Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Chiral symmetry restoration at high matter density observed in pionic atoms


According to quantum chromodynamics, vacuum is not an empty space, because it is filled with quark–antiquark pairs. The pair has the same quantum numbers as the vacuum and forms a condensate because the strong interaction of the quantum chromodynamics is too strong to leave the vacuum empty. This quark–antiquark condensation, the chiral condensate, breaks the chiral symmetry of the vacuum. The expectation value of the chiral condensate is an order parameter of the chiral symmetry, which is expected to decrease at high temperatures or high matter densities where the chiral symmetry is partially restored. Head-on collisions of nuclei at ultra-relativistic energies have explored the high-temperature regime, but experiments at high densities are rare. Here we measure the spectrum of pionic 121Sn atoms and study the interaction between the pion and the nucleus. We find that the expectation value of the chiral condensate is reduced at finite density compared to the value in vacuum. The reduction is linearly extrapolated to the nuclear saturation density and indicates that the chiral symmetry is partially restored due to the extremely high density of the nucleus.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Get just this article for as long as you need it


Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Experimental layout.
Fig. 2: Measured spectra.
Fig. 3: Deduced in-medium \(\langle \bar{{\boldsymbol{q}}}{\boldsymbol{q}}\rangle ({\boldsymbol{\rho}} )\) normalized to \(\langle \bar{{\boldsymbol{q}}}{\boldsymbol{q}}\rangle ({\boldsymbol{0}})\) in vacuum.

Data availability

Raw data were generated at the RI Beam Factory. Source data are provided with this paper.

Code availability

The computer codes used to generate results are available from the corresponding author upon reasonable request.


  1. Weise, W. Nuclear aspects of chiral symmetry. Nucl. Phys. A 553, 59–72 (1993).

    Article  ADS  Google Scholar 

  2. Brown, G. E. & Rho, M. Chiral restoration in hot matter. Phys. Rep. 269, 333–380 (1996).

    Article  ADS  Google Scholar 

  3. Higgs, P. W. Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett. 13, 508–509 (1964).

    Article  ADS  MathSciNet  Google Scholar 

  4. DeTar, C. E. & Heller, U. M. QCD thermodynamics from the lattice. Eur. Phys. J. A 41, 405–437 (2009).

    Article  ADS  Google Scholar 

  5. Fu, W. J., Pawlowski, J. M. & Rennecke, F. QCD phase structure at finite temperature and density. Phys. Rev. D. 101, 054032 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  6. Stock, R. (ed.) Relativistic Heavy Ion Physics. Landolt-Börnstein Group I Elementary Particles, Nuclei and Atoms Vol. 23 (Springer, 2010).

  7. Fukushima, K. & Hatsuda, T. The phase diagram of dense QCD. Rep. Prog. Phys. 74, 014001 (2010).

    Article  ADS  Google Scholar 

  8. Friman, B. et al. (ed.) The CBM Physics Book: Compressed Baryonic Matter in Laboratory Experiments. Lecture Notes in Physics Vol. 814 (Springer, 2011).

  9. Muto, R. et al. Evidence for in-medium modification of the meson at normal nuclear density. Phys. Rev. Lett. 98, 042501 (2007).

    Article  ADS  Google Scholar 

  10. Kim, H. J. & Gubler, P. The ϕ meson with finite momentum in a dense medium. Phys. Lett. B 805, 135412 (2020).

    Article  MathSciNet  Google Scholar 

  11. Jido, D., Hatsuda, T. & Kunihiro, T. In-medium pion and partial restoration of chiral symmetry. Phys. Lett. B 670, 109–113 (2008).

    Article  ADS  Google Scholar 

  12. Klimt, S., Lutz, M., Vogl, U. & Weise, W. Generalized SU(3) Nambu-Jona-Lasinio model. Nucl. Phys. A 516, 429–468 (1990).

    Article  ADS  Google Scholar 

  13. Nagahiro, H., Jido, D., Fujioka, H., Itahashi, K. & Hirenzaki, S. Formation of η′(958)-mesic nuclei by the (p, d) reaction. Phys. Rev. C 87, 045201 (2013).

    Article  ADS  Google Scholar 

  14. Friedman, E. & Gal, A. The pion-nucleon σ term from pionic atoms. Phys. Lett. B 792, 340–344 (2019).

    Article  ADS  Google Scholar 

  15. Ajimura, S. et al. "Kpp", a K-meson nuclear bound state, observed in 3He(K, Λp)n reactions. Phys. Lett. B 789, 620–625 (2019).

  16. Yamaga, T. et al. Observation of a \({\bar{K}NN}\) bound state in the 3He(K, Λp)n reaction. Phys. Rev. C 102, 044002 (2020).

    Article  ADS  Google Scholar 

  17. Ishikawa, T. et al. Low-energy scattering parameters between the η meson and nucleon from η photoproduction on the deuteron. Acta Phys. Pol. B 48, 1801–1806 (2017).

    Article  ADS  Google Scholar 

  18. Tanaka, Y. K. et al. Measurement of excitation spectra in the 12C(p, d) reaction near the η′ emission threshold. Phys. Rev. Lett. 117, 202501 (2016).

    Article  ADS  Google Scholar 

  19. Suzuki, K. et al. Precision spectroscopy of pionic 1s states of Sn nuclei and evidence for partial restoration of chiral symmetry in the nuclear medium. Phys. Rev. Lett. 92, 072302 (2004).

    Article  ADS  Google Scholar 

  20. Gilg, H. et al. Deeply bound π states in 207Pb formed in the 208Pb(d, 3He) reaction. I. Experimental method and results. Phys. Rev. C 62, 025201 (2000).

    Article  ADS  Google Scholar 

  21. Itahashi, K. et al. Deeply bound π states in 207Pb formed in the 208Pb(d, 3He) reaction. II. Deduced binding energies and widths and the pion-nucleus interaction. Phys. Rev. C 62, 025202 (2000).

    Article  ADS  Google Scholar 

  22. Ericson, M. & Ericson, T. E. O. Optical properties of low-energy pions in nuclei. Ann. Phys. 36, 323–362 (1966).

    Article  ADS  MATH  Google Scholar 

  23. Batty, C., Friedman, E. & Gal, A. Strong interaction physics from hadronic atoms. Phys. Rep. 287, 385–445 (1997).

    Article  ADS  Google Scholar 

  24. Friedman, E. & Gal, A. In-medium nuclear interactions of low-energy hadrons. Phys. Rep. 452, 89–153 (2007).

    Article  ADS  Google Scholar 

  25. Yamazaki, T. et al. Discovery of deeply bound π states in the 208Pb(d, 3He) reaction. Z. Phys. A 335, 219–221 (1996).

    ADS  Google Scholar 

  26. Geissel, H. et al. Experimental indication of a reduced chiral order parameter from the 1s π state in 205Pb. Phys. Lett. B 549, 64–71 (2002).

  27. Yamazaki, T., Hirenzaki, S., Hayano, R. S. & Toki, H. Deeply bound pionic states in heavy nuclei. Phys. Rep. 514, 1–87 (2012).

    Article  ADS  Google Scholar 

  28. Tomozawa, Y. Axial-vector coupling constant renormalization and the meson-baryon scattering lengths. Nuovo Cim. A 46, 707–717 (1966).

    Article  ADS  Google Scholar 

  29. Weinberg, S. Pion scattering lengths. Phys. Rev. Lett. 17, 616–621 (1966).

    Article  ADS  Google Scholar 

  30. Kolomeitsev, E., Kaiser, N. & Weise, W. Chiral dynamics of deeply bound pionic atoms. Phys. Rev. Lett. 90, 9–12 (2003).

    Article  Google Scholar 

  31. Friedman, E. et al. Elastic scattering of low energy pions by nuclei and the in-medium isovector πN amplitude. Phys. Rev. C 72, 034609 (2005).

    Article  ADS  Google Scholar 

  32. Kubo, T. In-flight RI beam separator BigRIPS at RIKEN and elsewhere in Japan. Nucl. Instrum. Methods Phys. Res. B 204, 97–113 (2003).

    Article  ADS  Google Scholar 

  33. Ohya, S. Nuclear data sheets for A = 121. Nucl. Data Sheets 111, 1619–1806 (2010).

    Article  ADS  Google Scholar 

  34. Nishi, T. et al. Spectroscopy of pionic atoms in 122Sn(d, 3He) reaction and angular dependence of the formation cross sections. Phys. Rev. Lett. 120, 152505 (2018).

    Article  ADS  Google Scholar 

  35. Nishi, T. et al. BigRIPS as a high resolution spectrometer for pionic atoms. Nucl. Instrum. Methods Phys. Res. B 317, 290–293 (2013).

    Article  ADS  Google Scholar 

  36. Fujita, H. et al. Realization of matching conditions for high-resolution spectrometers. Nucl. Instrum. Methods Phys. Res. A 484, 17–26 (2002).

    Article  ADS  Google Scholar 

  37. Ikeno, N., Yamagata-Sekihara, J., Nagahiro, H. & Hirenzaki, S. Formation spectra of pionic atoms in the Green’s function method. Prog. Theor. Exp. Phys. 2015, 033D01 (2015).

    Article  MATH  Google Scholar 

  38. Szwec, S. V. et al. Neutron occupancies and single-particle energies across the stable tin isotopes. Phys. Rev. C 104, 054308 (2021).

    Article  ADS  Google Scholar 

  39. Friedman, E. & Gal, A. Renormalization of the isovector πN amplitude in pionic atoms. Nucl. Phys. A 724, 143–156 (2003).

    Article  ADS  Google Scholar 

  40. Fricke, G. et al. Nuclear ground state charge radii from electromagnetic interactions. Data Nucl. Data Tables 60, 177–285 (1995).

    Article  ADS  Google Scholar 

  41. Terashima, S. et al. Proton elastic scattering from tin isotopes at 295 MeV and systematic change of neutron density distributions. Phys. Rev. C. 77, 024317 (2008).

    Article  ADS  Google Scholar 

  42. Ficenec, J. R., Fajardo, L. A., Trower, W. P. & Sick, I. Elastic electron-tin scattering. Phys. Lett. B 42, 213–215 (1972).

    Article  ADS  Google Scholar 

  43. Nose-Togawa, N., Nagahiro, H., Hirenzaki, S. & Kume, K. Residual interaction effects on deeply bound pionic states in Sn and Pb isotopes. Phys. Rev. C 71, 061601 (2005).

    Article  ADS  Google Scholar 

  44. Hirenzaki, S., Toki, H. & Yamazaki, T. (d, 3He) reactions for the formation of deeply bound pionic atoms. Phys. Rev. C 44, 2472–2479 (1991).

    Article  ADS  Google Scholar 

  45. Hirtl, A. et al. Redetermination of the strong-interaction width in pionic hydrogen. Eur. Phys. J. A 57, 70 (2021).

    Article  ADS  Google Scholar 

  46. Chanfray, G., Ericson, M. & Oertel, M. In-medium modification of the isovector pion-nucleon amplitude. Phys. Lett. B 563, 61–67 (2003).

    Article  ADS  Google Scholar 

  47. Hübsch, S. & Jido, D. Density dependence of the quark condensate in isospin-asymmetric nuclear matter. Phys. Rev. C 104, 015202 (2021).

    Article  ADS  Google Scholar 

  48. Kaiser, N., De Homont, P. & Weise, W. In-medium chiral condensate beyond linear density approximation. Phys. Rev. C 77, 025204 (2008).

    Article  ADS  Google Scholar 

  49. Goda, S. & Jido, D. Chiral condensate at finite density using the chiral Ward identity. Phys. Rev. C 88, 065204 (2013).

    Article  ADS  Google Scholar 

  50. Lacour, A., Oller, J. A. & Meißner, U.-G. The chiral quark condensate and pion decay constant in nuclear matter at next-to-leading order. J. Phys. G 37, 125002 (2010).

    Article  ADS  Google Scholar 

  51. Ikeno, N. et al. Precision spectroscopy of deeply bound pionic atoms and partial restoration of chiral symmetry in medium. Prog. Theor. Phys. 126, 483–509 (2011).

    Article  ADS  Google Scholar 

Download references


We thank the staff of the RI Beam Factory for stable operation of the facility. This experiment was performed at the RI Beam Factory, operated by RIKEN Nishina Center and CNS, University of Tokyo. This work is partly supported by the MEXT Grant-in-Aid for Scientific Research on Innovative Areas (nos. JP22105517, JP24105712 and JP15H00844 to K.I.), JSPS Grant-in-Aid for Scientific Research (B) (nos. JP16340083 and JP18H01242 to K.I.), (A) (no. JP16H02197 to K.I. and T.U.) and (C) (nos. JP16K05355 and JP24540274 to S.H.), Grant-in-Aid for Early-Career Scientists (no. JP19K14709 to N. Ikeno), Grant-in-Aid for JSPS Research Fellow (no. JP12J08538 to T.N.), JSPS Fund for the Promotion of Joint International Research (Fostering Joint International Research (B); no. JP20KK0070 to K.I.), Institute for Basic Science (IBS-R031-D1 to D.A.), the Bundesministerium für Bildung und Forschung (H.G., E.H. and H.W.), the National Science Foundation through grant no. Phys-0758100, and the Joint Institute for Nuclear Astrophysics (grants nos. Phys-0822648 and PHY-1430152; JINA Center for the Evolution of the Elements to G.P.A.B.).

Author information

Authors and Affiliations




T.N. and K.I. designed the experimental concepts and performed experiments, developed detectors and ion optics, analysed data, performed theoretical calculations, and wrote the paper. G.P.A.B., M.D., H.F., N. Fukuda, N. Fukunishi, H.G., E.H., K. Kusaka, N.S., H.S., K.S., H.T., Y.K.T., T.U., Y.W. and H.W. developed the ion optics and performed experiments. S.H. and N. Ikeno designed experimental concepts, performed theoretical calculations and wrote the paper. N.N.-T. performed theoretical calculations and wrote the paper. H.M. developed the detectors. H.N. and M.I. designed the experimental concepts. All other authors performed experiments.

Corresponding author

Correspondence to Kenta Itahashi.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Paul Indelicato and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Source data

Source Data Fig. 2

Measured spectra data in Figure 2.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nishi, T., Itahashi, K., Ahn, D. et al. Chiral symmetry restoration at high matter density observed in pionic atoms. Nat. Phys. (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing