Tomography of ultrarelativistic nuclei with polarized photon-gluon collisions

Background and contamination

In addition to the ρ⁰ that we study here, there are several other processes that potentially contaminate the selected sample. The first potential contamination results from misidentified $\mathrm{\varphi }\to K_L^0{K}_S^0$ and the subsequent $K_S^0\to {\mathrm{\pi }}^{+}{\mathrm{\pi }}^{-}$ decay. Such background from $K_S^0$ is removed by requiring M_ππ > 650 MeV. Similarly, the ϕ meson and ω meson can produce π⁺π⁻π⁰ final states, which contaminate the π⁺π⁻ sample because STAR does not detect π⁰. Additional contamination may result from ρ^′ [e.g., ρ(1450) or ρ(1570)] because it can decay to four-pion final states (π⁺π⁻π⁺π⁻ and π⁺π⁻π⁰π⁰). Because STAR does not measure π⁰, any final state with two neutral pions appears as a π⁺π⁻ event. Similarly, any final state with four charged pions may contaminate the π⁺π⁻ signal pairs when two of the charged pions are lost, either due to imperfect reconstruction efficiency or by traveling outside of the acceptance. All of these background processes are reconstructed predominately at low M_ππ and are hence easily removed by analyzing π⁺π⁻ with M_ππ > 650 MeV. For instance, analyses conducted at HERA (29) and Jefferson Laboratory (JLab) (46, 47) rejected these backgrounds by selecting candidates with M_ππ > 600 MeV. Despite the relatively long lifetime of the charged pion (cτ = 7.8 m) in comparison with the radius of STAR’s primary tracking detector (R ≈ 2 m), as many as 5% of all the π⁺π⁻ pairs will have at least one of the pions undergo decay to a muon and a neutrino within that volume. These events, with a final state of π⁺μ⁻X, μ⁺π⁻X, or μ⁺μ⁻X [where X denotes the unmeasured (anti-)neutrino], originate from the same primordial process (e.g., diffractive photoproduction of a ρ⁰ or direct π⁺π⁻ pair) that we study. However, they are undesirable for the measurement in question because the decay randomizes the angle and momentum of the measured final-state charged track. The use of precise particle time-of-flight (TOF) measurement (see Materials and Methods) helps reject pairs in which one (or both) of the charged pions decay, especially for pairs where either daughter track has low transverse momentum (p_T < 300 MeV). The lower invariant mass threshold (M_ππ > 650 MeV) is useful for further reducing the contamination from such events, because pairs with a daughter π^± that decays to a μ^± and an (anti-)neutrino are generally reconstructed with an invariant mass shifted to a smaller value (12).

Data analysis

An angular distribution, sensitive to photon polarization interference effects, is constructed from selected π⁺π⁻ pairs using the ϕ observable (32), defined as

where ${\overrightarrow{p}}_{\mathrm{T}1}$ and ${\overrightarrow{p}}_{\mathrm{T}2}$ are the 2D momentum vectors of the daughter pions in the plane transverse to the beam direction. At the relevant kinematics to this study with $\mid {\overrightarrow{p}}_{\mathrm{T}1}+{\overrightarrow{p}}_{\mathrm{T}2}\mid \ll \mid {\overrightarrow{p}}_{\mathrm{T}1}-{\overrightarrow{p}}_{\mathrm{T}2}\mid$, the ϕ angle from Eq. 1 is equivalent to the angle between the momentum of the parent momentum and the momentum of one of its daughters. On this ground, we use these descriptions interchangeably throughout this article. To remove any effect due to charge-dependent track reconstruction efficiency, ${\overrightarrow{p}}_{\mathrm{T}1}$ and ${\overrightarrow{p}}_{\mathrm{T}2}$ are randomly assigned from the daughter π⁺ and π⁻ in each event. This has the additional effect of naturally eliminating any odd harmonics of the distribution. However, recent calculations have suggested that odd harmonics of the ρ⁰ distribution may be sensitive to Coulomb-nuclear interference effects (48) and will therefore be pursued in detail in future work. The transverse momentum distribution of the parent ρ⁰ can be decomposed into two components: one parallel (P_x) to and one perpendicular (P_y) to the polarization direction

The result of this 2D representation of the ρ⁰ transverse momentum is shown in Fig. 2 (B and C) for Au + Au collisions and in Fig. 2D for U + U collisions. In all three cases, a notable interference pattern is observed in the P_x direction, while no such effect is observed in the P_y direction. As previously mentioned, the observation of an interference pattern along P_x is expected, because the photon polarization is almost perfectly aligned with the impact parameter as shown in Fig. 1C (26).

Figures 3 (A and B) shows the measured $P_{\mathrm{T}}^2\approx \mid t\mid$ distribution for signal π⁺π⁻ pairs from Au + Au (Fig. 3A) and U + U (Fig. 3B) collisions, with two distinct classic diffractive peaks. A clear difference between the shape of the two spectra is visible, especially for |t| < 0.01 (GeV)² due to the different size and shape of the two nuclei. The measured |t| spectra can be described by an empirical model, composed of a coherent contribution characterized by the form factor of a Woods-Saxon distribution and an incoherent contribution characterized by a dipole form factor. The Woods-Saxon distribution, ρ_A(r), is commonly used to describe the density distribution of spherically symmetric heavy nuclei. It has the form

where R is the nuclear radius, a is the surface thickness, and the factor ρ₀ is a normalization factor defined as ρ₀ = 3A/(4πR³), with A being the atomic mass number for the given nucleus. The nuclear form factor can be computed from the density distribution via the Fourier transform, denoted in Eq. 4, using the notation $\mathcal{F}$[…](k) (see Materials and Methods for more details). Following this approach, the best-fit Woods-Saxon radius for both gold and uranium can be extracted from the |t| spectra by fitting them to a function of the form

where A_c is the amount of coherent production and A_i is the amount of incoherent production characterized by a dipole form factor. The value of $Q_0^2$ is fixed to $Q_0^2=0.099$GeV⁻² from previous measurement of the incoherent contribution (3, 49). Figure 3 further shows the |t| distribution for ∣ϕ∣ < π/24 (Fig. 3C) and ∣ϕ − π/2∣ < π/24 (Fig. 3D) to illustrate the profound impact that the interference effect has on the |t| distribution. The extracted radius for the case with maximum interference (ϕ ≃ 0) is larger by 0.81 ± 0.05 (statistical) ± 0.03 (systematic) fm compared to the case with minimum interference (ϕ ≃ π/2).

One can also examine the ϕ distribution itself. The ϕ distribution for Au + Au, U + U, and p + Au events is shown in Fig. 4A for pairs with 650 < M_ππ < 900 MeV and with P_T < 60 MeV. Each distribution is scaled such that the average yield (integrated over ϕ) is unity, to improve comparison. The low P_T range is chosen to select a region where coherent production is dominant (in the case of Au + Au and U + U), with very little production from incoherent photonuclear interactions. The signal in this low P_T range is almost completely unaffected by the STAR detector acceptance and efficiency. In both the Au + Au and U + U datasets, a clear and prominent cos2ϕ modulation is visible, while the p + Au data display an isotropic distribution. The amplitude of the modulation is quantitatively determined by fitting the distribution to a function of the form

where A is the amplitude of the cos2ϕ modulation. The extracted amplitudes are

A_Au+Au =[29.2 ± 0.4 (statistical) ±0.4 (systematic)] × 10⁻² (χ²/ndf = 45/49), A_U+U =[23.7 ± 0.6 (statistical) ±0.4 (systematic)] × 10⁻² (χ²/ndf = 34/49), and A_p+Au =[ − 0.5 ± 1.2 (statistical) ±0.9 (systematic)] × 10⁻² (χ²/ndf = 16/19) for Au + Au, U + U, and p + Au collisions, respectively. The fully corrected 2〈 cos 2ϕ〉 distributions as a function of the ρ⁰ transverse momentum are shown in Fig. 4B for Au + Au, U + U, and p + Au collisions. In both the Au + Au and U + U cases, a prominent maximum is visible at low P_T, which falls rapidly toward zero, with another smaller peak visible at higher P_T. In contrast, the data from p + Au collisions show no structure, being consistent with zero at all P_T. At the highest reported P_T (~240 MeV), the modulation strength is consistent with zero in all three datasets.

To study the impact of photon momentum and interference, we perform another analysis by dividing the |t| distribution into different ϕ bins. In each ϕ bin, the |t| distribution is fit with Eq. 4. Examples of two ϕ slices (ϕ = 0, π/2) of the cross section as a function of |t| are shown in Fig. 3 (C and D), together with the empirical fit (using Eq. 4) and predictions from model II. In principle, Eq. 4 should be able to describe the entire |t| spectra, except the very low-|t| region where detector resolution, photon transverse momentum, and destructive interference effects are dominant. Calculations shown as model I and model II in Fig. 3, which take these effects into account, are able to describe the entire spectrum, including the low |t| dip quite well. Figure 5A presents the radius R extracted by fitting Eq. 4 to data as a function of ϕ. There appears to be a second-order modulation in the resulting R as a function of ϕ. Thence, the equation

is used to extract the minimum radius R₀ at ϕ = ± π/2 with a second-order angular modulation parameter σ_b, where the parameter ϵ_p = 〈 cos 2ϕ〉 = 1/2 is included to account for the resolution of the measurement when using the daughter momentum as a proxy for the polarization direction. The σ_b parameter quantifies the observed strength of the interference effect, taking into account multiple contributions from effects such as the range in impact parameter probed, the finite photon transverse momentum, and the resolution of the polarization measurement. The detailed contribution of each effect to the final value of σ_b can be ascertained from model I and model II but is out of scope for this article. These details, their finer effects on the shape of the |t| distribution, and the final strength of the modulation may be further explored in the future. Using this approach, which gives the value of the true diffractive radius from the nuclear form factor, we obtain R₀ = 6.62 ± 0.03 fm, a = 0.5 ± 0.1 fm, and σ_b = 2.38 ± 0.04 fm for Au + Au and R₀ = 7.37 ± 0.07 fm, a = 0.5 ± 0.1 fm, and σ_b = 1.9 ± 0.1 fm for U + U. Quoted uncertainties are those for Poisson statistics. Systematic uncertainties are computed for the fully corrected values discussed below (see Materials and Methods).

Extracting the nuclear radii

The extracted values of R₀ from the |t| distributions as a function of ϕ should have the least contribution from photon transverse momentum and interference effects. However, there are still contributions from the finite size ρ⁰ wave function (40), with $R_T^{\mathrm{\rho }}=1.03$ fm, and the finite angle between the photon polarization and impact parameter illustrated in Fig. 1C. The depolarization can be estimated from the average angle between the photon and the impact parameter as 1 − P = (R₀/〈b〉)²/4 ≃ 0.03. The resulting true nuclear radii are $R_A^2=R_0^2-R_{\mathrm{T}}^{\mathrm{\rho }2}-(1/P-1){\mathrm{\sigma }}_b^2$, resulting in R_Au = 6.53 ± 0.03 (statistical) ±0.05 (systematic) fm for ¹⁹⁷Au and R_U = 7.29 ± 0.06 (statistical) ±0.05 (systematic) fm for ²³⁸U. The systematic uncertainty is dominated by the difference resulting from the use of different form factors and fit ranges. This also shows that, unlike ρ⁰ photoproduction off nucleons or light nuclei, the details of the ρ⁰ wave function (size) actually play almost no role in determining the nuclear radii. The ratio of these two nuclear radii is 1.12 ± 0.01. Furthermore, these radii are systematically larger than the nuclear charge radii obtained from low-energy electron scattering (51). It should be noted that the strong-interaction nuclear radii have been measured at other facilities at lower energies (7, 8, 52) with diffractive photoproduction of ρ⁰ in photon-nucleus collisions. A scaling of R = (1.12 ± 0.02)A^1/3 was extracted from the experimental data at the German Electron Synchrotron (DESY) (7, 52) with R_Au = 6.45 ± 0.27 fm and R_U = 6.90 ± 0.14 fm, while another experiment at Cornell (8) obtained R_Au = 6.74 ± 0.06 fm. The neutron skin (53), a root mean square difference between the neutral and nuclear charge radii, has been measured in even-even stable nuclei and shown to be around 0.3 fm for ²⁰⁸Pb (54). When the strong-interaction radius is taken as a weighted average of neutral and nuclear charge radii, our measurements of the same quantity yield S = R_n − R_p = 0.17 ± 0.03 (statistical)±0.08 (systematic) fm for ¹⁹⁷Au and 0.44 ± 0.05 (statistical)±0.08 (systematic) fm for ²³⁸U [using previous measurements of R_p (51, 55)]. Our measurement of S for ¹⁹⁷Au seems to follow the trend of world measurements at low energies (56), while that of ²³⁸U is significantly nonzero and indicates a value larger than that expected for neutron skins of similar nuclei [in terms of the fraction of neutron excess (N − Z)/A] (56).

Comparison to theory

We also compare to theoretical calculations from model I and model II that take into account the interacting photon’s transverse linear polarization and the quantum interference effects. Figure 3 shows very good agreement between the data and both models over the entire |t| distribution, except for small disagreements in the structures at higher |t|. Figure 5B shows a comparison of the cos2ϕ modulation between the Au + Au data and two different theoretical calculations from model I and model II. Both of the model calculations are able to reproduce the qualitative features of the data, namely, the prominent peak at low P_T and the second peak at higher P_T. Model I predicts a peak modulation strength and position of the first peak in good agreement with the data, while model II does not reproduce the precisely measured magnitude or structure as observed. While neither model describes the precise location and amplitude of the second peak, both theoretical models indicate that the detailed structure of the modulation is sensitive to the distribution of gluons within the nucleus. Model II used gluon saturation and polarization with an effective nuclear radius of 6.9 fm to reproduce the data shown in Figs. 3 and 5B. Although CGC is an effective theory applicable to the nonperturbative regime of QCD, it usually requires an energy scale above ~1 GeV for reasonable leading-order calculation, as depicted in (32) and in Fig. 1A. Considering that the ρ⁰ mass is ~770 MeV and close to that energy scale, it may not provide the hard scale needed for reliable leading-order model calculation. Additional phenomenological approaches have been demonstrated in the modeling of color transparency (47, 57) and may be needed in this case to achieve the necessary precision for quantitative comparison to data. The full 2D |t| distribution as a function of ϕ explored in this study contains rich information about the nuclear geometry and gluon distribution at small x, which could be further investigated at RHIC, LHC, and EIC.

Origin of the interference

As illustrated in Fig. 1A, the observability of the ρ⁰ spin alignment is a result of the interference of two wave functions from two indistinguishable target and projectile nuclei at the distance of an impact parameter apart from each other. This scheme is very different from most of the fundamental particle interactions, where two wave functions are connected by a mediator or virtual particle in the corresponding Feynman diagram. In this case, the two potential ρ⁰ wave functions only overlap through the decay daughters that propagate to the detector. There are several possible scenarios that would result in interference effects. First, if the phases of these ρ⁰ are random, then this would be similar to the azimuthal Hanbury Brown and Twiss intensity interference effect (37). In this circumstance, both coherent and incoherent ρ⁰ production would create a correlation (32, 41). Second, one can also take an alternative view on this as an example of entanglement enabled intensity interferometry (E²I²) (58) between two nonidentical particles (π⁺ and π⁻). The components in the mass range of the ρ⁰ consist of the ρ⁰ particle and direct π⁺π⁻ (3, 12, 29) that are subject to the same interference effect. In the example of (58), the sources, labeled 1 and 2, are the two gold nuclei each emitting a pair of π⁺π⁻, and detectors A and B measure either π⁺ or π⁻. Because of the entanglement of the π⁺π⁻ at the source, there is a nontrivial interference term, as shown in equation 4 of (58). Last, there exists a third scenario in which the initial ρ⁰ wave functions are locked in phase through phase entanglements of the initial photons and Pomerons. In this case, the interference would only appear in the coherent process and would not produce any interference from the incoherent process (26, 31). In all three cases, the interference occurs over the characteristic distance of the average impact parameter of the collisions, about 20 fm, while the lifetime of the ρ⁰ is only about 1 fm. The decay daughter pions are spin zero particles. Our measurements of nonzero spin alignment {[29.2 ± 0.4 (statistical) ±0.4 (systematic)]% in Au + Au and [23.8 ± 0.6 (statistical) ±0.6 (systematic)]% in U + U} show a definite interference effect due to the nonlocality of the pion wave functions. Through this measurement, we can also set a limit on whether or not the wave functions experience decoherence due to the decay process or other activity in their vicinity. The prediction from model I matches well with data, while the prediction from model II is about 20% above the data, as shown in Fig. 5B. This implies that the coherence is at least 80%.

This measurement of photonuclear production in Au + Au and U + U collisions constitutes the first utilization of the interacting photon’s transverse linear polarization recently demonstrated by STAR in measurements of the γγ → e⁺e⁻ process. We observe a significant cos2ϕ modulation through the ρ⁰ → π⁺π⁻ production channel. The observed amplitude and structure of the cos2ϕ as a function of P_T distribution in Au + Au and U + U collisions is qualitatively consistent with theoretical calculations that include quantum interference effects due to the photon’s transverse linear polarization. We demonstrate that the cos2ϕ modulation is absent in p + Au collisions, which are expected to be free from interference effects, because only one amplitude predominantly contributes in those interactions. Independent of theoretical models, we quantify the interference patterns observed in Au + Au and U + U measurements by studying the polarization dependence of the |t| distribution in two dimensions. We are able to remove the effects of photon transverse momentum and two-source interference to extract the nuclear radius of gold and uranium. The resulting radii are systematically larger than the measured nuclear charge radii at lower energies. It further demonstrates that this spin-induced OAM interferometry offers a new avenue for studying nuclear geometry and gluon distribution within large nuclei at a quantitative level.

Signal selection

In total, ~41 × 10⁶ Au + Au collisions, ~23 × 10⁶ U + U collisions, and ~1.8 × 10⁸ p + Au collisions were recorded using the trigger system. From the triggered events, candidate exclusive diffractive events were further required to fall within ±100 cm of the center of STAR, contain an interaction vertex formed from exactly two oppositely charged particles, and have no more than two additional background charged particles elsewhere in STAR. The reconstruction and identification of π⁺π⁻ pairs (3, 22) is accomplished using the time projection chamber (TPC) and the TOF detectors. By measuring the track curvature, the TPC is capable of measuring the transverse momenta (p_T) of charged particles bent by a uniform 0.5-T solenoidal magnetic field. The TPC has acceptance for p_T > 100 MeV with a resolution σ_pT ∕ p_T better than 1% for tracks with momenta below 1 GeV. The 3D tracks reconstructed by the TPC are required to have at least 15 hit points out of a possible maximum of 45 to ensure good quality. In addition to tracking information, the TPC provides particle identification information via measurement of the mean specific ionization energy loss per unit length (〈dE/dx〉). The measured ⟨dE/dx⟩ can be expressed in terms of the measurement resolution as nσ, where nσ is the number of SDs from the expected ⟨dE/dx⟩ for a pion. The nσ measurement of each track is used to construct the ${\mathrm{\chi }}_{\mathrm{\pi }\mathrm{\pi }}^2$ for a π⁺π⁻ pair hypothesis, where ${\mathrm{\chi }}_{\mathrm{\pi }\mathrm{\pi }}^2=n{\mathrm{\sigma }}_1^2+n{\mathrm{\sigma }}_2^2$ and the subscripts 1 and 2 denote the two tracks in the pair.

While the ${\mathrm{\chi }}_{\mathrm{\pi }\mathrm{\pi }}^2$ variable alone can be used to select a relatively pure set of π⁺π⁻ pairs, those pairs may suffer from a small amount of contamination from e⁺e⁻, pp, and K⁺K⁻ pairs when the daughter tracks have momenta above 200 MeV. To further reject these possible types of contamination, the TOF detector, which provides precise timing measurement of charged particles with a time resolution of ~ 75 ps, can be used. The TOF identification technique is based on the measured time difference between the two tracks in a pair (ΔTOF = t₂ − t₁). Because the TPC provides measurement of the daughter particle momenta (p₁, p₂) and path lengths (s₁, s₂), the expected time difference for a π⁺π⁻ pair can be expressed as $\mathrm{\Delta }{\mathrm{TOF}}^E=t_2^E-t_1^E$ where $t_{1,2}^E=\sqrt{S_{1,2}^2/c^2\times (1+m_{\mathrm{\pi }}^2/p_{1,2}^2)}$ (where m_π is the mass of the charged pion ≈139 MeV/c²). Last, the double difference, ΔΔTOF = ΔTOF − ΔTOF^E, can be used to select π⁺π⁻ pairs while also rejecting contamination from e⁺e⁻, pp, and K⁺K⁻ pairs. Last, for the Au + Au and U + U datasets, candidate π⁺π⁻ pairs are selected from those pairs with ${\mathrm{\chi }}_{\mathrm{\pi }\mathrm{\pi }}^2<8$ and ΔΔTOF < 750 ps. Because the TOF information was not available in the p + Au dataset, only the ${\mathrm{\chi }}_{\mathrm{\pi }\mathrm{\pi }}^2<8$ condition is required. The effect of potential contamination from e⁺e⁻, pp, and K⁺K⁻ pairs in the p + Au dataset is found to be negligible and is included as a systematic uncertainty.

Detector acceptance and efficiency corrections

Instead of fitting to the ϕ distribution to extract the modulation strength, the cos 2ϕ distribution can be directly measured with respect to the various pair kinematics by taking the moments of the distribution. The strength of the modulation can then be determined directly as 2⟨cos 2ϕ⟩, without the need for a fit, where the ⟨…⟩ indicate the average value defined as

where α(ϕ) is the true modulation strength and w(ϕ) is the bias caused by imperfect detector efficiency and incomplete acceptance. Without loss of generality, both the signal and the detector effects can be expressed in terms of Fourier series

In general, the α and w are also functions of the pair kinematics, e.g., α = α(ϕ, p_T, …). For an ideal detector, there would be no bias [i.e., w(ϕ) = 1], and the measured distribution would directly reflect the true distribution α(ϕ). Although the STAR detector has full 2π coverage in azimuth, the limited acceptance in p_T (p_T > 200 MeV) leads to a bias [w(ϕ) ≠ 1], and therefore, the measured ϕ distribution must be corrected to remove this detector effect. This acceptance bias is estimated directly from the data by constructing pairs with π⁺ and π⁻ tracks from different events (so-called mixed-event method). In this way, the detailed acceptance and efficiency of the STAR detector can be reproduced to determine the acceptance effects with high precision. Using this method, the bias on the cos2ϕ modulation (w2) is estimated and found to be small ( ≪ 1 % ) for small pair P_T but grows larger with increasing P_T to a value of ≈ − 10% at a P_T of 200 MeV. In principle, the detector bias can also lead to mixing between the various Fourier components. On the other hand, in practice, this is not an issue, because all other components (w₁, w₃, w₄, …) are found to be consistent with zero in the P_T range of interest. In that case, Eq. 7 can be evaluated by substituting in the definitions of α and w from Eq. 8 and setting all w_n to zero except for n = 2. Evaluating and solving for the true signal yields

where γ₂ is the measured distribution, i.e., γ₂ = 2⟨ cos 2ϕ⟩_measured.

Systematic uncertainties

In addition to the statistical uncertainties reported in Figs. 2 and 3, we also evaluate possible sources of systematic bias for the ϕ measurement. The dominant sources of systematic uncertainty are track selection and quality, pion identification, pion-to-muon decay, detector efficiency effects, and kinematic acceptance corrections. At low P_T, all of these sources are found to be smaller than the statistical uncertainty leading to point-to-point variations at the level of 3% or less. At higher P_T, the pion-to-muon decay process is problematic because it potentially produces a self-correlation between $({\overrightarrow{p}}_1+{\overrightarrow{p}}_2)$ and $({\overrightarrow{p}}_1-{\overrightarrow{p}}_2)$. Because the muon tracks have approximately the correct azimuthal angle but are reconstructed with the wrong mass (charged pion mass instead of muon mass), the pairs with one or more muons result in a ϕ distribution roughly peaked at 0 and ±π. This peak structure appears as anomalous components in a Fourier expansion of the distribution. The effect of muon contamination is estimated using the Geometry and Tracking 3 (GEANT3) simulation of π⁺π⁻ pairs through the STAR detector. In the measured mass range, the maximum effect is ≈2% for P_T > 60 MeV and is subtracted from the measured signal with the possible variation in the muon decay contribution taken as a systematic uncertainty. The other sources of uncertainty all contribute approximately equally for P_T > 60 MeV, leading to a combined one sigma uncertainty on the amplitude of the cos2ϕ modulation of ~3 to 6%.

To study the systematical uncertainty of the choice of a form factor to extract the nuclear radius, we perform another analysis by dividing the |t| distribution into different ϕ bins. In each ϕ bin, the |t| distribution is fit with

where A_c is the amount of coherent production with an exponential shape and A_i is the amount of incoherent production characterized by a dipole form factor. The value of $Q_0^2$ is fixed to $Q_0^2=0.099{\mathrm{GeV}}^{-2}$ from previous measurement of the incoherent contribution (3). The resulting b_T as a function of ϕ is fitted with an empirical function to extract the nuclear radius at ϕ = π ∕ 2. Using an empirical second-order fit function of

we obtained R₀ = 6.72 ± 0.02 fm for Au + Au and R₀ = 7.37 ± 0.03 fm for U + U and corresponding σ_b = 2.90 ± 0.05 fm and 2.9 ± 0.1 fm. The fit to the Woods-Saxon form factor is superior because (i) a much larger fit range is possible (which includes the second diffractive peak) and (ii) Eq. 4 describes the non-Gaussian shape better than Eq. 10. For these reasons, the exponential shape is not used to characterize the coherent contribution but as an estimate of the systematical uncertainty (±0.05 fm) for the form factor choice.

Tabulated results

We compile all the results from the measurements of radii in Table 1 and on cos2ϕ to compare with models and nuclear charge radius in Table 2.

Woods-Saxon form factor calculation

As mentioned, in several cases, the dN/d ∣ t∣ distributions are fit to Eq. 4, which characterizes the coherent contribution via the form factor of a Woods-Saxon distribution. In general, the form factor can be computed as the Fourier transform of the density distribution (ρ_A)

where Z is the charge of the nucleus, e is the fundamental unit of charge, $\overrightarrow{k}$ is the momentum transfer, and $\overrightarrow{r}$ is the position vector. For a spherically symmetric Woods-Saxon distribution, ρ_A is a function of the radial distance only and is defined as

where R is the radial distance from the center of the distribution, R_A is the nuclear radius, and a is the surface thickness of the distribution. The factor ρ₀ is a normalization factor defined as ${\mathrm{\rho }}_0=3A/(4\mathrm{\pi }{R}_A^3)$, with A being the atomic mass number for the given nucleus. In the case of a spherically symmetric distribution, Eq. 12 can be further simplified to

Because Eq. 13 does not have an analytic Fourier transform, some authors choose to use an approximately equivalent analytic formula [see, for instance, equation 10 of (18)]. Alternatively, the form factor for the Woods-Saxon distribution can be computed numerically, as is done in this analysis. We use the QAG adaptive Monte Carlo integration technique (59) for computing the form factor, ensuring that the numerical accuracy is better than 0.1%.

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