Electromagnetic fields in ultra-peripheral relativistic heavy-ion collisions

Abstract

Ultra-peripheral heavy-ion collisions (UPCs) offer unique opportunities to study processes under strong electromagnetic fields. In these collisions, highly charged fast-moving ions carry strong electromagnetic fields that can be effectively treated as photon fluxes. The exchange of photons can induce photonuclear and two-photon interactions and excite ions. This excitation of the ions results in Coulomb dissociation with the emission of photons, neutrons, and other particles. Additionally, the electromagnetic fields generated by the ions can be sufficiently strong to enforce mutual interactions between the two colliding ions. Consequently, the two colliding ions experience an electromagnetic force that pushes them in opposite directions, causing a back-to-back correlation in the emitted neutrons. Using a Monte Carlo simulation, we qualitatively demonstrate that the above electromagnetic effect is large enough to be observed in UPCs, which would provide a clear means to study strong electromagnetic fields and their effects.

Appendix

The probability of a UPC event associated with neutron emission ($P_{x\text{n},x\text{n}}^\textrm{UPC}$) is calculated as follows:

$$\begin{aligned} P_{x\text{n},x\text{n}}^\textrm{UPC} = P_{0H}(b)\times P_{x\text{n},x\text{n}}(b), \end{aligned}$$

(9)

where $P_{0H}(b)$ denotes the probability of having no hadronic interactions and $P_{x\text{n},x\text{n}}(b)$ represents the probability of nuclear breakup with neutron emission in both colliding nuclei [12, 15, 37]. $P_{0H}(b)$ is given by:

$$\begin{aligned} \begin{aligned}&P_{0H}(b) = \exp [-T_{AA}(b)\sigma _\text{NN}], \\&T_{AA}(b) = \int \mathrm{d}^{2}{\varvec{r}}_\perp T_{A}({\varvec{r}}_\perp )T_{A}({\varvec{r}}_\perp -{\varvec{b}}), \end{aligned} \end{aligned}$$

(10)

where ${\varvec{b}}$ is the two-dimensional impact parameter vector (with $b=|{\varvec{b}}|$) 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:

$$\begin{aligned} T_{A}({\varvec{r}}_\perp ) = \int {\text{d}}z\; \rho \left( \sqrt{{\varvec{r}}_\perp ^{2}+z^{2}}\right) , \end{aligned}$$

(11)

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

Assuming an independent nuclear breakup, $P_{x\text{n},x\text{n}}(b)$ can be factorized as

$$\begin{aligned} P_{x\text{n},x\text{n}}(b)=P_{x\text{n}}(b)\times P_{x\text{n}}(b). \end{aligned}$$

(12)

Following the methodology of STARlight [12, 15, 29, 36], the probability of nuclear breakup with neutron emission ($P_{x\text{n}}(b)$) is given by

$$\begin{aligned} P_{x\text{n}}(b)\varpropto \int {\text{d}}k \frac{{\text{d}}^{3}N(b,k)}{{\text{d}}k{\text{d}}^{2}b} \sigma _{\gamma A\rightarrow A^{*}+x\text{n}}(k), \end{aligned}$$

(13)

where $\sigma _{\gamma A\rightarrow A^{*}+x\text{n}}$ is determined from experimental data [56, 57]. The photon flux is calculated using the Weizsäcker–Williams approach [12, 15, 58]

$$\begin{aligned} \frac{{\text{d}}^{3}N(r_\perp ,k)}{{\text{d}}k{\text{d}}^{2}r_\perp } = \frac{Z^{2}\alpha x^{2}}{{\pi }^{2}kr_\perp ^{2}} K_{1}^{2}(x). \end{aligned}$$

(14)

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 $P_{0H}(b)$, $P_{x\text{n},x\text{n}}(b)$, $P_{x\text{n},x\text{n}}^\textrm{UPC}$ obtained from the simulation.

Fig. 3

(Color online) The probability of a UPC event associated with neutron emission ($P_{x\text{n},x\text{n}}^\textrm{UPC}$, black line), the probability of having no hadronic interactions ($P_{0H}(b)$, red line), and the probability of nuclear breakup with neutron emission in both colliding nuclei ($P_{x\text{n},x\text{n}}(b)$, blue line) as a function of the impact parameter in Au+Au collisions at $\sqrt{s_{\text {NN}}}$ = 200 GeV

Fig. 4

Photon energy distribution in the target frame in an UPC event with neutron emission using the Weizsäcker–Williams approach with Eqs. (14) and (8)

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