$\displaystyle\frac{2}{3\pi^{2}}\frac{1}{k^{2}}\bar{\kappa}_{k}(\bar{V}^{{}^{\prime\prime}}_{k}(\bar{\kappa}_{k}))^{2}\mathcal{BB}_{(2,2)}(\tilde{m}_{\pi,k}^{2},\tilde{m}_{\sigma,k}^{2};T)$
$\displaystyle-\frac{N_{c}}{\pi^{2}}\bar{h}_{k}^{2}\int_{0}^{1}dx\bigg{[}(1-\eta_{q,k})\sqrt{x}+\eta_{q,k}x\bigg{]}$
$\displaystyle\times\int_{-1}^{1}d\cos\theta\Bigg{\\{}\bigg{[}\Big{(}\mathcal{FF}_{(1,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{q,k}^{2})-\mathcal{F}_{(2)}(\tilde{m}_{q,k}^{2})\Big{)}$
$\displaystyle-\Big{(}\mathcal{FF}_{(2,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{q,k}^{2})-\mathcal{F}_{(3)}(\tilde{m}_{q,k}^{2})\Big{)}\bigg{]}$
$\displaystyle+\bigg{[}\Big{(}\sqrt{x}-\cos\theta\Big{)}\Big{(}1+r_{F}(x^{\prime})\Big{)}\mathcal{FF}_{(2,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{q,k}^{2})$
$\displaystyle-\mathcal{F}_{(3)}(\tilde{m}_{q,k}^{2})\bigg{]}-\frac{1}{2}\bigg{[}\Big{(}\sqrt{x}-\cos\theta\Big{)}\Big{(}1+r_{F}(x^{\prime})\Big{)}$
$\displaystyle\times\mathcal{FF}_{(1,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{q,k}^{2})-\mathcal{F}_{(2)}(\tilde{m}_{q,k}^{2})\bigg{]}\Bigg{\\}}\,,$
(433)

with $x=\bm{q}^{2}/k^{2}$ and $x^{\prime}=(\bm{q}-\bm{p})^{2}/k^{2}$, where
$\bm{q}$ and $\bm{p}$ stand for the loop and external 3-momenta, respectively,
and $\theta$ is the angle between them. The external momentum is chosen to be
$|\bm{p}|=k$.

The contribution to the gluon anomalous dimension from the quark loop reads

$\displaystyle\eta_{A}^{q}$
$\displaystyle=-\frac{N_{f}}{\pi^{2}}g_{\bar{q}Aq,k}^{2}\int_{0}^{1}dx\bigg{[}(1-\eta_{q,k})\sqrt{x}+\eta_{q,k}x\bigg{]}$
$\displaystyle\times\int_{-1}^{1}d\cos\theta\bigg{[}\Big{(}\mathcal{FF}_{(1,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{q,k}^{2})$
$\displaystyle-\mathcal{FF}_{(2,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{q,k}^{2})\Big{)}+\Big{(}\sqrt{x}\cos^{2}\\!\theta-\cos\theta\Big{)}$
$\displaystyle\times\Big{(}1+r_{F}(x^{\prime})\Big{)}\Big{(}\mathcal{FF}_{(2,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{q,k}^{2})$
$\displaystyle-\frac{1}{2}\mathcal{FF}_{(1,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{q,k}^{2})\Big{)}\bigg{]}\,.$
(434)

As same as in Equation 433, the external momentum in Equation 434 is chosen to
be $|\bm{p}|=k$. The in-medium contribution from the light quarks, included in
Equation 203, is given by

$\displaystyle\Delta\eta_{A}^{q}$
$\displaystyle=\eta_{A}^{q}-\eta_{A}^{q}\big{|}_{T,\mu=0}\,,$ (435)

with $\bar{m}_{q}=\bar{m}_{l}$, $N_{f}=2$ and the light-quark–gluon coupling,
$g_{\bar{l}Al}$ in Equation 434. For the strange quark, since the vacuum
contribution is also presented in Equation 204, one needs

$\displaystyle\eta_{A}^{s}$
$\displaystyle=\eta^{s}_{A,\mathrm{vac}}+\Delta\eta_{A}^{s}\,,$ (436)

which is obtained from Equation 434 with $\bar{m}_{q}=\bar{m}_{s}$, $N_{f}=1$
and the strange-quark–gluon coupling $g_{\bar{s}As}$.

The second term on the r.h.s. of Equation 203,
$\Delta\eta_{A}^{\mathrm{glue}}$, denotes the contribution to the gluon
anomalous dimension from the thermal part of glue sector, which is taken into
account through the thermal screening mass of gluons. The modified gluon
anomalous dimension reads

$\displaystyle\bar{\eta}_{A}=$ $\displaystyle\eta_{A}+\frac{\Delta
m_{\textrm{scr}}^{2}(k,T)}{\bar{Z}_{A}k^{2}}(2-\eta_{A})$
$\displaystyle-\frac{1}{\bar{Z}_{A}k^{2}}\partial_{t}\big{(}\Delta
m_{\textrm{scr}}^{2}(k,T)\big{)}\,.$ (437)

with $\bar{\eta}_{A}=-\partial_{t}\bar{Z}_{A}/\bar{Z}_{A}$, where $\eta_{A}$
is the gluon anomalous dimension without the thermal screening mass, and see
[18] for a more detailed discussion. The screening mass reads

$\displaystyle\Delta m_{\textrm{scr}}^{2}(k,T)$
$\displaystyle=(c\,T)^{2}\exp\Big{[}-\Big{(}\frac{k}{\pi
T}\Big{)}^{n}\Big{]}\,.$ (438)

where $c=2$ is adopted for $N_{f}=2+1$, which is consistent with the result in
[57]. Furthermore, $n=2$ is chosen in Equation 438.

The flow of the quark-gluon couplings in Equation 214 and Equation 215 are
given by

$\displaystyle\overline{\textrm{Flow}}^{(3),A}_{(\bar{q}qA)}=$
$\displaystyle\frac{3}{8\pi^{2}N_{c}}\bar{g}_{\bar{q}qA,k}^{3}\tilde{m}_{q,k}^{2}\bigg{\\{}\frac{2}{15}(5-\eta_{A,k})\mathcal{FB}_{(2,2)}(\tilde{m}_{q,k}^{2},0)$
$\displaystyle+\frac{1}{3}(4-\eta_{q,k})\mathcal{FB}_{(3,1)}(\tilde{m}_{q,k}^{2},0)\bigg{\\}}$
$\displaystyle+\frac{3N_{c}}{8\pi^{2}}\bar{g}_{\bar{q}qA,k}^{2}\bar{g}_{A^{3},k}\bigg{\\{}\frac{1}{20}(5-\eta_{q,k})\mathcal{FB}_{(1,2)}(\tilde{m}_{q,k}^{2},0)$
$\displaystyle-\frac{1}{6}(4-\eta_{q,k})\mathcal{FB}_{(2,1)}(\tilde{m}_{q,k}^{2},0)+\frac{1}{30}(5-2\eta_{q,k})$
$\displaystyle\times\mathcal{FB}_{(2,2)}(\tilde{m}_{q,k}^{2},0)-\frac{4}{15}(5-\eta_{A,k})$
$\displaystyle\times\mathcal{FB}_{(1,2)}(\tilde{m}_{q,k}^{2},0)+\frac{1}{30}(10-3\eta_{A,k})$
$\displaystyle\times\mathcal{FB}_{(1,3)}(\tilde{m}_{q,k}^{2},0)\bigg{\\}}\,.$
(439)

and

$\displaystyle\overline{\textrm{Flow}}^{(3),\phi}_{{(\bar{q}qA)}}=$
$\displaystyle-\frac{1}{8\pi^{2}N_{f}}\bar{g}_{\bar{q}qA,k}\bar{h}_{k}^{2}\bigg{\\{}\frac{1}{6}(4-\eta_{q,k})$
$\displaystyle\times\Big{[}\mathcal{FB}_{(2,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\sigma,k}^{2})$
$\displaystyle+2\tilde{m}_{q,k}^{2}\mathcal{FB}_{(3,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\sigma,k}^{2})\Big{]}+\frac{2}{15}(5-\eta_{\phi,k})$
$\displaystyle\times\Big{[}\mathcal{FB}_{(1,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\sigma,k}^{2})$
$\displaystyle+\tilde{m}_{q,k}^{2}\mathcal{FB}_{(2,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\sigma,k}^{2})\Big{]}\bigg{\\}}$
$\displaystyle-\frac{N_{f}^{2}-1}{8\pi^{2}N_{f}}\bar{g}_{\bar{q}qA,k}\bar{h}_{k}^{2}\bigg{\\{}\frac{1}{6}(4-\eta_{q,k})$
$\displaystyle\times\Big{[}\mathcal{FB}_{(2,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2})$
$\displaystyle+2\tilde{m}_{q,k}^{2}\mathcal{FB}_{(3,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2})\Big{]}+\frac{2}{15}(5-\eta_{\phi,k})$
$\displaystyle\times\Big{[}\mathcal{FB}_{(1,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2})$
$\displaystyle+\tilde{m}_{q,k}^{2}\mathcal{FB}_{(2,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2})\Big{]}\bigg{\\}}\,.$
(440)

Here for the light-quark–gluon coupling in Equation 214 we use the light quark
mass and $N_{f}=2$, and for the strange-quark–gluon coupling in Equation 215
we use the strange quark mass and $N_{f}=1$.

The flow of the four-quark coupling in the $\sigma-\pi$ channel from two gluon
exchanges in Equation 223, as diagrammatically shown in the first line of
Figure 39, reads

$\displaystyle\overline{\mathrm{Flow}}^{(4),A}_{(\bar{q}\tau q)^{2}}$
$\displaystyle=-\frac{3}{2\pi^{2}}\frac{N_{c}^{2}-1}{2N_{c}}\Big{(}\frac{3}{4}-\frac{1}{N_{c}^{2}}\Big{)}\bar{g}_{\bar{q}Aq,k}^{4}$
$\displaystyle\quad\times\bigg{\\{}\frac{2}{15}(5-\eta_{A,k})\Big{[}\mathcal{FB}_{(1,3)}(\tilde{m}_{q,k}^{2},0)$
$\displaystyle\quad-\tilde{m}_{q,k}^{2}\mathcal{FB}_{(2,3)}(\tilde{m}_{q,k}^{2},0)\Big{]}$
$\displaystyle\quad+\frac{1}{12}(4-\eta_{q,k})\Big{[}\mathcal{FB}_{(2,2)}(\tilde{m}_{q,k}^{2},0)$
$\displaystyle\quad-2\tilde{m}_{q,k}^{2}\mathcal{FB}_{(3,2)}(\tilde{m}_{q,k}^{2},0)\Big{]}\bigg{\\}}\,.$
(441)

The contributions to the flow of four-quark coupling from two meson exchanges,
as shown in the second line of Figure 39, reads

$\displaystyle\overline{\mathrm{Flow}}^{(4),\phi}_{(\bar{q}\tau q)^{2}}$
$\displaystyle=\frac{1}{32\pi^{2}}\frac{N_{f}^{2}-2}{N_{f}N_{c}}\bar{h}_{k}^{4}\bigg{\\{}\frac{2}{15}(5-\eta_{\phi,k})$
$\displaystyle\quad\times\Big{[}\Big{(}\mathcal{FBB}_{(1,1,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2},\tilde{m}_{\sigma,k}^{2})$
$\displaystyle\quad+\mathcal{FBB}_{(1,1,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\sigma,k}^{2},\tilde{m}_{\pi,k}^{2})$
$\displaystyle\quad-2\mathcal{FB}_{(1,3)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2})\Big{)}$
$\displaystyle\quad-\tilde{m}_{q,k}^{2}\Big{(}\mathcal{FBB}_{(2,1,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2},\tilde{m}_{\sigma,k}^{2})$
$\displaystyle\quad+\mathcal{FBB}_{(2,2,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2},\tilde{m}_{\sigma,k}^{2})$
$\displaystyle\quad-2\mathcal{FB}_{(2,3)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2})\Big{)}\Big{]}+\frac{1}{6}(4-\eta_{q,k})$
$\displaystyle\quad\times\Big{[}\Big{(}\mathcal{FBB}_{(2,1,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2},\tilde{m}_{\sigma,k}^{2})$
$\displaystyle\quad-\mathcal{FB}_{(2,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2})\Big{)}$
$\displaystyle\quad-2\tilde{m}_{q,k}^{2}\Big{(}\mathcal{FBB}_{(3,1,1)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2},\tilde{m}_{\sigma,k}^{2})$
$\displaystyle\quad-\mathcal{FB}_{(3,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2})\Big{)}\Big{]}\bigg{\\}}\,.$
(442)

The flow of the Yukawa coupling in Equation 228 reads

$\displaystyle\overline{\mathrm{Flow}}^{(3)}_{(\bar{q}\bm{\tau}q)\bm{\pi}}$
$\displaystyle\quad=\frac{1}{4\pi^{2}N_{f}}\bar{h}_{k}^{3}\bigg{[}-(N_{f}^{2}-1)$
$\displaystyle\qquad\times
L_{(1,1)}^{(4)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2},\eta_{q,k},\eta_{\phi,k};T,\mu_{q},p_{0,\text{\tiny{ex}}})$
$\displaystyle\qquad+L_{(1,1)}^{(4)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\sigma,k}^{2},\eta_{q,k},\eta_{\phi,k};T,\mu_{q},p_{0,\text{\tiny{ex}}})\bigg{]}$
$\displaystyle\qquad-\frac{3}{2\pi^{2}}\frac{N_{c}^{2}-1}{2N_{c}}\bar{g}_{\bar{q}qA,k}^{2}\bar{h}_{k}$
$\displaystyle\qquad\times
L_{(1,1)}^{(4)}(\tilde{m}_{q,k}^{2},0,\eta_{q,k},\eta_{A,k};T,\mu_{q},p_{0,\text{\tiny{ex}}})\,.$
(443)

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