Charge symmetry breaking in $\Lambda$ hypernuclei revisited

The large charge symmetry breaking (CSB) implied by the $\Lambda$ binding energy difference $\Delta B^{4}_{\Lambda}(0^+_{\rm g.s.})\equiv B_{\Lambda}(_{\Lambda}^4$He)$-$$B_{\Lambda}(_{\Lambda}^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 $\Lambda-\Sigma^0$ mixing. Using a schematic strong-interaction $\Lambda N\leftrightarrow\Sigma N$ coupling model developed by Akaishi and collaborators for $s$-shell $\Lambda$ hypernuclei, we revisit the evaluation of CSB in the $A=4$ $\Lambda$ hypernuclei and extend it to $p$-shell mirror $\Lambda$ hypernuclei. The model yields values of $\Delta B^{4}_{\Lambda} (0^+_{\rm g.s.})\sim 0.25$ MeV. Smaller size and mostly negative $p$-shell binding energy differences are calculated for the $A=7-10$ mirror hypernuclei, in rough agreement with the few available data. CSB is found to reduce by almost 30 keV the 110 keV $_{~\Lambda}^{10}$B g.s. doublet splitting anticipated from the hyperon-nucleon strong-interaction spin dependence, thereby explaining the persistent experimental failure to observe the $2^-_{\rm exc}\to 1^-_{\rm g.s.}$ $\gamma$-ray transition.


Introduction
Charge symmetry breaking (CSB) in nuclear physics is primarily identified by considering the difference between nn and pp scattering lengths, or the binding-energy difference between the mirror nuclei 3 H and 3 He [1]. In these nuclei, about 70 keV out of the Coulomb-dominated 764 keV bindingenergy difference is commonly attributed to CSB which can be explained either by ρ 0 ω mixing in one-boson exchange models of the NN interaction, or by considering N∆ intermediate-state mass differences in models limited to pseudoscalar meson exchanges [2].
The large ∆B 4 Λ values reported for both 0 + g.s. and 1 + exc states have defied theoretical attempts to explain these differences in terms of hadronic or quark CSB mechanisms within four-body calculations [8,9,10,11,12]. Meson mixing, including ρ 0 ω mixing which explains CSB in the A = 3 nuclei, gives only small negative contributions about −30 and −10 keV for ∆B 4 Λ (0 + g.s. ) and ∆B 4 Λ (1 + exc ), respectively [9]. CSB contributions to ∆B 4 Λ (0 + g.s. ) from oneand two-pion exchange interactions in Y NNN coupled-channel calculations [10,11] with hyperons Y = Λ, Σ amount to as much as 100 keV; this holds for the OBE-based Nijmegen NSC97 models [13] which are widely used in Λ-hypernuclear structure calculations. 1 Binding energies of ground states in p-shell mirror Λ hypernuclei, determined from emulsion studies [3], suggest much weaker CSB effects than for A = 4, with values of ∆B A Λ ≡ B Λ (A, I, I z ) − B Λ (A, I, −I z ) (I z > 0) consistent with zero for A = 8 and somewhat negative beyond [3]. Accommodating ∆B Λ values in the p shell with ∆B 4 Λ by using reasonable phenomenological CSB interactions is impossible, as demonstrated in recent four-body clustermodel calculations of p-shell Λ hypernuclei [15]. This difficulty may be connected to the absence of explicit ΛN ↔ ΣN coupling in these cluster-model calculations, given that such explicit coupling was shown to generate nonnegligible CSB contributions to ∆B 4 Λ (0 + g.s. ) [12]. It is our purpose in this note to use a schematic ΛN ↔ ΣN coupling model, proposed by Akaishi et al. [16,17] for s-shell Λ hypernuclei and extended by Millener [18] to the p shell, for calculating ∆B Λ values in both s and p shells, thereby making predictions on CSB effects in p-shell Λ hypernuclei consistently with a relatively sizable value of ∆B 4 Λ (0 + g.s. ). The paper is organized as follows. In Sect. 2, we update the original treatment by Dalitz and Von Hippel [19] of the Λ − Σ 0 mixing mechanism for generating CSB one-pion exchange contributions in Λ hypernuclei, linking it to the strong-interaction ΛN ↔ ΣN coupling model employed in this work. Our CSB calculations for the A = 4 hypernuclei are sketched in Sect. 3 and their results are compared with those reported in several Y NNN four-body calculations [9,10,11,12]. Finally, CSB contributions in p-shell mirror Λ hypernuclei, evaluated here for the first time, are reported in Sect. 4.

Pionic CSB contributions in Λ hypernuclei
The I = 0 isoscalar nature of the Λ hyperon forbids it to emit or absorb a single pion, and hence there is no one-pion exchange (OPE) contribution to the ΛN strong interaction. However, by allowing for Λ − Σ 0 mixing in 1 Contradictory statements were made in Refs. [8,10] on the ability of the earlier Nijmegen model NSC89 [14] to reproduce ∆B 4 Λ (0 + g.s. ). We note that the strength of CSB contributions in this work, on p. 2236, is inflated erroneously by a factor g N N M , about 3.7 for pion exchange, which might have propagated into some of the calculations claiming to have resolved the CSB puzzle for ∆B 4 Λ (0 + g.s. ) using NSC89. [19] with ΛΛπ coupling constant

SU(3), a CSB OPE contribution arises
where the matrix element of the mass mixing operator δM is given by The resulting CSB OPE potential is given by where the z component of the isospin Pauli matrix τ N assumes the values τ N z = ±1 on protons and neutrons, respectively, Y (r) = exp(−m π r)/(m π r) is a Yukawa form, and the tensor contribution is specified by In Eq. (3), the transition g ΛΣπ → f N N π was made in accordance with NSC models, using f 2 N N π /4π = 0.0740 from NSC89, Table IV in Ref [14]. Since Pauli-spin S pp = 0 and S nn = 0 hold in 4 Λ He(0 + g.s. ) and in 4 Λ H(0 + g.s. ), respectively, the CSB potential (3) which is linear in the nucleon spin gives no contribution from these same-charge nucleons. Therefore τ N z = ∓1 owing to the odd nucleon in these hypernuclei, respectively, and since σ Λ · σ N = −3 holds for this odd nucleon, one gets a positive ∆B Λ contribution from the central spin-spin part which provides the only nonvanishing contribution for simple L = 0 wavefunctions. For the 0 + g.s. wavefunction used by Dalitz and Von Hippel [19], and updating the values of the coupling constants used in their work to those used here, one gets ∆B OPE Λ (0 + g.s. ) ≈ 95 keV, a substantial single contribution with respect to ∆B exp Λ (0 + g.s. ) = 350 ± 60 keV. The Λ − Σ 0 mixing mechanism gives rise also to a variety of (e.g. ρ) meson exchanges other than OPE. In baryon-baryon models that consider explicitly the strong-interaction ΛN ↔ ΣN coupling, the matrix element of V CSB ΛN is related to a suitably chosen strong-interaction isospin I N Y = 1/2 matrix element NΣ|V |NΛ by generalizing Eq. (1): where the isospin Clebsch-Gordan coefficient 1/ √ 3 accounts for the NΣ 0 amplitude in the I N Y = 1/2 NΣ state, and the space-spin structure of this NΣ state is taken identical with that of the NΛ state sandwiching V CSB ΛN . Following hyperon-core calculations of s-shell Λ hypernuclei by Akaishi et al. [16] we use G-matrix Y N effective interactions derived from NSC97 models to calculate CSB contributions from Eq. (5). The ΛΣ 0s N 0s Y effective interaction V ΛΣ is given in terms of a spin-dependent central interaction

CSB in the A = 4 hypernuclei
where t ΛΣ converts a Λ to Σ in isospace. The s-shell matrix elementsV 0s ΛΣ and ∆ 0s ΛΣ are listed in Table 1 for two such G-matrix models denoted (ΛΣ) e,f . Also listed are the calculated downward energy shifts δE ↓ (J π ) defined by δE ↓ (J π ) = v 2 (J π )/(80 MeV), where the 0s 3 N 0s Y matrix elements v(J π ) for A = 4 are given in terms of ΛΣ two-body matrix elements by We note that the diagonal 0s N 0s Σ interaction matrix elements have little effect in this coupled-channel model because of the large energy denominators of order M Σ − M Λ ≈ 80 MeV with which they appear. Finally, by listing ∆E ΛΣ (0 + g.s. −1 + exc ) from Refs. [16,17] we demonstrate the sizable contribution of ΛΣ coupling to the excitation energy ∆E(0 + g.s. − 1 + exc ) ≈ 1.1 MeV deduced from the γ-ray transition energies marked in Fig. 1. For comparison, the full ∆E(0 + g.s. − 1 + exc ) in these (ΛΣ) e,f models, and as calculated by Nogga [10] using the underlying Nijmegen models NSC97 e,f , are also listed in the table.
Having discussed the effect of strong-interaction ΛΣ coupling, we now discuss the CSB splittings ∆B 4 Λ (0 + g.s. ) and ∆B 4 Λ (1 + exc ). Results of our ΛΣ coupling model calculations, using Eq. (5) for one of several contributions, are listed in the last two lines of Table 2, preceded by results obtained in other models within genuine four-body calculations [9,10,11,12]. Partial contributions to ∆B 4 Λ (0 + g.s. ) are listed in columns 2-5, whereas for ∆B 4 Λ (1 + exc ) only its total value is listed.  All of the models listed in Table 2 except for [9] include ΛΣ coupling, with 0 + g.s. ΣNNN admixture probabilities P Σ ≈ P Σ ± + P Σ 0 in ( 4 Λ He, 4 Λ H) respectively, and P Σ ± ≈ 2 3 P Σ . The 1 + exc ΣNNN admixtures (unlisted) are considerably weaker than the listed 0 + g.s. admixtures. Charge asymmetric kinetic-energy contributions to ∆B Λ , dominated by ΣN intermediate-state mass differences, are marked ∆T Y N in the table. In the present ΛΣ coupling model these are given for the 0 + g.s. by [10] ∆T yielding as much as 50 keV, in agreement with those four-body calculations where such mass differences were introduced [10, 11,12]. The next column in the table, ∆V C = ∆V Λ C + ∆V Σ C , addresses contributions arising from nuclear-core Coulomb energy modifications induced by the hyperons. ∆V Λ C is negative, its size ranges from less than 10 keV [10,12] to about 40 keV [9]. ∆V Σ C which accounts for Σ ± p Coulomb energies in the ΣNNN admixed components is also negative and uniformly small with size of a few keV at most.
The values assigned to ∆V C in the ΛΣ model use values from Ref. [9] for ∆V Λ C and the estimate ∆V Σ C ≈ − 2 3 P Σ E C ( 3 He) for ∆V Σ C , where E C ( 3 He) = 644 keV is the Coulomb energy of 3 He.
The next contribution, ∆V Y N , is derived from V CSB ΛN . No ∆V Y N contributions are available from the coupled channels calculation by Hiyama et al. [20] (not listed here) and also from the recent chiral-model calculation in which CSB contributions are disregarded [12] in order to remain consistent with EFT power counting rules that exclude CSB from the NLO chiral version of the Y N interaction [21]. With the exception of the purely ΛNNN four-body calculation of Ref. [9], all those models for which a nonzero value is listed in the table effectively used Eq. (5) to evaluate ∆V Y N (0 + g.s. ). This ensures that meson exchanges arising from Λ − Σ 0 mixing beyond OPE are also included in the calculated CSB contribution. Generally, the CSB potential contribution ∆V Y N (0 + g.s. ) is not linked in any simple model-independent way to the Σ admixture probability P Σ (0 + g.s. ). For example, the calculations using NSC97 [10,11] produce too little CSB contributions, whereas the present ΛΣ model, in spite of its weaker Σ admixtures, gives sizable contributions which essentially resolve the CSB puzzle in the 0 + g.s. of the A = 4 hypernuclei. Indeed, using a typical ΛΣ strong-interaction matrix element NΣ|V (0 + g.s. )|NΛ ∼ 7 MeV in Eq. (5) one obtains P Σ (0 + g.s. ) = 0.77% and a CSB contribution of 240 keV to ∆B Λ (0 + g.s. ); this CSB contribution is proportional to √ P Σ in the present ΛΣ model. The resulting values of ∆B 4 Λ (0 + g.s. ) listed in Table 2 are smaller than 100 keV within the calculations presented in Refs. [9,10,11,12], leaving the A = 4 CSB puzzle unresolved, while being larger than 200 keV in the present ΛΣ model and thereby getting considerably closer to the experimentally reported 0 + g.s. CSB splitting. The main difference between these two groups of calculations arises from the difference in the CSB potential contributions ∆V Y N (0 + g.s. ). A similarly large difference also appears between the CSB potential negative contributions ∆V Y N (1 + exc ) in the calculations of Refs. [9,10,11] [9,10,11]. A common feature of all CSB model calculations so far is that none of them is able to generate values in excess of 50 keV for ∆B 4 Λ (1 + exc ). A direct comparison between the NCS97 models and the present ΛΣ model is not straightforward because the ΛΣ coupling in NSC97 models is dominated by tensor components, whereas no tensor components appear in present ΛΣ model. It is worth noting, however, that the ρ exchange contribution to the matrix element NΣ|V |NΛ in Eq. (5) is of opposite sign to that of OPE for the tensor ΛΣ coupling which dominates in NSC models, leading to cancellations, whereas both ρ exchange and OPE contribute constructively in the present central ΛΣ coupling model in agreement with the calculation by Coon et al. [9] which also has no tensor components. 2 This point deserves further study by modeling various input Y N interactions in future four-body calculations.

CSB in p-shell hypernuclei
Several few-body cluster-model calculations, of the A = 7, I = 1 isotriplet [22] and the A = 10, I = 1 2 isodoublet [23], have considered the issue of CSB contributions to Λ binding energy differences of p-shell mirror hypernuclei. It was verified in these calculations that the introduction of a ΛN phenomenological CSB interaction fitted to ∆B 4 Λ , for both 0 + g.s. and 1 + exc states, failed to reproduce the observed ∆B A Λ values in these p-shell hypernuclei; in fact, it only aggravated the discrepancy between experiment and calculations. Although it is possible to reproduce the observed values by introducing additional CSB components that hardly affect ∆B 4 Λ , this prescription lacks any physical origin and is therefore questionable, as acknowledged very recently by Hiyama [15]. Here we explore p-shell CSB contributions, extending the NSC97e model 0s N 0s Y effective interactions considered in Sect. 3, by providing (ΛΣ) e 0p N 0s Y central-interaction matrix elements which are consistent with the role ΛN ↔ ΣN coupling appears to play in a shell-model reproduction of hypernuclear γ-ray transition energies [24]: These p-shell matrix elements are smaller by roughly a factor of two from the corresponding s-shell matrix elements in Table 1, reflecting a reduced weight, about 1/2, with which the dominant relative s-wave matrix elements of V N Y appear in the p shell. This suggests that Σ admixtures which are quadratic in these matrix elements, are weaker roughly by a factor of 4 with respect to the s-shell calculation, and also smaller CSB interaction contributions in the p shell with respect to those in the A = 4 hypernuclei, although only by a factor of 2. To evaluate these CSB contributions, instead of applying the one-nucleon or nucleon-hole expression (5) valid in the s shell, we use in the p shell the general multi-nucleon expression for V CSB ΛN obtained by summing over p-shell nucleons: Results of applying the present (ΛΣ) e coupling model to several pairs of g.s. levels in p-shell hypernuclear isomultiplets are given in Table 3 [30] were not used for lack of similar data on their mirror partners.  Table 2 are also listed for comparison. The Σ admixture percentages P Σ in Table 3 follow from ΛΣ stronginteraction contributions to p-shell hypernuclear g.s. energies computed in Ref. [24], and the associated CSB kinetic-energy contributions ∆T Y N were calculated using a straightforward generalization of Eq. (8). These contributions, of order 10 keV and less, are considerably weaker than the ∆T Y N contributions to ∆B 4 Λ listed in Table 2, reflecting weaker Σ admixtures in the p shell as discussed following Eq. (9). The Coulomb-induced contributions ∆V C are dominated by their ∆V Λ C components which were taken from Hiyama's cluster-model calculations [22,26] for A = 7, 8 and from Millener's shell-model calculations [27] for A = 9, 10. The shell-model estimate of −156 keV adopted here for A = 10 is somewhat smaller than the −180 keV cluster-model result [23]. The ∆V Σ C components are negligible, with size of 1 keV at most (for A = 8,9). ∆V Λ C is always negative, as expected from the increased Coulomb repulsion owing to the increased proton separation energy in the Λ hypernucleus with respect to its core. The sizable negative p-shell ∆V C contributions, in distinction from their secondary role in forming the total ∆B 4 Λ (0 + g.s. ), exceed in size the positive p-shell ∆V Y N contributions by a large margin beginning with A = 9, thereby resulting in clearly negative values of ∆B A Λ (g.s.). The CSB ∆V Y N contributions listed in Table 3 were calculated using weak-coupling Λ-hypernuclear shell-model wavefunctions in terms of the corresponding nuclear-core g.s. leading SU(4) supermultiplet components, except for A = 8 where the first excited nuclear-core level had to be included. This proved to be a sound and useful approximation, yielding ΛΣ stronginteraction contributions close to those given in Figs. 1-3 of Ref. [24]. 3 Details will be given elsewhere. The listed A = 7 − 10 values of ∆V Y N exhibit strong SU(4) correlations, marked in particular by the enhanced value of 119 keV for the SU(4) nucleon-hole configuration in 8 Λ Be-8 Λ Li with respect to the modest value of 17 keV for the SU(4) nucleon-particle configuration in 10 Λ B-10 Λ Be. This enhancement follows from the relative magnitudes of the Fermi-like interaction termV 0p ΛΣ and its Gamow-Teller partner term ∆ 0p ΛΣ in Eq. (9). Noting that both A = 4 and A = 8 mirror hypernuclei correspond to SU(4) nucleon-hole configuration, the roughly factor two ratio of ∆V Y N (A = 4) = 232 keV to ∆V Y N (A = 8) = 119 keV reflects the approximate factor of two for the ratio between s-shell to p-shell ΛΣ matrix elements, as discussed following Eq. (9).
Comparing ∆B calc Λ with ∆B exp Λ in Table 3, we note the reasonable agreement reached between the present (ΛΣ) e coupling model calculation and experiment for all four pairs of p-shell hypernuclei, A = 7 − 10, considered in this work. Extrapolating to heavier hypernuclei, one might naively expect negative values of ∆B calc Λ , as suggested by the listed A = 9, 10 values. However, this rests on the assumption that the negative ∆V Λ C contribution remains as large upon increasing A as it is in the beginning of the p shell, which need not be the case. As nuclear cores beyond A = 9 become more tightly bound, the Λ hyperon is unlikely to compress these nuclear cores as much as it does in lighter hypernuclei, so that the additional Coulomb repulsion in 12 Λ C, for example, over that in 12 Λ B, while still negative, may not be sufficiently large to offset the attractive CSB contribution. In making this argument we rely on the expectation, based on SU(4) supermultiplet fragmentation patterns in the p shell, that ∆V Y N does not exceed ∼100 keV.
Before closing the discussion of CSB in p-shell hypernuclei, we wish to draw attention to the state dependence of CSB splittings, recalling the vast difference between the calculated ∆B 4 Λ (0 + g.s. ) and ∆B 4 Λ (1 + exc ) in the s shell. In Table 4 we list CSB contributions ∆E CSB ΛΣ to several g.s. doublet excitation energies, as well the excitation energies ∆E CS calculated by Millener [24] using charge symmetric (CS) YN spin-dependent interactions, including CS ΛΣ contributions ∆E CS ΛΣ (also listed). It is tacitly assumed that ∆V Λ C is state independent for the hypernuclear g.s. doublet members. As for the other, considerably smaller contributions, we checked that ∆V Σ C remains at the 1 keV level and that the difference between the appropriate Σ-dominated ∆T Y N values is less than 10 keV. Under these circumstances, it is sufficient to limit the discussion to the state dependence of ∆V Y N alone, although the splittings ∆E CSB ΛΣ listed in the table include these other tiny contributions. Inspection of Table 4 reveals that whereas CSB contributions ∆E CSB ΛΣ are negligible in 9 Λ Li, with respect to both ∆E CS ΛΣ and to the total CS splitting ∆E CS , they need to be incorporated in re-evaluating the g.s. doublet splittings in 8 Λ Li and in 10 Λ B.
• In 8 Λ Li, these ∆E CSB ΛΣ contributions spoil the perhaps fortuitous agreement between ∆E exp , deduced from a tentative assignment of a γ-ray transition observed in the 10 B(K − , π − ) 10 Λ B reaction continuum spectrum [32], and ∆E CS evaluated using the Y N spin-dependent interaction parameters deduced from well identified γ-ray transitions in other hypernuclei. The 50 keV discrepancy arising from adding ∆E CSB ΛΣ surpasses significantly the typical 20 keV theoretical uncertainty in fitting doublet splittings in p-shell hypernuclei (see Table 1, Ref. [24]).
• The inclusion of ∆E CSB ΛΣ in the calculated 10 Λ B g.s. doublet splitting helps solving the longstanding puzzle of not observing the 2 − exc → 1 − g.s.
γ-ray transition, thereby placing an upper limit of 100 keV on this transition energy [32,33]. Including our CSB calculated contribution would indeed lower the expected transition energy from 110 keV to about 85 keV, in accordance with the experimental upper limit. 4 It might appear unnatural that ∆E CSB ΛΣ is calculated to be a sizable fraction of ∆E CS ΛΣ in 8 Λ Li, or even exceed it in 10 Λ B. This may be understood noting that the evaluation of ∆E CSB ΛΣ involves a CSB small parameter of ∼0.03, see Eq. (5), whereas the evaluation of ∆E CS ΛΣ involves a small parameter of √ P Σ which is less than 0.05 for 8 Λ Li and less than 0.025 for 10 Λ B in our (ΛΣ) e coupling model, see Table 3.

Conclusion
It was shown in this work how a relatively large CSB contribution of order 250 keV arises in (ΛΣ) coupling models based on Akaishi's centralinteraction G-matrix calculations in s-shell hypernuclei [16,17], coming close to the binding energy difference B Λ ( 4 Λ He)−B Λ ( 4 Λ H) = 350 ± 60 keV deduced from emulsion studies [3]. It was also argued that the reason for most of the Y NNN coupled-channel calculations done so far to come out considerably behind, with 100 keV at most by using NSC97 f , is that their ΛΣ channel coupling is dominated by strong tensor interaction terms. In this sense, the 4 Adding CSB is not essential in the 10 Λ B ααpΛ cluster-model calculation [23] that results in 80 keV g.s. doublet excitation using CS ΛN interactions. However, to do a good job on the level of ∼20 keV, one needs to include α-breakup ΛN contributions which are missing in this calculation.
CSB-dominated large value of ∆B 4 Λ (0 + g.s. ) places a powerful constraint on the strong-interaction Y N dynamics.
In spite of the schematic nature of the present (ΛΣ) coupling model of the A = 4 hypernuclei, which undoubtedly does not match the high standards of solving coupled-channel four-body problems, this model has the invaluable advantage of enabling a fairly simple application to heavier hypernuclei, where it was shown to reproduce successfully the main CSB features as disclosed from the several measured binding energy differences in p-shell mirror hypernuclei. More quantitative work, particularly for the 12 Λ C-12 Λ B mirror hypernuclei, has to be done in order to confirm the trends established here in the beginning of the p shell upon relying exclusively on data reported from emulsion studies. Although the required calculations are rather straightforward, a major obstacle in reaching unambiguous conclusions is the unavailability of alternative comprehensive and accurate measurements of g.s. binding energies in mirror hypernuclei that may replace the existing old emulsion data.