Electromagnetic fields in ultra-peripheral relativistic heavy-ion collisions

Over the past two decades, several novel phenomena associated with strong electromagnetic fields in hot quantum chromodynamics (QCD) have been proposed, such as the chiral magnetic effect (CME) Kharzeev et al. (1998); Huang (2016); Wang and Zhao (2018); Zhao and Wang (2019); Fang et al. (2021); Liu and Huang (2022). Such strong electromagnetic fields are expected to be generated during relativistic heavy-ion collisions; however, they are extremely challenging to measure experimentally. Several attempts have been made to detect strong electromagnetic fields in heavy ion collisions Adamczyk et al. (2017a); Adam et al. (2021a); Liu and Huang (2020); Gao et al. (2020). However, electromagnetic processes are obscured by strong hadronic interactions when nuclei collide. Conversely, ultraperipheral heavy-ion collisions (UPCs) offer a distinctive advantage in observing electromagnetic processes, as the impact parameters in these collisions are more than twice the nuclear radius, preventing hadronic interactions Bertulani et al. (2005); Klein et al. (2017); Klein and Steinberg (2020). Therefore, UPCs provide a unique opportunity to study strong electromagnetic processes in relativistic heavy ion collisions Baur et al. (1998); Bertulani et al. (2005).

In UPCs, fast-moving heavy ions are accompanied by intense photon fluxes due to their large electric charges and strong Lorentz-contracted electromagnetic fields. These fields are sufficiently strong to induce photonuclear and photon-photon interactions Bertulani et al. (2005); Klein et al. (2017); Klein and Steinberg (2020). Many interesting results regarding particle production, including $e^{+}+e^{-}$ and $\pi^{+}+\pi^{-}$, and vector mesons, such as $\rho,\omega,J/\psi$ have been reported in UPCs Klein and Nystrand (1999); Adler et al. (2002); Afanasiev et al. (2009); Abelev et al. (2013); Khachatryan et al. (2017); Aaboud et al. (2017); Adam et al. (2021b). Thus, it is natural to assume that these strong electromagnetic fields could likewise have visible impacts on collision dynamics through the electromagnetic force.

The most important region for Coulomb dissociation is the giant dipole resonance (GDRs) Goldhaber and Teller (1948), approximately a few MeVs. The photon-nuclear interaction in UPCs has a large cross-section to produce GDRs or excite nuclei to high-energy states Berman and Fultz (1975); Chomaz and Frascaria (1995). The GDRs typically decay by emitting photons or neutrons. The excitation energy of the GDRs is approximately 10-14 MeV for heavy nuclei and higher for lighter nuclei Berman and Fultz (1975); Veyssiere et al. (1970); Tao et al. (2013); He et al. (2014); Huang and Ma (2021). The emitted neutrons have similar energies, which are considerably lower than the typical energy scales of relativistic heavy-ion collisions. Consequently, the emitted neutrons can be used to detect strong electromagnetic fields. In this study, we employed Monte (MC) simulations to demonstrate that the electromagnetic effect on neutron emission from Coulomb dissociation, primarily through GDRs decay, is sufficiently significant to be observed in UPCs. Our simulation incorporates the calculation of the strong electromagnetic field using Liénard-Wiechert potential, combined with neutron emission data from the existing experimental results and model simulations. Natural units $\hbar=c=1$ are used in this study.

When two ions collide head-on in the UPC, a strong electromagnetic field is generated. For instance, in a gold-gold (Au + Au) UPC with an impact parameter of $b=20$   fm Baltz et al. (2002) and a center-of-mass energy of $\sqrt{s_{\text{NN}}}=200$ GeV, as shown in Fig.  1, the electric and magnetic fields produced by one nucleus acting on another are approximately $eE\sim 0.2m_{\pi}^{2}$ and $eB\sim 0.2m_{\pi}^{2}$, respectively,  Deng and Huang (2012); Bzdak and Skokov (2012). Electric and Lorentz forces would push the two colliding ions in opposite directions in the transverse plane. Assuming that the nuclei have no initial transverse moment, the velocity of each nucleon can be calculated using the following equation:

where $m_{\text{N}}$ denotes nucleon mass. At the RHIC and LHC energies, where $v_{z}$ is very large, the nuclei are strongly Lorentz contracted and adopt a pancake-like shape. The effective collision time was estimated as $\Delta t=2R/(\gamma v_{z})$ Bertulani et al. (2005), where $R$ is the radius of the nucleus and $\gamma=1/\sqrt{(1-v_{z}^{2})}$ is the Lorentz factor, which is approximately 107 for Au + Au collisions at $\sqrt{s_{\text{NN}}}=200$ GeV. Hence, during the collision, the transverse momentum per nucleon due to the strong electromagnetic field is approximately 2 MeV by a rough estimate [$\Delta p=Z(eE+v_{z}eB)\Delta t/A\approx 0.4m_{\pi}^{2}\times 0.12\,{\rm fm}\times 79/197\sim 2$ MeV, with $Z=79,A=197$, and $R=6.38$ fm for the Au nucleus].

However, the excitation energy of a typical GDRs is about 10-14 MeV for heavy nuclei Berman and Fultz (1975). Owing to the large GDR cross-section in photonuclear interactions, the exchange of photons between two colliding nuclei in a UPC event has a high probability of exciting one or both ions into GDRs or even higher excitations. GDRs can decay by emitting a single neutron, whereas higher-excitation resonances can decay by emitting two or more neutrons Berman and Fultz (1975); Adamczyk et al. (2017b); Chen et al. (2018). These neutrons have low momenta, approximately $\sim 10$ MeV, comparable to the transverse momentum shift, $\Delta p$, induced by the electromagnetic force. Thus, strong electromagnetic fields may affect the momentum distribution of the emitted neutrons, such that their emission directions correlate with the direction of the impact parameter or the electromagnetic fields. Consequently, a back-to-back correlation may occur between the neutrons emitted from the two colliding nuclei. This provides a means of investigating strong electromagnetic fields in UPCs and offers a way to measure the direction of the impact parameter or electromagnetic field.

The electromagnetic fields can be calculated using the Liénard-Wiechert potentials, as described in Refs.  Deng and Huang (2012); Hattori and Huang (2017). The electric and magnetic fields are expressed as follows:

where $Z_{n}$ denotes the charge number of the $n$th particle. Here, ${\bm{R}}_{n}=\bm{r}-\bm{r}_{n}$ represents the relative coordinates of the field point $\bm{r}$ to the location $\bm{r}_{n}$ of the $n$th particle with velocity $\bm{v}_{n}$ at retarded time $t_{n}=t-|\bm{r}-\bm{r}_{n}|$. Summations were performed for all protons in the nucleus.

For UPCs near the collision time $t=0$, Eq. (2) can be approximated as

where ${\bm{e}}_{z}$ denotes the unit vector in $\pm z$ direction (depending on whether the source nucleus is the target or projectile nucleus), ${\bm{R}}_{\perp}$ denotes the transverse position of the center of the source nucleus, and $Z$ denotes the total charge number of the source nucleus.

To obtain more quantitative results, MC simulations were performed. We parameterized the nucleon density distribution of the nucleus using the Woods-Saxon function Woods and Saxon (1954):

where $a=0.54$ fm is the skin depth parameter, and $R=6.38$ fm is the Au nucleus in this calculation. The electromagnetic fields were calculated using Eq. (3) Deng and Huang (2012); Bzdak and Skokov (2012). We obtained the momentum per nucleon induced by strong electric and Lorentz forces through the following integration:

where $q=Ze/A$ denotes the average charge per nucleon. Due to the Lorentz contraction, the longitudinal dimension $L_{z}=2R/\gamma$ is significantly smaller than the transverse size $2R$, causing the two colliding nuclei to pass each other within a very short timescale $L_{z}/v_{z}$. In this study, we assumed that the electromagnetic fields remained constant during the collisions and were negligible before or after the two nuclei touched, a supposition qualitatively consistent with the findings in Ref.Deng and Huang (2012).

Following the methodology of STARlight Klein and Nystrand (1999); Baltz et al. (2002, 2009); Klein et al. (2017), the probability of an UPC event associated with neutron emission ($P_{xn,xn}^{\rm UPC}$) is calculated as follows:

where $P_{0H}(b)$ denotes the probability of having no hadronic interactions and $P_{xn,xn}(b)$ represents the probability of nuclear breakup with neutron emission in both colliding nuclei Klein and Nystrand (1999); Klein et al. (2017); Broz et al. . Assuming an independent nuclear breakup, $P_{xn,xn}(b)$ can be factorized as

Details of the Eq.6,7 can be found in Refs. Klein and Nystrand (1999); Baltz et al. (2002, 2009); Klein et al. (2017) and in the appendix.

Mutual Coulomb dissociation was measured at $\sqrt{s_{\text{NN}}}$ = 130 GeV Au+Au at the RHIC Chiu et al. (2002). The measured cross section of “Coulomb”-like events agrees well with the theoretical calculations. The measured neutron multiplicity from a “Coulomb”-like event is concentrated at $\sim 1$, and at most of $\sim 35$. Direct measurement of the neutron multiplicity in UPC events is also available for Au + Au collisions at $\sqrt{s_{\text{NN}}}$ = 200 GeV from the STAR experiment Abelev et al. (2008); Adamczyk et al. (2017b), where the neutron multiplicity distribution is comparable to the above result. The Coulomb dissociation of the halo nucleus ¹¹Be at 72$A$ MeV Nakamura et al. (1994) shows a cos$(2\phi)$ modulation between the emitted neutron direction and the impact parameter. A cos$(2\phi)$ modulation of the neutrons emitted from the two nuclei in the UPC was proposed to estimate the impact parameter direction Baur et al. (2003); STA . The current study primarily focuses on the electromagnetic field impact, which mostly manifests as a back-to-back correlation represented by the cos$(\phi)$ modulation. The fluctuation of electromagnetic fields that possess both $x$ and $y$ components may induce higher-order cos$(n\phi)$ modulations. Photoneutron experiments at an energy $\sim$20 MeV measured the polar angle of neutron emission, revealing definite anisotropy on Au and Pb targets Tagliabue and Goldemberg (1961). Similar results were obtained in other experiments on Coulomb dissociation Bakhtiari et al. (2022); Nakamura et al. (1994). Assuming that the photonuclear process is a “direct photoelectric effect“, an anisotropic polar angular distribution with a maximum at $\pi/2$ for the neutrons is expected Courant (1951). Nevertheless, because our study primarily focused on transverse neutron emission, the aforementioned polar angular effect was not considered.

In this study of the neutron emission, the Landau distribution was used to directly estimate the neutron multiplicity from the STAR experiment Abelev et al. (2008); Adamczyk et al. (2017b). As mentioned in Ref. Broz et al. , the measurement of spectra for secondary particles from photon-nuclear interactions is currently limited. Therefore, the neutron energies were generated using the same method as in the $\mathrm{n^{O}_{O}n}$ generator Broz et al. , utilizing the Evaluated Nuclear Data File (ENDF) tables Chadwick et al. (2011). ENDF tables for Au are even more limited; tables from ²⁰⁸Pb are currently used. The accuracy could be improved with future measurements; however, this was sufficient for our qualitative study. The energy distribution of the incident photons in a UPC with neutron emission was calculated as  Klein and Nystrand (1999); Klein et al. (2017); Broz et al.

Neutrons are then produced according to the photon energy and ENDF tables with an isotropic angular distribution Broz et al. .

Figure 2 presents the simulation results. The $\Psi^{\text{east}}_{n}$ and $\Psi^{\text{west}}_{n}$ are the first order harmonic planes Poskanzer and Voloshin (1998) constructed by the neutrons emitted from the two nuclei (east and west indicate the two nuclei going opposite directions) in $\sqrt{s_{NN}}$ = 200 GeV Au + Au UPCs. They are calculated using the sum of the momentum vectors ($q$ vector) Poskanzer and Voloshin (1998) of neutrons from each nucleus. $\Psi^{east}_{n}$ is rotated by $\pi$ following the experimental conversion. As mentioned in the previous section, the Lorentz force and Coulomb repulsion in such UPCs are sufficiently large to affect the momentum distribution of the emitted neutrons. The red line in Fig. 2 is a cos$(\phi)$ fit to the simulation, which shows that strong repulsive forces act in opposite directions to the two colliding nuclei. Owing to the fluctuation of electromagnetic fields that possess both $x$ and $y$ components, higher-order cos$(n\phi)$ modulations may also be present. The dashed line represents a fit that includes the cos$(2\phi)$-modulation, which better describes the distribution. Our calculation also demonstrates the sensitivity to the impact parameter direction or the electromagnetic field direction, which could be utilized to study spin-polarization-related physics in UPCs in the future Xiao et al. (2020); Xing et al. (2020); Zhang et al. (2020); Wu et al. (2022), as the photons are linearly polarized Adam et al. (2021b). This may provide insights into the chiral and spin-related effects in relativistic hadronic heavy-ion collisions, such as CME, hyperon spin polarization, and vector meson spin alignment Liang and Wang (2005); Adamczyk et al. (2017a); Wang (2023); Chen et al. (2023).

As discussed in Ref. Baur et al. (2003); Bertulani et al. (2005), neutrons have been measured using zero-degree calorimeters (ZDCs) Adler et al. (2001) in relativistic heavy-ion collisions. At the RHIC, the ZDCs are positioned $\pm 18$ m away from the nominal interaction point at the center of the detector, with dimensions of 10 cm width and 18.75 cm height Adler et al. (2001). For Au + Au collisions at $\sqrt{s_{\text{NN}}}$ = 200 GeV at the RHIC, where the neutron longitudinal momentum is $\sim$ 100 GeV, a maximum $p_{\rm T}$ of 140 MeV/c corresponds to a deflection of approximately $2.5$   cm, which is within the acceptance of the ZDCs. With ZDCs’ shower-maximum detectors STA , which provide position measurements of neutrons, the measurement of the neutron-neutron correlations between the ZDCs is feasible.

In summary, using Monte Carlo simulation, we demonstrated that the electromagnetic fields generated in UPCs are strong enough to induce measurable back-to-back emission of neutrons in the transverse plane. The effect UPCs discussed here provides a clean way for detecting strong electromagnetic fields, as no hadronic interactions are involved, which would be the strongest electromagnetic fields that can be detected to date. It also offers a method to measure the direction of the impact parameter or electromagnetic field. Thus, it may also shed light on chiral and spin-related effects in relativistic hadronic heavy-ion collisions to understand the fundamental features of hot QCD matter. Additionally, this may aid in understanding the initial conditions of hadronic/heavy-ion collisions with a small $b$, wherein the electromagnetic fields would be considerably stronger.

We thank Dr. Fu-Qiang Wang, Dr. Si-Min Wang, and Dr. Shi Pu for useful discussions.

The probability of a UPC event associated with neutron emission ($P_{xn,xn}^{\rm UPC}$) was calculated as follows:

where $P_{0H}(b)$ denotes the probability of having no hadronic interactions and $P_{xn,xn}(b)$ represents the probability of nuclear breakup with neutron emission in both colliding nuclei Klein and Nystrand (1999); Klein et al. (2017); Broz et al. . $P_{0H}(b)$ is given by:

where $\bm{b}$ is the two-dimensional impact parameter vector (with $b=|\bm{b}|$) in the transverse plane, $\sigma_{NN}$ is the total nucleon-nucleon interaction cross-section, and $T_{AA}$ is the overlap function. The number of nucleon-nucleon collisions follows a Poisson distribution with a mean of $T_{AA}(b)\sigma_{NN}$. The nuclear thickness function $T_{A}$ is calculated as follows:

where $\rho$ corresponds to the Woods-Saxon functions in Eq.  (4).

Assuming an independent nuclear breakup, $P_{xn,xn}(b)$ can be factorized as

Following the methodology of STARlight Klein and Nystrand (1999); Baltz et al. (2002, 2009); Klein et al. (2017), the probability of nuclear breakup with neutron emission ($P_{xn}(b)$) is given by

where $\sigma_{\gamma A\rightarrow A^{*}+xn}$ was determined from experimental data Baltz et al. (1996); Harland-Lang (2023). The photon flux was calculated using the Weizsäcker-Williams approach Krauss et al. (1997); Klein and Nystrand (1999); Klein et al. (2017)

where $k$ represents the photon energy, $Z$ is the nuclear charge, $K_{1}$ is the modified Bessel function, and $x=kr_{\perp}/\gamma$. Figure  3 shows $P_{0H}(b)$, $P_{xn,xn}(b)$, $P_{xn,xn}^{\rm UPC}$ obtained from the simulation.

Figure 4 shows the calculation of the photon energy distribution in the target frame for a UPC event with neutron emissions.