# Critical behaviors of the $O(4)$ and $Z(2)$ symmetries in the QCD phase
diagram

Yong-rui Chen  School of Physics, Dalian University of Technology, Dalian,
116024, P.R. China  Rui Wen  School of Physics, Dalian University of
Technology, Dalian, 116024, P.R. China  Wei-jie Fu<EMAIL_ADDRESS>School of
Physics, Dalian University of Technology, Dalian, 116024, P.R. China

###### Abstract

In this work we have studied the QCD phase structure and critical dynamics
related to the 3-$d$ $O(4)$ and $Z(2)$ symmetry universality classes in the
two-flavor quark-meson low energy effective theory within the functional
renormalization group approach. We have employed the expansion of Chebyshev
polynomials to solve the flow equation for the order-parameter potential. The
chiral phase transition line of $O(4)$ symmetry in the chiral limit, and the
$Z(2)$ line of critical end points related to the explicit chiral symmetry
breaking are depicted in the phase diagram. Various critical exponents related
to the order parameter, chiral susceptibilities and correlation lengths have
been calculated for the 3-$d$ $O(4)$ and $Z(2)$ universality classes in the
phase diagram, respectively. We find that the critical exponents obtained in
the computation, where a field-dependent mesonic nontrivial dispersion
relation is taken into account, are in quantitative agreement with results
from other approaches, e.g., the conformal bootstrap, Monte Carlo simulations
and $d=3$ perturbation expansion, etc. Moreover, the size of the critical
regime in the QCD phase diagram is found to be very small.

††preprint:

##  I Introduction

Significant progress has been made in studies of QCD phase structure over the
last decade, both from the experimental and theoretical sides; see, e.g.
Stephanov (2006); Friman  _et al._ (2011); Luo and Xu (2017); Andronic  _et
al._ (2018); Fischer (2019); Bzdak  _et al._ (2020); Fu  _et al._ (2020);
Bazavov  _et al._ (2020); Borsanyi  _et al._ (2020); Fu  _et al._ (2021). One
of the most prominent features of the QCD phase structure is the probable
presence of a second order critical end point (CEP) in the phase diagram
spanned by the temperature $T$ and baryon chemical potential $\mu_{B}$ or
densities, which separates the first order phase transition at high $\mu_{B}$
from the continuous crossover at low $\mu_{B}$ Stephanov (2006). The existence
and location of CEP are, however, still open questions, whose answers would
definitely help us to unravel the most mysterious veil related to the
properties of strongly interacting matter under extreme conditions. The Beam
Energy Scan (BES) Program at the Relativistic Heavy Ion Collider (RHIC) is
aimed at searching for and locating the critical end point, where fluctuation
observables sensitive to the critical dynamics, e.g., high-order cumulants of
net-proton, net-charge, net-kaon multiplicity distributions, have been
measured Adamczyk  _et al._ (2014a, b); Luo (2015); Adamczyk  _et al._ (2018).
Notably, a non-monotonic dependence of the kurtosis of the net-proton
multiplicity distribution on the beam energy with $3.1\sigma$ significance in
central collisions has been reported by the STAR collaboration recently Adam
_et al._ (2020).

On the other hand, lattice QCD simulations have provided us with a plethora of
knowledge about the QCD phase structure, e.g., the crossover nature of the
chiral phase transition at finite $T$ and vanishing $\mu_{B}$ with physical
current quark mass Aoki  _et al._ (2006), pseudo-critical temperature Borsanyi
_et al._ (2014); Bazavov  _et al._ (2014), curvature of the phase boundary
Bazavov  _et al._ (2019); Borsanyi  _et al._ (2020), etc. Because of the
notorious sign problem at finite chemical potential, the reliability regime of
lattice calculations is restricted to be $\mu_{B}/T\lesssim 2\sim 3$, where no
CEP has been found. Free from the sign problem, the first-principle functional
approaches, e.g, the functional renormalization group (fRG) and Dyson-
Schwinger equations (DSE), could potentially extend the regime of reliability
to $\mu_{B}/T\sim 4$ Fischer (2019); Fu  _et al._ (2020). With benchmark tests
of observables at finite $T$ and low $\mu_{B}$ in comparison to lattice
calculations, e.g., the quark condensate, curvature of the phase boundary,
etc., functional approaches, both fRG and DSE, have predicted a CEP located in
a region of $450\,\mathrm{MeV}\lesssim\mu_{B}\lesssim 650\,\mathrm{MeV}$
Fischer (2019); Fu  _et al._ (2020); Isserstedt  _et al._ (2019); Gao and
Pawlowski (2020a, b) recently.

An alternative method used to circumvent the possible location of CEP, is to
determine the critical temperature $T_{c}$ of the chiral phase transition in
the chiral limit, more specifically, i.e., massless light up and down quarks
and a physical strange quark mass. Since it is believed that the value of
$T_{c}$ sets an upper bound for the temperature of CEP Halasz  _et al._
(1998); Buballa and Carignano (2019). Very recently, the critical temperature
$T_{c}$ in the chiral limit has been investigated and its value is
extrapolated from both lattice simulations Ding  _et al._ (2019) and
functional approach Braun  _et al._ (2020). Moreover, further lattice
calculations indicate that axial anomaly remains manifested at $T\approx
1.6\,T_{c}$, which implies that the chiral phase transition of QCD in the
chiral limit is of 3-$d$ $O(4)$ universality class Ding  _et al._ (2020); see,
e.g., Pisarski and Wilczek (1984) for more discussions about the relation
between the axial anomaly and the symmetry universality classes.

In this work, we would like to study the QCD phase structure in the chiral
limit and finite current quark mass, i.e., with a finite pion mass, in the
two-flavor quark-meson low energy effective theory (LEFT) within the fRG
approach. For more discussions about the fRG approach, see, e.g., QCD related
reviews Berges  _et al._ (2002); Pawlowski (2007); Schaefer and Wambach
(2008); Gies (2012); Rosten (2012); Braun (2012); Pawlowski (2014); Dupuis
_et al._ (2020). In contrast with the lattice simulation and the first-
principle fRG-QCD calculation Ding  _et al._ (2019); Braun  _et al._ (2020),
the chiral limit could be accessed strictly in the LEFT. Furthermore, we would
also like to study the critical behaviors of the 3-$d$ $O(4)$ and $Z(2)$
universality classes, including various critical exponents, which belong to
the second-order chiral phase transitions in the chiral limit and at the
critical end point with finite quark mass, respectively. To that end, we
expand the effective potential of order parameter as a sum of Chebyshev
polynomials in the computation of fRG flow equations; see Risch (2013) for
more details. The Chebyshev expansion of solutions to a set of
integrodifferential equations is, in fact, a specific formalism of more
generic pseudo-spectral methods Boyd (2000), and see also, e.g., Borchardt and
Knorr (2015, 2016); Knorr (2020) for applications of pseudo-spectral methods
in the fRG.

In fact, another two numerical methods are more commonly used in solving the
flow equation for the effective potential: one is the Taylor expansion of the
effective potential around some value Pawlowski and Rennecke (2014); Yin  _et
al._ (2019), and the other discretization of the effective potential on a grid
Schaefer and Wambach (2005). The (dis)advantages of these two methods are
distinct. The former is liable to implementation of the numerical
calculations, but short of global properties of the effective potential, that
is, however, indispensable to studies of chiral phase transition in the chiral
limit or around CEP; the latter is encoded with global information on the
potential, but it loses numerical accuracy near the phase transition point
which is necessary especially for the computation of critical exponents. The
Chebyshev expansion used in this work combines the merits from both
approaches, i.e., the global potential and the numerical accuracy, and thus it
is very suitable for the studies of critical behaviors in the QCD phase
diagram. Remarkably, a discontinuous Galerkin scheme has been applied in the
context of fRG recently Grossi and Wink (2019), which is well-suited for
studies of the first-order phase transition.

This paper is organized as follows: In Sec. II we briefly introduce the flow
equations in the quark-meson LEFT and the method of the Chebyshev expansion
for the effective potential. The obtained phase diagram and QCD phase
structure are presented and discussed in Sec. III. In Sec. IV scaling analyses
for the the 3-$d$ $O(4)$ and $Z(2)$ universality classes are performed, and
various critical exponents are obtained. We also discuss the size of the
critical regime there. In Sec. V we give a summary and conclusion. Some
threshold functions and anomalous dimension in the flow equations, and some
relations for the Chebyshev polynomials are collected in Appendix A and
Appendix B, respectively.

##  II Functional renormalization group and the low energy effective theories

Thanks to the Wilson’s idea of the renormalization group (RG), see, e.g.,
Wilson and Kogut (1974), it has been well known that usually the active
degrees of freedom are quite different, when the energy scale of a system
evolves from a hierarchy into another. The relevant dynamics in different
hierarchies are connected with each other through the evolution of RG
equations. To be more specific, in QCD the partonic degrees of freedom, i.e.,
the quarks and gluons, in the high energy perturbative regime are transformed
into the collective hadronic ones in the nonperturbative region of low energy,
with the RG scale evolving from the ultraviolet (UV) to infrared (IR) limits
Weinberg (1979), and see also, e.g., Gies and Wetterich (2002, 2004);
Pawlowski (2007); Floerchinger and Wetterich (2009); Braun  _et al._ (2016);
Mitter  _et al._ (2015); Cyrol  _et al._ (2018a); Eser  _et al._ (2018); Fu
_et al._ (2020) for recent development of the relevant ideas within the fRG
approach. When the momentum or RG scale is below, say $\sim 1$ GeV, which is
related to a narrow transition region from the perturbative to nonperturbative
QCD, calculated results of Yang-Mills theory and QCD in Landau gauge indicate
that the gluons develop a finite mass gap and decouple from the system, and
see, e.g. Mitter  _et al._ (2015); Cyrol  _et al._ (2016); Fu  _et al._
(2020); Huber (2020) for more details. As a consequence, contributions to the
flow equations of effective action from the glue sector could be safely
neglected, if the initial evolution scale is set at a UV scale
$\Lambda\lesssim 1$ GeV.

Hence, within the fRG approach, one is left with the flow equation for the low
energy effective theory, which reads

$\displaystyle\partial_{t}\Gamma_{k}[\Phi]=$
$\displaystyle-\mathrm{Tr}\Big{(}G_{q\bar{q},k}\partial_{t}R_{q,k}\Big{)}+\frac{1}{2}\mathrm{Tr}\Big{(}G_{\phi\phi,k}\partial_{t}R_{\phi,k}\Big{)}\,,$
(1)

with the RG scale $k$ and the RG time defined as $t=\ln(k/\Lambda)$.
Apparently, Eq. (1) is an ordinary differential equation for the $k$-dependent
effective action, $\Gamma_{k}[\Phi]$, the arguments $\Phi=(q,\bar{q},\phi)$ of
which are the quark and mesonic fields in the LEFT. The equation in Eq. (1),
which describes the evolution of the effective action with the RG scale, is
also well known as the Wetterich equation Wetterich (1993), see also Ellwanger
(1994); Morris (1994). The flow receives contributions from both the quark and
mesonic degrees of freedom, as shown on the r.h.s. of Eq. (1), where
$G_{q\bar{q},k}$ and $G_{\phi\phi,k}$ are the $k$-dependent full quark and
meson propagators, respectively, and are related to the quadratic derivatives
of $\Gamma_{k}[\Phi]$ with respect to their respective fields, viz.

$\displaystyle
G_{\phi\phi/q\bar{q}}[\Phi]=\left(\frac{1}{\frac{\delta^{2}\Gamma_{k}[\Phi]}{\delta\Phi^{2}}+R_{\Phi,k}}\right)_{\phi\phi/q\bar{q}}\,.$
(2)

where $R_{q,k}$ and $R_{\phi,k}$ as well as in Eq. (1) are the IR regulators,
which are employed to suppress quantum fluctuations of momenta $q\lesssim k$,
and their explicit expressions used in the work are given in Eqs. (60) and
(61). Moreover, interested readers could refer to QCD related fRG review
articles Berges  _et al._ (2002); Pawlowski (2007); Schaefer and Wambach
(2008); Gies (2012); Rosten (2012); Braun (2012); Pawlowski (2014); Dupuis
_et al._ (2020) for more details about the formalism of fRG, and also Braun
_et al._ (2010); Braun (2009); Braun  _et al._ (2011a); Mitter  _et al._
(2015); Braun  _et al._ (2016); Cyrol  _et al._ (2016, 2018a, 2018b); Fu  _et
al._ (2020); Braun  _et al._ (2020); Fu  _et al._ (2021) for recent progress
on relevant studies.

In this work, we adopt a truncation for the effective action in Eq. (1) as
follows

$\displaystyle\Gamma_{k}[\Phi]=$
$\displaystyle\int_{x}\bigg{\\{}Z_{q,k}\bar{q}\big{(}\gamma_{\mu}\partial_{\mu}-\gamma_{0}\hat{\mu}\big{)}q+\frac{1}{2}Z_{\phi,k}(\rho)\big{(}\partial_{\mu}\phi\big{)}^{2}$
$\displaystyle+h_{y,k}\bar{q}\big{(}T^{0}\sigma+i\gamma_{5}\vec{T}\cdot\vec{\pi}\big{)}q+V_{k}(\rho)-c\sigma\bigg{\\}}\,,$
(3)

with the shorthand notation $\int_{x}=\int_{0}^{1/T}dx_{0}\int d^{3}x$, where
the quark field $q=(u\,,d)^{T}$ and the meson field
$\phi=\left(\sigma,\vec{\pi}\right)$ are in the fundamental and adjoint
representations of $SU(N_{f})$ in the flavor space with $N_{f}=2$,
respectively. They interact with each other via a Yukawa coupling with a
coupling strength $h_{Y,k}$, where the subscript Y is used to distinguish it
from the reduced external field $h$ in Eq. (15). Here $T^{i}$ ($i=1\,,2\,,3$)
are the generators of $SU(2)$ with
$\operatorname{Tr}(T^{i}T^{j})=\frac{1}{2}\delta^{ij}$ and
$T^{0}=\frac{1}{\sqrt{2N_{f}}}\mathbb{1}_{N_{f}\times N_{f}}$. Note that both
the effective potential $V_{k}(\rho)$ and the mesonic wave function
renormalization $Z_{\phi,k}(\rho)$ in Eq. (3) depend on the meson field by
means of $\rho=\phi^{2}/2$, which are $O(4)$ invariant. $Z_{q,k}$ is the quark
wave function renormalization. Notice that the term linear in the order
parameter field, i.e., $-c\sigma$ in Eq. (3), breaks the chiral symmetry
explicitly, and thus here $c$ is essentially an external “magnetic” field in
the language of magnetization. Moreover,
$\hat{\mu}=\mathrm{diag}(\mu_{u},\mu_{d})$ is the matrix of quark chemical
potentials in the flavor space, and $\mu=\mu_{u}=\mu_{d}$ is assumed
throughout this work, which is related to the baryon chemical potential via
$\mu=\mu_{B}/3$. For more discussions about the quark-meson LEFT in Eq. (3) or
its extensions, e.g., Polyakov-loop quark-meson LEFT, QCD assisted LEFT, etc.,
and their applications in calculations of QCD thermodynamics and phase
structure, fluctuations and correlations of conserved charges, etc., see,
e.g., Schaefer and Wambach (2005); Schaefer  _et al._ (2007); Skokov  _et al._
(2010); Herbst  _et al._ (2011); Skokov  _et al._ (2011); Karsch  _et al._
(2011); Morita  _et al._ (2011); Skokov  _et al._ (2012); Haas  _et al._
(2013); Herbst  _et al._ (2013, 2014); Fu and Pawlowski (2016, 2015); Fu  _et
al._ (2016); Sun  _et al._ (2018); Fu  _et al._ (2018, 2019); Wen  _et al._
(2019); Wen and Fu (2019); Yin  _et al._ (2019); Hansen  _et al._ (2020); Fu
_et al._ (2021).

###  II.1 Flow equations

Substituting the effective action in Eq. (3) into the Wetterich equation in
Eq. (1), one readily obtains the flow equation of the effective potential as
follows

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

with the threshold functions $l^{(B,4)}_{0}$ and $l^{(F,4)}_{0}$ given in Eq.
(64) and Eq. (65), respectively. Here, the scale-dependent meson and quark
masses read

$\displaystyle\bar{m}^{2}_{\pi,k}$
$\displaystyle=\frac{V^{\prime}_{k}(\rho)}{k^{2}Z_{\phi,k}}\,,\qquad\bar{m}^{2}_{\sigma,k}=\frac{V^{\prime}_{k}(\rho)+2\rho
V^{\prime\prime}_{k}(\rho)}{k^{2}Z_{\phi,k}}\,,$ (5)
$\displaystyle\bar{m}^{2}_{q,k}$
$\displaystyle=\frac{h_{y,k}^{2}\rho}{2k^{2}Z^{2}_{q,k}}\,,$ (6)

which are RG invariant and dimensionless.

The meson and quark anomalous dimensions in the threshold functions in Eq. (4)
are defined as follows

$\displaystyle\eta_{\phi,k}$
$\displaystyle=-\frac{\partial_{t}Z_{\phi,k}}{Z_{\phi,k}}\,,\quad\eta_{q,k}=-\frac{\partial_{t}Z_{q,k}}{Z_{q,k}}\,,$
(7)

where the meson anomalous dimension is obtained by projecting the flow
equation in Eq. (1) onto the inverse pion propagator, to wit,

$\displaystyle\eta_{\phi,k}(\rho)$
$\displaystyle=-\frac{1}{3Z_{\phi,k}}\delta_{ij}\frac{\partial}{\partial(|\bm{p}|^{2})}\frac{\delta^{2}\partial_{t}\Gamma_{k}}{\delta\pi_{i}(-p)\delta\pi_{j}(p)}\Bigg{|}_{\begin{subarray}{c}p_{0}=0\\\
\bm{p}=0\end{subarray}}\,,$ (8)

the explicit expression of which is presented in Eq. (68). Note that
$\eta_{\phi,k}$ is dependent on the meson field via $\rho$.

In comparison to the effects of the meson wave function renormalization on the
chiral phase transition at finite temperature and density, it has been found
that those of quark wave function renormalization and the running Yukawa
coupling are relatively milder, see, e.g., Pawlowski and Rennecke (2014); Fu
and Pawlowski (2015); Yin  _et al._ (2019). Therefore, in this work we adopt
the simplification as follows

$\displaystyle\eta_{q,k}$
$\displaystyle=0\,,\quad\quad\partial_{t}\bar{h}_{y,k}=0\,,$ (9)

with the renormalized Yukawa coupling given in Eq. (69), and use two different
truncations: one is the usual local potential approximation (LPA), where the
mesonic anomalous dimension is vanishing as well, and the $k$-dependent term
in Eq. (3) is just the effective potential; the other is the truncation with
the field-dependent mesonic anomalous dimension in Eq. (8) taken into account
besides the potential, which is denoted as LPA′ in this work. Note that the
notation LPA′ in literatures, e.g., Helmboldt  _et al._ (2015); Fu and
Pawlowski (2015), usually stands for the truncation with a field-independent
mesonic anomalous dimension which is, strictly speaking, different from the
case in this work.

Figure 1: Dependence of the mesonic wave function renormalization $Z_{\phi}$
on the order-parameter field $\bar{\sigma}$ at vanishing baryon chemical
potential $\mu_{B}=0$ and several values of temperature $T=\Delta T+T_{c}$.
See text for more details.

As an illustrative example, we show the mesonic wave function renormalization
$Z_{\phi}\equiv Z_{\phi,k=k_{\mathrm{IR}}}$ as a function of the renormalized
sigma field $\bar{\sigma}=Z_{\phi}^{1/2}\sigma$ obtained in LPA′ in Fig. 1,
where $k_{\mathrm{IR}}$ is the RG scale in the IR limit, and one would has
$k_{\mathrm{IR}}\rightarrow 0$ in principle, which, however, is impossible to
realize in numerical calculations. In our calculation the value of
$k_{\mathrm{IR}}$ is reduced as small as possible, and we find the convergence
is obtained when $k_{\mathrm{IR}}=1$ MeV. Note that the mesonic wave function
renormalization at the scale of UV cutoff $\Lambda$, see Sec. III in the
following, is assumed to be identical to unity, i.e., $Z_{\phi,k=\Lambda}=1$.
In Fig. 1, we choose several values of temperature $T=\Delta T+T_{c}$ at and
above the critical temperature that is $T_{c}=143.6$ MeV in the chiral limit
and at vanishing $\mu_{B}$. One observes that with the increase of the
temperature, the peak structure of $Z_{\phi}$ as a function of the
renormalized sigma field $\bar{\sigma}$ becomes smoother.

###  II.2 Chebyshev expansion of the effective potential

Figure 2: Phase diagrams in the plane of $T$ and $\mu_{B}$, obtained in the
quark-meson low energy effective theory within the fRG approach. Two
truncations for the fRG calculations have been employed: one is the local
potential approximation (LPA) and the other is that beyond the LPA, in which a
field-dependent mesonic wave function renormalization is taken into account,
i.e., the truncation LPA′, and see text for more details. The relevant results
are presented in the left and right panels, respectively.
The black dashed lines in both panels denote the $O(4)$ chiral phase
transition in the chiral limit, and the black circles indicate the location of
the tricritical point. The solid lines of different colors in the left panel
denote the first-order phase transitions with different pion masses in the
vacuum, i.e. different values of $c$ in Eq. (3), and the solid one in the
right panel is the first-order phase transition line in the chiral limit. The
red dashed lines in both panels stand for line composed of critical end points
(CEP) corresponding to continuously varying pion masses, which belong to the
$Z(2)$ symmetry class. The star in the left panel indicates the location of
CEP with physical pion mass. In both phase diagrams we use red and blue
crosses to label the locations where critical exponents in Sec. IV are
calculated for the $O(4)$ and $Z(2)$ universality classes, respectively.

In this work we solve the flow equation in Eq. (4) by expanding the effective
potential as a sum of Chebyshev polynomials up to an order $N_{v}$, to wit,

$\displaystyle\bar{V}_{k}(\bar{\rho})$
$\displaystyle=\sum^{N_{v}}_{n=1}c_{n,k}T_{n}(\bar{\rho})+\frac{1}{2}c_{0,k}\,,$
(10)

with $\bar{V}_{k}(\bar{\rho})=V_{k}(\rho)$, $\bar{\rho}=Z_{\phi,k}\rho$, where
quantities with a bar denote renormalized variables. The Chebyshev polynomial
$T_{n}(\bar{\rho})$ is given in Eq. (78), and the superscript
$[0,\bar{\rho}_{\mathrm{max}}]$ in Eq. (78) denoting the interval of
$\bar{\rho}$ is neglected for brevity here. Differentiating Eq. (10) with
respect to the RG time $t$ with $\rho$ fixed, one is led to

$\displaystyle\partial_{t}\big{|}_{\rho}\bar{V}_{k}(\bar{\rho})=$
$\displaystyle\sum^{N_{v}}_{n=1}\Big{(}\partial_{t}c_{n,k}-d_{n,k}\eta_{\phi,k}(\bar{\rho})\bar{\rho}\Big{)}T_{n}(\bar{\rho})$
$\displaystyle+\frac{1}{2}\Big{(}\partial_{t}c_{0,k}-d_{0,k}\eta_{\phi,k}(\bar{\rho})\bar{\rho}\Big{)}\,,$
(11)

where we have used the Chebyshev expansion for the derivative of the effective
potential as shown in Eq. (82) and coefficients $d_{n,k}$’s are the respective
expanding coefficients. Employing the discrete orthogonality relation in Eq.
(76) by summing up the $N+1$ zeros of $T_{N+1}(\bar{\rho})$ in Eq. (79), one
arrives at

$\displaystyle\partial_{t}c_{m,k}=$
$\displaystyle\frac{2}{N+1}\sum^{N}_{i=0}\Big{(}\partial_{t}\big{|}_{\rho}\bar{V}_{k}(\bar{\rho}_{i})\Big{)}T_{m}(\bar{\rho}_{i})$
$\displaystyle+\frac{2}{N+1}\sum^{N_{v}}_{n=1}\sum^{N}_{i=0}d_{n,k}T_{m}(\bar{\rho}_{i})T_{n}(\bar{\rho}_{i})\eta_{\phi,k}(\bar{\rho}_{i})\bar{\rho}_{i}$
$\displaystyle+\frac{1}{N+1}d_{0,k}\sum^{N}_{i=0}T_{m}(\bar{\rho}_{i})\eta_{\phi,k}(\bar{\rho}_{i})\bar{\rho}_{i}\,,$
(12)

which is the flow equation for the expansion coefficients in Eq. (10).

##  III Phase diagram

It is left to specify the parameters in the LEFT, prior to presenting our
calculated results. The UV cutoff of flow equations in the LEFT is chosen to
be $\Lambda=500$ MeV, and the effective potential in Eq. (3) at $k=\Lambda$
reads

$\displaystyle V_{\Lambda}(\rho)$
$\displaystyle=\frac{\lambda_{\Lambda}}{2}\rho^{2}+\nu_{\Lambda}\rho\,,$ (13)

with $\lambda_{\Lambda}=20$ and $\nu_{\Lambda}=0$ . The Yukawa coupling is
$k$-independent as shown in Eq. (9) and is given by $\bar{h}_{y}=6.4$.
Concerning the Chebyshev expansion, we choose $N=81$ for the number of zeros
and $N_{v}=21$ for the maximal order of Chebyshev polynomials. We have also
checked that there is no difference when the value of $N_{v}$ is increased.
Moreover, the upper bound of $\bar{\rho}$ is chosen to be
$\bar{\rho}_{\mathrm{max}}=9\times 10^{3}\,\mathrm{MeV}^{2}$, well above the
value of minimum of the potential in the IR. In the LPA, these values of
parameters lead to the pion decay constant $f_{\pi}=$87 MeV and the
constituent quark mass $m_{q}=$278.4 MeV in the vacuum and in the chiral
limit. While if the explicit breaking strength of the chiral symmetry in Eq.
(3) is increased up to $c=1.85\times 10^{-3}\,(\mathrm{GeV})^{3}$, one obtains
the physical pion mass $m_{\pi}=$138 MeV, as well as $f_{\pi}=$93 MeV and
$m_{q}=$297.6 MeV in the vacuum. Note that in order to facilitate the
comparison between the calculation with the truncation LPA and that with LPA′,
we use the same values of parameters above in the LPA′ computation as in LPA.

In Fig. 2 we show the phase diagrams of LEFT in the $T\\!-\\!\mu_{B}$ plane,
calculated within the fRG approach with the truncations LPA and LPA′, in the
left and right panels, respectively. The black dashed lines in both panels
denote the second-order $O(4)$ chiral phase transition of $N_{f}=2$ flavor in
the chiral limit. The black circles indicate the location of the tricritical
point, beyond which the second-order phase transition evolves into a
discontinuous first-order one, which are shown by the solid lines. Note that
the solid lines of different colors in the left panel denote the first-order
phase transitions with different pion masses in the vacuum, i.e. different
values of $c$ in Eq. (3), and in the right panel, we only give the first-order
phase transition line in the chiral limit, since numerical calculations become
quite difficult in the region of high $\mu_{B}$ and low $T$ with the
truncation LPA′. The red dashed lines in both panels are the trajectories of
the critical end points with the change of the strength of explicit chiral
symmetry breaking $c$, which belong to the 3-$d$ $Z(2)$ Ising university
class.

The critical temperature at vanishing baryon chemical potential is found to be
$T_{c}=144$ MeV in LPA and 143 MeV in LPA′ in the chiral limit. The
tricritical point is located at
$(T_{\mathrm{tri}},{\mu_{B}}_{\mathrm{tri}})_{{}_{\tiny{\mathrm{LPA}}}}=(50,764)$
MeV in the LPA and
$(T_{\mathrm{tri}},{\mu_{B}}_{\mathrm{tri}})_{{}_{\tiny{\mathrm{\mathrm{LPA}^{\prime}}}}}=(47,687)$
MeV in the LPA′, which are shown in the phase diagrams by the black circles.
The location of CEP corresponding to the physical pion mass in the LPA, shown
in the left panel of Fig. 2 by the star, is
$(T_{{}_{\tiny{\mathrm{CEP}}}},{\mu_{B}}_{{}_{\tiny{\mathrm{CEP}}}})_{{}_{\tiny{\mathrm{LPA}}}}=(8,885)$
MeV. In both phase diagrams in Fig. 2 we also use red and blue crosses to
label the locations where critical exponents in Sec. IV would be calculated
for the 3-$d$ $O(4)$ and $Z(2)$ universality classes, respectively. The
calculated points for the $O(4)$ and $Z(2)$ phase transition in the LPA are
given by
$(T_{{}_{O(4)}},{\mu_{B}}_{{}_{O(4)}})_{{}_{\tiny{\mathrm{LPA}}}}=(144,0)$ MeV
and
$(T_{{}_{Z(2)}},{\mu_{B}}_{{}_{Z(2)}})_{{}_{\tiny{\mathrm{LPA}}}}=(38,795)$
MeV, respectively; and the relevant values in the LPA′ read
$(T_{{}_{O(4)}},{\mu_{B}}_{{}_{O(4)}})_{{}_{\tiny{\mathrm{LPA}^{\prime}}}}=(143,0)$
MeV and
$(T_{{}_{Z(2)}},{\mu_{B}}_{{}_{Z(2)}})_{{}_{\tiny{\mathrm{LPA}^{\prime}}}}=(41,702)$
MeV.

##  IV Critical behavior and critical exponents

A variety of scaling analysis has been performed for the $O(4)$ universality
class, e.g., in the $O(N)$ model Toussaint (1997); Engels and Mendes (2000);
Parisen Toldin  _et al._ (2003); Engels  _et al._ (2003); Braun and Klein
(2008); Engels and Vogt (2010) and two-flavor quark-meson model Berges  _et
al._ (1999); Schaefer and Pirner (1999); Bohr  _et al._ (2001); Stokic  _et
al._ (2010). The dynamics of a system in the critical regime near a second-
order critical point is governed by long-wavelength fluctuations, and the
correlation length tends to be divergent as the system moves towards the
critical point. Critical exponents play a pivotal role in studies of the
critical dynamics, which are independent of micro interactions, but rather
universal for the same symmetry class, dimension of the system, etc., and see
Stokic  _et al._ (2010); Braun and Klein (2008) for more details. In the
following, we follow the standard procedure and give our notations for the
relevant various critical exponents.

To begin with, from the effective action in Eq. (3) one readily obtains the
thermodynamic potential density, which reads

$\displaystyle\Omega\big{(}T,\mu_{B},\,c\big{)}$
$\displaystyle=V_{k=0}(\rho)-c\sigma\,,$ (14)

where the order parameter field $\sigma\equiv\langle\sigma\rangle$ or
$\rho=\sigma^{2}/2$ is on its equation of motion. We then introduce the
reduced temperature and reduced external “magnetic” field as follows

$\displaystyle t$ $\displaystyle=\frac{T-T_{c}}{T_{0}}\,,\qquad
h=\frac{c}{c_{0}}\,,$ (15)

where $T_{c}$ is the critical temperature, and they are normalized by $T_{0}$
and $c_{0}$, i.e., some appropriate values of $T$ and $c$. In the language of
magnetization under an external magnetic field, the order parameter $\sigma$
here is just the corresponding magnetization density, i.e., $M\equiv\sigma$,
and the explicit chiral symmetry breaking parameter $c$ is equivalent to the
magnetic field strength $H\equiv c$. We will not distinguish them in the
following any more. In the critical regime the thermodynamic potential in Eq.
(14) is dominated by its singular part $f_{s}$, i.e.,

$\displaystyle\Omega\big{(}t,h\big{)}$
$\displaystyle=f_{s}(t,h)+f_{reg}(t,h)\,,$ (16)

where the second term on the r.h.s. is the regular one, and the notation for
the baryon chemical potential is suppressed. In what follows we adopt the
notations in Braun  _et al._ (2011b), and the scaling function $f_{s}(t,h)$ on
the r.h.s. of Eq. (16) satisfies the scale relation to leading order, viz.

$\displaystyle f_{s}(t,h)$
$\displaystyle=\ell^{-d}f_{s}(t\,\ell^{y_{t}},\,h\,\ell^{y_{h}})\,,$ (17)

where $\ell$ is a dimensionless rescaling factor. The scaling function in Eq.
(17) leads us to a variety of relations for various critical exponents Berges
_et al._ (1999); Tetradis (2003); Schaefer and Pirner (1999); Braun and Klein
(2008), e.g.,

$\displaystyle y_{t}$
$\displaystyle=\frac{1}{\nu}\,,\quad\\!\\!y_{h}=\frac{\beta\delta}{\nu}\,,\quad\\!\\!\beta=\frac{\nu}{2}(d-2+\eta)\,,\quad\\!\\!\gamma=\beta(\delta-1)\,,$
$\displaystyle\gamma$
$\displaystyle=(2-\eta)\nu\,,\quad\delta=\frac{d+2-\eta}{d-2+\eta}\,,\quad\nu
d=\beta(1+\delta)\,,$ (18)

with the spacial dimension $d$. The critical exponents $\beta$ and $\delta$
describe the critical behavior of the order parameter in the direction of $t$
or $h$, respectively, i.e.,

$\displaystyle M(t,h=0)$ $\displaystyle\sim(-t)^{\beta}\quad\mathrm{with}\quad
t<0\,,$ (19) $\displaystyle M(t=0,h)$ $\displaystyle\sim h^{1/\delta}\,.$ (20)

The exponent $\gamma$ is related to the susceptibility of order parameter
$\chi$, and $\nu$ to the correlation length $\xi$, which reads

$\displaystyle\chi$
$\displaystyle\sim|t|^{-\gamma}\,,\quad\mathrm{and}\quad\xi\sim|t|^{-\nu}\,,$
(21)

The scaling relation in Eq. (17) allows us to readily obtain the critical
behavior for various observables. For instance, the order parameter and its
susceptibilities read

$\displaystyle M$ $\displaystyle=-\frac{\partial f_{s}}{\partial
H}\,,\quad\chi_{\sigma}=\frac{\partial M}{\partial
H}\,,\quad\chi_{\pi}=\frac{M}{H}\,,$ (22)

where $\chi_{\sigma}\equiv\chi_{l}$ and $\chi_{\pi}\equiv\chi_{t}$ are also
called as the longitudinal and transverse susceptibilities, respectively.

Choosing an appropriate value of the rescaling factor such that
$h\,\ell^{y_{h}}=1$ in Eq. (17), one is led to

$\displaystyle f_{s}(t,h)$ $\displaystyle=h^{d/y_{h}}f_{s}(z,1)\,,$ (23)

with the scaling variable $z=t/h^{1/(\beta\delta)}$. Inserting Eq. (23) into
the first equation in Eq. (22), one arrives at

$\displaystyle M$ $\displaystyle=h^{1/\delta}f(z)\,,$ (24)

where we have introduced

$\displaystyle f(z)$
$\displaystyle\equiv\frac{1}{H_{0}}\Big{[}\frac{z}{\beta\delta}\frac{\partial
f_{s}(z,1)}{\partial z}-\frac{d\nu}{\beta\delta}f_{s}(z,1)\Big{]}\,,$ (25)

which is a scaling function dependent only on $z$. With appropriate values of
$H_{0}$ and $T_{0}$ in Eq. (15), it can be shown that the scaling function in
Eq. (25) has the properties $f(0)=1$ and $f(z)\simeq(-z)^{\beta}$ with
$z\rightarrow-\infty$ Braun  _et al._ (2011b).

Consequently, it is straightforward to express the longitudinal and transverse
susceptibilities in Eq. (22) in terms of the scaling function $f(z)$, to wit,

$\displaystyle\chi_{\sigma}$
$\displaystyle=\frac{1}{H_{0}}h^{1/\delta-1}f_{\chi}(z)\,,$ (26)

with

$\displaystyle f_{\chi}(z)$
$\displaystyle\equiv\frac{1}{\delta}\Big{[}f(z)-\frac{z}{\beta}f^{\prime}(z)\Big{]}\,,$
(27)

and

$\displaystyle\chi_{\pi}$ $\displaystyle=\frac{1}{H_{0}}h^{1/\delta-1}f(z)\,.$
(28)

Alternative to the choice of $h\,\ell^{y_{h}}=1$ in Eq. (17), one can also
employ $t\,\ell^{y_{t}}=1$, which is equivalent to the Widom-Griffiths
parametrization Widom (1965); Griffiths (1967) of the equation of state by
means of the scaling variables, as follows

$\displaystyle x$ $\displaystyle\equiv\frac{t}{M^{1/\beta}}\,,\qquad
y\equiv\frac{h}{M^{\delta}}\,,$ (29)

which are obviously related to the other parametrization by the relations
which read

$\displaystyle z$ $\displaystyle=\frac{x}{y^{1/(\beta\delta)}}\,,\qquad
f(z)=\frac{1}{y^{1/\delta}}\,.$ (30)

Hence the scaling function $y(x)$ has the properties $y(0)=1$ and $y(-1)=0$.
In the same way, one readily obtains the expressions of susceptibilities in
this parametrization, which read

$\displaystyle\chi_{\sigma}$
$\displaystyle=\frac{1}{H_{0}M^{\delta-1}}\Big{[}\delta
y(x)-\frac{1}{\beta}xy^{\prime}(x)\Big{]}^{-1}\,,$ (31)
$\displaystyle\chi_{\pi}$
$\displaystyle=\frac{1}{H_{0}M^{\delta-1}}\frac{1}{y}\,.$ (32)

###  IV.1 Order parameter

Figure 3: Logarithm of the reduced order parameter $\tilde{\sigma}$ in Eq. (33) as a function of $\ln(-t)$ (left panel) or $\ln(h)$ (right panel) for the second-order $O(4)$ and $Z(2)$ phase transitions with truncations LPA and LPA′, where the phase transition points are chosen to be the locations of the red and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes, respectively. The solid lines represent linear fits to the calculated discrete data points, from which values of the critical exponents $\beta$ and $\delta$ are extracted. $T_{c}-T$ (MeV) |  ($10^{-4}$, $5\\!\times\\!10^{-3}$) |  ($10^{-2}$, 0.1) |  (0.1, 0.5 ) |  (0.5, 1) |  (1, 5)
---|---|---|---|---|---
$\beta^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$ | 0.3989(41) | 0.5164(65) | 0.4374(36) | 0.4077(44) | 0.3921(43)
$\beta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$ | 0.3352(12) | 0.2830(26) | 0.2724(18) | 0.2689(17) | 0.247(17)
Table 1: Values of the critical exponent $\beta$ extracted from different
ranges of temperature, which are denoted by their distances to the
corresponding critical temperature, i.e., $T_{c}-T$. The calculations are
performed with the truncation LPA, and the phase transition points are chosen
to be the locations of the red and blue crosses in the phase diagrams in Fig.
2 for the $O(4)$ and $Z(2)$ universality classes, respectively.

The flow equation of effective potential in Eq. (4) is solved by the use of
the Chebyshev expansion as discussed in Sec. II.2, i.e., evolving the flow
equations of the expansion coefficients in Eq. (12) from the UV cutoff
$\Lambda$ to the infrared limit $k\rightarrow 0$, and then the expectation
value of the order parameter $\sigma$ is determined by minimizing the
thermodynamic potential in Eq. (14). Note that two different truncations,
i.e., LPA and LPA′ as shown in Sec. II.1, are employed in the calculations.

The critical exponents $\beta$ and $\delta$ are given in Eqs. (19) and (20),
which are related to the scaling behavior of the order parameter as the phase
transition is approached towards in the temperature or external field
direction, respectively. Note, however, that in the case of $Z(2)$ phase
transition as indicated by the blue cross in the phase diagram in Fig. 2, the
order parameter should be modified slightly and we introduce the reduced order
parameter which reads

$\displaystyle\tilde{\sigma}$
$\displaystyle=\frac{\sigma-\sigma^{\prime}}{f_{\pi}}\,,$ (33)

where $f_{\pi}$ is the pion decay constant in the vacuum and $\sigma^{\prime}$
is the expectation value of sigma field at the phase transition point, which
is nonvanishing on the red dashed lines of $Z(2)$ in the phase diagrams in
Fig. 2. Correspondingly, the reduced external field in Eq. (15) is modified
into

$\displaystyle h$ $\displaystyle=\frac{c-c^{\prime}}{c_{0}}\,,$ (34)

where $c^{\prime}$ is the $\sigma^{\prime}$-related external field on the
$Z(2)$ phase transition line. Notice that both $c^{\prime}$ and
$\sigma^{\prime}$ are vanishing on the $O(4)$ phase transition line, viz., the
black dashed lines in Fig. 2. In our calculations below, the normalized
external field strength $c_{0}$ in Eq. (34) is chosen to be the value
corresponding to the physical pion mass, and the normalized temperature in Eq.
(15) is to be the critical one $T_{0}=T_{c}$.

In Fig. 3 we show the log-log plots of the reduced order parameter
$\tilde{\sigma}$ versus the reduced temperature $-t$ or external field $h$ for
the second-order $O(4)$ and $Z(2)$ phase transitions. The calculations are
performed in the quark-meson LEFT with the fRG in both LPA and LPA′. The phase
transition points are chosen to be the locations of the red and blue crosses
in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality
classes, respectively. A linear relation is used to fit the calculated
discrete data points in Fig. 3, and as shown in Eq. (19) and Eq. (20), one
could extract the values of the critical exponents $\beta$ and $\delta$ from
the slope of these linear curves. This leads us to

$\displaystyle\beta^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=0.3989(41)\,,\qquad\beta^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.3832(31)\,,$
(35)

for the $O(4)$ universality class in LPA and LPA′, respectively. In the case
of the $Z(2)$ universality class, one arrives at

$\displaystyle\beta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=0.3352(12)\,,\qquad\beta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.3259(01)\,.$
(36)

In the same way, the values of $\delta$ are obtained as follows

$\displaystyle\delta^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=4.975(57)\,,\qquad\delta^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=4.859(37)\,,$
(37) $\displaystyle\delta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=4.941(22)\,,\qquad\delta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=4.808(14)\,.$
(38)

It is found that the critical exponents $\beta$ and $\delta$ of the $O(4)$ and
$Z(2)$ phase transitions in 3-$d$ systems calculated in this work are
consistent with previous results, e.g., Monte Carlo simulation of spin model
Kanaya and Kaya (1995) and $d=3$ expansion for $Z(2)$Zinn-Justin (2001).
Comparing the relevant results in LPA and LPA′, one observes that both $\beta$
and $\delta$ obtained in LPA′ are slightly smaller than those in LPA.

###  IV.2 Preliminary assessment of the size of the critical region

It is well known that critical exponents are universal for the same
universality classes. The size of the critical region is, however, non-
universal and depends on the interactions and other details of system
concerned. Furthermore, there has been a longstanding debate on the size of
the critical region in QCD. Lattice QCD simulations show that the chiral
condensate, i.e., the order parameter in Eq. (24), for physical quark masses
are well described by Eq. (24) plus a small analytic regular term Ejiri  _et
al._ (2009); Kaczmarek  _et al._ (2011); Ding  _et al._ (2019), which, in
another word, implies that the size of the critical regime of QCD is large
enough, such that QCD with physical quark mass is still in the chiral critical
regime. On the contrary, it is found in Braun and Klein (2008); Braun  _et
al._ (2011b); Klein (2017) that the pion mass required to observe the scaling
behavior is very small, at least one order of magnitude smaller than the
physical pion mass. Moreover, it is also found that the critical region around
the CEP in the QCD phase diagram is very small Schaefer and Wambach (2007). In
Tab. 1 we present the values of the critical exponent $\beta$ extracted from
different ranges of temperature. One observes that when the temperature range
is away from the critical temperature larger than $0.01$ MeV, the value of
$\beta$ deviates from its universal value pronouncedly. This applies for both
the $O(4)$ and $Z(2)$ universality classes. Given the systematic errors in the
computation of this work, one could safely conclude that our calculation
indicates that the critical region in the QCD phase diagram is probably very
small, and it is smaller than 1 MeV in the direction of temperature.

###  IV.3 Chiral susceptibility

Figure 4: Logarithm of the longitudinal susceptibility $\chi_{\sigma}$ as a
function of $\ln(t)$ in the chiral symmetric phase. The calculation is done in
the quark-meson LEFT within the fRG approach with truncations LPA and LPA′,
where the phase transition points are chosen to be at the locations of the red
and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$
universality classes, respectively. The solid lines represent linear fits to
the calculated discrete data points, from which value of the critical exponent
$\gamma$ is extracted. Figure 5: Longitudinal susceptibility of the order
parameter $\chi_{\sigma}$ as a function of the reduced temperature $t$ with
several different values of the reduced external field $h$, calculated in the
LPA (left panel) and LPA′ (right panel). The phase transition is chosen to be
near the location of the red cross in the phase diagrams in Fig. 2 for the
$O(4)$ symmetry universality class. Figure 6: Left panel: logarithm of the
reduced pseudo-critical temperature $t_{pc}$, defined by the peak of the
susceptibility $\chi_{\sigma}$ as shown in Fig. 5, as a function of the
logarithm of the reduced external field strength $h$. Right panel: logarithm
of the peak height of the susceptibility, $\chi_{\sigma}\big{|}_{t_{pc}}$,
versus the logarithm of the reduced pseudo-critical temperature.
Calculations are done within the fRG approach with the truncations LPA and
LPA′. The phase transition is chosen to be near the location of the red cross
in the phase diagrams in Fig. 2 for the $O(4)$ symmetry universality class.
Figure 7: Logarithms of the transverse (left panel) and longitudinal (right
panel) susceptibilities as functions of the logarithm of $-t$ with a fixed
value of the reduced external field $h=8.4\times 10^{-9}$ in the chiral broken
phase near the coexistence line. Calculations are performed within the fRG
approach with the truncations LPA and LPA′. The phase transition is chosen to
be near the location of the red cross in the phase diagrams in Fig. 2 for the
$O(4)$ symmetry universality class, where the baryon chemical potential is
vanishing. Figure 8: Left panel: logarithm of the correlation length as a
function of the logarithm of the reduced external field strength with $t=0$.
Right panel: logarithm of the correlation length as a function of the
logarithm of the reduced temperature with $h=0$.
Both calculations are performed in the quark-meson LEFT within the fRG
approach with truncations LPA and LPA′, where the phase transition points are
chosen to be at the locations of the red and blue crosses in the phase
diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes,
respectively. The solid lines represent linear fits to the calculated discrete
data points, from which values of the critical exponent $\nu_{c}$ and $\nu$
are yielded.

According to Eq. (29), the reduced order parameter reads

$\displaystyle\tilde{\sigma}$ $\displaystyle\sim
h^{1/\delta}y(x)^{-1/\delta}\,.$ (39)

Moreover, it has been shown in Griffiths (1967) that given $x>0$ and $M>M_{0}$
for some value $M_{0}$ in Eq. (29), the scaling function can be expanded as

$\displaystyle y(x)$
$\displaystyle=\sum_{n=1}^{\infty}c_{n}x^{\gamma-2\beta(n-1)}$
$\displaystyle=x^{\gamma}\big{(}c_{1}+c_{2}x^{-2\beta}+c_{3}x^{-4\beta}+\dots\big{)}\,.$
(40)

Inserting the leading term in Eq. (40) into Eq. (39) and utilizing the
relation $\gamma=\beta(\delta-1)$ as shown in Eq. (18), one is led to the
reduced order parameter with $t>0$ and $h\rightarrow 0$, which reads

$\displaystyle\tilde{\sigma}$ $\displaystyle\sim t^{-\gamma}h\,.$ (41)

Consequently, the longitudinal and transverse susceptibilities of the order
parameter as defined in Eq. (22) are readily obtained as follows

$\displaystyle\chi_{\sigma}$ $\displaystyle=\chi_{\pi}\sim t^{-\gamma}\,,$
(42)

which is in agreement with Eq. (21) in the limit $h\rightarrow 0$ and in the
symmetric phase, as it should be. Equation (41) also allows us to extract the
value of the exponent $\gamma$, by directly investigating the scaling relation
of $\tilde{\sigma}$ and $t$ in the chiral symmetric phase with a fixed, small
value of $h$. In Fig. 4 we show the logarithm of the longitudinal
susceptibility $\chi_{\sigma}$ versus that of the reduced temperature, where
$h=3.5\times 10^{-10}$ is chosen in the calculations. We have checked that
this value of $h$ is small enough to make sure that the value of $\gamma$
obtained from the linear fit of $\ln(\chi_{\sigma})$-$\ln(t)$ is convergent.
In the same way, the flow equations of fRG are resolved with two truncations
LPA and LPA′, and the phase transition points are chosen to be at the
locations of the red and blue crosses in the phase diagrams in Fig. 2 for the
$O(4)$ and $Z(2)$ universality classes, respectively. The values of the
exponent $\gamma$ are obtained as follows

$\displaystyle\gamma^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=1.5458(68)\,,\qquad\gamma^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=1.4765(76)\,,$
(43) $\displaystyle\gamma^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=1.3313(96)\,,\qquad\gamma^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=1.2362(77)\,.$
(44)

Once more, one observes that these values, in particular those obtained in the
LPA′, are in good agreement with the values of $\gamma$ for the $O(4)$ and
$Z(2)$ symmetry universality classes, respectively; see, e.g., Kanaya and Kaya
(1995); Zinn-Justin (2001).

In Fig. 5 the longitudinal susceptibility of the order parameter
$\chi_{\sigma}$, as shown in Eq. (22), is depicted versus the reduced
temperature with several different values of the reduced external field. Here
we only focus on the case of $O(4)$ symmetry, and thus choose the phase
transition to be near the location of the red cross in the phase diagrams in
Fig. 2, i.e., the phase transition with vanishing baryon chemical potential.
When the external field $h$ that breaks the chiral symmetry explicitly is
nonzero, the second-order phase transition becomes a continuous crossover, as
shown in Fig. 5. One can define a pseudo-critical temperature $T_{pc}$, which
is the peak position of the curve $\chi_{\sigma}$ versus $T$, and thus the
reduced pseudo-critical temperature reads

$\displaystyle t_{pc}$ $\displaystyle=\frac{T_{pc}-T_{c}}{T_{c}}\,.$ (45)

One observes from Fig. 5 that with the increasing $h$, the peak height of the
susceptibility decreases and the pseudo-critical temperature $t_{pc}$
increases. The rescaling relation between $t_{pc}$ and $h$ as well as that
between the peak height of $\chi_{\sigma}$ and $t_{pc}$ reads

$\displaystyle t_{pc}$ $\displaystyle\sim
h^{1/(\gamma+\beta)}\,,\qquad\chi_{\sigma}\big{|}_{t_{pc}}\sim
t_{pc}^{-\gamma}\,,$ (46)

and see, e.g., Pelissetto and Vicari (2002) for more details.

In Fig. 6 we show the logarithm of the reduced pseudo-critical temperature
versus the logarithm of the reduced external field strength, and the logarithm
of the peak height of the susceptibility versus the logarithm of the reduced
pseudo-critical temperature in the left and right panels, respectively. The
phase transition is also chosen to be near the location of the red cross in
the phase diagrams in Fig. 2 for the $O(4)$ symmetry universality class, where
the baryon chemical potential is vanishing. Linear fitting to the calculated
discrete data in Fig. 6 yields $\beta=0.403(19)$ and $\gamma=1.543(15)$ for
the LPA, and $\beta=0.405(22)$ and $\gamma=1.454(17)$ for the LPA′, which are
in agreement with the relevant values in Eq. (35) and Eq. (43) within errors
for the $O(4)$ second-order phase transition in 3-$d$ space. In turn, the
agreement of critical exponents obtained from different scaling relations also
provides us with the necessary check for the inner consistency of
computations. Note, however, that the critical exponents $\beta$ and $\gamma$
determined from the scaling relations in Eq. (46) are significantly less
accurate than those in Eq. (35) and Eq. (43).

As another check for the consistency, we consider the susceptibilities in the
chiral broken phase near the coexistence line, i.e., $x=-1$, with $t<0$ and
$h\rightarrow 0$. Inserting Eq. (39) into Eqs. (31) (32), one is led to

$\displaystyle\chi_{\sigma}$ $\displaystyle\sim h^{1/\delta-1}\frac{\beta
y(x)^{1-1/\delta}}{\beta\delta y(x)-xy^{\prime}(x)}\,,$ (47)
$\displaystyle\chi_{\pi}$ $\displaystyle\sim
h^{1/\delta-1}y(x)^{-1/\delta}\,.$ (48)

when the system is near the coexistence line, one has $x\rightarrow-1$ and
$y\sim h/(-t)^{\beta\delta}$. Hence, the transverse susceptibility is readily
obtained as follows

$\displaystyle\chi_{\pi}$ $\displaystyle\sim h^{-1}(-t)^{\beta}\,.$ (49)

In order to obtain a similar expression for the longitudinal susceptibility,
one needs further information on the equation of state $y(x)$. As the system
is located in the broken phase near the coexistence line, the dynamics is
dominated by Goldstone modes, which are massless in the chiral limit. The
relevant critical behavior in this regime is governed by a Gaussian fixed
point, and thus the corresponding exponents are as same as values of mean
fields Wallace and Zia (1975); Brezin and Wallace (1973), which leaves us with

$\displaystyle y(x)$ $\displaystyle\sim(1+x)^{2}\,,\qquad\mathrm{for}\qquad
x\rightarrow-1\,,$ (50)

and see, e.g., Braun and Klein (2008); Stokic  _et al._ (2010) for more
relevant discussions. Substituting equation above into Eq. (47), one arrives
at

$\displaystyle\chi_{\sigma}$ $\displaystyle\sim
h^{-1/2}(-t)^{\beta-(\beta\delta/2)}\,.$ (51)

As Eqs. (49) (51) show, the transverse and longitudinal susceptibilities are
proportional to the external field with different powers in the broken phase,
i.e., $-1$ and $-1/2$ for the former and latter, respectively.

In Fig. 7 we show $\ln(\chi_{\pi})$ and $\ln(\chi_{\sigma})$ versus $\ln(-t)$
with a fixed value of the reduced external field $h=8.4\times 10^{-9}$ in the
chiral broken phase near the coexistence line. Similarly, here we only
consider the phase transition of $O(4)$ symmetry with $\mu_{B}=0$ in the phase
diagrams in Fig. 2. As shown in Eqs. (49) (51), the ratios of the linear
fitting to $\ln(\chi_{\pi})$-$\ln(-t)$ and $\ln(\chi_{\sigma})$-$\ln(-t)$ are
just the values of $\beta$ and $\beta-(\beta\delta/2)$, respectively.
Consequently, one arrives at $\beta=0.3979(41)$ and $\delta=4.984(74)$ in LPA
and $\beta=0.3832(54)$ and $\delta=4.86(10)$ in LPA′, which agree very well
with the relevant values in Eq. (35) and Eq. (37).

###  IV.4 Correlation length

| Method | $\beta$ | $\delta$ | $\gamma$ | $\nu$ | $\nu_{c}$ | $\eta$
---|---|---|---|---|---|---|---
$O(4)$ QM LPA (this work) | fRG Chebyshev | 0.3989(41) | 4.975(57) | 1.5458(68) | 0.7878(25) | 0.3982(17) | 0
$O(4)$ QM LPA′ (this work) | fRG Chebyshev | 0.3832(31) | 4.859(37) | 1.4765(76) | 0.7475(27) | 0.4056(19) | 0.0252(91)*
$Z(2)$ QM LPA (this work) | fRG Chebyshev | 0.3352(12) | 4.941(22) | 1.3313(96) | 0.6635(17) | 0.4007(45) | 0
$Z(2)$ QM LPA′ (this work) | fRG Chebyshev | 0.3259(01) | 4.808(14) | 1.2362(77) | 0.6305(23) | 0.4021(43) | 0.0337(38)*
$O(4)$ scalar theories Tetradis and Wetterich (1994) | fRG Taylor | 0.409 | 4.80* | 1.556 | 0.791 |  | 0.034
$O(4)$ KT phase transition Von Gersdorff and Wetterich (2001) | fRG Taylor | 0.387* | 4.73* |  | 0.739 |  | 0.047
$Z(2)$ KT phase transition Von Gersdorff and Wetterich (2001) | fRG Taylor |  |  |  | 0.6307 |  | 0.0467
$O(4)$ scalar theories Litim and Pawlowski (2001) | fRG Taylor | 0.4022* | 5.00* |  | 0.8043 |  |
$O(4)$ scalar theories LPABraun and Klein (2008) | fRG Taylor | 0.4030(30) | 4.973(30) |  | 0.8053(60) |  |
$O(4)$ QM LPA Stokic  _et al._ (2010) | fRG Taylor | 0.402 | 4.818 | 1.575 | 0.787 | 0.396 |
$O(4)$ scalar theories Bohr  _et al._ (2001) | fRG Grid | 0.40 | 4.79 |  | 0.78 |  | 0.037
$Z(2)$ scalar theories Bohr  _et al._ (2001) | fRG Grid | 0.32 | 4.75 |  | 0.64 |  | 0.044
$O(4)$ scalar theories De Polsi  _et al._ (2020) | fRG DE $\mathcal{O}(\partial^{4})$ |  |  |  | 0.7478(9) |  | 0.0360(12)
$Z(2)$ scalar theories Balog  _et al._ (2019); De Polsi  _et al._ (2020) | fRG DE $\mathcal{O}(\partial^{6})$ |  |  |  | 0.63012(5) |  | 0.0361(3)
$O(4)$ CFTs Kos  _et al._ (2015) | conformal bootstrap |  |  |  | 0.7472(87) |  | 0.0378(32)
$Z(2)$ CFTs Kos  _et al._ (2014) | conformal bootstrap |  |  |  | 0.629971(4) |  | 0.0362978(20)
$O(4)$ spin model Kanaya and Kaya (1995) | Monte Carlo | 0.3836(46) | 4.851(22) | 1.477(18) | 0.7479(90) | 0.4019(71)* | 0.025(24)*
$Z(2)$ $d=3$ expansion Zinn-Justin (2001) | summed perturbation | 0.3258(14) | 4.805(17)* | 1.2396(13) | 0.6304(13) | 0.4027(23) | 0.0335(25)
Mean Field |  | 1/2 | 3 | 1 | 1/2 | 1/3 | 0
Table 2: Critical exponents for the $O(4)$ and $Z(2)$ symmetry universality
classes in 3-$d$ space, obtained in the quark-meson LEFT within the fRG
approach with truncations LPA and LPA′, where the effective potential is
expanded as a sum of Chebyshev polynomials. Our calculated results are also in
comparison to relevant results from previous fRG calculations, e.g., scalar
theories with the effective potential expanded in a Taylor series Tetradis and
Wetterich (1994); Von Gersdorff and Wetterich (2001); Litim and Pawlowski
(2001); Braun and Klein (2008), or discretized on a grid Bohr  _et al._
(2001), the quark-meson (QM) low energy effective theory with LPA Stokic  _et
al._ (2010), derivative expansions (DE) up to orders of
$\mathcal{O}(\partial^{4})$ and $\mathcal{O}(\partial^{6})$ Balog  _et al._
(2019); De Polsi  _et al._ (2020). Moreover, results from other approaches,
such as the conformal bootstrap for the 3-$d$ conformal field theories (CFTs)
Kos  _et al._ (2014, 2015), Monte Carlo simulation Kanaya and Kaya (1995), and
$d=3$ perturbation expansion Zinn-Justin (2001), as well as the mean-field
values of exponents are also presented. Note that values with an asterisk are
obtained with scaling laws in Eq. (18).

It is well known that the correlation length $\xi$, plays a pivotal role in
the critical dynamics, since fluctuations of wavelength $\sim\xi$ are
inevitably involved in the dynamics. As a system is approaching towards a
second-order phase transition, the most relevant degrees of freedom are the
long-wavelength modes of low energy, and the correlation length is divergent
at the phase transition Landau and Lifshitz (1980).

The critical behavior of correlation length is described by the critical
exponent $\nu$, as shown in Eq. (21). In the symmetric phase $t>0$, it reads

$\displaystyle\xi$ $\displaystyle\sim t^{-\nu}\,,\qquad\mathrm{with}\qquad
h=0\,,$ (52)

which illustrates the scaling relation between the correlation length and the
reduced temperature. Moreover, one can also define another critical exponent
$\nu_{c}$ related to the scaling relation between the correlation length and
the reduced external field, to wit,

$\displaystyle\xi$ $\displaystyle\sim h^{-\nu_{c}}\,,\qquad\mathrm{with}\qquad
t=0\,.$ (53)

In our setup in the quark-meson LEFT, cf. Sec. II, the correlation length is
proportional to the inverse of the renormalized $\sigma$-meson mass, viz.,

$\displaystyle\xi$ $\displaystyle\sim\frac{1}{m_{\sigma}}\,,$ (54)

where $m_{\sigma}$ is related to the dimensionless $k$-dependent sigma mass
$\bar{m}_{\sigma,k}$ in Eq. (5) via the relation as follows

$\displaystyle m_{\sigma}$
$\displaystyle=\bar{m}_{\sigma,k}(\sigma=\sigma_{{}_{\mathrm{EoM}}})k\,,\quad\mathrm{with}\quad
k\rightarrow 0\,,$ (55)

where the scale $k$ is chosen to be in the IR limit $k\rightarrow 0$, and the
mass is calculated on the equation of motion of the order parameter field. In
Fig. 8 we show the scale relation between the correlation length and the
reduced external field strength, and that between the correlation length and
the reduced temperature, respectively. In the same way, we adopt the two
different truncations: LPA and LPA′. The phase transition points are also
chosen to be at the locations of the red and blue crosses in the phase
diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes,
respectively. By the use of the linear fitting to the calculated data, one
obtains values of the critical exponent $\nu$ as follows

$\displaystyle\nu^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=0.7878(25)\,,\qquad\nu^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.7475(27)\,,$
(56) $\displaystyle\nu^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=0.6635(17)\,,\qquad\nu^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.6305(23)\,,$
(57)

as well as those of the critical exponent $\nu_{c}$, i.e.,

$\displaystyle{\nu_{c}}^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=0.3982(17)\,,\qquad{\nu_{c}}^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.4056(19)\,,$
(58) $\displaystyle{\nu_{c}}^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$
$\displaystyle=0.4007(45)\,,\qquad{\nu_{c}}^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.4021(43)\,.$
(59)

Finally, we close Sec. IV with a summary of various critical exponents
calculated in this work in Tab. 2. Respective results for the $O(4)$ and
$Z(2)$ symmetry universality classes with truncation LPA or LPA′ are presented
in the first several rows in Tab. 2. As we have discussed in Sec. II, the
effective potential is expanded as a sum of Chebyshev polynomials in our
calculations, which captures global properties of the order-parameter
potential very well. In Tab. 2 we also present values of critical exponents
obtained from other computations, e.g., scalar theories calculated within the
fRG with the effective potential expanded in a Taylor series Tetradis and
Wetterich (1994); Von Gersdorff and Wetterich (2001); Litim and Pawlowski
(2001); Braun and Klein (2008), or discretized on a grid Bohr  _et al._
(2001), quark-meson LEFT within the fRG in LPA Stokic  _et al._ (2010),
derivative expansion of the fRG up to orders of $\mathcal{O}(\partial^{4})$
and $\mathcal{O}(\partial^{6})$ Balog  _et al._ (2019); De Polsi  _et al._
(2020), the conformal bootstrap for the 3-$d$ conformal field theories Kos
_et al._ (2014, 2015), Monte Carlo simulation Kanaya and Kaya (1995), and the
$d=3$ perturbation expansion Zinn-Justin (2001). One observes that our
calculated results are in good agreement with the relevant results from
previous fRG calculations as well as those from the conformal bootstrap, Monte
Carlo simulation, and the $d=3$ perturbation expansion. Remarkably, the
calculation with the truncation LPA′ is superior to that with LPA, and the
former has already provided us with quantitative reliability for the
prediction of the critical exponents in comparison to other approaches.

##  V summary

QCD phase structure and related critical behaviors have been studied in the
two-flavor quark-meson low energy effective theory within the fRG approach in
this work. More specifically, we have expanded the effective potential as a
sum of Chebyshev polynomials to solve its flow equation. Consequently, both
the global properties of the effective potential and the numerical accuracy
necessary for the computation of critical exponents are retained in our
calculations. Moreover, we have employed two different truncations for the
effective action: one is the usually used local potential approximation and
the other is that beyond the local potential approximation, in which a field-
dependent mesonic wave function renormalization is encoded.

With the numerical setup within the fRG approach described above, we have
obtained the phase diagram in the plane of $T$ and $\mu_{B}$ for the two-
flavor quark-meson LEFT in the chiral limit, including the second-order phase
transition line of $O(4)$, the tricritical point and the first-order phase
transition line. Furthermore, we also show the $Z(2)$ line in the phase
diagram, which is the trajectory of the critical end point moving with the
successive variance of the strength of explicit chiral symmetry breaking, or
the varying pion mass.

In the phase diagram, we have performed detailed scaling analyses for the
3-$d$ $O(4)$ and $Z(2)$ symmetry universality classes, and investigated the
critical behaviors in the vicinity of phase transition both in the chiral
symmetric and broken phases. Moreover, the transverse and longitudinal
susceptibilities of the order parameter have been calculated in the chiral
broken phase near the coexistence line.

A variety of critical exponents related to the order parameter, chiral
susceptibilities and correlation lengths have been calculated for the 3-$d$
$O(4)$ and $Z(2)$ symmetry universality classes in the phase diagram,
respectively. The calculated results are also compared with those from
previous fRG calculations, either employing the Taylor expansion for the
order-parameter potential or discretizing it on a grid, derivative expansion
of the effective action, the conformal bootstrap, Monte Carlo simulations, and
the $d=3$ perturbation expansion. We find that the critical exponents obtained
in the quark-meson LEFT within the fRG approach, where the order-parameter
potential is expanded in terms of Chebyshev polynomials and a field-dependent
mesonic wave function renormalization is taken into account, are in
quantitative agreement with results from approaches aforementioned.
Furthermore, we have also investigated the size of the critical regime, and it
is found that the critical region in the QCD phase diagram is probably very
small, and it is smaller than 1 MeV in the direction of temperature.

###### Acknowledgements.

We thank Jan M. Pawlowski for illuminating discussions. We also would like to
thank other members of the fQCD collaboration Braun  _et al._ (2021) for work
on related subjects. The work was supported by the National Natural Science
Foundation of China under Contract No. 11775041, and the Fundamental Research
Funds for the Central Universities under Contract No. DUT20GJ212.

##  Appendix A Threshold functions and anomalous dimensions

We employ the $3d$ flat regulators Litim (2001, 2000) for quarks and mesons in
this paper

$\displaystyle R_{\phi,k}(q_{0},\bm{q})$
$\displaystyle=Z_{\phi,k}\bm{q}^{2}r_{B}(\bm{q}^{2}/k^{2})\,,$ (60)
$\displaystyle R_{q,k}(q_{0},\bm{q})$
$\displaystyle=Z_{q,k}i\bm{\gamma}\cdot\bm{q}r_{F}(\bm{q}^{2}/k^{2})\,,$ (61)

with

$\displaystyle r_{B}(x)$
$\displaystyle=\left(\frac{1}{x}-1\right)\Theta(1-x)\,,$ (62) $\displaystyle
r_{F}(x)$ $\displaystyle=\left(\frac{1}{\sqrt{x}}-1\right)\Theta(1-x)\,.$ (63)

The threshold functions in Eq. (4) are given by

$\displaystyle l_{0}^{(B,d)}(\bar{m}^{2}_{\phi,k},\eta_{\phi,k};T)$
$\displaystyle=$
$\displaystyle\frac{2}{d-1}\left(1-\frac{\eta_{\phi,k}}{d+1}\right)\frac{1}{\sqrt{1+\bar{m}^{2}_{\phi,k}}}$
$\displaystyle\times\bigg{(}\frac{1}{2}+n_{B}(\bar{m}^{2}_{\phi,k};T)\bigg{)}\,,$
(64)

and

$\displaystyle l_{0}^{(F,d)}(\bar{m}^{2}_{q,k},\eta_{q,k};T,\mu)$
$\displaystyle=$
$\displaystyle\frac{2}{d-1}\left(1-\frac{\eta_{q,k}}{d}\right)\frac{1}{2\sqrt{1+\bar{m}^{2}_{q,k}}}$
$\displaystyle\times\Big{(}1-n_{F}(\bar{m}^{2}_{q,k};T,\mu)-n_{F}(\bar{m}^{2}_{q,k};T,-\mu)\Big{)}\,.$
(65)

with the bosonic and fermionic distribution functions reading

$\displaystyle n_{B}(\bar{m}^{2}_{\phi,k};T)=$
$\displaystyle\frac{1}{\exp\bigg{\\{}\frac{k}{T}\sqrt{1+\bar{m}_{\phi,k}^{2}}\bigg{\\}}-1}\,,$
(66)

and

$\displaystyle n_{F}(\bar{m}^{2}_{q,k};T,\mu)=$
$\displaystyle\frac{1}{\exp\bigg{\\{}\frac{1}{T}\Big{[}k\sqrt{1+\bar{m}^{2}_{q,k}}-\mu\Big{]}\bigg{\\}}+1}\,,$
(67)

respectively.

The meson anomalous dimension in Eq. (8) is given by

$\displaystyle\eta_{\phi,k}(\rho)$
$\displaystyle=\frac{1}{6\pi^{2}}\Bigg{\\{}\frac{4}{k^{2}}\bar{\rho}(\bar{V}^{\prime\prime}_{k}(\bar{\rho}))^{2}\mathcal{BB}_{(2,2)}(\bar{m}^{2}_{\pi,k},\bar{m}^{2}_{\sigma,k};T)$
$\displaystyle+N_{c}\bar{h}^{2}_{y,k}\bigg{[}\mathcal{F}_{(2)}(\bar{m}^{2}_{q,k};T,\mu)(2\eta_{q,k}-3)$
$\displaystyle-4(\eta_{q,k}-2)\mathcal{F}_{(3)}(\bar{m}^{2}_{q,k};T,\mu)\bigg{]}\Bigg{\\}}\,,$
(68)

with

$\displaystyle\bar{h}_{y,k}$
$\displaystyle=\frac{h_{y,k}}{Z_{q,k}(Z_{\phi,k})^{1/2}}\,.$ (69)

Note that threshold functions $\mathcal{BB}_{(2,2)}$, $\mathcal{F}_{(2)}$ and
$\mathcal{F}_{(3)}$ in Eq. (68) can be found in e.g., Fu and Pawlowski (2015);
Yin  _et al._ (2019).

##  Appendix B Some relations for the Chebyshev polynomials

In this appendix we collect some relations for the Chebyshev polynomials,
which are used in solving the flow equation for the effective potential in Eq.
(10). The Chebyshev polynomial of order $n$ reads

$\displaystyle T_{n}(x)$ $\displaystyle=\cos\big{(}n\arccos(x)\big{)}\,,$ (70)

with nonnegative integers $n$’s and $x\in[-1,1]$. The explicit expressions for
the Chebyshev polynomials could be obtained by the recursion relation as
follows

$\displaystyle T_{n+2}(x)$ $\displaystyle=2xT_{n+1}(x)-T_{n}(x)\,,\quad n\geq
0\,,$ (71)

with $T_{0}(x)=1$ and $T_{1}(x)=x$.

The $N+1$ zeros of $T_{N+1}(x)$ in the region $-1\leq x\leq 1$ are given by

$\displaystyle x_{k}$
$\displaystyle=\cos\left(\frac{\pi(k+\frac{1}{2})}{N+1}\right)\,,\quad
k=0,\,1,\,\cdots N\,.$ (72)

A discrete orthogonality relation is fulfilled by the Chebyshev polynomials,
to wit,

$\displaystyle\sum_{k=0}^{N}T_{i}(x_{k})T_{j}(x_{k})$
$\displaystyle=\left\\{\begin{array}[]{l}0\qquad\qquad\qquad i\neq
j\\\\[4.30554pt] (N+1)/2\qquad i=j\neq 0\\\\[4.30554pt]
N+1\qquad\quad\;\;\;i=j=0\end{array}\right.\,,$ (76)

where $x_{k}$’s are the $N+1$ zeros of $T_{N+1}(x)$ in Eq. (72), and
$i,\,j\leq N$. The interval $[-1,1]$ for $x$ could be extended to an arbitrary
one $[y_{\mathrm{min}},y_{\mathrm{max}}]$ for $y$ via the linear relation as
follows

$\displaystyle x$
$\displaystyle=\frac{2y-(y_{\mathrm{max}}+y_{\mathrm{min}})}{y_{\mathrm{max}}-y_{\mathrm{min}}}\,,$
(77)

and the generalized Chebyshev polynomials are defined by

$\displaystyle T_{n}^{[y_{\mathrm{min}},y_{\mathrm{max}}]}(y)$
$\displaystyle\equiv T_{n}\big{(}x(y)\big{)}\,.$ (78)

Therefore, the zeros in $y$ corresponding to Eq. (72) read

$\displaystyle y_{k}$
$\displaystyle=\frac{y_{\max}-y_{\min}}{2}\cos\left(\frac{\pi(k+\frac{1}{2})}{N+1}\right)+\frac{y_{\max}+y_{\min}}{2}\,,$
(79)

with $k=0,\,1,\,\cdots N$. Then, a function $f(y)$ with
$y\in[y_{\mathrm{min}},y_{\mathrm{max}}]$ can be approximated as

$\displaystyle f(y)$
$\displaystyle\approx\left[\sum_{i=1}^{N}c_{i}T_{i}^{[y_{\mathrm{min}},y_{\mathrm{max}}]}(y)\right]+\frac{1}{2}c_{0}\,,$
(80)

where the coefficients could be readily obtained by the use of the
orthogonality relation in Eq. (76), which yields

$\displaystyle c_{i}$
$\displaystyle=\frac{2}{N+1}\sum_{k=0}^{N}f(y_{k})T_{i}^{[y_{\mathrm{min}},y_{\mathrm{max}}]}(y_{k})\,,$
(81)

with $i=0,\,1,\,\cdots N$.

With the Chebyshev approximation of the function $f(y)$ in Eq. (80), it is
straightforward to obtain its derivative, viz.

$\displaystyle f^{\prime}(y)$
$\displaystyle\approx\sum_{i=1}^{N}c_{i}\frac{d}{dy}T_{i}^{[y_{\mathrm{min}},y_{\mathrm{max}}]}(y)$
$\displaystyle=\left[\sum_{i=1}^{N}d_{i}T_{i}^{[y_{\mathrm{min}},y_{\mathrm{max}}]}(y)\right]+\frac{1}{2}d_{0}\,,$
(82)

where the coefficients $d_{i}$’s can be deduced by the recursion relation,
that reads

$\displaystyle d_{N}$ $\displaystyle=0\,,\qquad
d_{N-1}=\frac{2}{y_{\mathrm{max}}-y_{\mathrm{min}}}2Nc_{N}\,,$ $\displaystyle
d_{i-1}$
$\displaystyle=d_{i+1}+\frac{2}{y_{\mathrm{max}}-y_{\mathrm{min}}}2ic_{i}\quad(i=N-1,\cdots,1)\,.$
(83)

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