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Precision tests of the nonlinear mode coupling of anisotropic flow via high-energy collisions of isobars

Published 14 Jun 2022 in nucl-th, hep-ph, and nucl-ex | (2206.07184v2)

Abstract: Valuable information on the dynamics of expanding fluids can be inferred from the response of such systems to perturbations in their initial geometry. We apply this technique in high-energy ${96}$Ru+${96}$Ru and ${96}$Zr+${96}$Zr collisions to scrutinize the expansion dynamics of the quark-gluon plasma, where the initial geometry perturbations are sourced by the differences in deformations and radial profiles between ${96}$Ru and ${96}$Zr, and the collective response is captured by the change in anisotropic flow $V_n$ between the two collision systems. Using a transport model, we analyze how the nonlinear coupling between lower-order flow harmonics $V_2$ and $V_3$ to the higher-order flow harmonics $V_4$ and $V_5$, expected to scale as $V_{4\mathrm{NL}}=\chi_4 V_22$ and $V_{5\mathrm{NL}}=\chi_5 V_2V_3$, gets modified as one moves from ${96}$Ru+${96}$Ru to ${96}$Zr+${96}$Zr systems. We find that these scaling relations are valid to high precision: variations of order 20\% in $V_{4\mathrm{NL}}$ and $V_{5\mathrm{NL}}$ due to differences in quadrupole deformation, octupole deformation, and nuclear skin modify $\chi_{4}$ and $\chi_5$ by about 1--2\%. Percent-level deviations are however larger than the expected experimental uncertainties and could be measured. Therefore, collisions of isobars with different nuclear structures are a unique tool to isolate subtle nonlinear effects in the expansion of the quark-gluon plasma that would be otherwise impossible to access in a single collision system.

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(ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Peng Huo, Jiangyong Jia,  and Soumya Mohapatra, “Elucidating the event-by-event flow fluctuations in heavy-ion collisions via the event shape selection technique,” Phys. Rev. C 90, 024910 (2014), arXiv:1311.7091 [nucl-ex] . Adam et al. (2016) Jaroslav Adam et al. (ALICE), “Correlated event-by-event fluctuations of flow harmonics in Pb-Pb collisions at sNN=2.76subscript𝑠NN2.76\sqrt{s_{{}_{\rm NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT start_FLOATSUBSCRIPT roman_NN end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG = 2.76 TeV,” Phys. Rev. Lett. 117, 182301 (2016), arXiv:1604.07663 [nucl-ex] . Jia (2022) Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Jaroslav Adam et al. (ALICE), “Correlated event-by-event fluctuations of flow harmonics in Pb-Pb collisions at sNN=2.76subscript𝑠NN2.76\sqrt{s_{{}_{\rm NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT start_FLOATSUBSCRIPT roman_NN end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG = 2.76 TeV,” Phys. Rev. Lett. 117, 182301 (2016), arXiv:1604.07663 [nucl-ex] . Jia (2022) Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] .
  35. Peng Huo, Jiangyong Jia,  and Soumya Mohapatra, “Elucidating the event-by-event flow fluctuations in heavy-ion collisions via the event shape selection technique,” Phys. Rev. C 90, 024910 (2014), arXiv:1311.7091 [nucl-ex] . Adam et al. (2016) Jaroslav Adam et al. (ALICE), “Correlated event-by-event fluctuations of flow harmonics in Pb-Pb collisions at sNN=2.76subscript𝑠NN2.76\sqrt{s_{{}_{\rm NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT start_FLOATSUBSCRIPT roman_NN end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG = 2.76 TeV,” Phys. Rev. Lett. 117, 182301 (2016), arXiv:1604.07663 [nucl-ex] . Jia (2022) Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Jaroslav Adam et al. (ALICE), “Correlated event-by-event fluctuations of flow harmonics in Pb-Pb collisions at sNN=2.76subscript𝑠NN2.76\sqrt{s_{{}_{\rm NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT start_FLOATSUBSCRIPT roman_NN end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG = 2.76 TeV,” Phys. Rev. Lett. 117, 182301 (2016), arXiv:1604.07663 [nucl-ex] . Jia (2022) Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] .
  36. Jaroslav Adam et al. (ALICE), “Correlated event-by-event fluctuations of flow harmonics in Pb-Pb collisions at sNN=2.76subscript𝑠NN2.76\sqrt{s_{{}_{\rm NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT start_FLOATSUBSCRIPT roman_NN end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG = 2.76 TeV,” Phys. Rev. Lett. 117, 182301 (2016), arXiv:1604.07663 [nucl-ex] . Jia (2022) Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] .
  37. Jiangyong Jia, “Shape of atomic nuclei in heavy ion collisions,” Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . Note (1) In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] .
  38. In principle, the nonlinear mode V12⁢V2superscriptsubscript𝑉12subscript𝑉2V_{1}^{2}V_{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is also allowed. However, the pTsubscript𝑝Tp_{\mathrm{T}}italic_p start_POSTSUBSCRIPT roman_T end_POSTSUBSCRIPT-integrated V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known to be very small Aad et al. (2012). Thus, due to the presence of V12superscriptsubscript𝑉12V_{1}^{2}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this mode should be strongly suppressed. Aad et al. (2012) Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] . Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] .
  39. Georges Aad et al. (ATLAS), “Measurement of the azimuthal anisotropy for charged particle production in sNN=2.76subscript𝑠NN2.76\sqrt{s_{\mathrm{NN}}}=2.76square-root start_ARG italic_s start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT end_ARG = 2.76 TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86, 014907 (2012), arXiv:1203.3087 [hep-ex] .
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