The effect of initial nuclear deformation on dielectron photoproduction
in hadronic heavy-ion collisions

The charge density for a spherical heavy ion is typically given by the Woods-Saxon distribution:

$\displaystyle\rho_{sph}(r)=\frac{\rho_{0}}{1+e^{(r-R_{0})/a}}$

(1)

where $\rho_{0}$ represents the normalization factor and denotes the density at the center of the nucleus. The radius $R_{0}$ and skin depth $a$ are obtained from elastic electron scattering [20, 21]. However, for deformed nuclei, an alternative way to describe their charge density is to extend the two-parameter Fermi distribution by introducing deformation parameters:

$\displaystyle\rho(\vec{r})=\frac{\rho_{0}}{1+{\rm exp}[\frac{r-R_{0}[1+\beta_{2}Y_{2}^{0}(\theta)+\beta_{4}Y_{4}^{0}(\theta)]}{a}]}$

(2)

where $\beta_{2}$ and $\beta_{4}$ are quadrupole and hexadecupole deformation expressed in the spherical-harmonics expansion, respectively [22]. It is noteworthy that this charge density is independent of the azimuthal angle, allowing us to derive $\rho(\vec{r})=\rho(r,\theta)$.

Deformation parameters for the ${}_{92}^{238}\textrm{U}$ nucleus are taken from Ref. [23]. Nuclear density distributions are not clear for deformed ${}_{44}^{96}\textrm{Ru}$ and ${}_{40}^{96}\textrm{Zr}$ because e-A scattering experiments [24, 25] and comprehensive model deductions [26] present significantly different results. In this study, we adopt larger $\beta_{2}$ values to evaluate the maximum impact of initial nuclear deformation on $e^{+}e^{-}$ pair photoproduction in hadronic heavy-ion collisions. The parameters for both spherical and deformed nuclei used in our analysis are listed in Table 1. Additionally, the shape of the deformed nucleus is a prolate spheroid when $\beta_{2}>0$, and the direction of the major axis $\vec{v}$ in Eq. (2) is along the $z$ axis.

In deformed heavy-ion collisions, the directions of the major axis of colliding nuclei $\vec{v}$ are expected to be random and irrelevant. Our calculations adopt the following reference frame: where the beam direction corresponds to the $z$ axis, and the direction of the impact parameter corresponds to the $x$ axis.

The charge density of a deformed nucleus with a specific $\vec{v}$ can then be expressed as:

$$\rho_{\vec{v}}(\vec{r})=\rho[R_{z}^{-1}(-\varphi_{v})R_{y}^{-1}(\theta_{v})R_{z}^{-1}(\varphi_{v})\vec{r}]$$

(3)

$$\vec{v}=({\rm sin}\theta_{v}{\rm cos}\varphi_{v},{\rm sin}\theta_{v}{\rm sin}\varphi_{v},{\rm cos}\theta_{v})$$

(4)

$$R_{y}(\theta_{v})=\begin{pmatrix}{\rm cos}\theta_{v}&0&{\rm sin}\theta_{v}\\ 0&1&0\\ -{\rm sin}\theta_{v}&0&{\rm cos}\theta_{v}\end{pmatrix}$$

(5)

$$R_{z}(\varphi_{v})=\begin{pmatrix}{\rm cos}\varphi_{v}&-{\rm sin}\varphi_{v}&0\\ {\rm sin}\varphi_{v}&{\rm cos}\varphi_{v}&0\\ 0&0&1\end{pmatrix}$$

(6)

where $R_{y}(\theta_{v})$ and $R_{z}(\varphi_{v})$ are rotation matrices, and $\theta_{v}$ and $\varphi_{v}$ denote the polar angle and azimuthal angle of $\vec{v}$, respectively. We assume that $\vec{v}$ is isotropic in the surface of the unit sphere, which means that cos$\theta_{v}$ is uniform in $[-1,1]$ and $\varphi_{v}$ is uniform in $[0,2\pi]$. In our calculations, the surface of the unit sphere is divided into 400 bins, leading to $N$ = 160000 collision configurations when two deformed nuclei collide. Conventionally, configurations with $\vec{v_{1}}=\vec{v_{2}}=(0,0,\pm 1)$ and $\vec{v_{1}}=\vec{v_{2}}=(\pm 1,0,0)$ are referred to as tip-tip and body-body collisions, respectively [27, 28], where subscripts 1 and 2 represent the two colliding nuclei. Selecting central tip-tip events and central body-body events based on experimental observables is possible [29], so we will also present calculations for the two limiting cases in deformed U + U collisions.

In relativistic heavy-ion collisions, the electric and magnetic fields accompanied by nuclei are mutually perpendicular and have the same absolute magnitudes. These almost transverse electromagnetic fields are very similar to the electromagnetic fields of photons and can be viewed as an equivalent swarm of quasireal photons [8]. According to the equivalent photon approximation (EPA) method, the induced photon flux with energy $\omega$ at transverse position $\vec{x_{\perp}}$ from the center of the nucleus is given by [7]:

$$n(\omega,\vec{x_{\perp}})=\frac{4Z^{2}\alpha}{\omega}\left|\int\frac{{\rm d}^{2}\vec{q_{\perp}}}{(2\pi)^{2}}\vec{q_{\perp}}\frac{F(\vec{q})}{\left|\vec{q}\right|^{2}}e^{i\vec{x_{\perp}}\cdot\vec{q_{\perp}}}\right|^{2}$$

(7)

$$\vec{q}=(\vec{q_{\perp}},\frac{\omega}{\gamma})$$

(8)

where $\alpha=1/137$ is the fine-structure constant, $\gamma$ is the Lorentz factor of the nucleus, $Z$ is the nuclear charge number, and $\vec{q_{\perp}}$ is the transverse momentum of the photon. The form factor $F(q)$, carrying the information about the charge distribution inside the nucleus, can be obtained by performing a Fourier transformation to the charge density $\rho(\vec{r})$:

$$F(\vec{q})=\int{\rm d}^{3}\vec{r}\rho(\vec{r})e^{i\vec{q}\cdot\vec{r}}$$

(9)

For a spherical nucleus, the form factor can be expressed as follows:

$$F(q)=\frac{4\pi}{q}\int{\rm d}rr\rho(r){\rm sin}(qr)$$

(10)

For a spheroidal nucleus, the form factor depends on the direction of momentum transfer as well:

	$\displaystyle F(q,\eta)=\iiint{\rm d}r{\rm d}\theta{\rm d}\varphi r^{2}{\rm sin}\theta\rho(r,\theta){\rm cos}[qr$
	$\displaystyle\times({\rm sin}\theta{\rm sin}\eta{\rm cos}\varphi+{\rm cos}\theta{\rm cos}\eta)]$		(11)

$${\rm cos}\eta=\frac{\vec{q}\cdot\vec{v}}{\left|\vec{q}\right|}$$

(12)

where $\eta$ denotes the angle between momentum transfer $\vec{q}$ and major axis $\vec{v}$. Utilizing Eq. (7), we can calculate the photon flux $n_{\vec{v}}(\omega,\vec{x_{\perp}})$ for a deformed nucleus with a given $\vec{v}$.

Fig. 1 shows the photon flux distributions with energy $\omega$ = 1 GeV in U + U collisions at $\sqrt{s_{NN}}$ = 193 GeV as a function of transverse position $\vec{x_{\perp}}$ from the center of the nucleus. The photon flux for the spherical nucleus is shown in panel (a), and those in the case of body and tip orientations for the deformed nucleus are presented in panels (b) and (c). The results from different configurations as a function of distance $r$ from the center of the nucleus are illustrated in panel (d), and one can observe that the differences are concentrated around $R_{0}$. The photon flux from the tip orientation is greater than that for the spherical nucleus, while the maximum region (orange circular band) presents a smaller radius. The pattern from the body orientation exhibits an ellipse, where the extreme points of photon flux along the $x$-axis and $y$-axis differ, corresponding to the polar (major) radius and equatorial radius of the prolate spheroid, respectively.

According to the equivalent photon approximation, the cross section of the $e^{+}e^{-}$ pair produced by the two-photon process in relativistic heavy-ion collisions can be expressed as [13]:

	$\displaystyle\sigma(AA\rightarrow AAe^{+}e^{-})=\int{\rm d}\omega_{1}\int{\rm d}\omega_{1}n_{1}(\omega_{1})n_{2}(\omega_{2})$
	$\displaystyle\times\sigma(\gamma\gamma\rightarrow e^{+}e^{-})$		(13)

where $\sigma(\gamma\gamma\rightarrow e^{+}e^{-})$ is the photon-photon reaction cross section for the $e^{+}e^{-}$ pair. The energy of the produced particles is $E=\omega_{1}+\omega_{2}$, while their longitudinal momentum becomes $p_{z}=\omega_{1}-\omega_{2}$ as the velocity of the moving heavy ion approaches the speed of light. The final- state particles have a small transverse momentum, which can be negligible compared to longitudinal momentum. Consequently, the invariant mass $W$ and rapidity $y$ of the $e^{+}e^{-}$ pair can be obtained as follows:

$$W=\sqrt{E^{2}-p^{2}}=\sqrt{4\omega_{1}\omega_{2}}$$

(14)

$$y=\frac{1}{2}{\rm ln}\frac{E+p_{z}}{E-p_{z}}=\frac{1}{2}{\rm ln}\frac{\omega_{1}}{\omega_{2}}$$

(15)

Therefore, ${\rm d}\omega_{1}{\rm d}\omega_{2}$ in Eq. (II.3) can be converted to ${\rm d}W{\rm d}y$. The cross section for producing a pair of electrons with invariant mass $W$ is given by the Breit-Wheeler formula [30]:

	$\displaystyle\sigma(\gamma\gamma\rightarrow e^{+}e^{-})=\frac{4\pi^{2}\alpha^{2}}{W^{2}}[(2+\frac{8m_{e}^{2}}{W^{2}}-\frac{16m_{e}^{4}}{W^{4}})$
	$\displaystyle\times{\rm ln}(\frac{W+\sqrt{W^{2}-4m_{e}^{2}}}{2m_{e}})-\sqrt{1-\frac{4m_{e}^{2}}{W^{2}}}(1+\frac{4m_{e}^{2}}{W^{2}})]$		(16)

where $m_{e}$ is the mass of the electron.

The model calculations of $e^{+}e^{-}$ pair photoproduction have been presented in hadronic Au + Au collisions [15], and we utilize a similar approach to conduct model calculations of $e^{+}e^{-}$ pair photoproduction in randomly oriented collisions of deformed heavy ions. The yield for the photoproduced $e^{+}e^{-}$ pair with the orientation $(\vec{v_{1}},\vec{v_{2}})$ in a selected centrality bin can be expressed as:

$$\frac{{\rm d}^{2}N}{{\rm d}W{\rm d}y}=\frac{\frac{W}{2}\int_{b_{min}}^{b_{max}}{\rm d}^{2}\vec{b}\int{\rm d}^{2}\vec{x_{\perp}}n_{\vec{v_{1}}}(\omega_{1},\vec{x_{\perp}})n_{\vec{v_{2}}}(\omega_{2},\vec{x_{\perp}}-\vec{b})\sigma(\gamma\gamma\rightarrow e^{+}e^{-})P_{\vec{v_{1}},\vec{v_{2}}}^{B}(\vec{b})}{\int_{b_{min}}^{b_{max}}{\rm d}^{2}\vec{b}P_{\vec{v_{1}},\vec{v_{2}}}^{B}(\vec{b})}$$

(17)

where $b_{min}$ and $b_{max}$ are the minimum and maximum impact parameters for a given centrality class, and $P_{\vec{v1},\vec{v2}}^{B}(\vec{b})$ is the probability of hadronic interactions:

$$P_{\vec{v_{1}},\vec{v_{2}}}^{B}(\vec{b})=1-{\rm exp}[-A^{2}\sigma_{NN}\int{\rm d}^{2}\vec{s}T_{\vec{v_{1}}}(\vec{s})T_{\vec{v_{2}}}(\vec{s}-\vec{b})]$$

(18)

where $A$ is the nucleon number, $\sigma_{NN}$ is the inelastic nucleon-nucleon cross section, which is dependent on collision energy $\sqrt{s_{NN}}$ [31], and the nuclear thickness function $T_{\vec{v}}(\vec{s})$ is the projection of nuclear charge density with orientation $\vec{v}$ on the $x$-$y$ plane:

$$T_{\vec{v}}(\vec{s})=\int{\rm d}z\rho_{\vec{v}}(\vec{s},z)$$

(19)

In this way, we can directly obtain the $e^{+}e^{-}$ pair yields in tip-tip and body-body collisions, but the calculations of all $N$ = 160000 collision configurations are required to obtain the average yields in deformed heavy-ion collisions. Our results are filtered to match the fiducial acceptance ($p_{T}^{e}>$ 0.2 GeV/c, $\left|\eta^{e}\right|<$ 1, $\left|y^{ee}\right|<$ 1) to compare with experimental data from the STAR collaboration [12]. As discussed in Refs. [15, 32], the impact of violent hadronic interactions occurring in the overlap region on photoproduction is negligible for peripheral collisions. In central collisions, this effect on differences between spherical and deformed configurations should be small. Therefore, we neglect the possible disruptive effects from hadronic interactions in our calculations.

In deformed heavy-ion collisions, we will employ the Glauber model [31, 33] to define centrality and provide corresponding impact parameters. For a random collision configuration with the orientation $(\vec{v_{1}},\vec{v_{2}})$, the centrality can be expressed as a percentage of the interaction probability:

$$c_{i}(b)=\frac{\int_{0}^{b}{\rm d}^{2}\vec{{b}^{\prime}}P_{\vec{v_{1}},\vec{v_{2}}}^{B}(\vec{{b}^{\prime}})}{\int_{0}^{\infty}{\rm d}^{2}\vec{{b}^{\prime}}P_{\vec{v_{1}},\vec{v_{2}}}^{B}(\vec{{b}^{\prime}})}$$

(20)

In tip-tip and body-body collisions, this approach is suitable, but it is not sufficient when calculating average yields because all configurations occur with the same probability. Instead, the two-component approach $fN_{coll}+(1-f)N_{part}$ is a better choice [33, 34, 35], where $N_{part}$ is the number of participating nucleons, and $N_{coll}$ is the number of nucleon-nucleon collisions [33]:

	$\displaystyle N_{part}(b)$	$\displaystyle=A\int{\rm d}^{2}\vec{s}T_{\vec{v_{1}}}(\vec{s})\left\{1-[1-T_{\vec{v_{2}}}(\vec{s}-\vec{b})\sigma_{NN}]^{A}\right\}$
		$\displaystyle+A\int{\rm d}^{2}\vec{s}T_{\vec{v_{2}}}(\vec{s}-\vec{b})\left\{1-[1-T_{\vec{v_{1}}}(\vec{s})\sigma_{NN}]^{A}\right\}$		(21)

$$N_{coll}(b)=A^{2}\sigma_{NN}\int{\rm d}^{2}\vec{s}T_{\vec{v_{1}}}(\vec{s})T_{\vec{v_{2}}}(\vec{s}-\vec{b})$$

(22)

The relative weight $f$ is usually small [31, 34, 35, 36, 37], and thus we set $f$ = 0 for simplicity. Therefore, the centrality in deformed heavy-ion collisions is defined by the cumulative distribution function of $N_{part}$:

$$c=\int_{N_{part}}^{\infty}{\rm d}{N}^{\prime}_{part}P({N}^{\prime}_{part})$$

(23)

$$P(N_{part})=\frac{\sum_{i=1}^{N}P_{i}(N_{part})}{N}$$

(24)

where $P(N_{part})$ is the average probability distribution of $N_{part}$ and $P_{i}(N_{part})$ is the probability distribution for a special configuration, which can be calculated using Eqs. (20) and (II.4):

$$P_{i}(N_{part})=-\frac{{\rm d}c_{i}(b)}{{\rm d}N_{part}(b)}$$

(25)

It is noteworthy that $N_{part}(b)$ monotonically decreases with impact parameter $b$. Once the range of $N_{part}$ in a given centrality class is obtained from Eq. (23), the corresponding range of the impact parameter for a random configuration can be determined from Eq. (II.4). Then, the yield for the photoproduced $e^{+}e^{-}$ pair can be calculated using Eq. (17). Table 2 presents the centrality definitions in 40%–60% and 60%–80% for U + U collisions as well as the tip-tip and body-body configurations. The average number of participants $\left\langle N_{part}\right\rangle$ is also listed in the table. Tables 3 and 4 report the centrality definitions for Ru + Ru and Zr + Zr collisions, respectively, under both spherical and deformed configurations. Despite the systematic differences of $\left\langle N_{part}\right\rangle$ observed when compared with the Glauber Monte Carlo approach [33], the variation in impact parameter between the two calculations is found to be minor.