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\usepackage{wrapfig}
\usepackage[table,xcdraw]{xcolor}
\usepackage{braket}
\usepackage{ascmac}
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\newcommand{\helium}{{}^3{\rm He}}
\newcommand{\pim}{\pi^-}
\newcommand{\reaction}{(d,{}^3{\rm He})}
%\newcommand{\reaction}{d + {\rm HI} \rightarrow {}^3{\rm He} + X}
\newcommand{\inverse}{D({\rm HI},{}^3{\rm He})}
%\newcommand{\inverse}{{\rm HI} + d \rightarrow {}^3{\rm He} + X}
\newcommand{\qq}{\Braket{\bar{q}q}}
\newcommand{\sigmap}{\sigma_{\pi N}}
\newcommand{\Sn}[1]{{}^{#1}{\rm Sn}}
\newcommand{\C}[1]{{}^{#1}{\rm C}}
\newcommand{\etap}{\eta^\prime}
\newcommand{\micron}{$\mu$m}
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\title{Chiral symmetry restoration in nuclear medium observed in pionic atoms}
\author{
K.~Itahashi$^{1,2}$,
T.~Nishi$^{1}$,
D.~Ahn$^{1,3}$,
G.~P.A.~Berg$^{4}$,
M.~Dozono$^{1}$,
D.~Etoh$^{5}$,
H.~Fujioka$^{6}$,
N.~Fukuda$^{1}$,
N.~Fukunishi$^{1}$,
H.~Geissel$^{7}$,
E.~Haettner$^{7}$,
T.~Hashimoto$^{2,8}$,
R.~S.~Hayano$^{9}$,
S.~Hirenzaki$^{10}$,
H.~Horii$^{9}$,
N.~Ikeno$^{11}$,
N.~Inabe$^{1}$,
M.~Iwasaki$^{1,2}$,
D.~Kameda$^{1}$,
K.~Kisamori$^{12}$,
Y.~Kiyokawa$^{12}$,
T.~Kubo$^{1}$,
K.~Kusaka$^{1}$,
M.~Matsushita$^{12}$,
S.~Michimasa$^{12}$,
G.~Mishima$^{9}$,
H.~Miya$^{1}$,
D.~Murai$^{1}$,
H.~Nagahiro$^{10}$,
M.~Niikura$^{9}$,
N.~Nose-Togawa$^{13}$,
S.~Ota$^{12}$,
N.~Sakamoto$^{1}$,
K.~Sekiguchi$^{5}$,
Y.~Shiokawa$^{5}$,
H.~Suzuki$^{1}$,
K.~Suzuki$^{7,14}$,
M.~Takaki$^{12}$,
H.~Takeda$^{1}$,
Y.~K.~Tanaka$^{2}$,
T.~Uesaka$^{1}$,
Y.~Wada$^{5}$,
A.~Watanabe$^{5}$,
Y.~N.~Watanabe$^{9}$,
H.~Weick$^{7}$,
H.~Yamakami$^{6}$,
Y.~Yanagisawa$^{1}$,
K.~Yoshida$^{1}$}
\instlist{
\inst{\it $^{1}$ RIKEN Nishina Center for Accelerator-Based Science, RIKEN, Saitama, Japan}
\inst{\it $^{2}$ RIKEN Cluster for Pioneering Research, RIKEN, Saitama, Japan}
\inst{\it $^{3}$ Center for Exotic Nuclear Studies, Institute for Basic Science (IBS), Daejeon, Republic of Korea}
\inst{\it $^{4}$ Department of Physics and the Joint Institute for Nuclear Astrophysics Center for the Evolution of the Elements, University of Notre Dame, Indiana, USA}
\inst{\it $^{5}$ Department of Physics, Tohoku University, Sendai, Japan}
\inst{\it $^{6}$ Department of Physics, Kyoto University, Kyoto, Japan}
\inst{\it $^{7}$ GSI Helmholtzzentrum f\"{u}r Schwerionenforschung GmbH, Darmstadt, Germany}
\inst{\it $^{8}$ Advanced Science Research Center, Japan Atomic Energy Agency, Ibaraki, Japan}
\inst{\it $^{9}$ Department of Physics, School of Science, University of Tokyo, Tokyo, Japan}
\inst{\it $^{10}$ Department of Physics, Nara Women's University, Nara, Japan}
\inst{\it $^{11}$ Department of Life and Environmental Agricultural Sciences, Faculty of Agriculture, Tottori University, Tottori, Japan}
\inst{\it $^{12}$ Center for Nuclear Study, the University of Tokyo, Saitama, Japan}
\inst{\it $^{13}$ Research Center for Nuclear Physics, Osaka University, Osaka, Japan}
\inst{\it $^{14}$ Ruhr-Universit\"at Bochum, Bochum, Germany}
}
%\collaboration{piAF Collaboration}
%% When only one author is present, please do not use the command \from{} near the author name.
\begin{document}
\maketitle
\begin{abstract}
We interpret the spectral information of the
pionic $1s$ and $2p$ states in the $\Sn{121}$ nucleus
observed with unprecedented precision and resolution with respect to
the in-medium pion-nucleus interaction to deduce partial
restoration of the chiral symmetry in the high density of the nuclear matter.
Most recent theoretical and experimental results are integrated to obtain
the precision information on the partial restoration of the chiral symmetry.
We find reduction of the chiral condensate in the Sn nucleus by a factor
of $77 \pm 2$\% at the nucleon density of 0.098 fm$^{-3}$.
The result is compared with the chiral theories showing fairly good agreement.
\end{abstract}
% 5 pages
\section{Introduction}
Deeply bound pionic atoms have repulsive level shifts in the
lowest orbitals representing the dominance of the $s$-wave strong
interaction~\cite{Nishi23,Itahashi22}, which is subject to modification in
the nuclear medium effect~\cite{Weise93}.
Experimental and theoretical analyses of the the medium effect
contribute to the understanding of the chiral symmetry at the high density using
the pion as a probe~\cite{Suzuki04,Jido08}.
The $s$-wave interaction is modified due to the partial restoration of the chiral symmetry
in the high-density nuclear matter through
the wavefunction renormalization of the medium effects~\cite{Kolomeitsev03}.
The expectation value of the chiral condensate $\qq$ is
an order parameter of the chiral symmetry, and its absolute
value is known to be reduced at high temperature or
high density~\cite{Weise93}. The temperature dependence has been studied
in the lattice QCD calculations, which show a phase transition near
$T \sim$ 150 MeV~\cite{DeTar09,Fu20}. High-energy nuclear collision experiments
report the observation of the color deconfinement phase transition to
the quark-gluon-plasma phase~\cite{RHIC}, where the color degree of freedom is
manifested. It should be noted that the quark confinement transition must
have a strong correlation with the chiral transition, but the relationship
has not yet been elaborated~\cite{Suganuma17}.
There are experimental projects to measure
the mass modifications of the pseudoscalar or vector
mesons in the nuclear medium~\cite{Tanaka16,E325}.
However, our experimental knowledge of the density dependence of $\qq$
has been limited. The powerful tool of the
lattice QCD calculations faces the difficulties of the
sign problems in the high densities.
The repulsive $s$-wave interaction
provides information on $\qq$ at nuclear densities.
The interaction is phenomenologically described by
an optical potential in Ericson-Ericson formulation~\cite{Ericson66}.
%% with slight modification as
%% \begin{eqnarray*}
%% 2\mu U_{\rm opt}(r) &=& U_s(r) + U_p(r) \label{eqn:E-E opt}\\
%% U_s(r) &=& -4\pi [\epsilon_1\{b_0\rho(r)+b_1 \Delta\rho(r) \} + \frac{4}{3}\epsilon_2 B_0(\rho_p(r)^2+2\rho_n(r)\rho_p(r))]\\
%% \rho(r) &=& \rho_n(r)+\rho_p(r)\\
%% \Delta \rho(r) &=& \rho_n(r)-\rho_p(r),
%% \end{eqnarray*}
%% where $\epsilon_1 = 1 + \mu/M = 1.147, \epsilon_2 = 1 + \mu/2M =1.073$~\cite{Ericson66}.
%% $\mu$ and $M$ are the pion-nucleus reduced mass and
%% the nucleon mass, respectively. $U_s(r)$ and $U_p(r)$
%% denote the $s$-wave and $p$-wave parts, respectively.
The isoscalar $b_0\rho$ and Re$B_0\rho^2$ terms are correlated~\cite{Seki83} indicating
that the pion is sensitive to the
potential at the effective density $\rho_e \simeq 0.098$ fm$^{-3}$~\cite{Itahashi00},
which corresponds to the overlap between the pion wavefunction and the nuclear densities.
Since the isoscalar interaction is small, the leading term is
the isovector term $b_1(\rho_n-\rho_p)$. Theoretically, $b_1$
has $\rho$ dependence for the medium effect. Considering the
wavefunction renormalization, the $b_1$ parameter is model-independently
related to in-medium $\qq$~\cite{Jido08} at the density $\rho_e$ probed by the pionic atoms as
\begin{equation}
\frac{\qq(\rho_e)}{\qq(0)} \simeq \left(\frac{b_1^v}{b_1}\right)^{1/2}\left(1-\gamma\frac{\rho_e}{\rho_c}\right),
\label{eqn:qqb1}
\end{equation}
where $\rho_c \equiv 0.17$ fm$^{-3}$ is the normal nuclear density and the coefficient $\gamma = 0.184 \pm 0.003$.
$b_1^v=-0.0866 \pm 0.0010 m_\pi^{-1}$ denotes pion-nucleon isovector
interaction determined in pionic hydrogen and deuterium measurements~\cite{Hirtl:2021tb}.
\section{Experiment and Analysis}
We made a spectroscopy experiment
of pionic $\Sn{121}$ atoms to deduce $\qq(\rho_e)/\qq(0)$
through determination of $b_1$.
A key is to achieve high resolution and high statistical sensitivity.
Simultaneous observation of the pionic $1s$ and $2p$ states~\cite{Nishi18} contributes largely to
reduction of the systematic errors. However, this requires very high statistical sensitivity.
The experiment was performed at the RI Beam Factory (RIBF), RIKEN.
We employed deuteron beam of $10^{12}$/s with the energy of 250 MeV/nucleon.
The deuteron beam impinged on a $\Sn{122}$ target with the thickness of 12.5 $\pm$ 0.5 mg/cm$^2$
to make $\reaction$ reactions.
The emitted $\helium$ is momentum-analyzed by the fragment separator BigRIPS~\cite{Kubo03}.
We installed two sets of multiwire-drift-chambers to measure the $\helium$
tracks. The largest resolution contribution was the beam momentum
spread. We developed a dedicated ion-optics~\cite{Nishi13} to realize a dispersion
matching condition~\cite{Fujita02} for the primary deuteron beam, which reduced the contribution.
The produced pionic atoms are coupled with neutron hole states mainly of
$2d_{3/2}, 3s_{1/2}$ and $2d_{5/2}$~\cite{Ikeno11}. We measured the missing mass spectrum
near the pion emission threshold for the scattering angle $\theta < 1.5$ degrees
as depicted in Fig.~1. The vertical bars show the statistical errors. As seen in the error bars,
we achieved very high statistics.
The abscissa is the excitation energy $E_x$ of the reaction products. The $\pim$ emission threshold is
shown by the vertical line. The ordinate is the double differential cross sections.
Note that the absolute value of the differential cross sections is associated with
systematic uncertainty of about 30\%.
The energy resolution has a parabolic $E_x$ dependence, and the
best energy resolution is achieved to be 287 keV (FWHM) near $E_x \sim 138.5$ MeV.
\begin{figure}
\begin{center}
\includegraphics[width=9cm]{Fig2_Spectrum_v9f}
\label{Fig:2}
\caption{Measured excitation spectrum of the $\reaction$ reaction for $\theta < 1.5$ degrees.
The peak near $E_x = 135.7$ MeV is assigned to formation of pionic $\Sn{121}$ atoms in the $1s_\pi$ state and a smaller peak near $E_x = 137.3$ MeV
to the $2p_\pi$ state. The grey curve shows a fitting of the spectrum
in the $E_x$ region indicated by the arrows.}
\end{center}
\vskip-9mm
\end{figure}
The largest peak near $E_x=135.7$ MeV is assigned to formation of
pionic $1s_\pi$ state coupled with neutron hole states of $3s_{1/2}$.
The second largest peak near the excitation energy $E_x=137.3$ MeV is mainly contributed
from $2p_\pi$. We have made a fit of the spectrum with a sum of theoretical
spectra of each state calculated by the effective number approach~\cite{Ikeno11}.
We made use of recent data of the spectroscopic factors of $\Sn{122}$~\cite{Szwec21}
in calculation of the theoretical spectra.
Fitting parameters are the binding energies ($B_\pi$),
widths ($\Gamma_\pi$) and cross sections of the $1s_\pi$ and $2p_\pi$ states
and a linear background. The fitting region was $E_x = [132.0,137.8]$ MeV.
The fitting region is indicated by the arrows.
We achieved
$B_\pi(1s) = 3830 \pm 3 ({\it stat.}) ^{+78}_{-76} ({\it syst.})$ keV,
$B_\pi(2p) = 2265 \pm 3 ({\it stat.}) ^{+84}_{-83} ({\it syst.})$ keV,
$\Gamma_\pi(1s) = 1565 \pm 11 ({\it stat.}) ^{+43}_{-40} ({\it syst.})$ keV,
and $\Gamma_\pi(2p) = 314 \pm 12 ({\it stat.}) ^{+49}_{-28} ({\it syst.})$ keV,
with the fitting $\chi^2/{\rm n.d.f.} = 231.3/108$.
A remarkable fact is the accuracy of the differences
$B_\pi(1s)-B_\pi(2p) = 1565 \pm 4 \pm 11$ keV and
$\Gamma_\pi(1s)-\Gamma_\pi(2p) = 194 \pm 16 ^{+31}_{-42}$ keV,
where a large fraction of the systematic errors is eliminated
in the binding energies since the systematic errors are mainly contributed
from the $E_x$ calibration.
The spectroscopic information of the pionic atoms sets constraints on the
pion-nucleus interaction. Solving Klein-Gordon equation, we have calculated
theoretical binding energies and widths, and compared with the $B_\pi$ and $\Gamma_\pi$
obtained above to calculate the likelihood. In the likelihood calculation, we have taken
into consideration the statistical and systematic errors with correlations, and
combined pionic-atom information of light spherical nuclei of $^{16}{\rm O}$,
$^{20}{\rm Ne}$, and $^{28}{\rm Si}$~\cite{Batty97} to set constraints in the isoscalar terms.
The $p$-wave parameters are fixed to the ``Global-2'' parameters in Table 2 of Ref~\cite{Friedman03}.
We thus obtained a likelihood contour on the plane of the isovector parameter $b_1$
and the absorption parameter Im$B_0$, which has a maximum at $b_1 = -0.0952 m_\pi^{-1}$ and Im$B_0=0.0469 m_\pi^{-4}$.
This achieved value shows slight deviation from the value in a preceding
experiment $b_1 = -0.1149 \pm 0.0074 m_\pi^{-1}$ and Im$B_0 = 0.0472 \pm 0.0013 m_\pi^{-4}$.
Before deduction of in-medium $\qq$, we have introduced several corrections
to improve the accuracy. We have considered differences
in the theoretical spectra calculated by the effective number approach~\cite{Ikeno11} and Green's function
method~\cite{Ikeno15}, differences in the $\rho_n(r)$ between the two-parameter Fermi model and
high precision data of proton elastic scattering experiment~\cite{Terashima08}, and
the residual interaction between the
pion and the nucleus~\cite{Nose-Togawa05}.
These corrections affect the likelihood contour, and the
finally deduced values are $b_1 = -0.1163 \pm 0.0056 m_\pi^{-1}$ and Im$B_0= 0.0473 \pm 0.0013 m_\pi^{-4}$.
\section{Discussion and Conclusion}
The above deduced $b_1$ at $\rho = \rho_e$ is larger than in-vacuum $b_1^v=-0.0866 \pm 0.0010 m_\pi^{-1}$~\cite{Hirtl:2021tb}
showing clear enhancement of the pion-nucleus repulsive interaction due to the wavefunction renormalization.
The deduced $b_1$ coincides with the preceding value of $b_1 = -0.1149 \pm 0.0074 m_\pi^{-1}$. However,
applying the corrections listed above, they exhibit a discrepancy of about two sigmas.
Now we calculate eq.\ref{eqn:qqb1} and obtain $\qq(\rho_e)/\qq(0) = 0.77 \pm 0.02$.
Thus deduced in-medium $\qq$ is
compared with values in the chiral effective theories. Figure 2 shows the deduced value
in the red filled circle with the error bars. The red hatched region shows an extrapolation
in linear density approximation to obtain $\qq(\rho_c)/\qq(0) = 60\pm 3$\% at the normal nuclear density
$\rho_c = 0.17$ fm$^{-3}$. Theoretical values in Refs.~\cite{Jido08,Friedman19,Huebsch21,Kaiser08,Goda13,Lacour_2010}
are also presented showing fairly good agreement.
\begin{figure}
\begin{center}
\includegraphics[width=9cm]{Fig3g}
\label{Fig:3}
\caption{Deduced in-medium $\qq(\rho)/\qq(0)$ and comparison with theories.}
\end{center}
\vskip-9mm
\end{figure}
For further studies, we have performed a systematic measurements of pionic Sn atoms to deduce
the density derivative of $\qq$.
%% According to chiral effective theories, we expect that
%% the density dependent $\qq(\rho)$ at the nuclear density $\rho$ can
%% be described ~\cite{Kaiser08} as
%% \begin{equation}
%% %\[ \qq(\rho_\mathrm{e})/\qq_0 = 1 - \frac{\rho}{f_\pi^2}\frac{\sigma_{\pi N}}{m_\pi^2} + O(\rho^2) \]
%% %\[ \qq(\rho)/\qq(0) = 1 - \rho \sigmap/ f_\pi^2m_\pi^2 + O(\rho^{2}) \vspace{-1mm}\]
%% \frac{\qq(\rho)}{\qq(0)} \simeq 1 - \frac{\rho \sigmap}{f_\pi^2m_\pi^2}\left(1-\frac{3k_{\rm F}^2}{10M^2}+\frac{9k_{\rm F}^4}{56M^4} \right)
%% + O(\rho^{n > 1}),
%% \vspace{-1mm}
%% \label{eq:sigmap}
%% \end{equation}
%% where $f_\pi \sim 92.2$ MeV is the pion weak decay constant
%% and $k_{\rm F}=(3/2\pi^2\rho)^{1/3}$ is the Fermi momentum.
Experimental determination of the $d\qq/d\rho$ at $\rho \sim \rho_e$
serves not only as confirmation of the theories but elaborate the insight of the non-trivial structure of the QCD vacuum.
\acknowledgments
The authors thank the staff of the RIBF for
the stable operation.
This experiment was performed at RIBF
operated by RIKEN Nishina Center and CNS, University of
Tokyo. This work is partly supported by Grants-in-Aid
for Scientific Research (Nos. JP22105517, JP24105712, and JP15H00844,
JP16340083, JP18H01242, JP16H02197, JP20540273, JP24540274, JP19K14709, JP12J08538, JP20KK0070),
Institute for Basic Science (IBS-R031-D1),
the Bundesministerium f\"ur Bildung und Forschung, and the
National Science Foundation through Grant No. Phys-0758100,
and the Joint Institute for Nuclear Astrophysics
through Grants No. Phys-0822648 and No. PHY-1430152
(JINA Center for the Evolution of the Elements).
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\end{thebibliography}
\end{document}
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