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Measurement of energy-level splitting from Charge-Symmetry Breaking in A = 4 mirror hypernuclei

Using the STAR experiment at RHIC, researchers measured the binding energies of A=4 mirror hypernuclei (⁴ΛH and ⁴ΛHe) in Au+Au collisions to precisely quantify charge-symmetry breaking in Λ–nucleon interactions, revealing that the energy-level splitting effects in ground and excited states are comparable in magnitude but opposite in sign.

Original authors: Tianhao Shao

Published 2026-08-26

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CC BY 4.0

Original authors: Tianhao Shao

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the heart of every atom lies a nucleus, a tightly packed cluster of protons and neutrons held together by the strong nuclear force. For decades, physicists have relied on a guiding principle called charge symmetry, which suggests that if you were to swap every proton in a nucleus with a neutron, and every neutron with a proton, the resulting atom would behave almost exactly the same, provided you could turn off the electrical repulsion between protons. This symmetry is a cornerstone of our understanding of how matter holds together. However, nature is rarely perfect. When scientists look closely at pairs of mirror nuclei—atoms that are identical except for this swapped proton-neutron count—they find tiny differences in how tightly their parts are bound. These small deviations, known as charge-symmetry breaking, offer a rare glimpse into the subtle ways the fundamental forces of the universe distinguish between particles that otherwise seem identical. Understanding these differences is not just an academic exercise; it helps scientists refine the rules that govern the behavior of matter under extreme conditions, such as the crushing density found inside neutron stars.

A unique laboratory for testing these rules involves hypernuclei, which are exotic atoms containing not just protons and neutrons, but also a particle called a lambda hyperon. This particle is heavier than a neutron and carries no electric charge, making it an ideal probe for studying how the strong force acts without the interference of electrical repulsion. The focus of a recent study by the STAR Collaboration at the Relativistic Heavy Ion Collider is a specific pair of these mirror hypernuclei, both with a total mass number of four. One contains a core of three neutrons and one proton, while the other has three protons and one neutron. By measuring the binding energy—the amount of energy required to pull the lambda hyperon out of the nucleus—researchers can determine if the strong force treats these two mirror images exactly the same. If charge symmetry were perfect, the binding energies would be identical. Any difference would reveal the strength of the symmetry breaking.

To capture these fleeting particles, the researchers directed a beam of gold nuclei at a stationary gold target, creating collisions at an energy of three billion electron volts. These high-energy crashes produced a shower of new particles, including the rare mirror hypernuclei. The team then tracked the decay of these hypernuclei as they broke apart into lighter particles. By measuring the paths and speeds of the resulting fragments, they reconstructed the mass of the original hypernuclei with extreme precision. This process allowed them to calculate the binding energy for the ground states of both the hydrogen-like and helium-like mirror hypernuclei. The results showed a clear difference: the binding energy for the helium-like version was higher than that of the hydrogen-like version by approximately 0.15 million electron volts. This difference is a direct measurement of charge-symmetry breaking in the ground state of these systems.

The story becomes even more intriguing when the researchers considered the excited states of these same hypernuclei. Previous experiments had measured the energy of gamma rays emitted when these excited states dropped down to their ground states. By combining those older measurements with their new, highly precise data on the ground states, the team calculated the binding energy difference for the excited states. They found a result that was strikingly different: the difference was negative, roughly -0.17 million electron volts. This means that while the helium-like hypernucleus was more tightly bound in its ground state, it was less tightly bound in its excited state compared to its mirror partner. The magnitude of the difference was nearly the same in both cases, but the sign was opposite.

This finding provides a precise confirmation of a specific pattern in how charge symmetry breaks. The data suggests that the energy levels of these mirror systems split in a symmetric way, with the ground state and excited state shifting in opposite directions. This behavior aligns with theoretical expectations for a system where a symmetry is broken, but it contradicts several previous theoretical models that predicted different outcomes. Some earlier theories suggested the effect would be much smaller in the excited state, or that the signs would not be opposite. The new measurements, which are significantly more precise than previous attempts, rule out those older predictions and provide a solid benchmark for future theories.

The implications of this work extend beyond the specific atoms studied. The measurements offer a new, tight constraint on the interactions between hyperons and nucleons, which are critical for understanding the equation of state for neutron stars. Since the interior of a neutron star is so dense that it may contain hyperons, knowing exactly how these particles interact helps physicists predict the maximum mass a neutron star can have before collapsing. By pinning down the strength of charge-symmetry breaking in these light systems, the researchers have reduced the uncertainty in models of dense matter. The study does not claim to have solved the entire puzzle of hypernuclear interactions, but it has removed a major layer of ambiguity, offering a clearer path forward for understanding the fundamental forces that shape the universe.

Technical Summary: Measurement of Energy-Level Splitting from Charge-Symmetry Breaking in A = 4 Mirror Hypernuclei

Problem Statement
Charge symmetry, a fundamental feature of the strong interaction, posits that the nuclear force remains invariant under the interchange of protons and neutrons. Deviations from this symmetry, known as Charge-Symmetry Breaking (CSB), offer a direct probe into the underlying mechanisms of nucleon interactions, including quark mass differences and meson mixing. While CSB in nucleon-nucleon (NN) interactions is well-documented (e.g., in the $^3$ H- $^3$ He system), the magnitude and spin dependence of CSB in the $\Lambda$ -nucleon ( $\Lambda N$ ) interaction remain a longstanding puzzle. Specifically, the $\Lambda$ binding energy ( $B_\Lambda$ ) difference between the mirror hypernuclei $^4_\Lambda$ H and $^4_\Lambda$ He serves as a critical observable. Previous measurements, ranging from 1970s nuclear emulsion experiments to recent high-precision $\gamma$ -ray spectroscopy, have yielded conflicting or imprecise results regarding the CSB effect in both ground ( $0^+$ ) and excited ( $1^+$ ) states. Early emulsion data suggested a significant positive splitting in the ground state, while later $\gamma$ -ray measurements implied a much smaller or even negative splitting in the excited state, hinting at a spin-dependent CSB mechanism that theoretical models struggle to reconcile.

Methodology
This study presents a high-precision measurement utilizing data from the STAR experiment at the Relativistic Heavy Ion Collider (RHIC). The analysis is based on a dataset of approximately 2 billion Au+Au collision events collected at a nucleon-nucleon center-of-mass energy of $\sqrt{s_{NN}} = 3$ GeV in 2021. The experimental setup involved a stationary gold target (0.25 mm thickness) positioned within the beam pipe, approximately 2 meters upstream of the STAR Time Projection Chamber (TPC).

The reconstruction strategy focused on the decay channels of the mirror hypernuclei:

$^4_\Lambda$ H: Reconstructed via the two-body decay $^4_\Lambda$ H $\to$ $^4$ He + $\pi^-$ .
$^4_\Lambda$ He: Reconstructed via the three-body decay $^4_\Lambda$ He $\to$ $^3$ He + $p$ + $\pi^-$ .

Particle identification was achieved primarily through the measurement of mean ionization energy loss ( $\langle dE/dx \rangle$ ) in the TPC, supplemented by Time-of-Flight (TOF) detector data where available. A solenoidal magnetic field (0.5 T) facilitated momentum determination via track curvature. The decay topology was reconstructed using the KFParticle package, employing Kalman filtering to calculate the $\chi^2$ between tracks and the collision vertex.

To ensure precision, several rigorous corrections were applied:

Mass Scale Calibration: A scale factor of 99.8% was applied to decay daughter momenta, determined by matching the reconstructed mass of $K^0_S \to \pi^+ \pi^-$ to the Particle Data Group (PDG) value.
Energy Loss Corrections: While pion mass corrections were standard, specific corrections for heavier particles ( $^4$ He, $^3$ He, and protons) were evaluated using Monte Carlo simulations of the STAR detector.
Background Subtraction: Combinatorial backgrounds were estimated by randomly rotating the tracks of $^4$ He or $^3$ He particles to disrupt inter-particle correlations while preserving kinematic distributions.
Systematic Reduction: Momentum corrections were performed as a function of particle rapidity and momentum, reducing systematic uncertainties compared to previous STAR measurements.

The invariant mass distributions were fitted using a Gaussian function (dominated by detector resolution) plus a linear function for residual background. The $\Lambda$ binding energies were calculated using the measured hypernucleus masses, the PDG $\Lambda$ mass, and CODATA nucleon core masses.

Key Results
The analysis yielded precise measurements of the $\Lambda$ binding energies for the ground states of the mirror hypernuclei:

$B_\Lambda(^4_\Lambda\text{H}) = 2.24 \pm 0.02 \text{ (stat.)} \pm 0.04 \text{ (syst.) MeV}$
$B_\Lambda(^4_\Lambda\text{He}) = 2.39 \pm 0.05 \text{ (stat.)} \pm 0.05 \text{ (syst.) MeV}$

These results lead to a ground-state charge-symmetry breaking difference of:

$\Delta B^4_\Lambda(0^+) = 0.15 \pm 0.05 \text{ (stat.)} \pm 0.04 \text{ (syst.) MeV}$

By combining these new ground-state measurements with previously established $\gamma$ -ray transition energies for the excited states ( $E_\gamma(^4_\Lambda\text{H}) \approx 1.09$ MeV and $E_\gamma(^4_\Lambda\text{He}) \approx 1.41$ MeV), the excited-state CSB difference was determined to be:

$\Delta B^4_\Lambda(1^+) = -0.17 \pm 0.05 \text{ (stat.)} \pm 0.04 \text{ (syst.) MeV}$

The total uncertainty in these results represents a reduction of approximately 63% compared to the previous STAR measurement using 2018 data. The ground-state result deviates from a 1973 emulsion measurement by approximately $3\sigma$ , though it is consistent with all post-1973 measurements while offering superior precision.

Significance and Claims
The paper claims that these measurements provide a precise determination of CSB in the hypernuclear system, establishing a critical pattern: the $\Lambda$ binding energy differences in the ground and excited states are comparable in magnitude but opposite in sign ( $\Delta B^4_\Lambda(1^+) \approx -\Delta B^4_\Lambda(0^+) < 0$ ).

The author interprets this symmetric energy-level splitting as a manifestation of CSB in a two-level quantum system, analogous to a double-well potential where an off-diagonal perturbation induces shifts of opposite signs. This finding challenges several existing theoretical models:

Models based on $\Lambda$ - $\Sigma$ mixing (e.g., NSC97e/f) that predict large ground-state splitting with the same sign as the excited state fail to describe the data.
Ab-initio no-core shell model calculations (NCSM) with Bonn-Jülich potentials and a CSB $\Lambda$ - $\Sigma^0$ mixing vertex quantitatively describe the measurements. However, updated calculations incorporating the Dalitz-von Hippel one-pion exchange mechanism underestimate the excited-state splitting.

The author concludes that these results offer new insights into the spin dependence of the CSB effect in $\Lambda N$ interactions. The high-precision constraints on the $A=4$ system are expected to refine the scattering length of the $\Lambda$ -neutron interaction and improve predictions for the equation of state of dense matter, particularly regarding the "hyperon puzzle" in neutron stars. The study emphasizes that these empirical constraints reduce model ambiguities related to $\Lambda$ - $\Sigma$ mixing and the interplay of two- and three-body hyperon-nucleon forces.

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