Clustering Structure Effect on Hanbury-Brown-Twiss Correlation in $^{12} $ C+ $^{197} $ Au Collisions at 200

$Clustering Structure Effect on Hanbury-Brown-Twiss Correlation in $^{12} $ C+ $^{197} $ Au Collisions at 200$

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BT ,1, 3, cutrn ofiuain nlgtnce 1,1] Also, 15]. [14, nuclei light in configurations -clustering s + C ai ftetid otescn-re B radii HBT second-order the to third- the of ratio nutarltvsi ev-o olsosa N-HCan BNL-RHIC at collisions heavy-ion ultra-relativistic In eety eadMa and He Recently, α NN † cutrn nlgtnce ntergon tt,which state, ground their in nuclei light in -clustering )rdirltv otescn-adthird-order and second- the to relative radii T) ihiChen, Jinhui 208 = 0 e,adtecngrto of configuration the and GeV, 200 bcliin olcieflwcnb h signature the be can flow collective collision, Pb ine,Saga 080 China 201800, Shanghai ciences, ainin lation o,ti okpeet h hadronic the presents work this ion, 3 43 China 0433, α n hnZhong Chen and α cutrsrcue swl stespher- the as well as structures -cluster 12 cutrn nlgtnce.A spe- a As nuclei. light in -clustering (MOE), China 12 + C rtosof urations tal. et + C etknit account, into taken re 197 197 α on httegatdipole giant the that found cutrn tutr,which structure, -clustering ucliin t200 at collisions Au ucliin at collisions Au 12 12 3 + C ,namely C, 197 π - 12 π ucliin at collisions Au nldstri- includes C orlto as correlation tal. et √ s NN pro- s pro- lar- ure ent ely gh li- as = ly r- i- i- d - , 2 driven understanding of the freeze-out configuration is the II. THE AMPT MODEL AND ALGORITHM crucial first step to learn the system evolution and the physics INTRODUCTION of hot dense quark-gluon matter. For example, a detailed understanding of the space-time geometry and dynamics of fire- The AMPT model is a successful model for studying reac- ball is required. Because of the short lifetime and extension tion dynamics in relativistic heavy-ion collisions, consisting of the source, two-particle intensity interferometry, or the so- of four main components: the initial conditions, partonic in- called HBT method, provides a unique way to obtain direct teractions, hadronization and hadronic interactions. There are information about the space-time structure from the measured two versions of the AMPT model: the default version and the particle momenta. string melting version. The phase-space distributions of minijet partons and soft string excitations are initialized by the heavy ion jet inter- Two-particle intensity interferometry was proposed and de- action generator (HIJING) model [64]. Scatterings among veloped by Hanbury Brown and Twiss in the 1950s, who used partons are modeled by Zhang’s parton cascade (ZPC) [65]. two-photon intensity interferometry to measure the angular In the default version, minijet partons are recombined with diameter of Sirius and other astronomical objects [33, 34]. the parent strings to form new excited strings after partonic In particle physics, this effect was discovered by Goldhaber interactions. Hadronization of these strings is described by et al. in 1960 when they studied the angular correlations be- the Lund string fragmentation model [66]. In the version of tween identical pions in proton-antiproton collisions. They string melting, both excited strings and minijet partons are de- observed an enhancement of two-pion correlation function at composed into partons, whose evolution in time and space is small relative momenta and explained this effect (GGLP ef- modeled by the ZPC model. After partons stop interacting fect) in terms of the Bose-Einstein statistics [35]. Later, it with each other, they are converted to hadrons via a simple was gradually realized that the correlations of identical parti- quark coalescence model. In both versions, the interactionsof cles were sensitive not only to the geometry of the source, but the subsequent hadrons are described by a hadronic cascade, also to its lifetime [36, 37], and even that the pair momentum which is based on a relativistic transport (ART) model [67]. dependence of the correlations contained information about The default AMPT model is able to give a reasonable de- the collision dynamics [38, 39]. Other important progress in- scription of the rapidity distributions and pT spectra in heavy- cluded a more detailed analysis of the effect of final state in- ion collisions from CERN Super Proton Synchrotron (SPS) to teractions [40–42], the development of a parametrization con- RHIC energies. However, it underestimates the elliptic flow sidering the longitudinal expansion of the source [39] and the observed at RHIC, and the reason is that most of the energy implementation of the HBT effect in prescriptions for event produced in the overlap volume of heavy-ion collisions is in generator studies [43]. Throughout the last decades, the HBT hadronic strings and thus not included in the parton cascade in method has been extensively applied in heavy-ion collisions, the model [68]. The string melting AMPT model is employed see e.g. [44–49]. in this calculation and the hadronic rescattering time is set as different time, namely 30 fm/c, 7 fm/c and zero (with resonance decay but without hadronic rescattering) for the small On the other hand, in order to understand the extensive colliding system relative to 197Au+197Au collision. experimental results of different observables, many theoret- Several theoretical predictions have been made on α- ical models were introduced, ranging from thermal mod- clustering configuration of 12C. For an example, a triangle- els [50–52], hydrodynamic models [53–55] to transport mod- like configuration is predicted around the ground state by els [56–58]. Simulations of the space-time evolution of rela- Fermionic molecular dynamics [69], antisymmetrized molec- tivistic heavy-ion collisions have been performed extensively ular dynamics [70], and covariant density functional the- with hydrodynamic models, transport models, and hybrid ory [71], which is experimentally supported [72]. A three-α models that combine a hydrodynamic model with a trans- linear-chain configuration is predicted as an excited state in port model [59–63]. Many observables such as azimuthal time-dependent Hartree-Fock theory [73] and other different anisotropies have been successfully described by all three approaches [74]. Together with the situation of Woods-Saxon types of models in spite of the different physics foundations (W-S) distribution, we initialize the configurations of 12C by and assumptions. these three cases in HIJING process. Fig. 1 shows the participant nucleon distributions of 12C 197 12 197 + Au central collisions for these three cases. The initial In this paper, we simulate the high-energy C+ Au col- 12 lisions and discuss the influences of initial geometry on the nucleon distribution in C is configured as (a) nucleons in observables at freeze-out stage using a multiphase transport Woods-Saxon distribution from the HIJING model [64], (b) (AMPT) model [57]. The article is arranged as follows: in three α clusters in triangle structure, and (c) three α clus- Sect. II the AMPT Model and algorithm introduction for 12C ters in chain structure. The distribution of the radial center of the α clusters in 12C is assumed to be a Gaussian func- initialization are described. HBT correlation function and par- − 2 −0.5 r rc σr ticipant plane are introduced in Sect. III. Results and discus- tion, e c , here rc is the average radial center of an sion for the effect of cluster configuration on the second- and α cluster and σrc is the width of the distribution. And the nu- third-order HBT radii as well as eccentricities are then pre- cleons inside each α cluster are given by the three-parameter sented in Sect. IV. Finally, a summary is given in Sect. V. Woods-Saxon distribution in HIJING. The parameter rc is set 3

10 10 10 (a) 12C, Woods-Saxon (b) 12C, Triangle (c) 12C, Chain

5 5 5

(fm) 0 (fm) 0 (fm) 0 y y y r r r

−5 −5 −5

−10 −10 −10 −10 −5 0 5 10 −10 −5 0 5 10 −10 −5 0 5 10 rx(fm) rx(fm) rx(fm)

FIG. 1: Participant distributions of 12C + 197Au central collisions with different initial conﬁgurations of 12C.

to match the measured value of the rms nuclear charge ra- two, qs. Therefore, the correlation function can be expressed 12 dius of C. For triangle structure, rc = 1.79 fm and σrc = as [38, 77, 78]

0.1 fm. For chain structure, rc = 2.19 fm, σrc = 0.1 fm 12 ~ ~ 2 ~ for two clusters and the other one is at the center of C. C(~q, K)=1+ λ(K)exp( Rij (K)qiqj ). (2) α − Once the radial center of the α cluster is determined, the cen- i,jX=o,s,l ters of the three clusters will be placed in an equilateral tri- where λ is coherence parameter. It should be unity for a fully angle for triangle structure or in a line for chain structure. chaotic source and smaller than unity for a partially coherent From the parameters and initialization, we obtain that the rms source, although in practice many other effects, e.g. parti- radius WS 2.46 fm for Woods-Saxon distribution. R( ) cle misidentification, resonance decay contributions and final For theh other twoi ≈configurations, R(Chain) 2.47 fm and state Coulomb interactions, can decrease the measured λ sig- Triangle 2.47 fm. These valuesh are alli consistent≈ with R( ) nificantly. After taking the notation, hexperimentali≈ data 2.47 fm [75]. d4xξS(x, K) ξ = , (3) h i R d4xS(x, K) III. HBT CORRELATION FUNCTION R the explicit azimuthal angle dependence of the HBT radii can The identical two-particle HBT correlation function with- be seen from [79], out final state interactions, defined as the ratio of the two- 2 2 R (KT , Φ, Y )= (y cosΦ x sin Φ) particle probability divided by the product of the single- s h − i (4) particle probabilities, can be written as [76] y cosΦ x sin Φ 2, − h − i ~ 2 d4xei~q·(~x−βt)S(x, K) 2 2 ~ (1) Ro(KT , Φ, Y )= (x cosΦ + y sin Φ β⊥t) C(~q, K)=1 4 , h − i (5) ± R d xS(x, K) 2 x cosΦ + y sin Φ β⊥t , R − h − i where K is the average momentum of two particles, K = 1 EK +Kl where Y = ln( ), β⊥ = KT /K0, and K0 EK . 1 2 EK −Kl ≈ 2 (p1 + p2), q is the relative momentum between two parti- The other terms of the HBT radii will not be studied in this cles, q = p1 p2, and S(x, K) is the emission function. work and the expression can also be found in reference [79]. The positive sign− is for bosons and the negative one is for In heavy-ion collisions, the direction of impact parameter ~b fermions. With the mass-shell constraint q K = 0, the · is always random and along with the line between the centers temporal component q0 in the four-dimensional Fourier trans- ~ of target and projectile. The so-called reaction plane is formed form is substituted with β~ = K . And, both smoothness ap- K0 by~b and the beam direction, which results in random distribu- proximation and on-shell approximation are used. With the tion of reaction plane angle. Hence, the observables sensitive assumption of Gaussian source, the correlation function will to reaction plane angle should be corrected to that. Participant also take a Gaussian form. Further, in order to obtain infor- plane is always considered as a reasonable approximation to mation of the source in the beam and transverse directions, reaction plane, and the participant plane angle Ψn P P can Cartesian or Bertsch-Pratt parametrization is taken. In this be defined by [80–82], { } parametrization, the relative momentum vector of the pair ~q is 2 decomposed into a longitudinal direction along the beam axis, rpart sin(nφpart) atan2 h 2 i + π ql, an outward direction parallel to the pair transverse mo- rpart cos(nφpart) Ψn P P = h i , (6) mentum, qo, and a sideward direction perpendicular to those { } n 4 where Ψn P P is the nth-order participant plane angle, rpart For the azimuthal angle dependence of the HBT radii rela- { } and φpart are radial coordinate and azimuthal angle of par- tive to the third-orderparticipant plane angle Ψ3 P P , shown { 2 } ticipants in the collision zone in the initial state, and the av- in panel (b), (d), (f) in Figs. 2, 3, 4 and 5, Rs(n=3) and erage denotes density weighting. The system will be R2(n=3) oscillate in cos[3(Φ Ψ P P )] mode. R2(n=3) h· · · i o − 3{ } o rotated to the participant plane direction in transverse plane. shows valleys around Φ Ψ3 P P = 0 and 2π/3 for both 2 − { } ± In references [83, 84], the azimuthal angle dependence of the KT bins. However, Rs(n=3) has a different behavior. With 2 2 HBT radii was investigated and the effects of geometric defor- high KT bin, Rs(n=3) shows a peak at 0 as Rs(n=2), while it mation and anisotropic flow were discussed therein. And in is reverse with low KT bin. Similar behavior was observed in experiments, RHIC-STAR [85, 86], RHIC-PHENIX [87] and a toy model simulation, and was explained by flow anisotropy CERN-ALICE [88] reported the measurements of π-π cor- dominated source [83]. relation relative to the second and third harmonic event plane, From the above results, it can be seen that the initial geo- respectively. In this calculation we will focus on the azimuthal metric distribution from the nuclear structure can be observed angle dependence of the HBT radii relative to the second- and from the azimuthal angle dependence of the HBT radii rela- third-order participant plane and investigate the initial geom- tive to the second- and third-order participant plane. In order etry effect. to quantitatively describe the anisotropy of the above HBT radii and discuss the effect from exotic nuclear structure, we perform a Fourier expansion in azimuthal angle for the HBT IV. RESULTS AND DISCUSSION radii which can be found in references [79, 83, 84, 89],

2 2 2 ± Rs(Φ Ψn)=Rs, +2Rs,n cos[n(Φ Ψn)], π meson, which is the most produced particle in rela- − 0 − (7) 2 2 2 tivistic heavy-ion collision, is taken for the HBT correlation Ro(Φ Ψn)=Ro,0 +2Ro,n cos[n(Φ Ψn)],n =2, 3 calculation. The transverse momentum spectrum of pions − − was calculated and basically insensitive to the initial condi- 2 where Rs(o),n is the so-called zeroth-, second- (n=2) and tion used (Woods-Saxon distribution, chain structure and tri- third- (n=3) order HBT radii respectively. 12 angle structure of C), which was consistent with our pre- Tables I and II show the fit parameters R2 at differ- vious work [22] that the multiplicity was not so dependent s(o),n ent hadronic rescattering time t for three initial 12C con- on the initial configuration. And then it will be an interest- hs figurations. The amplitude of R2 takes the similar value ing question how the initial distribution affects the differential s(o),0 distribution of particles in momentum and coordinate space at for chain structure and triangle structure, while the amplitudes for Woods-Saxon distribution case is slightly smaller than the the final stage of the collision. Next we focused on the HBT 2 radii from Eqs. (4) and (5) relative to the second- and third- others. Rs,0 increases with the increase of ths for all config- 12 2 order participant plane angle. urations of C, while Rs,2 decreases. This ths dependence 2 2 Figures 2, 3, 4, 5 display the azimuthal angle dependence of Rs,2 and Rs,0 reflects the approximate description of the 2 2 eccentricity of the colliding system [83, 84], as the final ec- of the HBT radii, including Rs and Ro as functions of Φ − f 2 2 Ψ2 P P or Φ Ψ3 P P , from π-π correlation with three centricity ǫ2 2Rs,2/Rs,0. It is noted that the dependence different{ } initial− structures{ of} 12C as well as different hadronic of the source≈ eccentricity on the lifetime of the system has rescattering time. The calculations are performed with rapid- been investigated [90]. From the tables, it is obvious that f ity from -1 to 1 and two KT bins (0.3-0.5 GeV/c and 0.8-1.2 ǫ2 for chain structure is larger than that for other cases, al- 12 197 2 GeV/c) in the most central C+ Au collisions at √sNN = though the system undergoes hadronic rescattering. Ro,0 and 200 GeV. t = 0 fm/c means hadrons with resonance decay 2 hs Ro,2 more or less increase with the increase of ths, while but without hadronic rescattering and Φ is the azimuthal an- | 2 | R with high KT does not present an obvious ths depen- gle of pion. From the hadronic rescattering time evolu- | o,2| ths dence. R2 and R2 for three configurations of 12C also tion, it can be seen that the azimuthal angle dependence of the s,3 | o,3| have an increasing tendency with ths, and triangle structure HBT radii keeps the similar dependent structure and only the always has the largest absolute value. Even though the third- amplitude has minor changes. In other words, the hadronic order harmonic eccentricity coefficient f cannot be obtained rescattering does not destroy the azimuthal angle dependence ǫ3 f 2 2 of the HBT radii from the early stage of the fireball. directly as ǫ2 2Rs,2/Rs,0 as discussed in reference [83], ≈2 For the azimuthal angle dependence of the HBT radii rel- the value of Rs(o),3 for triangle structure can still reflect the ative to the second-order participant plane angle Ψ2 P P , initial geometry effect on final state. Since chain (triangle) 2 { } shown in panel (a), (c), (e) in Figs. 2, 3, 4 and 5, Rs(n=2) and structure always has the largest second (third)-order oscilla- 2 2 Ro(n=2) present cos[2(Φ Ψ2 P P )] oscillation. Rs(n=2) tion, the ratio of the third-orderto the second-ordercoefficient − { } 2 2 shows a peak and valleys around Φ Ψ2 P P = 0 and π/2, Rs(o),3/Rs(o),2 can differentiate these three configurations as −2 { } ± respectively, but it is reverse for Ro(n=2). This azimuthal an- shown in Fig. 6. And the effect of different configurations gle dependence of the HBT radii relative to Ψ2 P P is sim- is clearer and more stable with high KT . For example, with { } 2 2 ilar to experimental results in Au+Au collisions at √sNN = high KT , Ro,3/Ro,2 for Woods-Saxon distribution is always 200 GeV [85–87]. The three cases of initial 12C configura- around0.5. For chain structure, it is no morethan 0.2 although tion present the same oscillation pattern for the HBT radii as finally it gets larger than the value at the beginning. For trian- a function of azimuthal angle but with different amplitudes. gle structure, it is always larger than 0.6 and even up to 1. The 5

(a) t = 0 fm/c (c) t = 7 fm/c (e) t = 30 fm/c 10 hs 10 hs 10 hs ) 2 9 9 9 (n=2)(fm 2 s

R 8 8 8

7 7 7 −2 0 2 −2 0 2 −2 0 2

9.5 9.5 9.5 (b) 12C, Woods-Saxon (d) AMPT most-central (f) 12 π+π++π-π-, 0.3

8.5 8.5 8.5 (n=3)(fm 2 s R

8 8 8

−2 0 2 −2 0 2 −2 0 2 Φ Ψ - n{PP}(rad)

2 FIG. 2: Azimuthal angle dependence of the HBT radii Rs relative to the second (top row n = 2) and third (bottom row n = 3) harmonic 12 197 participant plane Ψn P P from π-π correlation with KT from 0.3 GeV/c to 0.5 GeV/c for the most central C + Au collisions. The results { } from left column to right column correspond to the hadronic rescattering time ths = 0 fm/c, ths = 7 fm/c and ths = 30 fm/c, respectively. The dots represent the calculated results and the curves are the ﬁts by Eq. (7).

(a) ths = 0 fm/c (c) ths = 7 fm/c (e) ths = 30 fm/c 16 16 16 ) 2

14 14 14 (n=2)(fm 2 o R 12 12 12

−2 0 2 −2 0 2 −2 0 2

15 15 15 (b) 12C, Woods-Saxon (d) AMPT most-central (f) 14.5 14.5 14.5 12 π+π++π-π-, 0.3

12 12 12 −2 0 2 −2 0 2 −2 0 2 Φ Ψ - n{PP}(rad)

2 FIG. 3: Same as Fig. 2 but for Ro. 6

6 6 6 (a) ths = 0 fm/c (c) ths = 7 fm/c (e) ths = 30 fm/c

) 5 5 5 2

4 4 4 (n=2)(fm 2 s R 3 3 3

−2 0 2 −2 0 2 −2 0 2

(b) 12C, Woods-Saxon (d) AMPT most-central (f) 4.5 4.5 - - 4.5 12 π+π++π π , 0.8

−2 0 2 −2 0 2 −2 0 2 Φ Ψ - n{PP}(rad)

FIG. 4: Same as Fig. 2 but with KT from 0.8 GeV/c to 1.2 GeV/c.

6 6 6 (a) ths = 0 fm/c (c) ths = 7 fm/c (e) ths = 30 fm/c 5 5 5 ) 2

4 4 4 (n=2)(fm 2 o

R 3 3 3

2 2 2 −2 0 2 −2 0 2 −2 0 2

5 (b) 12C, Woods-Saxon 5 (d) AMPT most-central 5 (f) 12 π+π++π-π-, 0.8

3.5 3.5 3.5 (n=3)(fm 2 o R 3 3 3 2.5 2.5 2.5 −2 0 2 −2 0 2 −2 0 2 Φ Ψ - n{PP}(rad)

FIG. 5: Same as Fig. 3 but with KT from 0.8 GeV/c to 1.2 GeV/c. 7

0 π+π++π-π-, 0.3

0.8

0 0 0 10 20 30 40 0 10 20 30 40 ths(fm/c)

2 2 FIG. 6: The time evolution of the ratio of the third- to the second-order HBT radii Rs(o),3/Rs(o),2 with two KT bins.

reason may be that pions with high pT are more likely to come Table I and II. And the final eccentricity estimated with high 2 from the center of the source at earlier stage. Other results KT is closer to the initial eccentricity. These results of Rs(o),n of R2 /R2 for the three cases also show a consistent (n=2,3) indicate the initial geometry can be reflected in the fi- | o(s),3 o(s),2| order, which from large to small is triangle, Woods-Saxon, nal observables sensitive to azimuthal angle such as HBT cor- chain. relation and collective flow [20–23]. Further, the system scan Furthermore, we investigated the eccentricity of the system experiments around 12C may provide the platform to examine at the initial stage, which is calculated by [80], if there are α clusters in the carbon nuclei and to distinguish the structure by comparing the HBT correlation and collective r2 cos(nφ) 2 + r2 sin(nφ) 2 12 ǫi = h i h i , (8) flow between C+A collisions and other colliding systems. n p r2 h i where r and φ are the polar coordinate positions in transverse plane of participant nucleons. Table III presents the 2 2 final eccentricity estimated by 2Rs,2/Rs,0 and the initial eccentricity for different configurations of 12C. It is expected that the second-order eccentricity takes the largest value for chain structure and the third-order eccentricity is larger for triangle structure than that for other cases as R2 listed in | s(o),3| 8

2 TABLE II: Same as Table I but with T from 0.8 GeV/ to 1.2 TABLE I: Rs(o),n extracted by Eq. (7) from azimuthal angle de- K c pendence of the HBT radii relative to the second- and third-order GeV/c. participant plane with KT from 0.3 GeV/c to 0.5 GeV/c for different configurations of 12C. ths (fm/c) 0 7 30 Woods-Saxon ths (fm/c) 0 7 30 2 Rs,0 3.29 3.32 3.41 Woods-Saxon 2 Rs,2 0.269 0.266 0.251 2 Rs,0 7.92 8.23 8.60 2 Rs,3 0.0296 0.0396 0.0478 2 Rs,2 0.260 0.228 0.197 2 Ro,0 2.81 2.96 3.42 2 Rs,3 -0.0490 -0.0448 -0.0449 2 Ro,2 -0.288 -0.281 -0.289 2 Ro,0 12.3 12.5 13.3 2 Ro,3 -0.123 -0.136 -0.151 2 Ro,2 -0.418 -0.441 -0.462 Triangle R2 -0.0447 -0.0806 -0.0881 o,3 2 Rs,0 3.85 3.86 3.94 Triangle 2 Rs,2 0.292 0.284 0.272 2 Rs,0 8.39 8.68 9.08 2 Rs,3 0.0514 0.0701 0.0767 2 Rs,2 0.280 0.245 0.212 2 Ro,0 3.39 3.53 4.06 2 Rs,3 -0.0693 -0.0670 -0.0508 2 Ro,2 -0.309 -0.300 -0.301 2 Ro,0 13.1 13.3 14.2 2 Ro,3 -0.211 -0.228 -0.262 2 Ro,2 -0.417 -0.439 -0.447 Chain R2 -0.0790 -0.163 -0.154 o,3 2 Rs,0 3.81 3.82 3.89 Chain 2 Rs,2 0.625 0.610 0.577 2 Rs,0 8.42 8.70 9.09 2 Rs,3 0.0175 0.0333 0.0379 2 Rs,2 0.605 0.532 0.445 2 Ro,0 3.41 3.54 4.03 2 Rs,3 -0.0480 -0.0414 -0.0393 2 Ro,2 -0.732 -0.708 -0.715 2 Ro,0 13.0 13.2 14.1 2 Ro,3 -0.114 -0.131 -0.139 2 Ro,2 -0.973 -1.07 -1.11 2 Ro,3 -0.0547 -0.0740 -0.0960 For quantitative description of the oscillation of the HBT radii and the system eccentricity, the final eccentricity were estimated by the zeroth- and second-order HBT radii, and com- TABLE III: The final eccentricity compared with the initial eccen- pared with the initial eccentricity. According to our calcu- tricity for different configurations of 12C. lations, the ratio of the third-order to the second-order HBT Woods-Saxon Triangle Chain radii is quite sensitive to different configurations of 12C. And 2 2 2Rs,2/Rs,0(low KT ) 0.0458 0.0467 0.0979 we suggest performing azimuthally differential femtoscopy 2 2 2Rs,2/Rs,0(high KT ) 0.147 0.138 0.297 with high pT identical particles. From the results, the az- i ǫ2 0.263 0.227 0.499 imuthal angle dependence of the HBT radii relative to the i ǫ3 0.219 0.296 0.179 second- and third-order participant plane can be taken as an effective probe to distinguish the exotic nuclear structure be- sides collective flow. V. SUMMARY

Conﬁguration of 12C is initialized as α-clustered triangle, Acknowledgments α-clustered chain and Woods-Saxon distribution of nucleons, 12 197 and then the C + Au central collisions at √sNN = This work was supported in part by the National Natural 200 GeV are simulated by the string melting AMPT model. Science Foundation of China under contract Nos. 11890714, The azimuthal dependences of the HBT radii relative to the 11875066, 11421505, and 11775288, National Key R&D second- and third-order participant plane from π-π correlation Program of China under Grant No. 2016YFE0100900 and are presented with different hadronic rescattering time, show- 2018YFE0104600, the Key Research Program of Frontier ing the possibility as a probe of different initial geometries. Sciences of the CAS under Grant No. QYZDJ-SSW-SLH002, 9 and the Key Research Program of the CAS under Grant NO. XDPB09.

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