Charge symmetry breaking in hypernuclei revisited
Abstract
The large charge symmetry breaking (CSB) implied by the binding energy difference (He)(H) MeV of the mirror hypernuclei ground states, determined from emulsion studies, has defied theoretical attempts to reproduce it in terms of CSB in hyperon masses and in hyperonnucleon interactions, including one pion exchange arising from mixing. Using a schematic stronginteraction coupling model developed by Akaishi and collaborators for shell hypernuclei, we revisit the evaluation of CSB in the hypernuclei and extend it to shell mirror hypernuclei. The model yields values of MeV. Smaller size and mostly negative shell binding energy differences are calculated for the mirror hypernuclei, in rough agreement with the few available data. CSB is found to reduce by almost 30 keV the 110 keV B g.s. doublet splitting anticipated from the hyperonnucleon stronginteraction spin dependence, thereby explaining the persistent experimental failure to observe the ray transition.
keywords:
charge symmetry breaking, hypernuclei, effective interactions in hadronic systems, shell model1 Introduction
Charge symmetry breaking (CSB) in nuclear physics is primarily identified by considering the difference between and scattering lengths, or the bindingenergy difference between the mirror nuclei H and He Miller06 (). In these nuclei, about 70 keV out of the Coulombdominated 764 keV bindingenergy difference is commonly attributed to CSB which can be explained either by mixing in oneboson exchange models of the interaction, or by considering intermediatestate mass differences in models limited to pseudoscalar meson exchanges Mach01 ().
In hypernuclei, in contrast, CSB appears to be considerably stronger, judging by the bindingenergy difference =0.350.06 MeV deduced from the level diagram of the mirror hypernuclei (H, He) in Fig. 1. A very recent measurement of HHe+ decay at MAMI MAMI15 () reduces to 0.270.10 MeV, consistent with its emulsion value Davis05 (). Fig. 1 also suggests that is almost as large as . However, the deduction of the excitation energy in He from the 1.15 MeV ray transition Bedj79 () is not as firm as the one for H Tamura13 (). In passing we mention the weak 1.42 MeV ray transition reported in Ref. Bamb73 () that would imply almost no CSB splitting of the states if its assignment to He gets confirmed.
The large values reported for both and states have defied theoretical attempts to explain these differences in terms of hadronic or quark CSB mechanisms within fourbody calculations Carlson91 (); Coon99 (); Nogga02 (); Nogga07 (); Nogga13 (). Meson mixing, including mixing which explains CSB in the nuclei, gives only small negative contributions about and keV for and , respectively Coon99 (). CSB contributions to from one and twopion exchange interactions in coupledchannel calculations Nogga02 (); Nogga07 () with hyperons amount to as much as 100 keV; this holds for the OBEbased Nijmegen NSC97 models NSC97 () which are widely used in hypernuclear structure calculations.^{1}^{1}1Contradictory statements were made in Refs. Carlson91 (); Nogga02 () on the ability of the earlier Nijmegen model NSC89 NSC89 () to reproduce . We note that the strength of CSB contributions in this work, on p. 2236, is inflated erroneously by a factor , about 3.7 for pion exchange, which might have propagated into some of the calculations claiming to have resolved the CSB puzzle for using NSC89.
Binding energies of ground states in shell mirror hypernuclei, determined from emulsion studies Davis05 (), suggest much weaker CSB effects than for , with values of () consistent with zero for and somewhat negative beyond Davis05 (). Accommodating values in the shell with by using reasonable phenomenological CSB interactions is impossible, as demonstrated in recent fourbody clustermodel calculations of shell hypernuclei Hiyama13 (). This difficulty may be connected to the absence of explicit coupling in these clustermodel calculations, given that such explicit coupling was shown to generate nonnegligible CSB contributions to Nogga13 (). It is our purpose in this note to use a schematic coupling model, proposed by Akaishi et al. Akaishi00 (); Akaishi02 () for shell hypernuclei and extended by Millener Millener08 () to the shell, for calculating values in both and shells, thereby making predictions on CSB effects in shell hypernuclei consistently with a relatively sizable value of .
The paper is organized as follows. In Sect. 2, we update the original treatment by Dalitz and Von Hippel DvH64 () of the mixing mechanism for generating CSB onepion exchange contributions in hypernuclei, linking it to the stronginteraction coupling model employed in this work. Our CSB calculations for the hypernuclei are sketched in Sect. 3 and their results are compared with those reported in several fourbody calculations Coon99 (); Nogga02 (); Nogga07 (); Nogga13 (). Finally, CSB contributions in shell mirror hypernuclei, evaluated here for the first time, are reported in Sect. 4.
2 Pionic CSB contributions in hypernuclei
The isoscalar nature of the hyperon forbids it to emit or absorb a single pion, and hence there is no onepion exchange (OPE) contribution to the strong interaction. However, by allowing for mixing in SU(3), a CSB OPE contribution arises DvH64 () with coupling constant
(1) 
where the matrix element of the mass mixing operator is given by
(2) 
The resulting CSB OPE potential is given by
(3) 
where the z component of the isospin Pauli matrix assumes the values on protons and neutrons, respectively, is a Yukawa form, and the tensor contribution is specified by
(4) 
In Eq. (3), the transition was made in accordance with NSC models, using from NSC89, Table IV in Ref NSC89 ().
Since Paulispin and hold in He() and in H(), respectively, the CSB potential (3) which is linear in the nucleon spin gives no contribution from these samecharge nucleons. Therefore owing to the odd nucleon in these hypernuclei, respectively, and since holds for this odd nucleon, one gets a positive contribution from the central spinspin part which provides the only nonvanishing contribution for simple wavefunctions. For the wavefunction used by Dalitz and Von Hippel DvH64 (), and updating the values of the coupling constants used in their work to those used here, one gets keV, a substantial single contribution with respect to keV.
The mixing mechanism gives rise also to a variety of (e.g. ) meson exchanges other than OPE. In baryonbaryon models that consider explicitly the stronginteraction coupling, the matrix element of is related to a suitably chosen stronginteraction isospin matrix element by generalizing Eq. (1):
(5) 
where the isospin ClebschGordan coefficient accounts for the amplitude in the state, and the spacespin structure of this state is taken identical with that of the state sandwiching .
3 CSB in the hypernuclei
NSC97  

model  as calculated in models Millener08 ()  Nogga02 ()  Akaishi00 ()  Akaishi02 ()  
NSC97  2.96  5.09  0.574  0.036  0.539  0.75  0.89  1.13 
NSC97  3.35  5.76  0.735  0.046  0.689  1.10  1.48  1.51 
Following hyperoncore calculations of shell hypernuclei by Akaishi et al. Akaishi00 () we use matrix effective interactions derived from NSC97 models to calculate CSB contributions from Eq. (5). The effective interaction is given in terms of a spindependent central interaction
(6) 
where converts a to in isospace. The shell matrix elements and are listed in Table 1 for two such matrix models denoted . Also listed are the calculated downward energy shifts defined by , where the matrix elements for are given in terms of twobody matrix elements by
(7) 
We note that the diagonal interaction matrix elements have little effect in this coupledchannel model because of the large energy denominators of order MeV with which they appear. Finally, by listing from Refs. Akaishi00 (); Akaishi02 () we demonstrate the sizable contribution of coupling to the excitation energy MeV deduced from the ray transition energies marked in Fig. 1. For comparison, the full in these models, and as calculated by Nogga Nogga02 () using the underlying Nijmegen models NSC97, are also listed in the table.
Having discussed the effect of stronginteraction coupling, we now discuss the CSB splittings and . Results of our 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 fourbody calculations Coon99 (); Nogga02 (); Nogga07 (); Nogga13 (). Partial contributions to are listed in columns 2–5, whereas for only its total value is listed.
He–H  

model  
Coon99 ()  –  –  42  91  49  61 
NSC97 Nogga02 ()  1.6  47  16  44  75  10 
NSC97 Nogga07 ()  1.8  100  10  
NLO chiral Nogga13 ()  2.1  55  9  –  46  
[present]  0.72  39  45  232  226  30 
[present]  0.92  49  46  263  266  39 
All of the models listed in Table 2 except for Coon99 () include coupling, with admixture probabilities in (He, H) respectively, and . The admixtures (unlisted) are considerably weaker than the listed admixtures. Charge asymmetric kineticenergy contributions to , dominated by intermediatestate mass differences, are marked in the table. In the present coupling model these are given for the by Nogga02 ()
(8) 
yielding as much as 50 keV, in agreement with those fourbody calculations where such mass differences were introduced Nogga02 (); Nogga07 (); Nogga13 (). The next column in the table, , addresses contributions arising from nuclearcore Coulomb energy modifications induced by the hyperons. is negative, its size ranges from less than 10 keV Nogga02 (); Nogga13 () to about 40 keV Coon99 (). which accounts for Coulomb energies in the admixed components is also negative and uniformly small with size of a few keV at most. The values assigned to in the model use values from Ref. Coon99 () for and the estimate for , where keV is the Coulomb energy of He.
The next contribution, , is derived from . No contributions are available from the coupled channels calculation by Hiyama et al. Hiyama02 () (not listed here) and also from the recent chiralmodel calculation in which CSB contributions are disregarded Nogga13 () in order to remain consistent with EFT power counting rules that exclude CSB from the NLO chiral version of the interaction NLO13 (). With the exception of the purely fourbody calculation of Ref. Coon99 (), all those models for which a nonzero value is listed in the table effectively used Eq. (5) to evaluate . This ensures that meson exchanges arising from mixing beyond OPE are also included in the calculated CSB contribution. Generally, the CSB potential contribution is not linked in any simple modelindependent way to the admixture probability . For example, the calculations using NSC97 Nogga02 (); Nogga07 () produce too little CSB contributions, whereas the present model, in spite of its weaker admixtures, gives sizable contributions which essentially resolve the CSB puzzle in the of the hypernuclei. Indeed, using a typical stronginteraction matrix element MeV in Eq. (5) one obtains and a CSB contribution of 240 keV to ; this CSB contribution is proportional to in the present model.
The resulting values of listed in Table 2 are smaller than 100 keV within the calculations presented in Refs. Coon99 (); Nogga02 (); Nogga07 (); Nogga13 (), leaving the CSB puzzle unresolved, while being larger than 200 keV in the present model and thereby getting considerably closer to the experimentally reported CSB splitting. The main difference between these two groups of calculations arises from the difference in the CSB potential contributions . A similarly large difference also appears between the CSB potential negative contributions in the calculations of Refs. Coon99 (); Nogga02 (); Nogga07 () and the positive contributions in the present model, resulting in large but different values, about 200 keV in the present model and about 100 keV for all other calculations Coon99 (); Nogga02 (); Nogga07 (). 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 .
A direct comparison between the NCS97 models and the present model is not straightforward because the coupling in NSC97 models is dominated by tensor components, whereas no tensor components appear in present model. It is worth noting, however, that the exchange contribution to the matrix element in Eq. (5) is of opposite sign to that of OPE for the tensor coupling which dominates in NSC models, leading to cancellations, whereas both exchange and OPE contribute constructively in the present central coupling model in agreement with the calculation by Coon et al. Coon99 () which also has no tensor components.^{2}^{2}2It is worth noting that the exchange CSB contribution calculated in Ref. Coon99 () is of the same sign and remarkably stronger than the OPE CSB contribution. This point deserves further study by modeling various input interactions in future fourbody calculations.
4 CSB in shell hypernuclei
Several fewbody clustermodel calculations, of the , isotriplet Hiyama09 () and the , isodoublet Hiyama12 (), have considered the issue of CSB contributions to binding energy differences of shell mirror hypernuclei. It was verified in these calculations that the introduction of a phenomenological CSB interaction fitted to , for both and states, failed to reproduce the observed values in these 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 , this prescription lacks any physical origin and is therefore questionable, as acknowledged very recently by Hiyama Hiyama13 (). Here we explore shell CSB contributions, extending the NSC97e model effective interactions considered in Sect. 3, by providing centralinteraction matrix elements which are consistent with the role coupling appears to play in a shellmodel reproduction of hypernuclear ray transition energies Millener12 ():
(9) 
These shell matrix elements are smaller by roughly a factor of two from the corresponding shell matrix elements in Table 1, reflecting a reduced weight, about 1/2, with which the dominant relative wave matrix elements of appear in the shell. This suggests that admixtures which are quadratic in these matrix elements, are weaker roughly by a factor of 4 with respect to the shell calculation, and also smaller CSB interaction contributions in the shell with respect to those in the hypernuclei, although only by a factor of 2. To evaluate these CSB contributions, instead of applying the onenucleon or nucleonhole expression (5) valid in the shell, we use in the shell the general multinucleon expression for obtained by summing over shell nucleons:
(10) 
Results of applying the present coupling model to several pairs of g.s. levels in shell hypernuclear isomultiplets are given in Table 3. All pairs except for are mirror hypernuclei identified in emulsion Davis05 () where binding energy systematic uncertainties are largely canceled out in forming the listed values. For we compensated for the unavailability of a reliable (He) value from emulsion by replacing it with (Li), established by observing the 3.88 MeV ray transition Li+Li Tamura00 (). Recent values determined in electroproduction experiments at JLab for He JLab13 (); Gogami14 (), Li JLab14 () and Be Gogami14 () were not used for lack of similar data on their mirror partners.
Z–Z  Davis05 ()  

pairs  (%)  (keV)  (keV)  (keV)  (keV)  (keV)  
He–H  0.72  39  45  232  226  35060  
Be–Li  0.12  3  70 Hiyama09 ()  50  17  10090  
Be–Li  0.20  11  81 Hiyama02a ()  119  49  4060  
B–Li  0.23  10  145 Millener15 ()  81  54  210220  
B–Be  0.053  3  156 Millener15 ()  17  136  220250 
The admixture percentages in Table 3 follow from stronginteraction contributions to shell hypernuclear g.s. energies computed in Ref. Millener12 (), and the associated CSB kineticenergy contributions were calculated using a straightforward generalization of Eq. (8). These contributions, of order 10 keV and less, are considerably weaker than the contributions to listed in Table 2, reflecting weaker admixtures in the shell as discussed following Eq. (9). The Coulombinduced contributions are dominated by their components which were taken from Hiyama’s clustermodel calculations Hiyama09 (); Hiyama02a () for and from Millener’s shellmodel calculations Millener15 () for . The shellmodel estimate of 156 keV adopted here for is somewhat smaller than the 180 keV clustermodel result Hiyama12 (). The components are negligible, with size of 1 keV at most (for ). is always negative, as expected from the increased Coulomb repulsion owing to the increased proton separation energy in the hypernucleus with respect to its core. The sizable negative shell contributions, in distinction from their secondary role in forming the total , exceed in size the positive shell contributions by a large margin beginning with , thereby resulting in clearly negative values of .
The CSB contributions listed in Table 3 were calculated using weakcoupling hypernuclear shellmodel wavefunctions in terms of the corresponding nuclearcore g.s. leading SU(4) supermultiplet components, except for where the first excited nuclearcore level had to be included. This proved to be a sound and useful approximation, yielding stronginteraction contributions close to those given in Figs. 1–3 of Ref. Millener12 ().^{3}^{3}3I am indebted to John Millener for providing me with some of the wavefunctions required here. Details will be given elsewhere. The listed values of exhibit strong SU(4) correlations, marked in particular by the enhanced value of 119 keV for the SU(4) nucleonhole configuration in Be–Li with respect to the modest value of 17 keV for the SU(4) nucleonparticle configuration in B–Be. This enhancement follows from the relative magnitudes of the Fermilike interaction term and its GamowTeller partner term in Eq. (9). Noting that both and mirror hypernuclei correspond to SU(4) nucleonhole configuration, the roughly factor two ratio of keV to keV reflects the approximate factor of two for the ratio between shell to shell matrix elements, as discussed following Eq. (9).
Comparing with in Table 3, we note the reasonable agreement reached between the present coupling model calculation and experiment for all four pairs of shell hypernuclei, , considered in this work. Extrapolating to heavier hypernuclei, one might naively expect negative values of , as suggested by the listed values. However, this rests on the assumption that the negative contribution remains as large upon increasing as it is in the beginning of the shell, which need not be the case. As nuclear cores beyond become more tightly bound, the hyperon is unlikely to compress these nuclear cores as much as it does in lighter hypernuclei, so that the additional Coulomb repulsion in C, for example, over that in 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 shell, that does not exceed 100 keV.
Before closing the discussion of CSB in shell hypernuclei, we wish to draw attention to the state dependence of CSB splittings, recalling the vast difference between the calculated and in the shell. In Table 4 we list CSB contributions to several g.s. doublet excitation energies, as well the excitation energies calculated by Millener Millener12 () using charge symmetric (CS) YN spindependent interactions, including CS contributions (also listed). It is tacitly assumed that is state independent for the hypernuclear g.s. doublet members. As for the other, considerably smaller contributions, we checked that remains at the 1 keV level and that the difference between the appropriate dominated values is less than 10 keV. Under these circumstances, it is sufficient to limit the discussion to the state dependence of alone, although the splittings listed in the table include these other tiny contributions.
Z  Millener12 ()  Millener12 ()  

Li  4422 Chrien90 ()  445  149  53.2  
Li  570120 JLab14 ()  590  116  12.5  
B  Chrien90 (); Tamura05 ()  110  10  26.5 
Inspection of Table 4 reveals that whereas CSB contributions are negligible in Li, with respect to both and to the total CS splitting , they need to be incorporated in reevaluating the g.s. doublet splittings in Li and in B.

In Li, these contributions spoil the perhaps fortuitous agreement between , deduced from a tentative assignment of a ray transition observed in the B reaction continuum spectrum Chrien90 (), and evaluated using the spindependent interaction parameters deduced from well identified ray transitions in other hypernuclei. The 50 keV discrepancy arising from adding surpasses significantly the typical 20 keV theoretical uncertainty in fitting doublet splittings in shell hypernuclei (see Table 1, Ref. Millener12 ()).

The inclusion of in the calculated B g.s. doublet splitting helps solving the longstanding puzzle of not observing the ray transition, thereby placing an upper limit of 100 keV on this transition energy Chrien90 (); Tamura05 (). 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}^{4}4Adding CSB is not essential in the B clustermodel calculation Hiyama12 () that results in 80 keV g.s. doublet excitation using CS interactions. However, to do a good job on the level of 20 keV, one needs to include breakup contributions which are missing in this calculation.
It might appear unnatural that is calculated to be a sizable fraction of in Li, or even exceed it in B. This may be understood noting that the evaluation of involves a CSB small parameter of 0.03, see Eq. (5), whereas the evaluation of involves a small parameter of which is less than 0.05 for Li and less than 0.025 for B in our coupling model, see Table 3.
5 Conclusion
It was shown in this work how a relatively large CSB contribution of order 250 keV arises in () coupling models based on Akaishi’s centralinteraction matrix calculations in shell hypernuclei Akaishi00 (); Akaishi02 (), coming close to the binding energy difference (He)(H) keV deduced from emulsion studies Davis05 (). It was also argued that the reason for most of the coupledchannel calculations done so far to come out considerably behind, with 100 keV at most by using NSC97, is that their channel coupling is dominated by strong tensor interaction terms. In this sense, the CSBdominated large value of places a powerful constraint on the stronginteraction dynamics.
In spite of the schematic nature of the present () coupling model of the hypernuclei, which undoubtedly does not match the high standards of solving coupledchannel fourbody 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 shell mirror hypernuclei. More quantitative work, particularly for the C–B mirror hypernuclei, has to be done in order to confirm the trends established here in the beginning of the 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.
Acknowledgements
Stimulating and useful exchanges with Patrick Achenbach, Ben Gibson, Johann Haidenbauer, Emiko Hiyama, Ruprecht Machleidt, John Millener, Andreas Nogga, Thomas Rijken and Hirokazu Tamura are gratefully acknowledged as well as financial support by the EU initiative FP7, HadronPhysics3, under the SPHERE and LEANNIS cooperation programs.
Note added after publication
A recent LQCD work QCD15 () evaluates the – mixing matrix element to be smaller by about factor of 2 than the one used in the present work upon substituting PDG PDG14 () baryon mass values in the DalitzVon Hippel DvH64 () SU(3)symmetry expression (2) which we copy here,
(11) 
The resolution of this apparent discrepancy is that this much smaller value of in Ref. QCD15 () is obtainable simply by substituting LQCDcalculated mass differences in Eq. (11), particularly MeV which is about twice larger than the PDG value 1.29 MeV and which comes with a negative sign in (11). We conclude that this LQCD determination of – mixing is no better than their determination of the neutronproton mass difference.
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