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From imaginary to real chemical potential QCD with functional methods

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Abstract

We investigate the quality of the extrapolation procedure employed in Ref. [1] to extract the crossover line at real chemical potential from lattice data at imaginary potential. To this end we employ a functional approach that does not suffer from the sign problem. We utilize a well-studied combination of lattice Yang–Mills theory with a truncated set of Dyson–Schwinger equations in Landau gauge for \(2 + 1\) quark flavors. This system predicts a critical endpoint at moderate temperatures and rather large (real) chemical potential with a curvature of the pseudo-critical transition line comparable to recent lattice extrapolations. We determine the light quark condensate and chiral susceptibility at imaginary chemical potentials and perform an analytic continuation along the lines described in Borsányi et al. (Phys Rev Lett 125:052001. https://doi.org/10.1103/PhysRevLett.125.052001. arXiv:2002.02821 [hep-lat], 2020). We find that the analytically continued crossover line agrees very well (within one percent) with the explicitly calculated one for chemical potentials up to about 80 % of the one of the critical end point. The method breaks down in the region where the chiral susceptibility as a function of the condensate cannot any longer be well described by a polynomial.

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Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: Data are stored in a local repository and are available on request from christian.fischer@theo.physik.uni-giessen.de].

Notes

  1. We work in Euclidean space-time with positive metric signature (++++). The Hermitian \(\gamma \)-matrices satisfy \(\{\gamma _{\nu }, \gamma _{\rho }\} = 2 \delta _{\nu \rho }\).

  2. In principle, one could also determine the derivative directly, but this is about an order of magnitude more involved in terms of CPU-time. We have tested the quality of the finite difference method in comparable cases and found it to be accurate on the sub-percent level.

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Acknowledgements

We thank both, Jana N. Guenther and Philipp Isserstedt for enlightening discussions. We furthermore thank Jana N. Guenther for providing the lattice data for comparison and Philipp Isserstedt for crosschecks of the numerical code at an early stage of this work. This work has been supported by the Helmholtz Graduate School for Hadron and Ion Research (HGS-HIRe) for FAIR, the GSI Helmholtzzentrum für Schwerionenforschung and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Center TransRegio CRC-TR 211 “Strong-interaction matter under extreme conditions”. This work is furthermore part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement STRONG - 2020 - No 824093. Feynman diagrams were drawn with JaxoDraw [43].

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  1. Institut für Theoretische Physik, Justus-Liebig-Universität Gießen, 35392, Gießen, Germany

    Julian Bernhardt & Christian S. Fischer

  2. Helmholtz Forschungsakademie Hessen für FAIR (HFHF), GSI Helmholtzzentrum für Schwerionenforschung, Campus Gießen, 35392, Gießen, Germany

    Julian Bernhardt & Christian S. Fischer

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Bernhardt, J., Fischer, C.S. From imaginary to real chemical potential QCD with functional methods. Eur. Phys. J. A 59, 181 (2023). https://doi.org/10.1140/epja/s10050-023-01098-1

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