AMPT model includes both initial partonic collisions and final hadronic interactions and the transition between these two phases of matter. It has four main components, namely, initialization of collisions, parton transport after initialization, hadronization mechanism, and hadron transport [31–47]. A brief discussion on the main components of AMPT can be found in the Appendix. Since the particle flow and spectra are well described at mid-pT region by quark coalescence mechanism for hadronization [48,49], we have used AMPT string melting (SM) mode (AMPT version 2.26t9b) for our study. The AMPT settings in the current study are the same as reported in Ref. [50]. Centrality selection has been done using impact parameter slicing and we have used the impact parameter values for different centralities from Ref. [51] for our current analysis.

The overlap region in a noncentral heavy-ion collision is not isotropic in space. If the pressure gradient of the hot and dense medium formed in the heavy-ion collisions is large enough, the anisotropy in initial geometry can be translated into final momentum space azimuthal anisotropy. The anisotropy in the initial spatial distribution of the nucleons in the overlap region can be quantified by the quantities such as eccentricity (ϵ2), triangularity (ϵ3), etc. As the name suggests, eccentricity refers to how elliptical the medium can be, and is generated due to anisotropy in the nuclear overlap region. Similarly, triangularity characterizes the triangular geometry of the overlap region during the collision of two heavy-ions and arises due to event-by-event fluctuations in the participant nucleon collision points [54]. In the current study, we have used the AMPT model to see the dependence of transverse spherocity on eccentricity (ϵ2) and triangularity (ϵ3) having followed the notations used in Ref. [55]. Eccentricity and triangularity of the participant nuclei can be generalized as follows [55]: ϵn=⟨rncos(nϕpart)⟩2+⟨rnsin(nϕpart)⟩2⟨rn⟩,(4)where r and ϕpart are the polar coordinates of the participant nucleons. n=2 corresponds to eccentricity and n=3 corresponds to triangularity. A higher value of n can be used to study higher-order spatial anisotropy of the collision overlap region, although the contributions from the higher order terms will be smaller. In Eq. (4), angular brackets, “⟨…⟩” represents the mean taken over all the participant nucleons in an event.

Figure 3 shows the event-average eccentricity (⟨ϵ2⟩) (top) and triangularity (⟨ϵ3⟩) (middle), and their ratio (bottom) as a function of centrality for different spherocity events. Both ⟨ϵ2⟩ and ⟨ϵ3⟩ have clear dependence on the centrality and are increasing from central to peripheral collisions. This behavior is expected since in central collisions, the nuclear overlap region is more spatially symmetric compared to that in peripheral collisions, resulting in low values of ⟨ϵ2⟩ and ⟨ϵ3⟩ in central collisions. However, as one moves towards peripheral collisions, the participating nucleon overlap region gets more and more spatially anisotropic resulting in higher values of ⟨ϵ2⟩ and ⟨ϵ3⟩. In a particular centrality, one notices lower ⟨ϵ2⟩ for high-S0 events than the low-S0 events. This indicates that transverse spherocity can also be used to distinguish events based on the initial geometry. However, since the contribution in ϵn for n>2 arises due to fluctuations in the density profile of the participating nucleons, one should not expect significant transverse spherocity dependence on v3, as shown in the middle plot of Fig. 3. The bottom plot of Fig. 3 represents the ratio of (⟨ϵ3⟩) to (⟨ϵ2⟩), plotted against different centrality classes for high-S0, S0 integrated, and low-S0 classes. From the ratio, it is clear that ⟨ϵ2⟩ is higher than ⟨ϵ3⟩ for low-S0, and S0 integrated events throughout all centrality classes except the most central case where, ⟨ϵ3⟩ seems to be higher. But the order is reversed for high-S0 events where ⟨ϵ3⟩ is always greater than ⟨ϵ2⟩, indicating the dominance of density fluctuations over the geometry of the overlap region.

To quantify the correlation between eccentricity and triangularity, we calculate the Pearson correlation coefficient (ρ(ϵ2,ϵ3)) using Eq. (5): ρ(ϵ2,ϵ3)=Nev∑i=1Nevϵ2,iϵ3,i-(∑i=1Nevϵ2,i)(∑i=1Nevϵ3,i)[Nev∑i=1Nevϵ2,i2-(∑i=1Nevϵ2,i)2][Nev∑i=1Nevϵ3,i2-(∑i=1Nevϵ3,i)2],(5)where, Nev represents the number of events for a given centrality and spherocity class. ϵ2,i and ϵ3,i represent the eccentricity and triangularity for the ith event, respectively. The value of ρ(ϵ2,ϵ3) lies between -1 to 1; positive value indicates positive correlation while negative value of ρ(ϵ2,ϵ3) implies anticorrelation. Figure 4 shows the Pearson correlation coefficient between eccentricity and triangularity, denoted as ρ(ϵ2,ϵ3), for different centrality and spherocity classes in Pb–Pb collisions at sNN=5.02  TeV. The correlation appears to be explicitly higher for the most central case, then suddenly decreases for the midcentral cases and starts to rise again towards the peripheral collisions. This peculiar behavior of the correlation between eccentricity and the triangularity is extended for different spherocity classes, and it is found that for high-S0 class of events, the correlation is comparatively higher and positive compared to low-S0, and S0-integrated events. Even though the transverse spherocity is a final state event shape classifier, it successfully separates the observables related to initial geometry and establishes a correlation between eccentricity and triangularity for the isotropic case. It should also be noted that this correlation could be affected by the initial partonic scatterings, which one finds in the AMPT-SM model [56].

Anisotropic flow is a measure of the azimuthal momentum anisotropy of the final state particles produced in a collision. Anisotropic flow depends upon initial spatial anisotropy in the nuclear overlap region, transport properties, and the equation of state of the system. Anisotropic flow can be characterized by the coefficients of the Fourier expansion of momentum distribution of the final state particles and is given by: Ed3Ndp3=d2N2πpTdpTdy(1+2∑n=1∞vncos[n(ϕ-ψn)]).(6)Here ϕ is the azimuthal angle of the particles in the transverse plane and ψn is the nth harmonic event plane angle [19]. n stands for the order of the anisotropic flow coefficient. n=2 stands for elliptic flow (v2) and n=3 refers to the triangular flow (v3). In general nth order anisotropic flow, vn can be defined as: vn=⟨cos[n(ϕ-ψn)]⟩.(7)

We are dealing with different kinds of spherocity events and, among them, low-S0 events are prone to have contributions from jets. Thus, in order to see the fair dependence of transverse spherocity, we use the two-particle correlation method to study the elliptic and triangular flow as done in Refs. [53,57,58]. The two-particle correlation method has an advantage because it deals with the nonflow effects caused by jets and resonance decays by implementing a proper pseudorapidity cut. It has an additional advantage since it does not require event plane angle (ψn) to calculate vn. In experiments, a pseudorapidity dependence of ψn is observed, which is not taken into consideration in the present study for simplicity, and it does not affect the performed studies. Here we assume ψn to be the global phase angle for all the particles irrespective of the selection of the bins in pseudorapidity [28,59,60]. Although in AMPT we can set the reaction plane angle, ψR=0, this is not trivial in experiments to determine the same. Thus, one follows a two-particle correlation approach. The steps to find the correlation function are as follows: (i)

We compose two sets of particles based on particle transverse momentum denoted by labels “a” and “b.”

Flow coefficients defined in Eq. (6) contribute to Eq. (9) as: C(Δϕ)∝[1+2∑n=1∞vn(pTa)vn(pTb)cosnΔϕ].(11)In Eq. (11), the event plane angle drops out during the convolution leaving only the dependence on an azimuthal angle. If one assumes that the collective flow is driven by azimuthal anisotropy, then the two-particle harmonic coefficient (vn,n) should be a product of two single-particle harmonic coefficients. Therefore, vn,n(pTa,pTb)=vn(pTa)vn(pTb).(12)Another way around, we can calculate vn from vn,n by using the following expression: vn(pTa)=vn,n(pTa,pTb)vn,n(pTb,pTb).(13)

A crucial point to note here is that negative value of vn,n(pTb,pTb) in Eq. (13) makes vn(pTa) imaginary which is not physical. The correlation function and the anisotropic flow coefficients such as elliptic and triangular flow are calculated in the pseudorapidity range |η|<2.5, with a relative pseudorapidity gap of 2<|Δη|<4.8 in the transverse momentum range of the particle pairs from 0.5 to 5  GeV/c [28]. Figure 5 shows the transverse spherocity dependence on one dimensional two-particle azimuthal correlation function plotted with respect to Δϕ for (0–10)% and (40–50)% centralities in Pb–Pb collisions at sNN=5.02  TeV. The larger correlation amplitude for midcentral collisions compared to the central collisions indicates more azimuthal anisotropy in midcentral than the central collisions. There is a strong dependence of the correlation function on the spherocity for any particular centrality. One can infer from Fig. 5 that low-S0 events have more azimuthal anisotropy than high-S0 events. Two peaks in the away side in Δϕ for high-S0 events manifest due to higher contribution from v3.

Figure 6 represents centrality dependence of elliptic flow (⟨v2⟩) for different spherocity classes in Pb–Pb collisions at sNN=5.02  TeV. In high-energy heavy-ion collisions, due to the presence of a pressure gradient formed in the medium, anisotropy in the geometry of the initial nuclear overlap region is transformed into final state azimuthal anisotropy. Since central collisions are almost isotropic in geometry, corresponding v2 is also less. However, if one moves towards midcentral collisions, the elliptic flow coefficient increases since the eccentricity is higher. However, in peripheral collisions, although eccentricity is remarkably high, due to the lack of the number of participants a smaller size and shorter lifetime of the fireball, spatial anisotropy can not be completely transformed into v2. Figure 6 also shows that the value of elliptic flow significantly depends on event selection-based transverse spherocity. Low-S0 events have higher ⟨v2⟩ while we observe almost negligible ⟨v2⟩ in high-S0 events.

Figure 7 shows triangular flow (⟨v3⟩) as a function of centrality for different spherocity classes. In midcentral collisions, one observes slightly higher v3 than the central collisions for a specific spherocity class. The dependence of v3 on centrality is weaker than that of v2, which can also be observed from Figs. 6 and 7. This weak dependence of v3 on centrality is also observed in experiments [53,61]. In addition, one can observe appreciable dependence of triangular flow on transverse spherocity. In Fig. 6, v2 contribution is maximum in case of low-S0 events while in Fig. 7 the trend is reversed and high-S0 events have dominating v3. This anticorrelation between v2-v3 for q2-selection as pointed out in Ref. [53] can also be deduced from this study with transverse spherocity. However the anticorrelation between ϵ2-ϵ3 is not observed in Fig. 3. One may infer that the source of this anticorrelation between v2-v3 with spherocity selection may not be propagated from the initial geometry but may have effects from the medium during its evolution due to the fact that the fluidity of the medium affects differently to different flow coefficients. This is an indication that the fluidity of the medium is spherocity-dependent. In Ref. [62], authors have shown the dependence of specific shear viscosity (η/s) on the correlation between triangularity and triangular flow, where the correlation decreases with an increase in η/s and the correlation between v3 and ϵ3 is not strong in an ideal or minimally viscous fluid. Only (65–70)% of the triangular flow is related to initial triangularity, conveying that a substantial part of the triangular flow is unrelated to the initial triangularity. Although the exact reason for this effect is not yet completely understood, however, effects from higher moments or products of moments (via v1-v2 coupling) may lead to such observations [62]. This may be the reason for the observed dependence of triangular flow on transverse spherocity, while triangularity of the overlap region does not have any spherocity dependence. This can be investigated in a future study as a function of spherocity and it is not covered in the manuscript.

Figure 8 represents the ratio ⟨v3⟩/⟨v2⟩ versus centrality for different spherocity events. One notices that ⟨v3⟩/⟨v2⟩ is decreasing when going from central to peripheral collisions and shows considerable spherocity dependence. This may be because of three reasons; due to the reminiscent centrality dependence of ⟨ϵ3⟩/⟨ϵ2⟩ and/or due to viscous effects [61], or have a contribution from both the effects. One observes that the ⟨ϵ3⟩/⟨ϵ2⟩ vs centrality trend is followed by ⟨v3⟩/⟨v2⟩ starting from most central until semicentral collisions. Here, with small changes in centrality, the viscous effect might cause a little change in ⟨v3⟩/⟨v2⟩; however, for the peripheral collisions, the viscous effects play a major role and the ⟨ϵ3⟩/⟨ϵ2⟩ trend is not followed by ⟨v3⟩/⟨v2⟩ for all spherocity classes. One can notice that ⟨v3⟩/⟨v2⟩ is highest and always greater than one for high-S0 events showing the dominance of v3 over v2 for all centrality classes and the low-S0 curve shows the dominance of v2 over v3. This effect is expected to have propagated from initial geometry, which is evident from ⟨ϵ3⟩/⟨ϵ2⟩ plot shown in Fig. 3.

Figure 9 represents the elliptic and triangular flow normalized with eccentricity and triangularity, respectively, as a function of centrality in Pb–Pb collisions at sNN=5.02  TeV using AMPT. The figure qualitatively tells about the response of the medium formed, and its evolution to different centrality and spherocity selections. In a perfect fluid, vn∝εn and thus vn/εn is expected to be a constant value. However, in Fig. 9, both ⟨v2⟩/⟨ε2⟩ and ⟨v3⟩/⟨ε3⟩ are found to be varying with spherocity and centrality. As we move from central to peripheral collisions, ⟨vn⟩/⟨εn⟩ decreases. This trend with respect to different centrality selection is also observed in experimental results [61].

Figure 10 shows single particle triangular flow [v3(pTa)] as a function of pT for different centrality classes for (40–50)% centrality in Pb–Pb collisions at sNN=5.02  TeV. v3(pTa) shows an increase with pT from low to intermediate pT region, attains a maximum, and starts decreasing towards higher pT. A similar trend is observed in Ref. [28] for v2,2(pTa,pTb) and v2(pTa). One observes a remarkable dependence of transverse spherocity on the triangular flow when compared to elliptic flow. High-S0 events have the highest v3(pTa) value, whereas low-S0 events have the least v3(pTa) value.

Figure 11 shows the comparison between elliptic and triangular flow as a function of pT for (0–10)% and (40–50)% centrality for different spherocity classes. For the midcentral collisions, v2 is higher than v3 except for the high-S0 case. However, in the most central case we observe the domination of v3 over v2 after a certain pT value. The crossing point in pT (pTcross), where the values of v2 and v3 become equal, appears to change for different centrality classes. This behavior is also observed and is in agreement with experimental data [61,63,64], where pTcross is found to be increasing with particles’ mass and centrality. As pointed out in Refs. [63,64], the crossing point between the flow coefficients is attributed to the interplay of these flow coefficients with the radial flow [64]. This is because, for central collisions, the contribution of fluctuations in the initial nuclear distribution is more than the influence of the overlap region on the development of the anisotropic flow. However, in the peripheral collisions, the collision geometry contributes higher than initial density fluctuations [63]. In this work, we go a step ahead and try to see the dependence of pTcross with event topology which is shown explicitly in Fig. 12. pTcross for different centrality and spherocity classes is extracted by fitting a polynomial function to the plots in Fig. 11. pTcross is found to be almost flat for high-S0 events but it is observed to be increasing with centrality for S0 integrated and low-S0 cases. The expected low pTcross for high-S0 events can be accounted for due to a higher contribution from the initial density fluctuations than the influence of initial collision geometry on the anisotropic flow as shown in Fig. 3.