2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})$
$\displaystyle+e^{+\mathrm{i}(p_{1}-p_{2})\pi/2}\mathbf{G}^{p_{1}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})\Big{]}$
$\displaystyle~{}+\displaystyle\frac{\,{-2\mathrm{i}\pi e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu]}e^{-\mathrm{i}(p_{1}+p_{2})\pi/2}\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})$
$\displaystyle~{}+\displaystyle\frac{\,{2\mathrm{i}\pi e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\Gamma[\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu]}e^{+\mathrm{i}(p_{1}+p_{2})\pi/2}\mathbf{G}^{p_{1}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})\bigg{\\}}.$
(107)

Note that the nonlocal signal is symmetric with respect to
$(u_{1},p_{1})\leftrightarrow(u_{2},p_{2})$ , and thus the result applies well
to the other side $k_{12}<k_{34}$. So, we do not put a subscript $>$ for the
nonlocal signal.

#### Same-sign integrals: time-ordered part.

Now we come to the time-ordered part:

$\displaystyle\mathcal{I}_{++,\text{TO},>}^{p_{1}p_{2}}(r_{1},r_{2})\equiv$
$\displaystyle-
k_{s}^{3+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{2}\int_{-\infty}^{\tau_{2}}{\mathrm{d}}\tau_{1}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\pm\mathrm{i}(k_{12}\tau_{1}+k_{34}\tau_{2})}$
$\displaystyle~{}\times\Big{[}D_{<}(k_{s};\tau_{1},\tau_{2})-D_{>}(k_{s};\tau_{1},\tau_{2})\Big{]}.$
(108)

We use the following MB representations for the Wightman functions $D_{>}$ and
$D_{<}$ to ensure that all results are expressed in terms of $u_{1,2}$:

$\displaystyle D_{>}^{(h)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{\pi\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu]}e^{+\mathrm{i}k(\tau_{1}+\tau_{2})}$
$\displaystyle\times\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,e^{\mathrm{i}\pi(s_{1}-s_{2})/2}\cos\pi(s_{1}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)(2k)^{-s_{12}}(-\tau_{1})^{-s_{1}+1/2}(-\tau_{2})^{-s_{2}+1/2}$
$\displaystyle\times\Gamma\Big{[}-s_{1}+\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-s_{2}+\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,s_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]},$ (109)
$\displaystyle D_{<}^{(h)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{\pi\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu]}e^{+\mathrm{i}k(\tau_{1}+\tau_{2})}$
$\displaystyle\times\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,e^{-\mathrm{i}\pi(s_{1}-s_{2})/2}\cos\pi(s_{2}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)(2k)^{-s_{12}}(-\tau_{1})^{-s_{1}+1/2}(-\tau_{2})^{-s_{2}+1/2}$
$\displaystyle\times\Gamma\Big{[}-s_{1}+\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-s_{2}+\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,s_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}.$ (110)

Substituting these MB representations into the time-ordered integral (4.1), we
can finish the time integral to get the following result:

$\displaystyle{\mathcal{I}}^{(h)p_{1}p_{2}}_{++,\text{TO},>}=$
$\displaystyle~{}\displaystyle\frac{\,{-\mathrm{i}e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{\pi\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu]}\displaystyle\frac{\,{e^{-\mathrm{i}\pi(p_{1}+p_{2})/2}}\,}{2^{3+p_{1}+p_{2}}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}$
$\displaystyle\times\Big{[}e^{\mathrm{i}\pi
s_{2}}\cos\pi(s_{2}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu)-e^{\mathrm{i}\pi
s_{1}}\cos\pi(s_{1}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)\Big{]}u_{1}^{-s_{12}+3+p_{1}+p_{2}}$
$\displaystyle\times\Gamma\Big{[}-s_{1}+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,-s_{2}+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,s_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,s_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\Big{]}$
$\displaystyle\times\mathcal{F}\left[\begin{matrix}p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-s_{2},p_{1}+p_{2}+3-s_{12}\\\
p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-s_{2}\end{matrix}\middle|-\displaystyle\frac{\,{u_{1}}\,}{u_{2}}\right],$
(111)

where $\mathcal{F}$ is the dressed hypergeometric function defined in (277).
The function $\mathcal{F}$ does possess poles in $s_{1}$ and $s_{2}$ but they
are right poles, and thus do not contribute to the result, since we should
close the contour from the left. (Notice that the contribution from the poles
in (95) vanishes.) The result of the time-ordered integral is fully analytic
in $u_{1}$ and $u_{2}$ at $u_{1,2}=0$ (when $p_{1}+p_{2}$ are integers no
smaller than $-3$), as opposed to the opposite-sign integral and the
factorized same-sign integral. Thus the time-ordered integral exactly gives
rise to the background piece of correlator. Let us write

${\mathcal{I}}^{(h)p_{1}p_{2}}_{\text{BG},>}={\mathcal{I}}^{(h)p_{1}p_{2}}_{++,\text{TO},>}+{\mathcal{I}}^{(h)p_{1}p_{2}}_{--,\text{TO},>}={\mathcal{I}}^{(h)p_{1}p_{2}}_{++,\text{TO},>}+\text{c.c.}.$
(112)

Then the result for the background integral can be directly obtained as:

$\displaystyle{\mathcal{I}}^{(h)p_{1}p_{2}}_{\text{BG},>}=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-\mathrm{i}\pi(p_{1}+p_{2})/2}}\,}{2^{3+p_{1}+p_{2}}}\sum_{n_{1},n_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{12}}}\,}{n_{1}!n_{2}!}u_{1}^{n_{12}+3+p_{1}+p_{2}}$
$\displaystyle\times\bigg{\\{}\displaystyle\frac{\,{\mathrm{i}}\,}{2\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\big{(}\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu\big{)}_{n_{1}}\big{(}\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu\big{)}_{n_{2}}\big{(}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\big{)}_{-n_{1}}\big{(}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{)}_{-n_{2}}$
$\displaystyle\times{\mathcal{F}}\left[\begin{matrix}n_{2}+p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,n_{12}+p_{1}+p_{2}+3\\\
n_{2}+p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,-\displaystyle\frac{\,{u_{1}}\,}{u_{2}}\right]+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigg{\\}}+\text{c.c.}.$ (113)

###  4.2 Longitudinal integral

Next we consider the longitudinal component ($\lambda=L$) of the vector seed
integral (43). The longitudinal polarization is independent of the chemical
potential, and the mode function is quite similar to that of a massive scalar
field. The computation thus follows exactly the same procedure as in Sec. 3,
and is also relatively easier than the transverse integral. Therefore we will
not spell out all the details, but only present main steps and the final
results.

The longitudinal vector seed integral reads:

${\mathcal{I}}^{(L)p_{1}p_{2}}_{\mathsf{ab}}(r_{1},r_{2})=-\mathsf{ab}k_{s}^{3+p_{1}+p_{2}}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}\,(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{ab}}^{(L)}(k_{s};\tau_{1},\tau_{2}),$
(114)

Comparing (35), (36), and (50), we see that the longitudinal propagator
$D^{(L)}_{>}$ is related to the massive scalar propagator $D_{>}$ of the same
mass parameter $\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu$ (but not the same mass) via:

$\displaystyle
D_{>}^{(L)}(k_{s};\tau_{1},\tau_{2})=\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$}}\Big{(}\partial_{\tau_{1}}-\displaystyle\frac{\,{2}\,}{\tau_{1}}\Big{)}\Big{(}\partial_{\tau_{2}}-\displaystyle\frac{\,{2}\,}{\tau_{2}}\Big{)}D_{>}(k_{s};\tau_{1},\tau_{2}),$
(115)

where we have explicitly expressed the mass as $m^{2}=\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}+1/4$. Therefore
we can recycle the MB representation for the scalar propagator, which leads us
to the following expression:

$\displaystyle D_{\lessgtr}^{(L)}(k;\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{4\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s_{1}}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{{\mathrm{d}}s_{2}}\,}{2\pi\mathrm{i}}\,e^{\mp\mathrm{i}\pi(s_{1}-s_{2})}\Big{(}\displaystyle\frac{\,{k}\,}{2}\Big{)}^{-2s_{12}}(-\tau_{1})^{-2s_{1}+1/2}(-\tau_{2})^{-2s_{2}+1/2}$
$\displaystyle\times\Big{(}2s_{1}+\displaystyle\frac{\,{1}\,}{2}\Big{)}\Big{(}2s_{2}+\displaystyle\frac{\,{1}\,}{2}\Big{)}\Gamma\Big{[}s_{1}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{1}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{2}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2},s_{2}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}\Big{]}.$ (116)

Then, following the same procedure as before, we calculate each component of
the longitudinal vector seed integral, and find the following results. Again,
we only present the result for $r_{1}<r_{2}$, and the result for $r_{1}>r_{2}$
can be obtained by switching the variables
$(r_{1},p_{1})\leftrightarrow(r_{2},p_{2})$.

First, the opposite-sign integral is:

$\displaystyle\mathcal{I}^{(L)p_{1}p_{2}}_{\pm\mp}=$
$\displaystyle~{}\displaystyle\frac{\,{e^{\mp\mathrm{i}\pi(p_{1}-p_{2})/2}}\,}{4\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})}\bigg{[}{\mathbf{H}}^{p_{1}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+{\mathbf{H}}^{p_{1}}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\bigg{]}$
$\displaystyle\times\bigg{[}{\mathbf{H}}^{p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{2})\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+{\mathbf{H}}^{p_{2}}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{2})\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\bigg{]},$ (117)

where we have defined:

$\displaystyle\mathbf{H}^{p}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r)\equiv$
$\displaystyle~{}r^{3/2+p}\bigg{\\{}\Big{(}\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}\Gamma\Big{[}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,p+\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{p}\,}{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{\,{p}\,}{2}$}+\mbox{$\frac{\,{5}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,r^{2}\right]$
$\displaystyle+\displaystyle\frac{\,{r^{2}}\,}{2}\Gamma\Big{[}-1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,p+\displaystyle\frac{\,{7}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{p}\,}{2}$}+\mbox{$\frac{\,{7}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{p}\,}{2}$}+\mbox{$\frac{\,{9}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
2+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,r^{2}\right]\bigg{\\}}.$ (118)

Second, the factorized part of the same-sign integral is:

$\displaystyle\mathcal{I}^{(L)p_{1}p_{2}}_{\pm\pm,\text{F},>}=$
$\displaystyle~{}\displaystyle\frac{\,{-\mathrm{i}e^{\mp\mathrm{i}\pi(p_{1}+p_{2})/2}}\,}{4\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})}\bigg{[}e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathbf{H}}^{p_{1}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}{\mathbf{H}}^{p_{1}}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\bigg{]}$
$\displaystyle\times\bigg{[}{\mathbf{H}}^{p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{2})\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}+{\mathbf{H}}^{p_{2}}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{2})\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\bigg{]}.$ (119)

Finally, the time-ordered part of the same-sign integral is:

$\displaystyle{\mathcal{I}}_{\pm\pm,\text{TO},>}^{(L)p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{\mp\mathrm{i}\pi(p_{1}+p_{2})/2}}\,}{2(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})}r_{1}^{3+p_{1}+p_{2}}\sum_{n_{1},n_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{12}}}\,}{n_{1}!n_{2}!}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{12}}$
$\displaystyle\times\displaystyle\frac{\,{\mathrm{i}}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}2n_{1}-\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}\Big{(}2n_{2}-\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{1}}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times{\mathcal{F}}\left[\begin{matrix}2n_{2}+p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,2n_{12}+p_{1}+p_{2}+3\\\
2n_{2}+p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\right]+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (120)

Again, we can sum up all SK components, and separate the result into a
background piece, a local-signal piece, and a nonlocal-signal piece:

$\displaystyle\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(L)p_{1}p_{2}}(r_{1},r_{2})=\mathcal{I}_{\text{BG},>}^{(L)p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\text{L},>}^{(L)p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\text{NL}}^{(L)p_{1}p_{2}}(r_{1},r_{2}).$
(121)

More explicitly, the local signal is given by:

${\mathcal{I}}^{(L)p_{1}p_{2}}_{\text{L},>}(r_{1},r_{2})=\mathcal{D}^{p_{1}p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{H}^{p_{1}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1})\mathbf{H}^{p_{2}}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{2})+\text{c.c.},$
(122)

where

$\mathcal{D}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p_{1}p_{2}}\equiv\displaystyle\frac{\,{1}\,}{2\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})}\Big{[}\cos\big{(}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}-p_{2})\big{)}-\sin\big{(}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+p_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\big{)}\Big{]}.$
(123)

The nonlocal signal is given by:

${\mathcal{I}}^{(L)p_{1}p_{2}}_{\text{NL}}(r_{1},r_{2})=\mathcal{D}^{p_{1}p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{H}^{p_{1}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1})\mathbf{H}^{p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{2})+\text{c.c.},$
(124)

Finally, the background is given by:

$\displaystyle{\mathcal{I}}_{\text{BG},>}^{(L)p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$}}\cos\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+p_{2})\big{]}r_{1}^{3+p_{1}+p_{2}}\sum_{n_{1},n_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{12}}}\,}{n_{1}!n_{2}!}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{12}}$
$\displaystyle\times\displaystyle\frac{\,{\mathrm{i}}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}2n_{1}-\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}\Big{(}2n_{2}-\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{1}}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times{\mathcal{F}}\left[\begin{matrix}2n_{2}+p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,2n_{12}+p_{1}+p_{2}+3\\\
2n_{2}+p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\right]+\text{c.c.}.$
(125)

###  4.3 Summary

After the rather long and technical calculations, it would be useful to
summarize and present the result we have obtained in a self-contained manner.
In this section, we have computed the vector seed integral (43) for all three
polarizations $\lambda=\pm,L$ and for arbitrary power indices $(p_{1},p_{2})$.
For general applications, it is more convenient to present these results with
the SK indices summed. As we have shown before, the summed integral can be
broken into three pieces, the background piece
$\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\lambda)p_{1}p_{2}}$, the local signal
$\mathcal{I}_{\text{L}}^{(\lambda)p_{1}p_{2}}$, and the nonlocal signal
$\mathcal{I}_{\text{NL}}^{(\lambda)p_{1}p_{2}}$, according to their analytic
properties at $r_{1,2}=0$:

$\displaystyle\sum_{\mathsf{a},\mathsf{b}=\pm}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{(\lambda)p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{I}_{\text{BG}}^{(\lambda)p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\text{L}}^{(\lambda)p_{1}p_{2}}(r_{1},r_{2})+\mathcal{I}_{\text{NL}}^{(\lambda)p_{1}p_{2}}(r_{1},r_{2}),$
(126)
$\displaystyle\mathcal{I}_{\text{BG}}^{(\lambda)p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{I}_{\text{BG},>}^{(\lambda)p_{1}p_{2}}(r_{1},r_{2})\theta(r_{2}-r_{1})+\mathcal{I}_{\text{BG},>}^{(\lambda)p_{2}p_{1}}(r_{2},r_{1})\theta(r_{1}-r_{2}),$
(127)
$\displaystyle\mathcal{I}_{\text{L}}^{(\lambda)p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{I}_{\text{L},>}^{(\lambda)p_{1}p_{2}}(r_{1},r_{2})\theta(r_{2}-r_{1})+\mathcal{I}_{\text{L},>}^{(\lambda)p_{2}p_{1}}(r_{2},r_{1})\theta(r_{1}-r_{2}),$
(128)

Below we summarize the results for each of the three polarizations.

#### Transverse integrals.

The two transverse integrals correspond to $\lambda=h=\pm$. They are dependent
on the chemical potential. For convenience, we introduced new variables
$u_{1}=2r_{1}/(1+r_{1})$ and $u_{2}=2r_{2}/(1+r_{2})$. Then, the three
integrals in (126) can be written as:

$\displaystyle{\mathcal{I}}^{(h)p_{1}p_{2}}_{\text{BG},>}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-\mathrm{i}\pi(p_{1}+p_{2})/2}}\,}{2^{3+p_{1}+p_{2}}}\sum_{n_{1},n_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{12}}}\,}{n_{1}!n_{2}!}u_{1}^{n_{12}+3+p_{1}+p_{2}}$
$\displaystyle\times\bigg{\\{}\displaystyle\frac{\,{\mathrm{i}}\,}{2\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\big{(}\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu\big{)}_{n_{1}}\big{(}\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu\big{)}_{n_{2}}\big{(}-2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\big{)}_{-n_{1}}\big{(}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{)}_{-n_{2}}$
$\displaystyle\times{\mathcal{F}}\left[\begin{matrix}n_{2}+p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,n_{12}+p_{1}+p_{2}+3\\\
n_{2}+p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,-\displaystyle\frac{\,{u_{1}}\,}{u_{2}}\right]+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigg{\\}}+\text{c.c.},$ (129)
$\displaystyle\mathcal{I}_{\text{L},>}^{(h)p_{1}p_{2}}(u_{1},u_{2})=$
$\displaystyle~{}\mathcal{G}^{(h)p_{1}p_{2}}_{\text{L}}(u_{1},u_{2})\Big{(}\displaystyle\frac{\,{u_{1}}\,}{u_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.},$ (130)
$\displaystyle\mathcal{I}_{\text{NL}}^{(h)p_{1}p_{2}}(u_{1},u_{2})=$
$\displaystyle~{}\mathcal{G}^{(h)p_{1}p_{2}}_{\text{NL}}(u_{1},u_{2})(u_{1}u_{2})^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.},$ (131)

where we have defined:

$\displaystyle\mathcal{G}_{\text{L}}^{(h)p_{1}p_{2}}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-\pi h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2\pi^{2}}\bigg{\\{}\big{(}\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\big{)}\Big{[}e^{-\mathrm{i}(p_{1}-p_{2})\pi/2}\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})$
$\displaystyle+e^{+\mathrm{i}(p_{1}-p_{2})\pi/2}\mathbf{G}^{p_{1}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})\Big{]}$
$\displaystyle~{}+\displaystyle\frac{\,{-2\mathrm{i}\pi e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu]}e^{-\mathrm{i}(p_{1}+p_{2})\pi/2}\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})$
$\displaystyle~{}+\displaystyle\frac{\,{2\mathrm{i}\pi e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\Gamma[\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu]}e^{+\mathrm{i}(p_{1}+p_{2})\pi/2}\mathbf{G}^{p_{1}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})\bigg{\\}},$
(132) $\displaystyle\mathcal{G}_{\text{NL}}^{(h)p_{1}p_{2}}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{e^{-\pi h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2\pi^{2}}\bigg{\\{}\big{(}\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\big{)}\Big{[}e^{-\mathrm{i}(p_{1}-p_{2})\pi/2}\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})$
$\displaystyle+e^{+\mathrm{i}(p_{1}-p_{2})\pi/2}\mathbf{G}^{p_{1}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})\Big{]}$
$\displaystyle~{}+\displaystyle\frac{\,{-2\mathrm{i}\pi e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu]}e^{-\mathrm{i}(p_{1}+p_{2})\pi/2}\mathbf{G}^{p_{1}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})$
$\displaystyle~{}+\displaystyle\frac{\,{2\mathrm{i}\pi e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)]}\,}{\Gamma[\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu]}e^{+\mathrm{i}(p_{1}+p_{2})\pi/2}\mathbf{G}^{p_{1}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\mathbf{G}^{p_{2}}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{2})\bigg{\\}}.$
(133)

and the function $\mathbf{G}$ is defined by

$\displaystyle\mathbf{G}^{p}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u)\equiv\mathrm{i}\pi\text{csch}(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{(}\displaystyle\frac{\,{u}\,}{2}\Big{)}^{3/2+p}\mathcal{F}\left[\begin{matrix}\mbox{$\frac{\,{3}\,}{2}$}+p+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1+2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|u\right].$ (134)

#### Longitudinal integrals.

The longitudinal integral correspond to $\lambda=L$. This part is independent
of the chemical potential, and the results for the three integrals in (126)
are:

$\displaystyle{\mathcal{I}}_{\text{BG},>}^{(L)p_{1}p_{2}}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$}}\cos\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+p_{2})\big{]}r_{1}^{3+p_{1}+p_{2}}\sum_{n_{1},n_{2}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{12}}}\,}{n_{1}!n_{2}!}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{12}}$
$\displaystyle\times\displaystyle\frac{\,{\mathrm{i}}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Big{(}2n_{1}-\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}\Big{(}2n_{2}-\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{-n_{1}}(\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{-n_{2}}$
$\displaystyle\times{\mathcal{F}}\left[\begin{matrix}2n_{2}+p_{2}+\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,2n_{12}+p_{1}+p_{2}+3\\\
2n_{2}+p_{2}+\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,-\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\right]+\text{c.c.},$
(135) $\displaystyle{\mathcal{I}}^{(L)p_{1}p_{2}}_{\text{L},>}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{G}^{(L)p_{1}p_{2}}_{\text{L}}(r_{1},r_{2})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.},$ (136)
$\displaystyle{\mathcal{I}}^{(L)p_{1}p_{2}}_{\text{NL}}(r_{1},r_{2})=$
$\displaystyle~{}\mathcal{G}^{(L)p_{1}p_{2}}_{\text{NL}}(r_{1},r_{2})\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.},$ (137)

where we have defined the following functions:

$\displaystyle\mathcal{G}^{(L)p_{1}p_{2}}_{\text{L}}(r_{1},r_{2})\equiv$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})}\Big{[}\cos\big{(}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}-p_{2})\big{)}-\sin\big{(}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+p_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\big{)}\Big{]}\mathbf{H}^{p_{1}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1})\mathbf{H}^{p_{2}}_{-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{2}),$ (138)
$\displaystyle\mathcal{G}^{(L)p_{1}p_{2}}_{\text{NL}}(r_{1},r_{2})\equiv$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\mbox{$\frac{\,{1}\,}{4}$})}\Big{[}\cos\big{(}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}-p_{2})\big{)}-\sin\big{(}\mbox{$\frac{\,{\pi}\,}{2}$}(p_{1}+p_{2}+2\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\big{)}\Big{]}\mathbf{H}^{p_{1}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(r_{1})\mathbf{H}^{p_{2}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{2}),$ (139)

and the function $\mathbf{H}$ is defined by

$\displaystyle\mathbf{H}^{p}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r)\equiv$
$\displaystyle~{}r^{3/2+p}\bigg{\\{}\Big{(}\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}\Gamma\Big{[}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,p+\displaystyle\frac{\,{3}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{p}\,}{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{\,{p}\,}{2}$}+\mbox{$\frac{\,{5}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,r^{2}\right]$
$\displaystyle+\displaystyle\frac{\,{r^{2}}\,}{2}\Gamma\Big{[}-1-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,p+\displaystyle\frac{\,{7}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{p}\,}{2}$}+\mbox{$\frac{\,{7}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{p}\,}{2}$}+\mbox{$\frac{\,{9}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
2+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,r^{2}\right]\bigg{\\}}.$ (140)

#### Tree-level signal cutting rule.

At the end of this section, we briefly comment on the relation between our
calculation and a recently proposed cutting rule for computing CC signals at
the tree level [40]. With our notations, it was proposed in [40] that the
oscillating signals in inflation correlators are contributed by two sources:
1) The opposite-sign integrals $\mathcal{I}_{\pm\mp}$; 2) the factorized part
of the same-sign integrals $\mathcal{I}_{\pm\pm,\text{F}}$. On the other hand,
the time-ordered part of the same-sign integrals
$\mathcal{I}_{\pm\pm,\text{TO}}$ do not contribute to the signal. The upshot
is that the computation of CC signals involves no genuine time-ordered
integral, hence the name cutting rule. A cutting rule of this sort can be
phenomenologically useful, since the computation of factorized integrals are
almost always much simpler than that of the time-ordered integrals. The
cutting rule proposed in [40] was based on physical arguments of particle
production in the bulk, and was further checked with numerical evaluations.
Our results in this paper agree with the physical observations made in [40],
and can be viewed as an analytical derivation for the cutting rule in the
presence of the chemical potential. (The dS covariant case without chemical
potential can be verified with previously known results [20].) Therefore, the
partial MB representation can be used to give a more rigorous and more
satisfactory derivation of the cutting rule for the CC signals. We shall
pursue such a proof in a future work.

## 5 Bootstrapping Helical Correlators

In this section we provide an alternative derivation for both the scalar seed
integral (55) introduced in Sec. 3 and the vector seed integral (43)
introduced in Sec. 2, using a “bootstrapping” method developed in recent years
by several groups of authors [20, 22, 23, 29, 30, 31, 25, 32, 33]. See also
[21] for a recent review.

At the conceptual level, the cosmological bootstrap program aims at building
inflation correlators directly at the future boundary without doing bulk time
integral at each interaction vertex. This is achieved by exploiting the
symmetry properties of the inflation correlators, together with a set of
constraints and boundary conditions from the bulk unitarity, locality, as well
as the bulk initial condition. When boiled down to a practical problem of
computing inflation correlators, the bootstrap program suggests that we can
derive a set of differential equations satisfied by the correlators, so that
we can find the correlators by solving the differential equations instead of
computing bulk integrals. The boundary conditions that are required to
uniquely pin down the solution can be found from physical considerations in
the bulk. For simplicity, we will collectively call such differential
equations satisfied by the correlators the _bootstrap equations_ , although
its meaning is a bit shifted from the original conformal bootstrap program in
conformal field theories.

The idea of solving the inflation correlators from a set of differential
equations has been put forward originally in [19]. There, the authors get the
signal part of the whole correlator, which corresponds to a solution to the
homogeneous equation. The background solution, corresponding to a particular
solution to the inhomogeneous equation, was later obtained in [20]. In [20] it
was further shown that the bootstrap equation for the 4-point tree diagram
with massive exchange can be derived from the Ward identity of dS isometries.
The bootstrap program was subsequently generalized to the case of boost
breaking theories for purely massless correlators [29, 30] and for correlators
involving massive exchange [32, 33].

Of course, when the dS boosts are broken by the background, the bootstrap
equations are not directly from the Ward identities, although we suspect that
they can be related to the Ward identity with additional symmetry breaking
terms. Nonetheless, one can still derive the bootstrap equations for the
correlators using the very observation originally made in [19]. The essential
idea is the following. The tree-level exchange digram for the 4-point
correlator can be expressed as SK integrals with the massive propagator in the
integrand. As the Green function for the corresponding massive field, the
massive propagator solves the field equation with either a $\delta$-function
source or no source. Therefore, if we insert the differential operator for the
field equation in front of the massive propagator in the SK integral, the
integral will be reduced into either 0 or a much simpler integral over a
single time variable. Then, given that the factors multiplied to the massive
propagators (including the bulk-to-boundary propagators and the coupling
factors) are simple power functions or exponential functions, it is very often
possible to commute the differential operator in front of the massive
propagator with the integral itself, resulting in an differential equation for
the whole integral, which is the desired bootstrap equation.

This bootstrap program was recently generalized to boost-breaking theories
with massive exchanges [32, 33]. These works allowed for non-unit sound speed
of the massive particle and the external massless states, as well as more
general boost-breaking couplings. However, their results are not directly
applicable to cases with helical chemical potential, owing to the more
complicated mode function for helical spinning fields, which require either a
direct manipulation of Whittaker functions, or a resummation of infinite
number of boost breaking insertions in the propagator. We highlight the
difference between the two cases in Figs. 4 and 5.

The presentation below will be separated into 3 subsections, corresponding to
the case of scalar change, transverse vector change, and longitudinal vector
change, respectively. The procedure will be identical: We first derive the
bootstrap equation, and then try to solve it. The final result will be the sum
of a particular solution to the inhomogeneous equation and a solution to the
homogeneous equation with proper boundary conditions. We determine the
coefficients of the homogeneous solutions by imposing boundary conditions at
the squeezed limit $r_{1,2}\rightarrow 0$. We then check our results by
looking at the folded limit $r_{1,2}\rightarrow 1$. Along the way, we will
also get new analytical results for the 3-point and 2-point functions in
completely closed analytical form.

Figure 4: An illustration of boostless bootstrap program developed in [32,
33], remade from a similar figure in [32]. In this plot, the upper blue line
represents the future boundary of dS and the space below represents the bulk.
The blob represents the summed vertices with arbitrary number of background
value $\dot{\phi}_{0}$ insertions, represented by crosses. Note that, while
the number of crosses are arbitrary, they are all inserted at the two
vertices. The propagators in these figures can have non-unit sound speed, but
this can be implemented by simple change of the variable $k\rightarrow
c_{s}k$.  Figure 5: An illustration of helical inflation correlators from
resumming chemical-potential insertions. The blue wiggle line on the left hand
side represents the massive spin-1 propagator dressed by the chemical
potential, and the black wiggle lines on the right hand side represent the dS
covariant spin-1 propagator without chemical potentials. While each gray blob
already contains arbitrary number of inflaton background insertions, there are
further boost-breaking insertions attached to the black propagator. A
nonperturbative treatment of chemical potential effects require summing over
all such insertions. Technically, this resummation is done by directly solving
the equation of motion.

###  5.1 Scalar

Again we begin with the known example of a massive scalar exchange. We do not
pursue the most general results in this section. Instead, we will use a
specific example to illustrate the idea of bootstrap equations. For this
purpose, we derive a set of differential equations satisfied by the scalar
seed integral (55) with $p_{1}=p_{2}=-2$. This corresponds to the four
external states being directly coupled conformal scalars. The results for
other values of $(p_{1},p_{2})$ can be obtained by either following the same
procedure, or acting appropriate differential operators on the result with
$p_{1}=p_{2}=-2$.

Recall that the scalar seed integral with $p_{1}=p_{2}=-2$ is:

$\displaystyle\mathcal{I}_{\mathsf{a}\mathsf{b}}^{-2,-2}(r_{1},r_{2})\equiv-\mathsf{ab}\,k_{s}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{\tau_{1}^{2}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{\tau_{2}^{2}}\,e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2}).$
(141)

One useful observation here is that the following combination,

$\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}(k_{s}\tau_{1},k_{s}\tau_{2})\equiv
k_{s}^{3}D_{\mathsf{a}\mathsf{b}}(k_{s};\tau_{1},\tau_{2}),$ (142)

depends on the three variables $k_{s}$, $\tau_{1}$, and $\tau_{2}$ only
through the combinations $k_{s}\tau_{1}$ and $k_{s}\tau_{2}$. Therefore, if we
further define $z_{1}=k_{12}\tau_{1}$ and $z_{2}=k_{34}\tau_{2}$. ,then the
seed integral (141) can be rewritten in the following form:

$\displaystyle
r_{1}r_{2}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{-2,-2}(r_{1},r_{2})\equiv-\mathsf{ab}\,\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}^{2}}\displaystyle\frac{\,{{\mathrm{d}}z_{2}}\,}{z_{2}^{2}}\,e^{\mathrm{i}\mathsf{a}z_{1}+\mathrm{i}\mathsf{b}z_{2}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2}),$ (143)

where again $r_{1}=k_{s}/k_{12}$ and $r_{2}=k_{s}/k_{34}$.

To derive an equation satisfied by the integral (143), we act the Klein-Gordon
operator to the massive scalar propagator
$D_{\mathsf{a}\mathsf{b}}(k_{s};\tau_{1},\tau_{2})$, which, by definition,
annihilates the opposite-sign propagator $D_{\pm\mp}$ and reduces the same-
sign propagator $D_{\pm\pm}$ to a $\delta$ function:

$\displaystyle(\tau_{1}^{2}\partial_{\tau_{1}}^{2}-2\tau_{1}\partial_{\tau_{1}}+k_{s}^{2}\tau_{1}^{2}+m^{2})D_{\pm\mp}(k_{s};\tau_{1},\tau_{2})=0,$
(144)
$\displaystyle(\tau_{1}^{2}\partial_{\tau_{1}}^{2}-2\tau_{1}\partial_{\tau_{1}}+k_{s}^{2}\tau_{1}^{2}+m^{2})D_{\pm\pm}(k_{s};\tau_{1},\tau_{2})=\mp\mathrm{i}\tau_{1}^{2}\tau_{2}^{2}\delta(\tau_{1}-\tau_{2}).$
(145)

These equations can be rewritten as equations satisfied by the “hatted”
propagator $\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}(r_{1}z_{1},r_{2}z_{2})$:

$\displaystyle(r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2})\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}(r_{1}z_{1},r_{2}z_{2})=0,$ (146)
$\displaystyle(r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2})\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}(r_{1}z_{1},r_{2}z_{2})=\mp\mathrm{i}(r_{1}z_{1})^{2}(r_{2}z_{2})^{2}\delta(r_{1}z_{1}-r_{2}z_{2}).$
(147)

Now we insert the operator
$r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2}$
in front of $\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{a}\mathsf{b}}$ in the integral (143), and use the above two
equations, which give rise to the following two equations:

$\displaystyle+\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}^{2}}\displaystyle\frac{\,{{\mathrm{d}}z_{2}}\,}{z_{2}^{2}}\,e^{\mathrm{i}\mathsf{a}z_{1}+\mathrm{i}\mathsf{b}z_{2}}\big{(}r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2}\big{)}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}(r_{1}z_{1},r_{2}z_{2})=0,$ (148)
$\displaystyle-\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}^{2}}\displaystyle\frac{\,{{\mathrm{d}}z_{2}}\,}{z_{2}^{2}}\,e^{\mathrm{i}\mathsf{a}z_{1}+\mathrm{i}\mathsf{b}z_{2}}\big{(}r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2}\big{)}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}(r_{1}z_{1},r_{2}z_{2})=\displaystyle\frac{\,{(r_{1}r_{2})^{2}}\,}{r_{1}+r_{2}}.$
(149)

At this point, if we can somehow pull the differential operator
$r_{1}^{2}\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+r_{1}^{2}z_{1}^{2}+m^{2}$
out of the integral, then we will get an differential equation for the whole
integral. This can be done by using the following two relations:

$\displaystyle\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}^{2}}\,z_{1}e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2})=$
$\displaystyle~{}\mathrm{i}\mathsf{a}(r_{1}\partial_{r_{1}}-1)\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}^{2}}\,e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2}),$ (150)
$\displaystyle\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}^{2}}\,z_{1}^{2}e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2})=$
$\displaystyle-(r_{1}\partial_{r_{1}})(r_{1}\partial_{r_{1}}-1)\int_{-\infty}^{0}{\mathrm{d}}z_{1}\,e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2}).$ (151)

These relations are direct consequences of integration by parts. For instance,
the first relation (150) can be derived as follows:

$\displaystyle 0=$
$\displaystyle\int_{-\infty}^{0}{\mathrm{d}}z_{1}\,\partial_{z_{1}}\Big{[}z_{1}^{-1}e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2})\Big{]}=\int_{-\infty}^{0}{\mathrm{d}}z_{1}\,\Big{[}-\displaystyle\frac{\,{1}\,}{z_{1}^{2}}+\displaystyle\frac{\,{\mathrm{i}\mathsf{a}}\,}{z}+\displaystyle\frac{\,{r_{1}\partial_{r_{1}}}\,}{z_{1}^{2}}\Big{]}e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2}),$ (152)

and the second relation (151) can be derived similarly.

Then, using (151), we find that the integral
$r_{1}r_{2}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{-2,-2}(r_{1},r_{2})$ in (143)
satisfies the equations:

$\displaystyle\bigg{[}(r_{1}^{2}-r_{1}^{4})\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{9}\,}{4}\Big{)}\bigg{]}\Big{(}r_{1}r_{2}\mathcal{I}_{\pm\mp}^{-2,-2}(r_{1},r_{2})\Big{)}=0,$
(153)
$\displaystyle\bigg{[}(r_{1}^{2}-r_{1}^{4})\partial_{r_{1}}^{2}-2r_{1}\partial_{r_{1}}+\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{9}\,}{4}\Big{)}\bigg{]}\Big{(}r_{1}r_{2}\mathcal{I}_{\pm\pm}^{-2,-2}(r_{1},r_{2})\Big{)}=\displaystyle\frac{\,{(r_{1}r_{2})^{2}}\,}{r_{1}+r_{2}}.$
(154)

From these equations it is direct to get the following equations for the seed
integral itself, which turns out a bit simpler than the above ones:

$\displaystyle\bigg{[}(r_{1}^{2}-r_{1}^{4})\partial_{r_{1}}^{2}-2r_{1}^{3}\partial_{r_{1}}+\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{1}\,}{4}\Big{)}\bigg{]}{\mathcal{I}}^{-2,-2}_{\pm\mp}(r_{1},r_{2})=0,$
(155)
$\displaystyle\bigg{[}(r_{1}^{2}-r_{1}^{4})\partial_{r_{1}}^{2}-2r_{1}^{3}\partial_{r_{1}}+\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{1}\,}{4}\Big{)}\bigg{]}{\mathcal{I}}^{-2,-2}_{\pm\pm}(r_{1},r_{2})=\displaystyle\frac{\,{r_{1}r_{2}}\,}{r_{1}+r_{2}}.$
(156)

These are the bootstrap equations satisfied by the scalar seed integral, to
which we shall derive a proper solution satisfying all required boundary
conditions.

#### Particular solution to the inhomogeneous equation.

We first try to find a particular solution to the inhomogeneous equation
(156). Following the method originally used in [20], we use double series
expansion in both $r_{1}$ and $r_{1}/r_{2}$ as an ansatz and try to solve the
series coefficients. To this end, we assume $r_{1}<r_{2}$, and Taylor expand
the right hand side of the inhomogeneous equation (156):

$\displaystyle\frac{\,{r_{1}r_{2}}\,}{r_{1}+r_{2}}=r_{1}\sum_{n=0}^{\infty}(-1)^{n}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{n}.$
(157)

The form of this series motivates us to try the following ansatz for the
inhomogeneous solution:

$\mathcal{Z}(r_{1},r_{2})=r_{1}\sum_{m,n=0}^{\infty}(-1)^{n}\mathcal{Z}_{m,n}r_{1}^{m}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{n},$
(158)

Substituting the ansatz (158) and the expanded source term (157) back into the
inhomogeneous equation (156), and matching the powers of $r_{1}$ and
$r_{1}/r_{2}$, we find the following set of recursion relations satisfied by
the series coefficients $\mathcal{Z}_{m,n}$:

$\displaystyle\Big{[}\big{(}n+\mbox{$\frac{\,{1}\,}{2}$}\big{)}^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}\Big{]}\mathcal{Z}_{0,n}=$ $\displaystyle~{}1;$ (159a)
$\displaystyle\mathcal{Z}_{1,n}=$ $\displaystyle~{}0;$ (159b)
$\displaystyle\Big{[}\big{(}m+n+\mbox{$\frac{\,{5}\,}{2}$}\big{)}^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}\Big{]}\mathcal{Z}_{m+2,n}=$
$\displaystyle~{}(m+n+1)(m+n+2)\mathcal{Z}_{m,n}.$ (159c)

This recursion relation is straightforward to solve with the general term
given by:

$\displaystyle\mathcal{Z}_{2m,n}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(n+\mbox{$\frac{\,{1}\,}{2}$})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}\displaystyle\frac{\,{(n+1)(n+2)}\,}{(n+\mbox{$\frac{\,{5}\,}{2}$})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}\cdots\displaystyle\frac{\,{(n+2m-1)(n+2m)}\,}{(n+2m+\mbox{$\frac{\,{1}\,}{2}$})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}},$ (160)
$\displaystyle\mathcal{Z}_{2m+1,n}=$ $\displaystyle~{}0,$ (161)

or equivalently,

$\displaystyle\mathcal{Z}_{2m,n}=\displaystyle\frac{\,{(n+1)_{2m}}\,}{4^{1+m}(\mbox{$\frac{\,{n}\,}{2}$}+\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$})_{m+1}(\mbox{$\frac{\,{n}\,}{2}$}+\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$})_{m+1}},$
$\displaystyle\mathcal{Z}_{2m+1,n}=0.$ (162)

Note that the inhomogeneous solution we have found applies to both
${\mathcal{I}}^{-2,-2}_{++}$ and ${\mathcal{I}}^{-2,-2}_{--}$. Furthermore,
this solution is analytic in both $r_{1}$ and $r_{2}$ as $r_{1,2}\rightarrow
0$, and thus contributes only to the background piece of the 4-point
correlator. As we shall see soon, the homogeneous solutions are all
nonanalytic in $r_{1,2}$ at $r_{1,2}=0$, and therefore the above particular
solution is the only contribution to the background. Therefore we conclude
that the background piece of the scalar seed integral
${\mathcal{I}}^{-2,-2}_{\text{BG},>}(r_{1},r_{2})$ is twice of the above
particular solution, namely:

$\displaystyle{\mathcal{I}}^{-2,-2}_{\text{BG},>}(r_{1},r_{2})=\sum_{m,n=0}^{\infty}\displaystyle\frac{\,{(-1)^{n}(n+1)_{2m}}\,}{2^{2m+1}(\mbox{$\frac{\,{n}\,}{2}$}+\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$})_{m+1}(\mbox{$\frac{\,{n}\,}{2}$}+\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$})_{m+1}}r_{1}^{2m+1}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{n}.$
(163)

This recovers the same result originally found in [20]. Following the same
procedure, we can also derive the background part for other choices of
$p_{1,2}$. In Fig. 6 we show the results for this series for both
$p_{1}=p_{2}=-2$ and $p_{1}=p_{2}=0$, together with the results (3.3) from the
partial MB representation, for fixed $r_{2}=0.5$. From this figure we can see
that both methods yield convergent results for most of values of $r_{1}$, and
they agree well with each other. We also observe that the results from partial
MB representation have better convergence speed in the case of
$p_{1}=p_{2}=0$, which corresponds to a realistic case with derivatively
coupled external massless scalars. Therefore, we see that the partial MB
result could be advantageous for numerical implementation at least in some
cases.

Figure 6: The summed scalar seed integrals
$\sum_{\mathsf{a},\mathsf{b}}\mathcal{I}_{\mathsf{a}\mathsf{b}}^{p_{1}p_{2}}(r_{1},r_{2})$
as functions of $r_{1}$ with fixed $r_{2}=0.5$ and $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu=2$. For the two cases
$p_{1}=p_{2}=-2$ (left) and $p_{1}=p_{2}=0$ (right), we show the convergences
of series from partial MB representation (3.3) and from the bootstrap method
(163), by including terms of $m$ and $n$ summations up to values indicated in
the plots.

#### Homogeneous solutions and boundary conditions.

The full result for the scalar integral should be the sum of the above
particular solution (163) with a solution to the corresponding homogeneous
equation (155). It is straightforward to solve (155) to get the following pair
of independent solutions:

$\mathcal{Y}_{\pm}(r_{1})=r_{1}^{1/2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma\Big{[}\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\displaystyle\frac{\,{1}\,}{2}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{]}{}_{2}\mathrm{F}_{1}\left[\begin{matrix}\mbox{$\frac{\,{1}\,}{4}$}\pm\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$},\mbox{$\frac{\,{3}\,}{4}$}\pm\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2}$}\\\
1\pm\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,r_{1}^{2}\right]=\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\mathbf{F}_{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{-2}(r_{1}),$ (164)

where $\mathbf{F}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}^{p}(r)$ is the function defined in (78). Here we have multiplied an
$r_{1}$-independent coefficient for later convenience. Since
$\mathcal{Y}_{-}(r_{1})=\mathcal{Y}_{+}^{*}(r_{1})$, the general symmetric and
real homogeneous solution can be written as:

$\displaystyle{\mathcal{I}}^{-2,-2}_{\text{S},>}(r_{1},r_{2})=\mathcal{Y}_{+}(r_{1})\Big{[}\alpha_{++}\mathcal{Y}_{+}(r_{2})+\alpha_{+-}\mathcal{Y}_{-}(r_{2})\Big{]}+\text{c.c.},$
(165)

where $\alpha_{++}$ and $\alpha_{+-}$ are two arbitrary complex coefficients
to be determined. This homogeneous solution is in fact the signal piece
because of the non-analytical behaviors around $r_{1,2}\rightarrow 0$ and thus
we add a subscript S to ${\mathcal{I}}_{\text{S},>}^{-2,-2}$. The full
correlator is then:

${\mathcal{I}}^{-2,-2}_{>}(r_{1},r_{2})={\mathcal{I}}^{-2,-2}_{\text{BG},>}(r_{1},r_{2})+{\mathcal{I}}^{-2,-2}_{\text{S},>}(r_{1},r_{2}).$
(166)

To determine the coefficients $\alpha_{++}$ and $\alpha_{+-}$, we need to
impose boundary conditions, which are usually found by considering some
special limits of ${\mathcal{I}}^{-2,-2}_{>}(r_{1},r_{2})$ in $r_{1}$ and
$r_{2}$. In previous works, the folded limits $r_{1,2}\rightarrow 1$ and the
factorized limits $r_{1,2}\rightarrow-1$ were used to impose boundary
conditions. Here let us adopt a different choice: We shall consider the
squeezed limit $r_{1,2}\rightarrow 0$, keeping $r_{1}\ll r_{2}$, and determine
the coefficients by comparing the leading terms with the bulk integral.

In this squeezed limit, it is easy to check that the leading behaviors:

$\displaystyle{\mathcal{I}}^{-2,-2}_{\text{BG},>}(r_{1},r_{2})\rightarrow$
$\displaystyle~{}\mathcal{O}(r_{1}),$ (167)
$\displaystyle{\mathcal{I}}^{-2,-2}_{\text{S},>}(r_{1},r_{2})\rightarrow$
$\displaystyle~{}(r_{1}r_{2})^{1/2}\bigg{\\{}\alpha_{++}\Gamma^{2}(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma^{2}\Big{(}\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\alpha_{+-}|\Gamma(-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)|^{2}\Big{|}\Gamma\Big{(}\displaystyle\frac{\,{1}\,}{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\Big{)}\Big{|}^{2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\bigg{\\}}+\text{c.c.},$ (168)

so the background contribution can be neglected.

On the other hand, we consider the same squeezed limit for the bulk integral,
which is equivalent to taking the late-time limits for the propagators:
$\tau_{1,2}\rightarrow 0$ with $\tau_{1}<\tau_{2}$. So the (anti-)time-ordered
propagators become much simpler:

$\displaystyle D_{\pm\pm}(k_{s};\tau_{1},\tau_{2})\rightarrow
D_{\mp\pm}(k_{s};\tau_{1},\tau_{2}).$ (169)

By comparing the result of the bulk integral (See Sec. 3), we can finally
determine the coefficients:

$\alpha_{++}=\alpha_{+-}=\displaystyle\frac{\,{1+\mathrm{i}\sinh\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2\pi}.$ (170)

With these two coefficients determined, we reach the full solution (166) to
the bootstrap equation with appropriate boundary conditions. This completes
the bootstrap for the scalar seed integral.

#### Folded limit and alternative version of scalar bootstrap equation.

Now let us check the folded limit $r_{1,2}\rightarrow 1^{-}$ of the above
bootstrapped solution (166). Our choice of Bunch-Davies initial condition for
all fluctuations require that the solution (166) is regular in the folded
limit. However, both the signal and the background pieces are divergent in
this limit. The cancellation of this superficial divergence is thus a useful
consistency check of our result.

It is relatively complicated to analyze the behavior of the background piece
in the folded limit with the expression (163). Thus, inspired by the
transverse vector integral derived in Sec. 4, we introduce two new variables
$u_{1,2}=2r_{1,2}/(1+r_{1,2})$ as in (44), and rewrite the (inhomogeneous)
scalar bootstrap equation (156) as:

$\bigg{[}(u_{1}^{2}-u_{1}^{3})\partial_{u_{1}}^{2}-u_{1}^{2}\partial_{u_{1}}+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{1}\,}{4}\Big{)}\bigg{]}{\mathcal{I}}^{-2,-2}_{\pm\pm}(u_{1},u_{2})=\displaystyle\frac{\,{1}\,}{2}\displaystyle\frac{\,{u_{1}u_{2}}\,}{u_{1}+u_{2}-u_{1}u_{2}},$
(171)

where we have rewritten ${\mathcal{I}}^{-2,-2}_{\pm\pm}$ as a function of
$u_{1,2}$. Again, we Taylor expand the right hand side:

$\displaystyle\frac{\,{1}\,}{2}\displaystyle\frac{\,{u_{1}u_{2}}\,}{u_{1}+u_{2}-u_{1}u_{2}}=\displaystyle\frac{\,{1}\,}{2}\sum_{n=0}^{\infty}u_{1}^{n+1}\Big{(}1-\displaystyle\frac{\,{1}\,}{u_{2}}\Big{)}^{n},$
(172)

and try the following ansatz which is analytical for both $u_{1}$ and $u_{2}$
(and thus both $r_{1}$ and $r_{2}$):

$\mathcal{X}_{\pm}(u_{1},u_{2})=\sum_{m,n=0}^{\infty}\mathcal{X}_{m,n}u_{1}^{m+n+1}\Big{(}1-\displaystyle\frac{\,{1}\,}{u_{2}}\Big{)}^{n}.$
(173)

Substituting the ansatz and the Taylor expansion back into the bootstrap
equation, and matching the coefficients for each term, we obtain the following
recursion relations:

$\displaystyle\Big{[}\big{(}n+\mbox{$\frac{\,{1}\,}{2}$}\big{)}^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}\Big{]}\mathcal{X}_{0,n}=\displaystyle\frac{\,{1}\,}{2};$ (174a)
$\displaystyle\Big{[}\big{(}m+n+\mbox{$\frac{\,{3}\,}{2}$}\big{)}^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}\Big{]}\mathcal{X}_{m+1,n}=(m+n+1)^{2}\mathcal{X}_{m,n},$ (174b)

and the solution can be expressed as:

$\mathcal{X}_{m,n}=\displaystyle\frac{\,{(n+1)_{m}^{2}}\,}{2(n+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}(n+\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{m+1}},$ (175)

and finally the background piece is the twice of our solution, but now in
terms of $u_{1}$ and $u_{2}$:

${\mathcal{I}}^{-2,-2}_{\text{BG},>}(u_{1},u_{2})=\sum_{m,n=0}^{\infty}\displaystyle\frac{\,{(-1)^{n}(n+1)_{m}^{2}}\,}{(n+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}(n+\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}}u_{1}^{m+1}(1-u_{2})^{n}\Big{(}\displaystyle\frac{\,{u_{1}}\,}{u_{2}}\Big{)}^{n}.$
(176)

It is worth noting that the expression (176) is equivalent to (163), and we
will use (176) to analyze the background behavior in the folded limit.

Without loss of generality, we first consider the limit $r_{2}\rightarrow
1^{-}$. For the background piece, notice that $u_{2}\rightarrow 1^{-}$ when
$r_{2}\rightarrow 1^{-}$ and thus only $n=0$ terms survive due to the factor
$(1-u_{2})^{n}$. Therefore, the background can be written in closed form in
this limit:

$\displaystyle\lim_{r_{2}\rightarrow
1^{-}}{\mathcal{I}}^{-2,-2}_{\text{BG},>}(u_{1},u_{2})=$
$\displaystyle\sum_{m=0}^{\infty}\displaystyle\frac{\,{(1)_{m}^{2}}\,}{(\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}(\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{m+1}}u_{1}^{m+1}$
$\displaystyle=$ $\displaystyle~{}\pi\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)u_{1}\times{}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{2}\left[\begin{matrix}1,1,1\\\
\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,u_{1}\right],$ (177)

which converges for all $0\leq r_{1}<1$. This implies the divergences in the
homogeneous solution (the signal part) must be canceled with each other.
Indeed, the divergent parts for the homogeneous solutions are:

$\lim_{r\rightarrow
1^{-}}\mathcal{Y}_{\pm}(r)\sim\mp\mathrm{i}\sqrt{\displaystyle\frac{\,{\pi}\,}{2}}\text{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\log(1-r),$ (178)

and thus these divergences get canceled in the combination:

$\lim_{r\rightarrow
1^{-}}\Big{[}\mathcal{Y}_{+}(r)+\mathcal{Y}_{-}(r)\Big{]}=\sqrt{2\pi^{3}}\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (179)

Therefore, the signal converges in the folded limit for $0\leq r_{1}<1$:

$\lim_{r_{2}\rightarrow
1^{-}}{\mathcal{I}}^{-2,-2}_{\text{S},>}(r_{1},r_{2})=\sqrt{2\pi^{3}}\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\alpha_{++}\mathcal{Y}_{+}(r_{1})+\text{c.c.}.$ (180)

Now we go on to take the second folded limit $r_{1}\rightarrow 1^{-}$. This
time, both the background and the signal diverge, but the divergences are
canceled in the finally result:

$\displaystyle\lim_{r_{1}\rightarrow
1^{-}}\mathcal{I}^{-2,-2}_{\text{BG},>}(u_{1},1^{-})\sim$
$\displaystyle-\pi\,\text{sech}\log(1-u_{1})\sim-\pi\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\log(1-r_{1}),$ (181)
$\displaystyle\lim_{r_{1}\rightarrow
1^{-}}{\mathcal{I}}^{-2,-2}_{\text{S},>}(r_{1},1^{-})\sim$
$\displaystyle~{}\pi\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)\log(1-r_{1}).$ (182)

After the cancellation of divergences, we get a finite result in the double
folded limit $r_{1,2}\rightarrow 1^{-}$:

$\displaystyle{\mathcal{I}}^{-2,-2}_{>}(1^{-},1^{-})=$
$\displaystyle~{}2\pi\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\text{Re}\,\Big{[}\big{(}1+\mathrm{i}\,\text{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\big{)}\psi\big{(}\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{)}+\gamma\Big{]}$
$\displaystyle~{}+\mathcal{F}\left[\begin{matrix}1,1,\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
2,2,2\end{matrix}\middle|\,1\right],$ (183)

where $\psi(z)={\mathrm{d}}\log\Gamma(z)/{\mathrm{d}}z$ is the digamma
function. When supplemented with appropriate $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$-independent
prefactors (including the coupling and the momentum), this result is nothing
but the 2-point correlator of massless scalar fields $\varphi$ corrected by an
intermediate massive scalar $\sigma$ of mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ with coupling
$a^{3}\varphi^{\prime}\sigma$. This result has been derived originally in
[15]; See also [57]. Although our result has a different look from the
expressions in [15, 57], we have checked numerically that the two results
perfectly agree with each other.

Thus we have shown that the full correlator is regular in both the single
folded limit ($0\leq r_{1}<1$, $r_{2}\rightarrow 1^{-}$) and the double folded
limit ($r_{1,2}\rightarrow 1^{-}$), as expected.

###  5.2 Transverse vector

Next we consider the transverse components $(\lambda=h=\pm)$ of the vector
seed integral (43), following the same procedure as before. Again we do not
pursue the general case, but only focus on an example with $p_{1}=p_{2}=-1$.
Generalizations to arbitrary $p_{1,2}$ should be straightforward.

The goal here is to derive a differential equation for the following seed
integral:

$\displaystyle{\mathcal{I}}^{(h)-1,-1}_{\mathsf{ab}}(r_{1},r_{2})=$
$\displaystyle-\mathsf{ab}k_{s}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{\tau_{1}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{\tau_{2}}\,e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D^{(h)}_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2})$
$\displaystyle=$
$\displaystyle-\mathsf{ab}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}}\displaystyle\frac{\,{{\mathrm{d}}z_{2}}\,}{z_{2}}\,e^{\mathrm{i}\mathsf{a}z_{1}+\mathrm{i}\mathsf{b}z_{2}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu^{(h)}_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2}),$ (184)

where we have defined $z_{1}\equiv k_{12}\tau_{1}$, $z_{2}\equiv
k_{34}\tau_{2}$. In the second line, we have introduced the hatted propagator
for the transverse vector fields, which is again a function of $k_{s}\tau_{1}$
and $k_{s}\tau_{2}$:

$\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu^{(h)}_{\mathsf{ab}}(k_{s}\tau_{1},k_{s}\tau_{2})\equiv
k_{s}D^{(h)}_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2}).$ (185)

Then, the equation of motion for the transverse spin-1 fields (14) gives rise
to the following equations satisfied by the propagators,

$\displaystyle(\tau_{1}^{2}\partial_{\tau_{1}}^{2}+k_{s}^{2}\tau_{1}^{2}-2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0muk_{s}\tau_{1}+m^{2})D^{(h)}_{\pm\mp}(k_{s};\tau_{1},\tau_{2})=0,$ (186)
$\displaystyle(\tau_{1}^{2}\partial_{\tau_{1}}^{2}+k_{s}^{2}\tau_{1}^{2}-2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0muk_{s}\tau_{1}+m^{2})D^{(h)}_{\pm\pm}(k_{s};\tau_{1},\tau_{2})=\mp\mathrm{i}\tau_{1}\tau_{2}\delta(\tau_{1}-\tau_{2}),$
(187)

which further imply the following equations for the hatted propagators:

$\displaystyle(r_{1}^{2}\partial_{r_{1}}^{2}+r_{1}^{2}z_{1}^{2}-2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mur_{1}z_{1}+m^{2})\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\mp}(r_{1}z_{1},r_{2}z_{2})=0,$ (188)
$\displaystyle(r_{1}^{2}\partial_{r_{1}}^{2}+r_{1}^{2}z_{1}^{2}-2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mur_{1}z_{1}+m^{2})\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\pm\pm}(r_{1}z_{1},r_{2}z_{2})=\mp\mathrm{i}(r_{1}z_{1})(r_{2}z_{2})\delta(r_{1}z_{1}-r_{2}z_{2}).$
(189)

We can now again insert the above differential operators in front of the
hatted propagator in (5.2), using the following relations similar to (150) and
(151) to commute the differential operator with the integral:

$\displaystyle\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}}\,z_{1}e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2})=$
$\displaystyle~{}\mathrm{i}\mathsf{a}(r_{1}\partial_{r_{1}})\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}}\,e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2}),$ (190)
$\displaystyle\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}}\,z_{1}^{2}e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2})=$
$\displaystyle-(r_{1}\partial_{r_{1}}+1)(r_{1}\partial_{r_{1}})\int_{-\infty}^{0}{\mathrm{d}}z_{1}\,e^{\mathrm{i}\mathsf{a}z_{1}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2}),$ (191)

and we find the following equations satisfied by the integral:

$\displaystyle\bigg{[}(r_{1}^{2}-r_{1}^{4})\partial_{r_{1}}^{2}-(\pm
2\mathrm{i}h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mur_{1}^{2}+2r_{1}^{3})\partial_{r_{1}}+\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{1}\,}{4}\Big{)}\bigg{]}{\mathcal{I}}^{(h)-1,-1}_{\pm\mp}(r_{1},r_{2})=0,$
(192) $\displaystyle\bigg{[}(r_{1}^{2}-r_{1}^{4})\partial_{r_{1}}^{2}-(\pm
2\mathrm{i}h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mur_{1}^{2}+2r_{1}^{3})\partial_{r_{1}}+\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{1}\,}{4}\Big{)}\bigg{]}{\mathcal{I}}^{(h)-1,-1}_{\pm\pm}(r_{1},r_{2})=\displaystyle\frac{\,{r_{1}r_{2}}\,}{r_{1}+r_{2}}.$
(193)

It is possible to solve this set of equation directly, with a set of
homogeneous solutions and a particular solution to the inhomogeneous equation.
The resulting inhomogeneous solution has a double-series representation, with
the coefficients satisfying a set of recursion relations that can be easily
found. However, it turns out not trivial to find a general term formula from
this set of recursion relations. Thus it is not easy to write down the
explicit series solution in the same way as we did in (163). Fortunately, the
previous computation with partial MB representation in Sec. 4 has hinted at a
solution: We have learnt that the change of variables
$u_{1,2}=2r_{1,2}/(1+r_{1,2})$ as in (44) can further simplify the equations.
With the new variables $u_{1,2}$, we finally reach the following set of vector
bootstrap equations that are much easier to solve:

$\displaystyle\bigg{[}(u_{1}^{2}-u_{1}^{3})\partial_{u_{1}}^{2}-(1\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)u_{1}^{2}\partial_{u_{1}}+\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{1}\,}{4}\Big{)}\bigg{]}{\mathcal{I}}_{\pm\mp}^{(h)-1,-1}(u_{1},u_{2})=0,$
(194)
$\displaystyle\bigg{[}(u_{1}^{2}-u_{1}^{3})\partial_{u_{1}}^{2}-(1\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)u_{1}^{2}\partial_{u_{1}}+\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+\displaystyle\frac{\,{1}\,}{4}\Big{)}\bigg{]}{\mathcal{I}}_{\pm\pm}^{(h)-1,-1}(u_{1},u_{2})=\displaystyle\frac{\,{1}\,}{2}\displaystyle\frac{\,{u_{1}u_{2}}\,}{u_{1}+u_{2}-u_{1}u_{2}}.$
(195)

#### Particular solution to the inhomogeneous equation.

Now we look for a particular solution to the inhomogeneous equation (195). As
before, we first Taylor expand the right hand side of (195) as a series of
$u_{1}$, already given in (172), and consider the following ansatz for the
particular solution:

$\mathcal{V}_{\pm}^{(h)}(u_{1},u_{2})=\sum_{m,n=0}^{\infty}\mathcal{V}_{\pm;m,n}^{(h)}u_{1}^{m+n+1}\Big{(}1-\displaystyle\frac{\,{1}\,}{u_{2}}\Big{)}^{n},$
(196)

where the subscript in $\mathcal{V}_{\pm}^{(h)}(u_{1},u_{2})$ corresponds to
the two choices of the SK indices in ${\mathcal{I}}_{\pm\pm}^{(h)}$. (Unlike
the scalar equation, the inhomogeneous equations for the vector seed integral
are different for ${\mathcal{I}}_{++}^{(h)}$ and ${\mathcal{I}}_{--}^{(h)}$.)
Compare the coefficient for fixed $n$ and we find:

$\displaystyle\Big{[}\big{(}n+\mbox{$\frac{\,{1}\,}{2}$}\big{)}^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}\Big{]}\mathcal{V}_{\pm;0,n}^{(h)}=\displaystyle\frac{\,{1}\,}{2};$
(197a)
$\displaystyle\Big{[}\big{(}m+n+\mbox{$\frac{\,{3}\,}{2}$}\big{)}^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}\Big{]}\mathcal{V}_{\pm;m+1,n}^{(h)}=(m+n+1)(m+n+1\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)\mathcal{V}_{\pm;m,n}^{(h)}.$ (197b)

This recursion relation is easy to solve, with the solution given by:

$\displaystyle\mathcal{V}_{\pm;m,n}^{(h)}=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{2}\displaystyle\frac{\,{1}\,}{(n+\mbox{$\frac{\,{1}\,}{2}$})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}\displaystyle\frac{\,{(n+1)(n+1\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)}\,}{(n+\mbox{$\frac{\,{3}\,}{2}$})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}}\cdots\displaystyle\frac{\,{(n+m)(n+m\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)}\,}{(n+m+\mbox{$\frac{\,{1}\,}{2}$})^{2}+\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{(n+1)_{m}(n+1\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)_{m}}\,}{2(n+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}(n+\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{m+1}}.$ (198)

As in the scalar solution considered before, this particular solution is the
sole contribution to the background of the vector seed integral. Therefore,
the full result for the background is obtained by adding up
$\mathcal{V}_{+}^{(h)}$ and $\mathcal{V}_{-}^{(h)}$:

$\displaystyle{\mathcal{I}}_{\text{BG}}^{(h)-1,-1}(u_{1},u_{2})=$
$\displaystyle\sum_{m,n=0}^{\infty}\displaystyle\frac{\,{(-1)^{n}(n+1)_{m}(n+1\pm\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)_{m}}\,}{2(n+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}(n+\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}}u_{1}^{m+1}(1-u_{2})^{n}\Big{(}\displaystyle\frac{\,{u_{1}}\,}{u_{2}}\Big{)}^{n}+\text{c.c.}.$
(199)

In Fig. 7, we show the background series $\mathcal{I}_{\text{BG}}^{(h)-1,-1}$
in (199) together with the partial MB results (4.3). Again we see that both
methods yield convergent results with good agreements. Note that, for the
parameters chosen in the figure, the signals are much greater than the
background. Therefore we only show the background part of the vector seed
integral in this figure, since the purpose here is to show the convergence of
the background series.

Figure 7: The background part of the vector seed integrals
$\mathcal{I}_{\text{BG}}^{(h)-1,-1}(u_{1},u_{2})$ as functions of $u_{1}$ with
fixed $u_{2}=0.5$. We show the convergences of series from partial MB
representation (4.3) and from the bootstrap method (199), by including terms
of $m$ and $n$ summations up to values indicated in the plots.

#### Homogeneous solutions and limits.

Next we consider the homogeneous solution. Note that, unlike the previous
scalar case, the homogeneous and inhomogeneous equations are different for
$\mathcal{I}_{+\pm}^{(h)-1,-1}$ and $\mathcal{I}_{-\mp}^{(h)-1,-1}$. The
general solution to the upper-sign equation in (194) is the linear combination
of $\mathcal{U}_{+}(u_{1})$ and $\mathcal{U}_{-}(u_{1})$, while the general
solution to the lower-sign equation is the linear combination of
$\mathcal{U}^{*}_{+}(u_{1})$ and $\mathcal{U}^{*}_{-}(u_{1})$, where
$\mathcal{U}_{\pm}(u_{1})$ is given by:

$\mathcal{U}_{\pm}(u_{1})=u_{1}^{1/2\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathcal{F}\left[\begin{matrix}\mbox{$\frac{\,{1}\,}{2}$}\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\ 1\pm
2\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,u_{1}\right]=\mp\mathrm{i}\displaystyle\frac{\,{\sqrt{2}}\,}{\pi}\sinh(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)u_{1}^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{G}^{-1}_{h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(u_{1}).$ (200)

Here $\mathbf{G}_{h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}^{p}(u)$
is the function defined in (98). Note that we have expressed these solutions
in terms of the dressed hypergeometric function (277) for later convenience.
Furthermore, since amplitudes with different SK indices obey different
equations, we can separately determine the homogeneous solutions for each
${\mathcal{I}}_{\mathsf{ab}}^{(h)-1,-1}$ by comparing the leading terms in the
squeezed limit $r_{1,2}\rightarrow 0$ with $r_{1}\ll r_{2}$.

We first focus on the opposite-sign integrals
$\mathcal{I}^{(h)-1,-1}_{\pm\mp}$. On one hand,
$\mathcal{I}^{(h)-1,-1}_{+-}(u_{1},u_{2})$ satisfies the homogeneous equation
(194) and thus must be a linear combination of $\mathcal{U}_{+}(u_{1})$ and
$\mathcal{U}_{-}(u_{1})$. On the other hand, from (5.2) we notice that
$\mathcal{I}^{(h)-1,-1}_{+-}(u_{1},u_{2})=\mathcal{I}^{(h)-1,-1}_{-+}(u_{2},u_{1})$,
which also satisfies (194) but with the variable $u_{2}$. This implies that
$\mathcal{I}^{(h)-1,-1}(u_{1},u_{2})$ should also be a linear combination of
$\mathcal{U}_{+}^{*}(u_{2})$ and $\mathcal{U}_{-}^{*}(u_{2})$. Therefore, we
can write $\mathcal{I}^{(h)-1,-1}_{\pm\mp}$ as:

$\displaystyle{\mathcal{I}}^{(h)-1,-1}_{+-}(u_{1},u_{2})=$
$\displaystyle~{}\mathcal{U}_{+}(u_{1})\Big{[}\beta_{++}\mathcal{U}^{*}_{-}(u_{2})+\beta_{+-}\mathcal{U}^{*}_{+}(u_{2})\Big{]}$
$\displaystyle~{}+\mathcal{U}_{-}(u_{1})\Big{[}\beta_{-+}\mathcal{U}^{*}_{-}(u_{2})+\beta_{--}\mathcal{U}^{*}_{+}(u_{2})\Big{]},$
(201) $\displaystyle{\mathcal{I}}^{(h)-1,-1}_{-+}(u_{1},u_{2})=$
$\displaystyle~{}{\mathcal{I}}^{(h)-1,-1,*}_{+-}(u_{1},u_{2}),$ (202)

We then consider the same-sign integrals $\mathcal{I}^{(h)-1,-1}_{\pm\pm}$.
Similar to the scalar case, $\mathcal{I}^{(h)-1,-1}_{++}$ is the combination
of an inhomogeneous solution (corresponding to the background piece) to (195),
and a factorized homogeneous solution denoted by
${\mathcal{I}}^{(h)-1,-1}_{\pm\pm,\text{F},>}$. Furthermore, from (5.2) we can
see that
${\mathcal{I}}^{(h)-1,-1}_{++}(u_{1},u_{2})={\mathcal{I}}^{(h)-1,-1}_{++}(u_{2},u_{1})$,
which further implies that
${\mathcal{I}}^{(h)-1,-1}_{++,\text{F},>}(u_{1},u_{2})$ can be written as a
linear combination of $\mathcal{U}_{\pm}(u_{1})$, as well as a linear
combination of $\mathcal{U}_{\pm}(u_{2})$. Therefore we can write:

$\displaystyle{\mathcal{I}}^{(h)-1,-1}_{++,\text{F},>}(u_{1},u_{2})=$
$\displaystyle~{}\mathcal{U}_{+}(u_{1})\Big{[}\gamma_{++}\mathcal{U}_{+}(u_{2})+\gamma_{+-}\mathcal{U}_{-}(u_{2})\Big{]}$
$\displaystyle~{}+\mathcal{U}_{-}(u_{1})\Big{[}\gamma_{-+}\mathcal{U}_{+}(u_{2})+\gamma_{--}\mathcal{U}_{-}(u_{2})\Big{]},$
(203) $\displaystyle{\mathcal{I}}^{(h)-1,-1}_{--,\text{F},>}(u_{1},u_{2})=$
$\displaystyle~{}{\mathcal{I}}^{(h)-1,-1,*}_{++,\text{F},>}(u_{1},u_{2}).$
(204)

To determine the coefficients, we again consider the squeezed limit
$r_{1,2}\rightarrow 0$ with $r_{1}\ll r_{2}$ and compare the behaviors of
homogeneous solutions with the late-time approximation of the bulk integrals.
Again, in the squeezed limit, the background piece
${\mathcal{I}}^{(h)-1,-1}_{\text{BG},>}(u_{1},u_{2})\rightarrow~{}\mathcal{O}(u_{1})$
and thus can be neglected. Comparing homogeneous solutions to the bulk
integrals (See Sec. 4), we find:

$\displaystyle\beta_{++}=-\beta_{+-}=-\beta_{-+}=\beta_{--}=$
$\displaystyle-\displaystyle\frac{\,{e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{4}\text{csch}^{2}(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)(\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\cosh 2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu),$ (205)
$\displaystyle\gamma_{++}=-\gamma_{+-}=$
$\displaystyle~{}\displaystyle\frac{\,{-\mathrm{i}\pi e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}\text{csch}^{2}(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu]}e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu)],$ (206)
$\displaystyle\gamma_{--}=-\gamma_{-+}=$
$\displaystyle~{}\displaystyle\frac{\,{-\mathrm{i}\pi e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}\text{csch}^{2}(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}\,}{2\Gamma[\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu]}e^{-\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\cosh[\pi(-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu)],$ (207)

and the full amplitude is given by:

$\displaystyle{\mathcal{I}}_{>}^{(h)-1,-1}(u_{1},u_{2})=$
$\displaystyle~{}\Big{[}{\mathcal{I}}_{+-}^{(h)-1,-1}(u_{1},u_{2})+{\mathcal{I}}^{(h)-1,-1}_{++,\text{F},>}(u_{1},u_{2})+\mathcal{V}_{+}(u_{1},u_{2})\Big{]}+\text{c.c.}$
$\displaystyle=$
$\displaystyle~{}\bigg{\\{}\Big{[}\big{(}\gamma_{++}\mathcal{U}_{+}(u_{1})-\gamma_{--}\mathcal{U}_{-}(u_{1})\big{)}-\beta_{++}\big{(}\mathcal{U}^{*}_{+}(u_{1})-\mathcal{U}^{*}_{-}(u_{1})\big{)}\Big{]}\Big{(}\mathcal{U}_{+}(u_{2})-\mathcal{U}_{-}(r_{2})\Big{)}$
$\displaystyle+\mathcal{V}_{+}(u_{1},u_{2})\bigg{\\}}+\text{c.c.}.$ (208)

This completes the bootstrap of the transverse vector seed integral.

#### Folded limit.

Now we consider the folded limit $u_{1,2}\rightarrow 1^{-}$. First we take
$u_{2}\rightarrow 1^{-}$. The background piece is regular in this limit:

$\displaystyle\lim_{u_{2}\rightarrow 1^{-}}\mathcal{V}_{+}(u_{1},u_{2})=$
$\displaystyle~{}\sum_{m=0}^{\infty}\mathcal{V}_{+;m,0}^{(h)}u_{1}^{m+1}=\displaystyle\frac{\,{\pi}\,}{2}\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)u_{1}\times{}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathrm{F}\mkern-2.0mu}\mkern
2.0mu_{2}\left[\begin{matrix}1,1,1+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu\\\
\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,u_{1}\right].$ (209)

So, any divergences in the signals must cancel among themselves. The divergent
parts for the homogeneous solutions are:

$\displaystyle\lim_{u\rightarrow
1^{-}}\mathcal{U}_{\pm}(u)\sim-\displaystyle\frac{\,{\mathrm{i}\pi\text{csch}(h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)}\,}{\Gamma(1-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)}(1-u)^{-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu},$ (210)

we find that this divergence is canceled in the following combination:

$\displaystyle\lim_{u_{2}\rightarrow
1^{-}}\Big{[}\mathcal{U}_{+}(r_{2})-\mathcal{U}_{-}(r_{2})\Big{]}=\displaystyle\frac{\,{\mathrm{i}\pi^{2}\text{csch}(h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)\big{(}\text{sech}[\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)]-\text{sech}[\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)]\big{)}}\,}{\Gamma(1+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)\Gamma(\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Gamma(\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}.$ (211)

On the other hand, the signal part in the folded limit $u_{2}\rightarrow
1^{-}$ can be written as

$\displaystyle\lim_{u_{2}\rightarrow
1^{-}}{\mathcal{I}}^{(h)-1,-1}_{\text{S},>}(u_{1},u_{2})$ $\displaystyle=$
$\displaystyle~{}\Big{[}\big{(}\gamma_{++}\mathcal{U}_{+}(u_{1})-\gamma_{--}\mathcal{U}_{-}(u_{1})\big{)}-\beta_{++}\big{(}\mathcal{U}^{*}_{+}(u_{1})-\mathcal{U}^{*}_{-}(u_{1})\big{)}\Big{]}\lim_{u_{2}\rightarrow
1^{-}}\Big{[}\mathcal{U}_{+}(u_{2})-\mathcal{U}_{-}(r_{2})\Big{]}+\text{c.c.},$
(212)

which is therefore convergent, as expected. For later use, we also give an
explicit expression for the signal in $u_{2}\rightarrow 1^{-}$ limit:

$\displaystyle~{}{\mathcal{I}}^{(h)-1,-1}_{\text{S},>}(u_{1},1^{-})$
$\displaystyle=$ $\displaystyle~{}\displaystyle\frac{\,{\sinh(h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu)\sinh(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\text{csch}(2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}\,}{\sqrt{2}\pi}$
$\displaystyle\times\Big{[}(e^{-2\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}+e^{-2h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu})\displaystyle\frac{\,{\Gamma(\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)}\,}{\Gamma(\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)}-\mathrm{i}e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(1+e^{-2\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu+h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)})\displaystyle\frac{\,{\Gamma(-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)}\,}{\Gamma(\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}\Big{]}$
$\displaystyle\times u_{1}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\bigg{\\{}\Gamma(\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathbf{G}^{-1}_{-h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})+\mathrm{i}e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\Gamma(\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\mathbf{G}^{-1}_{+h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(u_{1})\bigg{\\}}+\text{c.c.}.$ (213)

We then consider the double folded limit, by further taking $u_{1}\rightarrow
1^{-}$. Like the case of scalar correlator considered before, both the
background and the signal diverge in this limit. The background diverges as

$\displaystyle\lim_{r_{1}\rightarrow
1^{-}}{\mathcal{I}}^{(h)-1,-1}_{\text{BG},>}(u_{1},1^{-})\sim-\displaystyle\frac{\,{\mathrm{i}\pi\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}\,}{2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}(1-u_{1})^{-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.},$ (214)

while the signal diverges as

$\displaystyle\lim_{u_{1}\rightarrow
1^{-}}{\mathcal{I}}^{(h)-1,-1}_{\text{S},>}(u_{1},1^{-})\sim$
$\displaystyle~{}(\gamma_{++}-\gamma_{-+})\displaystyle\frac{\,{-\mathrm{i}\pi\text{csch}(h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)}\,}{\Gamma(1-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu)}(1-u_{1})^{-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}\times\lim_{r_{2}\rightarrow
1^{-}}\Big{[}\mathcal{U}_{+}(u_{2})-\mathcal{U}_{-}(u_{2})\Big{]}+\text{c.c.}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{\mathrm{i}\pi\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}\,}{2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}(1-u_{1})^{-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu}+\text{c.c.},$ (215)

so the full correlator is regular in the double folded limit as well. After
the cancellation, the finite result in the double folded limit is:

$\displaystyle{\mathcal{I}}^{(h)-1,-1}_{>}(1^{-},1^{-})$ $\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{\pi\text{sech}^{2}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)}\,}{2h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}\bigg{\\{}\displaystyle\frac{\,{1}\,}{2}(1-e^{-2h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu})+\displaystyle\frac{\,{\mathrm{i}\cosh\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{\Gamma[\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu]}$
$\displaystyle\times\bigg{(}\pi e^{-h\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}\Gamma(-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu)\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)-\mathcal{F}\left[\begin{matrix}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}h\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
1-\mathrm{i}h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu,1-\mathrm{i}h\mkern 2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,1\right]\bigg{)}\bigg{\\}}+\text{c.c.}.$ (216)

The readers might have found it a bit unphysical to take the folded limit of
the transverse seed integral. After all, the transverse component of the
massive spin-1 exchange as shown in Fig. 1 does not survive the folded limit.
(For instance, the $r_{2}\rightarrow 1$ limit means that either
$\mathbf{k}_{s}$ is parallel to $\mathbf{k}_{4}$ or $\mathbf{k}_{4}\rightarrow
0$. In either case, the contraction
$\mathbf{k}_{4}\cdot\mathbf{e}_{\mathbf{k}_{s}}^{(\pm)^{*}}$ in (2.2)
vanishes.) However, there do exist cases where the folded limit of transverse
integrals does not vanish, if we consider tensor external states. We will show
such an example in Sec. 6, where the above folded limit can be directly used.

###  5.3 Longitudinal vector

Finally we consider the longitudinal component $(\lambda=L)$ of the vector
seed integral (43) with $p_{1}=p_{2}=-1$:

${\mathcal{I}}^{(L)-1,-1}_{\mathsf{ab}}(r_{1},r_{2})=-\mathsf{ab}\,k_{s}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{\tau_{1}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{\tau_{2}}\,e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}D^{(L)}_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2}).$
(217)

Similar to our computation with partial MB representation in Sec. 4, here we
can recycle the result for the scalar bootstrap equation. Note again that the
longitudinal propagator $D^{(L)}_{>}$ is related to the massive scalar
propagator $D_{>}$ by (115). For convenience, we define the operator:

$\mathcal{D}_{i}\equiv\partial_{\tau_{i}}-\displaystyle\frac{\,{2}\,}{\tau_{i}}.\qquad(i=1,2)$
(218)

Then, (115) implies the following relations for the longitudinal propagators
$D_{\mathsf{a}\mathsf{b}}^{(L)}$ and the scalar propagator
$D_{\mathsf{a}\mathsf{b}}$ of the same mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ (but not the same
mass):

$\displaystyle D_{\pm\mp}^{(L)}(k_{s};\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{m^{2}}\mathcal{D}_{1}\mathcal{D}_{2}D_{\pm\mp}(k_{s};\tau_{1},\tau_{2}),$
(219) $\displaystyle D_{\pm\pm}^{(L)}(k_{s};\tau_{1},\tau_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{m^{2}}\Big{[}\theta(\tau_{1}-\tau_{2})\mathcal{D}_{1}\mathcal{D}_{2}D_{\gtrless}(k_{s};\tau_{1},\tau_{2})+\theta(\tau_{2}-\tau_{1})\mathcal{D}_{1}\mathcal{D}_{2}D_{\lessgtr}(k_{s};\tau_{1},\tau_{2})\Big{]}$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{m^{2}}\Big{[}\mathcal{D}_{1}\mathcal{D}_{2}D_{\pm\pm}(k_{s};\tau_{1},\tau_{2})\mp\mathrm{i}\delta(\tau_{1}-\tau_{2})\Big{]}.$
(220)

Note in particular the appearance of a contact term
$\delta(\tau_{1}-\tau_{2})$ in the last line of (5.3), which is a consequence
of commuting the operator $\mathcal{D}_{1,2}$ with the Heaviside
$\theta$-functions in $D_{\pm\pm}$. See (in particular, App. B of) [57] for
more discussions about this contact term.

Using the relations (219) and (5.3), we can directly obtain the longitudinal
vector seed integral by acting differential operators on the scalar seed
${\mathcal{I}}_{\mathsf{a}\mathsf{b}}^{-2,-2}$ in (143). More explicitly,

$\displaystyle{\mathcal{I}}^{(L)-1,-1}_{\pm\mp}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{(r_{1}r_{2})^{-1}}\,}{m^{2}}(r_{1}\partial_{r_{1}}-2)(r_{2}\partial_{r_{2}}-2)\Big{(}r_{1}r_{2}{\mathcal{I}}_{\pm\mp}^{-2,-2}(r_{1},r_{2})\Big{)},$
(221) $\displaystyle{\mathcal{I}}^{(L)-1,-1}_{\pm\pm}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{(r_{1}r_{2})^{-1}}\,}{m^{2}}(r_{1}\partial_{r_{1}}-2)(r_{2}\partial_{r_{2}}-2)\Big{(}r_{1}r_{2}{\mathcal{I}}_{\pm\pm}^{-2,-2}(r_{1},r_{2})\Big{)}+\displaystyle\frac{\,{1}\,}{m^{2}}\displaystyle\frac{\,{{r_{1}r_{2}}}\,}{r_{1}+r_{2}}.$
(222)

We can rewrite (221) and (222) as:

$\displaystyle{\mathcal{I}}^{(L)-1,-1}_{\pm\mp}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}+1/4}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern
2.0mu_{2}{\mathcal{I}}_{\pm\mp}^{-2,-2}(r_{1},r_{2}),$ (223)
$\displaystyle{\mathcal{I}}^{(L)-1,-1}_{\pm\pm}(r_{1},r_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1/4}\bigg{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern
2.0mu_{2}{\mathcal{I}}_{\pm\pm}^{-2,-2}(r_{1},r_{2})+\displaystyle\frac{\,{r_{1}r_{2}}\,}{r_{1}+r_{2}}\bigg{]}.$
(224)

In these equations we have used the relation $m^{2}=\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}+1/4$, and
defined:

$\mkern 2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern
2.0mu_{i}\equiv r_{i}\partial_{r_{i}}-1.~{}~{}~{}~{}~{}(i=1,2)$ (225)

Summing over all SK indices, we can collectively write
${\mathcal{I}}_{>}^{(L)-1,-1}\equiv\sum_{\mathsf{a},\mathsf{b}}{\mathcal{I}}_{\mathsf{a}\mathsf{b}}^{(L)-1,-1}$
in terms of the scalar hatted integral
${\mathcal{I}}^{-2,-2}_{>}\equiv\sum_{\mathsf{a},\mathsf{b}}{\mathcal{I}}^{-2,-2}_{\mathsf{a}\mathsf{b}}$,
as:

${\mathcal{I}}^{(L)-1,-1}_{>}(r_{1},r_{2})=\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1/4}\bigg{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern
2.0mu_{2}{\mathcal{I}}^{-2,-2}_{>}(r_{1},r_{2})+\displaystyle\frac{\,{2r_{1}r_{2}}\,}{r_{1}+r_{2}}\bigg{]}.$
(226)

#### Intermediate steps.

Here we provide the intermediate steps leading to (221) and (222).
Uninterested readers can skip this part. The strategy is to compute the
following integral:

$\displaystyle-\mathsf{ab}\times\displaystyle\frac{\,{k_{s}}\,}{m^{2}}\times\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{\tau_{1}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{\tau_{2}}\,e^{\mathrm{i}\mathsf{a}k_{12}\tau_{1}+\mathrm{i}\mathsf{b}k_{34}\tau_{2}}\Big{(}\partial_{\tau_{1}}-\displaystyle\frac{\,{2}\,}{\tau_{1}}\Big{)}\Big{(}\partial_{\tau_{2}}-\displaystyle\frac{\,{2}\,}{\tau_{2}}\Big{)}D_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2})$
$\displaystyle=$
$\displaystyle-\mathsf{ab}\times\displaystyle\frac{\,{(r_{1}r_{2})^{-1}}\,}{m^{2}}\times\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}}\displaystyle\frac{\,{{\mathrm{d}}z_{2}}\,}{z_{2}}\,e^{\mathrm{i}\mathsf{a}z_{1}+\mathrm{i}\mathsf{b}z_{2}}\displaystyle\frac{\,{r_{1}\partial_{r_{1}}-2}\,}{z_{1}}\displaystyle\frac{\,{r_{2}\partial_{r_{2}}-2}\,}{z_{2}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2})$ $\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{m^{2}}\times(r_{1}r_{2})^{-1}\times(r_{1}\partial_{r_{1}}-2)(r_{2}\partial_{r_{2}}-2)\times(-\mathsf{ab})\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}z_{1}}\,}{z_{1}^{2}}\displaystyle\frac{\,{{\mathrm{d}}z_{2}}\,}{z_{2}^{2}}\,e^{\mathrm{i}\mathsf{a}z_{1}+\mathrm{i}\mathsf{b}z_{2}}\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2}),$ (227)

where $D_{\mathsf{ab}}(k_{s};\tau_{1},\tau_{2})$ in the first line denotes the
scalar propagator, and the hatted propagator $\mkern
2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern 2.0mu_{\mathsf{a}\mathsf{b}}$
is defined in (142). Similar to previous cases, we use the variables
$z_{1}=k_{12}\tau_{1}$, $z_{2}=k_{34}\tau_{2}$. In the second line we
substitute $z_{i}\partial_{z_{i}}$ with $r_{i}\partial_{r_{i}}$ acting on
$\mkern 2.0mu\widehat{\mkern-2.0muD\mkern-2.0mu}\mkern
2.0mu_{\mathsf{ab}}(r_{1}z_{1},r_{2}z_{2})$. Finally, the contact term for
$\mathcal{I}^{(L)-1,-1}_{\pm\pm}$ arising from the delta-function is:

$-\displaystyle\frac{\,{k_{s}}\,}{m^{2}}\int_{-\infty}^{0}\displaystyle\frac{\,{{\mathrm{d}}\tau_{1}}\,}{\tau_{1}}\displaystyle\frac{\,{{\mathrm{d}}\tau_{2}}\,}{\tau_{2}}\,\Big{[}\mp\mathrm{i}\tau_{1}\tau_{2}\delta(\tau_{1}-\tau_{2})\Big{]}e^{\pm\mathrm{i}k_{12}\tau_{1}\pm\mathrm{i}k_{34}\tau_{2}}=-\displaystyle\frac{\,{1}\,}{m^{2}}\displaystyle\frac{\,{r_{1}r_{2}}\,}{r_{1}+r_{2}}.$
(228)

Combining all results above, together with the definition of seed integrals,
we can directly get (221) and (222).

#### Solutions.

The above derivations show that the longitudinal correlator can be obtained by
acting the operator (225) on the scalar correlator given in (166), together
with an extra contact term which is regular when $r_{1,2}\rightarrow 0$ and
thus contributes to the background piece. Again, the longitudinal correlator
can be separated into a homogeneous part (the signals) and a inhomogeneous
particular solution (the background). We work them out in turn. Acting (225)
on the homogeneous solution solutions to the scalar integral, we find a pair
of solutions:

$\mathcal{W}_{\pm}(r_{1})=\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern
2.0mu_{1}\mathcal{Y}_{\pm}(r_{1})=-\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{H}^{-1}_{\pm\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(r_{1}),$ (229)

where $\mathbf{H}^{p}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}$ is defined in
(4.2). The homogeneous solution to the longitudinal bootstrap equation is thus
a linear combination of these two solutions with appropriate boundary
conditions:

$\mathcal{I}^{(L)-1,-1}_{\text{S},>}=\mathcal{W}_{+}(r_{1})\Big{[}\delta_{++}\mathcal{W}_{+}(r_{2})+\delta_{+-}\mathcal{W}_{-}(r_{2})\Big{]}+\text{c.c.}.$
(230)

Similar to the scalar bootstrap, we compare the solution in the squeezed limit
$r_{1,2}\rightarrow 0$ with the late-time limit of the bulk integral. This
determines the coefficients to be:

$\delta_{++}=\delta_{+-}=\displaystyle\frac{\,{1+\mathrm{i}\sinh\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}\,}{2\pi(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}+1/4)}.$ (231)

One can also obtain the same result by simply acting the operator $\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern 2.0mu_{2}/(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}+1/4)$ on (165).
For the inhomogeneous solution (the background), we apply the differential
operator to (163), and add the contact term:

$\displaystyle{\mathcal{I}}^{(L)-1,-1}_{\text{BG},>}(r_{1},r_{2})=\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1/4}\bigg{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern
2.0mu_{2}{\mathcal{I}}^{-2,-2}_{\text{BG},>}(r_{1},r_{2})+\displaystyle\frac{\,{2r_{1}r_{2}}\,}{r_{1}+r_{2}}\bigg{]}$
$\displaystyle=$ $\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1/4}\bigg{[}\sum_{m,n=0}^{\infty}\displaystyle\frac{\,{(-1)^{n+1}(n+1)_{2m}(2m+n)(n+1)}\,}{2^{2m+1}(\mbox{$\frac{\,{n}\,}{2}$}+\mbox{$\frac{\,{1}\,}{4}$}+\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$})_{m+1}(\mbox{$\frac{\,{n}\,}{2}$}+\mbox{$\frac{\,{1}\,}{4}$}-\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\,}{2}$})_{m+1}}r_{1}^{2m+1}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{n}+\displaystyle\frac{\,{2r_{1}r_{2}}\,}{r_{1}+r_{2}}\bigg{]}.$
(232)

Note that the last contact term is crucial for us to obtain the correct result
for the background. This completes the derivation of the longitudinal
correlator from the bootstrap method.

#### Folded limit.

Finally let us check the folded limit of the longitudinal correlator. For
simplicity, we again express the background piece in terms of $u_{1,2}$,
namely inserting (176) into the first line of (5.3):

$\displaystyle~{}{\mathcal{I}}^{(L)-1,-1}_{\text{BG},>}(u_{1},u_{2})$
$\displaystyle=$ $\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1/4}\bigg{[}\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}_{1}\mkern-2.0mu}\mkern 2.0mu\mkern
2.0mu\widetilde{\mkern-2.0mu\mathcal{D}\mkern-2.0mu}\mkern
2.0mu_{2}\sum_{m,n=0}^{\infty}\displaystyle\frac{\,{[(n+1)_{m}]^{2}}\,}{(n+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}(n+\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}}u_{1}^{m+n+1}\Big{(}1-\displaystyle\frac{\,{1}\,}{u_{2}}\Big{)}^{n}+\displaystyle\frac{\,{2r_{1}r_{2}}\,}{r_{1}+r_{2}}\bigg{]}$
$\displaystyle=$ $\displaystyle~{}\displaystyle\frac{\,{1}\,}{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1/4}\bigg{[}\sum_{m,n=0}^{\infty}\displaystyle\frac{\,{-(n+1)_{m}^{2}}\,}{(n+\mbox{$\frac{\,{1}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}(n+\mbox{$\frac{\,{1}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{m+1}}\displaystyle\frac{\,{(m+n-r_{1})}\,}{(1+r_{1})}\displaystyle\frac{\,{n+1-r_{2}}\,}{1-r_{2}}$
$\displaystyle\times
u_{1}^{m+n+1}\Big{(}1-\displaystyle\frac{\,{1}\,}{u_{2}}\Big{)}^{n}+\displaystyle\frac{\,{2r_{1}r_{2}}\,}{r_{1}+r_{2}}\bigg{]}.$
(233)

We first take $r_{2}\rightarrow 1^{-}$, then only terms with $n=0$ and $n=1$
survive, giving:

$\displaystyle\lim_{r_{2}\rightarrow
1^{-}}{\mathcal{I}}^{(L)-1,-1}_{\text{BG},>}(u_{1},u_{2})=$
$\displaystyle~{}\displaystyle\frac{\,{u_{1}}\,}{4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1}\bigg{\\{}4+u_{1}-\Big{(}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-\displaystyle\frac{\,{7}\,}{4}\Big{)}\pi\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)u_{1}$
$\displaystyle\times\bigg{(}\mathcal{F}\left[\begin{matrix}1,1,1\\\
\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,u_{1}\right]-(2-u_{1})\mathcal{F}\left[\begin{matrix}2,2,2\\\
\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\end{matrix}\middle|\,u_{1}\right]\bigg{)}\bigg{\\}},$ (234)

where $\mathcal{F}$ is the dressed (generalized) hypergeometric function
defined in (277). We find the background piece converge for all $0\leq
r_{1}<1$. On the other hand, the homogeneous solutions diverge:

$\displaystyle\lim_{r\rightarrow
1^{-}}\mathcal{W}_{\pm}(r_{1})\sim\pm\mathrm{i}\sqrt{\displaystyle\frac{\,{\pi}\,}{2}}\text{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{[}\displaystyle\frac{\,{1}\,}{1-r}-\displaystyle\frac{\,{4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-7}\,}{8}\log(1-r)\Big{]},$ (235)

but the combination:

$\lim_{r_{2}\rightarrow
1^{-}}\Big{[}\mathcal{W}_{+}(r_{2})+\mathcal{W}_{-}(r_{2})\Big{]}=\displaystyle\frac{\,{\pi^{3/2}}\,}{2^{5/2}}(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-7)\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$ (236)

converges. Therefore,

$\lim_{r_{2}\rightarrow
1^{-}}{\mathcal{I}}^{(L)-1,-1}_{\text{S},>}(r_{1},r_{2})=\displaystyle\frac{\,{\pi^{3/2}}\,}{2^{5/2}}(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-7)\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\delta_{++}\mathcal{W}_{+}(r_{1})+\text{c.c.},$ (237)

and thus the signal also converges for all $0\leq r_{1}<1$.

We then take the double folded limit by also taking $r_{1}\rightarrow 1^{-}$.
Again, both the background and the signal diverge, but the divergences are
canceled:

$\displaystyle\lim_{r_{1}\rightarrow
1^{-}}{\mathcal{I}}^{(L)-1,-1}_{\text{BG},>}(u_{1},1^{-})\sim$
$\displaystyle~{}\displaystyle\frac{\,{\pi(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}-7)}\,}{4(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1)}\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{[}\displaystyle\frac{\,{1}\,}{1-u_{1}}-\displaystyle\frac{\,{4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-7}\,}{4}\log(1-u_{1})\Big{]},$ (238) $\displaystyle\sim$
$\displaystyle~{}\displaystyle\frac{\,{\pi(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}-7)}\,}{2(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1)}\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{[}\displaystyle\frac{\,{1}\,}{1-r_{1}}-\displaystyle\frac{\,{4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-7}\,}{8}\log(1-r_{1})\Big{]},$ (239)
$\displaystyle{\mathcal{I}}^{(L)-1,-1}_{\text{S},>}(r_{1},1^{-})\sim$
$\displaystyle-\displaystyle\frac{\,{\pi(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}-7)}\,}{2(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1)}\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\Big{[}\displaystyle\frac{\,{1}\,}{1-r_{1}}-\displaystyle\frac{\,{4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-7}\,}{8}\log(1-r_{1})\Big{]},$ (240)

which shows that the longitudinal correlator is regular in the folded limit,
as expected. More explicitly,

$\displaystyle{\mathcal{I}}^{(L)-1,-1}_{>}(1^{-},1^{-})$ $\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{5}\,}{4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1}-\displaystyle\frac{\,{3}\,}{16}(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-7)\pi\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)$
$\displaystyle+\displaystyle\frac{\,{(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}-7)^{2}}\,}{8(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1)}\pi\,\text{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\,\text{Re}\,\Big{[}\big{(}1+\mathrm{i}\,\text{csch}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\big{)}\psi\big{(}\displaystyle\frac{\,{1}\,}{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\big{)}+\gamma\Big{]}$
$\displaystyle-\displaystyle\frac{\,{4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu^{2}-7}\,}{4(4\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu^{2}+1)}\bigg{\\{}\mathcal{F}\left[\begin{matrix}1,1,\mbox{$\frac{\,{3}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{3}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
2,2,2\end{matrix}\middle|\,1\right]-\mathcal{F}\left[\begin{matrix}1,1,\mbox{$\frac{\,{5}\,}{2}$}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu,\mbox{$\frac{\,{5}\,}{2}$}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\\\
3,3,3\end{matrix}\middle|\,1\right]\bigg{\\}}.$ (241)

Again, this result can be directly used to express the 2-point function
mediated by a massive longitudinal spin-1 field.

## 6 Variations: Higher Spins and Bispectra

In this work, we have been mainly focusing on the 4-point correlators with
spin-1 exchange. Nevertheless, it is straightforward to generalize our methods
and results to other cases, such as the chemical potential for higher spin
fields $(s\geq 2)$ and the 3-point function. As an example, in this section,
we shall briefly consider the 3-point function mediated by a massive spin-2
field with a helicity-dependent chemical potential. Note that the 3-point
functions with massive exchange such as in Fig. 3 and Fig. 8 are formally
obtained by taking $k_{4}\rightarrow 0$ in corresponding 4-point functions. In
this limit, $k_{s}=k_{3}$, and thus $r_{2}=u_{2}=1$. Thus the 3-point limit in
the seed integrals are formally equivalent to taking the folded limit
$r_{2}\rightarrow 1$. The way we take folded limits of 4-point functions in
Sec. 5 allows us to find closed analytical results for the 3-point functions
without any series expansion. Thus we also take this chance to present a
similar closed-form analytical expression for the well-studied scale-exchange
diagram (Fig. 3), which seems new to us.

#### Helical chemical potential for higher spins.

Before presenting the details, we first make two general remarks.

The first remark is about arbitrary nonzero spin. In principle, the chemical
potential reviewed in Sec. 2 has a natural generalization to massive fields of
any nonzero spin. Let us consider integer spins for simplicity, and we can
describe massive spin-$s$ states by a rank-$s$ symmetric tensor
$\sigma_{\mu_{1}\cdots\mu_{s}}$. A helicity-dependent chemical potential for
this field can thus be introduced from the following dimension-5 coupling
between $\sigma_{\mu_{1}\cdots\mu_{s}}$ and a background scalar field $\phi$
[88]:

$\mathcal{O}^{(s)}=\displaystyle\frac{\,{1}\,}{2\Lambda}\epsilon^{\mu\nu\rho\sigma}\phi\nabla_{\mu}\sigma_{\nu\lambda_{2}\cdots\lambda_{s}}\nabla_{\rho}\sigma_{\sigma}{}^{\lambda_{2}\cdots\lambda_{s}}.$
(242)

In particular, for a massive spin-2 field $\sigma_{\mu\nu}$, we have:

$\mathcal{O}^{(2)}=\displaystyle\frac{\,{1}\,}{2\Lambda}\epsilon^{\mu\nu\rho\sigma}\phi\nabla_{\mu}\sigma_{\nu\lambda}\nabla_{\rho}\sigma_{\sigma}{}^{\lambda}.$
(243)

Then, with the rolling background
$\langle\phi\rangle=\dot{\phi}_{0}t+\text{const.}$, these operators generate
terms proportional to the particle number operator weighted by the helicity.
Unlike the chemical potential operator for spin-1 field in (2), the above
operators break the gauge symmetry for the corresponding massless spin-$s$
fields. This does not place any obstacle, since the gauge symmetries are
absent for massive fields anyway. In fact, for the spin-2 case, one might
consider an alternative way of introducing the chemical potential via the
operator
$\phi\epsilon^{\mu\nu\rho\sigma}R_{\mu\nu\kappa\lambda}R_{\rho\sigma}{}^{\kappa\lambda}$
where $R_{\mu\nu\rho\sigma}$ is the Riemann tensor associated with
$\sigma_{\mu\nu}$ (but not with the spacetime metric $g_{\mu\nu}$). This
operator is indeed capable of generating a helicity-dependent chemical
potential for the spin-2 field in the perturbative regime, as has been
extensively studied in the literature. However, it is known that this operator
is pathologic in the UV [110], and our mechanism to boost the particle
production (and thus the inflation correlators) works exactly at or beyond the
UV cutoff scale of this operator. Therefore, we shall only consider (243). See
[88] for more discussions.

The second remark is about the 3-point function. In a 3-point function with a
massive spinning particle exchange at the tree level, it is necessary that the
intermediate massive particle is attached to the external inflaton fluctuation
through a two-point mixing. Then the angular momentum conservation at this
two-point vertex implies that only the longitudinal component of the spinning
particle can contribute nonzero results. On the other hand, the helicity-
dependent chemical potential only boosts transverse components of the massive
spinning particles, and does absolutely nothing to the longitudinal component.
Therefore, we do not expect to see the chemical potential boosted helical
correlators in tree-level 3-point functions. (It is possible to generate
chemical potential boosted 3-point function at one-loop as has been studied in
the literature [60, 74, 69, 70, 78].) However, there is a unique chance where
we can see the effect of the helical chemical potential in tree-level 3-point
functions. That is, we can consider a mixed correlator
$\langle\varphi\varphi\gamma\rangle$ among two external inflaton fluctuations
$\varphi$ and a tensor mode $\gamma$. Then, we can form a two-point mixing
between the massless tensor mode $\gamma_{\mu\nu}$ and the massive spin-2
field $\sigma_{\mu\nu}$. Due to the spin-2 nature of the external tensor mode,
this two-point mixing will pick out the helicity-2 component of
$\sigma_{\mu\nu}$ rather than the longitudinal component. Therefore, there is
a chance that this mixed operator is boosted by the chemical potential. This
example has been considered in [88], in which the signal part of the mixed
3-point function has been calculated. With our results from previous sections,
we can now easily derive the full analytical expression for this process.

Figure 8: The helical 3-point correlator (248). The straight lines represent
external inflaton modes, the black doubly wiggly line represents the massless
tensor mode, and the blue doubly wiggly line represents the massive spin-2
field with helical chemical potential.

In this section, we shall focus only on the helicity-2 states of the massive
spin-2 field, which is the dominant contribution to the inflation correlators
when there is a nonzero chemical potential. For the massive spin-2 field
$\sigma_{\mu\nu}$ of mass $m$ and with a chemical potential term (243), the
mode equation corresponding to the helicity-2 component $\Sigma^{(\pm
2)}(k,\tau)$ reads:

$\Sigma^{(\pm 2)}{}^{\prime\prime}-2a\Sigma^{(\pm
2)}{}^{\prime}+\Big{[}k^{2}\pm 2a\mu k+a^{2}(m^{2}-2)\Big{]}\Sigma^{(\pm
2)}=0,$ (244)

where $\mu\equiv\dot{\phi}_{0}/\Lambda$ is the chemical potential. From this
equation, we find the properly normalized mode function as (Our convention for
normalizing the mode function and the polarization tensors is in accordance
with [88].)

$\Sigma^{(\pm 2)}(k,\tau)=-\displaystyle\frac{\,{e^{\mp\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu/2}}\,}{2\sqrt{k}\tau}\mathrm{W}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu}(2\mathrm{i}k\tau),$
(245)

where $\mathrm{W}_{\mu,\nu}(z)$ is again the Whittaker W function, and $\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu\equiv\mu/H$ and
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\equiv\sqrt{m^{2}/H^{2}-9/4}$. (Here we temporarily restore the Hubble
parameter $H$. Note that the definition of the mass parameter $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu$ differs from the
rest of the paper, due to the spin of the particle. In general, for a spin-$s$
state with $s\neq 0$, the mass parameter is defined by $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\equiv\sqrt{m^{2}/H^{2}-(s-1/2)^{2}}$, and for $s=0$, it is defined by
$\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu\equiv\sqrt{m^{2}/H^{2}-9/4}$.)

With the above mode function, we can construct the transverse spin-2
propagator as in previous sections:

$D^{(\pm
2)}_{>}(k;\tau_{1},\tau_{2})=\displaystyle\frac{\,{1}\,}{2\tau_{1}\tau_{2}}\displaystyle\frac{\,{e^{\mp\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern
2.0mu}}\,}{2k}\mathrm{W}_{\pm\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}(2\mathrm{i}k\tau_{1})\mathrm{W}_{\mp\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\mu\mkern-2.0mu}\mkern 2.0mu,-\mathrm{i}\mkern