Thermal photons as a sensitive probe of $\alpha$-cluster in C+Au collisions at the BNL Relativistic Heavy Ion Collider

We consider three typical configurations of collisions, i.e., $\theta=0$, $\pi/4$, and $\pi/2$ in this study. This can be imagined as a deformed nucleus hitting a wall of a gold nucleus at different orientation angles (see Fig. 1). Thus, the angle $\theta=0$ describes situation where the symmetry axis is aligned with the beam axis producing an initial triangular geometry in the deposited entropy distribution and the highest multiplicity amongst all orientations of collision. For $\theta=\pi/2$, the multiplicity is the lowest. While a triangular geometry vanishes for such a scenario, a large elliptic geometry appears. We consider an intermediate situation where the symmetry axis makes an angle $\pi/4$ with the beam axis producing both the intrinsic elliptic and triangular geometry in the deposited entropy distribution.

In Fig. 2 (a), (b), and (c) the event-averaged initial entropy density profile for three orientation angles 0, $\pi$/4, and $\pi$/2 are shown respectively, while Fig. 2(d) represents the entropy profile for the unclustered carbon. We see that the entropy density at the center of each hot spot region for the clustered case is much larger than the entropy density obtained at the center of the unclustered case. In Fig. 2(a) and (b), the entropy profiles are seen to have a triangular shape, while in Fig. 2(c) a peanut-like structure appears with no such triangular geometry. The triangular eccentricity ($\varepsilon_{3}$) of the initial entropy density distribution for the cases in Fig. 2(a) and (b) is $0.43$ and $0.34$ respectively. On the other hand, the elliptic eccentricity ($\varepsilon_{2}$) for the cases in Figs. 2(b) and (c) is $0.20$ and $0.54$ respectively, whereas $\varepsilon_{2}$ is vanishingly small for the other two cases.

The time evolution of average temperature $\langle T\rangle$ and average transverse flow velocity $\langle v_{T}\rangle$ for the three configurations of collision for the clustered carbon and the unclustered carbon is shown in Fig. 3(a) and (b) respectively. The $\langle T\rangle$ and $\langle v_{T}\rangle$ are obtained by Eq. (7) and Eq. (8), respectively, i.e.

The initial average temperature (see Fig. 3 (a)) is found to be largest ($\sim 440$ MeV) for the configuration $\theta=\pi/2$ and smallest for the unclustered case ($\sim 407$ MeV). A relatively smaller area of the overlapping region leads to a larger $\langle T\rangle$ for the configuration $\theta=\pi/2$ at the initial time. We see an almost similar time evolution of the average temperature for all the configurations. However, the evolution of the average transverse flow velocity for different configurations are distinct from each other (see Fig. 3(b)). We see that $\langle v_{T}\rangle$ sharply rises for the clustered cases, whereas the rise is relatively slower for the unclustered case. The rise is maximum for the orientation angle $\pi/2$. During the time interval of $2<\tau<4$ fm/c, we observe a sudden jerk for the configuration $\theta=0$. However, no such effect is found for $\theta=\pi/4$ and $\pi/2$. The radially outward ripples, originated at the boundary of each hotspot (see Fig. 2 (a) ), cause a flow cancellation when they collide, which effectively reduces the growth of the average transverse flow velocity.

In Fig. 4, the thermal photon spectra are presented for the clustered and unclustered ${\rm C+Au}$ collisions. The thermal photon spectra for all the cases are found to be almost similar in the region $p_{T}<2$ GeV. However, the spectra become sensitive to the orientation angle for $p_{T}>2$ GeV. The thermal spectrum for the configuration $\theta=\pi/2$ is found to be slightly above the other spectra in the larger $p_{T}$ region. On the other hand, photon production from the unclustered carbon is found to be smallest. We plot a thermal photon spectrum (i.e. maroon box+dotted line) considering an orientation-averaged initial profile (averaged over 50,000 events with random orientations) for the clustered carbon and find it similar to the spectrum obtained for the unclustered carbon. We estimate the prompt photon production in the region $p_{T}>2$ GeV from ${\rm C+Au}$ collisions by scaling the prompt photon spectrum obtained in the p+p collisions at 200A GeV using the Monte Carlo code JETPHOX (version 1.2.2) Aurenche:2006vj . We expect similar production of prompt photons for collisions of clustered as well as unclustered carbon as the $N_{\rm coll}$ for both are close to each other. The Fig. 4 shows the prompt photons contribute significantly in the region $p_{T}>3$ GeV (black filled circles).

The elliptic and triangular flow of thermal photons as a function of $p_{T}$ are presented in Fig. 5(a) and (b) respectively. We see a substantially large thermal photon $v_{3}$ for the configurations $\theta=\pi/4$ and $\theta=0$. The initial triangular geometry of the entropy density profile produced in the collision of $\alpha$-clustered carbon and gold nucleus gives rise to the triangular flow of thermal photons. As there exists no such triangular geometry for the configuration $\theta=\pi/2$ or the unclustered case, we obtain a vanishingly small thermal photon $v_{3}$. The thermal photon $v_{3}$ for the configuration $\theta=\pi/4$ is seen to be slightly larger compared to the same obtained from the orientation angle $\theta=0$ even though the triangular anisotropy is smaller for the former than the latter. The reason is attributed to the larger average transverse flow velocity during the evolution for the configuration $\theta=\pi/4$ than the same obtained for the configuration $\theta=0$. On the other hand, the elliptic flow parameter of thermal photons is found to be vanishingly small for the configuration $\theta=0$ and the unclustered case. However, the thermal photon $v_{2}$ is found to be very large for the orientation angle $\pi/2$. The initial elliptic eccentricity and the transverse flow velocity, both play significant role in producing a large $v_{2}$ of thermal photons for the configuration $\theta=\pi/2$. It is important to note that the $v_{2}$ obtained for the above configuration is comparable to the direct photon $v_{2}$ data obtained for the mid-central Au+Au collisions at 200A GeV at RHIC Adare:2015lcd .

To obtain a more precise idea about the origin of the anisotropic flow coefficients of photons, we plot the individual contributions from the QGP and hadronic phases to the total value of the flow coefficients in Fig 6. We calculate the contributions of thermal photon $v_{3}$ for the clustered case with orientation angles $\theta=0$ and $\pi/4$ in Fig 6(a). We find that the maximum value of $v_{3}$ (at $p_{T}\sim 2$ GeV) from the QGP phase is substantial and close to the maximum value of the total thermal photon $v_{3}$. Hence, the response to the initial-state triangular geometry is predominantly carried by the QGP evolution. If we compare the hadronic medium contribution to the total thermal photon $v_{3}$ for these cases, we find that unlike the QGP photon $v_{3}$, the hadronic photon $v_{3}$ for the orientation $\theta=0$ is larger compared to the same obtained at the orientation $\theta=\pi/4$. The reason can be understood from Fig. 2, where we see that the triangular geometry associated with the hottest region is more prominent in Fig 2(b) compared to Fig. 2(a). Whereas, the boundary region, which is the source of hadronic photons, has a larger triangular geometry as shown in Fig. 2(a) compared to Fig. 2(b). The contributions of thermal photon $v_{2}$ for the clustered case with the orientation angle $\theta=\pi/2$ are plotted in Fig. 6(b), and we see a similar response. The peak value of the thermal photon $v_{2}$ from the QGP phase is found to be about $90\%$ of the peak value of the total thermal photon $v_{2}$ and identically, it reflects that the most of the net thermal photon $v_{2}$ has built up during the early phase of QGP evolution.

It is to be noted that the value of $T_{f}$ is fixed for Au+Au collisions by reproducing the experimental data of charged particle spectra and anisotropic flow parameters. We can not do the same for C+Au collisions due to the unavailability of experimental data. Therefore, we use the same $T_{f}$ for C+Au collisions as in Au+Au collisions. A smaller $T_{f}$ would enhance the photon yield from the hadronic phase in the region $p_{T}<1$ GeV and the larger $p_{T}$ part of the direct photon spectrum will remain unaltered. However, the photon anisotropic flow is expected to be larger for a smaller $T_{f}$. To show the effect of a relatively longer hadronic phase evolution on the photon flow observables we have calculated thermal photon $v_{2}$ and $v_{3}$ considering an arbitrary small $T_{f}$ $(=120$ MeV, see Fig. 7) for orientation angles $\pi/2$ and $\pi/4$ respectively where, the flow coefficients are estimated to be largest. We see that $v_{2}$ and $v_{3}$ are enhanced for a smaller $T_{f}$, however, the qualitative difference between the photon anisotropic flow parameters remains unchanged even for a significant change in the value of the freeze-out temperature. A detailed study considering a larger formation time and dynamical freeze-out would be valuable in this regards and we postpone it for the future.

The anisotropic flow parameters of direct photons for the clustered configurations with the largest thermal photon $v_{2}$ and $v_{3}$ are shown in Fig. 8. By ‘direct photons’ we mean the combined contribution of thermal and the prompt photons (produced in initial hard scatterings). We find that the values of the anosotropic flow cofficients are substantially large even after inclusion of the prompt contribution (The prompt spectrum is generated in the region $p_{T}>2$ GeV (due to some constraints) using JETPHOX and as a result we could not show the anisotropic flow in the low $p_{T}$ region). The maximum value of direct photon $v_{3}$ and $v_{2}$ (at 2 GeV) are found to be about $70\%$ of their respective maximum thermal photon $v_{3}$ and $v_{2}$ values. It is to be noted that these values are comparable to the anisotropic flow parameter data obtained from the $0-20\%$ Au+Au and Pb+Pb collisions at RHIC and LHC energy respectively Adare:2015lcd ; Acharya:2018bdy . Thus, considering even a smooth initial condition, we can obtain a significantly large direct photon $v_{2}$ and $v_{3}$ depending upon the orientations of collision for the clustered carbon. We expect that the inclusion of initial state fluctuations would further increase the value of direct photon $v_{2}$ and $v_{3}$. On the other hand, photon $v_{2}$ and $v_{3}$ for unclustered carbon appear only due to initial state fluctuations and are expected to be significantly smaller than the clustered carbon.

Thus, the clustered carbon and gold collisions at the RHIC have the potential to produce considerably large anisotropic flow parameters of photons even for the most central events. Furthermore, the strong sensitivity of the anisotropic flow parameters to the initial geometry and early QGP evolution also can be valuable to understand the ‘direct photon puzzle’. Our main focus in the present calculation is only to estimate the effect of initial geometry on the production and anisotropic flow parameters of photons. A more realistic calculation, using an event-by-event (3+1) dimensional hydrodynamic framework, would be valuable to precisely explore the role of fluctuations and participant asymmetry on the evolution of thermodynamic quantities and anisotropic flow parameters of photons.

To get an idea of the experimental signature, we simulate a large number of clustered C+Au events with different configurations and compare the results obtained from the collisions of unclustered carbon. Fig. 9 shows the average orientation angle $\langle\theta\rangle$ as a function of the number of participants in the collisions of clustered carbon, where and henceforth $\langle...\rangle$ denotes an event average. We observe that with the increasing $N_{\rm part}$, the average angle reduces significantly. Thus, based on the multiplicity cut, it may be possible to select events with various configurations.

In Fig. 10(a) we present the $N_{\rm part}$ dependence of average eccentricity and triangularity in unclustered and $\alpha$-clustered ${\rm C+Au}$ collisions at 200A GeV. We consider the events having $N_{\rm part}>40$. The superscripts ‘U’ and ‘C’ stand for the unclustered and clustered carbon respectively. We see that $\langle\varepsilon_{2}^{U}\rangle$ and $\langle\varepsilon_{3}^{U}\rangle$ show similar behavior with increasing $N_{\rm part}$. Due to vanishing geometric deformation and smaller fluctuations at a larger $N_{\rm part}$, the average eccentricity decreases. We observe a different trend for clustered carbon. Due to the intrinsic triangular geometry of the clustered-carbon, $\langle\varepsilon_{3}^{C}\rangle$ increases with $N_{\rm part}$, whereas $\langle\varepsilon_{2}^{C}\rangle$ decreases. The difference becomes substantial at larger $N_{\rm part}$, which indicates an obvious anti-correlation between ellipticity and triangularity exists in $\alpha$-clustered ${\rm C}$ + ${\rm Au}$ collisions.

In Fig. 10(b) we plot the event density distribution in $\varepsilon_{2}-\varepsilon_{3}$ plane for the collisions of clustered carbon. To quantify the anti-correlation, we define a normalized symmetric cumulants coefficient NSC(n,m) as follows Giacalone:2016afq ,

The NSC(2,3) is expected to be similar to the quantity involving the anisotropic flow coefficients ($v_{2}$ and $v_{3}$) under a linear response assumption Noronha-Hostler:2015dbi ; Yan:2015jma . In Fig. 10(c), we show the normalized symmetric cumulants coefficient as function of $N_{\rm part}$ for both unclustered and $\alpha$-clustered cases. The negative values of coefficients signify the anti-correlation relation between the two variables. We see a larger anti-correlation for the clustered case compared to the unclustered case. The coefficients for both the cases do not show significant $N_{\rm part}$ dependence. The NSC(2,3) for the clustered carbon is found to be three times larger than the NSC(2,3) obtained for the unclustered carbon. We find a small minimum of NSC(2,3) at $N_{\rm part}\sim 57$, where the values of the $\langle\varepsilon_{2}^{C}\rangle$ and $\langle\varepsilon_{3}^{C}\rangle$ are close to each other. Exploring the large anti-correlation between ellipticity and triangularity (elliptic flow and triangular flow), we can detect the $\alpha$-cluster effects of carbon in ${\rm C+Au}$ collisions. It is to be noted that some earlier studies have explored the characteristics of the initial-state eccentricities and final hadron flow observables from the collisions of the clustered carbon and heavy-ions at various beam energies in the event-by-event framework Broniowski:2013dia ; Bozek:2014cva . In principle, we expect that the anti-correlation behavior should hold for both photons and hadrons, since their flow observables basically arise from the initial geometry.