Deblurring for nuclei: 3D characteristics of heavy-ion collisions

Observables from nuclear and high-energy experiments can be degraded by detector performance and/or methodology in extracting the observables, such as of the final-state characteristics of heavy-ion collisions in relation to a coarsely estimated reaction-plane direction. We propose the use of deblurring methods, such as in optics, to correct for observable degradation. Our main focus is the restoration of triple-differential particle distributions in heavy-ion collisions. We demonstrate that these could be extracted from collision measurements following the Richardson-Lucy deblurring method from optics. We illustrate basic features of the restoration methodology in a schematic model assuming either ideal or more realistic particle detection. The inferred three-dimensional (3D) distributions for collisions may be easier to interpret in terms of collision dynamics and sought properties of bulk matter than the currently employed Fourier coefficients, that combine information from different azimuthal angles relative to the reaction plane.

Figure 1

Probability density for system velocity $V^{\prime }$, estimated with the average velocity for $N-1=9$ particles sampled from a symmetric uniform velocity distribution centered around the true velocity $V$. The solid (red) line shows the distribution from a direct simulation, calculated with $H=0.1$ binning, and the dashed (blue) line shows the distribution estimated with the central-limit theorem.

Figure 2

Distribution of particles in velocity within the projectile frame for a 1D model: original is represented by the solid (red) line, inferred from measurements is represented by the dashed (blue) line, and restored through deblurring is represented by the dash-dot (green) line. In the simulation, $10000$ events with $N=10$ particles each were generated. A binning of $H=0.1$ was used for the distributions. The restoration was done assuming the central-limit blurring function.

Figure 3

Detection efficiency $E$ (solid line) as a function of particle laboratory velocity $v^{\text{lab}}$, in the 1D model with inefficiencies, with superimposed particle distribution $dN/d{v}^{\text{lab}}$ (dotted line) for a projectile moving at velocity $V=0.5$ (short vertical dashed line).

Figure 4

Distribution of particles in velocity within the projectile frame for 1D model with inefficiencies: original represented by the solid (red) line, inferred from measurements represented by the dashed (blue) line and restored through deblurring and represented by the dash-dot (green) line. The simulated events originally contained $N=30$ particles. In the measurement the particles were assumed to be accepted with the efficiency of Fig. 3 and the events were processed if the number of accepted particles exceeded a minimum: $n\ge n_{\text{min}}=10$. A binning of $H=0.1$ was used for the distributions and in the restoration matrices.

Figure 5

Conditional probability density for determining velocity $V^{\prime }$, from decay products in the 1D model with inefficiencies, at three indicated velocities $V$ of the projectile. Those original velocities $V$ are marked with the short vertical dashed lines by the abscissa. Following the strategy of using the remainder of the system as a reference for any single particle, in generating the illustrated density, $N-1=29$ particles can be sampled for an incomplete event and the set analysed if at least $n_{\text{min}}-1=9$ particles pass the detector acceptance.

Figure 6

Conditional probability density for finding velocity $v^{\prime }$ relative to the projectile, for a particle measured in an event in the 1D model with inefficiencies, at three indicated true velocities $v$ relative to the projectile. Those original velocities $v$ are marked with the short vertical dashed lines by the abscissa.

Figure 7

Characteristics of protons and deuterons within the local equilibrium model. The lines represent differential distributions in normalized rapidity, $d{N}_X/d{y}_R$, and symbols represent first-order azimuthal-asymmetry coefficients, $v_{1X}={\langle cos(\varphi -\mathrm{\Phi })\rangle }_X$, all vs $y_R$. Here, the rapidity is in the center of mass and scaled with the rapidity of the beam, $y_R=y/y_{\text{Beam}}$.

Figure 8

Probability density for the azimuthal angle ${\mathrm{\Phi }}^{\prime }$ of the reaction plane in the local equilibrium model, when estimated with the direction of the vector $\overline{\mathbf{q}}$ out of $N-1$ particles in an event, relative to the true angle $\mathrm{\Phi }$. Here, $N$ is the total particle number in an event. The solid (red) line shows the density from a direct simulation in the local equilibrium model and the dashed (blue) line shows the density estimated with the central-limit equation (31). In using the last equation, $\langle q_x\rangle$, ${\sigma }_x^2$, and ${\sigma }_y^2$ in $\alpha$ and $\beta$ were obtained from averages over the event simulations, following the strategy laid out in the text. In addition, the multiplicity $N$, actually varying from an event to event, was replaced with the single average value $\langle N\rangle$ over the simulated events.

Figure 9

Triple differential deuteron distribution at $y_R=0.5$, in the local equilibrium model. The main figure shows the distribution in the reaction plane, as a function of the momentum component in the reaction plane. The inset shows the distribution at $p_{\perp }=1.17\text{GeV}/c$, as a function of azimuthal angle about the reaction plane. In each case, three versions of the distribution are displayed: one determined directly using the known direction of the reaction plane in the modeled events (diamonds), one obtained by estimating the direction of the plane with the vector $\mathbf{Q}$ (crosses), out of momenta of remaining particles in the simulated events, Eq. (24), and one arrived by applying the deblurring to the measured triple-differential distribution, with the RL algorithm applied to Eq. (23) and the blurring function from the central-limit theorem, Eq. (31).

Figure 10

Triple differential distributions in the reaction plane at $y_R=0.5$, for neutrons (circles) and protons (diamonds), from the pBUU simulations of ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ collisions at $270\text{MeV}/\mathrm{nucleon}$ and $b=3.3\text{fm}$. The rapidity $y_R$ is here in the nucleon-nucleon center of mass and normalized to the beam.

Figure 11

Ratio of neutron yield to proton yield in $250\text{MeV}/\mathrm{nucleon}$ ${}^{208}\mathrm{Pb}+{}^{208}\mathrm{Pb}$ collisions at $b=4\text{fm}$, at midrapidity, in the direction perpendicular to the reaction plane, as a function of transverse momentum. The two symbol sets represent the results for two different parametrizations of symmetry energy: diamonds—stiff with $L=105.5\text{MeV}$; circles—moderately soft with $L=38.7\text{MeV}$. The two parametrizations are chosen so that they yield the symmetry energy at $\rho \sim 2{\rho }_0/3$ consistent with the conclusions from binding energies of heavy nuclei and other determinations pertaining to subnormal densities.