Skip to main content

Collectivity in large and small systems formed in ultrarelativistic collisions

Abstract

Collective flow of the final-state hadrons observed in ultrarelativistic heavy-ion collisions or even in smaller systems formed in high-multiplicity pp and p/d/\(^3\)He-nucleus collisions is one of the most important diagnostic tools to probe the initial state of the system and to shed light on the properties of the short-lived, strongly interacting many-body state formed in these collisions. Limited, in the initial years, to the study of mainly the directed and elliptic flows—the first two Fourier harmonics of the single-particle azimuthal distribution—this field has evolved in recent years into a much richer area of activity. This includes not only higher Fourier harmonics and multiparticle cumulants, but also a variety of other related observables, such as the ridge seen in two-particle correlations, flow decorrelation, symmetric cumulants and event-plane correlators which measure correlations between the magnitudes or phases of the complex flows in different harmonics, coefficients that measure the nonlinear hydrodynamic response, statistical properties, e.g. the non-Gaussianity of the flow fluctuations, etc. We present a Tutorial Review of the modern flow picture and the various aspects of the collectivity—an emergent phenomenon in quantum chromodynamics.

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

Fig. 1

Figure adapted from [3]

Fig. 2
Fig. 3

Figure from [22]

Fig. 4

Figure from [22]

Fig. 5

Figure from [24]

Fig. 6
Fig. 7
Fig. 8

Figure from [38]

Fig. 9
Fig. 10

Figure from [30]

Fig. 11

Figure from [62]

Fig. 12

Figure from [64]

Fig. 13

Figure from [77]

Fig. 14

Notes

  1. 1.

    See Eqs. (1) and (4) for definitions of PP (\(\varPhi _n\)) and EP (\(\varPsi _n\)) specific to harmonic n.

  2. 2.

    \(n=1\) forms a special case [5].

  3. 3.

    \(V_n\) is sometimes called a flow vector, and ‘flow’ and ‘azimuthal anisotropy’ are often used synonymously.

  4. 4.

    These names are suggestive of the shapes of the polar plots \(r=1+2v_n \cos n\phi \), for \(0 < v_n \ll 1\).

  5. 5.

    Nonflow correlations are not related to the initial-state geometry and hence not associated with the symmetry plane \(\varPsi _n\), but arise due to jets, particle decays, etc. They are of short range.

  6. 6.

    \(v_n\{m\}\) are also called multiparticle cumulants of order m of the flow \(v_n\).

  7. 7.

    Recall the discussion in the last paragraph of Sect. 2.3.

  8. 8.

    ATLAS collaboration denotes the event plane by \(\varPhi _n\), which is different from our convention here. In Ref. [43], we have used the ATLAS convention.

  9. 9.

    Note also that \(\left\langle V_{n\mathrm{{L}}} \right\rangle =0\), because \(V_{n\mathrm{{L}}}\), like \(V_n\), is expected to carry a random phase factor depending on the reaction-plane angle \(\varPhi _{\mathrm{RP}}\).

  10. 10.

    A recent experiment [12] has shown that the ordering persists up to \(n=7\), with some enhancement for \(n=8,9\).

  11. 11.

    For simplicity of notation, we use the same symbols \(\rho \) and C to denote 1-, 2-, 3- and multiparticle correlation functions and cumulants, respectively.

References

  1. 1.

    S.A. Chin, Phys. Lett. B 78, 552 (1978). https://doi.org/10.1016/0370-2693(78)90637-8

    ADS  Article  Google Scholar 

  2. 2.

    F.B. Yano, S.E. Koonin, Phys. Lett. B 78, 556 (1978). https://doi.org/10.1016/0370-2693(78)90638-X

    ADS  Article  Google Scholar 

  3. 3.

    S.A. Voloshin, A.M. Poskanzer, R. Snellings, Landolt-Bornstein 23, 293 (2010). arXiv:0809.2949 [nucl-ex]

  4. 4.

    B. Alver, G. Roland, Phys. Rev. C 81, 054905 (2010). (Erratum: [Phys. Rev. C 82, 039903 (2010)]). https://doi.org/10.1103/PhysRevC.82.039903. https://doi.org/10.1103/PhysRevC.81.054905. arXiv:1003.0194 [nucl-th]

  5. 5.

    U. Heinz, R. Snellings, Ann. Rev. Nucl. Part. Sci. 63, 123–151 (2013). https://doi.org/10.1146/annurev-nucl-102212-170540. arXiv:1301.2826 [nucl-th]

    ADS  Article  Google Scholar 

  6. 6.

    D. Teaney, L. Yan, Phys. Rev. C 90(2), 024902 (2014). https://doi.org/10.1103/PhysRevC.90.024902. arXiv:1312.3689 [nucl-th]

    ADS  Article  Google Scholar 

  7. 7.

    J.Y. Ollitrault, Phys. Rev. D 46, 229 (1992). https://doi.org/10.1103/PhysRevD.46.229

    ADS  Article  Google Scholar 

  8. 8.

    S. Voloshin, Y. Zhang, Z Phys. C 70, 665 (1996). https://doi.org/10.1007/s002880050141. arXiv:hep-ph/9407282

    Article  Google Scholar 

  9. 9.

    M. Luzum, J. Phys. G 38, 124026 (2011). https://doi.org/10.1088/0954-3899/38/12/124026. arXiv:1107.0592 [nucl-th]

    ADS  Article  Google Scholar 

  10. 10.

    A.M. Poskanzer, S.A. Voloshin, Phys. Rev. C 58, 1671 (1998). https://doi.org/10.1103/PhysRevC.58.1671. arXiv:nucl-ex/9805001

    ADS  Article  Google Scholar 

  11. 11.

    J.Y. Ollitrault, arXiv:nucl-ex/9711003

  12. 12.

    S. Acharya et al. [ALICE Collaboration], arXiv:2002.00633 [nucl-ex]

  13. 13.

    V. Khachatryan et al. [CMS Collaboration], Phys. Rev. C 92(3), 034911 (2015) https://doi.org/10.1103/PhysRevC.92.034911. arXiv:1503.01692 [nucl-ex]

  14. 14.

    P. Bozek, W. Broniowski, J. Moreira, Phys. Rev. C 83, 034911 (2011). https://doi.org/10.1103/PhysRevC.83.034911. arXiv:1011.3354 [nucl-th]

    ADS  Article  Google Scholar 

  15. 15.

    N. Borghini, P.M. Dinh, J.Y. Ollitrault, Phys. Rev. C 64, 054901 (2001). https://doi.org/10.1103/PhysRevC.64.054901. arXiv:nucl-th/0105040

    ADS  Article  Google Scholar 

  16. 16.

    R.S. Bhalerao, J.Y. Ollitrault, Phys. Lett. B 641, 260 (2006). https://doi.org/10.1016/j.physletb.2006.08.055. arXiv:nucl-th/0607009

    ADS  Article  Google Scholar 

  17. 17.

    P. Di Francesco, M. Guilbaud, M. Luzum, J.Y. Ollitrault, Phys. Rev. C 95(4), 044911 (2017). https://doi.org/10.1103/PhysRevC.95.044911. arXiv:1612.05634 [nucl-th]

    ADS  Article  Google Scholar 

  18. 18.

    Z. Moravcova, K. Gulbrandsen, Y. Zhou, arXiv:2005.07974 [nucl-th]

  19. 19.

    G. Aad et al., ATLAS Collaboration. JHEP 1311, 183 (2013). https://doi.org/10.1007/JHEP11(2013)183. arXiv:1305.2942 [hep-ex]

  20. 20.

    S. Acharya et al., ALICE Collaboration. JHEP 1807, 103 (2018). https://doi.org/10.1007/JHEP07(2018)103. arXiv:1804.02944 [nucl-ex]

  21. 21.

    J.Y. Ollitrault, A.M. Poskanzer, S.A. Voloshin, Phys. Rev. C 80, 014904 (2009). https://doi.org/10.1103/PhysRevC.80.014904. arXiv:0904.2315 [nucl-ex]

    ADS  Article  Google Scholar 

  22. 22.

    A.M. Sirunyan et al., CMS Collaboration. Phys. Lett. B 789, 643 (2019). https://doi.org/10.1016/j.physletb.2018.11.063. arXiv:1711.05594 [nucl-ex]

  23. 23.

    S.A. Voloshin, A.M. Poskanzer, A. Tang, G. Wang, Phys. Lett. B 659, 537 (2008). https://doi.org/10.1016/j.physletb.2007.11.043. arXiv:0708.0800 [nucl-th]

    ADS  Article  Google Scholar 

  24. 24.

    M. Aaboud et al., ATLAS. JHEP 01, 051 (2020). https://doi.org/10.1007/JHEP01(2020)051. arXiv:1904.04808 [nucl-ex]

  25. 25.

    L. Yan, J.Y. Ollitrault, A.M. Poskanzer, Phys. Rev. C 90(2), 024903 (2014). https://doi.org/10.1103/PhysRevC.90.024903. arXiv:1405.6595 [nucl-th]

    ADS  Article  Google Scholar 

  26. 26.

    G. Giacalone, L. Yan, J. Noronha-Hostler, J.Y. Ollitrault, Phys. Rev. C 95(1), 014913 (2017). https://doi.org/10.1103/PhysRevC.95.014913. arXiv:1608.01823 [nucl-th]

    ADS  Article  Google Scholar 

  27. 27.

    R.S. Bhalerao, G. Giacalone, J.Y. Ollitrault, Phys. Rev. C 99(1), 014907 (2019). https://doi.org/10.1103/PhysRevC.99.014907. arXiv:1811.00837 [nucl-th]

    ADS  Article  Google Scholar 

  28. 28.

    R.S. Bhalerao, G. Giacalone, J.Y. Ollitrault, Phys. Rev. C 100(1), 014909 (2019). https://doi.org/10.1103/PhysRevC.100.014909. arXiv:1904.10350 [nucl-th]

    ADS  Article  Google Scholar 

  29. 29.

    S. Chatrchyan et al., CMS Collaboration, Phys. Lett. B 724, 213 (2013). https://doi.org/10.1016/j.physletb.2013.06.028. arXiv:1305.0609 [nucl-ex]

  30. 30.

    V. Khachatryan et al., CMS Collaboration, Phys. Lett. B 765, 193 (2017). https://doi.org/10.1016/j.physletb.2016.12.009. arXiv:1606.06198 [nucl-ex]

  31. 31.

    S. Chatrchyan et al., CMS Collaboration, JHEP 1402, 088 (2014). https://doi.org/10.1007/JHEP02(2014)088. arXiv:1312.1845 [nucl-ex]

  32. 32.

    A. Dumitru, F. Gelis, L. McLerran, R. Venugopalan, Nucl. Phys. A 810, 91 (2008). https://doi.org/10.1016/j.nuclphysa.2008.06.012. arXiv:0804.3858 [hep-ph]

    ADS  Article  Google Scholar 

  33. 33.

    F.G. Gardim, F. Grassi, M. Luzum, J.Y. Ollitrault, Phys. Rev. C 87(3), 031901 (2013). https://doi.org/10.1103/PhysRevC.87.031901. arXiv:1211.0989 [nucl-th]

    ADS  Article  Google Scholar 

  34. 34.

    M. Aaboud et al. [ATLAS Collaboration], Eur. Phys. J. C 78(2), 142 (2018). https://doi.org/10.1140/epjc/s10052-018-5605-7. arXiv:1709.02301 [nucl-ex]

  35. 35.

    G. Aad et al. [ATLAS Collaboration]. arXiv:2001.04201 [nucl-ex]

  36. 36.

    R.S. Bhalerao, M. Luzum, J.Y. Ollitrault, Phys. Rev. C 84, 034910 (2011). https://doi.org/10.1103/PhysRevC.84.034910. arXiv:1104.4740 [nucl-th]

    ADS  Article  Google Scholar 

  37. 37.

    R.S. Bhalerao, M. Luzum, J.Y. Ollitrault, J. Phys. G 38, 124055 (2011). https://doi.org/10.1088/0954-3899/38/12/124055. arXiv:1106.4940 [nucl-ex]

    ADS  Article  Google Scholar 

  38. 38.

    J. Adam et al., ALICE Collaboration, Phys. Rev. Lett. 117, 182301 (2016). https://doi.org/10.1103/PhysRevLett.117.182301. arXiv:1604.07663 [nucl-ex]

  39. 39.

    S. Acharya et al. [ALICE Collaboration], Phys. Rev. C 97(2), 024906 (2018) https://doi.org/10.1103/PhysRevC.97.024906. arXiv:1709.01127 [nucl-ex]

  40. 40.

    A.M. Sirunyan et al. [CMS], Phys. Rev. Lett. 120(9), 092301 (2018). https://doi.org/10.1103/PhysRevLett.120.092301. arXiv:1709.09189 [nucl-ex]

  41. 41.

    S. Acharya et al. [ALICE], Phys. Rev. Lett. 123(14), 142301 (2019). https://doi.org/10.1103/PhysRevLett.123.142301. arXiv:1903.01790 [nucl-ex]

  42. 42.

    C. Mordasini, A. Bilandzic, D. Karakoç, S.F. Taghavi, Phys. Rev. C 102(2), 024907 (2020). https://doi.org/10.1103/PhysRevC.102.024907. arXiv:1901.06968 [nucl-ex]

  43. 43.

    R.S. Bhalerao, J.Y. Ollitrault, S. Pal, Phys. Rev. C 88, 024909 (2013). https://doi.org/10.1103/PhysRevC.88.024909. arXiv:1307.0980 [nucl-th]

    ADS  Article  Google Scholar 

  44. 44.

    G. Aad et al. [ATLAS Collaboration], Phys. Rev. C 90(2), 024905 (2014). https://doi.org/10.1103/PhysRevC.90.024905. arXiv:1403.0489 [hep-ex]

  45. 45.

    S. Acharya et al., ALICE Collaboration, Phys. Lett. B 773, 68 (2017). https://doi.org/10.1016/j.physletb.2017.07.060. arXiv:1705.04377 [nucl-ex]

  46. 46.

    A. Bilandzic, M. Lesch, S.F. Taghavi, Phys. Rev. C 102(2), 024910 (2020). https://doi.org/10.1103/PhysRevC.102.024910. arXiv:2004.01066 [nucl-ex]

    ADS  Article  Google Scholar 

  47. 47.

    D. Teaney, L. Yan, Phys. Rev. C 86, 044908 (2012). https://doi.org/10.1103/PhysRevC.86.044908. arXiv:1206.1905 [nucl-th]

    ADS  Article  Google Scholar 

  48. 48.

    J. Qian, U.W. Heinz, J. Liu, Phys. Rev. C 93(6), 064901 (2016). https://doi.org/10.1103/PhysRevC.93.064901. arXiv:1602.02813 [nucl-th]

    ADS  Article  Google Scholar 

  49. 49.

    R.S. Bhalerao, J.Y. Ollitrault, S. Pal, Phys. Lett. B 742, 94–98 (2015). https://doi.org/10.1016/j.physletb.2015.01.019. arXiv:1411.5160 [nucl-th]

  50. 50.

    L. Yan, J.Y. Ollitrault, Phys. Lett. B 744, 82–87 (2015). https://doi.org/10.1016/j.physletb.2015.03.040. arXiv:1502.02502 [nucl-th]

    ADS  Article  Google Scholar 

  51. 51.

    J. Qian, U. Heinz, R. He, L. Huo, Phys. Rev. C 95(5), 054908 (2017). https://doi.org/10.1103/PhysRevC.95.054908. arXiv:1703.04077 [nucl-th]

    ADS  Article  Google Scholar 

  52. 52.

    G. Aad et al. [ATLAS], Phys. Rev. C 92(3), 034903 (2015). https://doi.org/10.1103/PhysRevC.92.034903. arXiv:1504.01289 [hep-ex]

  53. 53.

    M. Aaboud et al. [ATLAS Collaboration], Phys. Rev. C 96(2), 024908 (2017). https://doi.org/10.1103/PhysRevC.96.024908. arXiv:1609.06213 [nucl-ex]

  54. 54.

    V. Pacík [ALICE Collaboration], Nucl. Phys. A 982, 451 (2019). https://doi.org/10.1016/j.nuclphysa.2018.09.020. arXiv:1807.04538 [nucl-ex]

  55. 55.

    R.S. Bhalerao, N. Borghini, J.Y. Ollitrault, Phys. Lett. B 580, 157–162 (2004). https://doi.org/10.1016/j.physletb.2003.11.056. arXiv:nucl-th/0307018

    ADS  Article  Google Scholar 

  56. 56.

    R.S. Bhalerao, N. Borghini, J.Y. Ollitrault, Nucl. Phys. A 727, 373–426 (2003). https://doi.org/10.1016/j.nuclphysa.2003.08.007. arXiv:nucl-th/0310016

    ADS  Article  Google Scholar 

  57. 57.

    C. Aidala et al. [PHENIX Collaboration], Nat. Phys. 15(3), 214 (2019). https://doi.org/10.1038/s41567-018-0360-0. arXiv:1805.02973 [nucl-ex]

  58. 58.

    J. Adam et al., ALICE Collaboration, Nat. Phys. 13, 535 (2017). https://doi.org/10.1038/nphys4111. arXiv:1606.07424 [nucl-ex]

  59. 59.

    R.D. Weller, P. Romatschke, Phys. Lett. B 774, 351 (2017). https://doi.org/10.1016/j.physletb.2017.09.077. arXiv:1701.07145 [nucl-th]

    ADS  Article  Google Scholar 

  60. 60.

    Y. Zhou, W. Zhao, K. Murase, H. Song, arXiv:2005.02684 [nucl-th]

  61. 61.

    S. Acharya et al., ALICE Collaboration, JHEP 1811, 013 (2018). https://doi.org/10.1007/JHEP11(2018)013. arXiv:1802.09145 [nucl-ex]

  62. 62.

    M. Strickland, Nucl. Phys. A 982, 92 (2019). https://doi.org/10.1016/j.nuclphysa.2018.09.071. arXiv:1807.07191 [nucl-th]

    ADS  Article  Google Scholar 

  63. 63.

    G. Aad et al., ATLAS Collaboration, Phys. Rev. C 86, 014907 (2012). https://doi.org/10.1103/PhysRevC.86.014907. [arXiv:1203.3087 [hep-ex]]

  64. 64.

    C. Gale, S. Jeon, B. Schenke, P. Tribedy, R. Venugopalan, Phys. Rev. Lett. 110(1), 012302 (2013). https://doi.org/10.1103/PhysRevLett.110.012302. arXiv:1209.6330 [nucl-th]

    ADS  Article  Google Scholar 

  65. 65.

    F.G. Gardim, F. Grassi, M. Luzum, J.Y. Ollitrault, Phys. Rev. Lett. 109, 202302 (2012). https://doi.org/10.1103/PhysRevLett.109.202302. arXiv:1203.2882 [nucl-th]

    ADS  Article  Google Scholar 

  66. 66.

    P. Romatschke, Phys. Rev. Lett. 120(1), 012301 (2018). https://doi.org/10.1103/PhysRevLett.120.012301. arXiv:1704.08699 [hep-th]

    ADS  Article  Google Scholar 

  67. 67.

    W. Florkowski, M.P. Heller, M. Spalinski, Rep. Prog. Phys. 81(4), 046001 (2018). https://doi.org/10.1088/1361-6633/aaa091. arXiv:1707.02282 [hep-ph]

    ADS  Article  Google Scholar 

  68. 68.

    L. He, T. Edmonds, Z.W. Lin, F. Liu, D. Molnar, F. Wang, Phys. Lett. B 753, 506 (2016). https://doi.org/10.1016/j.physletb.2015.12.051. arXiv:1502.05572 [nucl-th]

    ADS  Article  Google Scholar 

  69. 69.

    H. Li, L. He, Z.W. Lin, D. Molnar, F. Wang, W. Xie, Phys. Rev. C 93(5), 051901 (2016). https://doi.org/10.1103/PhysRevC.93.051901. arXiv:1601.05390 [nucl-th]

    ADS  Article  Google Scholar 

  70. 70.

    N. Borghini, C. Gombeaud, Eur. Phys. J. C 71, 1612 (2011). https://doi.org/10.1140/epjc/s10052-011-1612-7. arXiv:1012.0899 [nucl-th]

    ADS  Article  Google Scholar 

  71. 71.

    A. Kurkela, U.A. Wiedemann, B. Wu, Eur. Phys. J. C 79(9), 759 (2019). https://doi.org/10.1140/epjc/s10052-019-7262-x. arXiv:1805.04081 [hep-ph]

    ADS  Article  Google Scholar 

  72. 72.

    K. Dusling, M. Mace, R. Venugopalan, Phys. Rev. Lett. 120(4), 042002 (2018). https://doi.org/10.1103/PhysRevLett.120.042002. arXiv:1705.00745 [hep-ph]

    ADS  Article  Google Scholar 

  73. 73.

    K. Dusling, M. Mace, R. Venugopalan, Phys. Rev. D 97(1), 016014 (2018). https://doi.org/10.1103/PhysRevD.97.016014. arXiv:1706.06260 [hep-ph]

    ADS  Article  Google Scholar 

  74. 74.

    M. Mace, V.V. Skokov, P. Tribedy, R. Venugopalan, Phys. Rev. Lett. 121(5), 052301 (2018). (Erratum: [Phys. Rev. Lett. 123(3), 039901 (2019)]). https://doi.org/10.1103/PhysRevLett.123.039901. https://doi.org/10.1103/PhysRevLett.121.052301. arXiv:1805.09342 [hep-ph]

  75. 75.

    C. Zhang, C. Marquet, G.Y. Qin, S.Y. Wei, B.W. Xiao, Phys. Rev. Lett. 122(17), 172302 (2019). https://doi.org/10.1103/PhysRevLett.122.172302. arXiv:1901.10320 [hep-ph]

    ADS  Article  Google Scholar 

  76. 76.

    C. Zhang, C. Marquet, G.Y. Qin, Y. Shi, L. Wang, S.Y. Wei, B.W. Xiao, arXiv:2002.09878 [hep-ph]

  77. 77.

    L. McLerran, https://doi.org/10.3204/DESY-PROC-2009-01/26. arXiv:0812.4989 [hep-ph]

  78. 78.

    A. Dumitru, K. Dusling, F. Gelis, J. Jalilian-Marian, T. Lappi, R. Venugopalan, Phys. Lett. B 697, 21 (2011). https://doi.org/10.1016/j.physletb.2011.01.024. arXiv:1009.5295 [hep-ph]

    ADS  Article  Google Scholar 

  79. 79.

    B. Schenke, S. Schlichting, P. Tribedy, R. Venugopalan, Phys. Rev. Lett. 117(16), 162301 (2016). https://doi.org/10.1103/PhysRevLett.117.162301. arXiv:1607.02496 [hep-ph]

    ADS  Article  Google Scholar 

  80. 80.

    A. Ortiz [ALICE and ATLAS and CMS and LHCb Collaborations], PoS LHCP 2019, 091 (2019). https://doi.org/10.22323/1.350.0091. arXiv:1909.03937 [hep-ex]

  81. 81.

    S. Morrow [PHENIX Collaboration], arXiv:1810.05321 [nucl-ex]

  82. 82.

    J.L. Nagle, W.A. Zajc, Ann. Rev. Nucl. Part. Sci. 68, 211 (2018). https://doi.org/10.1146/annurev-nucl-101916-123209. arXiv:1801.03477 [nucl-ex]

    ADS  Article  Google Scholar 

  83. 83.

    B. Schenke, Nucl. Phys. A 967, 105 (2017). https://doi.org/10.1016/j.nuclphysa.2017.05.017. arXiv:1704.03914 [nucl-th]

    ADS  Article  Google Scholar 

  84. 84.

    W. Li, Nucl. Phys. A 967, 59 (2017). https://doi.org/10.1016/j.nuclphysa.2017.05.011. arXiv:1704.03576 [nucl-ex]

    ADS  Article  Google Scholar 

  85. 85.

    S. Schlichting, P. Tribedy, Adv. High Energy Phys. 2016, 8460349 (2016). https://doi.org/10.1155/2016/8460349. arXiv:1611.00329 [hep-ph]

    Article  Google Scholar 

  86. 86.

    C. Loizides, Nucl. Phys. A 956, 200 (2016). https://doi.org/10.1016/j.nuclphysa.2016.04.022. arXiv:1602.09138 [nucl-ex]

    ADS  Article  Google Scholar 

  87. 87.

    K. Dusling, W. Li, B. Schenke, Int. J. Mod. Phys. E 25(01), 1630002 (2016). https://doi.org/10.1142/S0218301316300022. arXiv:1509.07939 [nucl-ex]

    ADS  Article  Google Scholar 

  88. 88.

    P.H. Westfall, Am. Stat. 68(3), 191 (2014). https://doi.org/10.1080/00031305.2014.917055

    Article  Google Scholar 

  89. 89.

    A. Bzdak, P. Bozek, Phys. Rev. C 93(2), 024903 (2016). https://doi.org/10.1103/PhysRevC.93.024903. arXiv:1509.02967 [hep-ph]

    ADS  Article  Google Scholar 

Download references

Acknowledgements

I am very thankful to Jean-Yves Ollitrault for helpful comments on the manuscript. I also thank him for our long-term collaboration which allowed me to learn many things. I acknowledge the award of the Core Research Grant, by the Science and Engineering Research Board, Department of Science and Technology, Government of India. I thank Bhavya Bhatt for drawing Fig. 14.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Rajeev S. Bhalerao.

Appendices

Appendix A

Moments and cumulants of a probability distribution

The nth moment of a (real, continuous) function f(x), about a constant a, is defined as

$$\begin{aligned} \mu _n(a) \equiv \int _{-\infty }^\infty (x-a)^n f(x) \mathrm{{d}}x. \end{aligned}$$
(A1)

We shall assume f(x) to be the probability density function (PDF), normalized to unity. The two most interesting values of a are 0 and \(\mu \equiv \left\langle x \right\rangle \), the mean of the distribution. Usually one refers to \(\mu _n(a=0)\) simply as the “moment” and \(\mu _n(a=\mu )\) as the “central moment”. Henceforth, we denote moments \(\mu _n(a=0)\) by \(\mu '_n\) and central moments \(\mu _n(a=\mu )\) by \(\mu _n\). Obviously, \(\mu '_0=1=\mu _0\) and \(\mu '_n=\left\langle x^n \right\rangle \). The first four central moments are

$$\begin{aligned} \begin{aligned} \mu _1&=0, \\ \mu _2&=\left\langle x^2 \right\rangle -\mu ^2 \equiv {{\mathrm{variance}}} \,(\sigma ^2) \equiv \,\, ({\mathrm{standard}} {\mathrm{deviation}} = \sigma )^2, \\ \mu _3&= \left\langle x^3 \right\rangle -3\mu \left\langle x^2 \right\rangle +2\mu ^3, \\ \mu _4&= \left\langle x^4 \right\rangle -4\mu \left\langle x^3 \right\rangle +6\mu ^2 \left\langle x^2 \right\rangle -3\mu ^4. \end{aligned}\nonumber \\ \end{aligned}$$
(A2)

It is often convenient to define standardized central moments which are scale-invariant or dimensionless quantities: \(\mu _n/\sigma ^n\). The first four standardized central moments are

$$\begin{aligned} \begin{aligned} \mu _1/\sigma&= 0, \\ \mu _2/\sigma ^2&= 1, \\ \mu _3/\sigma ^3&\equiv {{\mathrm{skewness}}} \,\, (\gamma ), \\ \mu _4/\sigma ^4&\equiv {{\mathrm{kurtosis}}} \,\, (\kappa ). \end{aligned}\end{aligned}$$
(A3)

Excess kurtosis is defined as \(\kappa -3\). However, it is not uncommon to find the excess kurtosis itself termed as kurtosis. The reason for subtracting 3 will become clear when we discuss the Gaussian (or normal) distribution (Appendix C).

\(\bullet \) Variance is a measure of the spread of the random numbers about their mean value.

\(\bullet \) Skewness is a measure of the lopsidedness or asymmetry of the distribution about its mean. It is clear from the definition that a distribution that is symmetric about its mean has vanishing skewness. In general, skewness can be positive or negative. If the left (right) tail is drawn out, or in other words, is longer than the right (left) tail, the distribution is said to be left(right)-skewed and has a negative (positive) skewness; see Fig. 14.

\(\bullet \) Kurtosis is a measure of the heaviness of the tails of the distribution as compared to the normal distribution with the same variance. It is clear from the definition that kurtosis is a nonnegative number. (Excess kurtosis may be positive or negative.) Kurtosis is a measure of the “tailedness” and not the “peakedness” of a distribution (Fig. 14), because the proportion of the kurtosis that is determined by the central \(\mu {\pm } \sigma \) range is usually quite small [88].

Moment-generating function M(t)

$$\begin{aligned} M(t)\equiv \left\langle e^{tx} \right\rangle =1+t\left\langle x \right\rangle +\frac{t^2}{2!}\left\langle x^2 \right\rangle +\cdots = \sum _{n=0}^\infty \frac{t^n}{n!}\left\langle x^n \right\rangle . \end{aligned}$$
(A4)

Observe that the moments \(\mu '_n=\left\langle x^n \right\rangle \) appear in the above expansion. To isolate the mth moment \(\left\langle x^m \right\rangle \), for example, one uses

$$\begin{aligned} \left[ \frac{\mathrm{{d}}^m}{\mathrm{{d}}t^m} M(t) \right] _{t=0} =\left\langle x^m \right\rangle . \end{aligned}$$
(A5)

Differentiating M(t)   m times removes the first m terms, i.e., the terms containing \(1, \left\langle x \right\rangle , \ldots , \left\langle x^{m-1} \right\rangle \). Further, setting \(t=0\) removes terms containing \(\left\langle x^{m+1} \right\rangle ,\left\langle x^{m+2} \right\rangle , \ldots \), leaving behind only \(\left\langle x^m \right\rangle \). The moment-generating function provides an alternative description of the PDF. The central moment generating function is \(e^{-\mu t}M(t)\).

Cumulant-generating function K(t)

$$\begin{aligned} K(t)\equiv \ln M(t) = \ln \left\langle e^{tx} \right\rangle =t\kappa _1+\frac{t^2}{2!}\kappa _2+\frac{t^3}{3!}\kappa _3+\cdots = \sum _{n=1}^\infty \frac{t^n}{n!}\kappa _n, \end{aligned}$$
(A6)

where \(\kappa _n\) are the cumulants. To isolate the mth cumulant \(\kappa _m\), for example, one uses

$$\begin{aligned} \left[ \frac{\mathrm{{d}}^m}{\mathrm{{d}}t^m} K(t) \right] _{t=0} =\kappa _m. \end{aligned}$$
(A7)

Differentiating K(t)   m times removes the first \(m-1\) terms, i.e., the terms containing \(\kappa _1, \ldots , \kappa _{m-1}\). Further, setting \(t=0\) removes terms containing \(\kappa _{m+1},\kappa _{m+2}, \ldots \), leaving behind only \(\kappa _m\). It is straightforward to show that the first few cumulants are

$$\begin{aligned} \begin{aligned} \kappa _1&= \left\langle x \right\rangle = {{\mathrm{mean}}} \,\, \mu , \\ \kappa _2&= \left\langle x^2 \right\rangle -\left\langle x \right\rangle ^2 = \left\langle (x-\left\langle x \right\rangle )^2 \right\rangle ={\mathrm{variance}} \,\, (\sigma ^2)\\&= {\mathrm{second \,\,central\,\, moment}}, \\ \kappa _3&= \left\langle x^3 \right\rangle -3\left\langle x \right\rangle \left\langle x^2 \right\rangle +2\left\langle x \right\rangle ^3 \\&= \left\langle (x-\left\langle x \right\rangle )^3 \right\rangle = {{\mathrm{third \,\, central \,\, moment}}}, \\ \kappa _4&= \left\langle x^4 \right\rangle -4\left\langle x \right\rangle \left\langle x^3 \right\rangle -3\left\langle x^2 \right\rangle ^2+12\left\langle x \right\rangle ^2\left\langle x^2 \right\rangle -6\left\langle x \right\rangle ^4 \\&\ne {{\mathrm{fourth \,\, central \,\, moment}}} \,\,\left\langle (x-\left\langle x \right\rangle )^4 \right\rangle , \\ \kappa _5&= \left\langle x^5 \right\rangle -5\left\langle x \right\rangle \left\langle x^4 \right\rangle -10\left\langle x^2 \right\rangle \left\langle x^3 \right\rangle +20\left\langle x \right\rangle ^2\left\langle x^3 \right\rangle \\&+30\left\langle x \right\rangle \left\langle x^2 \right\rangle ^2 -60\left\langle x \right\rangle ^3\left\langle x^2 \right\rangle +24\left\langle x \right\rangle ^5, \\ \kappa _6&= \left\langle x^6 \right\rangle -6\left\langle x \right\rangle \left\langle x^5 \right\rangle -15\left\langle x^2 \right\rangle \left\langle x^4 \right\rangle +30\left\langle x \right\rangle ^2\left\langle x^4 \right\rangle \\&\quad -10\left\langle x^3 \right\rangle ^2 +120\left\langle x \right\rangle \left\langle x^2 \right\rangle \left\langle x^3 \right\rangle \\&\quad -120\left\langle x \right\rangle ^3\left\langle x^3 \right\rangle +30\left\langle x^2 \right\rangle ^3 - 270\left\langle x \right\rangle ^2\left\langle x^2 \right\rangle ^2\\&\quad +360\left\langle x \right\rangle ^4\left\langle x^2 \right\rangle -120\left\langle x \right\rangle ^6. \end{aligned} \end{aligned}$$
(A8)

Fourth- and higher order cumulants are not identical to the corresponding central moments. Thus cumulants are certain polynomial functions of the moments and provide an alternative to the moments of the distribution.

The above equations can be inverted to express moments in terms of cumulants:

$$\begin{aligned} \begin{aligned} \left\langle x \right\rangle&= \kappa _1, \\ \left\langle x^2 \right\rangle&= \kappa _1^2+\kappa _2, \\ \left\langle x^3 \right\rangle&= \kappa _1^3+3\kappa _1\kappa _2+\kappa _3, \\ \left\langle x^4 \right\rangle&= \kappa _1^4+6\kappa _1^2\kappa _2+4\kappa _1\kappa _3+3\kappa _2^2+\kappa _4, \\ \left\langle x^5 \right\rangle&= \kappa _1^5+10\kappa _1^3\kappa _2+10\kappa _1^2\kappa _3 +15\kappa _1\kappa _2^2+5\kappa _1\kappa _4\\&+10\kappa _2\kappa _3+\kappa _5, \\ \left\langle x^6 \right\rangle&= \kappa _1^6+6\kappa _1\kappa _5+15\kappa _1^2\kappa _4 +20\kappa _1^3\kappa _3+15\kappa _1^4\kappa _2\\&\quad +60\kappa _1\kappa _2\kappa _3 +45\kappa _1^2\kappa _2^2+15\kappa _2\kappa _4 \\&\quad +10\kappa _3^2+ 15\kappa _2^3+\kappa _6. \end{aligned} \end{aligned}$$
(A9)

Appendix B

Correlation functions and cumulants

The n-particle correlation function (or simply a correlator) \(\rho (1,2,3,\ldots ,n)\) consists of terms that represent combinations of lower order correlations and a term that represents a genuine or “true” n-particle correlation \(C(1,2,3,\ldots ,n)\) which is called a cumulant. For example, \(\rho (1,2)=\rho (1)\rho (2)+C(1,2)\), so that \(C(1,2)=\rho (1,2)-\rho (1)\rho (2)\). If the two particles are statistically independent, \(\rho (1,2)\) simply reduces to \(\rho (1)\rho (2)\), whereas C(1, 2) vanishes by construction.Footnote 11 The first few correlation functions are [89]

$$\begin{aligned} \rho (1)= & {} C(1), \nonumber \\ \rho (1,2)= & {} \rho (1)\rho (2)+C(1,2), \nonumber \\ \rho (1,2,3)= & {} \rho (1)\rho (2)\rho (3)+\rho (1)C(2,3)+\rho (2)C(3,1)\nonumber \\&+ \rho (3)C(1,2)+C(1,2,3),\nonumber \\\equiv & {} \rho (1)\rho (2)\rho (3)+\sum _{(3)}\rho (1)C(2,3)+C(1,2,3), \nonumber \\ \rho (1,2,3,4)= & {} \rho (1)\rho (2)\rho (3)\rho (4)+\sum _{(6)}\rho (1)\rho (2)C(3,4)\nonumber \\&+\sum _{(4)}\rho (1)C(2,3,4) \nonumber \\&+\sum _{(3)}C(1,2)C(3,4)+C(1,2,3,4), \nonumber \\ \rho (1,2,3,4,5)= & {} \rho (1)\rho (2)\rho (3)\rho (4)\rho (5) +\sum _{(10)}\rho (1)\rho (2)\rho (3)C(4,5) \nonumber \\&+\sum _{(10)}\rho (1)\rho (2)C(3,4,5)\nonumber \\&+\sum _{(15)}\rho (1)C(2,3)C(4,5) +\sum _{(5)}\rho (1)C(2,3,4,5)\nonumber \\&+\sum _{(10)}C(1,2)C(3,4,5) +C(1,2,3,4,5), \nonumber \\ \rho (1,2,3,4,5,6)= & {} \rho (1)\rho (2)\rho (3)\rho (4)\rho (5)\rho (6)\nonumber \\&+\sum _{(6)}\rho (1)C(2,3,4,5,6) \nonumber \\&+\sum _{(15)}\rho (1)\rho (2)C(3,4,5,6)\nonumber \\&+\sum _{(20)}\rho (1)\rho (2)\rho (3)C(4,5,6) \nonumber \\&+\sum _{(15)}\rho (1)\rho (2)\rho (3)\rho (4)C(5,6)\nonumber \\&+\sum _{(60)}\rho (1)C(2,3)C(4,5,6) \nonumber \\&+\sum _{(45)}\rho (1)\rho (2)C(3,4)C(5,6)\nonumber \\&+\sum _{(15)}C(1,2)C(3,4,5,6) \nonumber \\&+\sum _{(10)}C(1,2,3)C(4,5,6)\nonumber \\&+\sum _{(15)}C(1,2)C(3,4)C(5,6) \nonumber \\&+C(1,2,3,4,5,6), \end{aligned}$$
(B1)

where the numbers in parentheses under the summation signs indicate the number of possible permutations of the indices.

The above equations can be inverted to get the expressions for the cumulants in terms of the correlation functions:

$$\begin{aligned} C(1)= & {} \rho (1), \nonumber \\ C(1,2)= & {} \rho (1,2)-\rho (1)\rho (2), \nonumber \\ C(1,2,3)= & {} \rho (1,2,3)-\sum _{(3)}\rho (1)\rho (2,3)+2\rho (1)\rho (2)\rho (3), \nonumber \\ C(1,2,3,4)= & {} \rho (1,2,3,4)-\sum _{(4)}\rho (1)\rho (2,3,4)\nonumber \\&-\sum _{(3)}\rho (1,2)\rho (3,4) \nonumber \\&+2\sum _{(6)}\rho (1)\rho (2)\rho (3,4) -6\rho (1)\rho (2)\rho (3)\rho (4), \nonumber \\ C(1,2,3,4,5)= & {} \rho (1,2,3,4,5) -\sum _{(5)}\rho (1)\rho (2,3,4,5)\nonumber \\&-\sum _{(10)}\rho (1,2)\rho (3,4,5)\nonumber \\&+2\sum _{(10)}\rho (1)\rho (2)\rho (3,4,5)\nonumber \\&+2\sum _{(15)}\rho (1)\rho (2,3)\rho (4,5) \nonumber \\&-6\sum _{(10)}\rho (1)\rho (2)\rho (3)\rho (4,5)\nonumber \\&+24\rho (1)\rho (2)\rho (3)\rho (4)\rho (5), \nonumber \\ C(1,2,3,4,5,6)= & {} \rho (1,2,3,4,5,6)\nonumber \\&-\sum _{(6)}\rho (1)\rho (2,3,4,5,6)\nonumber \\&-\sum _{(15)}\rho (1,2)\rho (3,4,5,6) \nonumber \\&+2\sum _{(15)}\rho (1)\rho (2)\rho (3,4,5,6)\nonumber \\&-\sum _{(10)}\rho (1,2,3)\rho (4,5,6) \nonumber \\&+4\sum _{(30)}\rho (1)\rho (2,3)\rho (4,5,6)\nonumber \\&-6\sum _{(20)}\rho (1)\rho (2)\rho (3)\rho (4,5,6) \nonumber \\&+2\sum _{(15)}\rho (1,2)\rho (3,4)\rho (5,6)\nonumber \\&-3\sum _{(90)}\rho (1)\rho (2)\rho (3,4)\rho (5,6) \nonumber \\&+24\sum _{(15)}\rho (1)\rho (2)\rho (3)\rho (4)\rho (5,6)\nonumber \\&-\sum _{(120)}\rho (1)\rho (2)\rho (3)\rho (4)\rho (5)\rho (6). \end{aligned}$$
(B2)

It is easy to verify that the cumulant vanishes if any one or more particles is statistically independent of the others. Thus the n-particle cumulant measures the statistical dependence of the entire n-particle set. For this reason, cumulants are also called connected correlation functions.

figurek
figurel

Appendix C

Gaussian or normal distribution in 1D

One-dimensional Gaussian (centered at \(\mu \)) is

$$\begin{aligned} f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left[ -\frac{(x-\mu )^2}{2\sigma ^2}\right] , \end{aligned}$$
(C1)

where \(\mu =\left\langle x \right\rangle \) is the mean, \(\sigma ^2\) is the variance and f(x) is normalized to unity. The odd central moments are obviously zero. The even central moments are given by \((n-1)!! \,\sigma ^n \,\, (n ~{\mathrm{even}})\). The first few even central moments are given in Table 1. The skewness (\(\gamma \)) is zero, kurtosis (\(\kappa \)) is 3 and excess kurtosis (\(\kappa -3\)) is zero. A probability distribution with tails heavier (lighter) than those of the normal distribution shows a higher (lower) propensity to produce outliers and has a positive (negative) excess kurtosis (Fig 14).

Table 1 First few central and noncentral moments of the Gaussian distribution
Table 2 First few moments (\(\mu _n\)) of f(r), Eq. (C3)

For the normal distribution, the moment-generating function is \(M(t)= \exp \,(\mu t + (\sigma ^2 t^2 /2))\) and the cumulant-generating function is \(K(t) = \ln M(t)= \mu t + (\sigma ^2 t^2 /2)\). Since this is a quadratic in t, only the first two cumulants survive, namely the mean \(\mu \) and the variance \(\sigma ^2\). It can be shown that the normal distribution is the only one for which the third and higher cumulants vanish.

figurem

Gaussian or normal distribution in 2D

Two-dimensional Gaussian centered at the origin and normalized to unity is

$$\begin{aligned} f(x,y)=\frac{1}{2 \pi \sigma _x \sigma _y} \exp \left[ -\frac{x^2}{2\sigma _x^2}-\frac{y^2}{2\sigma _y^2} \right] , \end{aligned}$$
(C2)

where \(\sigma _x^2\) and \(\sigma _y^2\) are the variances.

figuren

It is often convenient to introduce the symmetric case (\(\sigma _x=\sigma _y\)) and write it in terms of \(r=\sqrt{x^2+y^2}\):

$$\begin{aligned} f(r) =\frac{1}{\pi \sigma ^2}\exp \left[ -\frac{r^2}{\sigma ^2}\right] , \end{aligned}$$
(C3)

where \(\sigma ^2\equiv \sigma _x^2+\sigma _y^2\). f(r) is also centered at the origin and normalized to unity. Note, however, that unlike \(\left\langle x \right\rangle \) and \(\left\langle y \right\rangle \), \(\left\langle r \right\rangle \) is not zero. The first few moments of f(r), \(\mu _n=\left\langle r^n \right\rangle =\sigma ^n \, \varGamma ((n/2)+1)\), are given in Table 2. Note also that \(\left\langle r^2 \right\rangle =\left\langle x^2 \right\rangle +\left\langle y^2 \right\rangle \); using \(\left\langle r^2 \right\rangle \) from Table 2, we get \(\sigma ^2=\sigma _x^2+\sigma _y^2\). The kurtosis in this case is 2, unlike the case of a 1D Gaussian discussed earlier where it was 3.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bhalerao, R.S. Collectivity in large and small systems formed in ultrarelativistic collisions. Eur. Phys. J. Spec. Top. 230, 635–654 (2021). https://doi.org/10.1140/epjs/s11734-021-00019-x

Download citation