Abstract
We formulate a method for calculating the hadron-hadron scattering amplitudes at nonzero chemical potential (μ) in the hadronic phase at zero temperature, where the baryon number symmetry remains to be violated. Although it is widely believed that the physical quantities do not change even if we turn on a small μ at zero temperature, the shape of correlation functions for a single hadron depends on μ. Then, the dispersion relation of the single hadron is modified to E(p, μ) = \( \sqrt{{\textbf{p}}^2+{m}^2} \) – μnO. Here, m and nO denote the hadron mass at μ = 0 and the quantum number, respectively. From this relation, it is possible that the effective mass of the hadron depends on μ. We extend the HAL QCD method at μ = 0 to the case of μ ≠ 0, which allows us to extract the scattering phase shifts via the interaction potential. We have found that the interaction potential can depend on μ only through the effective mass while the scattering phase shifts, obtained by solving the Schrödinger equation with the interaction potential, are independent of μ. We also numerically analyze the S-wave scatterings of two pions with isospin I = 2 and two scalar diquarks within the framework of QC2D at nonzero quark chemical potential. While the lattice is not exactly set to zero temperature, the μ-independence can be observed. Furthermore, we improve the results for the S-wave scatterings of two hadrons obtained above by taking the μ-independence for granted. Thanks to the asymmetric property of the correlation functions for diquarks at μ ≠ 0, we can access a long-τ regime and can reduce the systematic error coming from inelastic contributions.
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Acknowledgments
We would like to thank S. Aoki, T. Doi, T.M. Doi, T. Hatsuda, D. Suenaga, and Y. Tanizaki for their useful conversations. K. M. especially thanks S. Aoki for fruitful discussions about analytical calculation. The numerical simulations are supported by the HPCI-JHPCN System Research Project (Project ID: jh220021) and HOKUSAI in RIKEN. The work of K. I. is supported by JSPS KAKENHI with Grant Numbers 18H05406 and 23H01167. The work of E. I. is supported by JSPS KAKENHI with Grant Number 19K03875, JST PRESTO Grant Number JPMJPR2113, JSPS Grant-in-Aid for Transformative Research Areas (A) JP21H05190 and JST Grant Number JPMJPF2221. K. M. is supported in part by JST SPRING, Grant Number JPMJSP2110, by Grants-in-Aid for JSPS Fellows (Nos. JP22J14889, JP22KJ1870), and by JSPS KAKENHI Grant No. 22H04917. This work is also supported by JPMXP1020230411.
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Murakami, K., Itou, E. & Iida, K. Chemical potential (in)dependence of hadron scatterings in the hadronic phase of QCD-like theories and its applications. J. High Energ. Phys. 2024, 152 (2024). https://doi.org/10.1007/JHEP02(2024)152
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DOI: https://doi.org/10.1007/JHEP02(2024)152