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Non-Bessel–Gaussianity and Flow Harmonic Fine-Splitting Eur. Phys. J. C (2019) 79:88 https://doi.org/10.1140/epjc/s10052-019-6549-2 Regular Article - Theoretical Physics Non-Bessel–Gaussianity and flow harmonic fine-splitting Hadi Mehrabpour1,2,a, Seyed Farid Taghavi2,3,b 1 Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran 2 School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran 3 Physik Department, Technische Universität München, James Franck Str. 1, 85748 Garching, Germany Received: 10 June 2018 / Accepted: 27 December 2018 / Published online: 29 January 2019 © The Author(s) 2019 Abstract Both collision geometry and event-by-event fluc- (RHIC) and Large Hadron Collider (LHC) measure the flow tuations are encoded in the experimentally observed flow har- harmonics vn [1–8], the coefficients of the momentum distri- monic distribution p(vn) and 2k-particle cumulants cn{2k}. bution Fourier expansion in the azimuthal direction [9–11]. In the present study, we systematically connect these observ- All these observations can be explained by several models ables to each other by employing a Gram–Charlier A series. based on the collective picture. We quantify the deviation of p(vn) from Bessel–Gaussianity Finding the flow harmonics is not straightforward, because in terms of harmonic fine-splitting of the flow. Subsequently, the reaction-plane angle (the angle between the orienta- we show that the corrected Bessel–Gaussian distribution can tion of the impact parameter and a reference frame) is not fit the simulated data better than the Bessel–Gaussian dis- under control experimentally. Additionally, the Fourier coef- tribution in the more peripheral collisions. Inspired by the ficients cannot be found reliably due to the low statistic in Gram–Charlier A series, we introduce a new set of cumulants a single event. These issues enforce us to use an analysis qn{2k}, ones that are more natural to use to study near Bessel– more sophisticated than a Fourier analysis. There are several Gaussian distributions. These new cumulants are obtained methods to find the flow harmonics experimentally, namely from cn{2k} where the collision geometry effect is extracted the event-plane method [12], multiparticle correlation func- from it. By exploiting q2{2k}, we introduce a new set of tions [13,14] and Lee–Yang zeros [15,16]. The most recent estimators for averaged ellipticity v¯2, ones which are more technique to find the flow harmonics is using the distribu- accurate compared to v2{2k} for k > 1. As another applica- tion of the flow harmonic p(vn). This distribution has been tion of q2{2k}, we show that we are able to restrict the phase obtained experimentally by employing the unfolding tech- space of v2{4}, v2{6} and v2{8} by demanding the consis- nique [17,18]. tency of v¯2 and v2{2k} with q2{2k} equation. The allowed It is well known that the initial shape of the energy density phase space is a region such that v2{4}−v2{6} 0 and depends on the geometry of the collision and the quantum 12v2{6}−11v2{8}−v2{4} 0, which is compatible with fluctuations at the initial state. As a result, the observed flow the experimental observations. harmonics fluctuate event-by-event even if we fix the initial geometry of the collision. In fact, the event-by-event fluc- tuations are encoded in p(vn) and experimentally observed 1 Introduction flow harmonics as well. It is worthwhile to mention that the observed event-by-event fluctuations are a reflection of the It is a well-established picture that matter produced in a heavy initial state fluctuations entangled with the modifications dur- ion experiment shows collective behavior. Based on this pic- ing different stages of the matter evolution, namely collec- ture, the initial energy density manifests itself in the final tive expansion and hadronization. For that reason, exploring particle momentum distribution. Accordingly, as the main the exact interpretation of the flow harmonics is crucial to consequence of this collectivity, the final particle momen- understand the contribution of each stage of the evolution tum distribution has extensively been studied by different on the fluctuations. Moreover, there has not been found a experimental groups in the past years. As a matter of fact, well-established picture for the initial state of the heavy ion the experimental groups at Relativistic Heavy Ion Collider collision so far. The interpretations of the observed quantities contain information about the initial state. This information a e-mail: [email protected] can shed light upon the heavy ion initial state models too. b e-mail: [email protected] 123 88 Page 2 of 24 Eur. Phys. J. C (2019) 79 :88 According to the theoretical studies, the flow harmon- cept of multiple orthogonal polynomials in mathematics is ics vn{2k} obtained from 2k-particle correlation functions presented in Appendix D. are different, and their difference is due to non-flow effects [13,14] and event-by-event fluctuations [19]. We should point out that experimental observations show that v2{2} is 2 Flow harmonic distributions and 2k-particle considerably larger than v2{4}, v2{6} and v2{8}. Addition- cumulants ally, all the ratios v2{6}/v2{4}, v2{8}/v2{4} and v2{8}/v2{6} are different from unity [20,21]. Alternatively, the distribu- This section is devoted to an overview of already well-known tion p(vn) is approximated by a Bessel–Gaussian distribution concepts concerning the cumulant and its application to study (corresponding to a Gaussian distribution for vn fluctuations) the collectivity in the heavy ion physics. We present this as a simple model [10,22]. Based on this model, the differ- overview to smoothly move forward to the harmonic distri- ence between v2{2} and v2{4} is related to the width of the bution of the flow and its deviation from Bessel–Gaussianity. v2 fluctuations. However, this model cannot explain the dif- v { } ference between the other 2 2k . 2.1 Correlation functions vs. distribution In the past years, several interesting studies of the non- v Gaussian n fluctuations have been done [23–28]. Specifi- According to the collective picture in the heavy ion experi- cally, it has been shown in Ref. [27] that the skewness of ments, the final particle momentum distribution is a conse- v v { }−v { } the 2 fluctuations is related to the difference 2 4 2 6 . quence of the initial state after a collective evolution. In order v Also, the connection between kurtosis of 3 fluctuations and to study this picture quantitatively, the initial anisotropies and v { }/v { } the ratio 3 4 3 2 has been studied in Ref. [28]. It is worth flow harmonics are used extensively to quantify the initial v noting that the deviation of n fluctuations from a Gaus- energy density and final momentum distribution. (v ) sian distribution immediately leads to the deviation of p n The initial energy (or entropy) density of a single event from a Bessel–Gaussian distribution. In [29], the quantities can be written in terms of the initial anisotropies, namely the v { }−v { } v { }−v { } n 4 n 6 and n 6 n 8 for a generic narrow distri- ellipticity and triangularity. Specifically, ellipticity and tri- bution are computed. angularity (shown by ε2 and ε3, respectively) are cumulants In the present work, we will introduce a systematic method of the distribution indicating how much it is deviated from a v { } (v ) to connect n 2k to the distribution p n . In Sect. 2,we two-dimensional rotationally symmetric Gaussian distribu- have an overview of the known concepts of cumulants, flow tion [30]. harmonic distributions and their relation with the averaged The final momentum distribution is studied by its Fourier v¯ flow harmonics n. Section 3 is dedicated to the Gram– expansion in the azimuthal direction, Charlier A series in which we find an approximate flow ∞ harmonic distribution in terms of cn{2k}. Specifically for 1 dN 1 = 1 + 2vn cos n(φ − ψn) . (1) the second harmonics, we show that the deviation of p(v2) N dφ 2π n=1 from Bessel–Gaussianity is quantified by the fine-splitting In the above, v and ψ are unknown parameters, which can v2{2k}−v2{2} where k, ≥ 2 and k = . These studies n n vˆ ≡ v inψn =inφ guide us to defining a new set of cumulants q {2k} where they be found easily via n ne e . Here, the aver- n 1 dN depend on the event-by-event fluctuations only. In Sect. 4, aging is obtained by using the distribution N dφ in a given we use the new cumulants to introduce more accurate esti- event. The parameter vˆn is called flow harmonic. Instead of mations for the average ellipticity. As another application of a complex form, we occasionally use the flow harmonics in new cumulants, we use q2{2k} to constrain the v2{4}, v2{6} Cartesian coordinates, v { } and 2 8 phase space in Sect. 5. We show that the phase v = v ψ ,v= v ψ . 1 n,x n cos n n n,y n sin n n (2) space is restricted to a domain where v2{4}−v2{6} 0 and 12v2{6}−11v2{8}−v2{4} 0. We present the conclusion Low multiplicity in a single event, randomness of the reac- in Sect. 6. The supplementary materials can be found in the tion plane angle and non-flow effects are the main challenges appendices. We would like to emphasize that in Appendix C, in extracting the experimental values of the flow harmonics.
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