Characterization of hadron production in γ−A photonuclear interactions of ultra-peripheral collisions

The spectrum of virtual photons of the ¹⁹⁷Au nucleus was obtained in terms of the rapidity of the $J/\Psi$ meson:

The following expression appears in review papers for the photon flux of a fast moving charged particle:

where ${\alpha }_{\text{em}}$ is the fine structure constant, $F_Z\left(Q^2\right)$ is the charge form factor, w de energy of the emitted photon and ${\gamma }_L$ the Lorentz factor. The spectrum was calculated with the Weizsäcker- Williams classical analytical expression [1]:

were $\zeta =\frac{{\text{wb}}_{\text{min}}}{{\gamma }_L},b_{\text{min}}=2R_{\text{Au}}$ , K₀ and K₁ are the Modified Bessel functions. The maximum value for the energy of the photons is calculated with γ_lab =22 997 in the laboratory frame, obtaining w_max =312 GeV.

IV. Simulation of the ultra-peripheral asymmetric p- ²⁰⁸ Pb collision at $\sqrt{s_{\text{nn}}}=\mathrm{5,}\text{02 TeV}$

First, the fluxes of virtual photons from the proton and lead, reported in literature [4], that have a better precision, are compared with the calculus performed in this contribution (4). In the case of the proton, Guzey and Zhalov use the dipole approximation of F_Z(Q²), and with this:

Also, it is used the approximated expression:

where:

The following figure represents the flux of photons N_γ/p of the ultra-relativistic proton as a function of the rapidity of the J/ψ meson:

In the figure the exact calculation of the equation representing the flux of a relativistic charged particle (3) is given by the blue curve and the approximate result from (6) by the black curve. In order to add more accuracy to the determination of the flux of photons, Guzey and Zhalov took into account the suppression of the strong interaction between the colliding particles. The resulting flux of photons for the proton is calculated in the following way:

${\Gamma }_{\text{pA}}\left(\overrightarrow{b}\right)$ is the probability of suppressing the nuclear strong interaction for small impact parameters b, and it is calculated as:

where σ_nn is the total nucleon-nucleon cross-section (σ_nn=90mb at $\sqrt{s_{\text{nn}}}=\mathrm{5,}\text{02 TeV}$ ) and ${\rho }_A\left(\overrightarrow{r}\right)$ is the nuclear density. The flux of photons N_γ/p, calculated using the equations (8), (9) and (10), is presented in the red line in the figure. The green curve is obtained implementing the expression (4) in the program MATLAB. The program Engauge Digitizer is used to place it next to the three approximations reported in literature. This allow us observing that it agrees with the most precise calculations accomplished by Guzey and Zhalov, which is why their use is valid in the present contribution.

N_γ/Pb is also presented including the approximation (4) for the Pb nucleus, in green color. First, the blue curve correspond to the calculations performed turning down the suppression of the nuclear strong interaction by means of the placement of ${\Gamma }_{\text{pA}}\left(\overrightarrow{b}\right)=1$ in the equation (8) and using the upper limit: $b_{\text{min}}=R_{\text{Pb}}$ . Second, the black curve corresponds to the evaluation of N_γ/Pb (3), with the charge form factor of the lead nucleus. The range of the impact parameter b, considered for the calculation of the integral, is from b_min=8 fm to b_max=10⁵ fm. Here it is corroborated the implementation of the expression (4), more simple, for the calculation of the virtual photon’s flux for the lead nucleus, the same as in the case of the proton.

The cross-section of production of π⁰ mesons as a function of the incident photon’s energy for the reaction: γ+p→p+π⁰ is studied. A resonance is observed for 340 MeV’s energy of the incident photon in the laboratory frame, corresponding to 1232 MeV’s energy in the center of momentum frame for γ−p.

The study of the incoherent γ−p mechanism is worthy of notice because it initiates the process of intranuclear cascade. The same resonance obtained in the simulation confirms the importance of the ∆(1232 MeV) resonance in the production of pions, in agreement with the experimental results reported in literature [5].

B. $p_T$ distribution and invariant mass spectrum

The $p_T$ distribution obtained in the simulation is compared hereafter with the reported by CMS experiment for the p-²⁰⁸Pb run at $\sqrt{s_{\text{nn}}}=\mathrm{5,}\text{02 TeV}$ [6]. The $p_T$ spectrum corresponding to $<N_{tracks}>=8$ , that’s to say, a small value for the multiplicity of particles reported by CMS collaboration is selected, since the ultra-peripheral events are characterized by having a smaller multiplicity and < $p_T$ > with regard to central collisions.

It is observed that, the multicollisional dynamics implemented at CRISP, initiated by photonuclear interactions, is capable of reproducing with an acceptable precision the $p_T$ distribution of opposite charged pions.

Another magnitude of concern that allows the identification of particles when an experimental spectrum is analysed is the invariant mass. In a range between 100 MeV/c² and 200 MeV/c² the invariant mass was obtained in the simulation and the π resonance observed.

The K mesons, with a mass of approximately 500 MeV/c², are the lighter strange mesons, that can be produced in reactions of the type:

The cross-section of these processes is of the range of millibarn, typical of processes that involve strong nuclear interaction, however, K mesons decay with a life time of 10⁻⁸ s, characteristic of the weak interaction. This is why different values from zero of strangeness are assigned to them. The invariant mass spectrum eliminating the peak of the π mesons was obtained in the simulation.

Two resonances were observed: 770 MeV/c² and 498 MeV/c², corresponding to the formation of ρ and K mesons, respectively, in agreement with experimental results.

In the incoherent processes, nucleus splitting can occur, with preferential emission of neutrons in the same direction of the incident photons. The figure shows the angular distribution of neutrons obtained in the simulation. It can be observed the expected maximum in the forward direction.

The neutron, proton and pion γ +¹⁹⁷Au photonuclear cross-sections were calculated. The cross-sections increase with the energy of the incident photon until a maximum value. From this point on, the cross-sections decrease as a result of the opening of other channels at higher energies.

The following figure shows the differential cross-sections, including the weigh-in of the spectrum for ¹⁹⁷Au. The inclusion of the weigh-in of the spectrum enables affirming that it has been passed from exploring the production of particles making photons collide with target nucleus, to the study of a process of ultra-peripheral nature, where the electromagnetic field of one of the nuclei has excited the other nucleus, causing emission of particles. When the weigh-in is included the higher energies are strongly suppressed, which indicates that the weigh-in of the spectrum is crucial if one wants to do comparisons between simulation and experiment in ultra-peripheral collisions.