$\displaystyle\beta\left[\left(\chi_{\beta}-\frac{1}{3}\right)-\frac{m^{2}}{3}\cdot\frac{(\Xi^{(1)}_{1})^{2}-\Xi^{(1)}_{2}\Xi^{(1)}_{0}}{(\Xi^{(1)}_{2})^{2}-\Xi^{(1)}_{3}\Xi^{(1)}_{1}}\right]=0,$
(177) $\displaystyle C_{\alpha}$ $\displaystyle:=$
$\displaystyle\chi_{\alpha}+\frac{m^{2}\beta}{3}\cdot\frac{\Xi^{(1)}_{2}\Xi^{(1)}_{1}-\Xi^{(1)}_{3}\Xi^{(1)}_{0}}{(\Xi^{(1)}_{2})^{2}-\Xi^{(1)}_{3}\Xi^{(1)}_{1}},$
(178) $\displaystyle C_{u}$ $\displaystyle:=$ $\displaystyle
1-\frac{3\left({\cal E}_{0}+P\right)}{\beta\aleph^{(1,1)}_{3}}=0.$ (179)

Thus, only $C_{\alpha}$ is non-zero and the others are identically zero
through nontrivial relations among $\Xi_{k}^{(n)}$. From the result, the first
order distribution is obtained as

$\displaystyle f^{[1]}$ $\displaystyle=$ $\displaystyle
f_{*}^{[1]}+f^{[1]}_{\epsilon}$ (180) $\displaystyle=$
$\displaystyle\tau_{R}\left[\left\\{E_{\bf
p}\beta\left(\chi_{\beta}-\frac{1}{3}\right)+\chi_{\alpha}-C_{\alpha}+\frac{m^{2}\beta}{3}\frac{1}{E_{\bf
p}}\right\\}\theta\right.$ $\displaystyle\qquad\left.+\frac{\beta}{E_{\bf
p}}p^{\mu}p^{\nu}\sigma_{\mu\nu}+\left(\frac{1}{E_{\bf
p}}-\frac{\aleph^{(1,1)}_{2}}{\aleph^{(1,1)}_{3}}\right)p^{\mu}\nabla_{\mu}\alpha\right]f^{(1)}_{{\rm
eq}},$

and the variables can be expressed as

$\displaystyle n_{*}^{\mu}$ $\displaystyle=$ $\displaystyle 0,$ (181)
$\displaystyle n_{\epsilon}^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{\tau_{R}}{3}\left(\aleph^{(1,1)}_{1}-\frac{(\aleph^{(1,1)}_{2})^{2}}{\aleph^{(1,1)}_{3}}\right)\nabla^{\mu}\alpha,$
(182) $\displaystyle\Pi_{*}$ $\displaystyle=$
$\displaystyle-\frac{\tau_{R}}{3}\aleph^{(1,1)}_{2}C_{\alpha}\theta,$ (183)
$\displaystyle\Pi_{\epsilon}$ $\displaystyle=$
$\displaystyle\frac{\tau_{R}}{3}\left[\beta\left(\chi_{\beta}-\frac{1}{3}\right)\aleph^{(1,1)}_{3}+\chi_{\alpha}\aleph^{(1,1)}_{2}+\frac{m^{2}\beta}{3}\aleph^{(1,1)}_{1}\right]\theta,$
(184) $\displaystyle\pi^{\mu\nu}_{*}$ $\displaystyle=$ $\displaystyle 0,$
(185) $\displaystyle\pi^{\mu\nu}_{\epsilon}$ $\displaystyle=$
$\displaystyle\tau_{R}\frac{2\beta}{15}\aleph^{(1,2)}_{3}\sigma^{\mu\nu},$
(186)

and thus,

$\displaystyle
n^{\mu}=n^{\mu}_{*}+n^{\mu}_{\epsilon}=\tau_{R}\kappa_{n}\nabla^{\mu}\alpha,$
$\displaystyle\Pi=\Pi_{*}+\Pi_{\epsilon}=-\tau_{R}\beta_{\Pi}\theta,$
$\displaystyle\pi^{\mu\nu}=\pi^{\mu\nu}_{*}+\pi^{\mu\nu}_{\epsilon}=2\tau_{R}\beta_{\pi}\sigma^{\mu\nu}$
$\displaystyle\Rightarrow\quad$
$\displaystyle\nabla^{\mu}\alpha=\frac{n^{\mu}}{\tau_{R}\kappa_{n}},\qquad\theta=-\frac{\Pi}{\tau_{R}\beta_{\Pi}},\qquad\sigma^{\mu\nu}=\frac{\pi^{\mu\nu}}{2\tau_{R}\beta_{\pi}},$
(188)

where

$\displaystyle\kappa_{n}$ $\displaystyle:=$
$\displaystyle-\frac{1}{3}\left[\aleph^{(1,1)}_{1}-\frac{(\aleph^{(1,1)}_{2})^{2}}{\aleph^{(1,1)}_{3}}\right],$
(189) $\displaystyle\beta_{\Pi}$ $\displaystyle:=$
$\displaystyle-\frac{1}{3}\left[\beta\left(\chi_{\beta}-\frac{1}{3}\right)\aleph^{(1,1)}_{3}+(\chi_{\alpha}-C_{\alpha})\aleph^{(1,1)}_{2}+\frac{m^{2}\beta}{3}\aleph^{(1,1)}_{1}\right]$
(190) $\displaystyle=$
$\displaystyle\frac{m^{4}\beta}{9}\left[\frac{(\Xi^{(1)}_{1})^{3}-2\Xi^{(1)}_{2}\Xi^{(1)}_{1}\Xi^{(1)}_{0}+\Xi^{(1)}_{3}(\Xi^{(1)}_{0})^{2}}{(\Xi^{(1)}_{2})^{2}-\Xi^{(1)}_{3}\Xi^{(1)}_{1}}+\Xi^{(1)}_{-1}\right],$
$\displaystyle\beta_{\pi}$ $\displaystyle:=$
$\displaystyle\frac{\beta}{15}\aleph^{(1,2)}_{3}.$ (191)

The distribution (180) can be expressed by

$\displaystyle f^{[1]}$ $\displaystyle=$
$\displaystyle\left[\sum_{k=-1}^{1}\Phi_{\Pi,k}E_{\bf
p}^{k}\Pi+\Phi_{\pi,-1}E_{\bf
p}^{-1}p^{\mu}p^{\nu}\pi_{\mu\nu}+\sum_{k=-1}^{0}\Phi_{n,k}E_{\bf
p}^{k}p^{\mu}n_{\mu}\right]f^{(1)}_{{\rm eq}},$

where202020 The energy dimensions of $\Phi_{{\cal O},k}$ is given by
$\displaystyle[\Phi_{\Pi,k}]=-4-k,\qquad[\Phi_{\pi,-1}]=-5,\qquad[\Phi_{n,k}]=-4-k.$
(193)

$\displaystyle\Phi_{\Pi,1}=-\frac{\beta}{\beta_{\Pi}}\left(\chi_{\beta}-\frac{1}{3}\right)=-\frac{m^{2}\beta}{3\beta_{\Pi}}\cdot\frac{(\Xi^{(1)}_{1})^{2}-\Xi^{(1)}_{2}\Xi^{(1)}_{0}}{(\Xi^{(1)}_{2})^{2}-\Xi^{(1)}_{3}\Xi^{(1)}_{1}},$
$\displaystyle\Phi_{\Pi,0}=-\frac{\chi_{\alpha}-C_{\alpha}}{\beta_{\Pi}}=\frac{m^{2}\beta}{3\beta_{\Pi}}\cdot\frac{\Xi^{(1)}_{2}\Xi^{(1)}_{1}-\Xi^{(1)}_{3}\Xi^{(1)}_{0}}{(\Xi^{(1)}_{2})^{2}-\Xi^{(1)}_{3}\Xi^{(1)}_{1}},$
(194) $\displaystyle\Phi_{\Pi,-1}=-\frac{m^{2}\beta}{3\beta_{\Pi}},$
$\displaystyle\Phi_{\pi,-1}=\frac{\beta}{2\beta_{\pi}},$ (195)
$\displaystyle\Phi_{n,0}=-\frac{1}{\kappa_{n}}\cdot\frac{\aleph^{(1,1)}_{2}}{\aleph^{(1,1)}_{3}},\qquad\Phi_{n,-1}=\frac{1}{\kappa_{n}}.$
(196)

Then, we derive the ODE of dissipative quantities. By rewriting the Boltzmann
equation as

$\displaystyle p^{\mu}\partial_{\mu}f=-\frac{E_{\bf p}}{\tau_{R}}f^{[1]}$
$\displaystyle\quad\Rightarrow\quad$ $\displaystyle
Df^{[1]}=-\frac{1}{\tau_{R}}f^{[1]}-Df_{\rm eq}-\frac{1}{E_{\bf
p}}p^{\mu}\nabla_{\mu}f,$ (197)

and using the definitions in Eq.(150), one finds that[57],

$\displaystyle Dn^{\mu}=-\frac{n^{\mu}}{\tau_{R}}-\Delta^{\mu}_{\
\alpha}\int_{p}p^{\alpha}\left(Df_{\rm eq}+\frac{1}{E_{\bf
p}}p^{\nu}\nabla_{\nu}f\right),$ (198) $\displaystyle
D\Pi=-\frac{\Pi}{\tau_{R}}+\frac{\Delta_{\alpha\beta}}{3}\int_{p}p^{\alpha}p^{\beta}\left(Df_{\rm
eq}+\frac{1}{E_{\bf p}}p^{\mu}\nabla_{\mu}f\right),$ (199) $\displaystyle
D\pi^{\mu\nu}=-\frac{\pi^{\mu\nu}}{\tau_{R}}-\Delta^{\mu\nu}_{\alpha\beta}\int_{p}p^{\alpha}p^{\beta}\left(Df_{\rm
eq}+\frac{1}{E_{\bf p}}p^{\rho}\nabla_{\rho}f\right).$ (200)

The leading order is obtained as

$\displaystyle Df_{\rm eq}$ $\displaystyle=$ $\displaystyle-\left[\beta E_{\bf
p}\chi_{\beta}\theta+\chi_{\alpha}\theta+\beta
p^{\mu}Du_{\mu}\right]{f}^{(1)}_{{\rm eq}},$ (201)
$\displaystyle\frac{1}{E_{\bf p}}p^{\mu}\nabla_{\mu}f_{\rm eq}$
$\displaystyle=$ $\displaystyle\left[\frac{3\left({\cal
E}_{0}+P\right)}{\aleph^{(1,1)}_{3}}p^{\mu}Du_{\mu}-\left(\frac{1}{E_{\bf
p}}-\frac{\aleph^{(1,1)}_{2}}{\aleph^{(1,1)}_{3}}\right)p^{\mu}\nabla_{\mu}\alpha\right.$
(202) $\displaystyle\left.+\frac{\beta}{3}\left(E_{\bf p}-\frac{m^{2}}{E_{\bf
p}}\right)\theta-\frac{\beta}{E_{\bf
p}}p^{\mu}p^{\nu}\sigma_{\mu\nu}\right]f^{(1)}_{{\rm eq}},$

and

$\displaystyle Df_{\rm eq}+\frac{1}{E_{\bf p}}p^{\mu}\nabla_{\mu}f_{\rm eq}$
$\displaystyle=$ $\displaystyle-\left[\left\\{\beta E_{\bf
p}\left(\chi_{\beta}-\frac{1}{3}\right)+\chi_{\alpha}+\frac{m^{2}\beta}{3E_{\bf
p}}\right\\}\theta+\frac{\beta}{E_{\bf
p}}p^{\mu}p^{\nu}\sigma_{\mu\nu}\right.$
$\displaystyle\left.+\left(\beta-\frac{3\left({\cal
E}_{0}+P\right)}{\aleph^{(1,1)}_{3}}\right)p^{\mu}Du_{\mu}+\left(\frac{1}{E_{\bf
p}}-\frac{\aleph^{(1,1)}_{2}}{\aleph^{(1,1)}_{3}}\right)p^{\mu}\nabla_{\mu}\alpha\right]f^{(1)}_{{\rm
eq}}.$

Thus,

$\displaystyle-\Delta^{\mu}_{\ \alpha}\int_{p}p^{\alpha}\left(Df_{\rm
eq}+\frac{1}{E_{\bf p}}p^{\mu}\nabla_{\mu}f_{\rm eq}\right)$ $\displaystyle=$
$\displaystyle\kappa_{n}\nabla^{\mu}\alpha,$ (204)
$\displaystyle\frac{\Delta_{\alpha\beta}}{3}\int_{p}p^{\alpha}p^{\beta}\left(Df_{\rm
eq}+\frac{1}{E_{\bf p}}p^{\mu}\nabla_{\mu}f_{\rm eq}\right)$ $\displaystyle=$
$\displaystyle-\widetilde{\beta}_{\Pi}\theta,$ (205)
$\displaystyle-\Delta^{\mu\nu}_{\alpha\beta}\int_{p}p^{\alpha}p^{\beta}\left(Df_{\rm
eq}+\frac{1}{E_{\bf p}}p^{\mu}\nabla_{\mu}f_{\rm eq}\right)$ $\displaystyle=$
$\displaystyle 2\beta_{\pi}\sigma^{\mu\nu},$ (206)

where

$\displaystyle\widetilde{\beta}_{\Pi}:=-\frac{1}{3}\left[\beta\left(\chi_{\beta}-\frac{1}{3}\right)\aleph^{(1,1)}_{3}+\chi_{\alpha}\aleph^{(1,1)}_{2}+\frac{m^{2}\beta}{3}\aleph^{(1,1)}_{1}\right].$
(207)

The next leading order is obtained as

$\displaystyle\frac{1}{E_{\bf p}}p^{\mu}\nabla_{\mu}f^{[1]}$ (208)
$\displaystyle=$ $\displaystyle\left[\sum_{{\cal
O}\in\\{\beta,\alpha\\}}\left(\sum_{k=-1}^{1}\partial_{\cal
O}\Phi_{\Pi,k}\cdot\nabla_{\mu}{\cal O}\cdot E_{\bf
p}^{k-1}p^{\mu}\Pi+\partial_{\cal O}\Phi_{\pi,-1}\cdot\nabla_{\mu}{\cal
O}\cdot E_{\bf p}^{-2}p^{\mu}p^{\nu}p^{\rho}\pi_{\nu\rho}\right.\right.$
$\displaystyle\left.\left.+\sum_{k=-1}^{0}\partial_{\cal
O}\Phi_{n,k}\cdot\nabla_{\mu}{\cal O}\cdot E_{\bf
p}^{k-1}p^{\mu}p^{\nu}n_{\nu}\right)\right.$
$\displaystyle\left.+\left(\sum_{k=-1}^{1}k\Phi_{\Pi,k}E_{\bf
p}^{k-2}p^{\mu}p^{\lambda}\Pi-\Phi_{\pi,-1}E_{\bf
p}^{-3}p^{\mu}p^{\nu}p^{\rho}p^{\lambda}\pi_{\nu\rho}\right.\right.$
$\displaystyle\left.\left.-\Phi_{n,-1}E_{\bf
p}^{-3}p^{\mu}p^{\nu}p^{\lambda}n_{\nu}\right)\cdot\left(\sigma_{\mu\lambda}+\frac{1}{3}\Delta_{\mu\lambda}\theta\right)\right.$
$\displaystyle\left.+\sum_{k=-1}^{1}\Phi_{\Pi,k}E_{\bf
p}^{k-1}p^{\mu}\nabla_{\mu}\Pi+\Phi_{\pi,-1}E_{\bf
p}^{-2}p^{\mu}p^{\nu}p^{\rho}\nabla_{\mu}\pi_{\nu\rho}+\sum_{k=-1}^{0}\Phi_{n,k}E_{\bf
p}^{k-1}p^{\mu}p^{\nu}\nabla_{\mu}n_{\nu}\right]{f}^{(1)}_{{\rm eq}}$
$\displaystyle-\left[\sum_{k=-1}^{1}\Phi_{\Pi,k}E_{\bf
p}^{k-1}p^{\mu}\Pi+\Phi_{\pi,-1}E_{\bf
p}^{-2}p^{\mu}p^{\nu}p^{\rho}\pi_{\nu\rho}\right.$
$\displaystyle\left.+\sum_{k=-1}^{0}\Phi_{n,k}E_{\bf
p}^{k-1}p^{\mu}p^{\nu}n_{\nu}\right]\cdot\left[E_{\bf
p}\nabla_{\mu}\beta+\nabla_{\mu}\alpha+\beta
p^{\lambda}\left(\sigma_{\mu\lambda}+\frac{1}{3}\Delta_{\mu\lambda}\theta\right)\right]{f}^{(2)}_{{\rm
eq}},$

Thus, one finds that

$\displaystyle-\int_{p}\frac{p^{\langle\mu\rangle}}{E_{\bf
p}}p^{\mu}\nabla_{\mu}f^{[1]}$
$\displaystyle\xrightarrow{\scriptsize\mbox{B-sym}}$
$\displaystyle-\frac{2}{15}\left[\Phi_{\pi,-1}\aleph^{(1,2)}_{2}\left(\nabla^{\nu}\pi_{\nu}^{\
\mu}+u^{\mu}\pi_{\nu\rho}\sigma^{\nu\rho}\right)\right]=0,$
$\displaystyle\frac{\Delta_{\alpha\beta}}{3}\int_{p}\frac{p^{\alpha}p^{\beta}}{E_{\bf
p}}p^{\mu}\nabla_{\mu}f^{[1]}$
$\displaystyle\xrightarrow{\scriptsize\mbox{B-sym}}$
$\displaystyle\frac{1}{9}\left[\sum_{k=-1}^{1}\Phi_{\Pi,k}\left(k\aleph^{(1,2)}_{k+2}-\beta\aleph^{(2,2)}_{k+3}\right)\Pi\theta+\frac{2}{5}\Phi_{\pi,-1}\left(\aleph^{(1,3)}_{3}+\beta\aleph^{(2,3)}_{4}\right)\left(\pi_{\zeta}^{\
\zeta}/\tau\right)\right],$
$\displaystyle-\frac{p^{\langle\mu}p^{\nu\rangle}}{E_{\bf
p}}p^{\mu}\nabla_{\mu}f^{[1]}$
$\displaystyle\xrightarrow{\scriptsize\mbox{B-sym}}$
$\displaystyle-\frac{2}{15}\left[\sum_{k=-1}^{1}\Phi_{\Pi,k}\left(k\aleph^{(1,2)}_{k+2}-\beta\aleph^{(2,2)}_{k+3}\right)\Pi\sigma^{\mu\nu}+\frac{11}{14}\Phi_{\pi,-1}\left(\aleph^{(1,3)}_{3}+\beta\aleph^{(2,3)}_{4}\right)\pi_{\zeta}^{\
\zeta}\sigma^{\mu\nu}\right],$

where “B-sym” denotes the expression under the Bjorken symmetry. Therefore,
under the Bjorken symmetry, the ODEs of viscosities are obtained as

$\displaystyle\frac{d\Pi}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\Pi}{\tau_{R}}-\widetilde{\beta}_{\Pi}\theta-\delta_{\Pi\Pi}\Pi\theta+\lambda_{\Pi\pi}\pi_{\mu\nu}\sigma^{\mu\nu},$
(212) $\displaystyle\frac{d\pi^{\mu\nu}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\pi^{\mu\nu}}{\tau_{R}}+2\beta_{\pi}\sigma^{\mu\nu}-\tau_{\pi\pi}\pi_{\rho}^{\
\langle\mu}\sigma^{\nu\rangle\rho}-\delta_{\pi\pi}\pi^{\mu\nu}\theta+\lambda_{\pi\Pi}\Pi\sigma^{\mu\nu},$
(213)

where

$\displaystyle\delta_{\Pi\Pi}$ $\displaystyle:=$
$\displaystyle-\frac{1}{9}\sum_{k=-1}^{1}\Phi_{\Pi,k}\left(k\aleph^{(1,2)}_{k+2}-\beta\aleph^{(2,2)}_{k+3}\right),$
$\displaystyle\lambda_{\Pi\pi}$ $\displaystyle:=$
$\displaystyle\frac{2}{45}\Phi_{\pi,-1}\left(\aleph^{(1,3)}_{3}+\beta\aleph^{(2,3)}_{4}\right),$
$\displaystyle\tau_{\pi\pi}$ $\displaystyle:=$
$\displaystyle\frac{8}{105}\Phi_{\pi,-1}\left(\aleph^{(1,3)}_{3}+\beta\aleph^{(2,3)}_{4}\right),$
(214) $\displaystyle\delta_{\pi\pi}$ $\displaystyle:=$
$\displaystyle\frac{2}{45}\Phi_{\pi,-1}\left(\aleph^{(1,3)}_{3}+\beta\aleph^{(2,3)}_{4}\right),$
$\displaystyle\lambda_{\pi\Pi}$ $\displaystyle:=$
$\displaystyle-\frac{2}{15}\sum_{k=-1}^{1}\Phi_{\Pi,k}\left(k\aleph^{(1,2)}_{k+2}-\beta\aleph^{(2,2)}_{k+3}\right).$

Notice that the Bjorken symmetry gives

$\displaystyle n^{\mu}=0.$ (215)

In summary, the ODEs are given by $(\widehat{\pi}:=\pi_{\zeta}^{\ \zeta})$

$\displaystyle\frac{d\beta}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\beta}{\tau}\left[\chi_{\beta}+\gamma_{\beta}(\Pi-\widehat{\pi})\right],$
(216) $\displaystyle\frac{d\alpha}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau}\left[\chi_{\alpha}-\gamma_{\alpha}(\Pi-\widehat{\pi})\right],$
(217) $\displaystyle\frac{d\Pi}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\Pi}{\tau_{R}}-\frac{1}{\tau}\left[\widetilde{\beta}_{\Pi}+\delta_{\Pi\Pi}\Pi-\lambda_{\Pi\pi}\widehat{\pi}\right],$
(218) $\displaystyle\frac{d\widehat{\pi}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\widehat{\pi}}{\tau_{R}}+\frac{1}{\tau}\left[\frac{4}{3}\beta_{\pi}-\left(\frac{1}{3}\tau_{\pi\pi}+\delta_{\pi\pi}\right)\widehat{\pi}+\frac{2}{3}\lambda_{\pi\Pi}\Pi\right].$
(219)

For convenience, we redefine the above ODEs by using the following new
variables:

$\displaystyle\bar{\Pi}:=\gamma_{\beta}\Pi,\qquad\bar{\pi}:=\gamma_{\beta}\widehat{\pi}.$
(220)

The modified ODEs are obtained as212121 In order to derive the modified ODEs,
we used the following result:
$\displaystyle\frac{d\log\gamma_{\beta}}{d\tau}$ $\displaystyle=$
$\displaystyle\left(F_{\beta}\partial_{\beta}+F_{\alpha}\partial_{\alpha}\right)\log\gamma_{\beta}$
(221) $\displaystyle=$
$\displaystyle\left[\frac{\beta}{\tau}\left(\chi_{\beta}+\bar{\Pi}-\bar{\pi}\right)\partial_{\beta}+\frac{1}{\tau}\left(\chi_{\alpha}-\beta\frac{\Xi^{(1)}_{2}}{\Xi^{(1)}_{1}}(\bar{\Pi}-\bar{\pi})\right)\partial_{\alpha}\right]\log\gamma_{\beta}$
$\displaystyle=$
$\displaystyle\frac{1}{\tau}\left[\left(\beta\chi_{\beta}\partial_{\beta}+\chi_{\alpha}\partial_{\alpha}\right)\log\gamma_{\beta}+\left(\bar{\Pi}-\bar{\pi}\right)\beta\left(\partial_{\beta}-\frac{\Xi^{(1)}_{2}}{\Xi^{(1)}_{1}}\partial_{\alpha}\right)\log\gamma_{\beta}\right].$

$\displaystyle\frac{d\beta}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\beta}{\tau}\left[\chi_{\beta}+\bar{\Pi}-\bar{\pi}\right]\quad=:F_{\beta},$
(222) $\displaystyle\frac{d\alpha}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau}\left[\chi_{\alpha}-\beta\frac{\Xi^{(1)}_{2}}{\Xi^{(1)}_{1}}(\bar{\Pi}-\bar{\pi})\right]\quad=:F_{\alpha},$
(223) $\displaystyle\frac{d\bar{\Pi}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\bar{\Pi}}{\tau_{R}}-\frac{1}{\tau}\left[\bar{\beta}_{\Pi}+C_{\Pi}\bar{\Pi}-\lambda_{\Pi\pi}\bar{\pi}-D\left(\bar{\Pi}-\bar{\pi}\right)\bar{\Pi}\right]\quad=:F_{\bar{\Pi}},$
(224) $\displaystyle\frac{d\bar{\pi}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\bar{\pi}}{\tau_{R}}-\frac{1}{\tau}\left[-\frac{4}{3}\bar{\beta}_{\pi}+C_{\pi}\bar{\pi}-\frac{2}{3}\lambda_{\pi\Pi}\bar{\Pi}-D\left(\bar{\Pi}-\bar{\pi}\right)\bar{\pi}\right]\quad=:F_{\bar{\pi}},$
(225)

where

$\displaystyle\bar{\beta}_{\Pi}:=\gamma_{\beta}\widetilde{\beta}_{\Pi},\qquad\bar{\beta}_{\pi}:=\gamma_{\beta}\beta_{\pi},$
$\displaystyle
C_{\Pi}:=\delta_{\Pi\Pi}-\left(\beta\chi_{\beta}\partial_{\beta}+\chi_{\alpha}\partial_{\alpha}\right)\log\gamma_{\beta},$
$\displaystyle
C_{\pi}:=\frac{1}{3}\tau_{\pi\pi}+\delta_{\pi\pi}-\left(\beta\chi_{\beta}\partial_{\beta}+\chi_{\alpha}\partial_{\alpha}\right)\log\gamma_{\beta},$
$\displaystyle
D:=\beta\left(\partial_{\beta}-\frac{\Xi^{(1)}_{2}}{\Xi^{(1)}_{1}}\partial_{\alpha}\right)\log\gamma_{\beta}.$
(226)

###  A.3 Asymptotic expansion around $z=\infty$ for transport coefficients

The asymptotic expansion around $z=+\infty$ and ${\cal R}=0$ for the
coefficients in the ODEs Eqs.(222)-(225) are obtained as (${\cal
R}_{a}:=-a{\cal R}$)

$\displaystyle\chi_{\beta}$ $\displaystyle\sim$ $\displaystyle
0.6667-1.6667z^{-1}+6.6667z^{-2}$ (227) $\displaystyle+{\cal
R}_{a}\left(0.2578z^{-1}-1.9312z^{-2}\right)+O({\cal R}_{a}^{2},z^{-3}),$
$\displaystyle\chi_{\alpha}$ $\displaystyle\sim$
$\displaystyle-0.6667z+1.6667-5.4167z^{-1}+19.1667z^{-2}$ (228)
$\displaystyle+{\cal
R}_{a}\left(-0.2578+1.3524z^{-1}-6.7174z^{-2}\right)+O({\cal
R}_{a}^{2},z^{-3}),$ $\displaystyle\beta\frac{\Xi^{(1)}_{2}}{\Xi^{(1)}_{1}}$
$\displaystyle\sim$ $\displaystyle z+1.5000+1.8750z^{-1}-1.8750z^{-2}$ (229)
$\displaystyle+{\cal
R}_{a}\left(-0.5303-0.4972z^{-1}+1.4864z^{-2}\right)+O({\cal
R}_{a}^{2},z^{-3}),$ $\displaystyle\bar{\beta}_{\Pi}$ $\displaystyle\sim$
$\displaystyle 0.5556z^{-2}-{\cal R}_{a}\cdot 0.0522z^{-2}+O({\cal
R}_{a}^{2},z^{-3}),$ (230) $\displaystyle\bar{\beta}_{\pi}$
$\displaystyle\sim$ $\displaystyle 0.6667-2.3333z^{-1}+11.3333z^{-2}$ (231)
$\displaystyle+{\cal
R}_{a}\left(-0.1768+0.7697z^{-1}-4.7190z^{-2}\right)+O({\cal
R}_{a}^{2},z^{-3}),$ $\displaystyle\lambda_{\Pi\pi}$ $\displaystyle\sim$
$\displaystyle 2.3333-2.3333z^{-1}+15.1667z^{-2}$ (232) $\displaystyle+{\cal
R}_{a}\left(0.2062z^{-1}-2.1011z^{-2}\right)+O({\cal R}_{a}^{2},z^{-3}),$
$\displaystyle\lambda_{\pi\Pi}$ $\displaystyle\sim$ $\displaystyle
3.6000-16.8000z^{-1}+159.6000z^{-2}$ (233) $\displaystyle+{\cal
R}_{a}\left(1.5313z^{-1}-18.4455z^{-2}\right)+O({\cal R}_{a}^{2},z^{-3}),$
$\displaystyle C_{\Pi}$ $\displaystyle\sim$ $\displaystyle
1.3333-14.0000z^{-1}+139.6667z^{-2}$ (234) $\displaystyle+{\cal
R}_{a}\left(0.8894z^{-1}-14.8891z^{-2}\right)+O({\cal R}_{a}^{2},z^{-3}),$
$\displaystyle C_{\pi}$ $\displaystyle\sim$ $\displaystyle
2.0000-3.6667z^{-1}+30.5000z^{-2}$ (235) $\displaystyle+{\cal
R}_{a}\left(-0.0626z^{-1}-2.8195z^{-2}\right)+O({\cal R}_{a}^{2},z^{-3}),$
$\displaystyle D$ $\displaystyle\sim$ $\displaystyle
1.0000+2.5000z^{-1}-13.7500z^{-2}$ (236) $\displaystyle+{\cal
R}_{a}\left(-0.1626-0.1795z^{-1}+0.6064z^{-2}\right)+O({\cal
R}_{a}^{2},z^{-3}).$

In addition, the energy density, bulk pressure, and charge density behave as

$\displaystyle{\cal E}_{0}$ $\displaystyle\sim$ $\displaystyle
m^{4}\frac{{\cal
R}}{\sqrt{z^{3}}}\left[0.0635+0.2143z^{-1}+0.3497z^{-2}\right.$ (237)
$\displaystyle\left.+{\cal
R}_{a}\left(0.0224+0.0379z^{-1}+0.0309z^{-2}\right)+O({\cal
R}_{a}^{2},z^{-3})\right],$ $\displaystyle P$ $\displaystyle\sim$
$\displaystyle m^{4}\frac{{\cal
R}}{\sqrt{z^{3}}}\left[0.0635z^{-1}+0.1191z^{-2}\right.$ (238)
$\displaystyle\left.+{\cal
R}_{a}\left(0.0112z^{-1}+0.0105z^{-2}\right)+O({\cal
R}_{a}^{2},z^{-3})\right],$ $\displaystyle n_{0}$ $\displaystyle\sim$
$\displaystyle m^{3}\frac{{\cal
R}}{\sqrt{z^{3}}}\left[0.0635+0.1191z^{-1}+0.0521z^{-2}\right.$ (239)
$\displaystyle\left.+{\cal
R}_{a}\left(0.0224+0.0210z^{-1}+0.0046z^{-2}\right)+O({\cal
R}_{a}^{2},z^{-3})\right].$

###  A.4 The massless limit

We obtain ODEs of the massless Bjorken flow. Since the massless limit gives

$\displaystyle\Xi_{k}\rightarrow-\frac{\Gamma(k+2)\beta^{-(k+2)}}{2\pi^{2}a}{\rm
Li}_{k+2}\left({\cal R}_{a}\right),$
$\displaystyle\Xi^{(n)}_{k}\rightarrow-\frac{\Gamma(k+2)\beta^{-(k+2)}}{2\pi^{2}a}{\rm
Li}_{k+2-n}\left({\cal R}_{a}\right),$ (240)
$\displaystyle\aleph^{(n,s)}_{k}\rightarrow-\frac{\Gamma(k+2)\beta^{-(k+2)}}{2\pi^{2}a}{\rm
Li}_{k+2-n}\left({\cal R}_{a}\right),$ (241)

where ${\cal R}_{a}=-a{\cal R}$ with ${\cal R}:=e^{-\alpha}$, one can find

$\displaystyle{\cal E}_{0}\rightarrow-\frac{3\beta^{-4}}{\pi^{2}a}{\rm
Li}_{4}\left({\cal R}_{a}\right)\qquad
P\rightarrow-\frac{\beta^{-4}}{\pi^{2}a}{\rm Li}_{4}\left({\cal
R}_{a}\right),$
$\displaystyle\chi_{\beta}\rightarrow\frac{1}{3},\qquad\chi_{\alpha}\rightarrow
0,$
$\displaystyle\kappa_{n}\rightarrow\frac{\beta^{-3}}{\pi^{2}a}\left[\frac{{\rm
Li}_{2}\left({\cal R}_{a}\right)}{3}-\frac{{\rm Li}_{3}\left({\cal
R}_{a}\right)^{2}}{4{\rm Li}_{4}\left({\cal R}_{a}\right)}\right],$
$\displaystyle\beta_{\Pi}\rightarrow
0,\qquad\beta_{\pi}\rightarrow-\frac{4\beta^{-4}}{5\pi^{2}a}{\rm
Li}_{4}\left({\cal R}_{a}\right),$ (242)

and $\Pi\propto\beta_{\Pi}\rightarrow 0$. Thus, the well-defined construction
of ODEs for the massless case needs to begin with Eq(180) and reconstruct the
ODEs from the exactly massless distribution:

$\displaystyle f^{[1]}$ $\displaystyle=$ $\displaystyle
f_{*}^{[1]}+f^{[1]}_{\epsilon}$ (243) $\displaystyle\rightarrow$
$\displaystyle\tau_{R}\left[\frac{\beta}{E_{\bf
p}}p^{\mu}p^{\nu}\sigma_{\mu\nu}+\left(\frac{1}{E_{\bf p}}-\frac{\beta{\rm
Li}_{3}\left({\cal R}_{a}\right)}{4{\rm Li}_{4}\left({\cal
R}_{a}\right)}\right)p^{\mu}\nabla_{\mu}\alpha\right]f^{(1)}_{{\rm eq}}$
$\displaystyle=$ $\displaystyle\left[\Phi_{\pi,-1}E_{\bf
p}^{-1}p^{\mu}p^{\nu}\pi_{\mu\nu}+\sum_{k=-1}^{0}\Phi_{n,k}E_{\bf
p}^{k}p^{\mu}n_{\mu}\right]f^{(1)}_{{\rm eq}}.$

It is notable that in the massless limit $C_{\alpha}$ given in Eq.(178)
becomes

$\displaystyle C_{\alpha}\rightarrow 0,$ (244)

and thus, thermodynamic frame naturally vanishes, i.e.

$\displaystyle f^{[1]}_{*}\rightarrow 0.$ (245)

By repeating the same construction, the ODEs are obtained as

$\displaystyle\frac{d\beta}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\beta}{\tau}\left[\frac{1}{3}-\bar{\pi}\right],$ (246)
$\displaystyle\frac{d{\cal R}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{3{\cal R}}{\tau}\cdot\frac{{\rm Li}_{3}({\cal
R}_{a})}{{\rm Li}_{2}({\cal R}_{a})}\bar{\pi},$ (247)
$\displaystyle\frac{d\bar{\pi}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\bar{\pi}}{\tau_{R}}-\frac{1}{\tau}\left[-\frac{4}{3}\bar{\beta}_{\pi}({\cal
R}_{a})+C_{\pi}\bar{\pi}+D({\cal R}_{a})\bar{\pi}^{2}\right],$ (248)

where

$\displaystyle C_{\pi}:=\frac{38}{21},$ $\displaystyle\bar{\beta}_{\pi}({\cal
R}_{a}):=\frac{4}{15}\cdot\frac{{\rm Li}_{2}({\cal R}_{a}){\rm Li}_{4}({\cal
R}_{a})}{4{\rm Li}_{2}({\cal R}_{a}){\rm Li}_{4}({\cal R}_{a})-3{\rm
Li}_{3}({\cal R}_{a})^{2}},$ (249) $\displaystyle D({\cal
R}_{a}):=4-\frac{3{\rm Li}_{3}\left({\cal R}_{a}\right)^{2}\left(3{\rm
Li}_{1}\left({\cal R}_{a}\right){\rm Li}_{3}({\cal R}_{a})-2{\rm
Li}_{2}\left({\cal R}_{a}\right)^{2}\right)}{{\rm Li}_{2}\left({\cal
R}_{a}\right)^{2}\left(4{\rm Li}_{2}\left({\cal R}_{a}\right){\rm
Li}_{4}\left({\cal R}_{a}\right)-3{\rm Li}_{3}\left({\cal
R}_{a}\right)^{2}\right)}.$

Setting $\tau_{R}=\beta\theta_{0}$ and $w=\tau/\beta$ yields

$\displaystyle\frac{d\beta}{dw}$ $\displaystyle=$
$\displaystyle\frac{\beta}{w}\cdot\frac{\frac{1}{3}-\bar{\pi}}{\frac{2}{3}+\bar{\pi}}$
(250) $\displaystyle\frac{d{\cal R}}{dw}$ $\displaystyle=$
$\displaystyle-\frac{3{\cal R}}{w}\cdot\frac{{\rm Li}_{3}({\cal R}_{a})}{{\rm
Li}_{2}({\cal R}_{a})}\cdot\frac{\bar{\pi}}{\frac{2}{3}+\bar{\pi}},$ (251)
$\displaystyle\frac{d\bar{\pi}}{dw}$ $\displaystyle=$
$\displaystyle-\frac{\frac{\bar{\pi}}{\theta_{0}}+\frac{1}{w}\left[-\frac{4}{3}\bar{\beta}_{\pi}({\cal
R}_{a})+\frac{38}{21}\bar{\pi}+D({\cal
R}_{a})\bar{\pi}^{2}\right]}{\frac{2}{3}+\bar{\pi}}.$ (252)

##  Appendix B Transseries and Borel resummation

In this appendix, we summarize topics related to transseries, Borel
resummation, and resurgence. See Refs.[44, 8, 9, 10, 11, 12, 13, 6, 7], for
example, in more detail.

###  B.1 Construction of transmonomials

In this appendix, we derive transmonomials expanded around $w=+\infty$. We
take $m=1$ for simplicity. Let us start with ODEs given by Eqs.(19)-(22). We
firstly assume that
$\lim_{w\rightarrow+\infty}\bar{\Pi}=\lim_{w\rightarrow+\infty}\bar{\pi}=0$
and $\lim_{w\rightarrow+\infty}\beta=+\infty$. From Eqs.(19)(20), the leading
order of $(\beta,\alpha)$ is given by

$\displaystyle\frac{d\beta}{dw}\sim\frac{2}{w}\beta,\qquad\frac{d{\cal
R}}{dw}\sim\frac{1}{w}O(\beta^{-1})\quad\Rightarrow\quad\beta\sim\sigma_{\beta}w^{2},\quad{\cal
R}\sim\sigma_{\cal R}.$ (253)

The leading order of $(\beta,{\cal R})$ and Eqs.(21)(22) give the one of
$(\bar{\Pi},\bar{\pi})$ as

$\displaystyle\bar{\Pi}\sim\frac{28}{9}C^{\bar{\beta}_{\pi}}_{0}(\sigma_{{\cal
R},a})\left(\frac{\theta_{0}}{w}\right)^{2},\qquad\bar{\pi}\sim\frac{4}{3}C^{\bar{\beta}_{\pi}}_{0}(\sigma_{{\cal
R},a})\frac{\theta_{0}}{w}.$ (254)

The higher orders can be recursively obtained by each of the ODEs, and their
formal power expansion with respect to $w^{-1}$ is closed under all operations
in the ODEs. Notice that these are consistent with our assumption.

Then, we construct higher transmonomials in the NP sectors. It can be derived
from the linearized equation of $(\bar{\Pi},\bar{\pi})$ given by

$\displaystyle\frac{d\delta\bar{\bf
X}}{dw}=-\left[\Lambda+\frac{1}{w}{\mathfrak{B}}\right]\delta\bar{\bf
X}+O(\delta\bar{\bf X}^{2},\delta\bar{\bf X}w^{-2}),$ (255)

where

$\displaystyle\Lambda:=S{\mathbb{I}}_{2},\qquad S:=\frac{3}{\theta_{0}}$
$\displaystyle{\mathfrak{B}}:=\begin{bmatrix}4\left(1-3C^{\bar{\beta}_{\pi}}_{0}(\sigma_{{\cal
R},a})\right)&-7\\\
-\frac{36}{5}&6\left(1-2C^{\bar{\beta}_{\pi}}_{0}(\sigma_{{\cal
R},a})\right)\end{bmatrix},$ (256) $\displaystyle\delta\bar{\bf
X}:=(\delta\bar{\Pi},\delta\bar{\pi})^{\top}.$

Solving the linearized equation gives

$\displaystyle\delta\bar{\bf X}=U^{-1}\begin{pmatrix}\zeta_{+}\\\
\zeta_{-}\end{pmatrix},$ (257)

where

$\displaystyle\zeta_{\pm}:=\sigma_{\pm}\frac{e^{-S_{\pm}w}}{w^{\rho_{\pm}}},\qquad
S_{\pm}=S:=\frac{3}{\theta_{0}},\qquad\rho_{\pm}:=5-12C^{\bar{\beta}_{\pi}}_{0}(\sigma_{{\cal
R},a})\mp\frac{\sqrt{1285}}{5},$ (258) $\displaystyle
U=\begin{pmatrix}\frac{18}{\sqrt{1285}}&\frac{1}{2}-\frac{5}{2\sqrt{1285}}\\\
-\frac{18}{\sqrt{1285}}&\frac{1}{2}+\frac{5}{2\sqrt{1285}}\end{pmatrix},\qquad
U^{-1}=\begin{pmatrix}\frac{5+\sqrt{1285}}{36}&\frac{5-\sqrt{1285}}{36}\\\
1&1\end{pmatrix},$ (259)

and $\sigma_{\pm}\in{\mathbb{R}}$ is the integration constant.

###  B.2 Explicit form of transseries for other variables

####  B.2.1 Transseries of $\widetilde{X}_{\pm}$

$\displaystyle\widetilde{X}_{+{\rm pt}}$ $\displaystyle\sim$
$\displaystyle(0.3825-0.1014\sigma_{{\cal
R},a})\theta_{0}w^{-1}+(0.4040-0.5128\sigma_{{\cal R},a})\theta_{0}^{2}w^{-2}$
(260) $\displaystyle+\left\\{0.5615-0.1267\sigma_{{\cal
R},a}+(-1.3386+0.2514\sigma_{{\cal
R},a})(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{3}w^{-3}$
$\displaystyle+O(w^{-4},\sigma_{{\cal R},a}),$
$\displaystyle\widetilde{X}_{-{\rm pt}}$ $\displaystyle\sim$
$\displaystyle(0.5064-0.1343\sigma_{{\cal
R},a})\theta_{0}w^{-1}+(-1.8855-0.0372\sigma_{{\cal
R},a})\theta_{0}^{2}w^{-2}$ (261)
$\displaystyle+\left\\{6.3916+0.9019\sigma_{{\cal
R},a}+(-1.7725+0.3329\sigma_{{\cal
R},a})(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{3}w^{-3}$
$\displaystyle+O(w^{-4},\sigma_{{\cal R},a}),$
$\displaystyle\widetilde{X}_{+{\rm np}}$ $\displaystyle\sim$
$\displaystyle\zeta_{+}\left[1.0000\right.$ (262)
$\displaystyle\left.+\left\\{61.7850-13.9493\sigma_{{\cal
R},a}+(-15.0000+2.3202\sigma_{{\cal
R},a})(\sigma_{\beta}\theta^{2}_{0})^{-1}\right\\}\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{1713.7587-817.8193\sigma_{{\cal
R},a}+(-811.7237+311.5972\sigma_{{\cal
R},a})(\sigma_{\beta}\theta^{2}_{0})^{-1}\right.\right.$
$\displaystyle\left.\left.+(112.5000-34.8029\sigma_{{\cal
R},a})(\sigma_{\beta}\theta^{2}_{0})^{-2}\right\\}\theta_{0}^{2}w^{-2}+O(w^{-3},\sigma_{{\cal
R},a}^{2})\right]$ $\displaystyle+\zeta_{-}\left[0.0368\sigma_{{\cal
R},a}\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{-0.1378+0.7902\sigma_{{\cal
R},a}+(0.2733-0.5776\sigma_{{\cal
R},a})(\sigma_{\beta}\theta^{2}_{0})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+O(w^{-3},\sigma_{{\cal
R},a}^{2})\right]+O(\zeta_{\pm}^{2}),$ $\displaystyle\widetilde{X}_{-{\rm
np}}$ $\displaystyle\sim$
$\displaystyle\zeta_{+}\left[\left\\{0.0617+0.2641\sigma_{{\cal
R},a}+(-2.9069+0.2062\sigma_{{\cal
R},a})(\sigma_{\beta}\theta^{2}_{0})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+O(w^{-3},\sigma_{{\cal R},a}^{2})\right]$
$\displaystyle+\zeta_{-}\left[1.0000\right.$
$\displaystyle\left.+\left\\{214.9524-56.7540\sigma_{{\cal
R},a}+(-291.7763+66.1111\sigma_{{\cal
R},a})(\sigma_{\beta}\theta^{2}_{0})^{-1}\right.\right.$
$\displaystyle\left.\left.+(112.5000-34.8029\sigma_{{\cal
R},a})(\sigma_{\beta}\theta^{2}_{0})^{-2}\right\\}\theta_{0}^{2}w^{-2}+O(w^{-3},\sigma_{{\cal
R},a}^{2})\right]+O(\zeta_{\pm}^{2}).$

####  B.2.2 Transseries of $({\cal E}_{0},P,n_{0})$ and $(P_{\perp},P_{||})$

$\displaystyle{\cal E}_{0\,{\rm pt}}$ $\displaystyle\sim$
$\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{3/2}}w^{-3}\left[0.0635+0.0224\sigma_{{\cal
R},a}+\left(-0.5079-0.0449\sigma_{{\cal R},a}\right)\theta_{0}w^{-1}\right.$
(264) $\displaystyle\left.+\left\\{3.7250-0.1197\sigma_{{\cal
R},a}+\left(-0.3810-0.0779\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+O(w^{-3},\sigma_{{\cal R},a}^{2})\right],$ $\displaystyle
P_{{\rm pt}}$ $\displaystyle\sim$ $\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\left[0.0635+0.0112\sigma_{{\cal
R},a}+\left(-1.0159+0.0449\sigma_{{\cal R},a}\right)\theta_{0}w^{-1}\right.$
(265) $\displaystyle\left.+\left\\{11.5135-2.2149\sigma_{{\cal
R},a}+\left(-0.9524-0.0316\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+O(w^{-3},\sigma_{{\cal R},a}^{2})\right],$ $\displaystyle
n_{0\,{\rm pt}}$ $\displaystyle\sim$ $\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{3/2}}w^{-3}\left[\left(0.0635+0.0224\sigma_{{\cal
R},a}\right)+\left(-0.5079-0.0449\sigma_{{\cal
R},a}\right)\theta_{0}w^{-1}\right.$ (266)
$\displaystyle\left.+\left\\{3.7250-0.1197\sigma_{{\cal
R},a}+\left(-0.4762-0.0947\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+O(w^{-3},\sigma_{{\cal R},a}^{2})\right],$ $\displaystyle
P_{\perp\,{\rm pt}}$ $\displaystyle\sim$ $\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\left[0.0635+0.0112\sigma_{{\cal
R},a}+\left(-0.9736+0.0524\sigma_{{\cal R},a}\right)\theta_{0}w^{-1}\right.$
(267) $\displaystyle\left.+\left\\{10.9632-2.1625\sigma_{{\cal
R},a}+\left(-0.9524-0.0316\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+O(w^{-3},\sigma_{{\cal R},a}^{2})\right],$ $\displaystyle
P_{\parallel\,{\rm pt}}$ $\displaystyle\sim$ $\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\left[0.0635+0.0112\sigma_{{\cal
R},a}+\left(-1.1006+0.0299\sigma_{{\cal R},a}\right)\theta_{0}w^{-1}\right.$
(268) $\displaystyle\left.+\left\\{13.2067-2.2149\sigma_{{\cal
R},a}+\left(-0.9524-0.0316\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+O(w^{-3},\sigma_{{\cal R},a}^{2})\right],$
$\displaystyle{\cal E}_{0\,{\rm np}}$ $\displaystyle\sim$
$\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{3/2}}w^{-3}\zeta_{+}\left[\left(-0.3537-0.1251\sigma_{{\cal
R},a}\right)\theta_{0}w^{-1}\right.$ (269)
$\displaystyle\left.+\left\\{-3.0035-0.9928\sigma_{{\cal
R},a}+\left(5.3054+1.0551\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+\left\\{-8.7911-4.9825\sigma_{{\cal
R},a}+\left(26.6861+12.8558\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(-39.7908-1.7585\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{3}w^{-3}+O(w^{-4},\sigma_{{\cal
R},a}^{2})\right]$ $\displaystyle+\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{3/2}}w^{-3}\zeta_{-}\left[\left(0.0256+0.0091\sigma_{{\cal
R},a}\right)\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{1.3210+0.1804\sigma_{{\cal
R},a}+\left(-0.3847-0.0765\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+\left\\{30.0234-3.2081\sigma_{{\cal
R},a}+\left(-16.0793+0.5731\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(2.8851+0.1275\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{3}w^{-3}+O(w^{-4},\sigma_{{\cal
R},a}^{2})\right]+O(\zeta_{\pm}^{2}),$ $\displaystyle P_{{\rm np}}$
$\displaystyle\sim$ $\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\zeta_{+}\left[\left(-0.7074-0.1563\sigma_{{\cal
R},a}\right)\theta_{0}w^{-1}\right.$ (270)
$\displaystyle\left.+\left\\{-0.3479-1.0641\sigma_{{\cal
R},a}+\left(10.6109+0.7034\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+\left\\{-11.0259-4.3437\sigma_{{\cal
R},a}+\left(-24.0872+30.3412\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(-79.5816+7.0341\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{3}w^{-3}+O(w^{-4},\sigma_{{\cal
R},a}^{2})\right]$ $\displaystyle+\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\zeta_{-}\left[\left(0.0513+0.0113\sigma_{{\cal
R},a}\right)\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{2.2316+0.0014\sigma_{{\cal
R},a}+\left(-0.7694-0.0510\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+\left\\{41.9206-10.4999\sigma_{{\cal
R},a}+\left(-26.5421+4.5612\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(5.7703-0.5100\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{3}w^{-3}+O(w^{-4},\sigma_{{\cal
R},a}^{2})\right]+O(\zeta_{\pm}^{2}),$ $\displaystyle n_{0\,{\rm np}}$
$\displaystyle\sim$ $\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{3/2}}w^{-3}\zeta_{+}\left[\left(-0.3537-0.1251\sigma_{{\cal
R},a}\right)\theta_{0}w^{-1}\right.$ (271)
$\displaystyle\left.+\left\\{-3.0035-0.9928\sigma_{{\cal
R},a}+\left(5.3054+1.0551\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+\left\\{-8.7911-4.9825\sigma_{{\cal
R},a}+\left(27.7472+13.0902\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(-39.7908-1.7585\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{3}w^{-3}+O(w^{-4},\sigma_{{\cal
R},a}^{2})\right]$ $\displaystyle+\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{3/2}}w^{-3}\zeta_{-}\left[\left(0.0256+0.0091\sigma_{{\cal
R},a}\right)\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{1.3210+0.1804\sigma_{{\cal
R},a}+\left(-0.3847-0.0765\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}^{2}w^{-2}\right.$
$\displaystyle\left.+\left\\{30.0234-3.2081\sigma_{{\cal
R},a}+\left(-16.1562+0.5561\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(2.8851+0.1275\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{3}w^{-3}+O(w^{-4},\sigma_{{\cal
R},a}^{2})\right]+O(\zeta_{\pm}^{2}),$ $\displaystyle P_{\perp{\rm np}}$
$\displaystyle\sim$ $\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\zeta_{+}\left[-0.0340-0.0150\sigma_{{\cal
R},a}\right.$ (272) $\displaystyle\left.+\left\\{-0.9639-0.3175\sigma_{{\cal
R},a}+\left(0.5098+0.1464\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{-1.5042-1.9009\sigma_{{\cal
R},a}+\left(12.8380+2.8703\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(-3.8236-0.5069\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{2}w^{-2}+O(w^{-3},\sigma_{{\cal
R},a}^{2})\right]$ $\displaystyle+\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\zeta_{-}\left[0.1557+0.0688\sigma_{{\cal
R},a}\right.$ $\displaystyle\left.+\left\\{7.1793+1.7052\sigma_{{\cal
R},a}+\left(-2.3353-0.6708\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{143.3957+0.7239\sigma_{{\cal
R},a}+\left(-91.6245-7.8216\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(17.5144+2.3221\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{2}w^{-2}+O(w^{-3},\sigma_{{\cal
R},a}^{2})\right]+O(\zeta_{\pm}^{2}),$ $\displaystyle P_{\parallel{\rm np}}$
$\displaystyle\sim$ $\displaystyle\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\zeta_{+}\left[-0.1768-0.0782\sigma_{{\cal
R},a}\right.$ (273) $\displaystyle\left.+\left\\{-2.0423-1.0243\sigma_{{\cal
R},a}+\left(2.6527+0.7620\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{-2.8573-5.0047\sigma_{{\cal
R},a}+\left(21.9815+12.3426\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(-19.8954-2.6378\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{2}w^{-2}+O(w^{-3},\sigma_{{\cal
R},a}^{2})\right]$ $\displaystyle+\frac{\sigma_{\cal
R}}{\sigma_{\beta}^{5/2}}w^{-5}\zeta_{-}\left[0.0128+0.0057\sigma_{{\cal
R},a}\right.$ $\displaystyle\left.+\left\\{0.6384+0.1501\sigma_{{\cal
R},a}+\left(-0.1923-0.0553\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right\\}\theta_{0}w^{-1}\right.$
$\displaystyle\left.+\left\\{13.7766+0.0105\sigma_{{\cal
R},a}+\left(-7.7467-0.5033\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-1}\right.\right.$
$\displaystyle\left.\left.+\left(1.4426+0.1913\sigma_{{\cal
R},a}\right)(\sigma_{\beta}\theta_{0}^{2})^{-2}\right\\}\theta_{0}^{2}w^{-2}+O(w^{-3},\sigma_{{\cal
R},a}^{2})\right]+O(\zeta_{\pm}^{2}).$

###  B.3 Review of Borel resummation

In this appendix, we briefly review Borel resummation theory. We suppose the
following transseries expanded around $w=+\infty$:

$\displaystyle X(w)\sim\sum_{n\in{\mathbb{N}}_{0}}\zeta^{n}X^{[n]}(w),\qquad
X^{[n]}(w)\sim\sum_{k\in{\mathbb{N}}_{0}}a^{[n,k]}w^{-k},\qquad\zeta=\sigma\frac{e^{-Sw}}{w^{b}},$
(274)

where $w\in{\mathbb{R}}$,
$a^{[n,k]}\in{\mathbb{C}},b\in{\mathbb{C}}\setminus{\mathbb{Z}}_{0-},S\in\\{x\in{\mathbb{C}}\,|\,{\rm
Re}\,x>0\\},\sigma\in{\mathbb{C}}$. In the main text, we call the PT sector
for $n=0$ and the $n$-the NP sector if $n\neq 0$. For simplicity, we assume
that $a^{[0,0]}=0$. For the technical reason, we redefine Eq.(274) as

$\displaystyle\widecheck{X}^{[n]}(w)=w^{-nb}X^{[n]}(w)$ (275)

such that $\zeta^{n}X^{[n]}(w)=(\sigma e^{-Sw})^{n}\widecheck{X}^{[n]}(w)$.
The Borel transform, ${\cal B}$, to $\widecheck{X}^{[n]}(w)$ is defined as

$\displaystyle\widehat{X}^{[n]}(\xi):={\cal
B}[\widecheck{X}^{[n]}](\xi)=\sum_{k\in{\mathbb{N}}_{0}}\frac{a^{[n,k]}}{\Gamma(k+nb)}\xi^{k+nb-1}.\qquad(\xi\in{\mathbb{C}})$
(276)

In order to avoid confusion, we also define ${\mathbb{I}}_{\xi}:={\cal
B}[w^{-1}]$. It is a homomorphism, and the multiplication of ${\cal
B}[X](\xi)$ and ${\cal B}[Y](\xi)$ is defined using the convolution $*$ as

$\displaystyle{\cal B}[X\cdot Y](\xi)={\cal B}[X]*{\cal
B}[Y](\xi):=\int_{0}^{\xi}d\xi^{\prime}\,{\cal B}[X](\xi^{\prime})\,{\cal
B}[Y](\xi-\xi^{\prime}).$ (277)

The Laplace integral using the integration ray with a complex phase $\theta$,
${\cal L}_{\theta}$, is defined as

$\displaystyle\widecheck{X}^{[n]}_{{\rm ex},\theta}(w):={\cal
L}_{\theta}[\widehat{X}^{[n]}](w):=\int^{+\infty e^{i\theta}}_{0}d\xi e^{-\xi
w}\,\widehat{X}^{[n]}(\xi).$ (278)

In this paper, we consider the case that $\theta=0$ or $0_{\pm}$. Taking the
asymptotic limit, $|w|\rightarrow+\infty$, to $\widecheck{X}^{[n]}_{{\rm
ex},0}(w)$ reduces to $\widecheck{X}^{[n]}(w)$ in Eq.(275). The combination of
the Borel transform and the Laplace integral, ${\cal S}_{\theta}:={\cal
L}_{\theta}\circ{\cal B}$, is called as Borel resummation. It is important
that the Borel resummation is homomorphism, i.e. ${\cal S}_{\theta}[f\cdot
g]={\cal S}_{\theta}[f]\cdot{\cal S}_{\theta}[g]$.

If $\widecheck{X}^{[n]}(w)$ is Borel nonsummable along $\theta=0$, meaning
that the Laplace integral is not performable because of singular points on the
real positive axis on the Borel plane, we introduce the infinitesimal complex
phase, $\theta={0_{\pm}}$, to avoid them in the Laplace integral. In such a
case, due to the singular points, the resulting Borel resummed function (or
transseries) has discontinuity,

$\displaystyle{\cal S}_{0_{+}}[\widecheck{X}^{[n]}](w)\neq{\cal
S}_{0_{-}}[\widecheck{X}^{[n]}](w).$ (279)

Here, we formally introduce the Stokes automorphism,
${\mathfrak{S}}_{\theta}$, to make Eq.(279) to be equality as

$\displaystyle{\cal S}_{0_{+}}[\widecheck{X}^{[n]}](w)={\cal
S}_{0_{-}}\circ{\mathfrak{S}}_{0}[\widecheck{X}^{[n]}](w).$ (280)

If $\widecheck{X}^{[n]}(w)$ is Borel summable along $\theta=0$, then
${\mathfrak{S}}_{0}={\rm id}$ and ${\cal
S}_{0_{+}}[\widecheck{X}^{[n]}](w)={\cal S}_{0_{-}}[\widecheck{X}^{[n]}](w)$.
From now on, we omit the subscript $\theta$ in ${\mathfrak{S}}_{\theta}$
because we only consider the case that $\theta=0$. The Stoke automorphism is a
mapping from a transseries to a transseries generally including exponential
decay. So that it can be extended to group transformation by introducing a
real parameter, $\nu\in{\mathbb{R}}$, as
${\mathfrak{S}}\rightarrow{\mathfrak{S}}^{\nu}$ defined as

$\displaystyle{\mathfrak{S}}^{\nu}=\exp\left[\nu\sum_{\xi_{*}\in\Gamma}\overset{\underset{\bullet}{}}{\Delta}_{\xi_{*}}\right],\qquad\overset{\underset{\bullet}{}}{\Delta}_{\xi_{*}}:=e^{-\xi_{*}w}\Delta_{\xi_{*}},$
(281)

where $\Gamma$ is a set of singular points on the positive real axis of the
Borel plane and $\Delta_{\xi_{*}}$ is the alien derivative with respect to the
singular point, $\xi_{*}$. The alien derivative is a mapping from a formal
transseries to the same type of a formal transseries,

$\displaystyle\Delta_{\xi_{*}}:w^{-b}{\mathbb{C}}[[w^{-1}]]\rightarrow
w^{-b^{\prime}}{\mathbb{C}}[[w^{-1}]]$ (282)

with some $b^{\prime}\in{\mathbb{C}}$. Notice that
${\mathfrak{S}}^{\nu=0}={\rm id}$ and that ${\mathfrak{S}}$ in Eq.(280) is
identical to ${\mathfrak{S}}^{\nu=1}$. The transformation of Stokes
automorphism depends on a given problem. We construct the resurgent relation,
which is a relationship among sectors, by defining the transformation based on
the structure of our ODE in the next section.

For the $i$-dimensional vectorial expression, ${\bf X}=(X_{1},\cdots,X_{i})$,
the above definition is applicable by the extension such as $n\rightarrow{\bf
n}\in{\mathbb{N}}^{i},\zeta^{n}\rightarrow\bm{\zeta}^{\bf
n}=\prod_{j=1}^{i}\zeta_{j}^{n_{j}}$ with
$\zeta_{i}=\sigma_{i}e^{-S_{i}w}/w^{b_{i}}$ on the real positive axis.

###  B.4 Derivation of the resurgent relation

In this section, we review the construction of the resurgent relation. In this
paper, we only consider the case reproducing the PT sector from the NP
sectors, but other cases are also possible. See Refs.[12, 9, 11, 8, 10] in
details.

We define $\widetilde{\bf X}:=U\bar{\bf X}$, where $\widetilde{\bf
X}=(\widetilde{X}_{+},\widetilde{X}_{-})^{\top}$ and $\bar{\bf
X}=(\bar{\Pi},\bar{\pi})^{\top}$, which has the following asymptotic form:

$\displaystyle\widetilde{X}_{\pm}(w)\sim\sum_{{\bf
n}\in{\mathbb{N}}_{0}^{2}}\bm{\zeta}^{\bf n}\widetilde{X}_{\pm}^{[{\bf
n}]}(w),\qquad\widetilde{X}_{\pm}^{[{\bf
n}]}(w)=\sum_{k\in{\mathbb{N}}_{0}}a_{\widetilde{X}_{\pm}}^{[{\bf
n},k]}w^{-k}.$ (283)

Notice that
$a_{\widetilde{X}_{+}}^{[(0,1),0]}=a_{\widetilde{X}_{-}}^{[(1,0),0]}=0$. For
the simplified notation, we define

$\displaystyle\widecheck{X}_{\pm}^{[{\bf n}]}(w)=w^{-{\bf
n}\cdot\bm{\rho}}\widetilde{X}^{[{\bf n}]}_{\pm}(w).$ (284)

As is seen in App.B.1, $\widetilde{\bf X}$ is the asymptotic solution of the
type of ODE given by $\frac{d\widetilde{\bf X}}{dw}=-\Lambda\widetilde{\bf
X}-\frac{1}{w}\left[{\bf V}+{\mathfrak{B}}\widetilde{\bf
X}\right]+O(\widetilde{\bf X}^{2},w^{-2})$, where $\Lambda$ and
${\mathfrak{B}}$ are defined in Eq.(256) and ${\bf V}$ is a constant vector
with nonzero-components. Acting the Borel transform to the ODE yields

$\displaystyle-\xi\widetilde{{\bf X}}_{B}=-\Lambda\widetilde{{\bf
X}}_{B}-{\mathbb{I}}_{\xi}{\bf
V}-{\mathfrak{B}}{\mathbb{I}}_{\xi}*\widetilde{{\bf X}}_{B}+O(\widetilde{{\bf
X}}_{B}^{*2},\xi),$ (285)

where ${\cal B}[{\frac{d\widetilde{\bf X}}{dw}}]=-\xi\widetilde{\bf X}_{B}$,
$\widetilde{\bf X}_{B}:={\cal B}[\widetilde{\bf X}]$, $\widetilde{\bf
X}_{B}^{*2}:=\widetilde{\bf X}_{B}*\widetilde{\bf X}_{B}$, and
${\mathbb{I}}_{\xi}:={\cal B}[w^{-1}]$, and it can be written as

$\displaystyle\widetilde{{\bf
X}}_{B}=-(\Lambda-\xi)^{-1}\left[{\mathbb{I}}_{\xi}{\bf
V}+O({\mathbb{I}}_{\xi}*\widetilde{{\bf X}}_{B},\widetilde{{\bf
X}}_{B}^{*2},\xi)\right].$ (286)

As one can see immediately, the singularity appears at $\xi=S=3/\theta_{0}$.
In this type of problem, the information of the $n$-th sector propagate to the
$(n+1)$-th sector, but not directly to the $(n+2)$-th sector or higher. By
using the alien derivative, it can be expressed by

$\displaystyle{\Delta}_{\ell S}[\widecheck{X}_{\pm}^{[{\bf
0}]}](w)=\begin{cases}\sum_{s=\pm}A_{s}\widecheck{X}_{\pm}^{[\bm{\delta}_{s}]}(w)&\mbox{if
$\ell=1$}\\\
0&\mbox{otherwise}\end{cases},\qquad\bm{\delta}_{\pm}:=\begin{cases}(1,0)&\mbox{for
$\bm{\delta}_{+}$}\\\ (0,1)&\mbox{for $\bm{\delta}_{-}$}\end{cases},$ (287)

where $A_{\pm}\in i{\mathbb{R}}$ is the Stokes constant. For a generic ${\bf
n}=(n_{+},n_{-})$,

$\displaystyle{\Delta}_{\ell S}[\widecheck{X}_{\pm}^{[{\bf n}]}](w)$
$\displaystyle=$
$\displaystyle\begin{cases}\sum_{s=\pm}(n_{s}+1)A_{s}\widecheck{X}_{\pm}^{[{\bf
n}+\bm{\delta}_{s}]}(w)&\mbox{if $\ell=1$}\\\ 0&\mbox{otherwise}\end{cases},$
(288) $\displaystyle(\Delta_{S})^{p}[\widecheck{X}_{\pm}^{[{\bf n}]}](w)$
$\displaystyle=$
$\displaystyle\sum_{q=0}^{p}A_{+}^{q}A_{-}^{p-q}\frac{(n_{+}+q)!(n_{-}+p-q)!}{n_{+}!n_{-}!}\widecheck{X}_{\pm}^{[{\bf
n}+(q,p-q)]}(w).$ (289)

Therefore, the action of the Stokes automorphism is obtained as

$\displaystyle{\mathfrak{S}}^{\nu}[\widetilde{\bf
X}](w;\bm{\sigma})=\widetilde{\bf X}(w;\bm{\sigma}+\nu{\bf A}),$ (290)

where $\bm{\sigma}:=(\sigma_{+},\sigma_{-})$ and ${\bf A}:=(A_{+},A_{-})$.
Additionally, the discontinuity (imaginary ambiguity) cancellation is realized
by the median resummation given by

$\displaystyle{\cal S}_{0_{+}}\circ{\mathfrak{S}}^{-1/2}[\widetilde{\bf
X}](w;\bm{\sigma})={\cal S}_{0_{-}}\circ{\mathfrak{S}}^{+1/2}[\widetilde{\bf
X}](w;\bm{\sigma})$ (291) $\displaystyle\Rightarrow\quad$ $\displaystyle{\cal
S}_{0_{+}}[\widetilde{\bf X}](w;\bm{\sigma}-\tfrac{1}{2}{\bf A})={\cal
S}_{0_{-}}[\widetilde{\bf X}](w;\bm{\sigma}+\tfrac{1}{2}{\bf A}).$

Let us obtain the resurgent relation of $a^{[{\bf
0},k]}_{\widetilde{X}_{\pm}}$ for $k\gg 1$. In order to do it, we start with

$\displaystyle a^{[{\bf 0},k]}_{\widetilde{X}_{\pm}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(k)}{(2\pi i)^{2}}\oint_{|w|\ll
1}\frac{dw}{w}\oint_{|\xi|\ll
1}\frac{d\xi}{\xi^{k}}\widecheck{X}_{\pm,B}^{[{\bf 0}]}(\xi)e^{-\xi w}$ (292)
$\displaystyle=$ $\displaystyle\frac{\Gamma(k)}{(2\pi i)^{2}}\oint_{|w|\ll
1}\frac{dw}{w}(-{\rm Int}_{w})^{k}\oint_{|\xi|\ll
1}d\xi\widecheck{X}_{\pm,B}^{[{\bf 0}]}(\xi)e^{-\xi w},$

where ${\rm Int}_{w}$ is the integration with respect to $w$. Since222222We
assume that singular points exist only on the positive real axis on the Borel
plane.

$\displaystyle\oint_{|\xi|\ll 1}d\xi\widecheck{X}_{\pm,B}^{[{\bf
0}]}(\xi)e^{-\xi w}$ $\displaystyle=$ $\displaystyle({\cal S}_{0_{+}}-{\cal
S}_{0_{-}})[\widecheck{X}_{\pm}^{[{\bf 0}]}](w)$ (293) $\displaystyle=$
$\displaystyle\sum_{p\in{\mathbb{N}}}\frac{(\overset{\underset{\bullet}{}}{\Delta}_{\lambda})^{p}}{p!}[\widecheck{X}_{\pm}^{[{\bf
0}]}](w)\,=\,\sum_{\begin{subarray}{c}{\bf n}\in{\mathbb{N}}_{0}^{2}\\\ |{\bf
n}|>0\end{subarray}}\frac{A_{+}^{n_{+}}A_{-}^{n_{-}}}{C(|{\bf
n}|,n_{-})}e^{-{\bf n}\cdot{\bf S}w}\widecheck{X}_{\pm}^{[{\bf n}]}(w),$

where $|{\bf n}|:=n_{+}+n_{-}$, ${\bf n}\cdot{\bf S}=n_{+}S_{+}+n_{-}S_{-}$,
and $C(p,q)=\left(\begin{smallmatrix}p\\\ q\end{smallmatrix}\right)$ is the
binomial coefficient, one finds that

$\displaystyle(-{\rm Int}_{w})^{k}\oint_{|\xi|\ll
1}d\xi\widecheck{X}_{\pm,B}^{[{\bf 0}]}(\xi)e^{-\xi w}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}{\bf n}\in{\mathbb{N}}_{0}^{2}\\\ |{\bf
n}|>0\end{subarray}}\sum_{h\in{\mathbb{N}}_{0}}\frac{A_{+}^{n_{+}}A_{-}^{n_{-}}}{C(|{\bf
n}|,n_{-})}a_{\widetilde{X}_{\pm}}^{[{\bf n},h]}\frac{({\bf n}\cdot{\bf
S})^{{\bf n}\cdot\bm{\rho}+h-k}}{(k-1)!}\sum_{s=0}^{k-1}\begin{pmatrix}k-1\\\
s\end{pmatrix}\Gamma(k-{\bf n}\cdot\bm{\rho}-h-s)$
$\displaystyle\cdot(-1)^{s}\left[({\bf n}\cdot\bm{\rho}\,w)^{s}-e^{-{\bf
n}\cdot{\bf S}w}({\bf n}\cdot{\bf S}\,w)^{k-{\bf
n}\cdot\bm{\rho}-h}\sum_{r\in{\mathbb{N}}_{0}}\frac{({\bf n}\cdot{\bf
S}\,w)^{r}}{\Gamma(k-{\bf n}\cdot\bm{\rho}-h-s+r+1)}\right],$

where ${\bf n}\cdot\bm{\rho}=n_{+}\rho_{+}+n_{-}\rho_{-}$. Therefore, the
resurgent relation is obtained as

$\displaystyle a_{\widetilde{X}_{\pm}}^{[{\bf 0},k]}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2\pi i}\sum_{\begin{subarray}{c}{\bf
n}\in{\mathbb{N}}_{0}^{2}\\\ |{\bf
n}|>0\end{subarray}}\sum_{h\in{\mathbb{N}}_{0}}\frac{A_{+}^{n_{+}}A_{-}^{n_{-}}}{C(|{\bf
n}|,n_{-})}a_{\widetilde{X}_{\pm}}^{[{\bf n},h]}\frac{\Gamma(k-{\bf
n}\cdot\bm{\rho}-h)}{({\bf n}\cdot{\bf S})^{k-{\bf
n}\cdot\bm{\rho}-h}}.\qquad(k\gg 1)$ (295)

Here, we took the most dominant part of Eq.(LABEL:eq:-IntwInt) for the large
$k$.

##  Appendix C Convergent point in the UV limit

We briefly describe finding convergent points (CPs) the UV limit,
$\tau\rightarrow 0_{+}$, by beginning with ODEs in Eqs.(222)-(225) with
$\tau_{R}=\beta\theta_{0}$. We assume that the temperature is divergent
($\beta\rightarrow 0_{+}$) in this limit, so that the transport coefficients
has the following asymptotic forms:

$\displaystyle\chi_{\beta}\sim\frac{1}{3}+O(z^{2}),\qquad\chi_{\alpha}\sim
O(z^{2}),\qquad z\frac{\Xi^{(1)}_{2}}{\Xi^{(1)}_{1}}\sim\frac{3{\rm
Li}_{3}({\cal R}_{a})}{{\rm Li}_{2}({\cal R}_{a})}+O(z^{2}),$
$\displaystyle\bar{\beta}_{\Pi}\sim O(z^{4}),\qquad
C_{\Pi}\sim-\frac{1}{3}+O(z),\qquad\lambda_{\Pi\pi}\sim 2+O(z^{2}),$
$\displaystyle\bar{\beta}_{\pi}\sim C^{\bar{\beta}_{\pi}}_{0}({\cal
R}_{a})+O(z^{2}),\qquad
C_{\pi}\sim\frac{38}{21},\qquad\lambda_{\pi\Pi}\sim\frac{6}{5}+O(z),$
$\displaystyle D\sim C^{D}_{0}({\cal R}_{a})+O(z^{2}),$ (296)

where ${\cal R}_{a}:=-ae^{-\alpha}$, and

$\displaystyle C^{\bar{\beta}_{\pi}}_{0}({\cal R}_{a})=\frac{4{\rm
Li}_{2}({\cal R}_{a}){\rm Li}_{4}({\cal R}_{a})}{60{\rm Li}_{2}({\cal
R}_{a}){\rm Li}_{4}({\cal R}_{a})-45{\rm Li}_{3}({\cal R}_{a})^{2}},$ (297)
$\displaystyle C^{D}_{0}({\cal R}_{a})=4-\frac{3{\rm Li}_{3}\left({\cal
R}_{a}\right)^{2}\left(3{\rm Li}_{1}\left({\cal R}_{a}\right){\rm
Li}_{3}({\cal R}_{a})-2{\rm Li}_{2}\left({\cal R}_{a}\right)^{2}\right)}{{\rm
Li}_{2}\left({\cal R}_{a}\right)^{2}\left(4{\rm Li}_{2}\left({\cal
R}_{a}\right){\rm Li}_{4}\left({\cal R}_{a}\right)-3{\rm Li}_{3}\left({\cal
R}_{a}\right)^{2}\right)}.$ (298)

The approximated ODEs are given by

$\displaystyle\widehat{F}_{\beta}$ $\displaystyle\sim$
$\displaystyle-\frac{\beta}{s}\left[\frac{1}{3}+\bar{\Pi}-\bar{\pi}\right],$
(299) $\displaystyle\widehat{F}_{\alpha}$ $\displaystyle\sim$
$\displaystyle\frac{3}{s}\cdot\frac{{\rm Li}_{3}({\cal R}_{a})}{{\rm
Li}_{2}({\cal R}_{a})}(\bar{\Pi}-\bar{\pi}),$ (300)
$\displaystyle\widehat{F}_{\bar{\Pi}}$ $\displaystyle\sim$
$\displaystyle\frac{\bar{\Pi}}{\tau_{R}s^{2}}-\frac{1}{s}\left[\frac{\bar{\Pi}}{3}+2\bar{\pi}+C^{D}_{0}({\cal
R}_{a})\left(\bar{\Pi}-\bar{\pi}\right)\bar{\Pi}\right],$
$\displaystyle\widehat{F}_{\bar{\pi}}$ $\displaystyle\sim$
$\displaystyle\frac{\bar{\pi}}{\tau_{R}s^{2}}-\frac{1}{s}\left[\frac{4}{3}C^{\bar{\beta}_{\pi}}_{0}({\cal
R}_{a})-\frac{38}{21}\bar{\pi}+\frac{4}{5}\bar{\Pi}+C^{D}_{0}({\cal
R}_{a})\left(\bar{\Pi}-\bar{\pi}\right)\bar{\pi}\right],$ (302)

where $\widehat{F}_{\cal O}:=-F_{\cal O}/s^{2}$ with $s:=1/\tau$, and thus,
$\frac{d{\cal O}}{ds}=\widehat{F}_{\cal O}$.

#### Case (I): the non-collision part is dominant

We firstly consider the situation that the non-collision term is dominant in
this limit. In this case, the CPs are given by solving the following
equations:

$\displaystyle\frac{\bar{\Pi}_{\rm UV}}{3}+2\bar{\pi}_{\rm UV}+C^{D}_{0}({\cal
R}_{a})\left(\bar{\Pi}_{\rm UV}-\bar{\pi}_{\rm UV}\right)\bar{\Pi}_{\rm
UV}=0,$ (303) $\displaystyle\frac{4}{3}C^{\bar{\beta}_{\pi}}_{0}({\cal
R}_{a})-\frac{38}{21}\bar{\pi}_{\rm UV}+\frac{4}{5}\bar{\Pi}_{\rm
UV}+C^{D}_{0}({\cal R}_{a})\left(\bar{\Pi}_{\rm UV}-\bar{\pi}_{\rm
UV}\right)\bar{\pi}_{\rm UV}=0,$ (304)

where $(\bar{\Pi}_{\rm UV},\bar{\pi}_{\rm
UV}):=\lim_{s\rightarrow+\infty}(\bar{\Pi}(s),\bar{\pi}(s))$. Notice that the
possibility that $\bar{\Pi}_{\rm UV}=\bar{\pi}_{\rm UV}=0$ is rejected because
in this case $\lim_{s\rightarrow+\infty}\alpha(s)=\sigma_{\alpha}$ and
Eq.(304) does not give $\bar{\Pi}_{\rm UV}=\bar{\pi}_{\rm UV}=0$. In addition,
the case that $\alpha\rightarrow-\infty$ is also rejected because $|{\cal
R}_{a}|\rightarrow+\infty$ and $(\bar{\beta}_{\pi},D)$ becomes complex value
in this domain. Thus, the allowed situation is only $\alpha\rightarrow+\infty$
and ${\cal R}_{a}\rightarrow 0$. The real solution in this situation is
obtained as

$\displaystyle(\bar{\Pi}_{\rm UV},\bar{\pi}_{\rm UV})\approx(1.0518,-1.5367).$
(305)

From Eqs.(299)(300), the leading order of $(\beta,\alpha,{\cal R})$ is
obtained as

$\displaystyle\beta\sim\sigma_{\beta}s^{-2.9218},\qquad\alpha\sim
7.7655\cdot\log s+\sigma_{\alpha},\qquad{\cal R}\sim\sigma_{\cal
R}s^{-7.7655}.$ (306)

This result implies that $1/(\beta s^{2})\gg 1/s$ as $s\rightarrow\infty$, so
that it is contradiction to the assumption that the non-collision part is
dominant. Therefore, this possibility is rejected. When constructing
$w=1/(s\beta)$, then $w\sim\sigma_{\beta}^{-1}s^{1.9218}$ and
$\lim_{s\rightarrow+\infty}w(s)=+\infty$, which contradicts to our assumption.

#### Case (II): the collision part $\approx$ the non-collision part

In this case, $\beta\sim\sigma_{\beta}s^{-1}$, and thus $\bar{\Pi}_{\rm
UV}-\bar{\pi}_{\rm UV}=\frac{2}{3}$. The CP in this case is given by solving

$\displaystyle\frac{\bar{\Pi}_{\rm
UV}}{\sigma_{\beta}\theta_{0}}-\left[\frac{\bar{\Pi}_{\rm
UV}}{3}+2\bar{\pi}_{\rm UV}+\frac{2}{3}C^{D}_{0}({\cal R}_{a})\bar{\Pi}_{\rm
UV}\right]=0,$ (307) $\displaystyle\frac{\bar{\pi}_{\rm
UV}}{\sigma_{\beta}\theta_{0}}-\left[\frac{4}{3}C^{\bar{\beta}_{\pi}}_{0}({\cal
R}_{a})-\frac{38}{21}\bar{\pi}_{\rm UV}+\frac{4}{5}\bar{\Pi}_{\rm
UV}+\frac{2}{3}C^{D}_{0}({\cal R}_{a})\bar{\pi}_{\rm UV}\right]=0,$ (308)

with $\bar{\Pi}_{\rm UV}-\bar{\pi}_{\rm UV}=\frac{2}{3}$. The solution is
given by

$\displaystyle(\bar{\Pi}_{\rm
UV},\sigma_{\beta})=\left(1.0868,\frac{0.5640}{\theta_{0}}\right)\ \mbox{and}\
\left(0.24468,-\frac{0.4083}{\theta_{0}}\right),$ (309)

and the integration constant, $\sigma_{\beta}$, can not be taken freely. The
second solution is unphysical because of the negative $\sigma_{\beta}$. The
leading orders of $\alpha$ and ${\cal R}$ are obtained as

$\displaystyle\alpha\sim 2\log s+\sigma_{\alpha},\qquad{\cal
R}\sim\sigma_{\cal R}s^{-2},$ (310)

in both of the solutions. Then, we consider the stability around the CPs. It
can be obtained from the Jacobian defined as

$\displaystyle\widehat{J}_{\rm
UV}:=\left.\begin{pmatrix}\frac{\partial\widehat{F}_{\bar{\Pi}}}{\partial\bar{\Pi}}&\frac{\partial\widehat{F}_{\bar{\Pi}}}{\partial\bar{\pi}}\\\
\frac{\partial\widehat{F}_{\bar{\pi}}}{\partial\bar{\Pi}}&\frac{\partial\widehat{F}_{\bar{\pi}}}{\partial\bar{\pi}}\end{pmatrix}\right|_{(\bar{\Pi},\bar{\pi})\rightarrow(\bar{\Pi}_{\rm
UV},\bar{\pi}_{\rm UV})}\cdot s,$ (311)

where $s$ in the r.h.s. is the regularization. The eigenvalues are obtained as

$\displaystyle{\rm Eigen}(\widehat{J}_{\rm
UV})=(-0.3136,3.3361)\quad\mbox{and}\quad(-3.6940,-1.7284).$ (312)

Thus, on the $(\bar{\Pi},\bar{\pi})$-plane, the CPs are saddle and source,
respectively.

When using $w=1/(s\beta)$ for the flow time, these CPs do not appear because
$1-(\chi_{\beta}+\bar{\Pi}-\bar{\pi})\rightarrow 0$ and all of Eqs.(19)-(22)
are singular in the limit. More precisely, it comes from the fact that $\tau$
and $w$ are not one-to-one correspondence.

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