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\begin{document}
% Journal sections (see http://pkp.jinr.ru/index.php/PEPAN_LETTERS/about/editorialPolicies#focusAndScope)
%\issuearea{Physics of Elementary Particles and Atomic Nuclei. Theory}
% or in Russian
%\issuearea{ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА. ТЕОРИЯ}

\title{%Charged 
Pion condensation and color superconductivity%in dense quark matter
: different regularization schemes}
\maketitle
\authors{ K. G.\, Klimenko $^{a}$,
R. N.\, Zhokhov $^{b}$ \footnote{zhokhovr@gmail.com}}
\setcounter{footnote}{0}
\authors{Р. Н.\, Жохов $^{b}$,
К. Г.\, Клименко $^{a}$}
\from{$^{a}$\,State Research Center
of Russian Federation -- Institute for High Energy Physics,
NRC "Kurchatov Institute", 142281 Protvino, Moscow Region, Russia}
\from{$^{a}$\,Институт физики высоких энергий имени А.А.Логунова НИЦ "Курчатовский институт"    }
\from{$^{b}$\,Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN),
108840 Troitsk, Moscow, Russia}
\from{$^{b}$\,Федеральное государственное бюджетное учреждение науки Институт Земного магнетизма, ионосферы и распространения радиоволн им. Н.В. Пушкова }

\begin{abstract}
% Russian translation of the abstract
 Хорошо известно, что существует зависимость фазовой структуры, полученной в рамках модели NJL, от схемы регуляризации. В этой статье показано, что с хорошей точностью фазовая структура плотной кварковой материи с изоспиновым и киральным дисбалансом одинакова для разных схем регуляризации. Например, является универсальным недавнее предсказание генерации заряженной пионной конденсации в плотной кварковой среде с киральным дисбалансом и ее устойчивость к явлению цветовой сверхпроводимости, которое доминирует при высоких барионных плотностях.

\vspace{0.2cm}

The dependence of the phase structure obtained in the framework of effective NJL model on regularization scheme and fit used is rather well known. It has been shown in this letter that with good precision the phase structure of dense quark matter with isospin and chiral imbalances are the same for different regularization schemes. %Such as the
For example, recent prediction of pion condensation in dense quark matter with chiral imbalance and its resilience against color superconductivity phenomenon, which is dominant at high baryon densities.
\end{abstract}
\vspace*{6pt}

%\noindent
%PACS: 44.25.$+$f; 44.90.$+$c

\label{sec:intro}
%\section*{NJL model and regularization schemes}
The phase diagram of quark matter will be considered in the framework of NJL type model with two quark flavors and Lagrangian %possessing the interaction between quarks both in the quark--antiquark and diquark channels,
\begin{eqnarray}
 L=\bar q\Big [\gamma^\nu i\partial_\nu+ \gamma^0{\cal M}\Big ]q+ G\Big [(\bar qq)^2+
(\bar qi\gamma^5\vec\tau q)^2\Big ]+H\sum_{A=2,5,7}
[\overline{ q^c}i\gamma^5\tau_2\lambda_{A}q]
[\bar qi\gamma^5\tau_2\lambda_{A} q^c].\nonumber
\label{1}
\end{eqnarray}
\vspace{-14pt}

$q$ is Dirac spinor and color triplet, there are two
flavors, $u$ and $d$. Pauli matrices $\tau_a$ act in flavor space, while Gell-Mann matrices $\lambda_A$ in color one.

${\cal M}$ term possessing %incorporate 
chemical potentials has the following form
\vspace{-7pt}
$$
\gamma^0{\cal M}\equiv\gamma^0 \left [\frac{\mu_B}{3}+\frac{\mu_I}2\tau_3+\frac{\mu_{I5}}2\gamma^5\tau_3+
\mu_5\gamma^5\right ] ,
$$
\vspace{-15pt}

Baryon chemical potential $\mu_B$ (we will use quark chemical potential $\mu=\frac{\mu_B}{3}$), $\mu_I$ is isospin chemical potential (we use $\nu=\frac{\mu_I}{2}$), chiral isospin chemical potential (we use $\nu_5=\frac{\mu_{I5}}{2}$) and chiral chemical potential $\mu_5$


\begin{figure}[t]
%----figure 1
\begin{center}
\includegraphics[width=\textwidth]{fig5reg31.eps}
\end{center}
%\label{fig1}
%\parbox[t]{0.45\textwidth}{
\vspace{-7mm}
\caption{$(\nu, \nu_5)$-phase diagrams at $\nu=0$ and $\mu=0$. Left panel hard cutoff scheme, then soft cutoff scheme and right panel is comparison between the two (solid line is hard cutoff scheme, dashed line is soft cutoff scheme). %, where all symmetries are restored.
 } %}
%\label{fig2}
\end{figure}

The current quark mass is assumed to be zero, since the main results do not depend on that if at least one of the chemical potentials is larger than half the pion mass.

In order to study the phase diagram %of the system 
in the mean-field approximation, let us use linearized Lagrangian with auxiliary bosonic fields $\sigma(x)$, $\pi_a(x)$ and $\Delta_A(x)$, which equations of motion are given by 

\vspace{-20pt}
\begin{eqnarray}
&&\sigma (x)=-2G(\bar qq),~~~~\pi_a(x)=-2G(\bar qi\gamma^5\tau_a q),~~\nonumber\\
&&\Delta_{A}(x)=-2H(\overline{q^c}i\gamma^5\tau_2\lambda_{A}q),~~~~
\Delta^{*}_{A}(x)=-2H(\bar qi\gamma^5\tau_2\lambda_{A} q^c).\nonumber
\label{4}
\end{eqnarray}
\vspace{-20pt}

 %The expression for TDP of the model can be obtained in the mean field approximation. 
 We do not present the full expression for TDP of the model that can be found due to brevity (see \cite{Khunjua:2019lbv, Khunjua:2024qha}) but mention it has the following form

 \vspace{-15pt}
\begin{eqnarray}
\Omega(M,\pi_1,\Delta)
=\left[\frac{M^2+\pi^2_1}{4G}+
\frac{\Delta^2}{4H}\right]+i\int\frac{d^4p}{(2\pi)^4}
\Big[ F(p, M,\pi_1,\Delta, \mu, \nu, \nu_5, \mu_5)\Big].\nonumber
\end{eqnarray}
\vspace{-15pt}

Now in order to proceed with further calculations one has to regularize the expression for the TDP. And then fix the model and regularization scheme parameters to reproduce physical quantities such as pion decay constant ($f_{\pi}=93$ MeV) and quark-antiquark condensate ($\langle\bar{q}q\rangle=(-250\text{ MeV})^3)$.

In NJL model studies of phase structure the so called hard cutoff scheme is often used, where one cutoffs momentum at some energy scale $\Lambda$,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
%So one makes the following in the expression of the TDP.

\;\;\;\;\;\;\;\;\;\;\;\;\;\;
$
\int d^4p\equiv \int_{-\infty}^{\infty} dp_0 \,4\pi\int_{0}^{\infty} p^2d p\to \int_{-\infty}^{\infty}dp_0\,4\pi\int_{0}^{\Lambda} p^2d p.
$

\;\;\;\;\; and fit of parameters $G=5.01$ GeV$^{-2}$, $\Lambda=650$ MeV is used \cite{Khunjua:2019lbv} 


 If one studies, for example, the influence of magnetic field etc. it is often convenient to use the soft cut-off regularization scheme. 
 It is defined as
 
 $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int d^4p\equiv \int dp_0d^3\vec p\to \int dp_0d^3\vec p f_{\Lambda}(\vec p)$. 
 
where the cut-off function and fit used are
\begin{center}
$f_{\Lambda}(\vec p)=\sqrt{\frac{\Lambda^{2N}}{\Lambda^{2N}+|\vec p|^{2N}}}\;\;\; \text{and\;\;\; $G=4.79$ GeV$^{-2}$, $\Lambda=638.8$ MeV and $N=5$}.$
\end{center}
%and the parameter fit is $G=4.79$ GeV$^{-2}$, $\Lambda=638.8$ MeV and $N=5$.

%\section{Phase structure}
Let us consider the phase structure of quark matter in these two regularization schemes and start with simple case of $(\nu, \nu_5)$-phase diagrams with all other chemical potentials equal to zero, $\mu=0$ and $\mu_5=0$. One can see in Fig. 1 that the phase structures are the same, for rather large isospin chemical potential $\nu$ pion condensation (PC) phase ($\pi_1\neq0$) settles in, whereas for rather large $\nu_5$ chemical potential chiral symmetry breaking (CSB) phase is realized ($M\neq0$). One can see that the difference between two regularizations starts only from values of chemical potentials around $0.4$ GeV and first it is rather negligible %for interpreting results 
and the picture is the same. The drastic difference is that for hard cutoff there is a transition to symmetric phase at some energy scale. Though at such large values of chemical potentials the results of effective NJL model is not trustworthy and it is probably the artifact of the model, soft cutoff scheme tackles this region slightly better and there is no symmetric phase up to pretty large scale. 

\begin{figure}[t]
%----figure 1
\;\;\;\;\;\includegraphics[width=0.9\textwidth]{fig2reg.eps}
%\label{fig1}
%\parbox[t]{0.45\textwidth}{
\vspace{-5mm}
\caption{$(\nu, \nu_5)$-phase diagrams at $\mu=0.2$ GeV and $\mu_5=0$ GeV. Left panel hard cutoff scheme, right panel is soft cutoff scheme. %, where all symmetries are restored.
 } %}
%\label{fig2}
\end{figure}


\begin{figure}[t]
%----figure 1
\includegraphics[width=\textwidth]{fig1reg.eps}
%\label{fig1}
%\parbox[t]{0.45\textwidth}{
\vspace{-8mm}
\caption{$(\mu_5, \mu)$-phase diagrams at $\nu_5=-0.35$ GeV and $\nu=0.15$ GeV. Left panel hard cutoff scheme, right panel is soft cutoff scheme. %, where all symmetries are restored.
 } %}
%\label{fig2}
\end{figure}


\begin{figure}[t]
%----figure 1
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\includegraphics[width=0.5\textwidth]{fig3reg3.eps}
%\label{fig1}
%\parbox[t]{0.45\textwidth}{
\vspace{-3mm}
\caption{Pion condensate as a function of $\mu_5$ at $\nu_5=-0.35$ GeV, $\nu=0.15$ GeV and  $\mu=0.34$ GeV. Solid line is hard cutoff scheme, dashed line is soft cutoff scheme. %, where all symmetries are restored.
 } %}
%\label{fig2}
\end{figure}


If one considers more complicated case, when baryon chemical potential is non-zero, $\mu=0.2$ GeV and $\mu_5=0$ (Fig. 2), PC and CSB phases start to occupy the region of $\nu$ and $\nu_5$ around the value of $\mu$. Though the main features of the phase structure are the same, some details are different. These details are located at rather large values of chemical potentials and are also outside or at a edge of the values of applicability of the model. Let us only note that for soft cutoff there is no a sharp cut of the PC and CSB phases at large $\nu$ and $\nu_5$ respectively that looks pretty unphysical and the behaviour is more trustworthy and probably these phases are to be realized even for large values of $\nu$ and $\nu_5$. Also let us note that we picked the phase diagram, where the difference of the two schemes can be easily noticed on purpose to show that it does not change the conclusions anywhere.

Let us recall that it was shown that at non-zero chiral imbalances PC phase is generated in the quark matter with non-zero baryon/quark densities (denoted as PC$_{d}$ phase) \cite{Khunjua:2019lbv}, the phenomenon found not to be realized in absence of chiral imbalances in the media. One can see in Fig.3 that PC$_d$ phase is widely generated if $\nu_5,\,\mu_5\neq0$. It is rather interesting phenomenon and it is interesting to test it in various regularization schemes and shows that it is regularization scheme independent. %It is not straightforward to apply Pauli-Villars regularization scheme when pion condensation is present so we employ here hard cutoff and soft cutoff scheme.
One can see in Fig. 3 that the predictions of different regularization schemes are the same and the phases are almost identical and only the behaviour for rather large $\mu_5$ is somewhat different quantitatively. In Fig. 4 behavior of pion condensate at $\mu=0.34$ GeV in Fig. 3 is also shown. This shows that prediction of PC$_d$ phase is universal and does not depend on regularization scheme employed. The fact that values of pion condensate differ considerably at $\mu_5>500$ MeV is explained by the fact that the results of NJL model in this case are not trustworthy and artifacts start to appear. Since it is due to the proximity to cut-off value $\Lambda$ it is natural to expect that soft cut-off scheme describes the situation slightly better.

When PC$_{d}$ phase was first predicted \cite{Khunjua:2019lbv, eklim} color superconductivity (CSC) phenomenon was omitted and it was not obvious that it does not take over pion condensation in the non-zero baryon density region. Then it was shown in \cite{Khunjua:2024qha} that CSC phase does not exclude pion condensation phenomenon in dense matter with chiral imbalance. In first studies of PC$_{d}$ phase \cite{Khunjua:2019lbv, eklim} hard cutoff scheme was used, whereas in \cite{Khunjua:2024qha} soft cutoff regularization was employed. So it was important to make sure that all the results are universal and regularization independent. One can see in Fig. 3 that CSC phase does not meddle into PC$_{d}$ phase region for these baryon densities and chiral imbalances. %In another region of phase diagram in Fig. one can see that CSC phase is present around PC$_{d}$ phase but pion condensation is nowhere undermined by it. 
In another region of phase diagram one can observe in Fig. 5 that CSC phase is present around PC$_{d}$ phase and shape and the form of CSC phase is slightly different for different schemes but qualitatively absolutely the same and pion condensation is nowhere undermined by it. One can argue that details of PC phase in dense quark matter is independent of regularization procedure.

\begin{figure}[t]
%----figure 1
\includegraphics[width=\textwidth]{fig4reg.eps}
%\label{fig1}
%\parbox[t]{0.45\textwidth}{
\vspace{-5mm}
\caption{$(\nu, \mu)$-phase diagrams at $\nu_5=0.3$ GeV and $\mu_5=-0.2$ GeV, coupling constant $H=0.9$. Left panel hard cutoff scheme, right panel is soft cutoff scheme. %, where all symmetries are restored.
 } %}
%\label{fig2}
\end{figure}

%\vspace{-10pt}
%\section*{Summary}

%\vspace{-5pt}
%Recently it was predicted that in dense quark matter with chiral imbalance pion condensation can be generated. These prediction has been made in the framework of effective models since in the region of rather large baryon density lattice QCD simulations are impossible due to sign problem. 
%Regularization dependence of the results obtained in the framework of NJL model is rather well known. %Effective models are non-renormalizable and valid only up to some energy scale. The cutoff energy and the scheme of regularization %defines the model 
%is a part of definition of the model. Cutoff scale and coupling are chosen to reproduced known low-energy properties such as pion mass etc.
As a summary one can note that
in this letter in effective NJL model with hard and soft cutoff regularization schemes we showed that

\vspace{5pt}

1) qualitatively and with rather good precision quantitatively the results on the phase structure of dense quark matter with isospin and chiral imbalances are the same for different schemes

2) the prediction of pion condensation in dense quark matter is universal and its main features and details do not depend on regularization

3) the prediction of pion condensation in dense quark matter is not compromised by color superconductivity at large baryon density and it is independent of regularization scheme employed
%\label{sec:acknowledgement}
%\section*{Acknowledgement}
%Any information about any assistance received when carrying out the work and preparing the manuscript.

%\nocite{*}
%\bibliographystyle{pepan}
%\bibliography{maikbibl}
% Bibliography automatically generated via BibTeX (see template_bibtex.tex)

\begin{thebibliography}{1}
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\end{thebibliography}


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