this v4 version matches the version published in the NPA Special Issue on Recent Progress in Strangeness and Charm in Hadronic and Nuclear Physics, plus (i) an added footnote #4 on OPE CSB contributions, and (ii) an added Appendix A on the present NCSM application to hypernuclei
Charge symmetry breaking (CSB) in the $Λ$-nucleon strong interaction generates a charge dependence of $Λ$ separation energies in mirror hypernuclei, which in the case of the $A=4$ mirror hypernuclei $0^+$ ground states is sizable, $ΔB^{J=0}_Λ\equiv B^{J=0}_Λ (_Λ^4{\rm He}) -B^{J=0}_Λ(_Λ^4{\rm H})=230\pm 90$ keV, and of opposite sign to that induced by the Coulomb repulsion in light hypernuclei. Recent ab-initio calculations of the ($_Λ^4{\rm H},{_Λ^4{\rm He}}$) mirror hypernuclei $0^+_{\rm g.s.}$ and $1^+_{\rm exc}$ levels have demonstrated that a $Λ- Σ^0$ mixing CSB model due to Dalitz and von Hippel (1964) is capable of reproducing this large value of $ΔB^{J=0}_Λ$. These calculations are discussed here with emphasis placed on the leading-order chiral EFT hyperon-nucleon strong-interaction Bonn-Juelich potential model used and the no-core shell-model calculational scheme applied. The role of one-pion exchange in producing sizable CSB level splittings in the $A=4$ mirror hypernuclei is discussed.
Charge symmetry breaking (CSB) is particularly strong in the A=4 mirror hypernuclei $_Λ^4$H--$_Λ^4$He. Recent four-body no-core shell model calculations that confront this CSB by introducing $Λ$-$Σ^0$ mixing to leading-order chiral effective field theory hyperon-nucleon potentials are reviewed, and a shell-model approach to CSB in p-shell $Λ$ hypernuclei is outlined.
Ongoing progress in understanding and evaluating charge symmetry breaking in $Λ$ hypernuclei is discussed in connection to recent measurements of the $_Λ^{4}{\rm H}(0^+_{\rm g.s.})$ binding energy at MAMI [A1 Collaboration: PRL 114 (2015) 232501] and of the $_Λ^{4}{\rm He}(1^+_{\rm exc})$ excitation energy at J-PARC [E13 Collaboration: PRL 115 (2015) 222501].
The large charge symmetry breaking (CSB) implied by the $Λ$ binding energy difference $ΔB^{4}_Λ(0^+_{\rm g.s.})\equiv B_Λ(_Λ^4$He)$-$$B_Λ(_Λ^4$H) = 0.35$\pm$0.06 MeV of the $A=4$ mirror hypernuclei ground states, determined from emulsion studies, has defied theoretical attempts to reproduce it in terms of CSB in hyperon masses and in hyperon-nucleon interactions, including one pion exchange arising from $Λ-Σ^0$ mixing. Using a schematic strong-interaction $ΛN\leftrightarrowΣN$ coupling model developed by Akaishi and collaborators for $s$-shell $Λ$ hypernuclei, we revisit the evaluation of CSB …
We report on ab-initio NCSM calculations of the A=4 mirror Lambda hypernuclei Lambda-4H and Lambda-4He, using the Bonn-Juelich LO chiral EFT YN potentials plus a CSB Lambda0--Sigma0 mixing vertex. In addition to reproducing rather well the 0+ (g.s.) and 1+ (exc.) binding energies, these four-body calculations demonstrate for the first time that the observed CSB splitting of mirror levels, reaching hundreds of keV for 0+ (g.s.), can be reproduced using realistic theoretical interaction models, although with a non-negligible momentum cutoff dependence. Our results are discussed in relation to re…
The $Λ$ separation energies in the mirror hypernuclei ${_Λ^4}{\rm H}-{_Λ^4}{\rm He}$ exhibit large charge symmetry breaking (CSB). Analyzing this CSB within pionless effective field theory while using partially conserved baryon-baryon SU(3) flavor symmetry, we deduce a $Λ-Σ^0$ induced in-medium admixture amplitude ${\cal A}_{I=1}\approx 1.5\%$ in the dominantly isospin $I=0$ $Λ$ hyperon. Our results confirm the free-space value ${\cal A}^{(0)}_{I=1}$ inferred directly within the SU(3) baryon octet by Dalitz and von-Hippel in 1964 and reaffirmed in a recent QCD+QED lattice calculation. Furtherm…
Recent experiments have presented more accurate data on the $ΛΛ$-binding energies of a few $ΛΛ$- hypernuclei. This is important as the $ΛΛ$- bond energies ($ΔB_{ΛΛ}$) of double-$Λ$ hypernuclei provide a measure of the in-medium strength of the $ΛΛ$ interaction. A mass formula, optimized with the newly available $ΛΛ$ binding energy data, is used to estimate the binding energy and bond energy over a wide range of hypernuclei. The $ΔB_{ΛΛ}$ values calculated with this mass formula are in good agreement with the experimental data, predictions of the quark mean-field (QMF) model and the relativisti…
We include the leading charge symmetry breaking contributions into the hyperon-nucleon interactions derived within chiral effective field theory up to next-to-leading order. Two low energy constants are determined using the experimentally known differences of Lambda separation energies of 4-Lambda-He and 4-Lambda-H. This allows one to predict the Lambda-neutron scattering lengths for the first time based on data. Various sources of uncertainty are discussed.
We propose a novel method for producing very-neutron-rich hypernuclei and corresponding resonance states by employing charge-exchange reactions via pp($^{12}$C, $^{12}$N $K^+$)n$Λ$ with single-charge-exchange and ppp($^{9}$Be, $^{9}$C $K^+$)nn$Λ$ with double-charge-exchange, both of which produce $ΛK^+$ in a target nucleus. The feasibility of producing very-neutron-rich hypernuclei using the proposed method was analysed by applying an ultra-relativistic quantum molecular dynamics model to a $^6$Li+$^{12}$C reaction at 2 $A$ GeV. The yields of very-neutron-rich hypernuclei, signal-to-background…
A simultaneous description of non-strange nuclei and hypernuclei is provided by a single mass formula inspired by the spin-flavour SU(6) symmetry breaking. This formula is used to estimate the hyperon binding energies of Lambda, double Lambda, Sigma, Cascade and Theta hypernuclei. The results are found to be in good agreement with the available experimental data on 'bound' nuclei and relativistic as well as quark mean field calculations. This mass formula is useful to estimate binding energies over a wide range of masses including the light mass nuclei. It is not applicable for repulsive poten…
We utilize the experimentally known difference of the $Λ$ separation energies of the mirror hypernuclei ${^4_Λ\rm He}$ and ${^4_Λ\rm H}$ to constrain the $Λ$-neutron interaction. We include the leading charge-symmetry breaking (CSB) interaction into our hyperon-nucleon interaction derived within chiral effective field theory at next-to-leading order. In particular, we determine the strength of the two arising CSB contact terms by a fit to the differences of the separation energies of these hypernuclei in the $0^+$ and $1^+$ states, respectively. By construction, the resulting interaction descr…
The separation energies of the isospin triplet $^7_Λ\mathrm{He}$, $^7_Λ\mathrm{Li^{*}}$, $^7_Λ$Be, and the $T=1/2$ doublet $^8_Λ$Li, $^8_Λ$Be are investigated within the no-core shell model. Calculations are performed based on a hyperon-nucleon potential derived from chiral effective field theory at next-to-leading order. The potential includes the leading charge-symmetry breaking (CSB) interaction in the $Λ$N channel, whose strength has been fixed to the experimentally known difference of the $Λ$ separation energies of the mirror hypernuclei $^4_Λ\mathrm{He}$ and $^4_Λ\mathrm{H}$. It turns ou…
The study of hypernuclei in the quark mean-field model are extended from single-$Λ$ hypernuclei to double-$Λ$ hypernuclei as well as $Ξ$ hypernuclei (double strangeness nuclei), with a broken SU(3) symmetry for the quark confinement potential. The strength of the potential for $u,d$ quarks is constrained by the properties of finite nuclei, while the one for $s$ quark is fixed by the single $Λ$-nucleus potential at the nuclear saturation density, which has a slightly different value. Compared to our previous work, we find that the introduction of such symmetry breaking improves efficiently the …
We investigate the weak kaon-nucleon (NNK) S-wave and P-wave interactions using heavy baryon chiral perturbation theory. The leading 1-loop SU(3) breaking contributions to the $ppK$, $pnK$, and $nnK$ couplings are computed. We find that they suppress all NNK amplitudes by $30\%$ to $50\%$. The ratio of neutron-induced to proton-induced hypernuclear decay widths is sensitive to such reductions. It has been argued that the discrepancy between the predicted and observed P-wave amplitudes in $Δs=1$ hyperon decay results from an accidental cancellation between tree-level amplitudes, and is not a fu…
The hypernuclei $^4_Λ$He and $^4_Λ$H provide important information on the hyperon-nucleon interaction. We present accurate Faddeev-Yakubovsky calculations for the $Λ$ separation energies of the $0^+$ ground and the $1^+$ excited states based on the Nijmegen SC YN interactions. We explicitly take the $Σ$ admixture into account. Mass differences of the baryons and the charge-dependence of the interaction are considered. The results show that the Nijmegen models cannot predict all separation energies simultaneously hinting to failures of the current interaction models.
The question of the competition between $Λ$ and $Ξ^{0,-}$ in the ground-state of multi-strange hypernuclei is addressed within a non-relativistic density functional approach, partially constrained by ab-initio calculations and experimental data. The exploration of the nuclear chart for $10<Z<120$ as a function of the strangeness number is performed by adding hyperons to a nuclear core imposing either conserved total charge $Q$ or conserved proton number $Z$. We find that almost all $Λ$ hypernuclei present an instability with respect to the strong interaction decay of $Λ$ towards $Ξ^{0,-}$ and …
The structure of the isodoublet hypernuclei, $^{10}_Λ$B and $^{10}_Λ$Be within the framework of an $α+α+Λ+N$ four-body cluster model is studied. Interactions between the constituent subunits are determined so as to reproduce reasonably well the observed low-energy properties of the $αα$, $αN$, $αΛ$, $ααΛ$ and $ααN$ subsystems. Furthermore, the two-body $ΛN$ interaction is adjusted so as to reproduce the $0^+$-$1^+$ splitting of $^4_Λ$H. The $Λ$ binding energies of $^{10}_Λ$B and $^{10}_Λ$Be are 8.76 MeV and 8.94 MeV, respectively. The energy splitting of the $1^-$-$2^-$ levels in $^{10}_Λ$B is…
The structure of the T=1 iso-triplet hypernuclei, $^7_Λ$He, $^7_Λ$Li and $^7_Λ$Be within the framework of an $α+Λ+N+N$ four-body cluster model is studied. Interactions between the constituent subunits are determined so as to reproduce reasonably well the observed low-energy properties of the $αN$, $αΛ$, $αNN$ and $αΛN$ subsystems. Furthermore, the two-body $ΛN$ interaction is adjusted so as to reproduce the $0^+$-$1^+$ splitting of $^4_Λ$H. Also a phenomenological $ΛN$ charge symmetry breaking(CSB) interaction is introduced. The $Λ$ binding energy of the ground state in $^7_Λ$He is predicted t…
Machine learning the single-$Λ$ hypernuclei with neural-network quantum states
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University（复旦大学现代物理研究所核物理与离子束应用教育部重点实验室）；State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University（北京大学物理学院核物理与技术国家重点实验室）；Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生部）；Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan University（上海理论核物理研究中心（国家自然科学基金委与复旦大学））
A quantitative understanding of charge-symmetry breaking is an increasingly important ingredient for the extraction of the nucleon's strange vector form factors. We review the theoretical understanding of the charge-symmetry-breaking form factors, both for single nucleons and for Helium-4.
The consequences of the charge symmetry breaking effects of the mass difference between the up and down quarks and electromagnetic effects for searches for strangeness form factors in parity violating electron scattering from the proton are investigated. The formalism necessary to identify and compute the relevant observables is developed by separating the Hamiltonian into charge symmetry conserving and breaking terms. Using a set of SU(6) non-relativistic quark models, the effects of the charge symmetry breaking Hamiltonian are considered for experimentally relevant alues of the momentum tran…
The effects of charge symmetry breaking in nucleon electromagnetic form factors on parity- violating elastic electron-12C scattering is studied, and found to be much smaller than other known effects. The analysis of a planned experiment is discussed.
A formalism for the study of charge symmetry breaking (CSB) effects is discussed and used to analyze the effects of charge symmetry breaking on neutron beta decay. The effect of including CSB reduces the beta decay matrix element by an amount on the order of $10^{-4}$, a value much larger than the previous estimate. A much smaller contribution due to proton recoil is treated as well, which is found to be on the order of $10^{-6}$. The current uncertainty in the value of $V_{ud}$ is also of order $10^{-4}$. An improvement of that uncertainty by an order of magnitude would require that charge sy…
The contribution of axial anomaly to charge symmetry breaking in pion-nucleon coupling constants is calculated within instanton model for QCD vacuum. It has been demonstrated that the contribution is large and allows to explain Nolen-Schiffer anomaly.
Charge symmetry breaking contributions to the proton's neutral weak form factors must be understood in order for future measurements of parity violating electron-proton scattering to be definitively interpreted as evidence of proton strangeness. We calculate these charge symmetry breaking form factor contributions using chiral perturbation theory with resonance saturation estimates for unknown low-energy constants. The uncertainty of the leading-order resonance saturation estimates is reduced by incorporating nuclear physics constraints. Higher-order contributions are investigated through phen…
Charge symmetry breaking (CSB) in the strong interaction occurs because of the difference between the masses of the up and down quarks. The use of effective field theories allows us to follow this influence of confined quarks in hadronic and nuclear systems. The progress in observing and understanding CSB is reviewed with particular attention to the recent successful observations of CSB in measurements involving the production of a single neutral pion and to the related theoretical progress.
The basic facts of charge symmetry breaking (CSB) phenomena are reviewed. The relevance of CSB to parity-violating electron-proton scattering experiments that seek to extract strange elastic form factors is discussed. Experimentalists have stated and written that the current uncertainty in our knowledge of CSB limits the ability to push further on the strange form factors. I discuss recent calculations using relativistic chiral perturbation theory and realistic values of strong coupling constants which show that the uncertainties due to lack of knowledge of CSB are at least ten times smaller t…
The binding energy differences of the valence proton and neutron of the mirror nuclei, $^{15}$O -- $^{15}$N, $^{17}$F -- $^{17}$O, $^{39}$Ca -- $^{39}$K and $^{41}$Sc -- $^{41}$Ca, are calculated using the quark-meson coupling (QMC) model. The calculation involves nuclear structure and shell effects explicitly. It is shown that binding energy differences of a few hundred keV arise from the strong interaction, even after subtracting all electromagnetic corrections. The origin of these differences may be ascribed to the charge symmetry breaking effects set in the strong interaction through the u…
Medium modification of pion form factor has been evaluated in asymmetric nuclear matter. It is shown that both the shape and the pole position of the pion form factor in dense asymmetric nuclear matter is different from its vacuum counterpart with $ρ$-$ω$ mixing. This is due to the density and asymmetry dependent $ρ$-$ω$ mixing which could even dominate over its vacuum counterpart in matter. Effect of the in-medium pion factor on experimental observables {\it e.g.}, invariant mass distribution of lepton pairs has been demonstrated.
Using a quark model, we study the effect of charge symmetry breaking on the valence quark distributions of the nucleon. The effect due to quark mass differences and the Coulomb interaction of the electrically charged quarks is calculated and, in contrast to recent claims, found to be small. In addition, we investigate the effect of charge symmetry breaking in the confining interaction, and in the perturbative evolution equations used to relate the quark model distributions to experiment. We find that both these effects are small, and that the strong charge symmetry breaking effect included in …
We study the interplay between charge symmetry breaking and renormalization in the NN system for s-waves. We find a set of universality relations which disentangle explicitly the known long distance dynamics from low energy parameters and extend them to the Coulomb case. We analyze within such an approach the One-Boson-Exchange potential and the theoretical conditions which allow to relate the proton-neutron, proton-proton and neutron-neutron scattering observables without the introduction of extra new parameters and providing good phenomenological success.
Recent experimental results for neutron-neutron scattering length are reanalyzed from the point of view of three-nucleon force contribution. We found that the limiting value of $a_{nn}=- 15.8\pm 0.5$~fm must be free of any implicit three-body force contribution. We have also shown that the difference between the above experimental value of $a_{nn}$ and the well established value of neutron-proton scattering length $a_{np}$ can be explained by differences in the one-pion exchange potentials.
The isospin breaking in the nucleon isovector axial charge, $g_A^3$, are calculated within the external field QCD sum-rule approach. The isospin violations arising from the difference in up and down current quark masses and in up and down quark condensates are included; electromagnetic effects are not considered. We find $δg^3_A/g^3_A \approx (0.5-1.0)\times 10^{-2}$, where $δg^3_A = (g^3_A)_p + (g^3_A)_n$ and $ g^3_A = [(g^3_A)_p - (g^3_A)_n]/2$. Using the Goldberger-Treiman relation, we also obtain an estimate of the isospin breaking in the pion-nucleon coupling constant, $(g_{ppπ_0}-g_{nnπ_…
We consider charge-symmetry violations in the nucleon-nucleon force which result from isospin-violating meson-baryon coupling constants. The vector mesons are assumed to couple to the nucleon's electromagnetic current, which we decompose into isoscalar and isovector quark components. We compute these currents in the context of a constituent quark model. The isospin violations in the meson-baryon couplings arise from the difference in the up and down constituent quark masses. We show that class IV charge-symmetry-breaking potentials arise in the resulting $ω$ and $ρ$ exchange contributions to t…
We investigate the effects of charge independence and charge symmetry breaking in neutron-rich matter. We consider neutron and proton properties in isospin-asymmetric matter at normal densities as well as the high-density neutron matter equation of state and the bulk properties of neutron stars. We find charge symmetry and charge independence breaking effects to be very small.
We discuss charge symmetry and charge independence breaking in an effective field theory approach for few-nucleon systems. We systematically introduce strong isospin-violating and electromagnetic operators in the theory. The charge dependence observed in the nucleon-nucleon scattering lengths is due to one-pion exchange and one electromagnetic four-nucleon contact term. This gives a parameter free expression for the charge dependence of the corresponding effective ranges, which is in agreement with the rather small and uncertain empirical determinations. We also compare the low energy phase sh…
We perform a complete calculation of charge symmetry breaking effects for the reaction pn -> d pi0 at leading order in chiral perturbation theory. A new leading-order operator is included. From our analysis we extract δm_N^{str}, the strong contribution to the neutron-proton mass difference. The value obtained, δm_N^{str} = 1.5 \pm 0.8 (exp.) \pm 0.5 (th.) MeV, is consistent with the result based on the Cottingham sum rule. This agreement provides a non-trivial test of our current understanding of the chiral structure of QCD.
The Jülich-Bonn group aims at an extensive study of the baryon-baryon ($BB$) interaction involving strange baryons ($Λ$, $Σ$, $Ξ$) within SU(3) chiral effective field theory. An overview of achievements and new developments over the past few years is provided. The topics covered are: 1) Derivation of the leading charge-symmetry breaking (CSB) interaction for the $ΛN$ system and its application in a study of CSB effects in $A$=$4$ $Λ$-hypernuclei. 2) Updated results for the $ΞN$ interaction at NLO and predictions for $Ξ^- p$ correlation functions. 3) Extension of the $ΛN$-$ΣN$ interaction to ne…
The role of the off-shell dependence of $ρ-ω$ mixing in the charge symmetry breaking nucleon-nucleon potential is discussed. It is shown that models describing the off-shell dependence of $ρ-ω$ mixing are not sufficient to determine the charge symmetry breaking nucleon-nucleon potential.
Charge-symmetry breaking in the nucleon-nucleon force is investigated within an effective field theory, using a classification of isospin-violating interactions based on power-counting arguments. The relevant charge-symmetry-breaking interactions corresponding to the first two orders in the power counting are discussed, including their effects on the 3He-3H binding-energy difference. The static charge-symmetry-breaking potential linear in the nucleon-mass difference is constructed using chiral perturbation theory. Explicit formulae in momentum and configuration spaces are presented. The presen…
I review the effects of charge symmetry breaking CSB on electromagnetic form factors and how that influences extraction of information regarding nucleon strangeness content and the weak mixing angle. It seems that CSB effects are very modest and should not impact the analysis of experiments.
We present a determination, from lattice QCD, of charge symmetry violation in the spin- independent and spin-dependent parton distribution functions of the nucleon. This is done by chirally extrapolating recent QCDSF/UKQCD Collaboration lattice simulations of the first several Mellin moments of the parton distribution functions of octet baryons to the physical point. We find small chiral corrections for the polarized moments, while the corrections are quantitatively significant in the unpolarized case.
Leading-order three-nucleon forces that violate isospin symmetry are calculated in Chiral Perturbation Theory. The effect of the charge-symmetry-breaking three-nucleon force is investigated in the trinucleon systems using Faddeev calculations. We find that the contribution of this force to the 3He - 3H binding-energy difference is approximately 5 keV.
We show that using parameters consistent with the charge symmetry violating difference between the strong nn and pp scattering lengths provides significant constraints on the amplitude for the dd -> alpha pi0 reaction.
The effects of charge independence breaking of nucleon-nucleon interaction on the bulk properties of nuclear matter are investigated. Our results indicate that at high densities, the inclusion of charge dependence in the nucleon-nucleon potential affects the bulk properties of nuclear matter. However, at low densities, this effect is not considerable. It is seen that the change of our results for the nuclear matter calculations due to the breaking of the charge independence increases by increasing density. It is shown that the energy contribution of the $^1S_0$ channel is sensitive to consider…
In this proceeding we summarize results for baryonic contact terms derived within SU(3) chiral effective field theory. The four-baryon contact terms, necessary for the description of the hyperon-nucleon interaction, include SU(3) symmetric and explicit chiral symmetry breaking terms. They also include four-baryon contact terms involving pseudoscalar mesons, which become important for three-body forces. Furthermore we derive the leading order six-baryon contact terms in the non-relativistic limit and study their contribution to the $ΛNN$ three-body contact interaction. These results could play …
The effect of isospin breaking pion s-wave rescattering is included in elastic NN scattering at low energies using effective field theory. Although this mechanism gave a large contribution to charge symmetry breaking in np --> d pi0, the effect is rather small in pp vs. nn scattering parameters and in the 3H-3He binding energy difference. This smallness is caused by large cancellation of the up-down quark mass difference contribution and electromagnetic effects to the np mass difference.
The status of the theoretical effort to calculate charge-symmetry-breaking pion production reactions will be presented. The main emphasis will be on the d d -> alpha pi0 reaction, especially in relation to the recent IUCF experiment. Suggestions for future theoretical and experimental work will be made.
Elastic nucleon scattering from the 3He and 3H mirror nuclei is examined as a test of charge symmetry violation. The differential cross-sections are calculated at 500 MeV using a microsopic, momentum-space optical potential including the full coupling of two spin 1/2 particles and an exact treatment of the Coulomb force. The charge-symmetry-breaking effects investigated arise from a violation within the nuclear structure, from the p-nucleus Coulomb force, and from the mass-differences of the charge symmetric states. Measurements likely to reveal reliable information are noted.
Charge-independence breaking due to the pion-mass difference in the (chiral) two-pion-exchange nucleon-nucleon force is investigated. A general argument based on symmetries is presented that relates the charge-symmetric part of that force to the proton-proton case. The static potential linear in that mass difference is worked out as an explicit example by means of Feynman diagrams, and this confirms the general argument.
The asymmetry in the angular distribution of n p --> d pi^0 due to Charge Symmetry Breaking is calculated using Heavy Baryon Chiral Perturbation Theory. Recent developments in power counting have proven successful in describing total cross sections, and we apply them to the asymmetry calculation. Reducibility in one of the leading order diagrams is examined. We compare the updated theory with experimental results for a range of physically reasonable parameters and find over-prediction for the entire range.
Effects of charge symmetry breaking associated with the $u$ and $d$ quark mass difference in the elastic form factors of the pion and kaon are presented. We use a confining version of the Nambu--Jona-Lasinio model. The pion and kaon are described as a dressed quark and antiquark bound states governed by the Bethe-Salpeter equation, and exhibit the properties of Goldstone bosons, with the pion mass difference given by $m_{π^{+}}^2 - m_{π^{0}}^2 \propto (m_u -m_d)^2$ as demanded by dynamical chiral symmetry breaking. We found significant charge symmetry breaking effects for realistic current qua…
