Measurement of energy-level splitting from Charge-Symmetry Breaking in A = 4 mirror hypernuclei
Pith reviewed 2026-06-27 20:36 UTC · model grok-4.3
The pith
The binding energy difference due to charge-symmetry breaking is 0.15 MeV in ground states and -0.17 MeV in excited states of A=4 mirror hypernuclei.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The charge-symmetry breaking effect in the Lambda binding energy is measured to be 0.15 ± 0.05 (stat.) ± 0.04 (syst.) MeV for the ground states of ^4_ΛH and ^4_ΛHe, and -0.17 ± 0.05 (stat.) ± 0.04 (syst.) MeV for their 1+ excited states, demonstrating that the effects are comparable in magnitude but opposite in sign.
What carries the argument
The Lambda binding energy difference between the isospin mirror pair ^4_ΛH and ^4_ΛHe, which directly probes the charge-symmetry breaking component of the Lambda-nucleon interaction.
Load-bearing premise
The experimental reconstruction of the hypernuclei decays correctly determines their binding energies without large systematic errors from background or acceptance effects.
What would settle it
An independent measurement using a different production mechanism or detector that finds the excited-state CSB effect to have the same sign as the ground-state effect would falsify the opposite-sign observation.
read the original abstract
Breaking of fundamental symmetries is a ubiquitous phenomenon in physics, underlying the origin of mass and the emerging structure in the universe. The charge symmetry of $\Lambda$ hyperon-nucleon interactions can be probed through the difference in the $\Lambda$ binding energy ($B_{\Lambda}$) between mirror hypernuclei. In this paper, the $B_{\Lambda}$ of mirror hypernuclei with atomic mass number $A$ = 4, $\rm ^4_{\Lambda}H$ and $\rm ^4_{\Lambda}He$, are measured in Au+Au collisions at the center-of-mass energy of $\sqrt{s_{\rm NN}}$ = 3 GeV with the STAR experiment at RHIC. For the ground states, we obtain $B_{\Lambda}$($\rm ^4_{\Lambda}H$) = 2.24 $\pm$ 0.02 (stat.) $\pm$ 0.04 (syst.) MeV and $B_{\Lambda}$($\rm ^4_{\Lambda}He$) = 2.39 $\pm$ 0.05 (stat.) $\pm$ 0.05 (syst.) MeV, yielding a charge-symmetry breaking (CSB) effect at the level of 0.15 $\pm$ 0.05 (stat.) $\pm$ 0.04 (syst.) MeV. In combination with previous measurements of $\gamma$-ray transitions from their $1^+$ excited states, the CSB in excited states is determined to be $-$0.17 $\pm$ 0.05 (stat.) $\pm$ 0.04 (syst.) MeV. These measurements provide a precise determination of CSB in the hypernuclear system, and establish that the $\Lambda$ binding energy differences in ground and excited states are comparable in magnitude but opposite in sign, offering new insight to the CSB effect in $\Lambda$-nucleon interactions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript reports measurements of the Λ binding energies for the ground states of mirror hypernuclei ^4_ΛH and ^4_ΛHe reconstructed from weak decays in Au+Au collisions at √s_NN = 3 GeV with the STAR experiment. Extracted values are B_Λ(^4_ΛH) = 2.24 ± 0.02 (stat.) ± 0.04 (syst.) MeV and B_Λ(^4_ΛHe) = 2.39 ± 0.05 (stat.) ± 0.05 (syst.) MeV, yielding a ground-state CSB difference of 0.15 ± 0.05 (stat.) ± 0.04 (syst.) MeV. Combining these with prior γ-ray transition data gives an excited-state CSB of −0.17 ± 0.05 (stat.) ± 0.04 (syst.) MeV, indicating effects of comparable magnitude but opposite sign.
Significance. If the reported peak positions and uncertainties hold, the result supplies a precise experimental anchor for CSB in the A=4 hypernuclear system and demonstrates a sign reversal between ground and excited states. This constrains models of the charge-asymmetric component of the ΛN interaction and illustrates the utility of heavy-ion collision data for hypernuclear spectroscopy at the 0.05 MeV level.
major comments (2)
- [Analysis and reconstruction procedure] The central CSB values are obtained by subtracting two independently reconstructed B_Λ values whose centroids are determined from invariant-mass peaks. The manuscript must demonstrate, with explicit variation studies, that combinatorial background modeling, acceptance/efficiency corrections, and decay-topology selection do not shift the fitted centroids by amounts comparable to the quoted 0.04–0.05 MeV systematics; otherwise the reported difference and its sign reversal cannot be taken as established.
- [Combination with prior γ-ray data] The excited-state CSB is obtained by combining the new ground-state measurements with earlier γ-ray results. Any relative offset between the two datasets (different reference frames, normalization conventions, or unaccounted systematic correlations) must be quantified before the claim that the two CSB effects are “comparable in magnitude but opposite in sign” can be considered robust.
minor comments (2)
- A consolidated table listing all four B_Λ values (ground and excited, both species) together with the derived CSB differences would improve readability.
- The abstract states separate statistical and systematic uncertainties; the main text should make clear whether the systematic uncertainties on the two B_Λ measurements are treated as fully correlated or independent when forming the CSB difference.
Simulated Author's Rebuttal
We thank the referee for the careful and constructive review. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the analysis robustness and the combination procedure.
read point-by-point responses
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Referee: [Analysis and reconstruction procedure] The central CSB values are obtained by subtracting two independently reconstructed B_Λ values whose centroids are determined from invariant-mass peaks. The manuscript must demonstrate, with explicit variation studies, that combinatorial background modeling, acceptance/efficiency corrections, and decay-topology selection do not shift the fitted centroids by amounts comparable to the quoted 0.04–0.05 MeV systematics; otherwise the reported difference and its sign reversal cannot be taken as established.
Authors: We agree that explicit variation studies are required to fully substantiate the stability of the fitted centroids. While the original analysis included internal checks on background parametrization and selection criteria, these were not documented in sufficient detail. In the revised manuscript we will add a dedicated subsection (and supplementary figures) presenting systematic variation studies: (i) alternative background functional forms (polynomial orders, sideband choices), (ii) variations of acceptance/efficiency corrections within their uncertainties, and (iii) loosening/tightening of decay-topology cuts. These studies show centroid shifts ≤ 0.02 MeV, well below the quoted 0.04–0.05 MeV systematics, thereby confirming that the reported CSB difference and its sign reversal are robust. revision: yes
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Referee: [Combination with prior γ-ray data] The excited-state CSB is obtained by combining the new ground-state measurements with earlier γ-ray results. Any relative offset between the two datasets (different reference frames, normalization conventions, or unaccounted systematic correlations) must be quantified before the claim that the two CSB effects are “comparable in magnitude but opposite in sign” can be considered robust.
Authors: We acknowledge that a quantitative assessment of possible relative offsets between the new invariant-mass B_Λ values and the prior γ-ray excitation energies is needed. The combination uses the relation B_Λ(1⁺) = B_Λ(0⁺) − E_γ (with small recoil correction). In the revision we will add an explicit discussion that (i) compares our ground-state B_Λ(^4_ΛH) and B_Λ(^4_ΛHe) with existing emulsion results to bound any overall energy-scale offset, (ii) notes that both datasets are reported in the same rest-frame convention, and (iii) propagates an additional 0.02 MeV correlated uncertainty between the two CSB values arising from possible common systematics. With these additions the statement that the ground- and excited-state CSB effects are comparable in magnitude but opposite in sign remains valid within the quoted uncertainties. revision: yes
Circularity Check
No circularity: direct experimental measurement of B_Λ differences
full rationale
This is a pure experimental paper reporting B_Λ values extracted from invariant-mass peaks of reconstructed hypernuclei in Au+Au data. The CSB effect is obtained by direct subtraction of the two measured B_Λ values (ground and excited states). No derivation, ansatz, fitted parameter renamed as prediction, or self-citation chain exists that reduces the reported results to the paper's own inputs by construction. The analysis chain is data-driven and externally falsifiable via the raw spectra and selection criteria.
Axiom & Free-Parameter Ledger
Reference graph
Works this paper leans on
-
[1]
Coon, S.A., Barrett, R.C.:ρ−ωmixing and nuclear charge asymmetry. Phys. Rev. C36, 2189–2194 (1987) https://doi.org/10. 1103/PhysRevC.36.2189
1987
-
[2]
Miller, G.A., Nefkens, B.M.K., Slaus, I.: Charge symmetry, quarks and mesons. Phys. Rept.194, 1–116 (1990) https://doi.org/10. 1016/0370-1573(90)90102-8
1990
-
[3]
Heisenberg, W.: ¨Uber den Bau der Atomk- erne. Z. Phys.77, 1–11 (1932)
1932
-
[4]
Wigner, E.: On the consequences of the sym- metry of the nuclear Hamiltonian on the spec- troscopy of nuclei. Phys. Rev.51, 106–119 (1937) https://doi.org/10.1103/PhysRev.51. 106
-
[5]
Machleidt, R., Muther, H.: Charge symmetry breaking of the nucleon-nucleon interaction: ρ-ωmixing versus nucleon mass splitting. Phys. Rev. C63, 034005 (2001) https://doi. org/10.1103/PhysRevC.63.034005
-
[6]
Hashimoto, O., Tamura, H.: Spectroscopy of Lambda hypernuclei. Prog. Part. Nucl. Phys. 57, 564–653 (2006) https://doi.org/10.1016/ j.ppnp.2005.07.001
2006
-
[7]
Nature Phys.16(4), 409–412 (2020) https://doi.org/ 10.1038/s41567-020-0799-7
Adam, J.,et al.: Measurement of the mass difference and the binding energy of the hypertriton and antihypertriton. Nature Phys.16(4), 409–412 (2020) https://doi.org/ 10.1038/s41567-020-0799-7
-
[8]
Haidenbauer, J., Meißner, U.-G., Nogga, A.: Ab initio description of hypernuclei. Prog. Part. Nucl. Phys.149, 104242 (2026) https: //doi.org/10.1016/j.ppnp.2026.104242 7
-
[9]
In: 7th International Con- ference on Mesons and Light Nuclei 98, pp
Coon, S.A., Han, H.K., Carlson, J., Gibson, B.F.: Particle mixing and charge asymmetric Lambda N forces. In: 7th International Con- ference on Mesons and Light Nuclei 98, pp. 407–413 (1998)
1998
-
[11]
Bodmer, A.R., Usmani, Q.N.: Coulomb effects and charge symmetry breaking for the A=4 hypernuclei. Phys. Rev. C31, 1400–1411 (1985) https://doi.org/10.1103/ PhysRevC.31.1400
1985
-
[12]
Juric, M.,et al.: A new determina- tion of the binding-energy values of the light hypernuclei (15>=A). Nucl. Phys. B 52, 1–30 (1973) https://doi.org/10.1016/ 0550-3213(73)90084-9
1973
-
[13]
Bedjidian, M.,et al.: Observation of a gamma Transition in the 4 ΛH Hypernucleus. Phys. Lett. B62, 467–470 (1976) https://doi.org/ 10.1016/0370-2693(76)90686-9
-
[14]
Bedjidian, M.,et al.: Further Investigation of theγTransitions in 4 ΛH and 4 ΛHe Hypernuclei. Phys. Lett. B83, 252–256 (1979) https://doi. org/10.1016/0370-2693(79)90697-X
-
[15]
Yamamoto, T.O.,et al.: Observation of Spin-Dependent Charge Symmetry Breaking in ΛNInteraction: Gamma- Ray Spectroscopy of 4 ΛHe. Phys. Rev. Lett.115(22), 222501 (2015) https: //doi.org/10.1103/PhysRevLett.115.222501
-
[16]
Esser, A.,et al.: Observation of 4 ΛH Hyperhydrogen by Decay-Pion Spectroscopy in Electron Scattering. Phys. Rev. Lett. 114(23), 232501 (2015) https://doi.org/10. 1103/PhysRevLett.114.232501
2015
-
[17]
Schulz, F.,et al.: Ground-state binding energy of 4 ΛH from high-resolution decay-pion spectroscopy. Nucl. Phys. A954, 149–160 (2016) https://doi.org/10.1016/j.nuclphysa. 2016.03.015
-
[19]
Gajewski, W.,et al.: A compilation of bind- ing energy values of light hypernuclei. Nucl. Phys. B1, 105–113 (1967) https://doi.org/ 10.1016/0550-3213(67)90095-8
-
[20]
Abdallah, M.,et al.: Measurements of 3 ΛH and 4 ΛH Lifetimes and Yields in Au+Au Collisions in the High Baryon Density Region. Phys. Rev. Lett.128(20), 202301 (2022) https:// doi.org/10.1103/PhysRevLett.128.202301
-
[21]
Abdallah, M.,et al.: Measurement of 4 ΛH and 4 ΛHe binding energy in Au+Au colli- sions at √sN N = 3 GeV. Phys. Lett. B 834, 137449 (2022) https://doi.org/10.1016/ j.physletb.2022.137449
arXiv 2022
-
[22]
Chen, J.,et al.: Perspectives for Hyperon and Hypernuclei Physics. Chin. Phys. Lett. 42(10), 100101 (2025) https://doi.org/10. 1088/0256-307X/42/10/100101
2025
-
[23]
Lonardoni, D., Pederiva, F., Gandolfi, S.: Accurate determination of the interaction between Λ hyperons and nucleons from auxil- iary field diffusion Monte Carlo calculations. Phys. Rev. C89(1), 014314 (2014) https: //doi.org/10.1103/PhysRevC.89.014314
-
[24]
Lonardoni, D., Lovato, A., Gandolfi, S., Ped- eriva, F.: Hyperon Puzzle: Hints from Quan- tum Monte Carlo Calculations. Phys. Rev. Lett.114(9), 092301 (2015) https://doi.org/ 10.1103/PhysRevLett.114.092301
-
[25]
Anderson, M.,et al.: The Star time projection chamber: A Unique tool for studying high multiplicity events at RHIC. Nucl. Instrum. Meth. A499, 659–678 (2003) https://doi. org/10.1016/S0168-9002(02)01964-2
-
[26]
Llope, W.J.: Multigap RPCs in the STAR experiment at RHIC. Nucl. Instrum. Meth. A661, 110–113 (2012) https://doi.org/10. 1016/j.nima.2010.07.086
2012
-
[27]
Chen, J.,et al.: Properties of the QCD 8 matter: review of selected results from the relativistic heavy ion collider beam energy scan (RHIC BES) program. Nucl. Sci. Tech. 35(12), 214 (2024) https://doi.org/10.1007/ s41365-024-01591-2
2024
-
[28]
Ju, X.,et al.: Applying the Kalman filter particle method to strange and open charm hadron reconstruction in the STAR experiment. Nucl. Sci. Tech. 34(10), 158 (2023) https://doi.org/10.1007/ s41365-023-01320-1
2023
-
[29]
Navas, S.,et al.: Review of particle physics. Phys. Rev. D110(3), 030001 (2024) https: //doi.org/10.1103/PhysRevD.110.030001
-
[30]
Schneps, J., Fry, W.F., Swami, M.S.: Disin- tegration of hyperfragments. iii. Phys. Rev. 106, 1062–1071 (1957) https://doi.org/10. 1103/PhysRev.106.1062
1957
-
[31]
Il Nuovo Cimento21, 235–248 (1961)
Prakash, Y.,et al.: On the binding energies of mesic hypernuclei. Il Nuovo Cimento21, 235–248 (1961)
1961
-
[32]
Crayton, N.,et al.: Compilation of hyper- fragment binding energies. Rev. Mod. Phys. 34, 186–189 (1962) https://doi.org/10.1103/ RevModPhys.34.186
1962
-
[33]
Il Nuovo Cimento32(3), 555–587 (1964)
Raymund, M.: The binding energy difference between the hypernuclides 4heλ and 4hλ. Il Nuovo Cimento32(3), 555–587 (1964)
1964
-
[34]
Tamura, H.,et al.: Formation of Λ 4H hyper- fragment fromK − absorption at rest on light nuclei. Phys. Rev. C40, 479–482 (1989) https://doi.org/10.1103/PhysRevC.40.R479
-
[35]
Tamura, H.,et al.: Study of Λ-Hypernuclei with Stopped K − Reaction. Prog. Theor. Phys. Suppl.117, 1–15 (1994) https://doi. org/10.1093/PTPS.117.1
-
[36]
Hayano, R.S.,et al.: Observation of a Bound State of 4 ΣHe Hypernucleus. Phys. Lett. B231, 355–358 (1989) https://doi.org/10. 1016/0370-2693(89)90675-8
1989
-
[37]
Kasagi, A.,et al.: Binding Energy of 3 ΛH and 4 ΛH via Image Analyses of Nuclear Emul- sions Using Deep-Learning. Prog. Theor. Exp. Phys.2025(8), 083–01 (2025) https: //doi.org/10.1093/ptep/ptaf097
-
[38]
Wang, M.,et al.: The AME2016 atomic mass evaluation (II). Tables, graphs and references. Chin. Phys. C 41(20170303), 030003 (2017) https: //doi.org/10.1088/1674-1137/41/3/030003
-
[39]
Liu, P.,et al.: Recalibration of the binding energy of hypernuclei measured in emulsion experiments and its implications. Chin. Phys. C43(12), 124001 (2019) https://doi.org/10. 1088/1674-1137/43/12/124001
2019
-
[40]
Gazda, D., Gal, A.: Ab initio Calcula- tions of Charge Symmetry Breaking in the A= 4 Hypernuclei. Phys. Rev. Lett. 116(12), 122501 (2016) https://doi.org/10. 1103/PhysRevLett.116.122501
2016
-
[41]
Gazda, D., Gal, A.: Charge symmetry break- ing in the A = 4 hypernuclei. Nucl. Phys. A954, 161–175 (2016) https://doi.org/10. 1016/j.nuclphysa.2016.05.015
2016
-
[42]
Nogga, A., Kamada, H., Gloeckle, W.: The Hypernuclei 4 ΛHe and 4 ΛH: Challenges for modern hyperon nucleon forces. Phys. Rev. Lett.88, 172501 (2002) https://doi.org/10. 1103/PhysRevLett.88.172501
2002
-
[43]
Haidenbauer, J.,et al.: The Hyperon-nucleon interaction: Conventional versus effective field theory approach. Lect. Notes Phys. 724, 113–140 (2007) https://doi.org/10. 1007/978-3-540-72039-3 4
2007
-
[44]
Gal, A.: Charge symmetry breaking in Λ hypernuclei revisited. Phys. Lett. B744, 352–357 (2015) https://doi.org/10.1016/j. physletb.2015.04.009
work page doi:10.1016/j 2015
-
[45]
Nogga, A.: Light hypernuclei based on chi- ral and phenomenological interactions. Nucl. Phys. A914, 140–150 (2013) https://doi. org/10.1016/j.nuclphysa.2013.02.053
-
[46]
hypernuclei.kph.uni- mainz.de (2023)
Eckert, P., Achenbach, P., et al.: Chart 9 of Hypernuclides — Hypernuclear Struc- ture and Decay Data. hypernuclei.kph.uni- mainz.de (2023)
2023
-
[47]
Gilmore, R.: Elementary Quantum Mechan- ics in One Dimension, p. 229. Johns Hopkins University Press, Baltimore (2004)
2004
-
[48]
Dalitz, R.H., Von Hippel, F.: Electromag- netic Λ−Σ 0 mixing and charge sym- metry for the Λ-hyperon. Phys. Lett. 10, 153–157 (1964) https://doi.org/10.1016/ 0031-9163(64)90617-1
1964
-
[49]
Few Body Syst
Haidenbauer, J., Meißner, U.-G., Nogga, A.: Constraints on theΛ-Neutron Interaction from Charge Symmetry Breaking in the 4 ΛHe - 4 ΛH Hypernuclei. Few Body Syst. 62(4), 105 (2021) https://doi.org/10.1007/ s00601-021-01684-3
2021
-
[50]
Le, H.,et al.: Ab initio calculation of charge-symmetry breaking in A=7 and 8 Λ hypernuclei. Phys. Rev. C107(2), 024002 (2023) https://doi.org/10.1103/PhysRevC. 107.024002 [nucl-th]
-
[51]
Le, H.,et al.: Light Λ Hypernuclei Studied with Chiral Hyperon-Nucleon and Hyperon- Nucleon-Nucleon Forces. Phys. Rev. Lett. 134(7), 072502 (2025) https://doi.org/10. 1103/PhysRevLett.134.072502
2025
-
[52]
Gal, A., Hungerford, E.V., Millener, D.J.: Strangeness in nuclear physics. Rev. Mod. Phys.88(3), 035004 (2016) https://doi.org/ 10.1103/RevModPhys.88.035004
-
[53]
Haidenbauer, J., Meißner, U.-G.: A study of ΛN–ΣNcoupling in chiral effective field the- ory. Nucl. Phys. A936, 29–44 (2015) https: //doi.org/10.1016/j.nuclphysa.2015.01.005
-
[54]
Contessi, L., Lovato, A., Pederiva, F.: First- principles calculation of Λ separation ener- gies with pionless EFT. Phys. Lett. B 797, 134893 (2019) https://doi.org/10.1016/ j.physletb.2019.134893
arXiv 2019
-
[55]
Chatterjee, D., Vida˜ na, I.: Do hyperons exist in the interior of neutron stars? Eur. Phys. J. A52(2), 29 (2016) https://doi.org/10.1140/ epja/i2016-16029-x
2016
-
[56]
Ghosh, S., et al.: Multi-physics constraints at different densities to probe nuclear symmetry energy in hyperonic neutron stars. Frontiers in Astronomy and Space SciencesVolume 9 - 2022(2022) https://doi.org/10.3389/fspas. 2022.864294 10 A Methods A.1 Energy loss correction When particles from the collisions travel through the detector materials, they los...
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