Measurement of elliptic flow of light nuclei at \sqrt{s_{NN}} = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV at RHIC

Measurement of elliptic flow of light nuclei at = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV at RHIC

L. Adamczyk, J. K. Adkins, G. Agakishiev, M. M. Aggarwal, Z. Ahammed, I. Alekseev, A. Aparin, D. Arkhipkin, E. C. Aschenauer, A. Attri, G. S. Averichev, X. Bai, V. Bairathi, R. Bellwied, A. Bhasin, A. K. Bhati, P. Bhattarai, J. Bielcik, J. Bielcikova, L. C. Bland, I. G. Bordyuzhin, J. Bouchet, J. D. Brandenburg, A. V. Brandin, I. Bunzarov, J. Butterworth, H. Caines, M. Calderón de la Barca Sánchez, J. M. Campbell, D. Cebra, I. Chakaberia, P. Chaloupka, Z. Chang, A. Chatterjee, S. Chattopadhyay, J. H. Chen, X. Chen, J. Cheng, M. Cherney, W. Christie, G. Contin, H. J. Crawford, S. Das, L. C. De Silva, R. R. Debbe, T. G. Dedovich, J. Deng, A. A. Derevschikov, B. di Ruzza, L. Didenko, C. Dilks, X. Dong, J. L. Drachenberg, J. E. Draper, C. M. Du, L. E. Dunkelberger, J. C. Dunlop, L. G. Efimov, J. Engelage, G. Eppley, R. Esha, O. Evdokimov, O. Eyser, R. Fatemi, S. Fazio, P. Federic, J. Fedorisin, Z. Feng, P. Filip, Y. Fisyak, C. E. Flores, L. Fulek, C. A. Gagliardi, D.  Garand, F. Geurts, A. Gibson, M. Girard, L. Greiner, D. Grosnick, D. S. Gunarathne, Y. Guo, S. Gupta, A. Gupta, W. Guryn, A. I. Hamad, A. Hamed, R. Haque, J. W. Harris, L. He, S. Heppelmann, S. Heppelmann, A. Hirsch, G. W. Hoffmann, S. Horvat, T. Huang, X.  Huang, B. Huang, H. Z. Huang, P. Huck, T. J. Humanic, G. Igo, W. W. Jacobs, H. Jang, A. Jentsch, J. Jia, K. Jiang, E. G. Judd, S. Kabana, D. Kalinkin, K. Kang, K. Kauder, H. W. Ke, D. Keane, A. Kechechyan, Z. H. Khan, D. P. Kikoła , I. Kisel, A. Kisiel, L. Kochenda, D. D. Koetke, L. K. Kosarzewski, A. F. Kraishan, P. Kravtsov, K. Krueger, L. Kumar, M. A. C. Lamont, J. M. Landgraf, K. D.  Landry, J. Lauret, A. Lebedev, R. Lednicky, J. H. Lee, X. Li, C. Li, X. Li, Y. Li, W. Li, T. Lin, M. A. Lisa, F. Liu, T. Ljubicic, W. J. Llope, M. Lomnitz, R. S. Longacre, X. Luo, R. Ma, G. L. Ma, Y. G. Ma, L. Ma, N. Magdy, R. Majka, A. Manion, S. Margetis, C. Markert, H. S. Matis, D. McDonald, S. McKinzie, K. Meehan, J. C. Mei, N. G. Minaev, S. Mioduszewski, D. Mishra, B. Mohanty, M. M. Mondal, D. A. Morozov, M. K. Mustafa, B. K. Nandi, Md. Nasim, T. K. Nayak, G. Nigmatkulov, T. Niida, L. V. Nogach, S. Y. Noh, J. Novak, S. B. Nurushev, G. Odyniec, A. Ogawa, K. Oh, V. A. Okorokov, D. Olvitt Jr., B. S. Page, R. Pak, Y. X. Pan, Y. Pandit, Y. Panebratsev, B. Pawlik, H. Pei, C. Perkins, P.  Pile, J. Pluta, K. Poniatowska, J. Porter, M. Posik, A. M. Poskanzer, N. K. Pruthi, J. Putschke, H. Qiu, A. Quintero, S. Ramachandran, R. Raniwala, S. Raniwala, R. L. Ray, H. G. Ritter, J. B. Roberts, O. V. Rogachevskiy, J. L. Romero, L. Ruan, J. Rusnak, O. Rusnakova, N. R. Sahoo, P. K. Sahu, I. Sakrejda, S. Salur, J. Sandweiss, A.  Sarkar, J. Schambach, R. P. Scharenberg, A. M. Schmah, W. B. Schmidke, N. Schmitz, J. Seger, P. Seyboth, N. Shah, E. Shahaliev, P. V. Shanmuganathan, M. Shao, M. K. Sharma, B. Sharma, W. Q. Shen, Z. Shi, S. S. Shi, Q. Y. Shou, E. P. Sichtermann, R. Sikora, M. Simko, S. Singha, M. J. Skoby, N. Smirnov, D. Smirnov, W. Solyst, L. Song, P. Sorensen, H. M. Spinka, B. Srivastava, T. D. S. Stanislaus, M.  Stepanov, R. Stock, M. Strikhanov, B. Stringfellow, M. Sumbera, B. Summa, X. M. Sun, Z. Sun, Y. Sun, B. Surrow, D. N. Svirida, Z. Tang, A. H. Tang, T. Tarnowsky, A. Tawfik, J. Thäder, J. H. Thomas, A. R. Timmins, D. Tlusty, T. Todoroki, M. Tokarev, S. Trentalange, R. E. Tribble, P. Tribedy, S. K. Tripathy, O. D. Tsai, T. Ullrich, D. G. Underwood, I. Upsal, G. Van Buren, G. van Nieuwenhuizen, M. Vandenbroucke, R. Varma, A. N. Vasiliev, R. Vertesi, F. Videbæk, S. Vokal, S. A. Voloshin, A. Vossen, Y. Wang, G. Wang, J. S. Wang, H. Wang, Y. Wang, F. Wang, G. Webb, J. C. Webb, L. Wen, G. D. Westfall, H. Wieman, S. W. Wissink, R. Witt, Y. Wu, Z. G. Xiao, W. Xie, G. Xie, K. Xin, H. Xu, Z. Xu, J. Xu, Y. F. Xu, Q. H. Xu, N. Xu, Y. Yang, S. Yang, C. Yang, Y. Yang, Y. Yang, Q. Yang, Z. Ye, Z. Ye, P. Yepes, L. Yi, K. Yip, I. -K. Yoo, N. Yu, H. Zbroszczyk, W. Zha, J. Zhang, Y. Zhang, X. P. Zhang, Z. Zhang, J. B. Zhang, S. Zhang, S. Zhang, J. Zhang, J. Zhao, C. Zhong, L. Zhou, X. Zhu, Y. Zoulkarneeva, M. Zyzak AGH University of Science and Technology, FPACS, Cracow 30-059, Poland Argonne National Laboratory, Argonne, Illinois 60439 Brookhaven National Laboratory, Upton, New York 11973 University of California, Berkeley, California 94720 University of California, Davis, California 95616 University of California, Los Angeles, California 90095 Central China Normal University, Wuhan, Hubei 430079 University of Illinois at Chicago, Chicago, Illinois 60607 Creighton University, Omaha, Nebraska 68178 Czech Technical University in Prague, FNSPE, Prague, 115 19, Czech Republic Nuclear Physics Institute AS CR, 250 68 Prague, Czech Republic Frankfurt Institute for Advanced Studies FIAS, Frankfurt 60438, Germany Institute of Physics, Bhubaneswar 751005, India Indian Institute of Technology, Mumbai 400076, India Indiana University, Bloomington, Indiana 47408 Alikhanov Institute for Theoretical and Experimental Physics, Moscow 117218, Russia University of Jammu, Jammu 180001, India Joint Institute for Nuclear Research, Dubna, 141 980, Russia Kent State University, Kent, Ohio 44242 University of Kentucky, Lexington, Kentucky, 40506-0055 Korea Institute of Science and Technology Information, Daejeon 305-701, Korea Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu 730000 Lawrence Berkeley National Laboratory, Berkeley, California 94720 Max-Planck-Institut fur Physik, Munich 80805, Germany Michigan State University, East Lansing, Michigan 48824 National Research Nuclear Univeristy MEPhI, Moscow 115409, Russia National Institute of Science Education and Research, Bhubaneswar 751005, India National Cheng Kung University, Tainan 70101 Ohio State University, Columbus, Ohio 43210 Institute of Nuclear Physics PAN, Cracow 31-342, Poland Panjab University, Chandigarh 160014, India Pennsylvania State University, University Park, Pennsylvania 16802 Institute of High Energy Physics, Protvino 142281, Russia Purdue University, West Lafayette, Indiana 47907 Pusan National University, Pusan 46241, Korea University of Rajasthan, Jaipur 302004, India Rice University, Houston, Texas 77251 University of Science and Technology of China, Hefei, Anhui 230026 Shandong University, Jinan, Shandong 250100 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 State University Of New York, Stony Brook, NY 11794 Temple University, Philadelphia, Pennsylvania 19122 Texas A&M University, College Station, Texas 77843 University of Texas, Austin, Texas 78712 University of Houston, Houston, Texas 77204 Tsinghua University, Beijing 100084 United States Naval Academy, Annapolis, Maryland, 21402 Valparaiso University, Valparaiso, Indiana 46383 Variable Energy Cyclotron Centre, Kolkata 700064, India Warsaw University of Technology, Warsaw 00-661, Poland Wayne State University, Detroit, Michigan 48201 World Laboratory for Cosmology and Particle Physics (WLCAPP), Cairo 11571, Egypt Yale University, New Haven, Connecticut 06520
Abstract

We present measurements of 2 order azimuthal anisotropy () at mid-rapidity for light nuclei d, t, He (for = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV) and anti-nuclei ( = 200, 62.4, 39, 27, and 19.6 GeV) and ( = 200 GeV) in the STAR (Solenoidal Tracker at RHIC) experiment. The for these light nuclei produced in heavy-ion collisions is compared with those for p and . We observe mass ordering in nuclei at low transverse momenta ( GeV/). We also find a centrality dependence of for d and . The magnitude of for t and He agree within statistical errors. Light-nuclei are compared with predictions from a blast wave model. Atomic mass number () scaling of light-nuclei seems to hold for GeV/. Results on light-nuclei from a transport-plus-coalescence model are consistent with the experimental measurements.

pacs:
25.75.Ld

STAR Collaboration

I Introduction

One of the main goals of high energy heavy-ion collision experiments is to study phase structures in the QCD phase diagram QCD_theo (); whitepapers (). With this purpose the Relativistic Heavy Ion Collider (RHIC) has finished the first phase of the Beam Energy Scan (BES) program bes_res1 (); v2_BES_prc (); v2_BES_prl (); v2_PID_cent (); BES_PHENIX1 (); BES_PHENIX2 (); v2_200_QM (). It has been found that the identified hadron shows number-of-constituent-quark (NCQ) scaling at high at the higher beam energies. This scaling behavior is an expected signature of partonic collectivity via quark coalescence in the strongly interacting medium of quarks and gluons formed in heavy-ion collisions part_coll2 (); part_coll3 (); part_coll4 (); part_coll1 (); part_Lacey (). Such a scaling behavior also suggests partonic coalescence to be a mechanism for hadron formation part_coll2 (); part_coll3 (); hadr_form (). In a relativistic heavy-ion collision, light (anti-)nuclei can be formed by coalescence of produced (anti-)nucleons or from transported nucleons nucl_prod1 (); nucl_prod2 (); nucl_prod3 (). The binding energies of light nuclei are very small ( few MeV), making it likely that surviving light nuclei are formed at a later stage of the evolution. This phenomenon is called final-state coalescence nucl_prod1 (); nucl_prod4 (). The coalescence probability of two nucleons is related to the local nucleon density nucl_prod1 (); nucl_prod2 (); nucl_prod5 (). Since the coalescence mechanism works best at the low density limit, low relative production of nucleons in heavy-ion collisions offers an ideal situation to study light-nuclei production via coalescence. Measurements of azimuthal anisotropy of light nuclei offers a tool to understand the light-nuclei production mechanism and freeze-out properties at a later stage of the evolution. Unlike the case of quark coalescence, in a nucleon coalescence, the momentum space distributions of both the constituents and the products are measurable in heavy-ion collision experiments.

Prior measurements of elliptic flow of light nuclei have been carried out at the top RHIC energy ( 200 GeV) by the PHENIX Phenix_nucl () and the STAR zhangbu_nucl (); cjena_nucl () experiments. The PHENIX Collaboration has measured the of deuterons (d) and anti-deuterons () at intermediate transverse momenta (1.1 4.5 GeV/). How the of these light nuclei scale with those of (anti-)protons also has been reported Phenix_nucl (). The STAR collaboration has measured the of d, , He, and in Au+Au collisions at 200 GeV in the years 2004 zhangbu_nucl () and 2007 cjena_nucl (). Negative for at low has also been reported zhangbu_nucl ().

In this work we expand upon previous studies with a detailed investigation on the energy and centrality dependence of of light nuclei with more event statistics. During the BES program, the STAR experiment has taken data over a wide range of collision energies from = 7.7 GeV to 200 GeV. In this paper we present the measurement of at mid-rapidity for light nuclei d, t, He ( = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV), and anti-nuclei ( = 200, 62.4, 39, 27, and 19.6 GeV) and ( = 200 GeV)

The paper is organized as follows. Sec. II briefly describes the experimental setup, the detectors and the particle (and light-nuclei) identification (PID) techniques. The centrality definition, event selection, event plane reconstruction and the event plane resolution correction are also discussed, along with the extraction procedure of light-nuclei . Presented in Sec. III are the results for minimum bias collisions, the centrality dependence, and a physical interpretation of the results. A comparison between light-nuclei measured in this experiment and those calculated from blast wave and transport-plus-coalescence model is also shown. Sec. IV summarizes the physics observations and discusses the main conclusions from the results.

(a)(b)

Figure 1: (color online) (a) Specific energy loss (dE/dx) as a function of rigidity (momentum/charge). Theoretical dE/dx expectations, using the model in Ref. Bichsel () of d, t, He are shown by solid curves. (b) Mass squared () as a function of momentum for mid-rapidity charged particles. The dotted lines correspond to of different nuclei. Both results are from minimum bias Au+Au collisions at = 19.6 GeV.

Ii Experimental setup

STAR is a multipurpose experiment at the RHIC facility at Brookhaven National Laboratory. It consists of a longitudinally-oriented (beam direction) solenoidal magnet and a collection of detectors for triggering, PID, and event categorization STAR_nim (). The main detectors used for this analysis are the Time Projection Chamber (TPC) STAR_TPC () and the Time of Flight (TOF) detector STAR_ToF (). The following subsections briefly describe their operations and PID techniques.

ii.1 TPC Measurements

The TPC is the primary tracking device in the STAR experiment which uses ionisation in a large gas volume to detect trajectories of charged particles. Curvature in the solenoidal field enables determination of the charge sign and rigidity (momentum/charge). The TPC has full azimuthal coverage and a uniform pseudorapidity range of  STAR_TPC (). The TPC can record up to 45 hit positions and specific ionisation energy loss (dE/dx) samples along tracks. Truncated means of the dE/dx samples are used for PID by comparing to theoretical expectations, using improved Bethe-Bloch functions Bichsel (), at the measured rigidities to characterize the probability for being any particular species. PID consequently allows deduction of the particles’ charges and momenta. A representative plot of measured track dE/dx versus rigidity is shown in Fig. 1(a) for minimum bias (defined later) Au+Au collisions at 19.6 GeV. The theoretical curves are shown as solid lines. Primary collision vertices are found through fits involving candidate daughter tracks, and a typical central collision at the top RHIC energy (with perhaps 1000 reconstructed tracks) may achieve a vertex position resolution of 350 m. These daughter tracks are then refitted using their vertex as a constraint to create a collection of primary tracks.

ii.2 TOF Measurements

The TOF detector STAR_ToF () in STAR uses Multigap Resistive Plate Chambers (MRPCs) and was fully installed in the year 2010. It covers 2 in azimuth within the pseudorapidity interval 0.94. The TOF detector and the Vertex Position Detector (VPD) STAR_VPD () measure the time interval, , over which a particle travels from the primary collision vertex to a read-out cell of the TOF detector. This time interval information is combined with the total path length, , measured by the TPC to provide the inverse velocity, 1/, via 1/c/, where c is the speed of light. The track mass-squared is then given by (1/1). For collision energies below 39 GeV the VPD efficiency is too low to use in every event. Instead, for these data sets a start time for each collision is inferred by working backwards from the TOF-measured stop times of a very limited selection of particles which are very cleanly identified in the TPC. The total time interval resolution obtained of 90-110 ps results in PID capabilities that are complementary to those from the TPC dE/dx at low momenta and also extend to momenta of several GeV. A representative plot of as a function of the particle momentum is shown in Fig. 1(b) for minimum bias AuAu collisions at 19.6 GeV. As the mass of a particle is a constant quantity, we expect horizontal bands for individual (anti-)nuclei as shown by the dotted lines in Fig. 1(b). We have selected individual nuclei using the which lie within 3 from the constant mean (dotted line).

Figure 2: Resolution correction factor () of sub-event planes as a function of centrality for Au+Au collisions at = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV.

ii.3 Trigger and event selection

The minimum bias events for all of the collision energies are based on a coincidence of the signals from the Zero-Degree Calorimeters (ZDC) STAR_ZDC (), VPD, and/or Beam-Beam Counters (BBC) STAR_BBC (). Due to larger beam emittance at lower collision energies, AuAu-triggered events are contaminated with Aubeam-pipe events. The radius of the beam-pipe going through the center of the TPC is 3.95 cm. Therefore, such Aubeam-pipe events are removed by requiring the primary vertex position to be within a transverse radius of less than 2 cm in the XY-plane v2_BES_prc (). The -position of the primary vertices (vertex-) is limited to the values listed in Table 1 v2_BES_prc () to ensure good quality events.

(GeV) vertex- (cm) MB events ()
200 vertex- 30 241
62.4 vertex- 40 62
39 vertex- 40 119
27 vertex- 70 60
19.6 vertex- 70 33
11.5 vertex- 50 11
7.7 vertex- 70 4
Table 1: The vertex- acceptance and total number of minimum-bias (MB) events for each energy ().

Furthermore, an extensive quality assurance of the events was performed based on the mean transverse momenta, the mean vertex position, the mean interaction rate, and the mean multiplicity in the detector. Run periods were removed if one of those quantities was more than 3 away from the global mean value. The total number of minimum bias events used in this analysis after these quality assurance cuts for each collision energy are shown in Table 1.

ii.4 Centrality definition

The centrality of each event is defined based on the uncorrected charged particle multiplicity () distribution, where is the number of events and is the number of charged particles measured within 0.5 v2_BES_prc (). Thus, for example, 0%-5% central events correspond to the events in the top 5% of the multiplicity distribution. The charged particle multiplicity distributions for all energies can be described by a two-component model two_comp (). The two-component model is a Glauber Monte-Carlo simulation in which the multiplicity per unit pseudo-rapidity () depends on the two components, namely, number of participant nucleons () and number of binary collisions ():

(1)

The fitting parameter is the in minimum-bias pp collisions and is the fraction of produced charged particles from the hard component. The centrality class is defined by calculating the fraction of the total cross-section obtained from the simulated multiplicity. Due to trigger inefficiencies, many of the most peripheral events were not recorded. This results in a significant difference between the measured distribution of charged particle multiplicities and the Glauber Monte Carlo (MC) simulation for peripheral collisions. When determining in a bin of multiplicity wide enough to see variation in the trigger inefficiency across the bin (e.g. for a minimum bias measurement), it is necessary to compensate for this variation by weighting particle yields in each event by the inverse of the trigger efficiency at that event’s multiplicity v2_BES_prc (). The correction is about 5% for the peripheral (70%-80%) events, and becomes negligible for central events. However, the corrections are severe for 80%-100% central events. Therefore, 80%-100% central events are not included in the current analysis, and minimum bias is defined for all data presented here as 0%-80%. In addition to the trigger inefficiency, two additional corrections are also applied to account for the vertex- dependent inefficiencies. These corrections account for the acceptance and detector inefficiencies and the time-dependent changes in .

(a)(b)

Figure 3: (color online) (a) distributions for mid-rapidity , t, He. The different ranges are for acceptance-representative purpose. The distribution for each species is fitted with a two-Gaussian function. One Gaussian is used to describe the distribution for the species of interest (dashed line), and another Gaussian is used to describe the background (dot-dashed line). (b) distributions for mid-rapidity , t, and He. Solid lines are fitted 2 order Fourier functions. All plots use minimum bias Au+Au collisions at = 39 GeV.

ii.5 Event plane and resolution correction

The azimuthal distribution of produced particles with respect to reaction plane angle can be expressed in terms of a Fourier series,

(2)

where is the azimuthal angle of the produced particle. is defined as the angle between -axis in the laboratory frame and axis of the impact parameter. Because we cannot directly measure , we must use a proxy. The second order azimuthal anisotropy or elliptic flow is measured with respect to the order event plane angle () instead. is calculated using the azimuthal distribution of all reconstructed primary tracks (art1 ():

(3)

and are defined as

(4a)
(4b)

where are the weights which optimise the event plane resolution art1 (). In this analysis, the weights scale with track-, then saturate above 2.0 GeV/. To reduce biases due to short range correlation, we utilize the sub-event plane method art1 (). In this analysis, the two sub-events were defined in windows of (-1.0    -0.05) and (0.05    1.0). Event plane angles are calculated within each window, and respectively, and is calculated in each sub-event using the opposite sub-event’s event plane angle. The gap ( = 0.1) between the sub-events reduces the short range non-flow contributions and avoids the self-correlation. However, long range correlations may persist art2 ().

Due to the acceptance inefficiency of the detectors, the reconstructed event plane distributions are not uniform. Therefore, we apply event-by-event recenter recenter () and shift shift () corrections. Finite multiplicities also restrict the degree to which the found event plane angles coincide with the true reaction plane angle. Hence, a resolution correction is applied to the observed elliptic flow (): = . We determine the resolution correction factor () in the sub-event plane method as follows art1 ():

(5)

The resolution as a function of centrality for sub-event planes is shown in Fig. 2 for Au+Au collisions. grows with increasing multiplicity (which is small for peripheral collisions) and with increasing (which is small for the most central collisions), so its value peaks in mid-central (20%-30%) collisions where neither is small.

Figure 4: (color online) Mid-rapidity for d, t, He, and from minimum bias Au+Au collisions at = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV. For comparison, proton are also shown as open circles v2_BES_prc (); v2_200_QM (). Lines and boxes at each marker represent statistical and systematic errors respectively.

ii.6 Extraction of yield and of nuclei

To identify light nuclei, we define a variable such that

(6)

where is the energy loss of the light nuclei measured by the TPC detector in the experiment and is the theoretical energy loss as obtained from the modified Bethe-Bloch formula Bichsel (). After cutting on from TOF (see Fig. 1(b)) to reduce backgrounds under the signals, the yields are extracted from the distributions in various and bins for each species of interest with a two-Gaussian function (one for the signal, the other for the background). Figure 3(a) shows sample distributions for and He, respectively, within 0 for 1.3 1.9 GeV/, 2.1 3.4 GeV/, and 1.9 2.5 GeV/ for minimum bias Au+Au data at GeV. The azimuthal angle variation of this yield is then fitted with a 2 order Fourier function to get the elliptic flow coefficient (). Figure 3(b) shows the distributions for , t and He for the same ranges as shown for distributions in Fig. 3(a). As the () distribution is expected to be symmetric about 0 and , the data points have been folded onto 0- to reduce the statistical errors.

The fitted 2 order Fourier functions are shown in Fig. 3(b). Event plane resolution correction factors are determined in each centrality bin. For integrated over multiple centrality bins, species-yield-weighted mean of the individual centrality bins’ resolutions are used:  hiroshi_alex ().

ii.7 Calculation of systematic uncertainty and removal of beam-pipe contaminations

We have reduced light-nuclei contaminants from interactions with the beam pipe by cutting tightly on the projected distance of closest approach (DCA) to the primary vertex. Remaining contaminants from such interactions are removed statistically by fitting the DCA distribution of nuclei with that of anti-nuclei (which are expected to have no such background) in each () bin. Systematic uncertainties are determined by varying cuts used in particle identification and background rejection, and by varying fitting methods and ranges when measuring yields. The absolute magnitude of uncertainties range from 2%-5% for intermediate (1.0 3.0 GeV/c) and from 5%-8% for low and high .

Figure 5: (color online) Mid-rapidity for (squares), (triangles), p (open circles), and d (crosses) for minimum bias Au+Au collisions at = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV.
Figure 6: (color online) The difference in of d and as a function of for minimum bias Au+Au collisions at = 200, 62.4, 39, 27, and 19.6 GeV, along with differences between p and  v2_BES_prc (). Solid lines correspond to constants fit to the data (see text for details).
Figure 7: (color online) Centrality dependence of mid-rapidity of d (open markers) for Au+Au collisions at = 7.7 - 200 GeV and (solid markers) for = 27 - 200 GeV. For = 200 GeV, circles correspond to 0%-10%, triangles to 10%-40%, and squares to 40%-80% central events. For other collision energies, circles correspond to 0%-30% and squares to 30%-80% central events.

Iii Results and discussion

iii.1 General properties of

Figure 4 shows the energy dependence of the of the light (anti-)nuclei d, , t, He, and as a function of for minimum bias Au+Au collisions. Insufficient statistics preclude measuring differential anti-nuclei at several collision energies. The of all light-nuclei species and anti-nuclei species ( at = 19.6 200 GeV and at = 200 GeV) show monotonically increasing trend with increasing (Fig. 4). Mass ordering of for 2.0 GeV is clear in both Figs. 4 and 5, where the of , K, and p from Ref. v2_BES_prc (); v2_200_QM () are also included (heavier species have a lower in this range). Such ordering occurs naturally in a hydrodynamic coalescence model of heavy-ion collisions hydro (). The negative observed for some (anti)-nuclei could be the result of radial flow.

Figure 6 presents the difference of between d and (), along with the difference between p and for comparison v2_BES_prc (); v2_200_QM (). Statistical uncertainties are too large to draw conclusions about any collision energy dependence, but the data are qualitatively consistent with the (anti-)protons and the results of fitting a constant at each energy (solid lines in Fig. 6) are consistently positive: 0.00120.0014, 0.0090.005, 0.00440.0046, 0.0170.009, 0.0240.019 for = 200, 62.4, 39, 27, and 19.6 GeV respectively.

Figure 7 shows of d and in 0%-30% and 30%-80% central events for where they could be measured in Au+Au collisions at = 62.4 to 7.7 GeV. For 200 GeV are measured in three centralities: 0%-10%, 10%-40%, and 40%-80%. The observed centrality dependences are qualitatively similar to those seen in identified hadrons v2_BES_prc (); v2_PID_cent (), with d and showing similar behavior for all centralities measured.

iii.2 Blast wave model

The nuclear fireball model was first introduced by Westfall et al. to explain midrapidity proton-inclusive spectra Gary_fireball (). Later, Siemens and Rasmussen Siemens_BW () generalized a non-relativistic formula by Bondorf, Garpman, and Zimanyi Bondorf_BW () to explain nucleons and pions as they are produced in a blast wave of an exploding fireball. The blast wave model has evolved since then, with more parameters to describe both spectra and anisotropic flow of produced particles Schnedermann_BW (); Huovinen_BW (); STAR_BW (). The blast wave parametrization modeled by the STAR Collaboration STAR_BW () has been recently used to fit the of identified particles PID_BW_fit (). This version of blast wave has four parameters, namely kinetic freeze-out temperature (), transverse expansion rapidity (), amplitude of its azimuthal variation (), and the variation in the azimuthal density of the source elements (PID_BW_fit (). The fit parameters obtained from blast wave fits to the of identified particles are listed in Table 1 of PID_BW_fit (). We have used the same blast wave model and fit parameter values to check whether the bast wave model also reproduces the of light nuclei measured in the data. Figure 8 shows the blast wave model predictions for light nuclei, along with the measurements. As is evident from Fig. 8, blast wave model under-predicts the of d and at low ( 1.0 GeV/) for most of the collision energies. Similar conclusion for t, He () is difficult to make due to their large statistical uncertainty. However, for = 200 GeV, the blast wave model clearly fails to reproduce the measured of light nuclei of all species at low ( 1.0 GeV/).

Figure 8: (color online) Blast wave model predictions (lines) of for d, , t, He () compared with the data for minimum bias Au+Au collisions. The blast wave model and parameter values have been used from PID_BW_fit (). (Some data points in the lower panels are off-scale.)
Figure 9: (color online) Atomic mass number () scaling of the mid-rapidity of p, , d, , t, He, and from minimum bias Au+Au collisions at = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV. Grey Solid (black dotted) lines correspond to order polynomial fits to the p () data. The ratios of for d, , t, and He are shown in the lower panels at each corresponding collision energy. (Some data points in the lower panels are off-scale.)
Figure 10: (color online) Mid-rapidity of d, t, and He are compared with the results of AMPT+coalescence calculations (solid bands).

iii.3 Atomic mass number scaling and coalescence model

Figure 9 presents the light-nuclei as a function of , where is the atomic mass number of the corresponding light nuclei. The main goal of this study is to understand whether light (anti-)nuclei production is consistent with coalescence of (anti-)nucleons. The model predicts that if a composite particle is produced by coalescence of number of particles that are very close to each other in phase-space, then of the composite will be times that of the constituents YGMA (). In Fig. 9 it is observed that the (anti-)nuclei closely follows of p () for up to 1.5 GeV/. The scaling behavior of these nuclei suggest that d () within 3.0 GeV/ and t, He () within 4.5 GeV/ might have formed via the coalescence of nucleons (anti-nucleons). The low relative production of light nuclei seems to favor the coalescence formalism rather than other methods, such as thermal production which can reproduce the measured particle ratios in data CFO_nucl (). As protons and neutrons have the same , expected from NCQ scaling, then we can readily see that the of t and He will be the same as they have the same atomic mass number ( = 3). We find that, within statistical errors, our measurement of () for t and He confirms this assumption. Although simple scaling seems to hold for the collision energies presented, the actual mechanism might be a more dynamic process including production and coalescence of nucleons in the local rest frame of the fluid cell. This scenario might give rise to deviations from simple scaling.

It is arguable that light nuclei could have also formed via coalescence of quarks as the scaling behavior holds when and are scaled by number of constituent quarks (e.g. 6 for d, and 9 for t, He) instead of mass number. Although this process seems physically acceptable, the survival of light nuclei, with their low binding energies (few MeV), is highly unlikely under the high temperatures requisite for dissociating nucleons into quarks and gluons.

To further verify the applicability of nucleon coalescence into light nuclei in heavy-ion collisions, we have run the string-melting version of A Multi Phase Transport (AMPT, version v1.25t7d) AMPT () model of the collisions in conjunction with a dynamic coalescence model. The AMPT model has been used to reproduce charged particle multiplicity, transverse momentum spectra at RHIC and LHC, as well as of identified particles at RHIC AMPT (). The dynamic coalescence model has been used extensively at both intermediate coal_inter () and high energies coal_high (). In this model, the probability for producing a cluster is determined by the overlap of the cluster’s Wigner phase-space density with the nucleon phase-space distribution at freeze-out procured from AMPT. For light nuclei, the Wigner phase-space densities are obtained from their internal wave functions, which are taken to be those of a spherical harmonic oscillator spherical (). For the coalescence model we have used radii of 1.96, 1.61, and 1.74 fm for d, t, and He respectively rms_radii (). These parameters are kept fixed for the collision energy range presented. The model’s results for of d, t, and He are shown as solid bands in Fig. 10. The data and model agree within errors over nearly all energies and measured, supporting the theory that light nuclei are produced via nucleon coalescence in heavy-ion collisions. Recently the ALICE collaboration has measured production of d and in Pb+Pb collisions at = 2.76 TeV ALICE_ddbar (). In that study, light-nuclei spectra were found to exhibit a significant hardening with increasing centrality. The stiffening of light-nuclei spectra at ALICE could be the result of increased hard scattering, modified fragmentation, or increased radial flow. However, the analysis lacks conclusive evidence regarding the production mechanism of light nuclei in heavy-ion collisions. In the collision energy range presented in this paper, light-nuclei production favors the coalescence model.

Iv Summary

Measurements of the 2 order azimuthal anisotropy, at mid-rapidity () have been presented for light nuclei d, t, He (for = 200, 62.4, 39, 27, 19.6, 11.5, and 7.7 GeV), and anti-nuclei ( = 19.6200 GeV) and ( = 200 GeV). Similar to hadrons over the measured range, light (anti-)nuclei show a monotonic rise with increasing , mass ordering at low , and a reduction for more central collisions. It is observed that of nuclei and anti-nuclei are of similar magnitude for = 39 GeV and above. The difference between d and is found to follow the difference between p and as a function of collision energy. The blast wave model is found to under-predict the light-nuclei measured in data. He and t nuclei show similar for all collision energies, and in fact all the light-nuclei generally follow an atomic mass number scaling, which indicates that the coalescence of nucleons might be the underlying mechanism of light-nuclei formation in high energy heavy-ion collisions. This observation is further corroborated by carrying out a model-based study of nuclei using a transport-plus-coalescence model, which reproduces well the light-nuclei measured in the data.

Acknowledgements

We thank the RHIC Operations Group and RCF at BNL, the NERSC Center at LBNL, the KISTI Center in Korea, and the Open Science Grid consortium for providing resources and support. This work was supported in part by the Office of Nuclear Physics within the U.S. DOE Office of Science, the U.S. NSF, the Ministry of Education and Science of the Russian Federation, NSFC, CAS, MoST and MoE of China, the National Research Foundation of Korea, NCKU (Taiwan), GA and MSMT of the Czech Republic, FIAS of Germany, DAE, DST, and UGC of India, the National Science Centre of Poland, National Research Foundation, the Ministry of Science, Education and Sports of the Republic of Croatia, and RosAtom of Russia. This work is supported by the DAE-BRNS project Grant No. 2010/21/15-BRNS/2026.

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