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Measures of azimuthal anisotropy in high-energy collisions

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Abstract

Azimuthal anisotropy is a key observation made in ultrarelativistic heavy-ion collisions. This phenomenon has played a crucial role in the development of the field over the last two decades. In addition to its interest for studying the quark-gluon plasma, which was the original motivation, it is sensitive to the properties of incoming nuclei, in particular to the nuclear deformation and to the nuclear skin. The azimuthal anisotropy is therefore of crucial importance when relating low-energy nuclear structure to high-energy nuclear collisions. This article is an elementary introduction to the various observables used in order to characterize azimuthal anisotropy, which go under the names of \(v_2\{2\}\), \(v_3\{2\}\), \(v_2\{4\}\), etc. The intended audience is primarily physicists working in the field of nuclear structure.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data sharing not applicable to this article as no data sets were generated or analysed during the current study.]

Notes

  1. This comment applies to the spatial image of the nucleus. As we shall discuss in the next paragraph, however, nucleons are made of partons, and the longitudinal extension of their wavefunction can be significantly larger.

  2. The motivation for building ion colliders (RHIC, then LHC), with two beams accelerated in opposite directions, is that \(\sqrt{s_{NN}}\) is much larger than in a fixed-target experiment, by a factor \(\sqrt{2\gamma }\) in the limit \(\gamma \gg 1\). In fixed-target experiments at the LHC, \(\sqrt{s_{NN}}\) is reduced by a factor \(\sim 75\) relative to the collider mode [2].

  3. This prescription was initially inspired by descriptions using high-energy QCD to model the collision [4,5,6,7,8].

  4. ALICE evaluates the average number of charged particles for collisions in the 0–5% centrality window. One must in addition take into account that \(\sim \frac{1}{3}\) of the produced particles are neutral, and that the average multiplicity in \(b=0\) collisions is \(\sim 11\%\) larger than in the considered centrality window, which consists roughly of collisions with \(b<3.5\) fm [15].

  5. Multiplicity fluctuations at fixed b are partly due to statistical (Poisson) fluctuations, which result in a standard deviation of \(\sqrt{N}\) if the average multiplicity is N. Relative statistical fluctuations are reduced if the detector acceptance is larger, so that it sees more particles. Statistical fluctuations dominate at lower energies, but they contribute by less than \(20\%\) to the variance at the LHC [22], so that upgrading detectors should not significantly improve the centrality resolution.

  6. In a grossly simplified picture, the system expands first longitudinally, then in the transverse directions. In this picture, \(T_{\textrm{eff}}\) corresponds to the temperature after the longitudinal expansion, and before the transverse expansion.

  7. This description was later refined by taking viscosity into account [34]. Note also that back in 2000, the pressure of QCD was largely unknown. It has since then been accurately computed as a function of temperature [35], and all modern fluid-dynamical calculations use this equation of state as input [36].

  8. The sign of \(v_2\) was actually not measured in this first analysis, as will be explained below.

  9. Equivalent information is obtained in calorimeters or scintillators, which record an amplitude as a function of azimuthal angle \(\phi \).

  10. In practice, some of the pairs are excluded, as will be explained in detail in Sect. 3.

  11. I borrow some of the following words from the introduction of the PhD thesis of Burak Alver [56] who, together with his supervisor Gunther Roland, gets the credit for the modern picture of azimuthal anisotropy.

  12. The data shown in Fig. 7 correspond to specific cuts on the transverse momenta of the particles (\(3<p_t<3.5\) GeV/c for one particle and \(1<p_t<1.5\) GeV/c for the other). The vast majority of particles are pions with mass energy \(mc^2\simeq 0.14\) GeV. Momenta are much larger than mc, which implies that velocities are very close to c.

  13. As argued by Anderson [61], emergent phenomena are usually associated with broken symmetries. In the case of Fig. 7, we will see that the origin of the observed pattern lies in the breaking of azimuthal symmetry in every event.

  14. This term was coined by Joern Putschke at the Quark Matter 2006 conference [62] because the structure, with a long ridge surrounding a small peak, reminded him of the Weisshorn peak in the Alps, which is climbed via ridges [63].

  15. The notation \(v_n\) for Fourier coefficients was introduced in the context of the old flow picture by Voloshin and Zhang [37].

  16. In writing this equation, I have assumed for simplicity that the single-particle distribution (4) is the same for both particles in the pair. It is not the case for the CMS analysis in Fig. 7 since the two particles are chosen in separate \(p_t\) intervals, but this is a detail which does not alter the general picture.

  17. The angle between the particles is \(\left( \varDelta \phi ^2+\varDelta \eta ^2\right) ^{1/2}\) when \(\varDelta \phi \) and \(\varDelta \eta \) are both much smaller than unity.

  18. We have pointed out that CMS results in Fig. 7 are compatible with a \(v_3\) close to \(\frac{2}{3}v_2\). This is compatible with the findings of ALICE and ATLAS, if one averages them over the \(0-5\%\) centrality range used by CMS.

  19. The viscous suppression is roughly twice larger for \(v_3\) than for \(v_2\) [39].

  20. The Au + Au results are equivalent to those shown in Fig. 4, but they are of much better quality, as many more data were available, and the analysis methods had also been refined. The larger values of \(v_2\) are due to a combination of three factors: 1. The more recent analysis exludes particles with \(0.1<p_t<0.2\) GeV/c. Particles in this range have \(v_2\approx 0\), therefore, one increases the average \(v_2\) by excluding them. 2.The older analysis uses the event-plane method, which yields results slightly smaller than \(v_2\{2\}\) in the presence of flow fluctuations [53]. 3. The recent analysis is done at a slightly higher collision energy.

  21. Therefore, the events with the highest multiplicities can have any value of \(\theta \). The ellipticity \(\varepsilon _2\) is minimal for a tip–tip collision and maximal for a body–body collision, in which the deformation of the \(^{238}\)U nucleus shows up. The net effect, after averaging over \(\theta \), is that this deformation increases \(\varepsilon _2\).

  22. One generally expects that \(v_2\) is larger in central collisions if nuclei are smaller, due to larger fluctuations, as was observed in Cu + Cu collisions at RHIC (Sect. 3.1). The statement here is a quantitative one, that the observed increase of \(v_2\) in Xe+Xe collisions relative to Pb+Pb is larger than one would expect just from the different nuclear size.

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Acknowledgements

This article is based on two talks I gave, first at the Symposium on collective flow in nuclear matter (a celebration of Art Poskanzer’s life and career) in December 2022 at Berkeley, then in January 2023 under the program “Intersection of nuclear structure and high-energy nuclear collisions” hosted by the Institute for Nuclear Theory at the University of Washington, which I thank for support. I thank Giuliano Giacalone for useful input and detailed comments on the manuscript, and Johanna Stachel and Peter Braun-Munzinger for comments on the first version. This work is supported by the GLUODYNAMICS project funded by the “P2IO LabEx (ANR-10-LABX-0038)” in the framework “Investissements d’Avenir” (ANR-11-IDEX-0003-01) managed by the Agence Nationale de la Recherche (ANR), France.

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  1. Université Paris Saclay, CNRS, CEA, Institut de physique théorique, 91191, Gif-sur-Yvette, France

    Jean-Yves Ollitrault

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Communicated by Thomas Duguet

In memory of Art Poskanzer.

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Ollitrault, JY. Measures of azimuthal anisotropy in high-energy collisions. Eur. Phys. J. A 59, 236 (2023). https://doi.org/10.1140/epja/s10050-023-01157-7

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