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Investigations of symmetry breakings by multiple condensates in exotic chiral condensed phases

Joshua Murakami and Kentaro Hayashi*

Yasuhiko Tsue

  • Graduate School of Integrated Arts and Sciences, Kochi University, Kochi 780-8520, Japan

  • Department of Mathematics and Physics, Faculty of Science and Technology, Kochi University, Kochi 780-8520, Japan

  • *Present address: Department of Mathematics and Physics, Faculty of Science and Technology, Kochi University, Kochi 780-8520, Japan.
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Phys. Rev. D 112, 014024 – Published 15 July, 2025

DOI: https://doi.org/10.1103/j476-lc2f

Abstract

The Nambu-Goldstone modes on the exotic chiral condensed phase with chiral and tensor-type quark-antiquark condensates are investigated by using the two-point vertex functions. It is shown that one of the Nambu-Goldstone modes appears as a result of meson mixing. As is well known, another method to find the Nambu-Goldstone modes is given by the use of the algebraic commutation relations between broken generators and massless modes obtained through the spontaneous symmetry breaking. This method is adopted to the cases of the chiral symmetry breakings due to the tensor-type condensate and the inhomogeneous chiral condensate. The result obtained by the use of the meson two-point vertex functions is obviously reproduced in the case of the tensor-type condensate. Furthermore, we investigate the general rules for determining the broken symmetries and the Nambu-Goldstone modes algebraically. As examples, the symmetry breaking pattern and the Nambu-Goldstone modes due to the tensor-type condensate or the inhomogeneous chiral condensate are shown by adopting the general rules developed in this paper in the algebraic method.

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Phys. Rev. D 112, 014024– Published 15 July, 2025
  • Received 17 March 2025
  • Accepted 10 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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