Connecting Heaven and Earth: PREX and CREX Tell us About Neutron Stars

The Lead Radius and Calcium Radius Experiments (PREX and CREX, respectively) utilized the weak force, one of the fundamental forces of nature, to probe the neutron skin thickness in lead-208 and calcium-48 nuclei. These measurements offer insights into the intricacies of nuclear structure of Earth-based nuclei, as well as neutron stars. Despite employing similar techniques, the experiments produced divergent outcomes: a thick neutron skin for lead-208 and an unexpectedly thin skin for calcium-48. This disparity underscores the complexity of nuclear structure and presents a challenge to current theoretical frameworks. In this article, we summarize these results and discuss their implications for the equation of state of neutron-rich matter used to predict the structure and composition of neutron stars.

Overview

Most of the constraints on theoretical models of nuclear structure arise from the assumption that protons and neutrons are essentially identical particles (isoscalar), whereas models treating them as independent particles (isovector) predict properties with less stringent constraints. So, while charge radii and binding energies of nuclei are relatively well described, quantities that rely on the differences between protons and neutrons and how they interact can take on wildly different values. One can, for example, compare the matter and charge radii of nuclei to learn about the differences between protons and neutrons, because both protons and neutrons have mass, but only protons have electric charge. However, by measuring a property of neutrons that is more orthogonal to that of protons, we can gain a better understanding of properties that depend on differences between the behavior of neutrons and protons. One such quantity is the weak charge, where neutrons have a maximal value of about −1, whereas protons have nearly a zero value.

The symmetry energy quantifies the energy cost for converting protons to neutrons in symmetric nuclear matter, highlighting the distinction between neutrons and protons. There are several ways to probe isovector properties related to the symmetry energy. One way is to study infinite nuclear matter under extreme conditions, such as that found in neutron star interiors. Measurements of neutron star structure, like maximum mass, radius, and moments of inertia, however, can be difficult to interpret. Recent multimessenger detections, including relativistic Shapiro delay, X-ray observations of isolated neutron stars, gravitational waves, and simultaneous gamma rays from binary neutron star mergers provide a whole new window into the universe, allowing us to learn more about dense matter in neutron stars. In particular, it was possible to measure the maximum observed neutron star mass [1], simultaneous mass and radius measurements [2], as well as their so-called tidal polarizabilities [3]. This last describes how much a neutron star is mass-polarized by tidal forces of the companion when two neutron stars are in a binary system. Alternatively, we can study normal nuclear matter—albeit neutron-rich—here on Earth. The PREX [4, 5] and CREX [6] experiments, which ran in Hall A of the Thomas Jefferson National Accelerator Facility (JLab), in Newport News, Virginia, aimed to do just that.

PREX and CREX are long-envisioned experiments which used the second weakest fundamental force in nature, the weak force, to measure the neutron skin in ²⁰⁸Pb and ⁴⁸Ca. The weak force is the only known force that violates parity conservation, challenging the idea that there should be no preferred direction in space. Because the weak charge of the proton is $Q_W^p=1-4{\mathrm{\text{sin}}}^2{\theta }_W\approx 0.07$, where ${\theta }_W$ is the Weinberg angle, and the weak charge of the neutron is $Q_W^n\approx 0.99$, we can learn more about the neutron distributions in the nucleus by studying the exchange of the neutral

Z⁰ boson, one of the exchange particles in the weak interaction. Note that the proton distribution within nuclei has been accurately mapped since the advent of powerful electron accelerators in the 1950s by exchanging photons. Conversely, the neutron distribution within nuclei remained relatively less understood due to experimental limitations involving strongly interacting probes, such as pions and protons, which introduce numerous uncertainties. In the parity-violating experiment, a relatively clean measurement is achieved by measuring the asymmetry in the cross-sections of the scattering of positive and negative-helicity electrons (1) $$A=\frac{{\sigma }_{+}-{\sigma }_{-}}{{\sigma }_{+}+{\sigma }_{-}}$$(1) from the nucleus. The first run of PREX [4] measured a parity-violating asymmetry A_PV = 656 ± 62 ppm, corresponding to a neutron skin thickness of $R_n-R_p={0.33}_{-0.16}^{+0.18}$ fm. The central value of this measurement implied that neutron stars could have much larger radii than those already observed (see Figure 1). Conversely, the neutron star radius measurements at the time implied a neutron skin for ²⁰⁸Pb of only 0.15 ± 0.02 fm, with these measurements marginally in agreement [8]. The PREX uncertainty was 3$\times$ larger than originally proposed, necessitating a second run, PREX-2 [5], which along with CREX [6], is the subject of this article.

Figure 1. Mass-versus-Radius relation predicted by several nuclear models that are consistent with the value of the neutron skin measured by PREX. The figure is taken from Ref. [7].

Same Physics, a Whole Lot Closer to Home

As famously proposed by Landau in 1931, neutron stars are essentially like giant nuclei. The same particles and interactions that occur in the extreme environment of a neutron star have to govern the physics of nuclei available for Earth-based experiments. Most of the known properties—such as the density distribution of the nucleus, described by a form factor in momentum space—relate to the electromagnetic charges, which are carried only by the protons and not the neutrons. However, neutrons have a maximal weak charge, while the proton’s weak charge is very small. Measuring the weak interactions of electrons with nuclei can tell us about the neutron distribution in heavy nuclei.

Unfortunately, the weak interaction (exchange of a Z⁰ boson) is swamped by the electromagnetic (EM) interaction (exchange of a photon) in the electron–nucleus scattering process. Unlike charged weak boson exchange, the neutral weak interaction is indistinguishable from the exchange of a photon because the interacting particles do not change the sign of their charges. Luckily, however, the weak force is the only known force that violates parity. This provides us with a way to distinguish the weak neutral current interactions from EM interactions.

The neutron skin is the difference in the root-mean-squared radii of neutrons and protons that can be determined from the neutron and proton density distributions. It is the uppermost dilute layer of the nucleus populated primarily by neutrons. For a neutron-rich nucleus, such as ²⁰⁸Pb, the thickness of the neutron skin is determined by a tug-of-war between the surface tension and the difference between the symmetry energy at saturation density and at the lower surface density [9], known as the symmetry pressure, often denoted by the symbol L. The larger the symmetry pressure is, the thicker the neutron skin, and vice versa. It is the same pressure, although at a different density, that is responsible for the size of a neutron star. Thus, a precise measurement of the neutron skin would imply an accurate prediction for the radius of a neutron star. Figure 2 shows the weak distribution measured in PREX 2 with the well-known charge distribution and the total baryon density.

Figure 2. Plot of the baryon density from the weak and charge densities, using the well-known proton density and the combined results of the two runs of PREX [5].

The parity-violating asymmetry is defined as (2) $$A_{\mathit{\text{PV}}}\approx \frac{G_F{Q}^2}{2\pi \alpha \sqrt{2}}\frac{F_w\left(Q^2\right)}{F_{\mathit{\text{ch}}}\left(Q^2\right)}$$(2) where Q² is the four-momentum transfer squared, G_F is the Fermi constant, α is the fine structure constant, and F_w and F_ch are weak and charge form factors, respectively. Given that the charge form factors $F_{\mathit{\text{ch}}}\left(Q^2\right)$ are very well known, one can use the Standard Model value of sin² θ_W to obtain the weak form factor $F_w\left(Q^2\right)$. In order to go from the parity-violating asymmetry to the weak density, one must first extract the neutral form factor at a specific momentum transfer, q = Q². For this, it is necessary to take into account both the Coulomb distortion calculations, as well as small corrections due to the neutron and strange electric form factors and meson-exchange corrections. Because the weak form factor is the Fourier transform of the weak charge density, using the mean-field models one can extract the full neutron distribution within the nucleus. In combination with the well-known proton distributions, this allows us to measure the neutron skin. The knowledge of the neutron skin can be used to extract the symmetry pressure, which is highly correlated with both the neutron skin of ²⁰⁸Pb [9, 10] and a host of neutron star observables [11].

The Whole Accelerator Is Part of the Experiment

Jefferson Lab has a continuous electron beam accelerator that can provide a high-quality, highly polarized beam to four experimental halls. The PREX and CREX experiments ran in Hall A. For a detailed description of the experimental apparatus, see Refs. [12, 13]. PREX and CREX had similar experimental designs, with the notable difference being the target material—isotopically pure ²⁰⁸Pb and ⁴⁸Ca, respectively. These are stable, doubly magic nuclei with neutron excess and the ability to form solid targets that would not melt due to the incident beam. The experiments were also optimized at slightly different kinematics, so the beam energy and four-momentum transfer are different (E ≈ 1 GeV and q = 78.5 MeV/c for PREX, and E ≈ 2 GeV and q = 172.3 MeV/c for CREX).Essentially all parity-violating electron scattering (PVES) experiments, including PREX and CREX, use a strained GaAs cathode, longitudinally polarized beam, fixed target, spectrometer system, and integrating detectors to measure the light yield in the two helicity states, and polarimetry to determine the degree of polarization of the beam. Hall A has two high-resolution spectrometer (HRS) arms that can be rotated around a central pivot. A septum magnet pre-bent the electrons before they entered the spectrometer arms in order for the experiments to reach the ≤5° scattering angles needed (see Figure 3). In the detector hut at the end of each HRS was placed a set of thin and thick quartz pieces, precision placement of which ensured that the elastic peak was centered on the quartz while minimizing the contribution from inelastic states. The light yield in each helicity state is recorded and used to form the asymmetries. The measured asymmetry is corrected for dilutions from backgrounds, helicity correlated beam asymmetries, and overall beam polarization.

Figure 3. Schematic of the HRS and the plot of the rate at the focal plane along the focusing direction.

Implications for Neutron Stars

A canonical neutron star is approximately 57 orders of magnitude larger in baryon number than a typical atomic nucleus. While it is commonly perceived that a neutron star is composed primarily of closely packed neutrons, its actual composition is far more intricate and captivating. Although the original concept of neutron stars as unheimliche Sterne (weird stars) dates back to 1931 (Lev Landau), they were first discovered in 1967 by Jocelyn Bell Burnell as a radio pulsar. With over 3,500 stars discovered, they are incredibly compact, with radii ranging from 11 to 14 kilometers, yet possessing masses of up to slightly over twice that of the Sun. General Relativity, as demonstrated originally in 1939 by Oppenheimer and Volkoff, plays a crucial role in predicting their structure.

Nuclear physics also plays a significant role in elucidating the structure and composition of neutron stars. The pressure supporting neutron stars against gravitational collapse is determined solely by nuclear interactions. Under the assumption that neutron stars are charge-neutral, the standard model posits that they are made up of neutrons, protons, electrons, and muons in chemical equilibrium. The relative proportions of these constituents are determined by the equation of state of neutron-rich matter, which describes the relationship between energy density and pressure in nuclear matter. Consequently, despite being vastly different in size by 18 orders of magnitude, the determination of their radii relies on the same physics that governs the neutron skin thickness of neutron-rich nuclei.

Observationally, one can estimate the radius of a neutron star using the black body radiation emitted from its surface. Although determining neutron star radii via photometric methods is challenging due to significant systematic uncertainties stemming from distance measurements and distortions to the black body spectrum caused by a thin stellar atmosphere, nevertheless rapid progress is being made on this front through a better understanding of these uncertainties, and the implementation of robust statistical methods [2].

Alternatively, the moment of inertia of a neutron star, which scales with the square of its radius, can also be measured in the future through timing observations, offering a greater sensitivity to the nuclear equation of state [15]. Furthermore, recent advancements in gravitational wave astronomy have introduced a powerful new method for measuring the bulk properties of neutron stars. The first direct detection of gravitational waves from a binary neutron star merger, known as GW170817, by the LIGO and Virgo collaborations in 2017 [3], has provided fundamental insights into the nature of dense matter.

In particular, the gravitational wave signals contained information on the tidal polarizability of neutron stars, denoted by Λ. Tidal polarizability scales as the fifth power of the stellar radius and is therefore highly sensitive to symmetry pressure. The constraints on the neutron skin provided by the PREX experiments aligned with the values obtained through this observation (see Figure 4). In particular, a combined analysis showed overly large neutron skins and neutron stars are ruled out [14].

Figure 4. Neutron star observables as a function of neutron skin thickness in ²⁰⁸ Pb as predicted by the set of energy density functionals. The figure is taken from Ref. [14].

And the Plot (If Not the Neutron Skin) Thickens

The Coulomb corrections are well understood, and the other corrections for PREX are also well understood. But CREX has a bigger surface to volume ratio, and was also forced to run at a nonoptimal Q² where the model differences in the charge and weak form factors are smaller, making it more difficult to distinguish between models. In addition, CREX is accurate enough to be sensitive to spin-orbit corrections as well as meson exchange currents in ⁴⁸Ca. PREX-2 measured a parity-violating asymmetry (3) $$A_{\text{PV}}=550\pm 18\text{ppm}$$(3) at a momentum transfer squared of Q² = 0.00616 GeV²/c², corresponding to a neutron skin thickness of (4) $$R_{\mathrm{n}}-R_{\mathrm{p}}=0.278\pm 0.08\text{fm}$$(4) confirming the relatively large neutron skin measured in the first run, with a much smaller uncertainty. The central baryon density determined from the measurement of the neutron density distribution combined with the well-known proton density is (5) $${\rho }_b=0.1482\pm 0.0040{\text{fm}}^{-3}$$(5)

CREX measured a parity-violating asymmetry (6) $$A_{\text{PV}}=2658.6\pm 113.2\text{ppb}$$(6) at Q² = 0.0297 GeV²/c², corresponding to (with theory correction estimates) a surprisingly small neutron skin thickness of (7) $$R_n-R_p=0.121\pm 0.035\text{fm}$$(7) could go after a surprisingly small neutron skin thickness of [6]. This implied that the symmetry pressure should be much smaller than previously predicted. However, this interpretation is highly model-dependent and requires more accurate spin-orbit and meson exchange current corrections. It became immediately challenging for many energy density functionals (EDFs) to reproduce both the CREX result of a thin skin in ⁴⁸Ca and the PREX result of a relatively thick skin in ²⁰⁸Pb. <end > Thus, the physics output from PREX/CREX—the contrasting neutron skin thicknesses observed in two nuclei—highlights the nuanced complexities of nuclear structure and the ongoing endeavor to reconcile these divergent experimental observations within theoretical frameworks.

Conclusion

Of course, while these results are exciting, they are also perplexing. Not only do the astrophysical observations and PREX results show only marginal agreement, but the PREX-2 and CREX results, measured using the same methods, also seem to exhibit discrepancies. The nuclear community is actively discussing the implications of this interplay between ²⁰⁸Pb and ⁴⁸Ca, which could underscore rich dynamics. Full implications for the symmetry pressure, L, will require continued collaboration between various theoretical and experimental groups. The CREX result is strongly inconsistent with predictions of a very thick skin as measured by PREX, and is more consistent with a thin neutron skin prediction (e.g., coupled cluster calculations). So far, many nuclear EDFs failed to reproduce both results. Some attempts were recently made to reconcile these seemingly contradictory findings by proposing a modified EDF, particularly focusing on constraining the previously unknown higher-order coefficients of the nuclear symmetry energy (see Figure 5 and Ref. [16]). The future Mainz Radius experiment at Mainz may be able to help resolve some of these discrepancies [17].

Figure 5. Predictions for the weak form factor of ²⁰⁸Pb and ⁴⁸Ca for relativistic and nonrelativistic energy density functionals. The blue ellipses represent joint PREX-2 and CREX 67% and 90% probability contours (the figure is taken from Ref. [16]).

Juliette Mammei

F. J. Fattoyev

Disclosure statement

No potential conflict of interest was reported by the author(s).