2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\,}{2},s_{4}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}\,}{2}\Big{]}.$
(101)

The assignment of the four Mellin variables $s_{1},\cdots,s_{4}$ are shown in
Fig. 8.

Figure 8: The Mellin variables for the two massive propagators in the
computation of the three-vertex seed integral.

Then, the time integrals can be directly done for all branches using results
of the previous section. It then remains to finish the integrals over the four
Mellin variables $s_{1},\cdots,s_{4}$. We work in the region where the two
bulk momenta $\bm{\ell}_{1}$ and $\bm{\ell}_{2}$ are both softer than all
energies $E_{1}$, $E_{2}$, and $E_{3}$. In this region, we have $0<r_{i}<1$
($i=1,2,3,4$), which means that we should pick up all left poles to finish the
Mellin integrals.777Slightly stronger condition than $0<r_{i}<1$ may be
required for the convergence of the final result, but we expect that the
radius of convergence for the final result should be $\mathcal{O}({1})$. We
will not be concerned with the precise value of the radius of convergence.

As already explained in Sec. 2, the left poles of the Mellin integrand are all
from the $\Gamma$ factors in (4.2) and (4.2), and are given by:

$\displaystyle
s_{1}=-n_{1}-\displaystyle\frac{\,{\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2},$
$\displaystyle
s_{2}=-n_{2}-\displaystyle\frac{\,{\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2},$
$\displaystyle
s_{3}=-n_{3}-\displaystyle\frac{\,{\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}\,}{2},$
$\displaystyle
s_{4}=-n_{4}-\displaystyle\frac{\,{\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}\,}{2}.$ (102)

Here $n_{i}=0,1,2,\cdots$ $(i=1,\cdots,4)$, and $\mathsf{a}_{i}=\pm$ are not
SK indices. Thus, by collecting the residues of the integrand at all these
poles, we get the final answer for the three-vertex seed integral in (4.1).
Similar to the case of single-exchange graph studied in previous works, it
turns out convenient to express the final answer of
$\mathcal{I}_{\mathsf{a}\mathsf{b}\mathsf{c}}^{p_{1}p_{2}p_{3}}$ with all
indices $\mathsf{a},\mathsf{b},\mathsf{c}$ summed. Then, the result can be
written as a sum of four distinct terms, plus trivial permutations:

$\displaystyle\sum_{\mathsf{a},\mathsf{b},\mathsf{c}=\pm}\mathcal{I}^{p_{1}p_{2}p_{3}}_{\mathsf{a}\mathsf{b}\mathsf{c}}\big{(}\\{r_{i}\\}\big{)}=$
$\displaystyle~{}\bigg{\\{}\Big{[}\mathcal{I}_{\text{SS}}\big{(}\\{r_{i}\\}\big{)}+\mathcal{I}_{\text{SB}}\big{(}\\{r_{i}\\}\big{)}+\mathcal{I}_{\text{BS}}\big{(}\\{r_{i}\\}\big{)}+\mathcal{I}_{\text{BB}}\big{(}\\{r_{i}\\}\big{)}\Big{]}$
$\displaystyle+(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\rightarrow-\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})+(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}\rightarrow-\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})+\begin{pmatrix}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\\\ \mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}\end{pmatrix}\bigg{\\}}+\text{c.c.}.$ (103)

Here we use the shorthand notation
$\mathcal{I}^{p_{1}p_{2}p_{3}}_{\mathsf{a}\mathsf{b}\mathsf{c}}(\\{r_{i}\\})\equiv\mathcal{I}^{p_{1}p_{2}p_{3}}_{\mathsf{a}\mathsf{b}\mathsf{c}}(r_{1},r_{2},r_{3},r_{4})$.
Also, we are using the terminology of CC physics: The term
$\mathcal{I}_{\text{SS}}$, with the subscript SS denoting “signal-signal,” is
nonanalytic in all four momentum ratios $r_{1},\cdots,r_{4}$ when these ratios
go to 0, and thus corresponds to the CC signals generated by both bulk lines:

$\displaystyle\mathcal{I}_{\text{SS}}=$
$\displaystyle\sum_{\mathsf{a}_{1},\mathsf{a}_{2}=\pm}\mathbf{A}^{p_{1}p_{2}p_{3}}_{\mathsf{a}_{1}\mathsf{a}_{2}}(r_{1},r_{2},r_{3},r_{4})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}},$ (104)

where the function
$\mathbf{A}_{\mathsf{a}_{1}\mathsf{a}_{2}}^{p_{1}p_{2}p_{3}}(r_{1},r_{2},r_{3},r_{4})$
is fully analytic in the limit $r_{i}\rightarrow 0$ ($i=1,\cdots,4$), whose
explicit expression will be given below in (4.2).

Next, the term $\mathcal{I}_{\text{SB}}$ denotes the “signal-background,”
which contains CC signals from the $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}$-leg, and thus is
nonanalytic in both $r_{1}$ and $r_{2}$ as $r_{1,2}\rightarrow 0$, but is
fully analytic888There could be factors such as $(r_{3}/r_{4})^{p_{3}}$ which
we do not count as nonanalytic. in both $r_{3}$ and $r_{4}$ as
$r_{3,4}\rightarrow 0$. Likewise, the term $\mathcal{I}_{\text{SB}}$, denoting
the “background-signal,” is a trivial permutation of
$\mathcal{I}_{\text{SB}}$:

$\displaystyle\mathcal{I}_{\text{SB}}=$
$\displaystyle\sum_{\mathsf{a}=\pm}\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|\mathsf{a}}^{p_{1}p_{2}p_{3}}(r_{1},r_{2},r_{3},r_{4})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}},$ (105)
$\displaystyle\mathcal{I}_{\text{BS}}=$
$\displaystyle\sum_{\mathsf{a}=\pm}\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}|\mathsf{a}}^{p_{1}p_{2}p_{3}}(r_{4},r_{3},r_{2},r_{1})\Big{(}\displaystyle\frac{\,{r_{4}}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}.$ (106)

Again, the function $\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|\mathsf{a}}^{p_{1}p_{2}p_{3}}(r_{1},r_{2},r_{3},r_{4})$ is fully
analytic in all $r_{i}$ as $r_{i}\rightarrow 0$ ($i=1,\cdots,4$), whose
explicit expression will be given below in (4.2).

Finally, the term $\mathcal{I}_{\text{BB}}$ denotes the “background-
background,” and is fully analytic in all momentum ratios $r_{i}$ as
$r_{i}\rightarrow 0$.

$\displaystyle\mathcal{I}_{\text{BB}}=$
$\displaystyle~{}\displaystyle\frac{\,{\sin(\mathrm{i}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})\sin(\mathrm{i}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})e^{-\mathrm{i}p_{123}\pi/2}}\,}{4\pi^{2}}\sum_{n_{1},\cdots,n_{4}=0}^{\infty}\displaystyle\frac{\,{r_{2}^{3}r_{3}^{3}}\,}{n_{1}!n_{2}!n_{3}!n_{4}!}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\Big{)}^{p_{1}+1}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\Big{)}^{p_{3}+1}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{2n_{12}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{34}}$
$\displaystyle\times\Gamma\Big{[}-n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{4}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\Big{]}$
$\displaystyle\times\mathcal{F}_{2}\left[p_{123}+2n_{1234}+9\middle|\begin{matrix}p_{1}+2n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{5}\,}{2}$},p_{3}+2n_{4}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{5}\,}{2}$}\\\ p_{1}+2n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{7}\,}{2}$},p_{3}+2n_{4}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{7}\,}{2}$}\end{matrix}\middle|-\displaystyle\frac{\,{r_{2}}\,}{r_{1}},-\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\right],$
(107)

where the summation is taken over $n_{1},\cdots,n_{4}=0,1,\cdots$, and
$\mathcal{F}_{2}$ denotes the dressed Appell $F_{2}$ function, which is
defined in (141).

Now we give explicit expressions for the functions
$\mathbf{A}_{\mathsf{a}_{1}\mathsf{a}_{2}}^{p_{1}p_{2}p_{3}}$ and
$\mathbf{B}_{\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|\mathsf{a}}^{p_{1}p_{2}p_{3}}$. The function
$\mathbf{A}_{\mathsf{a}_{1}\mathsf{a}_{2}}^{p_{1}p_{2}p_{3}}$ is given by:

$\displaystyle\mathbf{A}_{\mathsf{a}_{1}\mathsf{a}_{2}}^{p_{1}p_{2}p_{3}}(r_{1},r_{2},r_{3},r_{4})\equiv$
$\displaystyle~{}\mathcal{C}^{p_{1}p_{2}p_{3}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|\mathsf{a}_{1}\mathsf{a}_{2}}(r_{1}r_{2}r_{3}r_{4})^{\frac{3}{2}}\mathbf{F}(p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{5}\,}{2}$},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1};r_{1}^{2})\mathbf{F}(p_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{5}\,}{2}$},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2};r_{4}^{2})$
$\displaystyle\times\mathcal{F}_{4}\left[\begin{matrix}\mbox{$\frac{\,{\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+p_{2}+4}\,}{2}$},\mbox{$\frac{\,{\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+p_{2}+5}\,}{2}$}\\\ 1+\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},1+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}\end{matrix}\middle|r_{2}^{2},r_{3}^{2}\right].$ (108)

Here the coefficient $\mathcal{C}^{p_{1}p_{2}p_{3}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|\mathsf{a}_{1}\mathsf{a}_{2}}$ is defined by:

$\displaystyle\mathcal{C}^{p_{1}p_{2}p_{3}}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|\mathsf{a}_{1}\mathsf{a}_{2}}\equiv\displaystyle\frac{\,{2^{p_{2}-1+\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}}\,}{\pi^{1/2}\operatorname{sin}(\mathrm{i}\mathsf{a}_{1}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})\operatorname{sin}(\mathrm{i}\mathsf{a}_{2}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2})}\Big{[}$
$\displaystyle-e^{-\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+p_{123}/2)}+\mathrm{i}e^{-\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+p_{12}/2-p_{3}/2)}$
$\displaystyle+\mathrm{i}e^{-\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}-p_{1}/2+p_{23}/2)}+e^{\mathrm{i}\pi(-p_{2}/2+p_{13}/2)}\Big{]}.$
(109)

The function $\mathbf{F}(a,b;z)$ in (4.2) is defined in terms of the Gaussian
hypergeometric function, given in (140), and the function $\mathcal{F}_{4}$
denotes the dressed Appell $F_{4}$ function, which is defined in (142).
Finally, the function $\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|\mathsf{a}}^{p_{1}p_{2}p_{3}}$ is given by:

$\displaystyle\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|\mathsf{a}}^{p_{1}p_{2}p_{3}}$
$\displaystyle\,(r_{1},r_{2},r_{3},r_{4})\equiv\displaystyle\frac{\,{e^{-\mathrm{i}\pi(p_{23}+\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-1/2)/2}}\,}{4\pi^{2}}\sin\big{[}\mbox{$\frac{\,{\pi}\,}{2}$}(\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+p_{1}-\mbox{$\frac{\,{3}\,}{2}$})\big{]}\sin(\mathrm{i}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})(r_{1}r_{2}r_{3}^{2})^{\frac{3}{2}}$
$\displaystyle\times\sum_{n_{1},n_{2},n_{3}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{123}}}\,}{n_{1}!n_{2}!n_{3}!}\displaystyle\frac{\,{\Gamma[-n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}]}\,}{p_{3}+n_{3}+2n_{2}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{5}\,}{2}$}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{12}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\Big{)}^{n_{3}+p_{3}+1}$
$\displaystyle\times\mathbf{F}(p_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{5}\,}{2}$},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1};r_{1}^{2})\mathbf{F}(n_{3}+2n_{12}+p_{23}+\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{13}\,}{2}$},-\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1};r_{2}^{2}).$
(110)

We can make a finer classification of terms with signals in (4.2), according
to whether the signal is local or nonlocal. Once again, the nonlocal signal
means a piece in the correlator which is nonanalytic in the bulk momentum
$\ell_{1}$ or $\ell_{2}$ as $\ell_{1,2}\rightarrow 0$. Using our momentum
ratios, this means that a nonlocal signal contains noninteger powers of either
$r_{1}r_{2}$ or $r_{3}r_{4}$. On the other hand, a local signal means a piece
analytic as $\ell_{1,2}\rightarrow 0$ but nonanalytic in other momentum
ratios. In our case, local signals come from noninteger powers of
$r_{1}/r_{2}$ or $r_{3}/r_{4}$. Thus, we can further write:

$\displaystyle\mathcal{I}_{\text{SS}}=$
$\displaystyle~{}\mathcal{I}_{\text{LL}}+\mathcal{I}_{\text{LN}}+\mathcal{I}_{\text{NL}}+\mathcal{I}_{\text{NN}};$
(111) $\displaystyle\mathcal{I}_{\text{SB}}=$
$\displaystyle~{}\mathcal{I}_{\text{LB}}+\mathcal{I}_{\text{NB}};$ (112)
$\displaystyle\mathcal{I}_{\text{BS}}=$
$\displaystyle~{}\mathcal{I}_{\text{BL}}+\mathcal{I}_{\text{BN}}.$ (113)

Then, explicitly, we have:

$\displaystyle\mathcal{I}_{\text{LL}}=$
$\displaystyle~{}\mathbf{A}^{p_{1}p_{2}p_{3}}_{--}(r_{1},r_{2},r_{3},r_{4})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{r_{3}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}},$ (114)
$\displaystyle\mathcal{I}_{\text{LN}}=$
$\displaystyle~{}\mathbf{A}^{p_{1}p_{2}p_{3}}_{-+}(r_{1},r_{2},r_{3},r_{4})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{3}r_{4}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}},$ (115)
$\displaystyle\mathcal{I}_{\text{NL}}=$
$\displaystyle~{}\mathbf{A}^{p_{1}p_{2}p_{3}}_{+-}(r_{1},r_{2},r_{3},r_{4})\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{r_{3}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}},$ (116)
$\displaystyle\mathcal{I}_{\text{NN}}=$
$\displaystyle~{}\mathbf{A}^{p_{1}p_{2}p_{3}}_{++}(r_{1},r_{2},r_{3},r_{4})\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{3}r_{4}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}},$ (117)
$\displaystyle\mathcal{I}_{\text{LB}}=$ $\displaystyle~{}\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|-}^{p_{1}p_{2}p_{3}}(r_{1},r_{2},r_{3},r_{4})\Big{(}\displaystyle\frac{\,{r_{1}}\,}{r_{2}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}},$ (118)
$\displaystyle\mathcal{I}_{\text{NB}}=$ $\displaystyle~{}\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}|+}^{p_{1}p_{2}p_{3}}(r_{1},r_{2},r_{3},r_{4})\Big{(}\displaystyle\frac{\,{r_{1}r_{2}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}},$ (119)
$\displaystyle\mathcal{I}_{\text{BL}}=$ $\displaystyle~{}\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}|-}^{p_{1}p_{2}p_{3}}(r_{4},r_{3},r_{2},r_{1})\Big{(}\displaystyle\frac{\,{r_{4}}\,}{r_{3}}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}},$ (120)
$\displaystyle\mathcal{I}_{\text{BN}}=$ $\displaystyle~{}\mathbf{B}_{\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}|+}^{p_{1}p_{2}p_{3}}(r_{4},r_{3},r_{2},r_{1})\Big{(}\displaystyle\frac{\,{r_{3}r_{4}}\,}{4}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}.$ (121)

We have checked numerically that our analytical result (4.2) agrees well with
a direct numerical integration of (4.1).

## 5 Four-Point Correlator with Two Massive Exchanges

Figure 9: The tree-level 4-point inflaton correlator with two massive
exchanges.

Now, we consider a special case of the general two massive exchanges from the
last section. We will compute a four-point graph in Fig. 9, in which we have
four external massless scalar $\varphi$ derivatively coupled to two principal
massive scalars in the bulk. The Lagrangian is given in (88). We use this
example to show how to take folded limits in the three-vertex seed integral.
The correlator corresponding to Fig. 9 has been given in (89), which shows
that all we need to do is to take the following folded limit of the three-
vertex seed integral (4.2):

$\lim_{r_{1},r_{4}\rightarrow
1}\mathcal{I}_{\mathsf{a}_{1}\mathsf{a}_{2}\mathsf{a}_{3}}^{-2,0,-2}(r_{1},r_{2},r_{3},r_{4}).$
(122)

In the folded limit $r_{1}\rightarrow 1$ and $r_{4}\rightarrow 1$, we expect
that various individual terms in (4.2) diverge, but the divergence must cancel
out in the full result, as a consequence of choosing the Bunch-Davies initial
condition.999Choosing the Bunch-Davies initial condition is implicit in our
choice for all the mode functions.

Specifically, all the hypergeometric functions in the $\mathbf{F}$ factors in
both (4.2) and (4.2) could develop divergent terms when we take their
arguments to 1. With the knowledge that these divergent terms must cancel
among themselves, we can directly throw them away when evaluating the function
$\mathbf{F}(a,b;z)$ at $z=1$. Using the expansion of hypergeometric function
at argument unity, we get the following finite result for $\mathbf{F}(a,b;z)$
as $z\rightarrow 1$ [91]:

$\displaystyle\operatorname{Fin}\left\\{\lim_{z\rightarrow
1}\mathbf{F}(a,b;z)\right\\}=\Gamma\left[\begin{matrix}a,b,1-b,1/2-a-b\\\
1-b-a/2,1/2-b-a/2\end{matrix}\right],~{}~{}~{}~{}~{}~{}(1/2-a-b\notin\mathbb{Z}),$
(123)

where $\operatorname{Fin}\\{\\}$ means the finite part of the expression
within. The case of $1/2-a-b\in\mathbb{Z}$ can be computed by taking the
limit. For example, the limit of a term in $\mathcal{I}^{\text{FF}}$ when
$p\rightarrow-2$ can be computed as:

$\displaystyle\operatorname{Fin}\Bigg{\\{}\lim_{\begin{subarray}{c}p\rightarrow-2\\\
r\rightarrow 1\end{subarray}}$
$\displaystyle\bigg{[}\Big{(}\displaystyle\frac{\,{r}\,}{2}\Big{)}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu}\mathbf{F}(p+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu+\mbox{$\frac{\,{5}\,}{2}$},-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu;r^{2})+(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu\rightarrow-\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)\bigg{]}\Bigg{\\}}=\sqrt{2}\pi^{3/2}\operatorname{sech}(\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu).$ (124)

With all limits of all $\mathbf{F}$ factors properly taken as above, we find a
finite result for (89), which can again be separated into several pieces
according their analytical properties at $k_{1}\rightarrow 0$ or
$k_{2}\rightarrow 0$:

$\displaystyle\sum_{\mathsf{a}_{1},\mathsf{a}_{2},\mathsf{a}_{3}=\pm}\mathcal{I}_{\mathsf{a}_{1}\mathsf{a}_{2}\mathsf{a}_{3}}^{-2,-2,-2}\big{(}1,r_{2},r_{3},1\big{)}$
$\displaystyle=$
$\displaystyle~{}\Big{\\{}\Big{[}\mathcal{T}_{\text{SS}}(r_{2},r_{3})+\mathcal{T}_{\text{SB}}(r_{2},r_{3})+\mathcal{T}_{\text{BS}}(r_{2},r_{3})+\mathcal{T}_{\text{SS}}(r_{2},r_{3})\Big{]}$
$\displaystyle+(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\rightarrow-\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})+(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}\rightarrow-\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})+(\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\rightarrow-\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}\rightarrow-\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})\Big{\\}}+\text{c.c.}.$ (125)

Here $r_{2}\equiv k_{1}/k_{34}$ and $r_{3}\equiv k_{2}/k_{34}$. The four
pieces
$\\{\mathcal{T}_{\text{SS}},\mathcal{T}_{\text{SB}},\mathcal{T}_{\text{BS}},\mathcal{T}_{\text{BB}}\\}$
are defined in a similar way as before, according to whether the expression is
analytic in $r_{2}\rightarrow 0$ or $r_{3}\rightarrow 0$ limit. In particular,
the signal-signal piece is nonanalytic when $r_{2}\rightarrow 0$ and when
$r_{3}\rightarrow 0$, whose explicit expression is:

$\displaystyle\mathcal{T}_{\text{SS}}(r_{2},r_{3})=$
$\displaystyle~{}-\displaystyle\frac{\,{4\pi^{5/2}(e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}-\mathrm{i})(e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}-\mathrm{i})}\,}{\sin(2\pi\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})\sin(2\pi\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}r_{2}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+3/2}r_{3}^{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+3/2}\mathcal{F}_{4}\left[\begin{matrix}\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+4}\,}{2}$},\mbox{$\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}+5}\,}{2}$}\\\
1+\mathrm{i}\mkern 2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},1+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}\end{matrix}\middle|r_{2}^{2},r_{3}^{2}\right],$ (126)

where $\mathcal{F}_{4}$ is again the dressed Appell $F_{4}$ function defined
in (142). Next, the signal-background piece $\mathcal{T}_{\text{SB}}$ is
nonanalytic in $r_{2}\rightarrow 0$ but analytic in $r_{3}\rightarrow 0$:

$\displaystyle\mathcal{T}_{\text{SB}}(r_{2},r_{3})=$
$\displaystyle~{}\displaystyle\frac{\,{\sin(\mathrm{i}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}\,}{\sqrt{\pi}(e^{\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}-\mathrm{i})}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{3/2-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\sum_{n_{1},n_{2},n_{3}=1}^{\infty}\displaystyle\frac{\,{(-1)^{n_{123}}}\,}{n_{1}!n_{2}!n_{3}!}r_{3}^{2+n_{3}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{12}}$
$\displaystyle\times\displaystyle\frac{\,{\Gamma[-n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}]}\,}{n_{3}+2n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{1}\,}{2}$}}\mathbf{F}(n_{3}+2n_{12}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{9}\,}{2}$},\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1};r_{2}^{2}).$
(127)

The background-signal piece $\mathcal{T}_{\text{BS}}$ is obtained from
$\mathcal{T}_{\text{SB}}$ by switching $\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}\leftrightarrow\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$ as well as
$r_{2}\leftrightarrow r_{3}$. Finally, the background-background piece is
analytic in both $r_{2}\rightarrow 0$ and $r_{3}\rightarrow 0$ limit, and its
expression is:

$\displaystyle\mathcal{T}_{\text{BB}}(r_{2},r_{3})=$
$\displaystyle~{}\displaystyle\frac{\,{\sin(\mathrm{i}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})\sin(\mathrm{i}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})r_{2}^{2}r_{3}^{2}}\,}{4\pi^{2}}\sum_{n_{1},\cdots,n_{4}=0}^{\infty}\displaystyle\frac{\,{(-1)^{n_{1234}}}\,}{n_{1}!n_{2}!n_{3}!n_{4}!}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{2n_{12}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{34}}$
$\displaystyle\times\Gamma\Big{[}-n_{1}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{2}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{3}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{4}-\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\Big{]}$
$\displaystyle\times\mathcal{F}_{2}\left[2n_{1234}+5\middle|\begin{matrix}2n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{1}\,}{2}$},2n_{4}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{1}\,}{2}$}\\\ 2n_{1}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{3}\,}{2}$},2n_{4}+\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{3}\,}{2}$}\end{matrix}\middle|-r_{2},-r_{3}\right].$
(128)

Again, we have checked that our analytical result (5) for the 4-point graph in
Fig. 9 agrees well with a direct numerical integration.

## 6 Conclusion and Outlooks

Inflation correlators are important theoretical data for QFTs in dS spacetime,
and are promising targets for current and future cosmological observations.
Inflation correlators mediated by massive fields are central objects in
Cosmological Collider physics. Thus, the analytical computation of inflation
correlators deserves a systematic investigation.

Very often in weakly coupled theories, inflation correlators are dominated by
tree-level exchanges. However, analytical computation of general tree graphs
remains challenging in dS, due to the multi-layer integrals with time
orderings in Schwinger-Keldysh formalism. In previous works, it has been shown
that the partial Mellin-Barnes representation is useful in the analytical
computation of inflation correlators. Even using this method, the complete
analytical evaluation of tree graphs is still hampered by the complication of
nested time integrals.

In this work, we computed arbitrarily nested time integrals in PMB
representation. The result is in general a multi-variable hypergeometric
series. With our family-tree decomposition procedure, we can find series
representation in terms of the inverse of any desired energy variable, or even
the sum of several energy variables. This result largely solves the problem of
analytical continuation of the nested time integrals in most physical regions.

With our results, the analytical computation of inflation correlators with
arbitrary massive exchanges at the tree level is reduced to a work of pole
collecting in the Mellin integrand, which is largely trivial. Thus, barring
possible issues with analytical continuation in special kinematics to be
commented below, we can say that the problem of analytical computation of
tree-level inflation correlators is solved.

At this point, we want to comment on the meaning of analytical computation. As
we have seen, most tree-level inflation correlators with massive exchanges
have to be expressed in terms of gigantic hypergeometric series which are not
yet named. One may say that we can systematically classify hypergeometric
functions with increasing number of variables and parameters and give each of
them a name. However, given that we know so little about these series in
general, this is not super meaningful, and also is not really different from
directly giving names to inflation correlators. The meaning of analytical
calculation is thus somewhat obscure. Normally, when we say that we obtain an
analytical answer, what we really mean is that we have good understanding of
this answer in at least two ways: First, we know its analytical properties.
This includes how does the answer change with parameters, and where does it
blow up or show other singular behavior. Second, we know how to find numerical
values of this answer for any choices of parameters with reasonable precision
and computation time. Therefore, we can claim that we get an analytical answer
only when we have gained sufficient knowledge about this answer. Having an
answer is not enough; we need to understand it.

From this viewpoint, it seems to us that our result for the nested time
integrals can be called an analytical answer: We know how to write down this
answer as Taylor series for most kinematics. As long as there is a largest
energy variable and we can use it to form small energy ratios, we can always
express our answer as power series of these given small numbers. This means,
on the one hand, we know the analytical properties of the answer at any soft
energy limit, and also know how to take analytical continuations to different
parameter space. On the other hand, having a convergent series often means
that we can do fast numerical evaluation of the answer. This is proved true in
our examples for two massive exchanges: In many cases, the numerical
evaluation of our series solution is way faster than direct numerical
integration of the original graph.

Our results have opened new possibilities in the analytical study of inflation
correlators. Many interesting problems can be pursued along this direction. We
mention some of them as below.

First, a main result of this work is a simple procedure to write down the
analytical answer for arbitrary nested time integrals in PMB representation.
Since the same time integral also appears in the loop computation, the result
here could be useful for the computation of loop correlators as well. Thus it
would be interesting to apply our method to make more complete loop
computation in PMB representation.

Second, a nice feature of PMB representation is that the energy dependence and
the momentum dependence of a graph are separated: The energy dependence is
fully in the time integral, while the momentum dependence is fully in the loop
momentum integral (or trivially factored out in a tree graph). Thus, our
result on the nested time integral will be useful for studying energy
dependence of a graph. This is particularly relevant to digging out the local
CC signal in a graph, since, by definition, the local signal is a nonanalytic
power in the energy ratios. We leave a more systematic study of local signals
to a future work.

Third, it is important that the PMB representation does not assume full dS
isometries of the problem. Therefore, it is straightforward to apply our
results here to fields with dS-breaking dispersions such as non-unit sound
speed, helical chemical potential, or even more exotic dispersion relations.
Our method is also applicable to correlation functions in more general FRW
background. We leave these generalizations to future studies.

Finally, it remains challenging to take analytical continuation of our series
expressions to the parameter region where no small energy ratios exist. A
pragmatic solution is using numerical interpolation to bridge different
parameter regions in which various series expression converge. While this can
indeed be implemented in some cases, it is not clear to us if this method
works for all possible kinematics. Analytically, we may need more
sophisticated methods to take analytical continuation for multi-layer series,
which sounds like a nontrivial mathematical problem. We leave these more
mathematically oriented problems for future studies as well.

#### Acknowledgments.

We thank Zhehan Qin for useful discussions. This work is supported by the
National Key R&D Program of China (2021YFC2203100), NSFC under Grant No.
12275146, an Open Research Fund of the Key Laboratory of Particle Astrophysics
and Cosmology, Ministry of Education of China, and the Dushi Program of
Tsinghua University.

##  Appendix A Mathematical Appendix

###  A.1 Mellin-Barnes representation

We use MB representation for quite a few special functions in the main text,
which we collect here. All expressions here can be found in standard
mathematical handbooks such as [114].

First, the Hankel functions $\text{H}_{\nu}^{(j)}(az)$ of $j$’th kind
($j=1,2$) frequently appear. Their MB representations are given by:

$\text{H}_{\nu}^{(j)}(az)=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{(az/2)^{-2s}}\,}{\pi}e^{(-1)^{j+1}(2s-\nu-1)\pi\mathrm{i}/2}\Gamma\Big{[}s-\displaystyle\frac{\,{\nu}\,}{2},s+\displaystyle\frac{\,{\nu}\,}{2}\Big{]}.~{}~{}~{}~{}(j=1,2)$
(129)

Next, we use an exponential integral $\text{E}_{p}(z)$ defined in the
following way:

$\mathrm{E}_{p}(z)=\int_{1}^{\infty}\displaystyle\frac{\,{e^{-zt}}\,}{t^{p}}{\mathrm{d}}t.$
(130)

This exponential integral is related to a confluent hypergeometric function
$\text{U}(a,b;z)$ via

$\mathrm{E}_{p}(z)=z^{p-1}e^{-z}\mathrm{U}(p,p;z),$ (131)

The confluent hypergeometric function $U(a,b;z)$ has the following MB
representations:

$\mathrm{U}(a,b;z)=\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\Gamma\begin{bmatrix}a+s,1+a-b+s,-s\\\
a,1+a-b\end{bmatrix}z^{-a-s},$ (132)
$\mathrm{U}(a,b,z)=z^{1-b}e^{z}\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\Gamma\begin{bmatrix}b-1+s,s\\\
a+s\end{bmatrix}z^{-s}.$ (133)

The validities of these expressions put constraints on the range of $z$, which
are always satisfied in the cases studied in this work, and thus we do not
spell them out. With these expressions, we can get two different MB
representations for the exponential integral $\text{E}_{p}(z)$. First, we have
a partially resolved representation:

$\displaystyle\mathrm{E}_{p}(z)=e^{-z}\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\Gamma\begin{bmatrix}p+s,1+s,-s\\\
p\end{bmatrix}z^{-s-1}.$ (134)

Second, we have the following completely resolved representation:

$\mathrm{E}_{p}(z)=\int_{-\mathrm{i}\infty}^{+\mathrm{i}\infty}\displaystyle\frac{\,{{\mathrm{d}}s}\,}{2\pi\mathrm{i}}\displaystyle\frac{\,{\Gamma(s)z^{-s}}\,}{s+p-1}.$
(135)

We note that the denominator $1/(s+p-1)$ in the integrand of the last
expression comes from the $\Gamma$ factors $\Gamma(b-1+s)/\Gamma(a+s)$ in
(133) with $a=b=p$ as required by (131). After taking $a=b=p$, most left poles
of $\Gamma(b-1+s)$ are canceled by the zeros of $1/\Gamma(a+s)$, with only one
pole left, which is exactly the denominator $1/(s+p-1)$ in (135). Thus we see
that the pole from $1/(s+p-1)$ should be treated as a left pole.

###  A.2 Special functions

Following our previous works on similar topics, we use shorthand notations for
the products and fractions of Euler $\Gamma$ functions:

$\displaystyle\Gamma\left[z_{1},\cdots,z_{m}\right]\equiv$
$\displaystyle~{}\Gamma(z_{1})\cdots\Gamma(z_{m}),$ (136)
$\displaystyle\Gamma\left[\begin{matrix}z_{1},\cdots,z_{m}\\\
w_{1},\cdots,w_{n}\end{matrix}\right]\equiv$
$\displaystyle~{}\displaystyle\frac{\,{\Gamma(z_{1})\cdots\Gamma(z_{m})}\,}{\Gamma(w_{1})\cdots\Gamma(w_{n})}.$
(137)

A number of hypergeometric series have been well studied and designated with
special names. Several of these hypergeometric functions are used in the main
text, and we collect their definitions here. More details about these
functions can be found in [115]. First, the (generalized) hypergeometric
function ${}_{p}F_{q}$ is defined by the following way when the series
converges:

$\displaystyle{}_{p}F_{q}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{matrix}\middle|z\right]=\sum_{n=0}^{\infty}\displaystyle\frac{\,{(a_{1})_{n}\cdots(a_{p})_{n}}\,}{(b_{1})_{n}\cdots(b_{q})_{n}}\displaystyle\frac{\,{z^{n}}\,}{n!},$
(138)

where $(a)_{n}\equiv\Gamma(a+n)/\Gamma(a)$ is the Pochhammer symbol. In most
cases, it turns out simpler to use the following dressed version of
hypergeometric function:

$\displaystyle{}_{p}\mathcal{F}_{q}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{matrix}\middle|z\right]=$
$\displaystyle~{}\Gamma\begin{bmatrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{bmatrix}{}_{p}F_{q}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
b_{1},\cdots,b_{q}\end{matrix}\middle|z\right]$ $\displaystyle=$
$\displaystyle~{}\sum_{n=0}^{\infty}\Gamma\left[\begin{matrix}a_{1}+n,\cdots,a_{p}+n\\\
b_{1}+n,\cdots,b_{q}+n\end{matrix}\right]\displaystyle\frac{\,{z^{n}}\,}{n!}.$
(139)

A special case of Gauss hypergeometric function is frequently used in the main
text, and thus we give a particular symbol to it:

$\displaystyle\mathbf{F}(a,b;z)\equiv\Gamma[a,b]\times{}_{2}F_{1}\left[\begin{matrix}a/2,(1+a)/2\\\
1-b\end{matrix}\middle|z\right].$ (140)

Next we come to hypergeometric functions of two variables. First, there are
four Appell functions $F_{1},\cdots,F_{4}$. Two of them are used in the main
text. We only present the definition of their dressed versions:

$\displaystyle\mathcal{F}_{2}\left[a\middle|\begin{matrix}b_{1},b_{2}\\\
c_{1},c_{2}\end{matrix}\middle|x,y\right]=\sum_{m,n=0}^{\infty}\Gamma\begin{bmatrix}a+m+n,b_{1}+m,b_{2}+n\\\
c_{1}+m,c_{2}+n\end{bmatrix}\displaystyle\frac{\,{x^{m}y^{n}}\,}{m!n!}.$ (141)
$\displaystyle\mathcal{F}_{4}\left[\begin{matrix}a,b\\\
c_{1},c_{2}\end{matrix}\middle|x,y\right]=\sum_{m,n=0}^{\infty}\Gamma\begin{bmatrix}a+m+n,b+m+n\\\
c_{1}+m,c_{2}+n\end{bmatrix}\displaystyle\frac{\,{x^{m}y^{n}}\,}{m!n!}.$ (142)

Second, a more general class of two-variable hypergeometric functions are
called Kampé de Fériet function in the literature, whose definition is:

$\displaystyle{}^{p+q}F_{r+s}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
c_{1},\cdots,c_{r}\end{matrix}\middle|\begin{matrix}b_{1},b_{1}^{\prime};\cdots;b_{q},b_{q}^{\prime}\\\
d_{1},d_{1}^{\prime};\cdots;d_{s},d_{s}^{\prime}\end{matrix}\middle|x,y\right]$
$\displaystyle=$
$\displaystyle\sum_{m,n=0}^{\infty}\displaystyle\frac{\,{(a_{1})_{m+n}\cdots(a_{p})_{m+n}}\,}{(c_{1})_{m+n}\cdots(c_{r})_{m+n}}\displaystyle\frac{\,{(b_{1})_{m}(b_{1}^{\prime})_{n}\cdots(b_{q})_{m}(b_{q}^{\prime})_{n}}\,}{(d_{1})_{m}(d_{1}^{\prime})_{n}\cdots(d_{s})_{m}(d_{s}^{\prime})_{n}}\displaystyle\frac{\,{x^{m}y^{n}}\,}{m!n!}.$
(143)

Again, we use the dressed version in the main text:

$\displaystyle{}^{p+q}\mathcal{F}_{r+s}\left[\begin{matrix}a_{1},\cdots,a_{p}\\\
c_{1},\cdots,c_{r}\end{matrix}\middle|\begin{matrix}b_{1},b_{1}^{\prime};\cdots;b_{q},b_{q}^{\prime}\\\
d_{1},d_{1}^{\prime};\cdots;d_{s},d_{s}^{\prime}\end{matrix}\middle|x,y\right]$
$\displaystyle=$
$\displaystyle\sum_{m,n=0}^{\infty}\Gamma\begin{bmatrix}a_{1}+m+n,\cdots,a_{p}+m+n\\\
c_{1}+m+n,\cdots,c_{r}+m+n\end{bmatrix}\Gamma\begin{bmatrix}b_{1}+m,\cdots,b_{q}+m\\\
d_{1}+m,\cdots,d_{s}+m\end{bmatrix}\Gamma\begin{bmatrix}b_{1}^{\prime}+n,\cdots,b_{q}^{\prime}+n\\\
d_{1}^{\prime}+n,\cdots,d_{s}^{\prime}+n\end{bmatrix}\displaystyle\frac{\,{x^{m}y^{n}}\,}{m!n!}.$
(144)

Finally, there is a particular $n$-variable hypergeometric function that
appears in the main text, called Lauricella’s $F_{A}$ function:

$\displaystyle F_{A}\left[a\middle|\begin{matrix}b_{1},\cdots,b_{N}\\\
c_{1},\cdots,c_{N}\end{matrix}\middle|z_{1},\cdots,z_{N}\right]=\sum_{m_{1},\cdots,m_{N}=0}^{\infty}\displaystyle\frac{\,{(a)_{m_{1}+\cdots+m_{N}}(b_{1})_{m_{1}}\cdots(b_{N})_{m_{N}}}\,}{(c_{1})_{m_{1}}\cdots(c_{N})_{m_{N}}}\displaystyle\frac{\,{z_{1}^{m_{1}}\cdots
z_{N}^{m_{N}}}\,}{m_{1}!\cdots m_{N}!}.$ (145)

Again, we use this function in its dressed form:

$\displaystyle\mathcal{F}_{A}\left[a\middle|\begin{matrix}b_{1},\cdots,b_{N}\\\
c_{1},\cdots,c_{N}\end{matrix}\middle|z_{1},\cdots,z_{N}\right]$
$\displaystyle=$
$\displaystyle\sum_{m_{1},\cdots,m_{N}=0}^{\infty}\Gamma\begin{bmatrix}a+m_{1}+\cdots+m_{N},b_{1}+m_{1},\cdots,b_{N}+m_{N}\\\
c_{1}+m_{1},\cdots,c_{N}+m_{N}\end{bmatrix}\displaystyle\frac{\,{z_{1}^{m_{1}}\cdots
z_{N}^{m_{N}}}\,}{m_{1}!\cdots m_{N}!}.$ (146)

##  Appendix B Details of Computing the Three-Vertex Seed Integral

In this appendix we collect some intermediate steps in the computation of the
three-vertex seed integral (4.1) in Sec. 4. First, there are four independent
branches in the seed integrals (4.1), shown below. The other four can be found
by taking complex conjugation.

$\displaystyle\mathcal{I}_{+++}=$
$\displaystyle-\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}+E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times\Big{[}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})+\Big{(}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})-D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})\Big{)}\theta(\tau_{1}-\tau_{2})\Big{]}$
$\displaystyle\times\Big{[}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})+\Big{(}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})-D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})\Big{)}\theta(\tau_{3}-\tau_{2})\Big{]}$
(147)
$\displaystyle\mathcal{I}_{++-}=+\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}+E_{2}\tau_{2}-E_{3}\tau_{3})}$
$\displaystyle\times\Big{[}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})+\Big{(}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})-D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})\Big{)}\theta(\tau_{1}-\tau_{2})\Big{]}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3}).$ (148)
$\displaystyle\mathcal{I}_{-++}=+\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(-E_{1}\tau_{1}+E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})\Big{[}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})+\Big{(}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})-D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})\Big{)}\theta(\tau_{3}-\tau_{2})\Big{]}.$
(149) $\displaystyle\mathcal{I}_{+-+}=$
$\displaystyle+\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}-E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3}).$ (150)

Then, we classify the terms in the above four integrals according to whether
the adjacent two time variables are time-ordered (T) or factorized (F). Thus,
we get the following eight different terms:

$\displaystyle\mathcal{I}_{+++}^{\text{(FF)}}=$
$\displaystyle-\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}+E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3}),$ (151)
$\displaystyle\mathcal{I}_{+++}^{\text{(TF)}}=$
$\displaystyle-\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}+E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times\Big{(}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})-D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})\Big{)}\theta(\tau_{1}-\tau_{2})D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3}),$ (152)
$\displaystyle\mathcal{I}_{+++}^{\text{(FT)}}=$
$\displaystyle-\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}+E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})\Big{(}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})-D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})\Big{)}\theta(\tau_{3}-\tau_{2}),$
(153) $\displaystyle\mathcal{I}_{+++}^{\text{(TT)}}=$
$\displaystyle-\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}+E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times\Big{(}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})-D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})\Big{)}\theta(\tau_{1}-\tau_{2})$
$\displaystyle\times\Big{(}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})-D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})\Big{)}\theta(\tau_{3}-\tau_{2}),$
(154) $\displaystyle\mathcal{I}_{++-}^{\text{(F)}}=$
$\displaystyle+\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}+E_{2}\tau_{2}-E_{3}\tau_{3})}$
$\displaystyle\times D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3}),$ (155)
$\displaystyle\mathcal{I}_{++-}^{\text{(T)}}=$
$\displaystyle+\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(E_{1}\tau_{1}+E_{2}\tau_{2}-E_{3}\tau_{3})}$
$\displaystyle\times\Big{(}D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})-D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})\Big{)}\theta(\tau_{1}-\tau_{2})D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3}),$ (156)
$\displaystyle\mathcal{I}_{-++}^{\text{(F)}}=$
$\displaystyle+\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(-E_{1}\tau_{1}+E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3}),$ (157)
$\displaystyle\mathcal{I}_{-++}^{\text{(T)}}=$
$\displaystyle+\mathrm{i}E_{1}^{p_{1}+1}E_{2}^{p_{2}+1}E_{3}^{p_{3}+1}\ell_{1}^{3}\ell_{2}^{3}\int_{-\infty}^{0}{\mathrm{d}}\tau_{1}{\mathrm{d}}\tau_{2}{\mathrm{d}}\tau_{3}(-\tau_{1})^{p_{1}}(-\tau_{2})^{p_{2}}(-\tau_{3})^{p_{3}}e^{\mathrm{i}(-E_{1}\tau_{1}+E_{2}\tau_{2}+E_{3}\tau_{3})}$
$\displaystyle\times D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})}(\ell_{1};\tau_{1},\tau_{2})\Big{(}D_{+-}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})-D_{-+}^{(\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})}(\ell_{2};\tau_{2},\tau_{3})\Big{)}\theta(\tau_{3}-\tau_{2}).$
(158)

We then take the PMB representation for all these terms and regroup them
according to (96)-(99). The time integrals can then be carried out using the
method in Sec. 3. To write down the result after the time integral, we note
that there is a common factor in all terms after taking the PMB
representation:

$\displaystyle\mathbb{H}(\\{s\\})\equiv$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(4\pi)^{2}}\Gamma\Big{[}s_{1}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},s_{1}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},s_{2}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\,}{2},s_{2}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1}}\,}{2}\Big{]}$
$\displaystyle\times\Gamma\Big{[}s_{3}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\,}{2},s_{3}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\,}{2},s_{4}-\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\,}{2},s_{4}+\displaystyle\frac{\,{\mathrm{i}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}\,}{2}\Big{]}.$
(159)

Then, we can write down the results of all terms with time integrals finished:

$\displaystyle\mathcal{I}_{+++}^{\text{(FF)}}$
$\displaystyle+\mathcal{I}_{++-}^{\text{(F)}}+\mathcal{I}_{-++}^{\text{(F)}}+\mathcal{I}_{+-+}$
$\displaystyle=-\mathrm{i}\int_{s_{1},\cdots,s_{4}}\,\Big{[}-\mathrm{i}e^{\mathrm{i}\pi(2s_{23}-p_{123}/2)}-e^{\mathrm{i}\pi(2s_{2}-p_{12}/2+p_{3}/2)}-e^{\mathrm{i}\pi(2s_{3}+p_{1}/2-p_{23}/2)}+\mathrm{i}e^{\mathrm{i}\pi(p_{2}/2-p_{13}/2)}\Big{]}$
$\displaystyle\times\mathbb{H}(\\{s\\})(r_{1}r_{2}r_{3}r_{4})^{3/2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-2s_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-2s_{2}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{-2s_{3}}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{2}\Big{)}^{-2s_{4}}$
$\displaystyle\times\Gamma\Big{[}p_{1}-2s_{1}+\mbox{$\frac{\,{5}\,}{2}$},p_{2}-2s_{23}+4,p_{3}-2s_{4}+\mbox{$\frac{\,{5}\,}{2}$}\Big{]}.$
(160)
$\displaystyle\mathcal{I}_{+++}^{\text{(FT)}}+\mathcal{I}_{-++}^{\text{(T)}}=$
$\displaystyle-4\mathrm{i}\int_{s_{1},\cdots,s_{4}}\sin[\pi(s_{3}-s_{4})]\sin[\pi(s_{2}-p_{1}/2+3/4)]e^{-\mathrm{i}(p_{23}-2s_{234}+13/2)\pi/2}\mathbb{H}(\\{s\\})$
$\displaystyle\times(r_{1}r_{2}r_{3}^{2})^{3/2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-2s_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-2s_{2}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{-2s_{34}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\Big{)}^{p_{3}+1}$
$\displaystyle\times\Gamma(p_{1}-2s_{1}+5/2){}_{2}\mathcal{F}_{1}\left[\begin{matrix}p_{3}-2s_{4}+5/2,p_{23}-2s_{234}+13/2\\\
p_{3}-2s_{4}+7/2\end{matrix}\middle|-\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\right].$
(161)
$\displaystyle\mathcal{I}_{+++}^{\text{(TF)}}+\mathcal{I}_{++-}^{\text{(T)}}=$
$\displaystyle-4\mathrm{i}\int_{s_{1},\cdots,s_{4}}\sin[\pi(s_{2}-s_{1})]\sin[\pi(s_{3}-p_{3}/2+3/4)]e^{-\mathrm{i}(p_{12}-2s_{123}+13/2)\pi/2}\mathbb{H}(\\{s\\})$
$\displaystyle\times(r_{2}^{2}r_{3}r_{4})^{3/2}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{2}\Big{)}^{-2s_{4}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{-2s_{3}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-2s_{12}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\Big{)}^{p_{1}+1}$
$\displaystyle\times\Gamma(p_{3}-2s_{4}+5/2){}_{2}\mathcal{F}_{1}\left[\begin{matrix}p_{1}-2s_{1}+5/2,p_{12}-2s_{123}+13/2\\\
p_{1}-2s_{1}+7/2\end{matrix}\middle|-\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\right].$
(162) $\displaystyle\mathcal{I}_{+++}^{\text{(TT)}}=$
$\displaystyle-4\int_{s_{1},\cdots,s_{4}}\sin[\pi(s_{1}-s_{2})]\sin[\pi(s_{3}-s_{4})]\mathbb{H}(\\{s\\})e^{\mathrm{i}\pi(s_{1234}-p_{123}/2)}$
$\displaystyle\times
r_{2}^{3}r_{3}^{3}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\Big{)}^{p_{1}+1}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\Big{)}^{p_{3}+1}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-2s_{12}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{-2s_{34}}$
$\displaystyle\times\mathcal{F}_{2}\left[p_{123}-2s_{1234}+9\middle|\begin{matrix}p_{1}-2s_{1}+5/2,p_{3}-2s_{4}+5/2\\\
p_{1}-2s_{1}+7/2,p_{3}-2s_{4}+7/2\end{matrix}\middle|-\displaystyle\frac{\,{r_{2}}\,}{r_{1}},-\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\right].$
(163)

It then remains to finish the Mellin integrals. As indicated in the main text,
we pick up all left poles given in (102). The residues of
$\mathbb{H}(\\{s\\})$ are given by

$\displaystyle\underset{\\{s\\}=\\{-n+\mathrm{i}\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2\\}}{\operatorname{Res}}\mathbb{H}(\\{s\\})$
$\displaystyle=\displaystyle\frac{\,{1}\,}{(4\pi)^{2}}\displaystyle\frac{\,{(-1)^{n_{1}+n_{2}+n_{3}+n_{4}}}\,}{n_{1}!n_{2}!n_{3}!n_{4}!}\Gamma\Big{[}-n_{1}+\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{2}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{3}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{4}+\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\Big{]}.$ (164)

Summing up all the left poles, we can finish the Mellin integral:

$\displaystyle\mathcal{I}_{+++}^{\text{(FF)}}$
$\displaystyle+\mathcal{I}_{++-}^{\text{(F)}}+\mathcal{I}_{-++}^{\text{(F)}}+\mathcal{I}_{+-+}$
$\displaystyle=\sum_{\\{n\\},\\{\mathsf{a}\\}}\displaystyle\frac{\,{1}\,}{(4\pi)^{2}}\Big{[}-e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}-p_{123}/2)}+\mathrm{i}e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-p_{12}/2+p_{3}/2)}+\mathrm{i}e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+p_{1}/2-p_{23}/2)}+e^{\mathrm{i}\pi(p_{2}/2-p_{13}/2)}\Big{]}$
$\displaystyle\times\displaystyle\frac{\,{(-1)^{n_{1}+n_{2}+n_{3}+n_{4}}}\,}{n_{1}!n_{2}!n_{3}!n_{4}!}(r_{1}r_{2}r_{3}r_{4})^{3/2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{2n_{2}-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{3}-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{2}\Big{)}^{2n_{4}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$
$\displaystyle\times\Gamma\Big{[}p_{1}+2n_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+{5}/{2},p_{2}+2n_{23}-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+4,p_{3}+2n_{4}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}+{5}/{2}\Big{]}$
$\displaystyle\times\Gamma\Big{[}-n_{1}+\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{2}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{3}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{4}+\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\Big{]}$
$\displaystyle=\sum_{n_{3},\\{\mathsf{a}\\}}\displaystyle\frac{\,{1}\,}{(4\pi)^{2}}\Big{[}-e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}-p_{123}/2)}+\mathrm{i}e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-p_{12}/2+p_{3}/2)}+\mathrm{i}e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+p_{1}/2-p_{23}/2)}+e^{\mathrm{i}\pi(p_{2}/2-p_{13}/2)}\Big{]}$
$\displaystyle\times\mathbf{F}(p_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+{5}/{2},\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},r_{1}^{2})\mathbf{F}(p_{3}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+{5}/{2},\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2},r_{4}^{2})$
$\displaystyle\times\displaystyle\frac{\,{(-1)^{n_{3}}}\,}{n_{3}!}(r_{1}r_{2}r_{3}r_{4})^{3/2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{3}-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$
$\displaystyle\times\Gamma(-n_{3}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})\mathbf{F}(p_{2}+2n_{3}-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+4,\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},r_{2}^{2})$
$\displaystyle=$
$\displaystyle\sum_{\\{\mathsf{a}\\}}\displaystyle\frac{\,{-2^{p_{2}-1}}\,}{\pi^{1/2}}\Big{[}-e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}-p_{123}/2)}+\mathrm{i}e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}-p_{12}/2+p_{3}/2)}+\mathrm{i}e^{\mathrm{i}\pi(\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+p_{1}/2-p_{23}/2)}+e^{\mathrm{i}\pi(p_{2}/2-p_{13}/2)}\Big{]}$
$\displaystyle\times\operatorname{csch}(\mathsf{a}_{2}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1})\operatorname{csch}(\mathsf{a}_{3}\pi\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2})(r_{1}r_{2}r_{3}r_{4})^{3/2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}r_{2}^{-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}r_{3}^{-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}}$
$\displaystyle\times\mathbf{F}(p_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+{5}/{2},\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},r_{1}^{2})\mathbf{F}(p_{3}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+{5}/{2},\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2},r_{4}^{2})$
$\displaystyle\times\mathcal{F}_{4}\left[\begin{matrix}-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}/2+2+p_{2}/2,-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}/2+5/2+p_{2}/2\\\
1-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},1-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}\end{matrix}\middle|r_{2}^{2},r_{3}^{2}\right],$ (165)

where

$\displaystyle\mathbf{F}(a,b,c)\equiv\Gamma[a,b]{}_{2}F_{1}\left[\begin{matrix}a/2,(1+a)/2\\\
1-b\end{matrix}\middle|c\right].$ (166)

Similarly,

$\displaystyle\mathcal{I}_{+++}^{\text{(FT)}}+$
$\displaystyle\mathcal{I}_{-++}^{\text{(T)}}$ $\displaystyle=$
$\displaystyle\sum_{\\{n\\},\\{\mathsf{a}\\}}\displaystyle\frac{\,{-4\mathrm{i}}\,}{(4\pi)^{2}}\sin[\pi(\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}/2-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}/2)]\sin[\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2-p_{1}/2+3/4)]e^{-\mathrm{i}(p_{23}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{234}+13/2)\pi/2}$
$\displaystyle\times\displaystyle\frac{\,{(-1)^{n_{1}+n_{2}+n_{3}+n_{4}}}\,}{n_{1}!n_{2}!n_{3}!n_{4}!}(r_{1}r_{2}r_{3}^{2})^{3/2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{2n_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{2n_{2}-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{3}-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\Big{)}^{p_{3}+1}$
$\displaystyle\times\Gamma(p_{1}+2n_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+5/2)\Gamma\Big{[}-n_{1}+\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{2}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{3}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{4}+\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\Big{]}$
$\displaystyle\times{}_{2}\mathcal{F}_{1}\left[\begin{matrix}p_{3}+2n_{4}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+5/2,p_{23}+2n_{234}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{234}+13/2\\\
p_{3}+2n_{4}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+7/2\end{matrix}\middle|-\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\right]$
$\displaystyle=$
$\displaystyle\sum_{m,n_{3},n_{4},\\{\mathsf{a}\\}}\displaystyle\frac{\,{-4\mathrm{i}}\,}{(4\pi)^{2}}\sin[\pi(\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}/2-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}/2)]\sin[\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2-p_{1}/2+3/4)]e^{-\mathrm{i}(p_{23}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{234}+13/2)\pi/2}$
$\displaystyle\times\displaystyle\frac{\,{(-1)^{n_{3}+n_{4}}}\,}{n_{3}!n_{4}!m!}(r_{1}r_{2}r_{3}^{2})^{3/2}\Big{(}\displaystyle\frac{\,{r_{1}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{34}-\mathrm{i}\mathsf{a}_{34}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}-\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\Big{)}^{m}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\Big{)}^{p_{3}+1}$
$\displaystyle\times\displaystyle\frac{\,{\Gamma[-n_{3}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{4}+\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}]}\,}{m+p_{3}+2n_{4}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+5/2}\mathbf{F}(p_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+{5}/{2},\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},r_{1}^{2})$
$\displaystyle\times\mathbf{F}(m+p_{23}+2n_{34}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{234}+13/2,\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{1},r_{2}^{2});$
(167)

Here in the last line we have used a shorthand notation $(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{234}\equiv\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$.

$\displaystyle\mathcal{I}_{+++}^{\text{(TF)}}$
$\displaystyle+\mathcal{I}_{++-}^{\text{(T)}}$ $\displaystyle=$
$\displaystyle\sum_{m,n_{1},n_{2},\\{\mathsf{a}\\}}\displaystyle\frac{\,{-4\mathrm{i}}\,}{(4\pi)^{2}}\sin[\pi(\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2)]\sin[\pi(\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}/2-p_{3}/2+3/4)]e^{-\mathrm{i}(p_{12}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{123}+13/2)\pi/2}$
$\displaystyle\times\displaystyle\frac{\,{(-1)^{n_{1}+n_{2}}}\,}{n_{1}!n_{2}!m!}(r_{2}^{2}r_{3}r_{4})^{3/2}\Big{(}\displaystyle\frac{\,{r_{4}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{-\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{2n_{12}-\mathrm{i}\mathsf{a}_{12}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}}\Big{(}-\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\Big{)}^{m}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\Big{)}^{p_{1}+1}$
$\displaystyle\times\displaystyle\frac{\,{\Gamma[-n_{2}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{1}+\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}]}\,}{m+p_{1}+2n_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+5/2}\mathbf{F}(p_{3}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+{5}/{2},\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2},r_{4}^{2})$
$\displaystyle\times\mathbf{F}(m+p_{12}+2n_{12}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{123}+13/2,\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2},r_{3}^{2}).$
(168)

Here in the last line we have used a shorthand notation $(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{123}\equiv\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$.

$\displaystyle\mathcal{I}_{+++}^{\text{(TT)}}=$
$\displaystyle\sum_{\\{n\\},\\{\mathsf{a}\\}}\displaystyle\frac{\,{-4}\,}{(4\pi)^{2}}\sin[\pi(\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2-\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}/2)]\sin[\pi(\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}/2-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}/2)]e^{\mathrm{i}\pi(\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{1234}/2-p_{123}/2)}$
$\displaystyle\times\displaystyle\frac{\,{(-1)^{n_{1}+n_{2}+n_{3}+n_{4}}}\,}{n_{1}!n_{2}!n_{3}!n_{4}!}r_{2}^{3}r_{3}^{3}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{r_{1}}\Big{)}^{p_{1}+1}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\Big{)}^{p_{3}+1}\Big{(}\displaystyle\frac{\,{r_{2}}\,}{2}\Big{)}^{2n_{12}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{12}}\Big{(}\displaystyle\frac{\,{r_{3}}\,}{2}\Big{)}^{2n_{34}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu)_{34}}$
$\displaystyle\times\Gamma\Big{[}-n_{1}+\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{2}+\mathrm{i}\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1},-n_{3}+\mathrm{i}\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2},-n_{4}+\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}\Big{]}$
$\displaystyle\times\mathcal{F}_{2}\left[p_{123}+2n_{1234}-\mathrm{i}(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{1234}+9\middle|\begin{matrix}p_{1}+2n_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{5}\,}{2}$},p_{3}+2n_{4}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{5}\,}{2}$}\\\
p_{1}+2n_{1}-\mathrm{i}\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mbox{$\frac{\,{7}\,}{2}$},p_{3}+2n_{4}-\mathrm{i}\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mbox{$\frac{\,{7}\,}{2}$}\end{matrix}\middle|-\displaystyle\frac{\,{r_{2}}\,}{r_{1}},-\displaystyle\frac{\,{r_{3}}\,}{r_{4}}\right].$
(169)

Here in the last line, we used another shorthand notation $(\mathsf{a}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu)_{1234}\equiv\mathsf{a}_{1}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{a}_{2}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{1}+\mathsf{a}_{3}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern
2.0mu_{2}+\mathsf{a}_{4}\mkern
2.0mu\widetilde{\mkern-2.0mu\nu\mkern-2.0mu}\mkern 2.0mu_{2}$. Then, we finish
the summation over
$\\{\mathsf{a}\\}=\\{\mathsf{a}_{1},\cdots,\mathsf{a}_{4}\\}$, regroup terms
according to their analytical properties at $r_{i}\rightarrow 0$, and find the
final result as summarized in (4.2).

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