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# Estimates for truncated area functionals on the Bloch space

Iason Efraimidis  Departamento de Matemáticas, Universidad Autónoma de Madrid,
28049 Madrid, Spain<EMAIL_ADDRESS>,  Alejandro Mas  Departamento de
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071
Málaga, Spain<EMAIL_ADDRESS>and  Dragan Vukotić  Departamento de
Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
<EMAIL_ADDRESS>
(Date: 08 December, 2022.)

###### Abstract.

Recently, Kayumov [4] obtained a sharp estimate for the $n$-th truncated area
functional for normalized functions in the Bloch space for $n\leq 5$ and then,
together with Wirths [5], extended the result for $n=6$. We prove that for the
functions with non-negative Taylor coefficients, the same sharp estimate is
valid for all $n$. For arbitrary functions, we obtain an estimate that is
asymptotically of the same order but slightly larger (roughly by a factor of
$4/e$). We also consider related weighted estimates for functionals involving
the powers $n^{t}$, $t>0$, and show that the exponent $t=1$ represents the
critical case for the expected sharp estimate.

###### Key words and phrases:

Bloch space, coefficient problems

###### 2010 Mathematics Subject Classification:

30H30, 30C50

All authors are partially supported by PID2019-106870GB-I00 from MICINN,
Spain. The first author is supported by a María Zambrano contract, reference
number CA3/RSUE/2021-00386, from UAM and Ministerio de Universidades, Spain
(Plan de Recuperación, Transformación y Resiliencia).

##  1\. Introduction

Let ${\mathbb{D}}$ denote the unit disc in the complex plane. The Bloch space
${\mathcal{B}}$ (see [1]) is the Banach space of all analytic functions in
${\mathbb{D}}$ with the finite Bloch norm:

$\|f\|_{\mathcal{B}}=|f(0)|+\sup_{z\in{\mathbb{D}}}(1-|z|^{2})|f^{\prime}(z)|\,.$

For a function in ${\mathcal{B}}$ that vanishes at the origin, written as a
power series $f(z)=\sum_{n=1}^{\infty}b_{n}z^{n}$, the following sharp
estimate for the $n$-th Taylor coefficient is known [8]:

$B_{n}=\sup\\{|b_{n}|\,\colon\,\|f\|_{\mathcal{B}}\leq
1\\}=\frac{n+1}{2n}\left(\frac{n+1}{n-1}\right)^{(n-1)/2}\,,\quad n\geq 2\,.$

In the case $n=1$ this should be interpreted as $B_{1}=1$ (and is easily
proved directly). It can be checked by elementary means that the sequence
$(B_{n})_{n}$ is strictly increasing.

The Dirichlet space ${\mathcal{D}}$ is the set of all analytic functions in
the unit disc ${\mathbb{D}}$ with the finite area integral

$\int_{\mathbb{D}}|f^{\prime}(z)|^{2}dA(z)=\sum_{n=1}^{\infty}n|b_{n}|^{2}\,.$

It is easy to see that ${\mathcal{D}}\subset{\mathcal{B}}$ but not the other
way around. Thus, for a function $f\in{\mathcal{B}}$, the area functional
$\sum_{k=1}^{\infty}k|b_{k}|^{2}$ need not be finite. However, any truncated
area functional is bounded:

$\Phi_{n}(f)=\sum_{k=1}^{n}k|b_{k}|^{2}\leq C_{n}\|f\|_{\mathcal{B}}^{2}\,,$

where the constant $C_{n}$ depends only on $n$ but not on $f$. This has led
Kayumov [4] to show that, for small values of $n$, the above truncated area
functional is dominated by the supremum of its last term only:

(1) $\sum_{k=1}^{n}k|b_{k}|^{2}\leq nB_{n}^{2}\,,\qquad 2\leq n\leq 5\,,$

for all $\|f\|_{\mathcal{B}}\leq 1$ with $f(0)=0$, with equality if and only
if $f$ is an appropriate constant multiple of $z^{n}$. Kayumov and Wirths [5]
then showed that the estimate also holds for $n=6$. Related extremal problems
have also been considered recently in other papers (cf. [3] and [7]).

A standard argument involving normal families can be used to show that, for
each fixed $n\geq 1$, there exists an extremal function for the problem

(2)
$M_{n}=\sup\left\\{\Phi_{n}(f)=\sum_{k=1}^{n}k|b_{k}|^{2}\,\colon\,\|f\|_{\mathcal{B}}\leq
1,f(z)=\sum_{k=1}^{\infty}b_{k}z^{k}\right\\}\,.$

Since $\|B_{n}z^{n}\|_{\mathcal{B}}=1$, it is clear that $M_{n}\geq
nB_{n}^{2}$. A natural question is whether estimate (1) extends for arbitrary
$n\geq 2$. The present note is devoted to this question.

We first display some evidence towards the correctness of the expected bound
(1). Specifically, as the main result of Section 2, we prove this inequality
for all $n$ for the functions with non-negative coefficients. This is the
content of Theorem 2.

Next, in Section 3 we consider a more general weighted functional
$\Phi^{t}_{n}(f)=\sum_{k=1}^{n}k^{t}|b_{k}|^{2}$ for $t\geq 0$ and the
corresponding question as to whether $\Phi^{t}_{n}(f)\leq n^{t}B_{n}^{2}$ for
all functions $f$ with $\|f\|_{\mathcal{B}}\leq 1$ and
$f(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$. It is easy to see that this inequality
holds for all $t\geq 2$. Other results obtained here imply that it is true for
all $t\geq 1$ for the functions with non-negative Taylor coefficients.
However, we show that, when $t<1$, this estimate fails for all sufficiently
large $n$, even for functions with non-negative coefficients (see Theorem 3).
Thus, the exponent $t=1$ considered by Kayumov and Wirths is the critical
value, which provides further motivation for studying generalizations of their
results.

In Section 4 of the paper we observe that
$\sum_{k=1}^{n}k|b_{k}|^{2}\leq\frac{n^{n}}{(n-1)^{n-1}}\leq\frac{4}{e}nB_{n}^{2}$,
for all $n\geq 2$ (Proposition 4). Although larger, this bound is
asymptotically of the same order as $nB_{n}^{2}$. We also note that a typical
Marty-type relation for the coefficients of extremal functions shows that at
least one of the coefficients $b_{n}$ and $b_{n+1}$ must be zero, as expected.
However, obtaining more detailed information from a number of possible
variational methods has proved surprisingly difficult for us. Even proving
that for any extremal function we must have $b_{n}\neq 0$ has eluded us so
far.

In the final Subsection 4.3 we include further discussion and remarks, showing
that our Theorem 2 does not imply directly the desired estimate in the general
case. We include a relevant example in this respect.

##  2\. Functions with non-negative coefficients

In this section, we will prove that the bound obtained in [4, 5] is correct
for all $n$ for the functions with non-negative Taylor coefficients.

###  2.1. On extremal functions in the general setting

The following observations will be useful. For any $f\in{\mathcal{B}}$ and any
constant $c$ we have $\Phi_{n}(cf)=|c|^{2}\Phi_{n}(f)$. Thus, if
$\|f\|_{\mathcal{B}}<1$, since the function $f/\|f\|_{\mathcal{B}}$ has unit
norm and $\Phi_{n}(f/\|f\|_{\mathcal{B}})>\Phi_{n}(f)$, it is clear that $f$
cannot be an extremal function. In other words, if $F$ is an extremal function
for (2), we must have $\|F\|_{\mathcal{B}}=1$.

Extremal functions are clearly not unique since for a function
$f\in{\mathcal{B}}$ its rotation $f_{\lambda}$, given by
$f_{\lambda}(z)=f(\lambda z)$, $|\lambda|=1$, has the same norm and
$\Phi_{n}(f_{\lambda})=\Phi_{n}(f)$. Another issue is whether such a function
is unique up to a rotation, which is what is expected by the findings from [4]
and [5] for $2\leq n\leq 6$.

###  2.2. Functions with non-negative coefficients

We now turn to the restricted extremal problem for functions with non-negative
coefficients:

(3)
$\sup\left\\{\Phi_{n}(f)=\sum_{k=1}^{n}k|b_{k}|^{2}\,\colon\,\|f\|_{\mathcal{B}}\leq
1,\,f(z)=\sum_{k=1}^{\infty}b_{k}z^{k},\,b_{j}\geq 0\mathrm{\ for\ all\ }j\geq
1\right\\}\,.$

A few simple observations are in order. If $\|f\|_{\mathcal{B}}\leq 1$ and
$f(z)=\sum_{j\geq 1}b_{j}z^{j}$ with $b_{j}\geq 0$ for all $j\geq 1$, it is
obvious that $|f^{\prime}(z)|\leq f^{\prime}(|z|)$, hence

$\|f\|_{\mathcal{B}}=\sup_{0\leq r<1}(1-r^{2})|f^{\prime}(r)|\,.$

Next, by the same argument as in Subsection 2.1, all extremal functions for
the restricted problem (3) have this property as well. If $f$ is such a
function, it is clear that the truncated polynomial
$p_{n}(z)=\sum_{j=1}^{n}b_{j}z^{j}$ (the Taylor polynomial of $f$ of order
$n$) has the property that

$\Phi_{n}(p_{n})=\Phi_{n}(f)\quad\mbox{and}\quad\|p_{n}\|_{\mathcal{B}}\leq\|f\|_{\mathcal{B}}=1\,.$

Moreover, if there exists at least one $k>n$ such that $b_{k}>0$, then clearly
$\|p_{n}\|_{\mathcal{B}}<1$. But then $f$ cannot be extremal for $\Phi_{n}$
since $p_{n}/\|p_{n}\|_{\mathcal{B}}$ and $f$ both have norm one and

$\Phi_{n}\left(\frac{p_{n}}{\|p_{n}\|_{\mathcal{B}}}\right)=\frac{1}{\|p_{n}\|_{\mathcal{B}}^{2}}\Phi_{n}(f)>\Phi_{n}(f)\,.$

Thus, the only candidates for extremal functions for (3) are polynomials of
degree at most $n$. Note, however, that this consideration is not strictly
necessary in the proof of the following lemma.

######  Lemma 1.

Let $f(z)=\sum_{j\geq 1}b_{j}z^{j}$ be an extremal function for the restricted
extremal problem (3), with $b_{j}\geq 0$ for all $j\geq 1$. If $1\leq k<m\leq
n$ and $b_{k}>0$, then

$b_{m}\leq\frac{m-k}{2m}b_{k}\,.$

###### Proof.

Let

$g(z)=f(z)-b_{k}z^{k}+\frac{k}{m}b_{k}z^{m}\,.$

It is again an analytic function with non-negative coefficients. Since

$g^{\prime}(r)=f^{\prime}(r)+kb_{k}(r^{m-1}-r^{k-1})\leq f^{\prime}(r)$

for $0\leq r<1$, it follows that $\|g\|_{\mathcal{B}}\leq\|f\|_{\mathcal{B}}$.
Hence $g$ is in competition with $f$ and therefore

$\Phi_{n}(g)=2kb_{k}b_{m}+\frac{k^{2}}{m}b_{k}^{2}-kb_{k}^{2}+\Phi_{n}(f)\leq\Phi_{n}(f).$

Since $b_{k}>0$ by assumption, we obtain $2b_{m}+\frac{k}{m}b_{k}\leq b_{k}$,
and the desired conclusion follows. ∎

We are now ready to prove the main result of this section.

######  Theorem 2.

If $\|f\|_{\mathcal{B}}\leq 1$ and $f(z)=\sum_{j\geq 1}b_{j}z^{j}$ with
$b_{j}\geq 0$ for all $j\geq 1$, then

$\Phi_{n}(f)=\sum_{k=1}^{n}k|b_{k}|^{2}\leq nB_{n}^{2},\quad\mathrm{for\ all\
}n\geq 2\,.$

Moreover, equality is only achieved for $f(z)=B_{n}z^{n}$.

###### Proof.

First of all, we observe that it suffices to prove the statement only for
extremal functions. We proceed by induction on $n$. As is well known, the
statement is true for $n=2$. Let $n\geq 3$ and assume the inequality is true
for all $k<n$. We will now show that it also holds for $n$.

If $f$ is extremal for (3), as observed before, $f$ is a polynomial of degree
at most $n$. Suppose that $b_{j}>0$ for some $j$ with $1\leq j<n$, and let
$k=\max\\{j\,:\,1\leq j<n\mathrm{\ and\ }b_{j}>0\\}$. Recalling that the
sequence $(B_{n})_{n}$ is increasing, by our inductive assumption, we clearly
have

$nB_{k}^{2}\leq
nB_{n}^{2}\leq\Phi_{n}(f)=\sum_{j=1}^{k}jb_{j}^{2}+nb_{n}^{2}=\Phi_{k}(f)+nb_{n}^{2}\leq
kB_{k}^{2}+nb_{n}^{2}\,.$

It follows that $nb_{n}^{2}\geq(n-k)B_{k}^{2}$ and therefore

$b_{n}\geq\sqrt{\frac{n-k}{n}}B_{k}>\frac{1}{2}\frac{n-k}{n}B_{k}\geq\frac{1}{2}\frac{n-k}{n}b_{k}\geq
b_{n}$

in view of Lemma 1 (with $m=n$), which is clearly absurd. Thus, it follows
that $b_{j}=0$ for all $j$ with $1\leq j<n$. As was already observed, only
functions of norm one can be extremal and it follows immediately that
$B_{n}z^{n}$ is the only possible extremal function in this case. ∎

It should be noted that Theorem 2 is also true for functions $f$ with
$f(z)=\sum_{k\geq 1}b_{k}z^{k}$ with $b_{k}=|b_{k}|e^{i(a+kb)}$, for some $a$,
$b\in\mathbb{R}$. This is clear since, given a function $f$ of this kind, the
function $e^{-ia}f\left(e^{-ib}z\right)$ also vanishes at the origin and has
the same norm as $f$ and non-negative coefficients.

##  3\. On more general weighted estimates

As was noted in [4, p. 467], it is quite simple to show that for any function
with $\|f\|_{\mathcal{B}}\leq 1$ and $f(z)=\sum_{k\geq 1}b_{k}z^{k}$ the sharp
estimate

$\sum_{k=1}^{n}k^{2}|b_{k}|^{2}\leq n^{2}B_{n}^{2}$

holds for all $n\geq 1$. This suggests the idea of studying the generalized
functional

$\Phi^{t}_{n}(f)=\sum_{k=1}^{n}k^{t}|b_{k}|^{2}$

with $t\geq 0$. A natural question is whether $\Phi^{t}_{n}(f)\leq
n^{t}B_{n}^{2}$, under the same hypotheses as above, and whether the
inequality is sharp. We already know that this is true when $t=2$ and the
problem studied previously constitutes the case $t=1$.

It is easy to see that if the above weighted estimate is valid for all $n$ for
the functional $\Phi^{t}_{n}$, then the analogous estimate is also satisfied
for the functional $\Phi^{s}_{n}$, whenever $s\geq t$. Indeed, if
$\|f\|_{\mathcal{B}}\leq 1$, $f(z)=\sum_{k\geq 1}b_{k}z^{k}$, the inequality

$\Phi^{t}_{n}(f)=\sum_{k=1}^{n}k^{t}|b_{k}|^{2}\leq n^{t}B_{n}^{2}$

is satisfied and $s\geq t$, then clearly

$\Phi^{s}_{n}(f)=\sum_{k=1}^{n}k^{s}|b_{k}|^{2}\leq
n^{s-t}\sum_{k=1}^{n}k^{t}|b_{k}|^{2}\leq
n^{s-t}n^{t}B_{n}^{2}=n^{s}B_{n}^{2}.$

Thus, for all functions $f$ such that $\|f\|_{\mathcal{B}}\leq 1$ and
$f(z)=\sum_{k\geq 1}b_{k}z^{k}$, it follows that

(4) $\sum_{k=1}^{n}k^{t}|b_{k}|^{2}\leq n^{t}B_{n}^{2},$

whenever $t\geq 2$. Additionally, if $f$ has non-negative Taylor coefficients,
inequality (4) actually holds whenever $t\geq 1$ in view of Theorem 2.

The following result will show that for all $t$ with $t<1$ the above estimate
(4) fails in a strong way. In fact, it cannot even be true for the functions
with non-negative Taylor coefficients when $n$ is sufficiently large. This
shows that the exponent $t=1$ (the basic case studied here and in [4, 5]) is
really critical.

######  Theorem 3.

Given $t<1$, there exist a positive integer $N$ that depends only on $t$ and
such that for all $n\geq N$ there exists a function $f_{n}\in\mathcal{B}$ with
non-negative Taylor coefficients, $f_{n}(0)=0$, $\|f_{n}\|_{\mathcal{B}}\leq
1$, and with the property that

$\Phi^{t}_{n}(f_{n})>n^{t}B_{n}^{2}\,.$

###### Proof.

Let $\varepsilon>0$ such that $t+\varepsilon<1$, for example,
$\varepsilon=(1-t)/2$. Choose an integer $N\geq(2e^{2})^{2/(1-t)}$ and, for
$n\geq N$, consider the functions

$f_{n}(z)=z+b_{n}z^{n},\quad
b_{n}=\sqrt{B_{n}^{2}-\frac{1}{n^{t+\varepsilon}}}.$

Clearly,

$\Phi^{t}_{n}(f_{n})=1+n^{t}\left(B_{n}^{2}-\frac{1}{n^{t+\varepsilon}}\right)>n^{t}B_{n}^{2},$

hence it is only left to check that $\|f_{n}\|_{\mathcal{B}}\leq 1$ for all
$n\geq N$. To this end, note that

$\|f_{n}\|_{\mathcal{B}}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|1+nb_{n}z^{n-1}|=\sup_{0\leq
r<1}(1-r^{2})\left(1+nb_{n}r^{n-1}\right).$

Thus, in order to finish the proof, we need only show that for $n\geq N$ and
for all $r\in(0,1)$ the inequality $(1-r^{2})\left(1+nb_{n}r^{n-1}\right)\leq
1$ holds. The last statement is easily seen to be equivalent to

$nb_{n}r^{n-3}(1-r^{2})\leq 1,\quad\mathrm{for\ all\
}r\in(0,1)\quad\mathrm{and\ }n\geq N\,.$

If we define $h(r)=nb_{n}r^{n-3}(1-r^{2})$, it is clear that

$\max_{0\leq r\leq
1}h(r)=\max\left\\{h(0),h\left(\sqrt{\frac{n-3}{n-1}}\right),h(1)\right\\}=nb_{n}\left(\frac{n-3}{n-1}\right)^{\frac{n-3}{2}}\frac{2}{n-1}.$

It is, thus, only left to show that
$nb_{n}\left(\frac{n-3}{n-1}\right)^{\frac{n-3}{2}}\frac{2}{n-1}\leq 1$ for
$n\geq N$. The following chain of equivalences is clear:

$\displaystyle
nb_{n}\left(\frac{n-3}{n-1}\right)^{\frac{n-3}{2}}\frac{2}{n-1}\leq 1$
$\displaystyle\Longleftrightarrow
b_{n}^{2}\leq\frac{(n-1)^{n-1}}{4n^{2}(n-3)^{n-3}}\Longleftrightarrow
B_{n}^{2}-\frac{1}{n^{t+\varepsilon}}\leq\frac{(n-1)^{n-1}}{4n^{2}(n-3)^{n-3}}$
$\displaystyle\Longleftrightarrow\frac{(n+1)^{n+1}}{4n^{2}(n-1)^{n-1}}-\frac{(n-1)^{n-1}}{4n^{2}(n-3)^{n-3}}\leq\frac{1}{n^{t+\varepsilon}}$
$\displaystyle\Longleftrightarrow\frac{(n+1)^{2}\left(1+\frac{2}{n-1}\right)^{n-1}-(n-1)^{2}\left(1+\frac{2}{n-3}\right)^{n-3}}{n}\leq
4n^{1-t-\varepsilon}\,.$

Obviously, the right-hand side in the last inequality tends to infinity when
$n$ tends to infinity. On the other hand, using the classical inequalities
$\left(1+\frac{1}{x}\right)^{x}<e<\left(1+\frac{1}{x}\right)^{x+1}$ for all
$x>0$, we see that the left-hand side in the last inequality in the chain
above satisfies

$\displaystyle\frac{(n+1)^{2}\left(1+\frac{2}{n-1}\right)^{n-1}-(n-1)^{2}\left(1+\frac{2}{n-3}\right)^{n-3}}{n}$
$\displaystyle\leq
e^{2}\frac{(n+1)^{2}-(n-1)^{2}\left(1+\frac{2}{n-3}\right)^{-2}}{n}$
$\displaystyle=e^{2}\frac{(n+1)^{2}-(n-3)^{2}}{n}$
$\displaystyle=e^{2}\frac{8n-8}{n}$ $\displaystyle\leq 8e^{2}\,.$

Since $N\geq(2e^{2})^{2/(1-t)}$ and $\varepsilon=(1-t)/2$ (by our choices), it
is clear that $4n^{1-t-\varepsilon}\geq 8e^{2}$ and the desired conclusion
follows. ∎

##  4\. Discussion of the general case

###  4.1. An estimate of correct asymptotic order

If the conjectured bound on the truncated functional is true for all $n$, the
following estimate cannot be sharp. However, it may still be useful so we
mention it here.

######  Proposition 4.

If $\|f\|_{\mathcal{B}}\leq 1$ and $f(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$, then

(5)
$\sum_{k=1}^{n}k|b_{k}|^{2}\leq\frac{n^{n}}{(n-1)^{n-1}}\,,\quad\mathrm{for\
all\ }n\geq 2\,.$

This estimate is of the same asymptotic order as the conjectured estimate
since

$\frac{32}{27}nB_{n}^{2}\leq\frac{n^{n}}{(n-1)^{n-1}}\leq\frac{4}{e}nB_{n}^{2}\,,\quad\mathrm{for\
all\ }n\geq 2\,.$

###### Proof.

Starting with the standard estimate based on Parseval’s equality [1, 4]:

(6)
$\sum_{k=1}^{n}k^{2}|b_{k}|^{2}\rho^{k-1}\leq\frac{1}{(1-\rho)^{2}}\,,\qquad
0\leq\rho<1\,,$

and integrating with respect to $\rho$ from $0$ to $x$, we obtain:

$\sum_{k=1}^{n}k|b_{k}|^{2}x^{k-1}\leq\frac{1}{1-x}\,,\qquad 0\leq x<1\,.$

Since $x^{k-1}\geq x^{n-1}$ whenever $1\leq k\leq n$, it follows that

$\sum_{k=1}^{n}k|b_{k}|^{2}\leq\frac{1}{x^{n-1}(1-x)}$

for all $x\in(0,1)$. The right-hand side attains its minimum value

(7) $\frac{n^{n}}{(n-1)^{n-1}}$

at the point $x=(n-1)/n$. This yields the desired estimate (5).

As for the order of the estimate obtained, this is easily seen by elementary
calculus since the sequence

$\frac{n^{n}}{(n-1)^{n-1}}\bigg{/}(nB_{n}^{2})=\frac{4}{\left(\frac{n+1}{n}\right)^{n+1}}$

is increasing and convergent to $\frac{4}{e}\simeq 1.4715$. ∎

###  4.2. Remarks on the $n$-th coefficient of an extremal function

It is desirable to obtain further partial information on the Taylor
coefficients of extremal functions for the functional $\Phi_{n}$. For example,
one would like to show that we must always have $b_{n}\neq 0$. However, this
seemingly simple point has eluded us so far and we have not been able to find
a satisfactory proof.

It is, however, relatively simple to show that for every extremal function
either $b_{n+1}=0$ or $b_{n}=0$ by working out an analogue of the Marty
variation as in the theory of univalent functions (cf. [2, pp. 59–60]). This
can be proved by using conformal invariance; more specifically, if $F$ is
extremal, the function

$F_{\lambda}(z)=F\left(\frac{z+\lambda}{1+\overline{\lambda}z}\right)-F(\lambda)=\sum_{k=1}^{\infty}B_{k}(\lambda)z^{k}$

has Bloch norm one and satisfies $F_{\lambda}(z)=0$, hence it competes with
$f$: $\sum_{k=1}^{n}k|B_{k}(\lambda)|^{2}\leq\sum_{k=1}^{n}k|b_{k}|^{2}$. By
considering the asymptotic expansion:

$|B_{k}(\lambda)|^{2}=|b_{k}|^{2}+2\mathrm{Re}\,\left\\{\left((k+1)b_{k+1}\overline{b_{k}}-(k-1)\overline{b_{k-1}}b_{k}\right)\lambda\right\\}+O(|\lambda|^{2})\,,\qquad
k\geq 1\,,$

for $\lambda$ close to zero, where $b_{0}=0$, after some algebra, this leads
to

$\mathrm{Re}\,\left\\{\left(n(n+1)b_{n+1}\overline{b_{n}}\right)\lambda\right\\}+O(|\lambda|^{2})\leq
0$

and, upon dividing by $|\lambda|$ and letting $|\lambda|\to 0$, since the
argument of $\lambda$ can be arbitrary, we conclude that
$n(n+1)b_{n+1}\overline{b_{n}}=0$, which proves the statement.

As a way of restating the desired property of the $n$-th coefficients of
extremal functions for problem (2), it is not difficult to show that the
sequence $(M_{n})_{n}$ of numbers defined by (2) is strictly increasing if and
only if for each extremal function $F(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$ we
have $b_{n}\neq 0$.

###  4.3. Concluding remarks

We end this note by pointing out some limitations of the method employed in
Subsection 2.2.

Regarding Theorem 2, one may ask whether in the general case we may restrict
our attention to polynomials only. However, knowing only that $\Phi_{n}(p)\leq
nB_{n}^{2}$ for all polynomials $p$ of unit norm does not imply in a direct
way the same inequality for general functions of norm one. This can be seen as
follows. While for a function $f$ in the unit ball and its $n$-th Taylor
polynomial $p_{n}$ we have $\Phi_{n}(f)=\Phi_{n}(p_{n})$, it can be seen that
there are functions $f$ in the unit ball for which
$\|p_{n}\|_{\mathcal{B}}>1$, hence for the normalized function
$p_{n}/\|p_{n}\|_{\mathcal{B}}$ we obtain
$\Phi_{n}(p_{n}/\|p_{n}\|_{\mathcal{B}})=\Phi_{n}(f)/\|p_{n}\|_{\mathcal{B}}^{2}<\Phi_{n}(f)$,
showing that the maximum of the functional over the normalized polynomials
could in theory be strictly smaller than the global maximum for problem (2).
Of course, showing that the conjectured bound is correct would amount to
proving that such functions $f$ cannot be extremal; hence, some additional
work is required.

Another natural question is whether from the estimate obtained only for
functions with non-negative coefficients in Theorem 2 we can immediately
deduce the result for arbitrary functions. However, this does not seem so
simple since, employing the usual terminology, the unit ball of
${\mathcal{B}}$ is not solid. In other words, if
$f(z)=\sum_{k=1}^{\infty}b_{k}z^{k}$ is in the ball of ${\mathcal{B}}$, the
function $F(z)=\sum_{k=1}^{\infty}|b_{k}|z^{k}$ need not belong to the ball of
${\mathcal{B}}$. It is clear that $F$ is also analytic in the disc and
$\|f\|_{\mathcal{B}}\leq\|F\|_{\mathcal{B}}$ by crude estimates based on the
triangle inequality, but there are examples showing that this inequality can
actually be strict. More generally speaking, a change of signs in some of the
coefficients of a function could enlarge its norm. Thus, if $f$ and $F$ are as
above (hence $F$ has non-negative coefficients) and also
$1=\|F\|_{\mathcal{B}}>\|f\|_{\mathcal{B}}$, then $f/\|f\|_{\mathcal{B}}$ is
also of unit norm and

$\Phi_{n}\left(\frac{f}{\|f\|_{\mathcal{B}}}\right)=\frac{\Phi_{n}(F)}{\|f\|_{\mathcal{B}}^{2}}>\Phi_{n}(F)\,,$

so $F$ cannot be extremal. Hence, a proof of the desired estimate in the
general case would show in particular that functions $F$ with the above
property cannot be extremal.

One single example can be used to illustrate both phenomena (after the
appropriate normalization).

######  Example 5.

Let $n\geq 2$ and let $\varepsilon$ be sufficiently small, say
$0<\varepsilon\leq 1/5$. Consider the function
$f(z)=z+B_{n}z^{n}-\frac{\varepsilon z^{2n-1}}{2n-1}$. The associated
functions discussed above are $F(z)=z+B_{n}z^{n}+\frac{\varepsilon
z^{2n-1}}{2n-1}$ and $p_{n}(z)=z+B_{n}z^{n}$ and we have the inequality

(8) $\left\lVert F\right\rVert_{\mathcal{B}}>\left\lVert
p_{n}\right\rVert_{\mathcal{B}}>\left\lVert f\right\rVert_{\mathcal{B}}\,.$

To check the first inequality in (8), recall from Subsection 2.2 that for
every function with positive coefficients in ${\mathcal{B}}$ the supremum
defining its norm is attained on the radius $[0,1)$. In particular, this shows
that

$\left\lVert F\right\rVert_{\mathcal{B}}=\sup_{0\leq
r<1}(1-r^{2})(1+nB_{n}r^{n-1}+\varepsilon r^{2n-2})>\sup_{0\leq
r<1}(1-r^{2})(1+nB_{n}r^{n-1})=\left\lVert p_{n}\right\rVert_{\mathcal{B}}\,.$

The above inequality holds because we have the strict inequality

$(1-r^{2})(1+nB_{n}r^{n-1}+\varepsilon
r^{2n-2})>(1-r^{2})(1+nB_{n}r^{n-1})\,,\quad\mathrm{for}\ 0<r<1\,,$

equality holds for $r=0$ and at $r=1$, and also because the smaller quantity
is strictly bigger than $1$ (the value of both sides at $r=0$) for at least
one $r\in(0,1)$. This last observation follows in view of the fact that
$\|B_{n}z^{n}\|_{\mathcal{B}}=1$ and this norm is attained at the value
$r_{n}=\sqrt{\frac{n-1}{n+1}}\in(0,1)$, hence

$\left\lVert p_{n}\right\rVert_{\mathcal{B}}=\sup_{0\leq
r<1}(1-r^{2})(1+nB_{n}r^{n-1})\geq(1-r_{n}^{2})(1+nB_{n}r_{n}^{n-1})>(1-r_{n}^{2})nB_{n}r_{n}^{n-1}=1\,.$

The second inequality in (8) requires more work. In order to determine
$\left\lVert z+B_{n}z^{n}-\frac{\varepsilon
z^{2n-1}}{2n-1}\right\rVert_{\mathcal{B}}$, we need to obtain an upper
estimate on $|1+nB_{n}z^{n-1}-\varepsilon z^{2n-2}|$. Writing
$z^{n-1}=Re^{it}$ and then $R=r^{n-1}$, $\cos t=x$, one easily computes

$\displaystyle|1+nB_{n}z^{n-1}-\varepsilon z^{2n-2}|^{2}$ $\displaystyle=$
$\displaystyle|1+nB_{n}Re^{it}-\varepsilon R^{2}e^{2it}|^{2}$ $\displaystyle=$
$\displaystyle
1+n^{2}B_{n}^{2}R^{2}+\varepsilon^{2}R^{4}+2nB_{n}Rx-2nB_{n}\varepsilon
R^{3}x-2\varepsilon R^{2}(2x^{2}-1)$ $\displaystyle=$
$\displaystyle-4\varepsilon R^{2}x^{2}+2nB_{n}R(1-\varepsilon
R^{2})x+1+n^{2}B_{n}^{2}R^{2}+2\varepsilon R^{2}+\varepsilon^{2}R^{4}$
$\displaystyle=$ $\displaystyle u(x)\,.$

Then $u^{\prime}(x)=2R\left(nB_{n}(1-\varepsilon R^{2})-4\varepsilon
Rx\right)\geq 0$ in view of the obvious inequalities $nB_{n}(1-\varepsilon
R^{2})\geq 1-\varepsilon R^{2}\geq 1-\varepsilon\geq 4\varepsilon$. Thus, $u$
attains its maximum at $x=1$ and the maximum value is

$u(1)=2nB_{n}R(1-\varepsilon R^{2})+1+n^{2}B_{n}^{2}R^{2}-2\varepsilon
R^{2}+\varepsilon^{2}R^{4}=(1+nB_{n}R)^{2}+\varepsilon R^{2}(\varepsilon
R^{2}-2-2nB_{n}R)\,.$

The second summand in the last expression is negative, hence $u(x)\leq
u(1)<(1+nB_{n}R)^{2}$. This yields

$\left\lVert z+B_{n}z^{n}-\frac{\varepsilon
z^{2n-1}}{2n-1}\right\rVert_{\mathcal{B}}\leq\sup_{0\leq
r<1}(1-r^{2})(1+nB_{n}r^{n-1})=\left\lVert p_{n}\right\rVert_{\mathcal{B}}\,.$

Again, the inequality is easily seen to be strict since $\|p_{n}\|>1$, hence
this norm is attained for some $r\in(0,1)$, while the inequality in
$u(x)<(1+nB_{n}R)^{2}=(1+nB_{n}r^{n-1})^{2}$ is strict for $r\in(0,1)$.

## References

  * [1] J.M. Anderson, J. Clunie, Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–-37.
  * [2] P.L. Duren, Univalent Functions, Springer-Verlag, New York 1983.
  * [3] O.V. Ivriĭ, I.R. Kayumov, Makarov’s principle for the unit ball in Bloch space, Mat. Sb. 208 (2017), no. 3, 96–-110; translation in Sb. Math. 208 (2017), no. 3-4, 399–-412.
  * [4] I.R. Kayumov, A note on an area-type functional of Bloch functions, Lobachevskii J. Math. 38 (2017), no. 3, 466–-468.
  * [5] I.R. Kayumov, K.-J. Wirths, Coefficient inequalities for Bloch functions, Lobachevskii J. Math. 40 (2019), No. 9, pp. 1319–1323.
  * [6] I.R. Kayumov, K.-J. Wirths, On the sum of squares of the coefficients of Bloch functions, Monatsh. Math. 190 (2019), no. 1, 123–-135.
  * [7] I.R. Kayumov, K.-J. Wirths, Coefficients problems for Bloch functions, Anal. Math. Physics 9 (2019), 1069–-1085.
  * [8] K.-J. Wirths, Über holomorphe Funktionen, die einer Wachstumsbeschränkung unterliegen (German), Arch. Math. (Basel) 30 (1978), no. 6, 606–-612.


# Effects of Symmetry Energy on the Equation of State
for Hybrid Neutron Stars

Parada T. P. Hutauruk  Department of Physics, Pukyong National University
(PKNU), Busan 48513, Korea  Department of Physics Education, Daegu University,
Gyeongsan 38453, Korea  Hana Gil  Department of Physics Education, Daegu
University, Gyeongsan 38453, Korea  Center for Extreme Nuclear Matters, Korea
University, Seoul 02841, Korea  Seung-il Nam  Department of Physics, Pukyong
National University (PKNU), Busan 48513, Korea  Center for Extreme Nuclear
Matters, Korea University, Seoul 02841, Korea  Asia Pacific Center for
Theoretical Physics (APCTP), Pohang 37673, Korea  Chang Ho Hyun  Department of
Physics Education, Daegu University, Gyeongsan 38453, Korea  Center for
Extreme Nuclear Matters, Korea University, Seoul 02841, Korea

###### Abstract

In this paper, the implications of the symmetry energy on the hadron and quark
phase transitions in the compact star, including the properties of the
possible configurations of the quark-hadron hybrid stars, are investigated in
the frameworks of the energy-density functional (EDF) models and the flavor
SU(2) Nambu–Jona-Lasinio (NJL) model with the help of the Schwinger’s
covariant proper-time regularization (PTR) scheme. In this theoretical setup,
the equations of states (EoSs) of hadronic matter for various values of
symmetry energies obtained from the EDF models are employed to describe the
hadronic matter, and the flavor SU(2) NJL model with various repulsive-vector
interaction strengths are used to describe the quark matter. We then observe
the obtained EoS in the mass-radius properties of the hybrid star
configurations for various vector interactions and nuclear symmetry energies
by solving the Tolman-Oppenheimer-Volkoff equation. We obtain that the
critical density at which the phase transition occurs varies over the density
(3.6–6.7)$\rho_{0}$ depending on the symmetry energy and the strength of the
vector coupling $G_{v}$. The maximum mass of the neutron star (NS) is
susceptible to $G_{v}$. When there is no repulsive force, the NS maximum mass
is only about $1.5M_{\odot}$, but it becomes larger than $2.0M_{\odot}$ when
the vector coupling constant is about half of the attractive scalar coupling
constant. Surprisingly, the presence of the quark matter does not affect the
canonical mass of NS ($1.4M_{\odot}$), so observing the canonical mass of NSs
can provide unique constraints to the EoS of hadronic matter at high
densities.

##  I Introduction

A key question about the composition of matter at densities several times the
nuclear saturation density ($\rho_{0}$) is how it differs from the state that
constitutes the atomic nuclei. It was widely known that several exotic states
are candidates for the states of matter at ultra-high densities: The creation
of hyperons, the onset of Bose-Einstein condensates, and transformation to
quark matter. Using theories of subatomic physics, one can estimate in what
conditions the new state of matter will emerge. However, the existence of the
new state matter is not yet established, for instance, the appearance of the
hyperons and $\Delta$ baryon in the matter and how they interact with other
matter constituents are still open questions.

In a recent work of Ref. epja2022 , the in-medium interaction of the $\Lambda$
hyperon was investigated within non-relativistic nuclear density functional
theory. Parameters for the interaction of $\Lambda$ hyperon were determined to
reproduce the single-$\Lambda$ hypernuclear data, and the effect of the
density dependence of the symmetry energy was examined in more systematic
detail. It was shown that the density at which $\Lambda$ hyperons appear
depends strongly on the symmetry energy. However, the appearance density of
hyperons is always larger than $3\rho_{0}$, so the hyperon is not likely to
affect the properties of a canonical mass ($1.4M_{\odot}$) of NS because the
density at the center of the NS canonical mass is generally less than
$3\rho_{0}$. Kaon condensation was also considered in the relativistic and
non-relativistic models kcond1 ; kcond2 . Unless the interaction of the kaon
in dense nuclear matter is super-strongly attractive, the formation of the
kaon condensation is less probable.

Analogously, the matter composition in NSs naively can be illustrated as the
quarks in the baryons on a microscopic scale. In the bag model, baryons are
described as spherical bags in which quarks are assumed to be confined. In the
bag model, the charge radius of the proton is expected around 0.6 – 0.8 fm
(corresponding to the volume 0.9 – 2.1 fm3). The inverse of the number density
($\rho$) is the volume occupied by a nucleon. If $\rho_{0}=0.16\,{\rm
fm}^{-3}$, one nucleon occupies 6.25 fm3 at the saturation density. This
volume is larger than the volume of a nucleon, so nucleons are separated
spatially. At densities in the range of 3 – 6 $\rho_{0}$, the volume per
nucleon becomes equal to the volume of a nucleon, so the nucleons begin to
overlap each other. As the matter density evolves above the onset of
overlapping, nucleons overlap more, and it becomes vague to define the
confinement of a quark in a specific bag. Matter transforms to the phase of
deconfined quarks. A similar condition is expected to occur in NS.

Given the equation of state (EoS) of dense matter, one can calculate the bulk
properties of the NSs such as mass, radius, and particle distribution in the
core by solving the Tolman-Oppenheimer-Volkoff (TOV) equations. With the
nuclear models that are well constrained by the nuclear properties and nuclear
matter properties from the heavy ion collision experiments, ab initio
calculations, and modern neutron star observation data
Papakonstantinou:2016zpe ; kids-nuclei1 ; kids-nuclei2 ; kids-sym ; kids-k0 ;
kids-ksym , the density at the center of a canonical mass of NS is obtained to
be about or less than $3\rho_{0}$, and that of the heaviest stars ($\geq
2M_{\odot}$) is around $6\rho_{0}$. The degrees of freedom that constitute the
core of NSs might likely change from hadron to quarks in NSs with masses in
the range ($1.4-2.0$)$M_{\odot}$.

The main issue of the present work is to explore the uncertainty of the
critical density ($\rho_{c}$) at which the transition from the hadronic phase
to the deconfined quark matter begins and its impact on the properties of the
NS. For this purpose, for the EoS of hadronic matter (HM), we employ the KIDS
(Korea-IBS-Daegu-SKKU) energy density functional (EDF) in which the density
dependence of the symmetry energy is calibrated to the radius of
$1.4M_{\odot}$ mass NSs in the range $R_{1.4}=11.8-12.5$ km kids-k0 . The
KIDS0, KIDS-A, KIDS-B, KIDS-C, and KIDS-D models, as well as the standard
Skyrme SLy4 model used in the work, have distinctive density dependence of the
symmetry energy, so the results will provide the range of $\rho_{c}$ within
the uncertainty of the nuclear matter EoS. On the quark matter (QM) side, the
proper time NJL model is used for the EoS of the deconfined quark state. As
discussed above, massive NSs are likely to have a QM phase in the core, so we
investigate the range of parameters in the NJL model that are compatible with
the NS mass larger than $2M_{\odot}$. As a result, we can determine the ranges
of the parameters in the QM EoS. Combining the EoSs of the KIDS model with
those of the NJL model, we determine the range of $\rho_{c}$ from the
condition $P_{\rm HM}(\rho_{c})=P_{\rm QM}(\rho_{c})$ at $\rho_{\rm
HM}=\rho_{\rm QM}=\rho_{c}$, where $P_{\rm HM}$, $P_{\rm QM}$, $\rho_{\rm
HM}$, and $\rho_{\rm QM}$ represent the pressure in the hadronic phase,
pressure in quark phase, density in hadronic phase, and density in quark
phase, respectively. The results will shed light on (i) the effect of the
symmetry energy on the change of phase in the NS core, (ii) consistency of the
hybrid models with the NS properties determined from astronomical
observations, and (iii) uncertainty of the EoS in the hadronic and quark
phases.

This work is organized as follows. In Sec. II, models for the hadronic matter
and deconfined quark matter are briefly introduced. The hadronic matter is
described in the KIDS-EDF model, whereas the quark matter is described in the
flavor SU(2) Nambu–Jona-Lasinio model with the help of the Schwinger proper-
time regularization scheme, that is so-called NJLPTR model. We present and
discuss the numerical result of the current work in Sec. III, and we summarize
the work in Sec. IV.

##  II Formalism for the hadron and quark phases

###  II.1 Hadronic matter from the KIDS functional

In this section, we first present the description of HM in the KIDS-EDF model
Papakonstantinou:2016zpe ; Hutauruk:2022bii ; Hutauruk:2022bso . This model
was constructed based on the Fermi momentum expansion, which is relevant for
infinite nuclear matter (INM) and an NS. The energy per particle for
homogeneous hadronic matter in the powers of the cubic root of the density is
given by

$\displaystyle\mathcal{E}_{\rm{HM}}(\rho,\delta)$ $\displaystyle=$
$\displaystyle\mathcal{T}(\rho,\delta)+\sum_{j=0}^{3}(\alpha_{j}+\beta_{j}\delta^{2})\rho^{(1+j/3)},$
(1)

where $\mathcal{T}(\rho,\delta)$ is the kinetic energy.
$\delta=(\rho_{n}-\rho_{p})/\rho$ and $\rho=\rho_{n}+\rho_{p}$ are
respectively the isospin asymmetry and baryon density. The $\alpha_{j}$ and
$\beta_{j}$ are respectively the coefficients for the symmetric and
antisymmetric nuclear matter.

The symmetry energy $S(\rho)$ is straightforwardly determined through the
second derivative of the energy per particle with respect to $\delta$ at the
nuclear saturation density. It then gives

$\displaystyle\mathcal{E}_{\rm{HM}}(\rho,\delta)$ $\displaystyle=$
$\displaystyle\mathcal{E}(\rho,0)+S(\rho)\delta^{2}+\mathcal{O}(\delta^{4}),$
(2) $\displaystyle S(\rho)$ $\displaystyle=$
$\displaystyle\frac{\hbar}{6M}\left(\frac{3\pi^{2}}{2}\right)^{2/3}\rho^{2/3}+\sum_{j=0}^{3}\beta_{j}\rho^{(1+j/3)},$
(3)

where $M\equiv(M_{n}+M_{p})/2$ is the nucleon mass in free space. The pressure
for the HM can be calculated from the energy per particle with respect to
baryon density. It then has the form

$\displaystyle P_{\rm{HM}}$ $\displaystyle=$
$\displaystyle\rho^{2}\frac{\partial\mathcal{E}_{\rm{HM}}(\rho,\delta)}{\partial\rho}.$
(4)

Here we emphasize again that, in this work, the HM is represented by the five
KIDS-EDF models: KIDS0, KIDS-A, KIDS-B, KIDS-C, and KIDS-D models, and the
Skyrme force SLy4 model. Those various EDF models have different nuclear
symmetry energies, which cover soft to stiff nuclear symmetry energies.

Figure 1:  From the KIDS EDFs and the SLy4 model: (a) Symmetry energy, (b)
Pressure of HM as a function of $\rho/\rho_{0}$, (c) Energy density of HM as a
function of $\rho/\rho_{0}$ and (d) EoS of HM.

The nuclear symmetry energy results for the KIDS0, KIDS-A, KIDS-B, KIDS-C,
KIDS-D, and SLy4 models are given in Fig. 1 (a). It clearly shows different
ranges of symmetry energies covering soft and stiff symmetry energies that
will be used to construct the EoS of the hybrid model. Figure 1 (a) shows, for
HM, the stiffest symmetry energy is given by the KIDS-A model, and the softest
is given by the SLy4 model. We also provide the pressure of the NS matter as a
function of the densities in Fig. 1 (b). The pressures of the symmetric matter
obtained from the EDF models are consistent with the heavy-ion collision (HIC)
reaction data at high density Danielewicz:2002pu  as well as the chiral
perturbation theory (ChPT) at low density Tews:2012fj .

The result for the energy densities $E_{\rm HM}$ of the NS matter for the
hadronic models with respect to density is given in Fig. 1 (c). Results for
the $P_{\mathrm{HM}}$-$E_{\mathrm{HM}}$ relation for the hadronic models are
shown in Fig. 1 (d). This EoS relation is required as input for calculating
the TOV equation to obtain the properties of the NSs. In the HM model, we
found that the $M_{\mathrm{NS}}/M_{\odot}$ results for the KIDS0 and SLy4
models are rather similar because both models have soft symmetry energies. In
contrast, the KIDS-A and KIDS-B models have stiff symmetry energies giving a
larger radius of the NS. Overall, the mass and radius of NS results for all
KIDS models fit well with the recent observations Antoniadis:2013pzd ;
NANOGrav:2019jur ; Demorest:2010bx .

###  II.2 Pure quark matter from the NJLPTR model

In this section, we describe the pure (nonstrange) quark matter (PQM) in the
framework of the flavor SU(2) Nambu–Jona-Lasino (NJL) model Hutauruk:2021kej ;
Tanimoto:2019tsl ; Hutauruk:2021dgv ; Bentz:2001vc ; Baym:2017whm ;
Buballa:2003qv ; Glendenning:1992vb ; Hutauruk:2022sbm ; Hutauruk:2016sug ;
Hutauruk:2018zfk . This model is built in terms of the quark degrees of
freedom in the form of the four-fermion contact interactions. Therefore, this
model is very suitable and powerful for the description of quark matters. The
general NJL Lagrangian density for two quark flavors can be written as

$\displaystyle\mathscr{L}_{\mathrm{NJL}}$ $\displaystyle=$
$\displaystyle\bar{\psi}_{q}\left(i\partial\\!\\!\\!/-\hat{m}_{q}\right)\psi_{q}+G_{s}\left[(\bar{\psi}_{q}\psi_{q})^{2}+(\bar{\psi}_{q}\gamma_{5}\vec{\tau}\psi_{q})^{2}\right]-G_{\omega}(\bar{\psi}_{q}\gamma_{\mu}\psi_{q})^{2}-G_{\rho}(\bar{\psi}_{q}\gamma_{\mu}\vec{\tau}\psi_{q})^{2},$

where the quark field is given by $\psi_{q}=(\psi_{u},\psi_{d})^{T}$,
$\vec{\tau}$ stands for the Pauli isospin matrices, and
$\hat{m}_{q}=\textrm{diag}[m_{u},m_{d}]$ is the current (bare) quark mass,
where, in this work, we use the isospin symmetry, giving $m_{u}\simeq m_{d}$.
The $G_{s}$, $G_{\omega}$, and $G_{\rho}$ are respectively scalar, isoscalar-
vector, and isovector-vector coupling constants. Note that the NJL model
suffers the divergence in the quark propagator, therefore, we must apply a
regularization scheme to cure the divergence problem. In this work, we perform
the PTR scheme to tame the divergence Schwinger:1951nm . In the standard NJL
formalism, the effective quark mass can be easily obtained in the mean-field
approximation (MFA), which is explicitly given in the PTR scheme as

$\displaystyle M_{q}$ $\displaystyle=$ $\displaystyle
m_{q}-2G_{s}\langle\bar{\psi}_{q}\psi_{q}\rangle=m_{q}+\frac{3G_{s}M_{q}}{\pi^{2}}\int_{\Lambda_{\mathrm{UV}}^{-2}}^{\infty}d\tau\frac{\exp[-\tau
M_{q}^{2}]}{\tau^{2}},$ (6)

where $\langle\bar{\psi}_{q}\psi_{q}\rangle$ is the chiral condensate for
flavor $q$, which relates to the order parameter of the spontaneous breaking
of chiral symmetry (SB$\chi$S). It is worth noting that the effective quark
mass in Eq. (6) has an additional correction in the quark matter, which comes
from the density term in the quark propagator.

In the QM, the NJL Lagrangian density in Eq. (II.2) is modified by adding an
extra term of quark density or quark operator chemical potential. It then has
the form

$\displaystyle\mathscr{L}_{\mathrm{NJL}}$ $\displaystyle\rightarrow$
$\displaystyle\mathscr{L}_{\mathrm{NJL}}+\bar{\psi}_{q}\hat{\mu}_{q}\gamma^{0}\psi_{q},$
(7)

where $\hat{\mu}_{q}=\textrm{diag}[\mu_{u},\mu_{d}]$ is the quark chemical
potential matrix. Thus, considering the density effect, the quark propagator
for flavor $q$ in momentum space can be written by

$\displaystyle S_{q}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{\left[\left(p_{0}+\tilde{\mu}_{q}\right)\gamma^{0}-\vec{p}\cdot\vec{\gamma}-M_{q}\right]},$
(8)

where the $\tilde{\mu}_{q}$ is the so-called the reduced quark chemical
potential for flavor $q$ and it can can be written as

$\displaystyle\tilde{\mu}_{q}$ $\displaystyle=$
$\displaystyle\mu_{q}-2G_{v}\rho_{q}^{v},$ (9)

where
$\rho_{q}^{v}\simeq\langle\psi_{q}^{\dagger}\psi_{q}\rangle=p_{F_{i}}^{3}/\pi^{2}$
stands for the individual quark number densities of flavor $q$, where
$p_{F_{i}}$ is the Fermi momentum for quark flavor $i=(u,d)$. For simplicity,
in the present work, we opt the vector coupling $G_{v}=G_{\omega}=G_{\rho}$ by
considering vector dominance and small mass differences between $\rho$ and
$\omega$ masses Hell:2014xva ; Klimt:1990ws . Also, it is worth noting that
the $G_{v}$ will be treated as a free parameter in this work.

In the standard method, the effective potential for the nonstrange QM in the
NJL model is given by Lawley:2005ru

$\displaystyle\mathscr{V}^{\textrm{QM}}_{\textrm{NJL}}(M_{q},\mu_{q})$
$\displaystyle=$ $\displaystyle
2iN_{c}\sum_{q=[u,d]}\int\frac{d^{4}k}{(2\pi)^{4}}\log\left[\frac{k^{2}-M_{q}^{2}+i\epsilon}{k^{2}-M_{0}^{2}-i\epsilon}\right]$
(10) $\displaystyle+$
$\displaystyle\sum_{q=[u,d]}\frac{(M_{q}-m_{q})^{2}}{8G_{s}}-\sum_{q=[u,d]}\frac{(M_{0}-m_{q})^{2}}{8G_{s}}$
$\displaystyle-$ $\displaystyle
2N_{c}\sum_{q=[u,d]}\int\frac{d^{3}k}{(2\pi)^{3}}\Theta\left(\tilde{\mu}_{q}-E_{q}(k)\right)\left[\tilde{\mu}_{q}-E_{q}(k)\right]-\sum_{q=[u,d]}\frac{V_{0}^{2}}{8G_{v}},$

where the first term has divergence, therefore it has to be regularized using
the PTR scheme. The reduced quark chemical potentials for $u$ and $d$ quarks
are defined by $\tilde{\mu}_{u}=\tilde{\mu}_{d}=\mu_{u}-2G_{v}\rho^{v}_{u}$,
as also given in Eq. (9).
$V_{0}=2G_{v}\langle\psi_{q}^{\dagger}\psi_{q}\rangle$,
$E_{u}(k)=E_{d}(k)=\sqrt{\vec{k}^{2}+M^{2}_{0}}$, and $M_{0}$ are the vector
field (potential), quark energy, and the constituent quark mass in free space,
respectively. We note that we have subtracted a constant (free space)
contribution ($M_{q}=M_{0}$) in the effective potential in Eq. (10). Therefore
pressure becomes zero in free space. Along with this effective potential
definition and the so-called Gibbs-Duhem relation, the energy density will
also vanish in free space Ripka:1997zb .

Using the effective potential in Eq. (10) the pressure and the energy density
for the normal (nonstrange) QM are respectively expressed by

$\displaystyle P_{\rm QM}$ $\displaystyle=$
$\displaystyle-\mathscr{V}^{\textrm{QM}}_{\textrm{NJL}}(M_{q},\mu_{q})-\mathscr{V}_{l}(\mu_{l}),\,\,\,\,\,\,E_{\rm
QM}=\mathscr{V}^{\textrm{QM}}_{\textrm{NJL}}(M_{q},\mu_{q})+\sum_{q=[u,d]}\mu_{q}\rho^{v}_{q},$
(11) $\displaystyle\rho_{q}$ $\displaystyle=$
$\displaystyle-\frac{\partial\mathscr{V}^{\textrm{QM}}_{\textrm{NJL}}\left(M_{q},\mu_{q}\right)}{\partial\mu_{q}}.$
(12)

In the calculation of the QM EoS, the NJL parameters have been fitted to the
pion mass $m_{\pi}=0.14$ GeV, and the pion weak decay constant $f_{\pi}=$
0.093 GeV, giving $G_{s}=$ 3.17 GeV-2 and $\Lambda_{\mathrm{UV}}=$ 1.0789 GeV.
The constituent quark mass $M_{0}$ = 0.20 GeV and the current quark mass
$m_{q}=$ 0.0055 GeV, which is similar to PDG values ParticleDataGroup:2022pth
. Results for the effective quark mass, pressure, and energy density of the
quark matter for vector interaction couplings $G_{v}=(0.00,0.25,0.50)G_{s}$
are shown in Fig. 2. As expected, the effective quark masses decrease as the
density increases as shown in Fig. 2 (a). Additionally, the effective quark
mass is not sensitive to the change of the values of the vector repulsive
interaction coupling, where the rate of change is slightly low. This effective
quark mass result is consistent with the result obtained in Refs. Baym:2017whm
; Lawley:2005ru .

Figure 2: From the NJLPTR model: (a) Effective quark mass for different values
of $G_{v}$ as a function of $\rho/\rho_{0}$, (b) Pressure of QM as a function
of $\rho/\rho_{0}$, (c) Energy density of QM as a function of $\rho/\rho_{0}$,
and (d) EoS of PQM.

Results for the pressure of the PQM are given in Fig. 2 (b). It shows that the
pressure is stiffer with a larger value of $G_{v}$. Such behavior is a natural
and expected result because a larger $G_{v}$ value makes the repulsive force
become stronger. In addition, we show the results for the $E$-$\rho$ relation
of the PQM for different values of $G_{v}$ in Fig. 2 (c). A similar result
with the pressure is found: The energy density increases as the $G_{v}$ value
and density increase, which can be clearly understood from the relation of
$P_{\rm QM}=-\mathscr{V}^{\rm QM}_{\rm NJL}-\mathscr{V}_{l}$ and $E_{\rm
QM}=\mathscr{V}^{\rm QM}_{\rm NJL}+\sum_{q}\mu_{q}\rho_{q}^{\nu}$, as given in
Eq. (11). It implies that increasing the value of pressure is only given by
increasing the energy density (increasing $-E$) and vice versa, as explained
in the visualization of Fig. 12 of Ref. Baym:2017whm . Figure 2 (d) shows EoS
$P$-$E$ relation for different values of $G_{v}$. We found that the pressure
increases as the energy density and $G_{v}$ increase. This also indicates that
the EoS of the QM is stiffer for higher values of $G_{v}$.

###  II.3 Properties of the static hybrid star

Here, we input the obtained EoSs for hadronic and quark matters for different
values of $G_{v}$ and symmetry energies into the TOV equation to numerically
compute the properties of a non-rotating NS, obtaining the mass-radius (M-R)
relation (structure of the hybrid star). The TOV equations are given by
Oppenheimer:1939ne ; Tolman:1939jz ; Tolman:1934za

$\displaystyle\frac{dP(r)}{dr}$ $\displaystyle=$
$\displaystyle-\frac{G[E(r)+P(r)][M(r)+4\pi
r^{3}P(r)]}{r[r-2GM(r)]},\,\,\,\,\,\,\,\frac{dM(r)}{dr}=4\pi r^{2}E(r),$ (13)

where $P(r)$ and $E(r)$ are respectively the pressure and energy density at
radial position $r$. $G$ and $M(r)$ are the gravitational constant and mass
within the sphere of radius $r$, respectively. Using the EoSs of the
constructed hybrid stars as input, we obtain the $M$-$R$ relation as a result.
Note that the EoSs of the hybrid stars are depicted in Fig. 3.

##  III Numerical result

Figure 3: Pressure as a function of density for QM with different values of
$G_{v}$ and for HM in the different EDF models: (a) KIDS0, (b) KIDS-A, (c)
KIDS-B, (d) KIDS-C, (e) KIDS-D, and (f) SLy4. The critical density $\rho_{c}$
is given by the cross point between $P_{\rm HM}$ and $P_{\rm QM}$ with
different values of $G_{v}$. The thick solid line is $P$-$\rho$ relation for
HM. The thin solid line is $P$-$\rho$ relation for QM with $G_{v}=0.00G_{s}$,
the dashed line is $P$-$\rho$ relation for QM with $G_{v}=0.25G_{s}$, and the
dashed line is $P$-$\rho$ relation for QM with $G_{v}=0.50G_{s}$.

###  III.1 Critical density of the hybrid star

In this section, we present the results of the critical densities of the phase
transition of the hadronic to the quark matter for different KIDS models and
the Skyrme model. The critical density $\rho_{c}$ is simply determined by
finding the cross point between $P_{\rm HM}$ and $P_{\rm QM}$. It is also
worth noting that the $\rho_{c}$ is the phase transition from the hadronic and
quark phases that occurs when $P_{\rm HM}(\rho_{c})=P_{\rm QM}(\rho_{c})$ and
$\rho_{\rm HM}=\rho_{\rm QM}$. In Fig. 3, it shows $P_{\rm HM}$ for each
hadronic model and $P_{\rm QM}$ with three values of the vector coupling
constant $G_{v}=0$, $0.25G_{s}$, and $0.5G_{s}$. Figure 3 (a) shows $P_{\rm
HM}$ for the KIDS0 model and $P_{\rm QM}$ for the NJL model with different
values of $G_{v}$. The critical density for $P_{\rm HM}$ and $P_{\rm QM}$ for
$G_{v}=0.00G_{s}$ is around $\rho_{c}=3.60\rho_{0}$. The $\rho_{c}$ increases
as the value of $G_{v}$ increases. The $\rho_{c}$ for the KIDS-A model is
shown in Fig. 3 (b). Note that the KIDS-A model has stiffer symmetry energy
than that for the KIDS0 model. The critical density for the KIDS-A model and
the NJL model with $G_{v}=0.00G_{s}$ and $G_{v}=0.25G_{s}$ are smaller than
those for the KIDS0 model. However, the critical density for the KIDS-A model
and the NJL model with $G_{v}=0.5G_{s}$ is larger than that obtained for the
KIDS0 model. This feature is followed by the KIDS-B, KIDS-C, and KIDS-D models
as shown in Figs. 3 (c)-(e), while the SLy4 model and the NJL model with
different values of $G_{v}$ almost have similar critical density values as
those for the KIDS0 model and the NJL model with the corresponding values of
$G_{v}$. This can be understood because both KIDS0 and SLy4 models have softer
nuclear symmetry energies in comparison to other KIDS models as shown in Fig.
1. For all the hadronic models and the NJL model with different $G_{v}$
values, the EoS is stable in the hadron phase at low densities, and the QM
becomes more stable than the HM at high densities.

Again, comparing the $\rho_{c}$ values for larger values of $G_{v}=0.5G_{s}$
among the KIDS-A, KIDS-B, KIDS-C, and KIDS-D models, the highest value of
$\rho_{c}$ is given by the KIDS-A model, and it decreases in the order of
KIDS-B, KIDS-C, and KIDS-D models. The ordering of $\rho_{c}$ could be
understood from the stiffness of symmetry energy. With a soft symmetry energy
(small $L$ and $K_{\rm sym}$ values, where $L$ and $K_{\rm sym}$ are
respectively the slope and the curvature of symmetry energies), the energy to
create a neutron becomes small, so the $\beta$-equilibrium condition allows
easy creation of the neutron, which eventually leads to a large fraction of
the neutron. Pauli blocking makes the pressure exerted by the neutron strong
when there are more neutrons within the neutron star. For this reason, the
pressure becomes stiff from KIDS-A to KIDS-D while the symmetry energy is
stiffened in the reverse order. Exact values of $\rho_{c}$ are summarized in
Tab. 1. One can see that, regardless of $G_{v}$ value, the transition to the
quark phase occurs at low densities with soft symmetry energy. Values of the
pressure at $\rho_{c}$ ($P_{c}$) are also summarized in the table.

Overall, it indicates that the phase transition is highly sensitive to both
the nuclear symmetry energy in the hadron phase and the vector repulsion in
the quark phase. Their effect on physical observables can be probed by solving
the TOV equations and obtaining the mass and radius of neutron stars.

Table 1:  Transition point obtained from the $P$-$\rho/\rho_{0}$ relation for the hybrid KIDS-NJLPTR models with different values of $G_{v}$. The units of the coupling constants of $G_{v}$ and $G_{s}$ are MeV-2, and $P$ is MeV$\cdot$fm-3. Note that $\rho_{c}$ is in the unit of $\rho_{0}$. $G_{v}$ | KIDS0 | KIDS-A | KIDS-B | KIDS-C | KIDS-D | SLy4
---|---|---|---|---|---|---
| ($P_{c},\rho_{c}$) | ($P_{c},\rho_{c}$) | ($P_{c},\rho_{c}$) | ($P_{c},\rho_{c}$) | ($P_{c},\rho_{c}$) | ($P_{c},\rho_{c}$)
$0.00G_{s}$ | ($110.23,3.60$) | ($89.98,3.21$) | ($88.21,3.18$) | ($73.00,2.86$) | ($78.15,2.97$) | ($119.73,3.78$)
$0.25G_{s}$ | ($226.44,4.55$) | ($203.39,4.29$) | ($181.44,4.02$) | ($159.28,3.74$) | ($155.71,3.69$) | ($261.06,4.95$)
$0.50G_{s}$ | ($464.75,5.81$) | ($592.96,6.65$) | ($417.06,5.47$) | ($365.00,5.09$) | ($309.08,4.65$) | ($574.79,6.54$)

###  III.2  $M$-$\rho$ and $M$-$R$ relations of the hybrid star

Figure 4: NS mass-density relations of different hadron-quark matter EoS with
different values of $G_{v}$ for different EDF models: (a) KIDS0, (b) KIDS-A,
(c) KIDS-B, (d) KIDS-C, (e) KIDS-D, and (f) SLy4. The thick solid line
represents the NS mass for the HM. The thin solid line is the NS mass for the
HM+QM with $G_{v}=0.00G_{s}$, the dashed line is the NS mass for the HM+QM
with $G_{v}=0.25G_{s}$, and the dotted line represents the NS mass for the
HM+QM with $G_{v}=0.50G_{s}$.

Figure 4 displays the mass of an NS as a function of the density at the
center. For the pure hadronic matter, all six models produce the NS maximum
masses ($M_{\rm max}$) greater than $2M_{\odot}$. The NS maximum mass for the
KIDS-D model is the largest as shown in Fig. 4 (e), and it decreases in the
order of KIDS-C, KIDS-B, and KIDS-A models as in Figs. 4 (b)-(d). This result
is consistent with the stiffness of EoS in Fig. 1. The result changes
dramatically if QM exists in the core of the star.

The thick solid lines denote the NS mass results of the hadronic matter as in
Fig. 4 (a)-(f). In all the hybrid KIDS-NJL models, the density at which the
thin solid lines begin is equal to $\rho_{c}$ in Tab. 1, which means that as
the transition to quark matter occurs, the EoS becomes soft then the matter
cannot support strong gravity. As a result, $M_{\rm max}$ is determined at the
density of phase transition. For $G_{v}=0$, $\rho_{c}$ is not much sensitive
to the models, so the NS maximum masses are obtained in a narrow range
(1.5–1.6)$M_{\odot}$. These values are substantially low in comparison to
$2M_{\odot}$, so the result confirms that vector repulsion must be necessarily
accounted for in the QM to reproduce the observation of the large mass of NS.
The dashed lines correspond to the result of $G_{v}=0.25G_{s}$ as shown in
Fig. 4 (a)-(f). Similar to $G_{v}=0$, the density at which a dashed line
begins is the same with $\rho_{c}$ in Tab. 1, and $M_{\rm max}$ is determined
at the density where the mass is below $2M_{\odot}$ for all EDF models. The
result demands that the $G_{v}$ must be higher than $0.25G_{s}$.

The results of $G_{v}=0.5G_{s}$ which are shown with dotted lines are now
consistent with the $2M_{\odot}$ constraint. The density at which the QM curve
begins is the highest is given by the KIDS-A model, and it then decreases in
the order of KIDS-B, KIDS-C, and KIDS-D models. This ordering is understood
easily in terms of the stiffness of the EoS of each model. Contrary to the
results of $G_{v}=0$ and $0.25G_{s}$, the density at which the dotted curves
begin is higher than the $\rho_{c}$ values in Tab. 1. The difference means
that even after the QM is formed in the core, its EoS is stiff enough that it
can resist gravitational contraction up to a certain density, at which the NS
star reaches the maximum value. However, the interval between $\rho_{c}$ and
the density at the maximum mass of NS is narrow, so the fraction of QM in the
core of the NS is not likely to be significant. Summarizing the result, to
satisfy the $2M_{\odot}$ condition, the $G_{v}$ value must be larger than a
certain value to obtain the QM EoS stiff. The stiff QM EoS increases
$\rho_{c}$, and the high $\rho_{c}$ value constrains the existence of QM in a
limited range.

Figure 5: Same as Fig. 4, but for NS $M$-$R$ relations. Horizontal error bars
show the radius of $1.4M_{\odot}$ and $2.08M_{\odot}$ stars determined in the
NICER analysis apjl918 .

Figure 5 presents the mass-radius ($M$-$R$) relation obtained from the TOV
equations. Since the data from GW and NICER provide information on both mass
and radius of NS, they can constrain the EoS of HM and QM more strictly than
the data of mass alone. The radius of NS canonical mass ($1.4M_{\odot}$)
$R_{1.4}$ is of particular interest and importance. According to the NICER
analysis apjl918 , the NS radius is $R_{1.4}=12.45\pm 0.65$ km within
$1\sigma$ credible interval apjl918 . On the theoretical side, the density at
the center of $1.4M_{\odot}$ NS is obtained not to exceed $3\rho_{0}$ in
general. The density at which exotic states such as the mixture of the hyperon
or transition to QM appear is above $2\rho_{0}$ as a whole, though the
uncertainty is non-negligible. Therefore, the properties of $1.4M_{\odot}$ NS
are especially important to probe (i) the EoS of HM at high densities if there
is no transition to QM or hyperon, or (ii) consistency of the EoS of exotic
phases with the data of $1.4M_{\odot}$ NS if these exotic states exist in the
interior of the $1.4M_{\odot}$ NS. The $M$-$R$ result shows that the effect of
QM appears at NS masses higher than $1.5M_{\odot}$, so the radius of
$1.4M_{\odot}$ NS is unaffected by the transition to QM. Hyperon is another
source that can soften the EoS of $1.4M_{\odot}$ NS. According to work in
progress kidsll , if the $\Lambda\Lambda$ interaction determined from the
experimental data of double-$\Lambda$ hypernuclei is included in the EoS of
hyperon matter, it makes the EoS stiff. As a consequence, the maximum mass of
the NS becomes larger than $2M_{\odot}$, and the creation of the hyperon gives
negligible effect to the radius of $1.4M_{\odot}$ star. Therefore, accurate
measurement of $R_{1.4}$ can provide stringent constraints on the EoS of HM at
densities up to $3\rho_{0}$.

In comparison with other results, our results show the difference to some
extent, but qualitatively both results are compatible. In works of Refs.
prd100 ; apj934 , they reported that transition or crossover to QM is
considered. In both works, QM was described in terms of the NJL model, but for
the HM, an RMF model is used in prd100 , and a chiral effective field theory
($\chi$EFT) and Togashi EoSs are employed in apj934 .

In Ref. prd100 , the result of the MS-B+vNJL model is comparable to our work.
Three values are assumed for the vector repulsion of QM, $G_{v}/G_{s}=1.5$,
2.0, and 2.5. The result shows that the NS maximum mass is below $2M_{\odot}$
when $G_{v}=1.5G_{s}$, and it becomes obviously above $2M_{\odot}$ when
$G_{v}=2.0G_{s}$. It is notable that, in our calculation, $M_{\rm max}\geq
2M_{\odot}$ is obtained if $G_{v}\geq 0.5G_{s}$, so there is a huge difference
in the strength of the vector repulsion. $M$-$R$ curve of MS-B+vNJL is also
very different from what we obtain. Transition to QM occurs at
$\rho=1.5\rho_{0}$ or $\rho=2.0\rho_{0}$ in the MS-B+vNJL model, and then the
radius becomes smaller than the radius of the HM. The radius of the
$1.4M_{\odot}$ NS is about 12.5 km without QM, but it falls in the range
11.7–12.2 km depending on the $G_{v}$ and $\rho_{c}$ values. Radii of the
maximum mass NSs are located below 11 km, which is much smaller than the NICER
estimation $R_{2.08}=12.35\pm 0.75$ km, where $R_{2.08}$ represents the radius
of $2.08M_{\odot}$ NS apjl918 .

In Ref. apj934 , the crossover between HM to QM is assumed to begin at
$1.5\rho_{0}$, and the dependence on $G_{v}$ is presented. Compared to the
$R_{1.4}$ of Togashi EoS in which there are only nucleons, the NS radius
becomes larger if the QM is included in the core by about 0.2–0.3 km. This
behavior is opposite to the result of MS-B+vNJL in prd100 . The radius of
$2.08M_{\odot}$ NS is in the range 11.5–12 km for the $G_{v}$ values that have
maximum masses larger than $2.08M_{\odot}$. It is shown in the paper that the
values of $G_{v}$ must be larger than $0.84G_{s}$ to satisfy the condition
$M_{\rm max}\geq 2.08M_{\odot}$. Also, this shows that the $G_{v}$ value in
Ref. apj934  is much larger than the value of $G_{v}$ that is obtained in our
work to satisfy the $2M_{\odot}$. However, some recent references in the
iterature Alaverdyan:2022foz ; Kumar:2023lhv  also reported that the strength
of $G_{v}$ is in the same order as that used in this work to give
$2M_{\odot}$. It is worth noting that there is no stringent constraint for the
values of $G_{v}$ available nowadays. More precise data are needed to
constrain $G_{v}$.

It is informative to compare our result of NS radius around the NS maximum
mass with the NICER result in $R_{2.08}=11.6$–$13.1$ km. With HM only, the NS
radii at NS maximum mass are in the range of 9.8–11 km. The NS radii tend to
increase in a stiffer EoS, giving the largest NS radius for the KIDS-C model.
However, even the largest value of NS radius is relatively smaller compared to
the lower limit of NICER $R_{2.08}$. The inconsistency could be understood as
a signal for the existence of phases other than the hadrons. Comparing the
results of KIDS-A, KIDS-B, KIDS-C, and KIDS-D models, a soft EoS of HM gives
larger values of $\rho_{c}$, so even if the QM is formed in the core, $R_{\rm
max}$ (radius of the NS maximum mass) is too small to be consistent with the
range of NICER $R_{2.08}$. The KIDS-A model clearly shows such behavior. On
the other hand, stiff EoSs like the KIDS-C and D models show that the phase
transition occurs at a relatively large NS radius, so the NS radius of the
maximum mass is shifted to the values larger than those of pure HM. As a
result, the NS with QM in the core can explain the NICER $R_{2.08}$ data
better than the NSs with nucleons only.

While the KIDS-A, KIDS-B, KIDS-C, and KIDS-D models are determined to satisfy
the constraint $R_{1.4}=$ 11.8-12.5 km, which is more restrictive than the
NICER $R_{1.4}$, the SLy4 and KIDS0 models are fitted to the APR pure neutron
matter EoS apr  in the isovector component of the functional. For this reason,
the $M$-$R$ behaviors of the SLy4 and KIDS0 models are similar to each other,
which are not consistent with NICER $R_{1.4}$ and $R_{2.08}$.

The result of this work clearly shows that the symmetry energy plays an
important role in not only controlling the stiffness of EoS of HM, but also
affecting the bulk properties of the NSs with QM core. The KIDS-C and D models
with $G_{v}$ values larger than $0.5G_{s}$ satisfy the NS mass data of
$1.4M_{\odot}$ and $2.08M_{\odot}$. The values of the nuclear symmetry energy
parameters $L$ and $K_{\rm sym}$ are respectively 58 and $-91.5$ MeV for the
KIDS-C model, and they are 47 and $-134.5$ MeV for the KIDS-D model,
respectively. These values are consistent with the 95 % credible range
obtained from the KIDS-R14 model set npsm2022 , $L=49.8\pm 10.4$ MeV, and
$K_{\rm sym}=-82.4\pm 67.4$ MeV.

##  IV Summary

In summary, the role of nuclear symmetry energy has been investigated in the
transition from hadron matter to deconfined quark phase in the NS core. Major
concerns are (i) the dependence of the critical density for the phase
transition on the symmetry energy and (ii) the role of the repulsive vector
coupling in the EoS of QM and NS properties. The nuclear symmetry energy is
determined to satisfy the data of the canonical mass of NSs. The effect of the
uncertainty due to the symmetry energy becomes obvious in the $M$-$R$ relation
of the NSs with masses close to $2M_{\odot}$.

Quark matter EoS is described in the NJL model. For a rigorous treatment of
the ultraviolet divergence, we employed the PTR scheme. It is known that
repulsive vector coupling is essential to reach the maximum mass of the hybrid
star to be consistent with the $2M_{\odot}$ astrophysical observation.
Dependence on the vector coupling is examined by using three different values
of the vector coupling constants $G_{v}=$ 0.00, 0.25, and 0.50 in the unit of
the $G_{s}$. Critical density for the phase transition is determined from the
condition $P_{\rm HM}(\rho_{c})=P_{\rm QM}(\rho_{c})$ at $\rho_{\rm
HM}=\rho_{\rm QM}=\rho_{c}$.

We found that the critical density is highly sensitive to symmetry energy.
Comparing the $\rho_{c}$ values among the four models: KIDS-A, KIDS-B, KIDS-C,
and KIDS-D models, the value of $\rho_{c}$ tends to decrease with softer
symmetry energy, i.e., $\rho_{c}({\rm A})>\rho_{c}({\rm B})>\rho_{c}({\rm
C})>\rho_{c}({\rm D})$ regardless of the vector coupling constant. When the
vector repulsion is turned off ($G_{v}=0$), the maximum mass NS is obtained
about (1.5–1.6)$M_{\odot}$. This confirms that repulsive vector coupling is
really needed for obtaining the consistency result with $2M_{\odot}$
observation. Since the NS maximum masses are obtained at $\rho_{c}$ for
$G_{v}=0$, properties of the $1.4M_{\odot}$ NSs are not affected by the
transition to QM. Therefore, the EoS of $1.4M_{\odot}$ NS could be determined
accurately in terms of the hadronic degrees of freedom.

For $G_{v}=0.25G_{s}$, once the QM is created in the NS core, the NS reaches
the maximum mass around $1.9M_{\odot}$, which is quite independent of the
model. To satisfy the observation, a stronger repulsion is required. With
$G_{v}=0.50G_{s}$, we found that the NS maximum mass is consistent with
astrophysical observation, but the NS radius becomes inconsistent with the
observation findings. For the KIDS-A and B models, the phase transition does
not improve the NS $M$-$R$ relation, which is out of the range of the NS
radius determined by the NICER analysis. On the other hand, the KIDS-C, and D
models agree with the NICER range if the QM exists in the NS core.

Consequently, we have shown that the density dependence of the symmetry energy
and $G_{v}$ play a critical role in the phase transition from hadron to quark
matters. Critical density is sensitive to both symmetry energy and the vector
coupling constant. Their effects are crucial to the $M$-$R$ behavior of the
hybrid stars whose masses are above the NS canonical mass. Precise measurement
of the large mass of the NSs will offer a unique opportunity to constrain the
symmetry energy in the HM and repulsion vector coupling in the QM
simultaneously.

## Acknowledgments

P.T.P.H. thanks to Daniel Whittenbury for the discussion. This work was
supported by the National Research Foundation of Korea (NRF) Grant Nos.
2018R1A5A1025563, 2022R1A2C1003964, 2022K2A9A1A0609176, and 2023R1A2C1003177.

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