Hypergeometric Functions.  Cambridge University Press, 1966.
  * [94] “NIST Digital Library of Mathematical Functions.”  https://dlmf.nist.gov.


# A direct GHz-clocked phase and intensity modulated transmitter applied to
quantum key distribution

G. L. Roberts<EMAIL_ADDRESS>Toshiba Research Europe Ltd,
208 Cambridge Science Park, Milton Road, Cambridge, CB4 0GZ, United Kingdom
Cambridge University Engineering Department, 9 JJ Thomson Avenue, Cambridge,
CB3 0FA, United Kingdom  M. Lucamarini  Toshiba Research Europe Ltd, 208
Cambridge Science Park, Milton Road, Cambridge, CB4 0GZ, United Kingdom  J. F.
Dynes  Toshiba Research Europe Ltd, 208 Cambridge Science Park, Milton Road,
Cambridge, CB4 0GZ, United Kingdom  S. J. Savory  Cambridge University
Engineering Department, 9 JJ Thomson Avenue, Cambridge, CB3 0FA, United
Kingdom  Z. L. Yuan  Toshiba Research Europe Ltd, 208 Cambridge Science Park,
Milton Road, Cambridge, CB4 0GZ, United Kingdom  A. J. Shields  Toshiba
Research Europe Ltd, 208 Cambridge Science Park, Milton Road, Cambridge, CB4
0GZ, United Kingdom

###### Abstract

Quantum key distribution (QKD), a technology that enables perfectly secure
communication, has evolved to the stage where many different protocols are
being used in real-world implementations. Each protocol has its own
advantages, meaning that users can choose the one best-suited to their
application, however each often requires different hardware. This complicates
multi-user networks, in which users may need multiple transmitters to
communicate with one another. Here, we demonstrate a direct-modulation based
transmitter that can be used to implement most weak coherent pulse based QKD
protocols with simple changes to the driving signals. This also has the
potential to extend to classical communications, providing a low chirp
transmitter with simple driving requirements that combines phase shift keying
with amplitude shift keying. We perform QKD with concurrent time-bin and phase
modulation, alongside phase randomisation. The acquired data is used to
evaluate secure key rates for time-bin encoded BB84 with decoy states and a
finite key-size analysis, giving megabit per second secure key rates, 1.60
times higher than if purely phase-encoded BB84 was used.

##  I Introduction

Quantum key distribution (QKD) allows users to communicate with information
theoretic security Gisin  _et al._ (2002); Scarani  _et al._ (2009). This is
possible by encoding the key on single photons so that a malign party trying
to measure a key bit will alter its state in a manner observable to the
legitimate parties. The security provided is of great value to anyone wishing
to future-proof the secrecy of their information transfer. The technology is
also practical and is currently implemented in a number of metropolitan
networks Peev  _et al._ (2009); Sasaki  _et al._ (2011) and even in ground-
satellite links Vallone  _et al._ (2015); Liao  _et al._ (2017); Takenaka  _et
al._ (2017).

Research developments tend to aim at improving the secure key rate and the
achievable distance of QKD systems Ma  _et al._ (2005); Dixon  _et al._
(2008); Bacco  _et al._ (2016); Yin  _et al._ (2016). For example, the decoy-
state BB84 protocol has security against coherent attacks, is able to reach
distances of hundreds of kilometres and can achieve megabit per second secure
key rates Lim  _et al._ (2014); Fröhlich  _et al._ (2017). However, this often
makes systems more complex, requiring stabilisation routines and extra
consideration to protect against side channels, where Eve attacks the
practical implementation Dixon  _et al._ (2017); Lo  _et al._ (2012).

QKD can be carried out using orthogonal states within two or more non-
orthogonal bases. This means the result is non-deterministic if the state
encoded in a certain basis is measured in a different basis. Time-bin qubits,
prepared with the setup in Figure 1a, are the natural choice in optical fibres
because the pulses travel along the fibre with their phase reference, meaning
that perturbations apply to both pulses. Their state can be conveniently
represented using the Bloch sphere, as depicted in Figure 1b. The equatorial
bases, X and Y, correspond to two equal intensity pulses with a phase
difference between them. States in X and Y can be realised by separating a
single phase-randomised pulse into a signal and reference pulse using an
asymmetric Mach-Zehnder interferometer (AMZI), then encoding a phase
difference using a phase modulator. This can be decoded using an identical
AMZI. The polar basis, Z, corresponds to a pulse in just one of the two
potential time bins. States in this basis can be decoded by measuring the
arrival time of the time-bins in the receiver’s detectors.

In this manuscript, we refer to the QKD protocol using the Z basis alongside
either the X or Y bases as polar BB84, and the protocol using the X and Y
bases as phase-encoded BB84. Polar BB84 could theoretically be implemented
using the BS/switch component shown in Figure 1a. However this is not a
commonly available component and it is challenging to build a device that can
act reliably as a high-speed switch and beamsplitter at the rates required by
QKD systems. One of the most practical setups to implement polar BB84 Yoshino
_et al._ (2018) uses a phase-randomised pulsed laser diode as the source,
separated into a reference and signal pulse by an AMZI and then encoded using
another intensity modulator (IM) and a phase modulator (PM). This setup is
bulky and would require stabilisation routines to ensure the AMZI delay line
is matched to that in Bob’s receiver for a real-world implementation. Another
drawback is that the transmitter is not versatile, requiring modifications if
one wishes to implement another QKD protocol, for example differential phase
shift Shibata  _et al._ (2014); Inoue (2015), coherent one way Korzh  _et al._
(2015); Branciard  _et al._ (2008) or differential quadrature phase shift
Inoue and Iwai (2009); Kawakami  _et al._ (2016).

Figure 1: X, Y, Z QKD bases a) A possible schematic to produce all the
necessary QKD states using a phase-randomised pulse source and a device that
can act as a beamsplitter (BS) and a high-speed switch, plus a phase modulator
(PM) producing a phase shift $\phi$. To produce the X and Y bases, the device
acts as a beamsplitter (BS), phase modulating one half of the pulse using the
PM and delaying the other half with an optical delay line. To produce the Z
basis, the device acts as a switch, routing the pulse down one path to place
it in the desired time bin and b) Bloch sphere representation of qubit states.

A promising QKD transmitter that mitigates these aforementioned drawbacks
modulates the phase of one laser, which is then inherited by the pulses of
another laser via optical injection locking (OIL) Yuan  _et al._ (2016). This
enables the precise control of the output phase of pulses, as well as the
ability to perform on-demand phase randomisation Roberts  _et al._ (2017a).
OIL also gives an enhanced modulation bandwidth, allowing time-bin encoding to
be carried out on gain-switched pulses at 2 GHz, whilst maintaining a coherent
phase Roberts  _et al._ (2017b). However, it has not yet been possible to
simultaneously directly modulate the phase and intensity of a light source
with the high purity necessary for QKD.

Here, we use direct laser modulation to concurrently modulate the phase and
intensity of the transmitter to provide six states that can be used to perform
QKD without the need for an interferometer in the transmitter. The directly-
modulated system produces signal and vacuum states, allowing a single IM to be
used to prepare the decoy states and to equalise the mean photon number in the
phase bases. We use the Z and X bases to implement the polar BB84 protocol and
compare the results to phase-encoded BB84 implemented with the Y and X bases.
The low quantum bit error rate (QBER) of the polar basis relative to the
equatorial bases means that its use as the signal state allows for fewer bits
to be lost to error correction, enhancing the secure key rate.

##  II Experimental Realization

The transmitter we use is based on OIL and is shown in Figure 2. The protocols
implemented are decoy-state polar BB84 and decoy-state phase-encoded BB84
Hwang (2003); Wang (2005); Lo  _et al._ (2005).

Figure 2: Experimental design. a) Light from the phase preparation laser diode
(LD) is injected into the pulse preparation laser diode via a circulator. The
applied electrical signals are shown beside each laser and the pulse
intensities are shown along the fibre. An intensity modulator (IM) can then be
used to prepare decoy states and to equalise the polar and equatorial state
intensities before attenuation and transmission through the quantum channel.
b) The Z basis is measured through direct detection whereas X/Y basis
detection uses an asymmetric Mach-Zehnder interferometer (AMZI) with a phase
modulator (PM) on one arm. An attenuator (Att) on the Z detection arm balances
the losses of the X/Y detection arm.

The phase preparation laser encodes a relative phase between pulse pairs using
a 750 ps signal to bring the laser above threshold and to coincide temporally
with two pulse preparation laser pulses 500 ps apart. A 250 ps modulation is
applied in the middle of this signal to control the relative phase between the
two pulses. The laser is then driven below threshold for 250 ps to ensure the
global phase of every pulse pair is completely random due to the random phase
of spontaneous emission photons Yuan  _et al._ (2016); Roberts  _et al._
(2017a). The optical signal is injected into the pulse preparation laser via a
circulator, where pulses adopt the phase of the phase preparation laser. This
removes the need for a high speed phase modulator and an extra random number
generator for phase-encoding and randomisation. A 1550 nm DFB laser diode with
a 10 GHz bandwidth (Gooch & Housego AA0701) is used as the phase preparation
laser and a custom-made laser without an optical isolator as the pulse
preparation laser.

The electrical signal into the pulse preparation laser is patterned to produce
intensity-modulated gain-switched pulses Roberts  _et al._ (2017b). For polar
BB84 with decoy states, empty pulses are required to prepare Z basis states
and vacuum states. 250 ps electrical pulses are input to the pulse preparation
laser at a frequency of 2 GHz and the DC is set to below the lasing threshold.
When a signal or decoy pulse is required, the electrical signal is above the
lasing threshold. To prepare a vacuum state the electrical signal is below the
lasing threshold, as shown in Figure 2. The X and Y bases can then be
attenuated by 3 dB so that they contain the same mean photon number as the Z
basis. Although only two bases are used for BB84, we take experimental data
for the X, Y and Z bases, allowing us to demonstrate the versatility of the
transmitter.

The transmission basis probabilities are set to $P_{Z}=0.8$, $P_{X}=P_{Y}=0.1$
and the probabilities of sending a signal (photon flux s), decoy (photon flux
v) and vacuum (photon flux w) state are $P_{s}=0.8$, $P_{v}=P_{w}=0.1$
respectively. The photon fluxes are 0.5, 0.038 and 0.001 for s, v and w
respectively. A proof of principle experiment is then carried out, where data
is measured for 20 minutes per basis at each distance, giving 40 minutes of
key time for both the polar BB84 protocol and phase-encoded BB84. This allows
us to maximise the number of key bits, whilst providing a sufficient number of
bits in the check basis to keep the fluctuations low.

The pulse preparation laser is clocked at 2 GHz, giving an effective system
clock rate of 1 GHz. This is because two time bins are required to encode a
single qubit. A 210-bit pseudorandom sequence is generated as Alice’s pattern,
allowing the corresponding electrical signals to be input to drive the laser
diodes. A fixed 12 GHz spectral filter (Advanced Optics Solutions – ASE
Filter) at 1550.12 nm is placed at Alice’s output to reduce any amplified
spontaneous emission. The pulses are then attenuated to the required photon
number before being sent through the quantum channel to Bob.

The X and Y data are collected using an AMZI with a 500 ps time delay on one
arm to interfere consecutive pulses. A polariser is placed at the output of
the AMZI to clean the signal, necessitating the use of a polarisation
controller in Alice for alignment. The AMZI has a 1.7 dB loss and half of the
photons (the reference pulses) contain no information so are discarded. A
fixed attenuation of 4.7 dB must be placed on the Z measurement arm to balance
the detection efficiencies for each measurement basis. This is because the
security of BB84 relies on identical basis-independent detection probabilities
Koashi (2006); Scarani  _et al._ (2009); Tomamichel  _et al._ (2012).  The
detected counts are tagged using a digitizer and binned into 211-bin
histograms for extraction of the counts and error rates. A 10 ns subset of
this histogram before creation of decoy states and equalisation of the X and Y
bases intensities can be seen in Figure 3. Random interference occurs in the X
and Y bases when two pulses from separate blocks interfere, giving an average
interference intensity of half the maximum intensity. An empty pulse followed
by a full pulse has no interference, thus producing a quarter of the maximum
intensity.

Figure 3: Directly-modulated traces. Measurement traces before decoy state
preparation and basis intensity equalisation. The Z basis (top) has only a
single trace, whereas the X and Y bases (middle and bottom) have two AMZI
outputs. The corresponding input pattern values are displayed at the top,
labelled as Bb, where B is the basis and b is the logical bit value inside
that basis. A red (grey) peak in the X and Y bases corresponds to a ‘0’ (‘1’)
logical bit, where the photon exits through the upper (lower) AMZI port. Peaks
in output 1 (2) counts of the AMZI are complemented by small counts in output
2 (1) (middle inset), showing the high distinguishability between bits.

The detectors used are superconducting nanowire SPDs with a detection
efficiency of 34 %, a dark count rate of 30 Hz and a deadtime of $<$20 ns. The
jitter increases and efficiency decreases with increasing count rate, so
measurements are not taken at count rates over 10 MCounts/s.  A digitizer with
100 ps time bins and constant factor discrimination then processes the counts.
The detectors have a polarisation dependence, so a polarisation controller is
used for optimal detection efficiency. Although the receiver is adapted to
each specific protocol, the transmitter remains entirely unchanged. This is a
necessary feature for a multi-protocol QKD transmitter.

##  III Results

The number of signal and decoy counts measured in each basis is shown in
Figure 4a). These decrease exponentially because the measurement time remains
constant at 20 minutes, regardless of the distance, whereas received counts
scales exponentially with channel loss. Figure 4b) highlights the 2.6
percentage point drop in QBER between the XY bases and Z basis. Simulations
using the experimental parameters and the predicted count rate based on the
system losses are also shown. The finite key-size analysis is detailed by Lim
et al Lim  _et al._ (2014), which quantifies the security and correctness of
the protocol through the parameters $\varepsilon_{\textrm{sec}}$ and
$\varepsilon_{\textrm{cor}}$. In this implementation, these values are set to
$2\times 10^{-11}$ and $1\times 10^{-15}$ respectively. The key rate, RL is
calculated using

$R_{L}=\left[s_{ZZ;0}+s_{ZZ;1}(1-\textrm{h}(\phi_{z}))-\lambda_{EC}-\Delta(\varepsilon_{\textrm{sec}},\varepsilon_{\textrm{cor}})\right]/t,$
(1)

where $s_{XX,ZZ;n}$ is the number of counts measured by Bob in the X or Z
basis, given that Alice prepared an n-photon state in the X or Z basis
respectively, $\phi_{Z}$ is the single photon phase error rate in Z,
$\lambda_{EC}$ is the error correction information, $\Delta$ is the finite
key-size correction term and t is the time used to collect the experimental
data block Lim  _et al._ (2014). The key rates displayed in 4c) show an
experimental secure key rate of 1.26 megabits per second at an equivalent
distance of 40 km (assuming optical fibre with a 0.2 dB/km loss) using an
attenuator and 246 kilobits per second in real fibre of length 75 km. A
positive secure key rate could be achieved up to 250 km in the asymptotic
limit and up to 180 km with the finite key-size analysis of Eq 1 for just 40
minutes of key time.

Figure 4: Polar BB84 results with decoy states, secure against coherent
attacks. Lines are theoretical results and symbols are experimentally measured
or calculated values. a) Number of counts measured in each basis, b) QBER for
each basis, c) Secure key rate (SKR) in the asymptotic (filled symbols, dotted
line) and finite key-size regimes (empty symbols, solid line). The star
symbols are data measured with real fibre as the quantum channel.

To compare between polar BB84 and phase-encoded BB84, we looked at the secure
key rate in the asymptotic limit using the experimental parameters obtained in
the ZX and the YX bases respectively. The transmission basis probabilities are
renormalised to allow for a fair comparison. The key rate is improved by 1.60
times when using the ZX bases compared to the YX bases. Also, phase-encoded
BB84 is able to reach an attenuation of 48.5 dB with a positive secure key
rate, whereas polar BB84 can reach slightly further, at 50.1 dB.

##  IV Discussion

Our implementation produced six states in order to show the versatility of the
source, however only used four for the QKD implementations. Six-state QKD is
possible and has a slightly higher tolerance to noise than its four-state
counterpart, leading to the ability to share secure keys at slightly longer
distances Scarani  _et al._ (2009). The main drawback is found in the
receiver. Polar BB84 can be implemented with three SPDs (phase-encoded BB84
would require two SPDs in an active receiver and four SPDs in a passive
receiver). Six-state QKD, on the other hand, requires an extra AMZI and two
extra SPDs if it is to remain passive, or a high speed PM in the AMZI to
choose the basis in an active implementation. Both of these options add
significant complexity when compared to their meager increase in secure key
distance. Indeed, four-state QKD could be carried out for a longer time-period
to reduce statistical fluctuations and increase the achievable distance.
Reference frame independent-QKD Laing  _et al._ (2010) is another protocol
that requires three bases, allowing two bases to drift in time while the other
stays constant. This basis drift is a problem for polarisation-encoded systems
in real fibre, however is not an issue for phase encoded systems like the one
demonstrated here because the signal travels along the fibre with the phase
reference. Multi-protocol transmitters have also been demonstrated in Sibson
_et al._ (2017); Korzh  _et al._ (2013), although these have the drawbacks of
being complex and not offering phase randomisation, respectively.

As well as the aforementioned benefit of requiring one fewer SPD for polar
BB84 compared to phase-encoded BB84, the key rates are also improved by a
factor of 1.60 times. This is made possible by the reduced QBER of the signal
basis from 2.8% to 0.2%, which reduces the bits lost to error correction,
hence improving the secure key rate.

Direct preparation of decoy states can be realised by driving the pulse
preparation laser at different levels above the lasing threshold to reach
different intensities. This is ideal because no external hardware, for example
an intensity modulator, is required. This would also be useful for classical
communications, increasing the number of bits encoded per symbol Noguchi  _et
al._ (1986). We have achieved intensity modulation using this method. This
creates a patterning effect for the decoy states, however, where the intensity
of a pulse is correlated with the intensity of the preceding pulse, opening
the door to side-channel attacks Yoshino  _et al._ (2018). To avoid this
security loophole, the QKD decoy states are instead produced using a two-level
Sagnac-based IM Roberts  _et al._ (2018).

The transmitter also shows promise to be useful in classical communications.
The patterning effects that proved prohibitive for QKD when directly producing
multiple intensity states are not a major concern here, it will just add a
slight degradation to the distinguishability between states. The direct
modulation means that the system is not reliant on multiple external
modulators, making it cheaper, less complex and also easier to integrate with
other components. The optical injection locking ensures that all pulses have
the same wavelength. This removes a side channel for QKD, but also means the
system has low chirp, reducing the inter-symbol interference caused by
dispersion effects. In this paper we have shown the accurate production of
four phase states, however this can easily be increased by using more phase-
preparation levels. Different intensities can be produced directly, and also a
vacuum state can be produced, further increasing the amount of information
encoded per symbol.

##  V Conclusion

In this paper, we have demonstrated a transmitter capable of performing all
weak coherent pulse-based QKD protocols, performing phase and intensity
encoding simultaneously. With this system, we have demonstrated the decoy-
state polar BB84 protocol and the decoy-state phase-encoded BB84 protocol in a
single experiment, preparing the six states in three different bases required
by the simultaneous execution of these two protocols from a single
transmitter. In both bases we found a secure key rate in the order of 1
megabit per second at 8 dB attenuation, with the decoy-state polar BB84
protocol providing a 1.6 times larger secure key rate on average and a
slightly higher tolerance to losses.

The ability to adapt to different protocols with simple software changes makes
the transmitter more robust in network scenarios where all the users could
potentially have different receivers. Alongside this, the system is more
simple than many other transmitters that are dedicated to a single protocol,
which is appealing for real world implementations. Also the relatively few
components ensure it has a good power efficiency and make it ideal for on-chip
implementations. The versatility, low power consumption and stability of this
transmitter make it the natural choice for use in metropolitan multi-user
quantum networks.

## Acknowledgments

G. L. R gratefully acknowledges financial support from the EPSRC CDT in
Integrated Photonic and Electronics Systems, Toshiba Research Europe Limited
and an industrial fellowship with The Royal Commission for the Exhibition of
1851. This work has been supported by funding through the EPSRC Quantum
Communications Hub EP/M013472/1.

## References

  * Gisin  _et al._ (2002) N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
  * Scarani  _et al._ (2009) V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009).
  * Peev  _et al._ (2009) M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, New Journal of Physics 11, 075001 (2009).
  * Sasaki  _et al._ (2011) M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, Optics Express 19, 10387 (2011).
  * Vallone  _et al._ (2015) G. Vallone, D. Bacco, D. Dequal, S. Gaiarin, V. Luceri, G. Bianco, and P. Villoresi, Physical Review Letters 115, 040502 (2015).
  * Liao  _et al._ (2017) S.-K. Liao, H.-L. Yong, C. Liu, G.-L. Shentu, D.-D. Li, J. Lin, H. Dai, S.-Q. Zhao, B. Li, J.-Y. Guan, W. Chen, Y.-H. Gong, Y. Li, Z.-H. Lin, G.-S. Pan, J. S. Pelc, M. M. Fejer, W.-Z. Zhang, W.-Y. Liu, J. Yin, J.-G. Ren, X.-B. Wang, Q. Zhang, C.-Z. Peng, and J.-W. Pan, Nature Photonics 11, 509 (2017).
  * Takenaka  _et al._ (2017) H. Takenaka, A. Carrasco-Casado, M. Fujiwara, M. Kitamura, M. Sasaki, and M. Toyoshima, Nature Photonics 11, 502 (2017).
  * Ma  _et al._ (2005) X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, Physical Review A 72, 012326 (2005).
  * Dixon  _et al._ (2008) A. R. Dixon, Z. L. Yuan, J. F. Dynes, A. W. Sharpe, and A. J. Shields, Optics Express 16, 18790 (2008).
  * Bacco  _et al._ (2016) D. Bacco, J. B. Christensen, M. A. U. Castaneda, Y. Ding, S. Forchhammer, K. Rottwitt, and L. K. Oxenløwe, Scientific Reports 6, 36756 (2016).
  * Yin  _et al._ (2016) H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, Physical Review Letters 117, 190501 (2016).
  * Lim  _et al._ (2014) C. C. W. Lim, M. Curty, N. Walenta, F. Xu, and H. Zbinden, Physical Review A 89, 022307 (2014).
  * Fröhlich  _et al._ (2017) B. Fröhlich, M. Lucamarini, J. F. Dynes, L. C. Comandar, W. W.-S. Tam, A. Plews, A. W. Sharpe, Z. Yuan, and A. J. Shields, Optica 4, 163 (2017).
  * Dixon  _et al._ (2017) A. R. Dixon, J. F. Dynes, M. Lucamarini, B. Fröhlich, A. W. Sharpe, A. Plews, W. Tam, Z. L. Yuan, Y. Tanizawa, H. Sato, S. Kawamura, M. Fujiwara, M. Sasaki, and A. J. Shields, Scientific Reports 7 (2017), 10.1038/s41598-017-01884-0.
  * Lo  _et al._ (2012) H.-K. Lo, M. Curty, and B. Qi, Physical Review Letters 108 (2012), 10.1103/PhysRevLett.108.130503.
  * Yoshino  _et al._ (2018) K.-i. Yoshino, M. Fujiwara, K. Nakata, T. Sumiya, T. Sasaki, M. Takeoka, M. Sasaki, A. Tajima, M. Koashi, and A. Tomita, npj Quantum Information 4 (2018), 10.1038/s41534-017-0057-8.
  * Shibata  _et al._ (2014) H. Shibata, T. Honjo, and K. Shimizu, Optics Letters 39, 5078 (2014).
  * Inoue (2015) K. Inoue, IEEE Journal of Selected Topics in Quantum Electronics 21, 109 (2015).
  * Korzh  _et al._ (2015) B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, Nature Photonics 9, 163 (2015).
  * Branciard  _et al._ (2008) C. Branciard, N. Gisin, and V. Scarani, New Journal of Physics 10, 013031 (2008).
  * Inoue and Iwai (2009) K. Inoue and Y. Iwai, Physical Review A 79 (2009), 10.1103/PhysRevA.79.022319.
  * Kawakami  _et al._ (2016) S. Kawakami, T. Sasaki, and M. Koashi, Physical Review A 94 (2016), 10.1103/PhysRevA.94.022332.
  * Yuan  _et al._ (2016) Z. L. Yuan, B. Fröhlich, M. Lucamarini, G. L. Roberts, J. F. Dynes, and A. J. Shields, Physical Review X 6, 031044 (2016).
  * Roberts  _et al._ (2017a) G. L. Roberts, M. Lucamarini, J. F. Dynes, S. J. Savory, Z. Yuan, and A. J. Shields, Applied Physics Letters 111, 261106 (2017a).
  * Roberts  _et al._ (2017b) G. L. Roberts, M. Lucamarini, J. F. Dynes, S. J. Savory, Z. L. Yuan, and A. J. Shields, Laser & Photonics Reviews 11, 1700067 (2017b).
  * Hwang (2003) W.-Y. Hwang, Physical Review Letters 91, 057901 (2003).
  * Wang (2005) X.-B. Wang, Physical Review Letters 94, 230503 (2005).
  * Lo  _et al._ (2005) H.-K. Lo, X. Ma, and K. Chen, Physical Review Letters 94, 230504 (2005).
  * Koashi (2006) M. Koashi, arXiv:quant-ph/0609180 (2006).
  * Tomamichel  _et al._ (2012) M. Tomamichel, C. C. W. Lim, N. Gisin, and R. Renner, Nature Communications 3, 634 (2012).
  * Laing  _et al._ (2010) A. Laing, V. Scarani, J. G. Rarity, and J. L. O’Brien, Physical Review A 82, 012304 (2010).
  * Sibson  _et al._ (2017) P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson, Nature Communications 8, 13984 (2017).
  * Korzh  _et al._ (2013) B. Korzh, N. Walenta, R. Houlmann, and H. Zbinden, Optics Express 21, 19579 (2013).
  * Noguchi  _et al._ (1986) T. Noguchi, Y. Daido, and J. A. Nossek, IEEE Communications Magazine 24, 21 (1986).
  * Roberts  _et al._ (2018) G. L. Roberts, M. Pittaluga, M. Minder, M. Lucamarini, J. F. Dynes, Z. L. Yuan, and A. J. Shields, arXiv:1807.07414 [quant-ph] (2018).