0.01$ MeV, the fRG results of both the $O(4)$ and $Z(2)$ symmetry universality
classes are consistent with the conformal bootstrap results. This indicates
that the size of the critical region is extremely small, far smaller than
$\sim 1$ MeV. Similar estimates for the size of the critical region are also
found in, e.g., [98, 190, 199, 200].

##  IV QCD at finite temperature and density

In this section we would like to present a brief review on recent progresses
in studies of QCD at finite temperature and densities within the fRG approach
to the first-principle QCD, which are mainly based on the work in [18, 62].
The relevant fRG flows at finite temperature and densities there have been
developed from the counterparts in the vacuum in [59, 60]. Furthermore, these
flows are also built upon the state-of-the-art quantitative fRG results for
Yang-Mills theory in the vacuum [56] and at finite temperature [57], QCD in
the vacuum [58, 61].

In the following we focus on the chiral phase transition. As mentioned above,
the QCD phase transitions involve both the chiral phase transition and the
confinement-deconfinement phase transition. In fact, related subjects of the
deconfinement phase transition, such as the Wilson loop, the Polyakov loop,
the flows of the effective action of a background gauge field, the relation
between the confinement and correlation functions, etc., have been widely
studied within the fRG approach to Yang-Mills theory and QCD. Significant
progresses have been made in relevant studies, see [55] for a recent overview,
and also e.g. [201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212,
213, 108, 70, 214, 215, 216].

###  IV.1 Propagators and anomalous dimensions

Figure 31: Diagrammatic representation of the flow equations for the inverse
quark, gluon and meson propagators, respectively, where the dashed lines stand
for the mesons and the dotted lines denote the ghost. Figure 32: Quark (left
panel, $Z_{q}$) and mesonic (right panel, $\bar{Z}_{\phi}$) wave function
renormalizations of $N_{f}=2+1$ flavor QCD obtained in the fRG with RG scale
$k=0$, as functions of the temperature with different values of the baryon
chemical potential. The plot is adopted from [18]. Figure 33: Gluon dressing
functions $1/Z_{A}(p)$ of $N_{f}=2$ and $N_{f}=2+1$ flavor QCD in the vacuum
as functions of the momentum, where the fRG results are in comparison to
lattice calculations of $N_{f}=2$ flavors [217] and $N_{f}=2+1$ flavors [218,
219].The plot is adopted from [18]. Figure 34: Gluon dressing function
$1/Z_{A}(p)$ and the gluon propagator $G_{A}=1/[Z_{A}(p)\,p^{2}]$ (inlay) as
functions of momenta for QCD of $N_{f}=2+1$ flavors in the vacuum, where the
fRG results denoted by the black lines are compared with the continuum
extrapolated lattice results by the RBC/UKQCD collaboration, see e.g., [220,
218, 219]. For the fRG results the momentum dependence is given by
$p^{2}=k^{2}$. The plot is adopted from [18]. Figure 35: Gluon dressing
function $1/Z_{A}$ of the $N_{f}=2+1$ QCD as a function of spatial momenta
$|{\bm{p}}|$ with several different values of temperature (left panel) and
baryon chemical potential (right panel), where the Matsubara frequency $p_{0}$
is chosen to be vanishing. The identification $\bm{p}^{2}=k^{2}$ is used. The
plot is adopted from [18]. Figure 36: Gluon dressing function $1/Z_{A}$ of
$N_{f}=2+1$ flavors as a function of spatial momenta $|{\bm{p}}|$ at several
values of temperature obtained in fRG, which is also compared with the lattice
results of $N_{f}=2+1+1$ in [221]. The plot is adopted from [18].

The flow equations of inverse propagators are obtained by taking the second
derivative of the Wetterich equation in Equation 86 with respect to respective
fields. The resulting flow equations of the inverse quark, gluon and meson
propagators are shown diagrammatically in Figure 31. Making appropriate
projections for these flow equations, one is able to arrive at the flow of the
wave function renormalization $Z_{\Phi,k}$ as shown in Equation 53 for a given
field $\Phi$, which includes nontrivial dispersion relation for the field, and
is more conveniently reformulated as the anomalous dimension as follows

$\displaystyle\eta_{\Phi,k}$
$\displaystyle=-\frac{\partial_{t}Z_{\Phi,k}}{Z_{\Phi,k}}\,.$ (191)

The quark two-point correlation function defined in Equation 90, see also
Equation 350, reads

$\displaystyle\Gamma^{(2)\bar{q}q}_{k}(p)=$
$\displaystyle-\frac{\delta^{2}\Gamma_{k}[\Phi]}{\delta\bar{q}(-p)\delta
q(p)}\bigg{|}_{\Phi=\Phi_{\mathrm{EoM}}}$ $\displaystyle=$ $\displaystyle
Z_{q,k}(p)i\gamma\cdot p+m_{q,k}(p)\,,$ (192)

where $\Phi_{\mathrm{EoM}}$ denotes the fields in Equation 68 on their
respective equations of motion, that are vanishing except the $\sigma$ field
and the temporal gluon field $A_{0}$. Note that the minus sign on the r.h.s.
of Equation 192, while not appearing in Equation 350, is due to the fact that
the right derivative is used in Equation 350. Moreover, the quark chemical
potentials in Equation 52 have been assumed to be vanishing in Equation 192.
Apparently, the quark wave function renormalization and mass are readily
obtained by projecting Equation 192 onto the vector and scalar channels,
respectively, which read

$\displaystyle Z_{q,k}(p)=$
$\displaystyle\frac{1}{4i}\frac{1}{p^{2}}\operatorname{tr}\Big{(}\gamma\cdot
p\,\Gamma^{(2)\bar{q}q}_{k}(p)\Big{)}$ $\displaystyle m_{q,k}(p)=$
$\displaystyle\frac{1}{4}\operatorname{tr}\Big{(}\Gamma^{(2)\bar{q}q}_{k}(p)\Big{)}\,,$
(193)

where the trace runs only in the Dirac space. Neglecting the dependence of the
quark wave function renormalization on the spacial momentum $\bm{p}$, one
arrives at the quark anomalous dimension as follows

$\displaystyle\eta_{q,k}(p_{0})=$ $\displaystyle\frac{1}{4Z_{q,k}(p_{0})}$
$\displaystyle\times\mathrm{Re}\left[\frac{\partial}{\partial(|\bm{p}|^{2})}\mathrm{tr}\left(i\bm{\gamma}\cdot\bm{p}\big{(}\partial_{t}\Gamma^{(2)\bar{q}q}_{k}(p)\big{)}\right)\right]_{\bm{p}=0}\,,$
(194)

where the computation is done at $\bm{p}=0$, and $p_{0}$ is a small, but
nonvanishing frequency. The nontrivial choice of $p_{0}$ and the fact that the
expression in the square bracket in Equation 194 is complex-valued at nonzero
chemical potentials, are both related to constraints from the Silver-Blaze
property for the correlation functions at finite chemical potentials, and see,
e.g., [222, 120, 119, 127, 18] for more details. In Equation 194 Re denotes
that the real part of the expression in the square bracket is adopted. Note
that in Equation 194 the difference between the parts of the quark wave
function renormalization longitudinal and transversal to the heat bath is
ignored, where the projection is done on the spacial component, and thus the
transversal anomalous dimension is used. The explicit expression of
$\eta_{q,k}$ is given in Equation 431. In the left panel of Figure 32, the
quark wave function renormalization $Z_{q,k=0}(p_{0},\bm{p}=0)$ is depicted as
a function of the temperature with different values of the baryon chemical
potential.

The mesonic two-point correlation functions, i.e., those of the $\pi$ and
$\sigma$ fields, are given by

$\displaystyle\Gamma^{(2)\pi\pi}_{k}(p)=$
$\displaystyle\frac{\delta^{2}\Gamma_{k}[\Phi]}{\delta\pi_{i}(-p)\delta\pi_{j}(p)}\bigg{|}_{\Phi=\Phi_{\mathrm{EoM}}}$
$\displaystyle=$
$\displaystyle\big{(}Z_{\pi,k}(p)p^{2}+m_{\pi,k}^{2}\big{)}\delta_{ij}\,,$
(195)

and

$\displaystyle\Gamma^{(2)\sigma\sigma}_{k}(p)=$
$\displaystyle\frac{\delta^{2}\Gamma_{k}[\Phi]}{\delta\sigma(-p)\delta\sigma(p)}\bigg{|}_{\Phi=\Phi_{\mathrm{EoM}}}$
$\displaystyle=$ $\displaystyle Z_{\sigma,k}(p)p^{2}+m_{\sigma,k}^{2}\,,$
(196)

respectively, where the curvature masses of the mesons read

$\displaystyle m_{\pi,k}^{2}=$ $\displaystyle V_{k}^{\prime}(\rho)\,,\qquad
m_{\sigma,k}^{2}=$ $\displaystyle V_{k}^{\prime}(\rho)+2\rho
V_{k}^{(2)}(\rho)\,.$ (197)

As same as the quark wave function renormalization, the thermal splitting of
the mesonic wave function renormalization in parts longitudinal and
transversal to the heat bath is neglected. Moreover, one has assumed a unique
renormalization for both the pion and sigma fields, viz.,
$Z_{\phi,k}=Z_{\pi,k}=Z_{\sigma,k}$. The validity of these approximations have
been verified seriously in [148]. The anomalous dimension of mesons at
vanishing frequency $p_{0}=0$ and a finite spacial momentum $\bm{p}$ is given
by

$\displaystyle\eta_{\phi}(0,\bm{p})=$
$\displaystyle-\frac{\delta_{ij}}{3Z_{\phi}(0,\bm{p})}\frac{\partial_{t}\Gamma_{\pi_{i}\pi_{j}}^{(2)}(0,\bm{p})-\partial_{t}\Gamma_{\pi_{i}\pi_{j}}^{(2)}(0,0)}{\bm{p}^{2}}\,,$
(198)

Two cases are of special interest: One is the choice $\bm{p}^{2}=k^{2}$, i.e.,
$\eta_{\phi}(0,k)$, which captures the most momentum dependence of the mesonic
two-point correlation functions, and thus is usually used in the r.h.s. of
flow equations involving mesonic degrees of freedom, for more detailed
discussions see [18]. The anomalous dimension $\eta_{\phi}(0,k)$ leaves us
with the renormalization constant defined as follows

$\displaystyle\frac{1}{\bar{Z}_{\phi,k}}\partial_{t}\bar{Z}_{\phi,k}\equiv-\eta_{\phi}(0,k)\,,\quad\textrm{with}\quad\bar{Z}_{\phi,k=\Lambda}=1\,.$
(199)

In the right panel of Figure 32, $\bar{Z}_{\phi,k=0}$ is depicted as a
function of the temperature with different values of the baryon chemical
potentials. The other is the case $\bm{p}=0$, and Equation 198 is reduced the
equation as follows

$\displaystyle\eta_{\phi}(0)$
$\displaystyle=-\frac{1}{3Z_{\phi}(0)}\delta_{ij}\left[\frac{\partial}{\partial\bm{p}^{2}}\partial_{t}\Gamma^{(2)}_{\pi_{i}\pi_{j}}\right]_{p=0}\,.$
(200)

The related wave function renormalization $Z_{\phi,k}(0)$ is used to extract
the renormalized meson mass as

$\displaystyle\bar{m}_{\pi}$
$\displaystyle=\frac{m_{\pi,k=0}}{\sqrt{Z_{\phi,k=0}(0)}}\,,\qquad\bar{m}_{\sigma}=\frac{m_{\sigma,k=0}}{\sqrt{Z_{\phi,k=0}(0)}}\,,$
(201)

where $\bar{m}_{\pi}$ and $\bar{m}_{\sigma}$ are approximately equal to their
pole masses, respectively. The explicit expressions for Equation 200 and
Equation 198 are presented in Equation 432 and Equation 433, respectively.

For the ghost anomalous dimension, one extracts it from the relation as
follows

$\displaystyle\eta_{c}=$
$\displaystyle-\left.\frac{p\partial_{p}Z^{\mathrm{QCD}}_{c,k=0}(p)}{Z^{\mathrm{QCD}}_{c,k=0}(p)}\right|_{p=k}\,,$
(202)

where $Z^{\mathrm{QCD}}_{c,k=0}(p)$ denotes the momentum-dependent ghost wave
function renormalization in QCD with $N_{f}=2$ flavors in the vacuum obtained
in [61]. Note that the ghost propagator is found to be very insensitive to the
effects of finite temperature as well as the quark contributions for $N_{f}=2$
and $N_{f}=2+1$ flavors, see e.g. [57].

The gluon anomalous dimension is decomposed into a sum of three parts, as
follows

$\displaystyle\eta_{A}=$
$\displaystyle\eta_{A,\mathrm{vac}}^{\mathrm{QCD}}+\Delta\eta_{A}^{\mathrm{glue}}+\Delta\eta_{A}^{q}\,,$
(203)

where the first term on the r.h.s. denotes the gluon anomalous dimension in
the vacuum, and the other two terms stand for the medium contributions to the
gluon anomalous dimension from the glue (gluons and ghosts) and quark loops,
respectively. The vacuum contribution in Equation 203 is further expressed as

$\displaystyle\eta_{A,\mathrm{vac}}^{\mathrm{QCD}}=$
$\displaystyle\left.\eta_{A,\mathrm{vac}}^{\mathrm{QCD}}\right|_{N_{f}=2}+\eta^{s}_{A,\mathrm{vac}}\,,$
(204)

where the two terms on the r.h.s. denote the contributions from the light
quarks of $N_{f}=2$ flavors and the strange quark. The former one is inferred
from the gluon dressing function $Z^{\textrm{QCD}}_{A,k=0}(p)$ for the
$N_{f}=2$ flavor QCD in [61], which reads

$\displaystyle\left.\eta_{A,\mathrm{vac}}^{\mathrm{QCD}}\right|_{N_{f}=2}=$
$\displaystyle-\left.\frac{p\partial_{p}Z^{\mathrm{QCD}}_{A,k=0}(p)}{Z^{\mathrm{QCD}}_{A,k=0}(p)}\right|_{p=k}\,.$
(205)

The explicit expressions of $\Delta\eta_{A}^{q}$ in Equation 203 and
$\eta^{s}_{A,\mathrm{vac}}$ in Equation 204 can be found in Equation 434,
Equation 435, Equation 436. Moreover, the in-medium contribution to the gluon
anomalous dimension resulting from the glue sector, i.e., the second term on
the r.h.s. of Equation 203, is taken into account in Equation 437.

In Figure 33 the gluon dressing functions $1/Z_{A}(p)$ of $N_{f}=2$ and
$N_{f}=2+1$ flavor QCD in the vacuum are shown. The fRG and lattice results
are presented. Here the fRG gluon dressing of $N_{f}=2$ flavors is inputted
from [61], as shown in Equation 205, which is also in quantitative agreement
with the lattice result in [217]. Note that the gluon dressing of $N_{f}=2+1$
flavors here is a genuine prediction, which is in good agreement with the
respective lattice results [218, 219]. In Figure 34 both the gluon dressing
function and the gluon propagator of $N_{f}=2+1$ flavors are presented. The
calculated gluon dressing functions at finite temperature and baryon chemical
potential in fRG are shown in Figure 35. In the left panel of Figure 35
several different values of temperature are chosen with $\mu_{B}=0$, and it is
found that the gluon dressing function $1/Z_{A}$ decreases with the increase
of temperature. In the right panel, several different values of $\mu_{B}$ are
adopted while with $T=150$ MeV fixed. It is observed that the dependence of
the gluon dressing function on the baryon chemical potential is very small.
The gluon dressing functions at finite temperature obtained in fRG are
compared with the relevant lattice results from [221] in Figure 36, where
several different values of temperature are chosen. One can see that the gluon
dressing at finite temperature in fRG is comparable to that of lattice QCD.

###  IV.2 Strong couplings

Figure 37: Diagrammatic representation of the flow equations for the quark-
gluon, three-gluon, and the quark-meson vertices, respectively. Figure 38:
Quark-gluon couplings for light quarks ($\alpha_{\bar{l}lA}$) and strange
quarks ($\alpha_{\bar{s}sA}$), and the three-gluon coupling ($\alpha_{A^{3}}$)
of $N_{f}=2+1$ flavor QCD as functions of the RG scale $k$ at several values
of the temperature and vanishing baryon chemical potential. The plot is
adopted from [18].

The quark-gluon coupling in Equation 57, the ghost-gluon coupling in Equation
58, and the three-gluon or four-gluon coupling in Equation 54, etc., are
consistent with one another in the perturbative regime of high energy, due to
the Slavnov-Taylor identities (STIs) resulting from the gauge symmetry.
However, when the momenta or the RG scale are decreased to $k\lesssim 1\sim
3\,$ GeV, the gluon mass gap begins to affect the running of strong couplings,
and also the transversal strong couplings can not be identified with the
longitudinal ones via the modified STIs, see, e.g., [223, 224, 56, 216] for
more details. Consequently, different strong couplings deviate from one
another in the low energy regime of $k\lesssim 1\sim 3\,$ GeV [58, 56, 61,
57], and thus it is necessary to distinguish them by adding appropriate
suffixes. The couplings of the purely gluonic sector read

$\displaystyle\alpha_{A^{3},k}=$
$\displaystyle\frac{1}{4\pi}\frac{\lambda_{A^{3},k}^{2}}{Z_{A,k}^{3}}\,,\qquad\alpha_{A^{4},k}=\frac{1}{4\pi}\frac{\lambda_{A^{4},k}}{Z_{A,k}^{2}}\,,$
(206) $\displaystyle\alpha_{\bar{c}cA,k}=$
$\displaystyle\frac{1}{4\pi}\frac{\lambda_{\bar{c}cA,k}^{2}}{Z_{A,k}\,Z_{c,k}^{2}}\,,$
(207)

where $\lambda_{A^{3},k}$, $\lambda_{A^{4},k}$ and $\lambda_{\bar{c}cA,k}$
denote the three-gluon, four-gluon, ghost-gluon dressing functions,
respectively, as shown in Equation 368, Equation 374, Equation 362. The
couplings of the matter sector read

$\displaystyle\alpha_{\bar{l}lA,k}=$
$\displaystyle\frac{1}{4\pi}\frac{\lambda_{\bar{l}lA,k}^{2}}{Z_{A,k}\,Z_{q,k}^{2}}\,,\qquad\alpha_{\bar{s}sA,k}=\frac{1}{4\pi}\frac{\lambda_{\bar{s}sA,k}^{2}}{Z_{A,k}\,Z_{q,k}^{2}}\,,$
(208)

where the light-quark-gluon coupling and the strange-quark-gluon coupling have
been distinguished. From the calculated results in, e.g., [59, 61], it is
reasonable to adopt the approximation as follows

$\displaystyle\alpha_{A^{4},k}=\alpha_{A^{3},k}\,,\qquad\alpha_{\bar{c}cA,k}\simeq\alpha_{\bar{l}lA,k}\,.$
(209)

The flow equation of the quark-gluon vertex is presented in the first line of
Figure 37. Projecting onto the classical tensor structure of the quark-gluon
vertex, denoted by

$\displaystyle\big{(}S^{(3)}_{\bar{q}qA}\big{)}^{a}_{\mu}$
$\displaystyle\equiv-i\gamma_{\mu}t^{a}\,,$ (210)

one is led to the flow of the quark-gluon coupling as follows

$\displaystyle\partial_{t}\bar{g}_{\bar{q}qA,k}=$
$\displaystyle\left(\frac{1}{2}\eta_{A}+\eta_{q}\right)\bar{g}_{\bar{q}qA,k}+\frac{1}{8(N_{c}^{2}-1)}$
$\displaystyle\times\operatorname{tr}\,\left[\left(\overline{{\mathrm{Flow}}}^{(3)}_{\bar{q}qA}\right)^{a}_{\mu}\left(S^{{(3)}}_{\bar{q}qA}\right)^{a}_{\mu}\right]\big{(}\\{p\\})\,,$
(211)

where the trace sums over the Dirac and color spaces, and $\\{p\\}$ stands for
the set of external momenta for the vertex. Here
$g_{\bar{q}qA,k}\equiv\lambda_{\bar{q}qA,k}$ is used. In Equation 211 the flow
of quark-gluon vertex, i.e., the r.h.s. of the first line in Figure 37, is
denoted by

$\displaystyle\left(\overline{{\mathrm{Flow}}}^{(3)}_{\bar{q}qA}\right)^{a}_{\mu}=$
$\displaystyle-\frac{1}{Z_{A,k}^{1/2}Z_{q,k}}\left(\partial_{t}\Gamma_{k}^{(3)\bar{q}qA}\right)^{a}_{\mu}\,,$
(212)

with

$\displaystyle\left(\Gamma_{k}^{(3)\bar{q}qA}\right)^{a}_{\mu}\equiv\frac{\delta}{\delta
A^{c}_{\mu}}\frac{\overrightarrow{\delta}}{\delta\bar{q}}\Gamma_{k}\frac{\overleftarrow{\delta}}{\delta
q}\,.$ (213)

Note that besides the classical tensor structure of the quark-gluon vertex in
Equation 210, some nonclassical tensor structures also play a sizable role in
the dynamical breaking of the chiral symmetry [58, 61, 225], see, e.g., [18]
for more detailed discussions. From Equation 211 one formulates the flow of
the light-quark–gluon coupling as

$\displaystyle\partial_{t}\bar{g}_{\bar{l}lA,k}=$
$\displaystyle\left(\frac{1}{2}\eta_{A}+\eta_{q}\right)\bar{g}_{\bar{l}lA,k}+\overline{\textrm{Flow}}^{(3),A}_{(\bar{l}lA)}+\overline{\textrm{Flow}}^{(3),\phi}_{(\bar{l}lA)}\,,$
(214)

where the second term on the r.h.s. corresponds to the two diagrams on the
r.h.s. of the flow equation for the quark-gluon vertex as shown in the first
line of Figure 37, and the last term to the last diagram, which arise from the
quark-gluon, quark-meson fluctuations, respectively. For the strange-
quark–gluon coupling, one has

$\displaystyle\partial_{t}\bar{g}_{\bar{s}sA,k}=$
$\displaystyle\left(\frac{1}{2}\eta_{A}+\eta_{q}\right)\bar{g}_{\bar{s}sA,k}+\overline{\textrm{Flow}}^{(3),A}_{(\bar{s}sA)}\,,$
(215)

where the contribution from the strange-quark–meson interactions are
neglected, that is reasonable due to relatively larger masses of the strange
quark and strange mesons. The explicit expressions in Eqs. (214) and (215) can
be found in Equation 439 and Equation 440.

In the same way, the flow equation of the three-gluon coupling,
$g_{A^{3},k}\equiv\lambda_{A^{3},k}$, is readily obtained from the flow of the
three-gluon vertex in the second line of Figure 37. The flow of the three-
gluon coupling is decomposed into a sum of the vacuum part and the in-medium
contribution, as follows

$\displaystyle\partial_{t}\bar{g}_{A^{3},k}=$
$\displaystyle\partial_{t}\bar{g}_{A^{3},k}^{\mathrm{vac}}+\partial_{t}\Delta\bar{g}_{A^{3},k}\,,$
(216)

where the vacuum part, i.e., the first term on the r.h.s. is computed in [59],
and the second term is identified with the in-medium contribution of the
quark-gluon coupling, to wit,

$\displaystyle\partial_{t}\Delta\bar{g}_{A^{3},k}=$
$\displaystyle\partial_{t}\Delta\bar{g}_{\bar{l}lA,k}\,.$ (217)

In Figure 38 the light-quark–gluon coupling, the strange-quark–gluon coupling,
and the three-gluon coupling are depicted as functions of the RG scale for
several different values of the temperature. It is observed that different
couplings are consistent with one another in the regime of $k\gtrsim 3$ GeV,
while deviations develop in the nonperturbative or even semiperturbative
scale. Moreover, one can see that the strong couplings decrease with the
increasing temperature.

###  IV.3 Dynamical hadronization, four-quark couplings and Yukawa couplings

Figure 39: Diagrammatic representation of the flow equation for the four-quark
vertex. The first line on the r.h.s. denotes the contributions from two gluon
exchanges and the second line those from two meson exchanges. The last line
stands for the contributions from the mixed diagrams with one gluon exchange
and one meson exchange. Figure 40: Left panel: Comparison between the flow of
the four-quark coupling in the $\sigma-\pi$ channel from two gluon exchanges,
$\overline{\mathrm{Flow}}^{(4),A}_{(\bar{q}\tau q)^{2}}$ in Equation 223, as
shown in the first line in Figure 39, and that from two meson exchanges,
$\overline{\mathrm{Flow}}^{(4),\phi}_{(\bar{q}\tau q)^{2}}$, in the second
line. The flows are depicted as functions of the RG scale with $T=0$ and
$\mu_{B}=0$. Right panel: Absolute values of the ratios
$\overline{\mathrm{Flow}}^{(4),A}_{(\bar{q}\tau
q)^{2}}/\overline{\mathrm{Flow}}^{(4)}_{(\bar{q}\tau q)^{2}}$ and
$\overline{\mathrm{Flow}}^{(4),\phi}_{(\bar{q}\tau
q)^{2}}/\overline{\mathrm{Flow}}^{(4)}_{(\bar{q}\tau q)^{2}}$ as functions of
the RG scale. The position where
$\overline{\mathrm{Flow}}^{(4),\phi}_{(\bar{q}\tau q)^{2}}$ changes sign is
labeled by the blue vertical line in both plots. The plots are adopted from
[18]. Figure 41: Left panel: Renormalized Yukawa coupling of $N_{f}=2+1$
flavor QCD at vanishing RG scale $k=0$ as a function of the temperature with
several different values of $\mu_{B}$ obtained in fRG. Right panel:
Renormalized Yukawa coupling of $N_{f}=2+1$ flavor QCD as a function of the RG
scale with several different values of $T$ and $\mu_{B}=0$. The plots are
adopted from [18]. Figure 42: Effective four-quark coupling of $N_{f}=2+1$
flavor QCD in the pseudoscalar channel
$\bar{h}_{k}^{2}/(2\bar{m}_{\pi,k}^{2})$ as a function of the RG scale with
several different values of temperature and $\mu_{B}=0$. The plot is adopted
from [18]. Figure 43: Effective four-quark coupling of $N_{f}=2+1$ flavor QCD
in the pseudoscalar channel at vanishing RG scale
$\bar{h}_{k=0}^{2}/(2\bar{m}_{\pi,k=0}^{2})$ as a function of the temperature
with several different values of baryon chemical potential. The plot is
adopted from [18].

Following Section II.5, one performs the dynamical hadronization for the
$\sigma-\pi$ channel, and use

$\displaystyle\langle\partial_{t}\hat{\phi}_{k}\rangle=$
$\displaystyle\dot{A}_{k}\,\bar{q}\tau q\,,$ (218)

with $\tau=(T^{0},i\gamma_{5}\bm{T})$ as shown in Equation 52, where
$\dot{A}_{k}$ is called as the hadronization function. It is more convenient
to adopt the dimensionless, renormalized hadronization function and four-quark
coupling

$\displaystyle\dot{\tilde{A}}=$
$\displaystyle\frac{Z_{\phi,k}^{1/2}}{Z_{q,k}}k^{2}\dot{A}_{k}\,,\qquad\tilde{\lambda}_{q,k}=\frac{k^{2}\lambda_{q,k}}{Z_{q,k}^{2}}\,,$
(219)

and the renormalized Yukawa coupling,

$\displaystyle\bar{h}_{k}=$
$\displaystyle\frac{h_{k}}{Z_{\phi,k}^{1/2}Z_{q,k}}\,.$ (220)

Inserting the effective action in Equation 52 into Equation 86 and performing
a projection on the four-quark interaction in the $\sigma-\pi$ channel, one is
left with

$\displaystyle\partial_{t}\tilde{\lambda}_{q,k}=$ $\displaystyle
2(1+\eta_{q,k})\tilde{\lambda}_{q,k}+\overline{\mathrm{Flow}}^{(4)}_{(\bar{q}\tau
q)^{2}}+\dot{\tilde{A}}\,\bar{h}_{k}\,,$ (221)

where

$\displaystyle\overline{\mathrm{Flow}}^{(4)}_{(\bar{q}\tau q)^{2}}\equiv$
$\displaystyle-\frac{k^{2}}{Z_{q,k}^{2}}\left(\partial_{t}\Gamma_{k,(\bar{q}\tau
q)^{2}}^{(4)}\right)_{\dot{A}_{k}=0}\,,$ (222)

is the flow of the four-quark coupling in the $\sigma-\pi$ channel, whose
contributions have been depicted in Figure 39. One can see that the
contributed diagrams can be classified into three sets, which correspond to
the three lines on the r.h.s. of the flow equation in Figure 39. The first
line is comprised of diagrams with two gluon exchanges, and the second line
with two meson exchanges. The last line denotes the contributions from the
mixed diagrams with one gluon exchange and one meson exchange. The mixed
diagrams are negligible, since the dynamics of gluons and mesons dominate in
different regimes of the RG scale, as shown in Figure 40. Therefore, the four-
quark flow in Equation 222 can be further written as

$\displaystyle\overline{\mathrm{Flow}}^{(4)}_{(\bar{q}\tau q)^{2}}=$
$\displaystyle\overline{\mathrm{Flow}}^{(4),A}_{(\bar{q}\tau
q)^{2}}+\overline{\mathrm{Flow}}^{(4),\phi}_{(\bar{q}\tau q)^{2}}\,,$ (223)

where the two terms on the r.h.s. correspond to the contributions from the two
gluon exchanges and two meson exchanges, respectivley. Their explicit
expressions are presented in Equation 441 and Equation 442. In Figure 40 the
two terms on the r.h.s. of Equation 223 and their ratios with respect to the
total flow of the four-quark coupling are depicted as functions of the RG
scale with $T=0$ and $\mu_{B}=0$. It is observed that the flow resulting from
the gluon exchange is dominant in the region of high energy, while that from
the meson exchange plays a significant role in lower energy.

The dynamical hadronization is done by demanding

$\displaystyle\tilde{\lambda}_{q,k}=$ $\displaystyle 0\,,$ (224)

for every value of the RG scale $k$, which is equivalent to the fact that the
Hubbard-Stratonovich transformation is performed for every value of $k$. Thus
from Equation 221 one arrives at the hadronization function as follows

$\displaystyle\dot{\tilde{A}}=$
$\displaystyle-\frac{1}{\bar{h}_{k}}\overline{\mathrm{Flow}}^{(4)}_{(\bar{q}\tau
q)^{2}}\,.$ (225)

Inserting the effective action in Equation 52 into Equation 86 and performing
a projection on the Yukawa interactions between quarks and $\sigma$-$\pi$
mesons, i.e., $\bar{q}\tau\cdot\phi q$, one is led to

$\displaystyle\partial_{t}\bar{h}_{k}=$
$\displaystyle\left(\frac{1}{2}\eta_{\phi,k}+\eta_{q,k}\right)\bar{h}_{k}+\overline{\mathrm{Flow}}^{(3)}_{(\bar{q}\bm{\tau}q)\bm{\pi}}-\tilde{m}^{2}_{\pi,k}\dot{\tilde{A}}\,,$
(226)

with the dimensionless and renormalized pion mass as follows

$\displaystyle\tilde{m}^{2}_{\pi,k}=$
$\displaystyle\frac{m^{2}_{\pi,k}}{Z_{\phi,k}k^{2}}\,.$ (227)

Here in Equation 226

$\displaystyle\overline{\mathrm{Flow}}^{(3)}_{(\bar{q}\bm{\tau}q)\bm{\pi}}\equiv$
$\displaystyle\frac{1}{Z_{\phi,k}^{1/2}Z_{q,k}}\left(\partial_{t}\Gamma_{k,(\bar{q}\bm{\tau}q)\bm{\pi}}^{(3)}\right)_{\dot{A}_{k}=0}\,,$
(228)

is the flow of the Yukawa coupling between the pion and quarks, which is shown
in the third line of Figure 37. Its explicit expression is presented in
Equation 443. Therefore, substituting the hadronization function in Equation
225 into Equation 226, one obtains the total flow of the Yukawa coupling
finally. Evidently, the dynamics of resonance of quarks are stored in the
interactions between quarks and mesons through the dynamic hadronization.

In the left panel of Figure 41 the renormalized Yukawa coupling with $k=0$ is
shown as a function of the temperature with several different values of baryon
chemical potential. It is observed that with the increase of $T$ or $\mu_{B}$,
the Yukawa coupling decays rapidly due to the restoration of the chiral
symmetry. In the right panel of Figure 41 the dependence of the renormalized
Yukawa coupling on the RG scale for different values of temperature is
investigated. One can see that the Yukawa coupling is stable in the region of
$k\gtrsim 1$ GeV, and the effects of temperature play a role approximately in
$k\lesssim 2\pi T$. The effective four-quark coupling in the $\sigma$-$\pi$
channel can be described by the ratio $\bar{h}_{k}^{2}/(2\bar{m}_{\pi,k}^{2})$
[59, 18], which is shown as a function of the RG scale with several different
values of temperature and $\mu_{B}=0$ in Figure 42. It is observed that with
the decrease of $k$ and entering the regime of the chiral symmetry breaking, a
resonance occurs in the scalar-pseudoscalar channel, which results in a rapid
increase of the effective four-quark coupling. Moreover, the dependence of the
effective four-quark coupling on the baryon chemical potential is shown in
Figure 43. One finds that the effective four-quark coupling decrease with the
increasing temperature or baryon chemical potential.

###  IV.4 Natural emergence of LEFTs from QCD

Figure 44: Four-quark single gluon exchange coupling for light quarks
$\bar{g}^{2}_{\bar{l}lA,k}$ and strange quarks $\bar{g}^{2}_{\bar{s}sA,k}$,
dimensionless four-quark single meson exchange coupling
$\bar{h}_{k}^{2}/(1+\tilde{m}_{\pi,k}^{2})$ and
$\bar{h}_{k}^{2}/(1+\tilde{m}_{\sigma,k}^{2})$ as functions of the RG scale,
obtained in fRG for the $N_{f}=2+1$ flavor QCD in the vacuum. The plot is
adopted from [18]. Figure 45: Dimensionless propagator gappings
$1/(1+\tilde{m}^{2}_{\Phi_{i},k})$ for $\Phi_{i}=l,s,\sigma,\pi$ as functions
of the RG scale. The gluon dressing function $1/Z_{A,k}$ is also shown for
comparison, which is normalized by its peak value $1/Z_{A}^{\mathrm{peak}}$ at
$k=k_{\mathrm{peak}}$. The plot is adopted from [18].

The fRG approach to QCD with the dynamical hadronization discussed above,
provide us with a method to study the transition of degrees of freedom from
fundamental to composite ones. One is also able to observe a natural emergence
of LEFTs from original QCD. To that end, in Figure 44 one shows the four-quark
single gluon exchange coupling for light quarks $\bar{g}^{2}_{\bar{l}lA,k}$
and strange quarks $\bar{g}^{2}_{\bar{s}sA,k}$, dimensionless four-quark
single meson exchange coupling $\bar{h}_{k}^{2}/(1+\tilde{m}_{\pi,k}^{2})$ and
$\bar{h}_{k}^{2}/(1+\tilde{m}_{\sigma,k}^{2})$ as functions of the RG scale in
the vacuum. Evidently, it is found that the gluonic exchange couplings are
dominant in the perturbative regime of $k\gtrsim 1$ GeV. However, with the
decrease of the RG scale the active dynamic is taken over gradually by the
mesonic degrees of freedom, and one can see that the gluonic couplings and the
Yukawa couplings are comparable to each other at $k\approx 600$. In Figure 45
dimensionless propagator gappings $1/(1+\tilde{m}^{2}_{\Phi_{i},k})$ for
$\Phi_{i}=l,s,\sigma,\pi$ are shown as functions of the RG scale. The gluon
dressing function is also show there for comparison. In Figure 45 one observes
the same information on decouplings as in Figure 44, that is, as the RG scale
evolves from the UV towards IR, the gluons decouple from the matter at first,
then the quarks, and the mesons finally. For more related discussions see [18]
.

###  IV.5 Chiral condensate

Figure 46: Left panel: Renormalized light quark chiral condensate
$\Delta_{l,R}$ of $N_{f}=2+1$ flavor QCD as a function of the temperature with
several different values of baryon chemical potential obtained in fRG, in
comparison to the lattice results at vanishing $\mu_{B}$ [160]. Right panel:
Derivative of $\Delta_{l,R}$ with respect to the temperature, i.e., the
thermal susceptibility of the renormalized light quark condensate, as a
function of the temperature with several different values of baryon chemical
potential. The plots are adopted from [18]. Figure 47: Reduced condensate of
$N_{f}=2+1$ flavor QCD as a function of the temperature with $\mu_{B}=0$
obtained in fRG, where the constituent quark mass differences are
$\Delta\bar{m}_{sl}=\bar{m}_{s}-\bar{m}_{l}=120,150,155,160$ MeV,
corresponding to the current quark mass ratios of
$m_{s}^{0}/m_{l}^{0}=c_{\sigma_{s}}/c_{\sigma}\approx 14,27,30,34$. The fRG
results are compared with the lattice ones in [160]. The plot is adopted from
[18].

The chiral condensate of quark $q_{i}=u,d,s$ reads

$\displaystyle\Delta_{q_{i}}=$ $\displaystyle-
m_{q_{i}}^{0}T\sum_{n\in\mathbb{Z}}\int\frac{d^{3}q}{(2\pi)^{3}}\operatorname{tr}\,G_{q_{i}\bar{q}_{i}}(q)\,,$
(229)

up to a renormalization term, where no summation for the index $i$ is assumed,
and $m_{q_{i}}^{0}$ is the current quark mass. Obviously, the light quark
condensate is given by $\Delta_{l}=\Delta_{u}=\Delta_{d}$ with
$m_{l}^{0}=m_{u}^{0}=m_{d}^{0}$. The renormalized condensate as follows

$\displaystyle\Delta_{q_{i},R}=$ $\displaystyle\frac{1}{{\cal
N}_{R}}\left[\Delta_{q_{i}}(T,\mu_{q})-\Delta_{q_{i}}(0,0)\right]\,,$ (230)

renders the vacuum part of the chiral condensate in Equation 229 to be
subtracted, where the dimensionless $\Delta_{q_{i},R}$ is normalized by a
constant ${\cal N}_{R}$, e.g., ${\cal N}_{R}\sim f_{\pi}^{4}$. In the present
fRG approach to QCD, the light quark condensate is given by

$\displaystyle\Delta_{l}=$
$\displaystyle\frac{1}{2}m_{l}^{0}\frac{\partial\Omega[\Phi_{\mathrm{EoM}};T,\mu_{q}]}{\partial{m_{l}^{0}}}=\frac{1}{2}c_{\sigma}\frac{\partial\Omega[\Phi_{\mathrm{EoM}};T,\mu_{q}]}{\partial{c_{\sigma}}}\,,$
(231)

where $\Omega$ is the thermodynamic potential with the field
$\Phi_{\mathrm{EoM}}$ being on is equation of motion. From Equation 52, one is
led to

$\displaystyle\Delta_{l}(T,\mu_{q})=$
$\displaystyle-\frac{1}{2}c_{\sigma}\,\sigma_{\mathrm{EoM}}(T,\mu_{q})\,,$
(232)

and

$\displaystyle\Delta_{l,R}(T,\mu_{q})=$
$\displaystyle-\frac{c_{\sigma}}{{2\,\cal
N}_{R}}\Big{[}\sigma_{\mathrm{EoM}}(T,\mu_{q})-\sigma_{\mathrm{EoM}}(0,0)\Big{]}\,.$
(233)

The reduced condensate $\Delta_{l,s}$ is defined as the weighted difference
between the light and strange quark condensates as follows,

$\displaystyle\Delta_{l,s}(T,\mu_{q})=$ $\displaystyle\frac{1}{{\cal
N}_{l,s}}\left[\Delta_{l}(T,\mu_{q})-\left(\frac{m_{l}^{0}}{m_{s}^{0}}\right)^{2}\Delta_{s}(T,\mu_{q})\right]\,,$
(234)

which is usually normalized with its value in the vacuum, i.e.,

$\displaystyle\Delta_{l,s}(T,\mu_{q})=$
$\displaystyle\frac{\Delta_{l}(T,\mu_{q})-\left(\frac{m_{l}^{0}}{m_{s}^{0}}\right)^{2}\Delta_{s}(T,\mu_{q})}{\Delta_{l}(0,0)-\left(\frac{m_{l}^{0}}{m_{s}^{0}}\right)^{2}\Delta_{s}(0,0)}\,.$
(235)

Similar with Equation 231, one arrives at

$\displaystyle\Delta_{s}=$ $\displaystyle
m_{s}^{0}\frac{\partial\Omega[\Phi_{\mathrm{EoM}};T,\mu_{q}]}{\partial{m_{s}^{0}}}=c_{\sigma_{s}}\frac{\partial\Omega[\Phi_{\mathrm{EoM}};T,\mu_{q}]}{\partial{c_{\sigma_{s}}}}\,,$
(236)

and thus

$\displaystyle\Delta_{s}(T,\mu_{q})=$
$\displaystyle-\frac{1}{\sqrt{2}}c_{\sigma_{s}}\,\sigma_{s,\textrm{EoM}}(T,\mu_{q})\,.$
(237)

Finally, one is led to

$\displaystyle\Delta_{l,s}(T,\mu_{q})=$
$\displaystyle\frac{\left(\sigma-\sqrt{2}\frac{c_{\sigma}}{c_{\sigma_{s}}}\sigma_{s}\right)_{T,\mu_{q}}}{\left(\sigma-\sqrt{2}\frac{c_{\sigma}}{c_{\sigma_{s}}}\sigma_{s}\right)_{0,0}}\,,$
(238)

where one has used

$\displaystyle\frac{m_{l}^{0}}{m^{0}_{s}}=$
$\displaystyle\frac{c_{\sigma}}{c_{\sigma_{s}}}\,,$ (239)

In the left panel Figure 46, the renormalized light quark condensate is shown
as a function of the temperature with several different values of baryon
chemical potential. In the case of vanishing baryon chemical potential, the
fRG result is compared with the continuum extrapolated result of lattice QCD
[160], and excellent agreement is found. In the right panel of Figure 46, the
derivative of $\Delta_{l,R}$ with respect to the temperature, i.e., the
thermal susceptibility of the renormalized light quark condensate, are shown.
The peak position of the thermal susceptibility can be used to define the
pseudocritcal temperature, which is found to be $T_{c}=156$ MeV for
$\mu_{B}=0$, in good agreement with the lattice result. In Figure 47 one shows
the reduced condensate as a function of the temperature with $\mu_{B}=0$, in
comparison to the lattice simulations. One finds that for the physical current
quark mass ratio, i.e., $m_{s}^{0}/m_{l}^{0}=c_{\sigma_{s}}/c_{\sigma}\approx
27$ [226], the fRG results are in quantitative agreement with the lattice
ones.

###  IV.6 Phase structure

| fRG: [18] | DSE: [19] | DSE: [20] | Lattice (HotQCD): [12] | Lattice (WB): [8] | Lattice (WB): [16] | Lattice: [227]
---|---|---|---|---|---|---|---
$\kappa$ | 0.0142(2) | 0.0147(5) | 0.0173 | 0.015(4) | 0.0149(21) | 0.0153(18) | 0.0144(26)
$(T,{\mu_{B}})_{{}_{\tiny{\mathrm{CEP}}}}$ [MeV] | (107, 635) | (109, 610) | (112, 636) |  |  |  |
Table 3: Curvature $\kappa$ of the phase boundary (second line) and location
of CEP (third line) for $N_{f}=2+1$ flavor QCD, obtained from different
approaches. fRG: [18] (Fu et al.); DSE: [19] (Gao et al.), [20] (Gunkel et
al.); Lattice QCD: [12] (HotQCD), [8, 16] (WB), [227] (Bonati et al.); .
Figure 48: Phase diagram of $N_{f}=2+1$ flavor QCD in the plane of the
temperature and the baryon chemical potential. The fRG results [18] are
compared with those from Dyson-Schwinger Equations [19, 20], lattice QCD [8,
12]. The hatched red area denotes the region of inhomogeneous instability for
the chiral condensate found in the calculations of fRG. Some freeze-out data
are also shown in the phase diagram: [175] (STAR), [228] (Alba et al.), [4]
(Andronic et al.), [229] (Becattini et al.), [230] (Vovchenko et al.), and
[231] (Sagun et al.).

The QCD phase diagram in the plane of $T$ and $\mu_{B}$ is shown in Figure 48.
The first-principle fRG results in [18] are in comparison to state-of-the-art
calculations of other functional approach, e.g., the Dyson-Schwinger Equations
(DSE) [19, 20], and lattice QCD [8, 12]. The phase boundary in the regime of
continuous crossover at small or medium $\mu_{B}$ shows the dependence of the
pseudocritical temperature on the value of the baryon chemical potential. In
the calculations of fRG, this pseudocritical temperature is determined by the
thermal susceptibility of the renormalized light quark chiral condensate,
$\partial\Delta_{l,R}/\partial T$, as discussed in Section IV.5. Expanding the
pseudocritical temperature around $\mu_{B}=0$, one arrives at

$\displaystyle\frac{T_{c}(\mu_{B})}{T_{c}}$
$\displaystyle=1-\kappa\left(\frac{\mu_{B}}{T_{c}}\right)^{2}+\lambda\left(\frac{\mu_{B}}{T_{c}}\right)^{4}+\cdots\,,$
(240)

with $T_{c}=T_{c}(\mu_{B}=0)$, where the quadratic expansion coefficient
$\kappa$ is usually called as the curvature of the phase boundary. It is a
sensitive measure for the QCD dynamics at finite temperature and densities.
Therefore, it provides a benchmark test for functional approaches to make a
comparison of the curvature at small baryon chemical potential with lattice
simulations. The curvature values of the phase boundary obtained from fRG,
DSE, and lattice QCD are summarized in the second line of Table 3. It is found
recent results of the curvature from state-of-the-art functional approaches,
e.g., fRG [18], DSE [19, 20], have already been comparable to the lattice
results. By contrast, those obtained from relatively former calculations of
functional methods, e.g., [232, 233, 21], are significantly larger than the
values of the curvature obtained from lattice simulations.

Besides the curvature of the phase boundary, another key ingredient of the QCD
phase structure is the critical end point (CEP) of the first-order phase
transition line at large baryon chemical potential, which is currently
searched for with lots of efforts in experiments [37, 38, 39, 22, 40, 41, 176,
43]. Lattice simulations, however, are restricted in the region of
$\mu_{B}/T\lesssim 2\sim 3$ because of the sign problem at finite chemical
potentials, and in this region no signal of CEP is observed [17]. Passing
lattice benchmark tests at small baryon chemical potentials, functional
approaches are able to provide relatively reliable estimates for the location
of the CEP. The recent results of CEP after benchmark testing are presented in
the third line of Table 3, and also depicted in the phase diagram in Figure
48. Remarkably, estimates of CEP from fRG and DSE converge in a rather small
region at baryon chemical potentials of about 600 MeV. Note that results of
[20] in Table 3 are obtained without the dynamics of sigma and pion, and the
relevant results are $\kappa=0.0167$ and
$(T,{\mu_{B}})_{{}_{\tiny{\mathrm{CEP}}}}=(117,600)$ MeV when they are
included, and see also, e.g., [234, 232, 21] for related discussions. It
should be reminded that errors of functional approaches increase significantly
when $\mu_{B}/T\gtrsim 4$, for a detailed discussion see [18, 235], and thus
one arrives at a more reasonable estimation for the location of CEP as
$450\,\mathrm{MeV}\lesssim{\mu_{B}}_{\tiny{\mathrm{CEP}}}\lesssim
650\,\mathrm{MeV}$.

####  IV.6.1  Region of inhomogeneous instability at large baryon chemical
potential

Figure 49: Left panel: Mesonic wave function renormalization at vanishing
external momenta $Z_{\phi,k=0}(0)$ as a function of the temperature with
several different values of the baryon chemical potential, obtained in the fRG
approach to $N_{f}=2+1$ flavor QCD. Right panel:
$\big{|}1/Z_{\phi,k=0}(0)\big{|}$ for the data in the left panel. The spikes
are used to locate the region of negative $Z_{\phi,k=0}(0)$. The plots are
adopted from [18]. Figure 50: Phase diagram of $N_{f}=2+1$ flavor QCD obtained
in the fRG approach to QCD in comparison to freeze-out data. See also the
caption of Figure 48. The blue area denotes the region of negative mesonic
wave function renormalization at vanishing external momenta $Z_{\phi,k=0}(0)$,
which is an indicator for the inhomogeneous instability. The red hatched area
stands for the regime with negative $Z_{\phi,k=0}(0)$ and also sizable chiral
condensate. The plot is adopted from [18].

At large baryon chemical potentials, it is found within the fRG approach to
QCD, that the mesonic wave function renormalization in Equation 195 and
Equation 196 develops negative values at small momenta [18]. As shown in
Figure 49 the mesonic wave function renormalization at vanishing external
momenta $Z_{\phi,k=0}(0)$ is depicted as a function of the temperature with
several different values of $\mu_{B}$. One can see a negative
$Z_{\phi,k=0}(0)$ begins to appear for a temperature region when
$\mu_{B}\gtrsim 420$ MeV. The negative regime is clearly shown in the right
plot of $\big{|}1/Z_{\phi,k=0}(0)\big{|}$ between the two spikes, and it
widens with the increase of the baryon chemical potential. From the two-point
correlation functions of mesons as shown in Eqs. (195) and (196), the negative
$Z_{\phi,k=0}(0)$ implies that, for the dispersion relations of mesons, there
is a minimum at a finite $\bm{p}^{2}\neq 0$. This nontrivial behavior that the
minimum of dispersion is located at a finite momentum is indicative of an
inhomogeneous instability, for instance, the formation of a spatially
modulated chiral condensate. However, it should be reminded that a negative
$Z_{\phi,k=0}(0)$ is not bound to the formation of an inhomogeneous phase, it
can also serve as an precursor for the inhomogeneous phase. See, e.g., [236,
237, 238, 239, 240, 241, 242, 137, 135, 243, 244, 245] for more related
discussions. Most notably, very recently consequence of the inhomogeneous
instability indicated by the negative wave function renormalization on the
phenomenology of heavy-ion collisions has been studied. It is found that this
inhomogeneous instability would result in a moat regime in both the particle
$p_{T}$ spectrum and the two-particle correlation, where the peaks are located
at nonzero momenta [244, 245]. The region of negative $Z_{\phi,k=0}(0)$ is
shown in Figure 50 by the blue area, while the red hatched area shows where
this region overlaps with a sizable chiral condensate.

###  IV.7 Magnetic equation of state

Figure 51: Susceptibility for the reduced condensate obtained in the fRG
approach to $N_{f}=2+1$ flavor QCD (fQCD) as a function of the temperature, in
comparison to the lattice results in [13, 246]. The plot is adopted from [62].
Figure 52: Pseudocritical temperature as a function of the pion mass obtained
in fRG (fQCD) in comparison to the lattice results [13]. The critical
temperature in the chiral limit $T_{\mathrm{c}}$ has been obtained from an
extrapolation of the pseudocritical temperature $T_{\mathrm{pc}}$ to
$m_{\pi}\to 0$. The plot is adopted from [62].

From Equation 231, Equation 233, and Equation 234, it is convenient to define
the corresponding susceptibilities for the light quark condensate, the
renormalized light quark condensate, and the reduced condensate, i.e.,

$\displaystyle\chi_{M}^{(i)}(T)=-\frac{\partial}{\partial
m_{l}^{0}}\left(\frac{\Delta_{i}(T)}{m_{l}^{0}}\right)\,,$ (241)

with $(i)=(l),\,(l,R),\,(l,s)$ [62]. Similar with the difference between the
light quark condensate and the renormalized light quark condensate, the
susceptibilities for them differ by the vacuum contribution. Among these three
susceptibilities, the reduce susceptibility for the reduced condensate defined
in fRG and lattice QCD, e.g., [13], matches better.

In Figure 51 the susceptibility obtained from the reduced condensate as a
function of the temperature for different pion masses in the fRG approach
[62], is compared with the lattice results [13]. One can see that the fRG
results are in good agreement with the lattice ones for pion masses
$m_{\pi}\gtrsim 100$ MeV. There are some slight deviations from the two
approaches for small pion masses. From the dependence of the reduced
susceptibility on the temperature for a fixed pion mass in Figure 51, one is
able to define the pseudocritical temperature for the chiral crossover at such
value of pion mass, via the peak position of the curves, denoted by
$T_{\mathrm{pc}}$.

In Figure 52 one shows the dependence of the pseudocritical temperature on the
pion mass, obtained both from the fRG approach and the lattice QCD. The values
of pseudocritical temperature at finite pion masses are also extrapolated to
the chiral limit, i.e., $m_{\pi}\to 0$, which leaves us with the critical
temperature $T_{\mathrm{c}}$ in the chiral limit. One obtains
$T_{c}^{\mathrm{fRG}}\approx 142$ MeV from the fRG approach [62], and
$T_{c}^{\mathrm{lattice}}=132_{-6}^{+3}$ MeV from the lattice QCD [13], that
is, the critical temperature obtained in fRG is a bit larger than the lattice
result. Recently, the critical temperature in the chiral limit from DSE is
found to be $T_{c}^{\mathrm{DSE}}\approx 141$ MeV [247].

From the general scaling hypothesis, see e.g., [66], in the critical regime
the pseudocritical temperature scales with the pion mass as

$\displaystyle T_{\mathrm{pc}}(m_{\pi})\approx
T_{\mathrm{c}}+c\,m_{\pi}^{p}\,,$ (242)

where $c$ is a non-universal coefficient, while the exponent $p$ is related to
the critical exponents $\beta$ and $\delta$ through $p=2/(\beta\delta)$ [62].
Inputting the values of $\beta$ and $\delta$ for the 3-$d$ $O(4)$ universality
class [197, 248, 249], one arrives at $p\approx 1.08$. On the contrary,
fitting of the fRG data in Figure 52 leads to $p\approx 0.91_{-0.03}^{+0.03}$
[62]. This discrepancy indicates that the pion masses under investigation in
[62], i.e., $m_{\pi}\gtrsim 30$ MeV are beyond the critical regime. In fact,
the critical regime is found to be extremely small, $m_{\pi}\lesssim 1$ MeV,
in fRG studies of LEFTs, and see, e.g., [98, 190, 199, 151] for a more
detailed discussion.

##  V Real-time fRG

In this section we would like to give a brief introduction about the real-time
fRG, based on a combination of the fRG approach and the formalism of
Schwinger-Keldysh path integral, where the flow equations are formulated on
the closed time path. The Schwinger-Keldysh path integral is devised to study
real-time quantum dynamics [250, 251], which has been proved be very powerful
to cope with problems of both equilibrium and nonequilibrium many-body
systems, see, e.g., [252, 253, 254, 255] for relevant reviews.

Within the fRG approach on the closed time path, nonthermal fixed points of
the $O(N)$ scalar theory are investigated [256, 257]. The transition from
unitary to dissipative dynamics is studied [258]. Spectral functions in a
scalar field theory with $d$=0+1 dimensions are computed within the real-time
fRG approach [259]. Moreover, it has also been employed to study the
nonequilibrium transport, dynamical critical behavior, etc. in open quantum
systems [260, 261], see also [255, 55] for related reviews. Recently, the
real-time fRG is compared with other real-time methods [262].

Furthermore, it should be mentioned that another conceptually new fRG with the
Keldysh functional integral is put forward in [263, 264, 265, 266], and the
regulation of the RG scale there is replaced with that of the time, which is
also called as the temporal renormalization group. Noteworthily, recently the
Källén-Lehmann spectral representation of correlation functions has been used
in the approach of DSE, to study the spectral functions in the
$\phi^{4}$-theory and Yang-Mills theory [267, 268, 269]. The spectral
functions provide important informations on the time-like properties of
correlation functions, see, e.g., [270, 271, 272, 273]. Besides the real-time
fRG, another commonly adopted approach is the analytically continued
functional renormalization group, where the Euclidean flow equations are
analytically continued into the Minkowski spacetime on the level of analytic
expressions with some specific truncations, and see, e.g., [112, 274, 275,
129, 276] for more details.

In this section we would like to discuss the fRG formulated on the Keldysh
path at the example of the $O(N)$ scalar theory [277].

###  V.1 fRG with the Keldysh functional integral

On the closed time path the classical action in Equation 1 for a closed system
reads

$\displaystyle S[\hat{\Phi}]$
$\displaystyle=\int_{x}\Big{(}\mathcal{L}[\hat{\Phi}_{+}]-\mathcal{L}[\hat{\Phi}_{-}]\Big{)}\,,$
(243)

with $\int_{x}\equiv\int_{-\infty}^{\infty}dt\int d^{3}x$, where the
subscripts $\pm$ denote variables on the forward and backward time branches,
respectively [252]. Now the Keldysh generating functional reads

$\displaystyle Z[J_{+},J_{-}]$ $\displaystyle=$
$\displaystyle\int\big{(}\mathcal{D}\hat{\Phi}_{+}\mathcal{D}\hat{\Phi}_{-}\big{)}\exp\Big{\\{}i\Big{(}S[\hat{\Phi}]+\big{(}J_{+}^{i}\hat{\Phi}_{i,+}-J_{-}^{i}\hat{\Phi}_{i,-}\big{)}\Big{)}\Big{\\}}\,.$
(244)

It is more convenient to adopt the physical representation in terms of the
“classical” and “quantum” variables, denoted by subscripts $c$ and $q$,
respectively, which are related to variables on the two time branches by a
Keldysh rotation, that is,

$\displaystyle\left\\{\begin{array}[]{l}\hat{\Phi}_{i,+}=\frac{1}{\sqrt{2}}(\hat{\Phi}_{i,c}+\hat{\Phi}_{i,q}),\\\\[12.91663pt]
\hat{\Phi}_{i,-}=\frac{1}{\sqrt{2}}(\hat{\Phi}_{i,c}-\hat{\Phi}_{i,q}),\end{array}\right.$
(247)

and

$\displaystyle\left\\{\begin{array}[]{l}J_{+}^{i}=\frac{1}{\sqrt{2}}(J_{c}^{i}+J_{q}^{i}),\\\\[12.91663pt]
J_{-}^{i}=\frac{1}{\sqrt{2}}(J_{c}^{i}-J_{q}^{i}),\end{array}\right.$ (250)

Then, Equation 244 reads

$\displaystyle Z[J_{c},J_{q}]$ $\displaystyle=$
$\displaystyle\int\big{(}\mathcal{D}\hat{\Phi}_{c}\mathcal{D}\hat{\Phi}_{q}\big{)}\exp\Big{\\{}i\Big{(}S[\hat{\Phi}]+\big{(}J_{q}^{i}\hat{\Phi}_{i,c}+J_{c}^{i}\hat{\Phi}_{i,q}\big{)}\Big{)}\Big{\\}}\,.$
(251)

The bilinear regulator term in Equation 2 can be chosen as

$\displaystyle\Delta S_{k}[\hat{\Phi}]=$
$\displaystyle\frac{1}{2}(\hat{\Phi}_{i,c},\hat{\Phi}_{i,q})\begin{pmatrix}0&R_{k}^{ij}\\\
(R_{k}^{ij})^{*}&0\end{pmatrix}\begin{pmatrix}\hat{\Phi}_{j,c}\\\
\hat{\Phi}_{j,q}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\hat{\Phi}_{i,c}R_{k}^{ij}\hat{\Phi}_{j,q}+\hat{\Phi}_{i,q}(R_{k}^{ij})^{*}\hat{\Phi}_{j,c}\Big{)}\,.$
(252)

Then the RG scale $k$-dependent generating function Equation 1 with the
Keldysh functional integral reads

$\displaystyle Z_{k}[J_{c},J_{q}]=$
$\displaystyle\int\big{(}\mathcal{D}\hat{\Phi}_{c}\mathcal{D}\hat{\Phi}_{q}\big{)}\exp\Big{\\{}i\Big{(}S[\hat{\Phi}]+\Delta
S_{k}[\hat{\Phi}]$
$\displaystyle+(J_{q}^{i}\hat{\Phi}_{i,c}+J_{c}^{i}\hat{\Phi}_{i,q})\Big{)}\Big{\\}}\,.$
(253)

The Schwinger functional in Equation 14 reads

$\displaystyle W_{k}[J_{c},J_{q}]$ $\displaystyle=-i\ln Z_{k}[J_{c},J_{q}]\,.$
(254)

Combining the indices $c,\,q$ and $i$ for other degrees of freedom into one
single label, say $a$, i.e.,

$\displaystyle\\{\hat{\Phi}_{a}\\}$
$\displaystyle=\big{\\{}\\{\hat{\Phi}_{i,c}\\},\\{\hat{\Phi}_{i,q}\\}\big{\\}}\,,$
(255) $\displaystyle\\{J^{a}\\}$
$\displaystyle=\big{\\{}\\{J_{q}^{i}\\},\\{J_{c}^{i}\\}\big{\\}}\,.$ (256)

one is able to simplify notations significantly, for instance, the regulator
term in Equation 252 now reading

$\displaystyle\Delta S_{k}[\varphi]=$
$\displaystyle\frac{1}{2}\hat{\Phi}_{a}R_{k}^{ab}\hat{\Phi}_{b}\,,$ (257)

with

$\displaystyle R_{k}^{ab}\equiv$ $\displaystyle\begin{pmatrix}0&R_{k}^{ij}\\\
(R_{k}^{ij})^{*}&0\end{pmatrix}\,.$ (258)

In the same way, the expectation value of the field reads

$\displaystyle\Phi_{a}$
$\displaystyle\equiv\langle\hat{\Phi}_{a}\rangle=\frac{\delta W_{k}[J]}{\delta
J^{a}}\,,$ (259)

and the propagator is given by

$\displaystyle G_{k,ab}\equiv$
$\displaystyle-i\langle\hat{\Phi}_{a}\hat{\Phi}_{b}\rangle_{c}=-i\big{[}\langle\hat{\Phi}_{a}\hat{\Phi}_{b}\rangle-\langle\hat{\Phi}_{a}\rangle\langle\hat{\Phi}_{b}\rangle\big{]}$
$\displaystyle=$ $\displaystyle-\frac{\delta^{2}W_{k}[J]}{\delta
J^{a}J^{b}}\,.$ (260)

The effective action is obtained from the Schwinger functional upon a Legendre
transformation, viz.,

$\displaystyle\Gamma_{k}[\Phi]$ $\displaystyle=W_{k}[J]-\Delta
S_{k}[\Phi]-J^{a}\Phi_{a}\,.$ (261)

Similar with Equation 22 and Equation 23, one has

$\displaystyle\frac{\delta(\Gamma_{k}[\Phi]+\Delta
S_{k}[\Phi])}{\delta\Phi_{a}}$ $\displaystyle=-\gamma^{a}_{\hskip
4.26773ptb}J^{b}\,,$ (262)

and

$\displaystyle G_{k,ab}$ $\displaystyle=\gamma^{c}_{\hskip
4.26773pta}\Big{[}\big{(}\Gamma_{k}^{(2)}[\Phi]+\Delta
S_{k}^{(2)}[\Phi]\big{)}^{-1}\Big{]}_{cb}\,.$ (263)

with

$\displaystyle\Big{(}\Gamma_{k}^{(2)}[\Phi]+\Delta
S_{k}^{(2)}[\Phi]\Big{)}^{ab}$
$\displaystyle\equiv\frac{\delta^{2}(\Gamma_{k}[\Phi]+\Delta
S_{k}[\Phi])}{\delta\Phi_{a}\delta\Phi_{b}}\,.$ (264)

Then it is left to specify the flow equation of Schwinger functional with the
Keldysh functional integral, that reads

$\displaystyle\partial_{\tau}W_{k}[J]=$
$\displaystyle\frac{1}{2}iG_{k,ab}\partial_{\tau}R^{ab}_{k}+\frac{1}{2}\Phi_{a}\big{(}\partial_{\tau}R^{ab}_{k}\big{)}\Phi_{b}$
$\displaystyle=$
$\displaystyle\frac{i}{2}\mathrm{STr}\Big{[}\big{(}\partial_{\tau}R^{\mathrm{T}}_{k}\big{)}G_{k}\Big{]}+\frac{1}{2}\Phi_{a}\big{(}\partial_{\tau}R^{ab}_{k}\big{)}\Phi_{b}\,,$
(265)

where the RG time is denoted by $\tau\equiv\ln(k/\Lambda)$, and
$R^{\mathrm{T}}_{k}$ stands for the transpose of the regulator only in the
$c$-$q$ space as shown in Equation 258. Finally, the flow equation of the
effective action reads

$\displaystyle\partial_{\tau}\Gamma_{k}[\Phi]$
$\displaystyle=\partial_{\tau}W_{k}[J]-\partial_{\tau}\Delta S_{k}[\Phi]$
$\displaystyle=\frac{i}{2}\mathrm{STr}\Big{[}\big{(}\partial_{\tau}R^{\mathrm{T}}_{k}\big{)}G_{k}\Big{]}\,.$
(266)

###  V.2 Real-time $O(N)$ scalar theory

Figure 53: Diagrammatic representation of the three different propagators for
the sigma and pion mesons. A line associated with two end points labelled with
“$c,q$” denotes the retarded propagator, while that with “$q,c$” denotes the
advanced propagator. A line with an empty circle inserted in the middle is
used to denote the Keldysh propagator, which results from Equation 290. The
plot is adopted from [277].

The Keldysh effective action in Equation 261 for the $O(N)$ scalar theory
reads

$\displaystyle\Gamma_{k}[\phi_{c},\phi_{q}]=$
$\displaystyle\int_{x}\Big{[}Z_{\phi,k}(\partial_{\mu}\phi_{q})\cdot(\partial^{\mu}\phi_{c})-U_{k}(\phi_{c},\phi_{q})\Big{]}\,,$
(267)

where the potential is given by

$\displaystyle U_{k}(\phi_{c},\phi_{q})=$ $\displaystyle
V_{k}(\rho_{+})-V_{k}(\rho_{-})\,,$ (268)

with $\rho_{\pm}=\phi_{\pm}^{2}/2$. Here $\phi_{i,\pm}$ and $\phi_{i,c/q}$
($i=0,1,...N-1$) denote the fields of $N$ components on the closed time
branches or in the physical representation, respectively, and their relations
are given in Equation 247. Note that in Equation 267 a local potential
approximation with a $k$-dependent wave function renormalization $Z_{\phi,k}$
has been adopted, which is usually called the LPA’ approximation.

When the $O(N)$ symmetry is broken into the $O(N-1)$ one in the direction of
component $i=0$, the expectation values of the fields read

$\displaystyle\bar{\phi}_{i,c}=$
$\displaystyle\bar{\phi}_{c}\delta_{i0}\,,\qquad\bar{\phi}_{i,q}=0\,.$ (269)

Then, the sigma and pion fields are give by

$\displaystyle\sigma_{c}=$
$\displaystyle\phi_{0,c}-\bar{\phi}_{c}\,,\qquad\sigma_{q}=\phi_{0,q}\,,$
(270)

and

$\displaystyle\pi_{i,c}=$
$\displaystyle\phi_{i,c}\,,\qquad\pi_{i,q}=\phi_{i,q}\,,\qquad(i\neq 0)\,.$
(271)

The sigma and pion masses read

$\displaystyle m_{\sigma,k}^{2}\equiv$ $\displaystyle
V^{\prime}_{k}(\bar{\rho}_{c})+2\bar{\rho}_{c}V^{(2)}_{k}(\bar{\rho}_{c})\,,$
(272) $\displaystyle m_{\pi,k}^{2}\equiv$ $\displaystyle
V^{\prime}_{k}(\bar{\rho}_{c})\,,$ (273)

with $\bar{\rho}_{c}\equiv{\bar{\phi}_{c}}^{2}/4$.

The regulator in Equation 258 in the case of the $O(N)$ scalar theory reads

$\displaystyle R_{k}(q)=$
$\displaystyle\begin{pmatrix}0&R_{k}^{A}(q)\\\\[8.61108pt]
R_{k}^{R}(q)&0\end{pmatrix}\,,$ (274)

with

$\displaystyle R_{k}^{R}(q)$
$\displaystyle=R_{k}^{A}(q)=\begin{pmatrix}R_{\sigma,k}(q)&0\\\\[8.61108pt]
0&R_{\pi,k}(q)\end{pmatrix}\,.$ (275)

where one has

$\displaystyle R_{\sigma,k}(q)=$ $\displaystyle
R_{\phi,k}(q)\,,\qquad\big{(}R_{\pi,k}\big{)}_{ij}(q)=R_{\phi,k}(q)\delta_{ij}\,,$
(276)

with

$\displaystyle R_{\phi,k}(q)=$ $\displaystyle
Z_{\phi,k}\Big{(}-{\bm{q}}^{2}r_{B}\big{(}\frac{{\bm{q}}^{2}}{k^{2}}\big{)}\Big{)}\,.$
(277)

Here, the $3d$ flat regulator is used.

The flow equation of effective action in Equation 266 with the regulator in
Equation 274 can be reformulated as

$\displaystyle\partial_{\tau}\Gamma_{k}[\Phi]$
$\displaystyle=\frac{i}{2}\mathrm{STr}\Big{[}\tilde{\partial}_{\tau}\ln\big{(}\Gamma_{k}^{(2)}[\Phi]+R_{k}\big{)}\Big{]}\,,$
(278)

with

$\displaystyle\Big{(}\Gamma_{k}^{(2)}[\Phi]\Big{)}_{ab}$
$\displaystyle\equiv\frac{\delta^{2}\Gamma_{k}[\Phi]}{\delta\Phi_{a}\delta\Phi_{b}}\,,$
(279)

where the fluctuation matrix can be decomposed into a sum of the inverse
propagator $\mathcal{P}_{k}$ and the interaction $\mathcal{F}_{k}$ as shown in
Equation 62. In thermal equilibrium at a temperature $T$, the
$\mathcal{P}_{k}$ matrix reads

$\displaystyle\mathcal{P}_{k}=$
$\displaystyle\begin{pmatrix}0&\mathcal{P}^{A}_{k}\\\\[4.30554pt]
\mathcal{P}^{R}_{k}&\mathcal{P}^{K}_{k}\end{pmatrix}\,,$ (280)

where the inverse retarded propagator is given by

$\displaystyle\mathcal{P}^{R}_{k}=$
$\displaystyle\begin{pmatrix}\mathcal{P}^{R}_{\sigma,k}&0\\\\[8.61108pt]
0&\mathcal{P}^{R}_{\pi,k}\end{pmatrix}\,,$ (281)

with

$\displaystyle\mathcal{P}^{R}_{\sigma,k}=$ $\displaystyle
Z_{\phi,k}\bigg{[}q_{0}^{2}-{\bm{q}}^{2}\Big{(}1+r_{B}\big{(}\frac{{\bm{q}}^{2}}{k^{2}}\big{)}\Big{)}\bigg{]}-m_{\sigma,k}^{2}$
$\displaystyle+\mathrm{sgn}(q_{0})i\epsilon\,,$ (282)
$\displaystyle\big{(}\mathcal{P}^{R}_{\pi,k}\big{)}_{ij}=$
$\displaystyle\Bigg{\\{}Z_{\phi,k}\bigg{[}q_{0}^{2}-{\bm{q}}^{2}\Big{(}1+r_{B}\big{(}\frac{{\bm{q}}^{2}}{k^{2}}\big{)}\Big{)}\bigg{]}-m_{\pi,k}^{2}$
$\displaystyle+\mathrm{sgn}(q_{0})i\epsilon\Bigg{\\}}\delta_{ij}\,.$ (283)

The inverse advanced propagator is related to the retarded one through a
complex conjugate, to wit,

$\displaystyle\mathcal{P}^{A}_{k}=$
$\displaystyle(\mathcal{P}^{R}_{k})^{*}\,.$ (284)

The $qq$ component of the inverse propagator in Equation 280, i.e.,
$\mathcal{P}^{K}_{k}$, reads

$\displaystyle\mathcal{P}^{K}_{k}=$
$\displaystyle\begin{pmatrix}\mathcal{P}^{K}_{\sigma,k}&0\\\\[8.61108pt]
0&\mathcal{P}^{K}_{\pi,k}\end{pmatrix}\,,$ (285)

with

$\displaystyle\mathcal{P}^{K}_{\sigma,k}=$ $\displaystyle
2i\epsilon\,\mathrm{sgn}(q_{0})\coth\Big{(}\frac{q_{0}}{2T}\Big{)}\,,$ (286)
$\displaystyle\big{(}\mathcal{P}^{K}_{\pi,k}\big{)}_{ij}=$
$\displaystyle\bigg{[}2i\epsilon\,\mathrm{sgn}(q_{0})\coth\Big{(}\frac{q_{0}}{2T}\Big{)}\bigg{]}\delta_{ij}\,.$
(287)

Then, one can obtain the propagator matrix as follows

$\displaystyle G_{k}=$
$\displaystyle\big{(}\mathcal{P}_{k}\big{)}^{-1}=\begin{pmatrix}G^{K}_{k}&G^{R}_{k}\\\\[4.30554pt]
G^{A}_{k}&0\end{pmatrix}\,,$ (288)

with the retarded and advanced propagator being

$\displaystyle G^{R}_{k}=$ $\displaystyle(\mathcal{P}^{R}_{k})^{-1}\,,\qquad
G^{A}_{k}=(\mathcal{P}^{A}_{k})^{-1}\,,$ (289)

and the correlation function or Keldysh propagator being

$\displaystyle G^{K}_{k}=$
$\displaystyle-(\mathcal{P}^{R}_{k})^{-1}\mathcal{P}^{K}_{k}(\mathcal{P}^{A}_{k})^{-1}=-G^{R}_{k}\mathcal{P}^{K}_{k}G^{A}_{k}\,.$
(290)

Finally, one can verify the fluctuation-dissipation relation in thermal
equilibrium, as follows

$\displaystyle G^{K}_{k}=$
$\displaystyle\big{(}G^{R}_{k}-G^{A}_{k}\big{)}\coth\Big{(}\frac{q_{0}}{2T}\Big{)}\,.$
(291)

From Equation 288 one arrives at the three different propagators which read

$\displaystyle iG^{R}_{\phi,k}=$ $\displaystyle\langle
T_{p}\phi_{c}(x)\phi_{q}(y)\rangle\,,\quad iG^{A}_{\phi,k}=\langle
T_{p}\phi_{q}(x)\phi_{c}(y)\rangle\,,$ (292) $\displaystyle iG^{K}_{\phi,k}=$
$\displaystyle\langle
T_{p}\phi_{c}(x)\phi_{c}(y)\rangle=\big{(}iG^{R}_{\phi,k}\big{)}\big{(}i\mathcal{P}^{K}_{\phi,k}\big{)}\big{(}iG^{A}_{\phi,k}\big{)}\,,$
(293)

where $T_{p}$ denotes the time ordering from the positive to negative time
branch in the Keldysh closed time path, and Equation 293 follows directly from
Equation 290. Diagrammatic representation of these propagators is shown in
Figure 53.

Figure 54: Diagrammatic representation of the flow equation of the effective
potential in the real-time $O(N)$ scalar theory. The vertices are denoted by
gray blobs, and their legs are distinguished for the “$q$” and “$c$” fields.
the crossed circles stand for the regulator insertion. The plot is adopted
from [277].

Projecting the flow equation of the effective action in Equation 278 onto a
derivative with respect to the quantum sigma field $\sigma_{q}$, i.e.,

$\displaystyle\left.\partial_{\tau}\left(\frac{i\delta\Gamma_{k}[\Phi]}{\delta\sigma_{q}}\right)\right|_{\Phi=0}\,,$
(294)

with $\Phi=(\sigma_{c},\\{\pi_{i,c}\\},\sigma_{q},\\{\pi_{i,q}\\})$, one is
led to

$\displaystyle\partial_{\tau}V^{\prime}_{k}(\bar{\rho}_{c})$ $\displaystyle=$
$\displaystyle\frac{i}{4}\int\frac{d^{4}q}{(2\pi)^{4}}\tilde{\partial}_{\tau}\Big{[}\frac{\partial
m_{\sigma,k}^{2}}{\partial\bar{\rho}_{c}}G^{K}_{\sigma,k}(q)+\frac{\partial
m_{\pi,k}^{2}}{\partial\bar{\rho}_{c}}\big{(}G^{K}_{\pi,k}\big{)}_{ii}(q)\Big{]}$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\bar{\rho}_{c}}\bigg{\\{}-\frac{i}{4}\int\frac{d^{4}q}{(2\pi)^{4}}\big{(}\partial_{\tau}R_{\phi,k}(q)\big{)}\Big{[}G^{K}_{\sigma,k}(q)$
$\displaystyle+\big{(}G^{K}_{\pi,k}\big{)}_{ii}(q)\Big{]}\bigg{\\}}\,,$ (295)

which is diagrammatically shown in Figure 54. Integrating both sides of
Equation 295 over $\bar{\rho}_{c}$, one arrives at

$\displaystyle\partial_{\tau}V_{k}(\bar{\rho}_{c})$ $\displaystyle=$
$\displaystyle-\frac{i}{4}\int\frac{d^{4}q}{(2\pi)^{4}}\big{(}\partial_{\tau}R_{\phi,k}(q)\big{)}\Big{[}G^{K}_{\sigma,k}(q)+\big{(}G^{K}_{\pi,k}\big{)}_{ii}(q)\Big{]}\,,$
(296)

up to a term independent of $\bar{\rho}_{c}$. Inserting Equation 282 and
Equation 283, one has

$\displaystyle\partial_{\tau}V_{k}(\bar{\rho}_{c})=$
$\displaystyle\frac{k^{4}}{4\pi^{2}}\bigg{[}l^{(B,4)}_{0}(\tilde{m}^{2}_{\sigma,k},\eta_{\phi,k};T)$
$\displaystyle+\big{(}N-1\big{)}l^{(B,4)}_{0}(\tilde{m}^{2}_{\pi,k},\eta_{\phi,k};T)\bigg{]}\,,$
(297)

with the threshold function given by

$\displaystyle l^{(B,4)}_{0}(\tilde{m}^{2}_{\phi,k},\eta_{\phi,k};T)$
$\displaystyle=$
$\displaystyle\frac{2}{3}\left(1-\frac{\eta_{\phi,k}}{5}\right)\frac{1}{\sqrt{1+\tilde{m}^{2}_{\phi,k}}}\left(\frac{1}{2}+n_{B}(\tilde{m}^{2}_{\phi,k};T)\right)\,,$
(298)

and the renormalized dimensionless meson masses reading

$\displaystyle\tilde{m}^{2}_{\sigma,k}=$
$\displaystyle\frac{m^{2}_{\sigma,k}}{k^{2}Z_{\phi,k}}\,,\qquad\tilde{m}^{2}_{\pi,k}=\frac{m^{2}_{\pi,k}}{k^{2}Z_{\phi,k}}\,,$
(299)

where the bosonic distribution function reads

$\displaystyle n_{B}(\tilde{m}^{2}_{\phi,k};T)$
$\displaystyle=\frac{1}{\exp\bigg{\\{}\frac{k}{T}\sqrt{1+\tilde{m}^{2}_{\phi,k}}\bigg{\\}}-1}\,.$
(300)

Note that Equation 297 is just the flow equation of the effective potential in
the LPA’ approximation obtained in the conventional Euclidean formalism of
fRG.

###  V.3 Flows of the two- and four-point correlation functions

Figure 55: Diagrammatic representation of the flow equation of the four-point
vertex in the symmetric phase. The plot is adopted from [277].

The 1PI $n$-point correlation function or vertex in the Euclidean field theory
is given in Equation 64, and the counterpart in the Keldysh field theory reads

$\displaystyle i\Gamma_{k,\Phi_{a_{1}}\cdots\Phi_{a_{n}}}^{(n)}$
$\displaystyle=\left.\left(\frac{i\delta^{n}\Gamma_{k}[\Phi]}{\delta\Phi_{a_{1}}\cdots\delta\Phi_{a_{n}}}\right)\right|_{\Phi=\langle\Phi\rangle}\,.$
(301)

Note that a label “$c$” or “$q$” is associated with an external leg of the
vertex to distinguish the classical or quantum field, as shown in Figure 54.
In the following we consider the symmetric phase with $\bar{\phi}_{c}=0$ in
Equation 269. In this case the pion and sigma fields are degenerate, and they
are collectively denoted by $\phi_{i}$ ($i=0,1,...N-1$). The diagrammatic
representation of the four-point vertex
$i\Gamma_{\phi_{q}\phi_{c}\phi_{c}\phi_{c}}^{(4)}$ reads

$\displaystyle
i\Gamma_{k,\phi_{i,q}\phi_{j,c}\phi_{k,c}\phi_{l,c}}^{(4)}(p_{i},p_{j},p_{k},p_{l})$
$\displaystyle\equiv\parbox[c]{65.04034pt}{\includegraphics[width=52.03227pt]{Gam4-piq1pic3-labels}}\,,$
(302)

whose flow equation is given in Figure 55. According to the generic
interchange symmetry of the external legs, the four-point vertex can be
parameterized as

$\displaystyle
i\Gamma_{k,\phi_{i,q}\phi_{j,c}\phi_{k,c}\phi_{l,c}}^{(4)}(p_{i},p_{j},p_{k},p_{l})$
$\displaystyle=$
$\displaystyle-\frac{i}{3}\Big{[}\lambda_{4\pi,k}^{\mathrm{eff}}(p_{i},p_{j},p_{k},p_{l})\delta_{il}\delta_{jk}+\lambda_{4\pi,k}^{\mathrm{eff}}(p_{i},p_{k},p_{l},p_{j})\delta_{ij}\delta_{kl}$
$\displaystyle+\lambda_{4\pi,k}^{\mathrm{eff}}(p_{i},p_{l},p_{j},p_{k})\delta_{ik}\delta_{jl}\Big{]}\,,$
(303)

where an effective four-point coupling $\lambda_{4\pi,k}^{\mathrm{eff}}$ is
introduced. In the following, one adopts the truncation that the momentum
dependence of the four-point vertex on the r.h.s. of the flow equation in
Figure 55 is neglected, and the four-point vertex there is identified as

$\displaystyle\lambda_{4\pi,k}$
$\displaystyle=\lambda_{4\pi,k}^{\mathrm{eff}}(0)\,.$ (304)

Then, from the flow of the four-point vertex in Figure 55, one arrives at

$\displaystyle\partial_{\tau}\lambda_{4\pi,k}^{\mathrm{eff}}(p_{i},p_{j},p_{k},p_{l})$
$\displaystyle=$
$\displaystyle\frac{\lambda_{4\pi,k}^{2}}{3}\Big{[}(N+4)\tilde{\partial}_{\tau}I_{k}(-p_{i}-p_{l})+2\,\tilde{\partial}_{\tau}I_{k}(-p_{i}-p_{k})$
$\displaystyle+2\,\tilde{\partial}_{\tau}I_{k}(-p_{i}-p_{j})\Big{]}\,,$ (305)

with

$\displaystyle I_{k}(p)$ $\displaystyle\equiv
i\int\frac{d^{4}q}{(2\pi)^{4}}G^{K}_{\pi,k}(q)G^{A}_{\pi,k}(q-p)\,.$ (306)

Note that the wave function renormalization $Z_{\phi,k}=1$ is adopted to
simplify the calculation of $\tilde{\partial}_{\tau}I_{k}(p)$ in Equation 305.
The explicit expression of $I_{k}(p)$ can be found in [277].

With the four-point vertex in Equation 303 one is able to obtain the self-
energy as follows

$\displaystyle-i\Sigma_{k,ij}(p)$
$\displaystyle\equiv\frac{1}{2}\parbox[c]{65.04034pt}{\includegraphics[width=65.04034pt]{Gam2-piqpic-
piloop-labels}}\,,$ (307)

which reads

$\displaystyle-i\Sigma_{k,ij}(p)$
$\displaystyle=\delta_{ij}(-\frac{i}{6})(N+2)$
$\displaystyle\times\int\frac{d^{4}q}{(2\pi)^{4}}iG^{K}_{\pi,k}(q)\bar{\lambda}_{4\pi,k}^{\mathrm{eff}}(p_{0},|\bm{p}|,q_{0},|\bm{q}|,\cos\theta)\,.$
(308)

Here, the function $\bar{\lambda}_{4\pi,k}^{\mathrm{eff}}$ is given by

$\displaystyle\bar{\lambda}_{4\pi,k}^{\mathrm{eff}}(p_{0},|\bm{p}|,q_{0},|\bm{q}|,\cos\theta)$
$\displaystyle=$
$\displaystyle\frac{1}{N+2}\Big{[}N\lambda_{4\pi,k}^{\mathrm{eff}}(-p,-q,q,p)+\lambda_{4\pi,k}^{\mathrm{eff}}(-p,p,-q,q)$
$\displaystyle+\lambda_{4\pi,k}^{\mathrm{eff}}(-p,q,p,-q)\Big{]}\,,$ (309)

with $\theta$ being the angle between the two momenta $\bm{p}$ and $\bm{q}$.
The inverse retarded propagator, i.e., the two-point correlation function with
one $q$ field and one $c$ field, is given by

$\displaystyle i\Gamma_{k,\phi_{i,q}\phi_{j,c}}^{(2)}(p)\equiv$ $\displaystyle
i\frac{\delta^{2}\Gamma_{k}[\phi]}{\delta\phi_{i,q}(p)\delta\phi_{j,c}(-p)}$
$\displaystyle=$ $\displaystyle
i\delta_{ij}\Big{(}Z_{\phi,k}(p^{2})p^{2}-m_{\pi,k}^{2}\Big{)}\,,$ (310)

and its flow equation reads

$\displaystyle\partial_{\tau}\Gamma_{k,\phi_{q}\phi_{c}}^{(2)}(p)=-\tilde{\partial}_{\tau}\Sigma_{k}(p)$
$\displaystyle=$
$\displaystyle(-\frac{i}{6})(N+2)\int\frac{d^{4}q}{(2\pi)^{4}}\tilde{\partial}_{\tau}\Big{(}G^{K}_{\pi,k}(q)\Big{)}$
$\displaystyle\times\bar{\lambda}_{4\pi,k}^{\mathrm{eff}}(p_{0},|\bm{p}|,q_{0},|\bm{q}|,\cos\theta)\,.$
(311)

Substituting the Keldysh propagator in Equation 290, one arrives at

$\displaystyle\partial_{\tau}\Gamma_{k,\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)$
$\displaystyle=$
$\displaystyle\partial_{\tau}\Gamma_{k,\phi_{q}\phi_{c}}^{(2)\mathrm{I}}(p_{0},|\bm{p}|)+\partial_{\tau}\Gamma_{k,\phi_{q}\phi_{c}}^{(2)\mathrm{II}}(p_{0},|\bm{p}|)\,.$
(312)

The first part on the r.h.s. of Equation 312 reads

$\displaystyle\partial_{\tau}\Gamma_{k,\phi_{q}\phi_{c}}^{(2)\mathrm{I}}(p_{0},|\bm{p}|))$
$\displaystyle=$
$\displaystyle-\frac{1}{24}\frac{(N+2)}{(2\pi)^{2}}\bigg{[}-\frac{\coth\Big{(}\frac{E_{\pi,k}(k)}{2T}\Big{)}}{\big{(}E_{\pi,k}(k)\big{)}^{3}}-\frac{\mathrm{csch}^{2}\Big{(}\frac{E_{\pi,k}(k)}{2T}\Big{)}}{2T\big{(}E_{\pi,k}(k)\big{)}^{2}}\bigg{]}$
$\displaystyle\times(2k^{2})\int_{0}^{k}d|\bm{q}||\bm{q}|^{2}\int_{-1}^{1}d\cos\theta\Big{[}\bar{\lambda}_{4\pi,k}^{\mathrm{eff}}\big{|}_{q_{0}=E_{\pi,k}(k)}$
$\displaystyle+\bar{\lambda}_{4\pi,k}^{\mathrm{eff}}\big{|}_{q_{0}=-E_{\pi,k}(k)}\Big{]}\,,$
(313)

and the second part is given by

$\displaystyle\partial_{\tau}\Gamma_{k,\phi_{q}\phi_{c}}^{(2)\mathrm{II}}(p_{0},|\bm{p}|)$
$\displaystyle=$
$\displaystyle-\frac{1}{24}\frac{(N+2)}{(2\pi)^{2}}\frac{\coth\Big{(}\frac{E_{\pi,k}(k)}{2T}\Big{)}}{\big{(}E_{\pi,k}(k)\big{)}^{2}}(2k^{2})\int_{0}^{k}d|\bm{q}||\bm{q}|^{2}$
$\displaystyle\times\int_{-1}^{1}d\cos\theta\Big{[}\frac{\partial}{\partial
q_{0}}\bar{\lambda}_{4\pi,k}^{\mathrm{eff}}\big{|}_{q_{0}=E_{\pi,k}(k)}$
$\displaystyle-\frac{\partial}{\partial
q_{0}}\bar{\lambda}_{4\pi,k}^{\mathrm{eff}}\big{|}_{q_{0}=-E_{\pi,k}(k)}\Big{]}\,.$
(314)

with

$\displaystyle E_{\pi,k}(k)=$
$\displaystyle\Big{(}k^{2}+m^{2}_{\pi,k}\Big{)}^{1/2}\,,$ (315)

where $m_{\pi,k}^{2}$ can be extracted from the two-point correlation function
at vanishing momentum, viz.,

$\displaystyle m_{\pi,k}^{2}=$
$\displaystyle-\Gamma_{k,\phi_{q}\phi_{c}}^{(2)}(0)\,.$ (316)

Note that the second part in Equation 314 is negligible if the momentum
dependence of the vertex is mild.

###  V.4 Spectral functions and dynamical critical exponent

Figure 56: Spectral function $\rho(p_{0},|\bm{p}|=0)$ as a function of $p_{0}$
with several small (left panel, close to the critical temperature $T_{c}=20.4$
MeV) and large (right panel) values of temperature obtained in the real-time
$O(4)$ scalar theory within the fRG approach. The plots are adopted from
[277]. Figure 57: 3D plots of the spectral function as a function of $p_{0}$
and $|\bm{p}|$ at temperature $T=54$ MeV (left panel) and $145$ MeV (right
panel), obtained in the real-time $O(4)$ scalar theory within the fRG
approach. The plots are adopted from [277]. Figure 58: Double logarithm plot
of the relaxation rate $\omega(|\bm{p}|)$ in Equation 326 as a function of the
spacial momentum with $p_{0}=0$ at the critical temperature $T_{c}=20.4$ MeV,
obtained in the real-time 3$d$ $O(4)$ scalar theory within the fRG approach.
The plot is adopted from [277].

The retarded propagator in the Källén-Lehmann spectral representation is
related to the spectral function $\rho$ via a relation as follows

$\displaystyle G_{R}(p_{0},|\bm{p}|)=$
$\displaystyle-\int_{-\infty}^{\infty}\frac{dp_{0}^{\prime}}{2\pi}\frac{\rho(p_{0}^{\prime},|\bm{p}|)}{p_{0}^{\prime}-(p_{0}+i\epsilon)}\,.$
(317)

Thus, the spectral function is proportional to the imaginary part of the
retarded propagator, i.e.,

$\displaystyle\rho(p_{0},|\bm{p}|)=$
$\displaystyle-2\Im\,G_{R}(p_{0},|\bm{p}|)\,.$ (318)

Notice that here the IR limit $k\to 0$ is tacitly assumed. The retarded
propagator is just the inverse of the two-point correlation function, to wit,

$\displaystyle G_{R}(p_{0},|\bm{p}|)=$
$\displaystyle\left[\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)\right]^{-1}\,.$
(319)

Consequently, the spectral function can be expressed in terms of the real and
imaginary parts of the two-point correlation function, that is,

$\displaystyle\rho(p_{0},|\bm{p}|)=$
$\displaystyle\frac{2\Im\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)}{\left[\Re\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)\right]^{2}+\left[\Im\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)\right]^{2}}\,.$
(320)

Evidently, one has

$\displaystyle\rho(-p_{0},|\bm{p}|)=$ $\displaystyle-\rho(p_{0},|\bm{p}|)\,.$
(321)

In Figure 56, the spectral function $\rho(p_{0},|\bm{p}|=0)$ is shown as a
function of $p_{0}$ with different values of the temperature. In the left
panel, the temperatures are above but close to the critical temperature
$T_{c}=20.4$ MeV for the phase transition, where the symmetry is broken from
$O(N)$ to $O(N-1)$ when $T<T_{c}$, while in the right panel, the temperatures
are far larger than $T_{c}$. One can see that when temperature is large, one
has a negative spectral function in the regime of small $p_{0}$, and there is
a minus peak structure around the pole mass. It is found that the negative
spectral function results from the contributions of the Landau damping, and
see [277] for a more detailed discussion. Nonetheless, when the temperature is
decreased below about 60 MeV, the Landau damping is dominated by the process
of creation and annihilation of particles, and the spectral function is
positive. Moreover, when the temperature is more and more close to the
critical temperature, the peak structure on the spectral function becomes more
and more wider, and finally disappears at the critical temperature. The 3D
plots of the spectral function as a function of $p_{0}$ and $|\bm{p}|$ are
shown in Figure 57 with two different values of the temperature.

We proceed to the discussion of the dynamical critical exponent, and begin
with the kinetic coefficient $\Gamma(|\bm{p}|)$ that is defined as

$\displaystyle\frac{1}{\Gamma(|\bm{p}|)}=$
$\displaystyle-i\frac{\partial\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)}{\partial
p_{0}}\bigg{|}_{p_{0}=0}$ $\displaystyle=$
$\displaystyle\frac{\partial\Im\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)}{\partial
p_{0}}\bigg{|}_{p_{0}=0}\,,$ (322)

where the parity properties of the real and imaginary parts of the two-point
correlation function have been used, i.e.,

$\displaystyle\Re\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(-p_{0},|\bm{p}|)=$
$\displaystyle\Re\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)\,,$ (323)
$\displaystyle\Im\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(-p_{0},|\bm{p}|)=$
$\displaystyle-\Im\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0},|\bm{p}|)\,.$ (324)

The relaxation rate, or dissipative characteristic frequency, is given by

$\displaystyle\omega(|\bm{p}|)=$
$\displaystyle\Gamma(|\bm{p}|)\left(-\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0}=0,|\bm{p}|)\right)$
$\displaystyle=$
$\displaystyle-\Gamma(|\bm{p}|)\Re\,\Gamma_{\phi_{q}\phi_{c}}^{(2)}(p_{0}=0,|\bm{p}|)\,.$
(325)

When the momentum is larger than the correlation length $|\bm{p}|>\xi^{-1}$,
with $\xi\sim m_{\phi}^{-1}$, the relaxation rate scales as

$\displaystyle\omega(|\bm{p}|)\propto$ $\displaystyle|\bm{p}|^{z}\,,$ (326)

through which one can extract the dynamical critical exponent $z$ [278]. From
Figure 58 the value of the dynamical critical exponent in the real-time 3$d$
$O(4)$ scalar theory is extracted, and one arrives at $z=2.02284(6)$, where
the numerical error is shown in the bracket.

According to the standard classification for universalities of the critical
dynamics [278], it is argued that the critical dynamics of the relativistic
$O(4)$ scalar theory belongs to the universality of Model G [279], see also
[280], which leaves us with $z=3/2$ in three dimensions. The dynamic critical
exponent for the $O(4)$ model is also calculated in real-time classical-
statistical lattice simulations, and it is found that $z$ is in favor of 2,
but there is still a sizable numerical error [280]. The dynamic critical
exponent in a relativistic $O(N)$ vector model is also found to be close to 2
[258]. Similar result is found in a $O(3)$ model in [281]. Moreover,
$z=1.92(11)$ is found for Model A in three spatial dimensions from real-time
classical-statistical lattice simulations [282]. In short, the dynamic
critical exponent is far from clear and conclusive in comparison to the static
critical exponent, and more studies of the critical dynamics from different
approaches, including the real-time fRG, are necessary and desirable in the
future.

##  VI Conclusions

In this paper we present an overview on recent progress in studies of QCD at
finite temperature and densities within the fRG approach. After a brief
introduction of the formalism, the fRG approach is applied in low energy
effective field theories (LEFTs) and the first-principle QCD. The mechanism of
quark mass production and natural emergence of bound states is well
illustrated within the fRG approach, and a set of self-consistent flow
equations of different correlation functions provide the necessary
resummations, and plays the same role as the quark gap equation and Bethe-
Salpeter equation of bound states.

We present results for the QCD phase structure and the location of the
critical end point (CEP), the QCD equation of state (EoS), the magnetic EoS,
baryon number fluctuations confronted with recent experimental measurements,
various critical exponents, etc. It is found that the non-monotonic dependence
of the kurtosis of the net-baryon or net-proton (proxy for net-baryon in
experiments) number distributions could arise from the increasingly sharp
crossover with the decrease of the beam collision energy, which in turn
indicates that the non-monotonicity observed in experiments is highly non-
trivial. Furthermore, recent estimates of the location of the CEP from first-
principle QCD calculations within fRG and Dyson-Schwinger Equations, which
passes through lattice benchmark tests at small baryon chemical potentials,
converge in a rather small region at baryon chemical potentials of about 600
MeV. But it should be reminded that errors of functional approaches increase
significantly in the regime of $\mu_{B}/T\gtrsim 4$, and thus one arrives at a
more reasonable estimation for the location of CEP as
$450\,\mathrm{MeV}\lesssim{\mu_{B}}_{\tiny{\mathrm{CEP}}}\lesssim
650\,\mathrm{MeV}$. Moreover, a region of inhomogeneous instability indicated
by a negative wave function renormalization is found with $\mu_{B}\gtrsim 420$
MeV, and its consequence on the phenomenology of heavy-ion collisions has been
investigated very recently [244, 245]. It is found that this inhomogeneous
instability would result in a moat regime in both the particle $p_{T}$
spectrum and the two-particle correlation. By investigating the critical
behavior and extracting critical exponents in the vicinity of the CEP, it is
found that the size of the critical region is extremely small, quite smaller
than $\sim 1$ MeV.

In this review we also discuss the real-time fRG, in which the flow equations
are formulated on the Schwinger-Keldysh closed time path. By organizing the
effective action and the flow equations in terms of the “classical” and
“quantum” fields, one is able to give a concise diagrammatic representation
for the flow equations of propagators and vertices in the real-time fRG. The
spectral functions of the $O(N)$ scalar theory in the critical regime in the
proximity of the critical temperature are obtained. The dynamical critical
exponent in the $O(4)$ scalar theory in $3+1$ dimensions is found to be
$z\simeq 2.023$.

###### Acknowledgements.

I would like to thank Heng-Tong Ding, Xiaofeng Luo, Jan M. Pawlowski, Bernd-
Jochen Schaefer, Yue-Liang Wu for reading this manuscript and providing
helpful comments. I thank Jens Braun, Yong-rui Chen, Chuang Huang, Zi-Wei Lin,
Yu-xin Liu, Xiaofeng Luo, Guo-Liang Ma, Jan M. Pawlowski, Fabian Rennecke,
Bernd-Jochen Schaefer, Yang-yang Tan, Rui Wen, Yue-Liang Wu, Shi Yin for
collaborating on related subjects. I am also grateful to Jens Braun, Yong-rui
Chen, Heng-Tong Ding, Fei Gao, De-fu Hou, Chuang Huang, Mei Huang, Zi-Wei Lin,
Yu-xin Liu, Xiaofeng Luo, Guo-Liang Ma, Jan M. Pawlowski, Fabian Rennecke,
Bernd-Jochen Schaefer, Huichao Song, Yang-yang Tan, Rui Wen, Nicolas Wink,
Yue-Liang Wu, Shi Yin, Yu Zhang, Peng-fei Zhuang for discussions on related
subjects. I am very grateful to the authors of [159] for providing me with the
unpublished results in Figure 16. I would like to thank the authors of [92]
for providing unpublished results in Figure 7, Figure 8, Figure 9, Figure 10
and Figure 11. Moreover, I also would like to thank the members of the fQCD
collaboration [283] for discussions and collaborations. This work is supported
by the National Natural Science Foundation of China under Grant No. 12175030.

##  Appendix A Flow equations of the gluon and ghost self-energies in Yang-
Mills theory at finite temperature

###  A.1 Feynman rules

First of all, we present the Feynman rules for relevant propagators and
vertices at finite temperature and densities.

####  A.1.1 Gluon propagator

The kinetic term for the gluon field in Equation 52 in momentum space is given
by

$\displaystyle\Gamma_{k,2A}\equiv$
$\displaystyle\frac{1}{2}\int\frac{d^{4}q}{(2\pi)^{4}}A^{a}_{\mu}(-q)q^{2}\Big{[}\big{(}Z_{A,k}^{\mathrm{M}}(q)\Pi_{\mu\nu}^{\mathrm{M}}(q)$
$\displaystyle+Z_{A,k}^{\mathrm{E}}(q)\Pi_{\mu\nu}^{\mathrm{E}}(q)\big{)}+\frac{1}{\xi}\Pi_{\mu\nu}^{\parallel}(q)\Big{]}A^{a}_{\nu}(q)\,,$
(327)

where the magnetic projection operator reads

$\displaystyle\Pi_{\mu\nu}^{\mathrm{M}}(q)=$ $\displaystyle(1-\delta_{\mu
0})(1-\delta_{\nu
0})\Big{(}\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{\bm{q}^{2}}\Big{)}\,,$ (328)

and the electric projection operator

$\displaystyle\Pi_{\mu\nu}^{\mathrm{E}}(q)=$
$\displaystyle\Pi_{\mu\nu}^{\perp}(q)-\Pi_{\mu\nu}^{\mathrm{M}}(q)\,,$ (329)

with the transverse and longitudinal tensors given by

$\displaystyle\Pi_{\mu\nu}^{\perp}(q)=$
$\displaystyle\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}},\quad\mathrm{and}\quad\Pi_{\mu\nu}^{\parallel}(q)=\frac{q_{\mu}q_{\nu}}{q^{2}}\,,$
(330)

respectively. Note that in Equation 327 different magnetic and electric
gluonic dressing functions, i.e., $Z_{A,k}^{\mathrm{M}}(q)\neq
Z_{A,k}^{\mathrm{E}}(q)$, are assumed at finite temperature. Differentiating
Equation 327 w.r.t. the gluon field twice, one obtains

$\displaystyle\big{(}\Gamma_{k,2A}^{(2)AA}\big{)}^{ab}_{\mu\nu}(q^{\prime},q)\equiv\frac{\delta^{2}\Gamma_{k,2A}}{\delta
A^{a}_{\mu}(q^{\prime})\delta A^{b}_{\nu}(q)}$ $\displaystyle=$ $\displaystyle
q^{2}\Big{[}\big{(}Z_{A,k}^{\mathrm{M}}(q)\Pi_{\mu\nu}^{\mathrm{M}}(q)+Z_{A,k}^{\mathrm{E}}(q)\Pi_{\mu\nu}^{\mathrm{E}}(q)\big{)}+\frac{1}{\xi}\Pi_{\mu\nu}^{\parallel}(q)\Big{]}$
$\displaystyle\times\delta^{ab}(2\pi)^{4}\delta^{4}(q^{\prime}+q)\,.$ (331)

Moreover, the regulator for the gluon reads

$\displaystyle\big{(}R_{k}^{AA}\big{)}^{ab}_{\mu\nu}(q^{\prime},q)$
$\displaystyle=$
$\displaystyle\Big{[}\big{(}Z_{A,k}^{\mathrm{M}}q^{2}r_{B}(q^{2}/k^{2})\Pi_{\mu\nu}^{\mathrm{M}}(q)+Z_{A,k}^{\mathrm{E}}q^{2}r_{B}(q^{2}/k^{2})\Pi_{\mu\nu}^{\mathrm{E}}(q)\big{)}$
$\displaystyle+\frac{1}{\xi}q^{2}r_{B}(q^{2}/k^{2})\Pi_{\mu\nu}^{\parallel}(q)\Big{]}\delta^{ab}(2\pi)^{4}\delta^{4}(q^{\prime}+q)\,,$
(332)

where $Z_{A,k}^{\mathrm{M}}$ and $Z_{A,k}^{\mathrm{E}}$ are independent of
momentum $q$ by contrast to those in Equation 331. The threshold function
$r_{B}(x)$ can be chosen to be that in Equation 10 or Equation 12. Therefore,
the gluon propagator is readily obtained from Equation 23, as follows

$\displaystyle\big{(}G_{k}^{AA}\big{)}^{ab}_{\mu\nu}(q,q^{\prime})$
$\displaystyle=(2\pi)^{4}\delta^{4}(q^{\prime}+q)\big{(}G_{k}^{AA}\big{)}^{ab}_{\mu\nu}(q)\,,$
(333)

with

$\displaystyle\big{(}G_{k}^{AA}\big{)}^{ab}_{\mu\nu}(q)$
$\displaystyle=\Big{(}G_{A,k}^{\mathrm{M}}(q)\Pi_{\mu\nu}^{\mathrm{M}}(q)+G_{A,k}^{\mathrm{E}}(q)\Pi_{\mu\nu}^{\mathrm{E}}(q)\Big{)}\delta^{ab}\,,$
(334)

where one has

$\displaystyle G_{A,k}^{\mathrm{M}}(q)$
$\displaystyle=\frac{1}{q^{2}\big{[}Z_{A,k}^{\mathrm{M}}(q)+Z_{A,k}^{\mathrm{M}}r_{B}(q^{2}/k^{2})\big{]}}\,,$
(335) $\displaystyle G_{A,k}^{\mathrm{E}}(q)$
$\displaystyle=\frac{1}{q^{2}\big{[}Z_{A,k}^{\mathrm{E}}(q)+Z_{A,k}^{\mathrm{E}}r_{B}(q^{2}/k^{2})\big{]}}\,.$
(336)

Here the gauge parameter is chosen to be $\xi=0$.

####  A.1.2 Ghost propagator

In the same way the kinetic term for the ghost field in Equation 52 reads

$\displaystyle\Gamma_{k,2c}$
$\displaystyle\equiv\int\frac{d^{4}q}{(2\pi)^{4}}Z_{c,k}(q)\bar{c}^{a}(-q)q^{2}c^{a}(q)\,,$
(337)

where we have used the convention of Fourier transformation for the Grassmann
fields as follows

$\displaystyle\bar{c}^{a}(x)=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}\bar{c}^{a}(q)e^{iqx}\,,$ (338)
$\displaystyle c^{a}(x)=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}c^{a}(q)e^{iqx}\,.$ (339)

Then, it follows that

$\displaystyle\big{(}\Gamma_{k,2c}^{(2)\bar{c}c}\big{)}^{ab}(q^{\prime},q)\equiv\frac{\overrightarrow{\delta}}{\delta\bar{c}^{a}(q^{\prime})}\Gamma_{k,2c}\frac{\overleftarrow{\delta}}{\delta
c^{b}(q)}$ $\displaystyle=$ $\displaystyle
Z_{c,k}(q)q^{2}\delta^{ab}(2\pi)^{4}\delta^{4}(q^{\prime}+q)\,,$ (340)

and

$\displaystyle\big{(}\Gamma_{k,2c}^{(2)c\bar{c}}\big{)}^{ab}(q^{\prime},q)\equiv\frac{\overrightarrow{\delta}}{\delta
c^{a}(q^{\prime})}\Gamma_{k,2c}\frac{\overleftarrow{\delta}}{\delta\bar{c}^{b}(q)}$
$\displaystyle=$ $\displaystyle-
Z_{c,k}(q^{\prime}){q^{\prime}}^{2}\delta^{ab}(2\pi)^{4}\delta^{4}(q^{\prime}+q)$
$\displaystyle=$
$\displaystyle-\big{(}\Gamma_{k,2c}^{(2)\bar{c}c}\big{)}^{ba}(q,q^{\prime})=-\Big{[}\big{(}\Gamma_{k,2c}^{(2)\bar{c}c}\big{)}^{T}\Big{]}^{ab}(q^{\prime},q)\,,$
(341)

Note that, for a Grassmann field, the relation such as

$\displaystyle\Gamma_{k,2c}^{(2)c\bar{c}}$
$\displaystyle=-\big{(}\Gamma_{k,2c}^{(2)\bar{c}c}\big{)}^{T}\,,$ (342)

always holds. The regulator for the ghost field reads

$\displaystyle\big{(}R_{k}^{\bar{c}c}\big{)}^{ab}(q^{\prime},q)$
$\displaystyle=Z_{c,k}q^{2}r_{B}(q^{2}/k^{2})\delta^{ab}(2\pi)^{4}\delta^{4}(q^{\prime}+q)\,,$
(343)

and $R_{k}^{c\bar{c}}=-\big{(}R_{k}^{\bar{c}c}\big{)}^{T}$. As a consequence,
the $\mathcal{P}$ matrix in Equation 62 for the ghost field reads

$\displaystyle\mathcal{P}_{c}$
$\displaystyle=\begin{pmatrix}0&-(\Gamma_{k,2c}^{(2)\bar{c}c}+R_{k}^{\bar{c}c})^{T}\\\
\Gamma_{k,2c}^{(2)\bar{c}c}+R_{k}^{\bar{c}c}&0\end{pmatrix}\,,$ (344)

whose inverse matrix is just the ghost propagator which is given by

$\displaystyle\left(\frac{1}{\mathcal{P}}\right)_{c}$
$\displaystyle=\begin{pmatrix}0&G_{k}^{c\bar{c}}\\\
-(G_{k}^{c\bar{c}})^{T}&0\end{pmatrix}\,,$ (345)

with

$\displaystyle G_{k}^{c\bar{c}}$
$\displaystyle=\frac{1}{\Gamma_{k,2c}^{(2)\bar{c}c}+R_{k}^{\bar{c}c}}\,.$
(346)

Recovering the indices, one obtains the ghost propagator as follows

$\displaystyle\big{(}G_{k}^{c\bar{c}}\big{)}^{ab}(q,q^{\prime})$
$\displaystyle=\big{(}G_{k}^{c\bar{c}}\big{)}^{ab}(q)(2\pi)^{4}\delta^{4}(q+q^{\prime})$
$\displaystyle=G_{k}^{c}(q)\delta^{ab}(2\pi)^{4}\delta^{4}(q+q^{\prime})\,,$
(347)

with

$\displaystyle G_{k}^{c}(q)$
$\displaystyle=\frac{1}{q^{2}\big{[}Z_{c,k}(q)+Z_{c,k}r_{B}(q^{2}/k^{2})\big{]}}\,.$
(348)

####  A.1.3 Quark propagator

The kinetic term for the quark field in Equation 52 reads

$\displaystyle\Gamma_{k,2q}$
$\displaystyle\equiv\int\frac{d^{4}q}{(2\pi)^{4}}\bar{q}(-q)\Big{[}Z_{q,k}(q)i\gamma\cdot
q+m_{q,k}(q)\Big{]}q(q)\,.$ (349)

It follows that

$\displaystyle\big{(}\Gamma_{k,2q}^{(2)\bar{q}q}\big{)}_{ij}(q^{\prime},q)=\frac{\overrightarrow{\delta}}{\delta\bar{q}_{i}(q^{\prime})}\Gamma_{k,2q}\frac{\overleftarrow{\delta}}{\delta
q_{j}(q)}$ $\displaystyle=$
$\displaystyle\Big{[}Z_{q,k}(q)iq_{\mu}(\gamma_{\mu})_{ij}+m_{q,k}(q)\delta_{ij}\Big{]}(2\pi)^{4}\delta^{4}(q^{\prime}+q)\,,$
(350)

and

$\displaystyle\big{(}\Gamma_{k,2q}^{(2)q\bar{q}}\big{)}_{ij}(q^{\prime},q)=\frac{\overrightarrow{\delta}}{\delta
q_{i}(q^{\prime})}\Gamma_{k,2q}\frac{\overleftarrow{\delta}}{\delta\bar{q}_{j}(q)}$
$\displaystyle=$
$\displaystyle-\Big{[}Z_{q,k}(q^{\prime})iq^{\prime}_{\mu}(\gamma_{\mu})_{ji}+m_{q,k}(q^{\prime})\delta_{ji}\Big{]}(2\pi)^{4}\delta^{4}(q+q^{\prime})$
$\displaystyle=-\Big{[}\big{(}Z_{q,k}(q^{\prime})iq^{\prime}_{\mu}\gamma_{\mu}+m_{q,k}(q^{\prime})\big{)}^{T}\Big{]}_{ij}(2\pi)^{4}\delta^{4}(q+q^{\prime})$
$\displaystyle=-\Big{[}\big{(}\Gamma_{k,2q}^{(2)\bar{q}q}\big{)}^{T}\Big{]}_{ij}(q^{\prime},q)\,.$
(351)

The regulator reads

$\displaystyle\big{(}R_{k}^{\bar{q}q}\big{)}_{ij}(q^{\prime},q)$
$\displaystyle=Z_{q,k}iq\cdot\gamma_{ij}r_{F}(q^{2}/k^{2})(2\pi)^{4}\delta^{4}(q^{\prime}+q)\,.$
(352)

The $\mathcal{P}$ matrix for the quark field reads

$\displaystyle\mathcal{P}_{q}$
$\displaystyle=\begin{pmatrix}0&-(\Gamma_{k,2q}^{(2)\bar{q}q}+R_{k}^{\bar{q}q})^{T}\\\
\Gamma_{k,2q}^{(2)\bar{q}q}+R_{k}^{\bar{q}q}&0\end{pmatrix}\,,$ (353)

and the relevant propagator is given by

$\displaystyle\left(\frac{1}{\mathcal{P}}\right)_{q}$
$\displaystyle=\begin{pmatrix}0&G_{k}^{q\bar{q}}\\\
-(G_{k}^{q\bar{q}})^{T}&0\end{pmatrix}\,,$ (354)

with

$\displaystyle G_{k}^{q\bar{q}}$
$\displaystyle=\frac{1}{\Gamma_{k,2q}^{(2)\bar{q}q}+R_{k}^{\bar{q}q}}\,.$
(355)

Displaying the momentum explicitly, one arrives at the quark propagator as
follows

$\displaystyle G_{k}^{q\bar{q}}(q,q^{\prime})$
$\displaystyle=G_{k}^{q\bar{q}}(q)(2\pi)^{4}\delta^{4}(q+q^{\prime})$
$\displaystyle=G_{k}^{q}(q)(2\pi)^{4}\delta^{4}(q+q^{\prime})\,,$ (356)

with

$\displaystyle G^{q}_{k}(q)$ $\displaystyle=\frac{1}{Z_{q,k}(q)i\gamma\cdot
q+Z_{q,k}r_{F}(q^{2}/k^{2})i\gamma\cdot q+m_{q,k}(q)}\,.$ (357)

####  A.1.4 Ghost-gluon vertex

First of all, we consider the case in the vacuum. The action relevant to the
ghost-gluon interaction in Equation 52 reads

$\displaystyle\Gamma_{k,2cA}\equiv\int_{x}(-g)f^{abc}\partial_{\mu}\bar{c}^{a}(x)c^{b}(x)A^{c}_{\mu}(x)$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\frac{d^{4}q}{(2\pi)^{4}}(-g)f^{abc}ip_{\mu}\bar{c}^{a}(p)c^{b}(q)A^{c}_{\mu}(-p-q)\,,$
(358)

which yields

$\displaystyle\big{(}\Gamma_{k,2cA}^{(3)\bar{c}cA}\big{)}^{abc}_{\mu}(p,q,k)\equiv\frac{\delta}{\delta
A^{c}_{\mu}(k)}\frac{\overrightarrow{\delta}}{\delta\bar{c}^{a}(p)}\Gamma_{k,2cA}\frac{\overleftarrow{\delta}}{\delta
c^{b}(q)}$ $\displaystyle=$ $\displaystyle-
gf^{abc}ip_{\mu}(2\pi)^{4}\delta^{4}(p+q+k)\,.$ (359)

Thus, the Feynman rule for the ghost-gluon vertex in the vacuum is given as
follows

$\displaystyle\parbox[c]{52.03227pt}{\includegraphics[width=52.03227pt]{ghost-
gluon-
vert}}\equiv-\big{(}\Gamma_{k,2cA}^{(3)\bar{c}cA}\big{)}^{abc}_{\mu}(p,q,k)$
$\displaystyle=$
$\displaystyle-g\big{(}S^{(3)}_{\bar{c}cA}\big{)}^{abc}_{\mu}(p,q,k)\,.$ (360)

with the classical tensor of the ghost-gluon vertex defined by

$\displaystyle\big{(}S^{(3)}_{\bar{c}cA}\big{)}^{abc}_{\mu}(p,q,k)$
$\displaystyle\equiv-ip_{\mu}f^{abc}\,,$ (361)

At finite temperature, the external leg of gluon would be split into the
magnetic and electric sectors, and thus the general form for the ghost-gluon
vertex at finite temperature reads

$\displaystyle=-\big{(}S^{(3)}_{\bar{c}cA}\big{)}^{abc}_{\mu^{\prime}}(p,q,k)\big{(}\lambda_{\bar{c}cA}\big{)}_{\mu^{\prime}\mu}(k)\,.$
(362)

with

$\displaystyle\big{(}\lambda_{\bar{c}cA}\big{)}_{\mu^{\prime}\mu}(k)$
$\displaystyle\equiv\lambda^{\mathrm{M}}_{\bar{c}cA}\Pi_{\mu^{\prime}\mu}^{\mathrm{M}}(k)+\lambda^{\mathrm{E}}_{\bar{c}cA}\Pi_{\mu^{\prime}\mu}^{\mathrm{E}}(k)\,,$
(363)

where $\lambda^{\mathrm{M}}_{\bar{c}cA}$ and
$\lambda^{\mathrm{E}}_{\bar{c}cA}$ are the dressing couplings of the ghost-
gluon vertex for the magnetic and electric components, respectively. Note that
the couplings are dependent on the RG scale, and their subscript $k$ has been
suppressed.

####  A.1.5 Three- and four-gluon vertices

From the effective action in Equation 52, one is able to obtain the three-
gluon vertex in the vacuum, as follows

$\displaystyle\big{(}\Gamma_{k}^{(3)A^{3}}\big{)}^{a_{1}a_{2}a_{3}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle\equiv$ $\displaystyle\frac{\delta^{3}\Gamma_{k}}{\delta
A^{a_{1}}_{\mu_{1}}(q_{1})\delta A^{a_{2}}_{\mu_{2}}(q_{2})\delta
A^{a_{3}}_{\mu_{3}}(q_{3})}\bigg{|}_{A=0}$ $\displaystyle=$
$\displaystyle(2\pi)^{4}\delta^{4}(q_{1}+q_{2}+q_{3})g\big{(}S^{(3)}_{A^{3}}\big{)}^{a_{1}a_{2}a_{3}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})\,,$
(364)

with the classical three-gluon tensor

$\displaystyle\big{(}S^{(3)}_{A^{3}}\big{)}^{a_{1}a_{2}a_{3}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle\equiv$ $\displaystyle-
if^{a_{1}a_{2}a_{3}}\Big{[}\delta_{\mu_{1}\mu_{2}}(q_{1}-q_{2})_{\mu_{3}}+\delta_{\mu_{2}\mu_{3}}(q_{2}-q_{3})_{\mu_{1}}$
$\displaystyle+\delta_{\mu_{3}\mu_{1}}(q_{3}-q_{1})_{\mu_{2}}\Big{]}\,.$ (365)

In the same way, the four-gluon vertex reads

$\displaystyle\big{(}\Gamma_{k}^{(4)A^{4}}\big{)}^{a_{1}a_{2}a_{3}a_{4}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle\equiv$ $\displaystyle\frac{\delta^{4}\Gamma_{k,A}}{\delta
A^{a_{1}}_{\mu_{1}}(q_{1})\delta A^{a_{2}}_{\mu_{2}}(q_{2})\delta
A^{a_{3}}_{\mu_{3}}(q_{3})\delta A^{a_{4}}_{\mu_{4}}(q_{4})}\bigg{|}_{A=0}$
$\displaystyle=$
$\displaystyle(2\pi)^{4}\delta^{4}(q_{1}+q_{2}+q_{3}+q_{4})g^{2}\big{(}S^{(4)}_{A^{4}}\big{)}^{a_{1}a_{2}a_{3}a_{4}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}\,,$
(366)

with the classical four-gluon tensor

$\displaystyle\big{(}S^{(4)}_{A^{4}}\big{)}^{a_{1}a_{2}a_{3}a_{4}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}$
$\displaystyle\equiv
f^{ea_{1}a_{2}}f^{ea_{3}a_{4}}(\delta_{\mu_{1}\mu_{3}}\delta_{\mu_{2}\mu_{4}}-\delta_{\mu_{1}\mu_{4}}\delta_{\mu_{2}\mu_{3}})$
$\displaystyle+f^{ea_{1}a_{3}}f^{ea_{2}a_{4}}(\delta_{\mu_{1}\mu_{2}}\delta_{\mu_{3}\mu_{4}}-\delta_{\mu_{1}\mu_{4}}\delta_{\mu_{2}\mu_{3}})$
$\displaystyle+f^{ea_{1}a_{4}}f^{ea_{2}a_{3}}(\delta_{\mu_{1}\mu_{2}}\delta_{\mu_{4}\mu_{3}}-\delta_{\mu_{1}\mu_{3}}\delta_{\mu_{4}\mu_{2}})\,.$
(367)

Similar with the case of ghost-gluon vertex, each external leg of the three-
and four-gluon vertices should be split into a sum of the magnetic and
electric components at finite temperature, and see, e.g., [57] for more
details. Consequently, the three-gluon vertex at finite temperature reads

$\displaystyle\parbox[c]{52.03227pt}{\includegraphics[width=52.03227pt]{A3-vert}}\equiv-\big{(}\Gamma_{k}^{(3)A^{3}}\big{)}^{a_{1}a_{2}a_{3}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle=-\big{(}S^{(3)}_{A^{3}}\big{)}^{a_{1}a_{2}a_{3}}_{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}(q_{1},q_{2},q_{3})\big{(}\lambda_{A^{3}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})\,,$
(368)

with

$\displaystyle\big{(}\lambda_{A^{3}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle\equiv$
$\displaystyle\lambda^{\mathrm{MMM}}_{A^{3}}\big{(}\Pi^{\mathrm{MMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle+\lambda^{\mathrm{EMM}}_{A^{3}}\big{(}\Pi^{\mathrm{EMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle+\lambda^{\mathrm{EEM}}_{A^{3}}\big{(}\Pi^{\mathrm{EEM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle+\lambda^{\mathrm{EEE}}_{A^{3}}\big{(}\Pi^{\mathrm{EEE}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})\,,$
(369)

where $\lambda^{\mathrm{MMM}}_{A^{3}}$, $\lambda^{\mathrm{EMM}}_{A^{3}}$,
$\lambda^{\mathrm{EEM}}_{A^{3}}$, $\lambda^{\mathrm{EEE}}_{A^{3}}$ are the
dressing three-gluon couplings for different components, and the relevant
projectors reads

$\displaystyle\big{(}\Pi^{\mathrm{MMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})\equiv\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\,,$
(370)
$\displaystyle\big{(}\Pi^{\mathrm{EEE}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})\equiv\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\,,$
(371)

the projector with one electric gluon and two magnetic gluons

$\displaystyle\big{(}\Pi^{\mathrm{EMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle\equiv$
$\displaystyle\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})$
$\displaystyle\times\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\,,$
(372)

the projector with two electric gluons and one magnetic gluon

$\displaystyle\big{(}\Pi^{\mathrm{EEM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})$
$\displaystyle\equiv$
$\displaystyle\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})$
$\displaystyle\times\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\,.$
(373)

The four-gluon vertex at finite temperature reads

$\displaystyle\parbox[c]{52.03227pt}{\includegraphics[width=52.03227pt]{A4-vert}}\equiv-\big{(}\Gamma_{k}^{(4)A^{4}}\big{)}^{a_{1}a_{2}a_{3}a_{4}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle=-\big{(}S^{(4)}_{A^{4}}\big{)}^{a_{1}a_{2}a_{3}a_{4}}_{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}\big{(}\lambda_{A^{4}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})\,,$
(374)

with

$\displaystyle\big{(}\lambda_{A^{4}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle\equiv$
$\displaystyle\lambda^{\mathrm{MMMM}}_{A^{4}}\big{(}\Pi^{\mathrm{MMMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle+\lambda^{\mathrm{EMMM}}_{A^{4}}\big{(}\Pi^{\mathrm{EMMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle+\lambda^{\mathrm{EEMM}}_{A^{4}}\big{(}\Pi^{\mathrm{EEMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle+\lambda^{\mathrm{EEEM}}_{A^{4}}\big{(}\Pi^{\mathrm{EEEM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle+\lambda^{\mathrm{EEEE}}_{A^{4}}\big{(}\Pi^{\mathrm{EEEE}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})\,,$
(375)

where $\lambda^{\mathrm{MMMM}}_{A^{4}}$, $\lambda^{\mathrm{EMMM}}_{A^{4}}$,
$\lambda^{\mathrm{EEMM}}_{A^{4}}$, $\lambda^{\mathrm{EEEM}}_{A^{4}}$, and
$\lambda^{\mathrm{EEEE}}_{A^{4}}$ are the dressing four-gluon couplings for
different components, and the relevant projectors reads

$\displaystyle\big{(}\Pi^{\mathrm{MMMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle\equiv$
$\displaystyle\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{M}}(q_{4})\,,$
(376)

and

$\displaystyle\big{(}\Pi^{\mathrm{EEEE}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle\equiv$
$\displaystyle\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{E}}(q_{4})\,,$
(377)

the projector with one electric gluon and three magnetic gluons

$\displaystyle\big{(}\Pi^{\mathrm{EMMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle\equiv$
$\displaystyle\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{M}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{M}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{M}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{E}}(q_{4})\,,$
(378)

the projector with three electric gluon and one magnetic gluons

$\displaystyle\big{(}\Pi^{\mathrm{EEEM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle\equiv$
$\displaystyle\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{M}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{E}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{E}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{E}}(q_{4})\,,$
(379)

the projector with two electric gluons and two magnetic gluons

$\displaystyle\big{(}\Pi^{\mathrm{EEMM}}\big{)}^{\mu_{1}^{\prime}\mu_{2}^{\prime}\mu_{3}^{\prime}\mu_{4}^{\prime}}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(q_{1},q_{2},q_{3},q_{4})$
$\displaystyle\equiv$
$\displaystyle\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{M}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{M}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{E}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{E}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{M}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{E}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{M}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{E}}(q_{4})$
$\displaystyle+\Pi_{\mu_{1}^{\prime}\mu_{1}}^{\mathrm{M}}(q_{1})\Pi_{\mu_{2}^{\prime}\mu_{2}}^{\mathrm{M}}(q_{2})\Pi_{\mu_{3}^{\prime}\mu_{3}}^{\mathrm{E}}(q_{3})\Pi_{\mu_{4}^{\prime}\mu_{4}}^{\mathrm{E}}(q_{4})\,.$
(380)

###  A.2 Gluon self-energy

With the Feynman rules discussed in Section A.1, it is straightforward to
write down expressions for the loop diagrams of the gluon self-energy as shown
on the r.h.s. of flow equation in Figure 3. The first gluon loop reads

$\displaystyle\equiv\big{(}\Sigma^{AA}_{a}\big{)}_{\mu\nu}^{ab}(p)\,.$ (381)

with

$\displaystyle\big{(}\Sigma^{AA}_{a}\big{)}_{\mu\nu}^{ab}(p)$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}(-)\big{(}S^{(3)}_{A^{3}}\big{)}^{aa_{1}a_{2}}_{\mu^{\prime}\mu_{1}^{\prime}\mu_{2}^{\prime}}(-p,-q+p,q)$
$\displaystyle\times\big{(}\lambda_{A^{3}}\big{)}^{\mu^{\prime}\mu_{1}^{\prime}\mu_{2}^{\prime}}_{\mu\mu_{1}\mu_{2}}(-p,-q+p,q)\Big{[}G_{A}^{\mathrm{M}}(q)\Pi_{\mu_{2}\nu_{2}}^{\mathrm{M}}(q)$
$\displaystyle+G_{A}^{\mathrm{E}}(q)\Pi_{\mu_{2}\nu_{2}}^{\mathrm{E}}(q)\Big{]}\delta^{a_{2}b_{2}}(-)\big{(}S^{(3)}_{A^{3}}\big{)}^{bb_{1}b_{2}}_{\nu^{\prime}\nu_{1}^{\prime}\nu_{2}^{\prime}}(p,q-p,-q)$
$\displaystyle\times\big{(}\lambda_{A^{3}}\big{)}^{\nu^{\prime}\nu_{1}^{\prime}\nu_{2}^{\prime}}_{\nu\nu_{1}\nu_{2}}(p,q-p,-q)\Big{[}G_{A}^{\mathrm{M}}(q-p)\Pi_{\nu_{1}\mu_{1}}^{\mathrm{M}}(q-p)$
$\displaystyle+G_{A}^{\mathrm{E}}(q-p)\Pi_{\nu_{1}\mu_{1}}^{\mathrm{E}}(q-p)\Big{]}\delta^{b_{1}a_{1}}\,.$
(382)

Projecting Equation 382 onto the magnetic component, one arrives at

$\displaystyle\big{(}\Sigma^{AA}_{a}\big{)}_{\mu\nu}^{ab}(p)\big{(}\delta^{ab}\Pi_{\mu\nu}^{\mathrm{M}}(p)\big{)}$
$\displaystyle=$ $\displaystyle
N_{c}(N_{c}^{2}-1)\int\frac{d^{4}q}{(2\pi)^{4}}\bigg{[}{\lambda^{\mathrm{MMM}}_{A^{3}}}^{2}G_{A}^{\mathrm{M}}(q)G_{A}^{\mathrm{M}}(q-p)C^{M}_{MM}$
$\displaystyle+{\lambda^{\mathrm{EEM}}_{A^{3}}}^{2}G_{A}^{\mathrm{E}}(q)G_{A}^{\mathrm{E}}(q-p)C^{M}_{EE}+{\lambda^{\mathrm{EMM}}_{A^{3}}}^{2}$
$\displaystyle\times\Big{(}G_{A}^{\mathrm{M}}(q)G_{A}^{\mathrm{E}}(q-p)C^{M}_{ME}+G_{A}^{\mathrm{E}}(q)G_{A}^{\mathrm{M}}(q-p)C^{M}_{EM}\Big{)}\bigg{]}\,.$
(383)

Here we have defined several coefficients, which reads

$\displaystyle C^{M}_{MM}=$
$\displaystyle\frac{2\sin^{2}\theta}{p_{s}^{2}+q_{s}^{2}-2p_{s}q_{s}\cos\theta}\bigg{\\{}11p_{s}^{2}q_{s}^{2}+4p_{s}^{4}+4q_{s}^{4}$
$\displaystyle+p_{s}q_{s}\left[p_{s}q_{s}\cos
2\theta-8\left(p_{s}^{2}+q_{s}^{2}\right)\cos\theta\right]\bigg{\\}}\,,$ (384)

where the two 4-momenta are $p=(p_{0},\bm{p})$ and $q=(q_{0},\bm{q})$, with
$p_{s}=|\bm{p}|$, $q_{s}=|\bm{q}|$ and
$\cos\theta=\bm{p}\cdot\bm{q}/(p_{s}q_{s})$. In the same way, we have

$\displaystyle C^{M}_{ME}=$ $\displaystyle\frac{2(\cos
2\theta+3)}{p_{s}^{2}+q_{s}^{2}-2p_{s}q_{s}\cos\theta}$
$\displaystyle\times\frac{\Big{[}q_{0}p_{s}^{2}+p_{0}q_{s}^{2}-p_{s}q_{s}\left(p_{0}+q_{0}\right)\cos\theta\Big{]}^{2}}{\left(p_{0}-q_{0}\right)^{2}+p_{s}^{2}+q_{s}^{2}-2p_{s}q_{s}\cos\theta}\,.$
(385)

The coefficient $C^{M}_{EM}$ could be deduced from $C^{M}_{ME}$ through the
replacement as follows

$\displaystyle C^{M}_{EM}=C^{M}_{ME}\Big{|}_{q\rightarrow-q+p}$
$\displaystyle=$ $\displaystyle\frac{2\left(p_{s}q_{0}\cos\theta-
p_{0}q_{s}\right)^{2}}{\left(q_{0}^{2}+q_{s}^{2}\right)\left(p_{s}^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}\right)}$
$\displaystyle\times\bigg{[}4p_{s}^{2}-8p_{s}q_{s}\cos\theta+(3+\cos
2\theta)q_{s}^{2}\bigg{]}\,.$ (386)

Finally, the last one is given by

$\displaystyle C^{M}_{EE}=$
$\displaystyle\frac{4\sin^{2}\theta}{\left(q_{0}^{2}+q_{s}^{2}\right)\left(p_{s}^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}\right)}$
$\displaystyle\times\frac{1}{p_{s}^{2}+\left(p_{0}-q_{0}\right)^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}}$
$\displaystyle\times\Big{\\{}q_{s}^{4}+p_{s}^{2}q_{0}^{2}+\left(p_{0}^{2}+p_{s}^{2}-p_{0}q_{0}+q_{0}^{2}\right)q_{s}^{2}$
$\displaystyle-
p_{s}q_{s}\left[q_{0}\left(p_{0}+q_{0}\right)+2q_{s}^{2}\right]\cos\theta\Big{\\}}^{2}\,.$
(387)

Projection of Equation 382 onto the electric component yields

$\displaystyle\big{(}\Sigma^{AA}_{a}\big{)}_{\mu\nu}^{ab}(p)\left(\delta^{ab}\Pi_{\mu\nu}^{\mathrm{E}}(p)\right)$
$\displaystyle=$ $\displaystyle
N_{c}(N_{c}^{2}-1)\int\frac{d^{4}q}{(2\pi)^{4}}\bigg{[}{\lambda^{\mathrm{EEE}}_{A^{3}}}^{2}G_{A}^{\mathrm{E}}(q)G_{A}^{\mathrm{E}}(q-p)C^{E}_{EE}$
$\displaystyle+{\lambda^{\mathrm{EMM}}_{A^{3}}}^{2}G_{A}^{\mathrm{M}}(q)G_{A}^{\mathrm{M}}(q-p)C^{E}_{MM}+{\lambda^{\mathrm{EEM}}_{A^{3}}}^{2}$
$\displaystyle\times\Big{(}G_{A}^{\mathrm{M}}(q)G_{A}^{\mathrm{E}}(q-p)C^{E}_{ME}+G_{A}^{\mathrm{E}}(q)G_{A}^{\mathrm{M}}(q-p)C^{E}_{EM}\Big{)}\bigg{]}\,.$
(388)

There are four coefficients as well, and their explicit expressions are given
as follows,

$\displaystyle C^{E}_{EE}=$
$\displaystyle\frac{1}{\left(p_{0}^{2}+p_{s}^{2}\right)\left(q_{0}^{2}+q_{s}^{2}\right)}\frac{1}{p_{s}^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}}$
$\displaystyle\times\frac{1}{p_{s}^{2}+\left(p_{0}-q_{0}\right)^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}}$
$\displaystyle\times\Bigg{\\{}p_{s}q_{s}\bigg{[}\Big{(}2p_{s}^{2}q_{0}+p_{0}q_{0}\left(p_{0}+q_{0}\right)+2p_{0}q_{s}^{2}\Big{)}\cos
2\theta$
$\displaystyle+\left(p_{0}+q_{0}\right)\Big{(}2p_{0}^{2}-3p_{0}q_{0}+2\left(p_{s}^{2}+q_{0}^{2}+q_{s}^{2}\right)\Big{)}\bigg{]}$
$\displaystyle-2\bigg{[}p_{s}^{2}q_{0}\Big{(}p_{0}^{2}+p_{s}^{2}-p_{0}q_{0}+q_{0}^{2}\Big{)}+\Big{(}p_{0}^{3}-p_{0}^{2}q_{0}$
$\displaystyle+2p_{s}^{2}q_{0}+p_{0}\left(2p_{s}^{2}+q_{0}^{2}\right)\Big{)}q_{s}^{2}+p_{0}q_{s}^{4}\bigg{]}\cos\theta\Bigg{\\}}^{2}\,,$
(389)

coefficient $C^{E}_{ME}$

$\displaystyle C^{E}_{ME}=$
$\displaystyle\frac{4\sin^{2}\theta}{\left(p_{0}^{2}+p_{s}^{2}\right)\left(p_{s}^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}\right)}$
$\displaystyle\times\frac{1}{p_{s}^{2}+\left(p_{0}-q_{0}\right)^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}}$
$\displaystyle\times\bigg{[}p_{s}^{2}\left(p_{0}^{2}+p_{s}^{2}-p_{0}q_{0}+q_{0}^{2}\right)$
$\displaystyle-
p_{s}q_{s}\Big{(}2p_{s}^{2}+p_{0}\left(p_{0}+q_{0}\right)\Big{)}\cos\theta+\left(p_{0}^{2}+p_{s}^{2}\right)q_{s}^{2}\bigg{]}^{2}\,,$
(390)

coefficient $C^{E}_{EM}$

$\displaystyle C^{E}_{EM}=C^{E}_{ME}\Big{|}_{q\rightarrow-q+p}$
$\displaystyle=$
$\displaystyle\frac{4\sin^{2}\theta}{\left(p_{0}^{2}+p_{s}^{2}\right)\left(q_{0}^{2}+q_{s}^{2}\right)}\frac{1}{p_{s}^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}}$
$\displaystyle\times\bigg{[}p_{s}^{2}\left(q_{0}^{2}+q_{s}^{2}\right)+p_{0}^{2}q_{s}^{2}-p_{0}p_{s}q_{0}q_{s}\cos\theta\bigg{]}^{2}\,,$
(391)

and coefficient $C^{E}_{MM}$

$\displaystyle C^{E}_{MM}=$
$\displaystyle\frac{2\left(p_{s}q_{0}-p_{0}q_{s}\cos\theta\right)^{2}}{\left(p_{0}^{2}+p_{s}^{2}\right)\left(p_{s}^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}\right)}$
$\displaystyle\times\left(3p_{s}^{2}+p_{s}^{2}\cos
2\theta-8p_{s}q_{s}\cos\theta+4q_{s}^{2}\right)\,.$ (392)

The ghost loop of the gluon self-energy, i.e., the second diagram on the
r.h.s. of flow equation in Figure 3, reads

$\displaystyle\equiv\big{(}\Sigma^{AA}_{b}\big{)}_{\mu\nu}^{ab}(p)\,.$ (393)

with

$\displaystyle\big{(}\Sigma^{AA}_{b}\big{)}_{\mu\nu}^{ab}(p)$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}(-)\big{(}S^{(3)}_{\bar{c}cA}\big{)}^{a_{1}a_{2}a}_{\mu^{\prime}}(-q+p,q,-p)\big{(}\lambda_{\bar{c}cA}\big{)}_{\mu^{\prime}\mu}(-p)$
$\displaystyle\times
G_{c}(q)\delta^{a_{2}b_{2}}(-)\big{(}S^{(3)}_{\bar{c}cA}\big{)}^{b_{2}b_{1}b}_{\nu^{\prime}}(-q,q-p,p)\big{(}\lambda_{\bar{c}cA}\big{)}_{\nu^{\prime}\nu}(p)$
$\displaystyle\times G_{c}(q-p)\delta^{b_{1}a_{1}}\,.$ (394)

Projecting Equation 394 onto the magnetic component leads us to

$\displaystyle\big{(}\Sigma^{AA}_{b}\big{)}_{\mu\nu}^{ab}(p)\left(\delta^{ab}\Pi_{\mu\nu}^{\mathrm{M}}(p)\right)$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}N_{c}(N_{c}^{2}-1)(\lambda^{\mathrm{M}}_{\bar{c}cA})^{2}G_{c}(q)G_{c}(q-p)C^{g}_{M}\,,$
(395)

with

$\displaystyle C^{g}_{M}$ $\displaystyle=q_{s}^{2}\sin^{2}\theta\,.$ (396)

And the electric component is given by

$\displaystyle\big{(}\Sigma^{AA}_{b}\big{)}_{\mu\nu}^{ab}(p)\left(\delta^{ab}\Pi_{\mu\nu}^{\mathrm{E}}(p)\right)$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}N_{c}(N_{c}^{2}-1)(\lambda^{\mathrm{E}}_{\bar{c}cA})^{2}G_{c}(q)G_{c}(q-p)C^{g}_{E}\,,$
(397)

with

$\displaystyle C^{g}_{E}$
$\displaystyle=\frac{\big{(}p_{s}q_{0}-p_{0}q_{s}\cos\theta\big{)}^{2}}{p_{0}^{2}+p_{s}^{2}}\,.$
(398)

The tadpole diagram of the gluon self-energy due to the four-gluon vertex,
i.e., the third diagram on the r.h.s. of flow equation in Figure 3, reads

$\displaystyle\equiv\big{(}\Sigma^{AA}_{c}\big{)}_{\mu\nu}^{ab}(p)\,.$ (399)

with

$\displaystyle\big{(}\Sigma^{AA}_{c}\big{)}_{\mu\nu}^{ab}(p)$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}(-)\big{(}S^{(4)}_{A^{4}}\big{)}^{aa_{1}b_{1}b}_{\mu^{\prime}\mu_{1}^{\prime}\nu_{1}^{\prime}\nu^{\prime}}\big{(}\lambda_{A^{4}}\big{)}^{\mu^{\prime}\mu_{1}^{\prime}\nu_{1}^{\prime}\nu^{\prime}}_{\mu\mu_{1}\nu_{1}\nu}(-p,-q,q,p)$
$\displaystyle\times\Big{(}G_{A}^{\mathrm{M}}(q)\Pi_{\mu_{1}\nu_{1}}^{\mathrm{M}}(q)+G_{A}^{\mathrm{E}}(q)\Pi_{\mu_{1}\nu_{1}}^{\mathrm{E}}(q)\Big{)}\delta^{a_{1}b_{1}}\,.$
(400)

Projecting Equation 400 onto the magnetic component leaves us with

$\displaystyle\big{(}\Sigma^{AA}_{c}\big{)}_{\mu\nu}^{ab}(p)\left(\delta^{ab}\Pi_{\mu\nu}^{\mathrm{M}}(p)\right)$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}N_{c}(N_{c}^{2}-1)\Big{[}\lambda^{\mathrm{MMMM}}_{A^{4}}G_{A}^{\mathrm{M}}(q)C^{t}_{MM}$
$\displaystyle+\lambda^{\mathrm{EEMM}}_{A^{4}}G_{A}^{\mathrm{E}}(q)C^{t}_{ME}\Big{]}\,,$
(401)

with

$\displaystyle C^{t}_{MM}$ $\displaystyle=\cos(2\theta)-5\,,$ (402)
$\displaystyle C^{t}_{ME}$
$\displaystyle=-\frac{2\big{(}q_{0}^{2}+q_{0}^{2}\cos^{2}\theta+2q_{s}^{2}\big{)}}{q_{0}^{2}+q_{s}^{2}}\,.$
(403)

The electric component reads

$\displaystyle\big{(}\Sigma^{AA}_{c}\big{)}_{\mu\nu}^{ab}(p)\left(\delta^{ab}\Pi_{\mu\nu}^{\mathrm{E}}(p)\right)$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}N_{c}(N_{c}^{2}-1)\Big{[}\lambda^{\mathrm{EEMM}}_{A^{4}}G_{A}^{\mathrm{M}}(q)C^{t}_{EM}$
$\displaystyle+\lambda^{\mathrm{EEEE}}_{A^{4}}G_{A}^{\mathrm{E}}(q)C^{t}_{EE}\Big{]}\,,$
(404)

with

$\displaystyle C^{t}_{EM}$
$\displaystyle=-\frac{2\big{(}p_{0}^{2}+p_{0}^{2}\cos^{2}\theta+2p_{s}^{2}\big{)}}{p_{0}^{2}+p_{s}^{2}}\,,$
(405) $\displaystyle C^{t}_{EE}$
$\displaystyle=-\frac{2\left[p_{s}^{2}q_{0}^{2}-2p_{0}p_{s}q_{0}q_{s}\cos\theta+p_{0}^{2}\left(q_{0}^{2}\sin^{2}\theta+q_{s}^{2}\right)\right]}{\left(p_{0}^{2}+p_{s}^{2}\right)\left(q_{0}^{2}+q_{s}^{2}\right)}\,,$
(406)

Substituting Eqs. (383), (388), (395), (397), (401), (404) into the flow
equation in Figure 3, one obtains the flow equation of the magnetic gluon
dressing function

$\displaystyle\partial_{t}Z_{A,k}^{\mathrm{M}}(p)$ $\displaystyle=$
$\displaystyle-\frac{N_{c}}{2p^{2}}\int\frac{d^{4}q}{(2\pi)^{4}}\Bigg{\\{}\bigg{[}{\lambda^{\mathrm{MMM}}_{A^{3}}}^{2}(\tilde{\partial}_{t}G_{A}^{\mathrm{M}}(q))G_{A}^{\mathrm{M}}(q-p)C^{M}_{MM}$
$\displaystyle+{\lambda^{\mathrm{EEM}}_{A^{3}}}^{2}(\tilde{\partial}_{t}G_{A}^{\mathrm{E}}(q))G_{A}^{\mathrm{E}}(q-p)C^{M}_{EE}+{\lambda^{\mathrm{EMM}}_{A^{3}}}^{2}\Big{(}(\tilde{\partial}_{t}G_{A}^{\mathrm{M}}(q))$
$\displaystyle\times
G_{A}^{\mathrm{E}}(q-p)C^{M}_{ME}+(\tilde{\partial}_{t}G_{A}^{\mathrm{E}}(q))G_{A}^{\mathrm{M}}(q-p)C^{M}_{EM}\Big{)}\bigg{]}$
$\displaystyle-2(\lambda^{\mathrm{M}}_{\bar{c}cA})^{2}(\tilde{\partial}_{t}G_{c}(q))G_{c}(q-p)C^{g}_{M}+\frac{1}{2}\Big{[}\lambda^{\mathrm{MMMM}}_{A^{4}}$
$\displaystyle\times(\tilde{\partial}_{t}G_{A}^{\mathrm{M}}(q))C^{t}_{MM}+\lambda^{\mathrm{EEMM}}_{A^{4}}(\tilde{\partial}_{t}G_{A}^{\mathrm{E}}(q))C^{t}_{ME}\Big{]}\Bigg{\\}}\,,$
(407)

and the flow equation of the electric gluon dressing function

$\displaystyle\partial_{t}Z_{A,k}^{\mathrm{E}}(p)$ $\displaystyle=$
$\displaystyle-\frac{N_{c}}{p^{2}}\int\frac{d^{4}q}{(2\pi)^{4}}\Bigg{\\{}\bigg{[}{\lambda^{\mathrm{EEE}}_{A^{3}}}^{2}(\tilde{\partial}_{t}G_{A}^{\mathrm{E}}(q))G_{A}^{\mathrm{E}}(q-p)C^{E}_{EE}$
$\displaystyle+{\lambda^{\mathrm{EMM}}_{A^{3}}}^{2}(\tilde{\partial}_{t}G_{A}^{\mathrm{M}}(q))G_{A}^{\mathrm{M}}(q-p)C^{E}_{MM}+{\lambda^{\mathrm{EEM}}_{A^{3}}}^{2}\Big{(}(\tilde{\partial}_{t}G_{A}^{\mathrm{M}}(q))$
$\displaystyle\times
G_{A}^{\mathrm{E}}(q-p)C^{E}_{ME}+(\tilde{\partial}_{t}G_{A}^{\mathrm{E}}(q))G_{A}^{\mathrm{M}}(q-p)C^{E}_{EM}\Big{)}\bigg{]}$
$\displaystyle-2(\lambda^{\mathrm{E}}_{\bar{c}cA})^{2}(\tilde{\partial}_{t}G_{c}(q))G_{c}(q-p)C^{g}_{E}+\frac{1}{2}\Big{[}\lambda^{\mathrm{EEMM}}_{A^{4}}$
$\displaystyle\times(\tilde{\partial}_{t}G_{A}^{\mathrm{M}}(q))C^{t}_{EM}+\lambda^{\mathrm{EEEE}}_{A^{4}}(\tilde{\partial}_{t}G_{A}^{\mathrm{E}}(q))C^{t}_{EE}\Big{]}\Bigg{\\}}\,,$
(408)

###  A.3 Ghost self-energy

Figure 59: Diagrammatic representation of the flow equation for the ghost
self-energy in Yang-Mills theory.

The flow equation of the ghost self-energy in Yang-Mills theory is depicted in
Figure 59. The one-loop diagram on the r.h.s. reads

$\displaystyle\equiv\big{(}\Sigma^{\bar{c}c}\big{)}^{ab}(p)\,.$ (409)

with

$\displaystyle\big{(}\Sigma^{\bar{c}c}\big{)}^{ab}(p)$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}(-)\big{(}S^{(3)}_{\bar{c}cA}\big{)}^{aa_{1}a_{2}}_{\mu_{2}^{\prime}}(-p,q,-q+p)$
$\displaystyle\times\big{(}\lambda_{\bar{c}cA}\big{)}_{\mu_{2}^{\prime}\mu_{2}}(-q+p)G_{c}(q)\delta^{a_{1}b_{1}}$
$\displaystyle\times(-)\big{(}S^{(3)}_{\bar{c}cA}\big{)}^{b_{1}bb_{2}}_{\nu_{2}^{\prime}}(-q,p,q-p)$
$\displaystyle\times\big{(}\lambda_{\bar{c}cA}\big{)}_{\nu_{2}^{\prime}\nu_{2}}(q-p)\Big{[}G_{A}^{\mathrm{M}}(q-p)\Pi_{\mu_{2}\nu_{2}}^{\mathrm{M}}(q-p)$
$\displaystyle+G_{A}^{\mathrm{E}}(q-p)\Pi_{\mu_{2}\nu_{2}}^{\mathrm{E}}(q-p)\Big{]}\delta^{a_{2}b_{2}}\,.$
(410)

Tracing the color indices, one arrives at

$\displaystyle\big{(}\Sigma^{\bar{c}c}\big{)}^{ab}(p)\delta^{ab}$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}N_{c}(N_{c}^{2}-1)\Big{[}(\lambda^{\mathrm{M}}_{\bar{c}cA})^{2}G_{A}^{\mathrm{M}}(q-p)G_{c}(q)C^{\bar{c}c}_{M}$
$\displaystyle+(\lambda^{\mathrm{E}}_{\bar{c}cA})^{2}G_{A}^{\mathrm{E}}(q-p)G_{c}(q)C^{\bar{c}c}_{E}\Big{]}\,,$
(411)

with

$\displaystyle C^{\bar{c}c}_{M}=$
$\displaystyle\frac{p_{s}^{2}q_{s}^{2}\sin^{2}\theta}{p_{s}^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}}\,,$
(412) $\displaystyle C^{\bar{c}c}_{E}=$
$\displaystyle\frac{1}{\Big{(}p_{s}^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}\Big{)}}$
$\displaystyle\times\frac{\Big{[}p_{s}^{2}q_{0}-p_{s}q_{s}\left(p_{0}+q_{0}\right)\cos\theta+p_{0}q_{s}^{2}\Big{]}^{2}}{p_{s}^{2}+\left(p_{0}-q_{0}\right)^{2}-2p_{s}q_{s}\cos\theta+q_{s}^{2}}\,.$
(413)

In the following we also need the expression in Equation 411 with the
replacement of the internal momentum $q\rightarrow-q+p$, which reads

$\displaystyle\big{(}\Sigma^{\bar{c}c}\big{)}^{ab}(p)\delta^{ab}$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}N_{c}(N_{c}^{2}-1)\Big{[}(\lambda^{\mathrm{M}}_{\bar{c}cA})^{2}G_{A}^{\mathrm{M}}(q)G_{c}(q-p)C^{\prime\bar{c}c}_{M}$
$\displaystyle+(\lambda^{\mathrm{E}}_{\bar{c}cA})^{2}G_{A}^{\mathrm{E}}(q)G_{c}(q-p)C^{\prime\bar{c}c}_{E}\Big{]}\,,$
(414)

with

$\displaystyle C^{\prime\bar{c}c}_{M}$
$\displaystyle=C^{\bar{c}c}_{M}\Big{|}_{q\rightarrow-q+p}\,,\quad
C^{\prime\bar{c}c}_{E}$
$\displaystyle=C^{\bar{c}c}_{E}\Big{|}_{q\rightarrow-q+p}\,.$ (415)

Finally, inserting Eqs. (411) and (414) into the flow equation in Figure 59,
one is led to the flow equation of the ghost dressing function, as follows

$\displaystyle\partial_{t}Z_{c,k}(p)$ $\displaystyle=$
$\displaystyle-\frac{N_{c}}{p^{2}}\int\frac{d^{4}q}{(2\pi)^{4}}\Bigg{\\{}\Big{[}(\lambda^{\mathrm{M}}_{\bar{c}cA})^{2}G_{A}^{\mathrm{M}}(q-p)(\tilde{\partial}_{t}G_{c}(q))C^{\bar{c}c}_{M}$
$\displaystyle+(\lambda^{\mathrm{E}}_{\bar{c}cA})^{2}G_{A}^{\mathrm{E}}(q-p)(\tilde{\partial}_{t}G_{c}(q))C^{\bar{c}c}_{E}\Big{]}$
$\displaystyle+\Big{[}(\lambda^{\mathrm{M}}_{\bar{c}cA})^{2}(\tilde{\partial}_{t}G_{A}^{\mathrm{M}}(q))G_{c}(q-p)C^{\prime\bar{c}c}_{M}$
$\displaystyle+(\lambda^{\mathrm{E}}_{\bar{c}cA})^{2}(\tilde{\partial}_{t}G_{A}^{\mathrm{E}}(q))G_{c}(q-p)C^{\prime\bar{c}c}_{E}\Big{]}\Bigg{\\}}\,.$
(416)

##  Appendix B Fierz-complete basis of four-quark interactions of $N_{f}=2$
flavors

Here we list the Fierz-complete basis of four-quark interactions of $N_{f}=2$
flavors Equation 88, see also e.g., [58, 61, 92]. The ten different channels
can be classified into four different subsets according to their invariance or
not, under the global transformations of the groups $SU_{\mathrm{V}}(N_{f})$,
$U_{\mathrm{V}}(1)$, $SU_{\mathrm{A}}(N_{f})$, and $U_{\mathrm{A}}(1)$. Of the
ten channels, several are invariant under all the transformations mentioned
above, which read

$\displaystyle\mathcal{O}^{(V-A)}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\gamma_{\mu}T^{0}q)^{2}-(\bar{q}i\gamma_{\mu}\gamma_{5}T^{0}q)^{2}\,,$
(417)
$\displaystyle\mathcal{O}^{(V+A)}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\gamma_{\mu}T^{0}q)^{2}+(\bar{q}i\gamma_{\mu}\gamma_{5}T^{0}q)^{2}\,,$
(418)
$\displaystyle\mathcal{O}^{(S-P)_{+}}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{0}q)^{2}-(\bar{q}\,\gamma_{5}T^{0}q)^{2}$
$\displaystyle+(\bar{q}\,T^{a}q)^{2}-(\bar{q}\,\gamma_{5}T^{a}q)^{2}\,,$ (419)
$\displaystyle\mathcal{O}^{(V-A)^{\mathrm{adj}}}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\gamma_{\mu}T^{0}t^{a}q)^{2}-(\bar{q}i\gamma_{\mu}\gamma_{5}T^{0}t^{a}q)^{2}\,,$
(420)

where generators of the flavor $SU(N_{f})$ group and the color $SU(N_{c})$
group are denoted by $T^{a}$’s and $t^{a}$’s, respectively, and summation for
the indices is assumed. Furthermore, one has
$T^{0}=1/\sqrt{2N_{f}}\mathbb{1}_{N_{f}\times N_{f}}$. Another two channels,
given by

$\displaystyle\mathcal{O}^{(S+P)_{-}}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{0}q)^{2}+(\bar{q}\,\gamma_{5}T^{0}q)^{2}$
$\displaystyle-(\bar{q}\,T^{a}q)^{2}-(\bar{q}\,\gamma_{5}T^{a}q)^{2}\,,$ (421)
$\displaystyle\mathcal{O}^{(S+P)_{-}^{\mathrm{adj}}}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{0}t^{a}q)^{2}+(\bar{q}\,\gamma_{5}T^{0}t^{a}q)^{2}$
$\displaystyle-(\bar{q}\,T^{a}t^{b}q)^{2}-(\bar{q}\,\gamma_{5}T^{a}t^{b}q)^{2}\,,$
(422)

break the symmetry of $U_{\mathrm{A}}(1)$ while preserve
$SU_{\mathrm{V}}(N_{f})\otimes U_{\mathrm{V}}(1)\otimes
SU_{\mathrm{A}}(N_{f})$. The two channels that read

$\displaystyle\mathcal{O}^{(S-P)_{-}}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{0}q)^{2}-(\bar{q}\,\gamma_{5}T^{0}q)^{2}$
$\displaystyle-(\bar{q}\,T^{a}q)^{2}+(\bar{q}\,\gamma_{5}T^{a}q)^{2}\,,$ (423)
$\displaystyle\mathcal{O}^{(S-P)_{-}^{\mathrm{adj}}}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{0}t^{a}q)^{2}-(\bar{q}\,\gamma_{5}T^{0}t^{a}q)^{2}$
$\displaystyle-(\bar{q}\,T^{a}t^{b}q)^{2}+(\bar{q}\,\gamma_{5}T^{a}t^{b}q)^{2}\,,$
(424)

break $SU_{\mathrm{A}}(N_{f})$ while preserve $SU_{\mathrm{V}}(N_{f})\otimes
U_{\mathrm{V}}(1)\otimes U_{\mathrm{A}}(1)$. The last two independent
channels, viz.,

$\displaystyle\mathcal{O}^{(S+P)_{+}}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{0}q)^{2}+(\bar{q}\,\gamma_{5}T^{0}q)^{2}$
$\displaystyle+(\bar{q}\,T^{a}q)^{2}+(\bar{q}\,\gamma_{5}T^{a}q)^{2}\,,$ (425)
$\displaystyle\mathcal{O}^{(S+P)_{+}^{\mathrm{adj}}}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{0}t^{a}q)^{2}+(\bar{q}\,\gamma_{5}T^{0}t^{a}q)^{2}$
$\displaystyle+(\bar{q}\,T^{a}t^{b}q)^{2}+(\bar{q}\,\gamma_{5}T^{a}t^{b}q)^{2}\,.$
(426)

break both $U_{\mathrm{A}}(1)$ and $SU_{\mathrm{A}}(N_{f})$, while preserve
$SU_{\mathrm{V}}(N_{f})\otimes U_{\mathrm{V}}(1)$.

Moreover, it is also useful to combine linearly several different channels of
the four-quark couplings to form new independent elements of basis. For
instance, the four scalar-pseudoscalar channels as follows

$\displaystyle\mathcal{O}^{\sigma}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{0}q)^{2}\,,$ (427)
$\displaystyle\mathcal{O}^{\pi}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle-(\bar{q}\,\gamma_{5}T^{a}q)^{2}\,,$ (428)
$\displaystyle\mathcal{O}^{a}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle(\bar{q}\,T^{a}q)^{2}\,,$ (429)
$\displaystyle\mathcal{O}^{\eta}_{ijlm}{\bar{q}}_{i}q_{j}{\bar{q}}_{l}q_{m}=$
$\displaystyle-(\bar{q}\,\gamma_{5}T^{0}q)^{2}\,,$ (430)

are obtained from linear combinations of $\mathcal{O}^{(S-P)_{+}}$ in Equation
419, $\mathcal{O}^{(S+P)_{-}}$ in Equation 421, $\mathcal{O}^{(S-P)_{-}}$ in
Equation 423, and $\mathcal{O}^{(S+P)_{+}}$ in Equation 425.

##  Appendix C Some flow functions

In this appendix we present explicit expressions for some flow equations. The
threshold functions involved in this appendix can be found in, e.g., [18,
148].

The anomalous dimension of quarks $\eta_{q,k}$ in Equation 194 reads

$\displaystyle\eta_{q,k}=$
$\displaystyle\frac{1}{24\pi^{2}N_{f}}(4-\eta_{\phi,k})\bar{h}_{k}^{2}$
$\displaystyle\times\bigg{\\{}(N_{f}^{2}-1)\mathcal{FB}_{(1,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\pi,k}^{2};T,\mu_{q},p_{0,\text{\tiny{ex}}})$
$\displaystyle+\mathcal{FB}_{(1,2)}(\tilde{m}_{q,k}^{2},\tilde{m}_{\sigma,k}^{2};T,\mu_{q},p_{0,\text{\tiny{ex}}})\bigg{\\}}$
$\displaystyle+\frac{1}{24\pi^{2}}\frac{N_{c}^{2}-1}{2N_{c}}g_{\bar{q}Aq,k}^{2}$
$\displaystyle\times\bigg{\\{}2(4-\eta_{A,k})\mathcal{FB}_{(1,2)}(\tilde{m}_{q,k}^{2},0;T,\mu_{q},p_{0,\text{\tiny{ex}}})$
$\displaystyle+3(3-\eta_{q,k})\Big{(}\mathcal{FB}_{(1,1)}(\tilde{m}_{q,k}^{2},0;T,\mu_{q},p_{0,\text{\tiny{ex}}})$
$\displaystyle-2\mathcal{FB}_{(2,1)}(\tilde{m}_{q,k}^{2},0;T,\mu_{q},p_{0,\text{\tiny{ex}}})\Big{)}\bigg{\\}},$
(431)

with $N_{f}=2$ and $N_{c}=3$. The anomalous dimension of mesons at $p=0$ in
Equation 200 reads

$\displaystyle\eta_{\phi,k}(0)=$
$\displaystyle\,\frac{\bar{Z}_{\phi,k}}{Z_{\phi,k}(0)}$
$\displaystyle\,\times\frac{1}{6\pi^{2}}\bigg{\\{}\frac{4}{k^{2}}\bar{\kappa}_{k}(\bar{V}^{{}^{\prime\prime}}_{k}(\bar{\kappa}_{k}))^{2}\mathcal{BB}_{(2,2)}(\tilde{m}_{\pi,k}^{2},\tilde{m}_{\sigma,k}^{2};T)$
$\displaystyle\,+N_{c}\bar{h}_{k}^{2}\Big{[}(2\eta_{q,k}-3)\mathcal{F}_{(2)}(\tilde{m}_{q,k}^{2};T,\mu_{q})$
$\displaystyle\,-4(\eta_{q,k}-2)\mathcal{F}_{(3)}(\tilde{m}_{q,k}^{2};T,\mu_{q})\Big{]}\bigg{\\}}\,,$
(432)

and that at $p_{0}=0$ and $\bm{p}^{2}=k^{2}$ in Equation 198 reads

$\displaystyle\eta_{\phi,k}(0,k)$ $\displaystyle=$