1 0
E
ex[M eV ]
1/2+3
He
4He
-0.66
NLO13
0.22
-0.44
NLO19
0.15
-0.83
FIT A
0.28
-0.97
FIT B
0.32
-1.09
FIT C
0.36
-1.05 0+ Exp
0.35 1+
Figure 4.31: Energy spectrum of4ΛHe. Calculations are based on chiral SMS N4LO+(450) withλNN=1.6 fm-1 and the YN NLO13, NLO19 interactions and FIT A-C for a wide range of chiral cutoffsΛY =500−650 MeV.
The experimental values are taken from [8,123].
possibly indicates a large influence of chiral 3BFs on the levels. Also, the NLO13 and NLO19 lead to slightly different predictions further reinforcing that 3BFs are non-negligible for the excitation energies. Let us further stress that the P-wave interactions of all considered NLO forces are by construction identical and small. We found that neglecting P- and higher partial waves in the interactions changes the energies only marginally, well within our regulator dependence.
In general all of the considered interactions qualitatively reproduce the7ΛLi spectrum. Quant-itatively, however, none of the interactions is able to describe the experiment. For example, we find that the predicted 5/2+state of7ΛLi is located above the 3+state of6Li whereas the ordering is opposite for the experimental values. While the two potentials NLO13 and NLO19 predicts the correct splittings of the two lowest doublets, the new fits A to C, however, further increase the splittings bringing them away from the experimental values. Nevertheless the deviations are minor when taking into account the possible contributions from 3BFs. In any case, the result show that changes of singlet scattering length (and consequently the hypertriton binding energies) indeed affect the spectra of p-shell hypernuclei. However, the changes are moderate and, therefore, the separation energy and spectrum remains qualitatively consistent with experiment for the new fits.
4.7 A=7 isotriplet and CSB splittings
It has been demonstrated in Fig.4.26that without CSB terms included in the YN interactions, the CSB splittings of theA = 4 hypernuclei (4ΛHe and 4ΛH) are almost invisible. Let us also remind that the s-shell CSB effect arising from the Coulomb interaction and different masses within the isospin multiplet of nucleons and Σ’s are tiny [129, 130]. However, these calculations are an important baseline for starting investigations of CSB in hypernuclei (see works by Gazda [142]
and Nogga [130] for recent progresses). Obviously, such investigations need to be carried out into the p-shell hypernuclei. Undoubtedly, the A = 7 isotriplet hypernuclei,7ΛHe(12+,1), 7ΛLi*(12+,1)
Chapter 4 Results forA=4−7Hypernuclei
1 0 1 2 3 4
E
ex[M eV ]
1+
Exp
6
Li
7Li
3+ 2.19 0+1 3.56
J-NCSM
2.91 4.14
-0.48
NLO13
0.24 2.56 3.02 3.94
-0.34
NLO19
0.17 2.69 2.94 3.98
-0.71
FIT A
0.35 2.42 3.17 3.93
-0.83
FIT B
0.41 2.32 3.24 3.93
-0.93
FIT C
0.47 2.21 3.30 3.95
-0.46 1/2+ Exp
0.23 3/2+ 5/2+
1.59 2.06 7/2+
3.42 1/2+1
Figure 4.32: Energy spectra of7ΛLi and6Li. Same NN and YN interactions as in ]. Calculations are based on chiral SMS N4LO+(450) withλNN =1.6 fm-1and the YN NLO13, NLO19 and Fit A-C interactions for a range of regulatorΛY =500−650 MeV regulator. The experimental values are taken from [8,11,126].
and7ΛBe(12+,1) will be the most perfect candidates to start such a study. These systems are both experimentally and theoretically very interesting. Generally, hypernuclear binding energies are measured with rather large uncertainties, the situation is even worser for p-shell hypernuclei in particular 7ΛHe with only 16 events from emulsion experiments being reported so far [143].
Recently, the separation energy BΛ(7ΛHe) has been measured at JLab but with an uncomfortably large uncertainty, BΛ(7ΛHe)= 5.55±0.1 ±0.11 MeV [144] in comparison with that for 7ΛLi*: BΛ(7ΛLi*)=5.26±0.03 MeV [133,137] and7ΛBe: BΛ(7ΛBe)=5.16±0.08 MeV [137]. Theoretically, theΛ-separation energiesBΛof the isotriplet together with their energy level splittings can serve as key elements for studying the spin-dependence YN interactions [7,8] as well as the origin of CSB splitting in combination with the CSB observed in the s-shell hypernuclei [113]. This motivates us to also study theA= 7 isotriplet within the NCSM approach. With regard to the convergence of resonance states, we should stress that although the ground state of6Be(0+,1) is not a particle-stable state (decaying through the channel6Be→4He+p+p), it exhibits a very narrow resonance with the width ofΓ =92±6 keV [126]. Therefore, we can still obtain well-converged results for the ground states of both the nuclear core6Be and the corresponding hypernucleus7ΛBe within the J-NCSM approach even for model space size up toN =10.
Overall, we observe again the strong dependence of theΛ-separation energies for theA = 7 isotriplet on the regulator and on the SRG evolution, like for the other states of7ΛLi. There are also strong correlations between theBΛof the multiplet and that of the5ΛHe, as exemplified in Fig.4.33for the NLO13 potential. Furthermore, at the magic SRG cutoffs, the results predicted by the NLO13 and NLO19 are quite similar to each other, and importantly, comparable to the experiments. We provide in Table4.11theΛ-separation energies computed with the two potentials for a regulator ofΛY =700 MeV as an illustration. One sees that within the numerical uncertainties, the separation energies of the A=7 multiplet are actually the same. We also remark that our estimated errors are, however, rather large, of the same size as of the CSB estimated by Gal,∆BΛ(7ΛBe,7ΛLi∗)=−17 keV [113] or
4.7 A=7 isotriplet and CSB splittings
YN-NLO13 YN-NLO19 Exp
λmY N =0.904 fm-1 λmY N =0.87 fm-1
7ΛLi* 5.45±0.10 5.38±0.09 5.26±0.03
7ΛBe 5.44±0.09 5.38±0.08 5.16±0.08
7ΛHe 5.37±0.09 5.30±0.09 5.55±0.1±0.11
Table 4.11:Λ-separation energies of the A=7 isotriplet calculated with the NLO13 and NLO19 potentials for a regulator of 700 MeV in combination with the chiral SMS N4LO+(450) evolved to an SRG cutoffof λNN =1.6 fm-1. The NLO13 and NLO19 are also evolved to their magic cutoffs at which5ΛHe is properly described. The experiment values are from [133,137,144]. All energies are given in MeV.
4 5 6 7 8 9 10 11
B (7X)
2 3 4 5 6 7
B(5He)
Bex(7Li*)
Bex(7He) Bex(7Be)
7He
7Be
7Li*
5.25 5.50 3.050
3.075 3.100 3.125 3.150
4 5 6 7 8 9 10
B (7Li, (1/2+, 1))
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
B(5He)
Expt 500600 700550
5.0 5.5
2.8 3.0 3.2 3.4 Expt
Figure 4.33: Correlations ofΛ-separation energies for a wide range of flow parametersΛY of5ΛHe and (a) BΛ(7ΛX), where X=He, Li∗ and Be using the NLO13(700) interaction, (b)BΛ(7ΛLi∗) using the NLO13 for several regulators ofΛY =500−700 MeV. Same NN interaction as described in Table4.11.
deduced from the experimentalΛ-separation energies, e.g. ∆BexpΛ (7ΛBe,7ΛLi∗)= −100±90 keV [133, 137]. It is therefore more meaningfull to extract the CSB splittings of the multiplet from the separation energies calculated for a specific model space size (for example, the largest one i.e.N =10) instead of using the final extrapolatedBΛ. Before extracting CSB, let us first study the binding energiesEhypof the triplet for the largest model spaceN =10 in some details. To clarify the importance of different contributions to the binding (separation) energies, it is instructive to decomposeEhypinto the kinetic energyT, strong NN potentialVNNnucl, strong YN potentialVY Nnucland Coulomb interactions between a proton-proton (pp) pairVNNC and between a YN pairVY NC . Note that similar decompositions have also been considered by Nogga when studied the CSB effects of the s-shell hypernuclei [145]. The decomposition for7ΛLi*using the NLO13(700) potential evolved to several SRG flow parameters is tabulated in Table4.12. Similar tables are also obtained for the other two hypernuclei of the isotriplet which we refrain from showing here since they exhibit very
Chapter 4 Results forA=4−7Hypernuclei
λYN T VNNC VY NC VNNnucl VY Nnucl VΛnucl VΛΣnucl VΣnucl Ehyp [fm-1] [MeV] [MeV] [keV] [MeV] [MeV] [MeV] [MeV] [MeV] [MeV]
0.836 89.851 1.841 -2 -113.903 -11.767 -10.277 -1.122 -0.370 -33.981 1.0 93.944 1.871 -1 -116.880 -15.415 -14.861 -0.482 -0.072 -36.480 1.6 101.315 1.917 -1 -120.827 -22.012 -20.560 -1.374 -0.080 -39.608 2.0 102.399 1.920 -1 -120.938 -23.053 -20.167 -2.824 -0.063 -39.672 2.6 101.392 1.901 -2 -119.647 -22.109 -17.062 -4.894 0.065 -38.457 3.0 99.604 1.895 -2 -118.300 -20.205 -14.200 -6.192 0.186 -37.007 4.0 93.765 1.857 -2 -114.901 -13.218 -7.337 -6.288 0.405 -32.499 Table 4.12: Difference contributions to the binding energy of7ΛLi*for model spaceN =10. The calculations are based on the SMS N4LO+(450) with an SRG cutoffofλNN =1.6 fm-1and the YN NLO13(700) evolved to severalλY Nflow parameters.
similar trends to that of7ΛLi*.
Overall, the NN Coulomb interaction is small (about 1.9 MeV) and almost independent of the evolution. The similar Coulomb interaction (between a proton and aΣ)VY NC is much smaller, of order of several keV only, and it is attractive in7ΛLi*and7ΛHe but becomes repulsive in7ΛBe due to the difference betweenΣ+- andΣ−-probabilities in each system. In addition, at low SRG parameters (λY N ≤3.0 fm-1), theΛN−ΛN interaction,VΛnucl, dominantly contributes to the total YN potential, VY Nnucl, as compared to the total contributions from theΛN−ΣN andΣN−ΣN interactions,VΛΣnucland VΣnucl, respectively. However, at higher SRG cutoffs, the two potentialsVΛnuclandVΛΣnuclare practically of the same size. TheVΣnucl potential is general small (about several hundreds keV at most) but changes sign during the SRG evolution. Interestingly, we observe noticeable growths inVΣnucland VΛΣnuclwhen evolving from the flow parameter ofλY N = 1.0 toλY N =0.836 fm-1. Note that this is also consistent with the slight increase inPΣ that was observed before. During the evolution, the kinetic energyT and the strong NN interactionVNNnuclalso vary, but their effects largely cancel each other. As a result, the change in the total binding energyEhyp is mainly driven by the dependence of theVYnuclon the evolution parameter.
Analogously, the nuclear-core binding energies can also be decomposed into the kinetic energy T(core), strong NN potentialVNNnucl(core), and Coulomb interactions between a proton-proton (pp) pairVNNC (core). Now, based on the decompositions of the hypernuclear and nuclear-core binding energies, we can compute the CSB,∆CS B, between every two components of the multiplet, saying
4.7 A=7 isotriplet and CSB splittings
7ΛHe and7ΛLi*as follows
∆CS B(7ΛHe, 7ΛLi∗)= BΛ(7ΛHe)−BΛ(7ΛLi∗)
= E(6He)−E(6Li∗)− E(7ΛHe)−E(7ΛLi∗)
= T(6He)−T(6Li∗)− T(7ΛHe)−T(7ΛLi∗)
+ VNNnucl(6He)−VNNnucl(6Li∗)− VNNnucl(7ΛHe)−VNNnucl(7ΛLi∗) + VNNC (6He)−VNNC (6Li∗)− VNNC (7ΛHe)−VCNN(7ΛLi∗)
− VY Nnucl(7ΛHe)−VY Nnucl(7ΛLi∗)
− VY NC (7ΛHe)−VY NC (7ΛLi∗)
≡∆CSBT + ∆CSBVNN,nucl+ ∆CSBVNN,C+ ∆CSBVY N,nucl+ ∆CSBVY N,C.
(4.8)
We further estimate perturbatively the individual CSB contributions in Eq. (4.8), employing the two wavefunctions7ΛLi*and6Li*. At this exploratory stage, let us stick with specific NN and YN interactions, namely the SMS N4LO+(450) for the NN interaction evolved to an SRG parameter of λNN =1.6 fm-1and the NLO13(700) interaction. The YN potential is also SRG-evolved to a range of flow parameters, 0.836≤λY N ≤4.0 fm-1. The influence of the interaction models on the results will be investigated in future. Table4.13displays various contributions to the perturbative∆CSBand the exact CSB∆exactCSB splittings (the last two columns) between every two members of the isotriplet.
Clearly, our perturbative estimations of∆CSBagree perfectly (within less than 5 keV) with the full calculations. This accuracy is definitely sufficient for our purpose of studying the origin of p-shell CSB splittings.
It apparently sticks out from Table4.13that our estimated CSB for7ΛBe -7ΛLi*actually agrees both in sign and magnitude with the experimental value. It is, however, difficult to compare the CSB results for7ΛHe−7ΛLi* with the experiments. Our computed values even change sign with respect to the flow parameters while experimental estimation,∆exp =20±230 keV, comes with the very large uncertainty. Nevertheless, it can be deduced that the CSB splitting for the pair7ΛBe−7ΛLi* is insignificantly small. Overall, one can read offthat the∆CSBsplitting in A=7 isotriplet is fairly small, and driven predominantly by the NN Coulomb modification∆CSBNN,C. The latter is mostly a result of the contraction of nuclear cores induced by the presence of a hyperon, therefore, strongly influenced by the SRG-YN evolutions as it can be seen from the table. The contributions due to the NN and YN interactions and the Σ+/Σ−mass differences,∆CSBNN,nucl, ∆CSBY N,nucland∆CSBT , respectively, are of the same order of magnitude. However these contributions largely cancel each other in7ΛBe−7ΛLi* leaving a sole contribution from∆CSBNN,C. As a result, the∆CS Bbetween the two members,7ΛBe and
7ΛLi*, about 80 keV at the SRG cutoffofλY N =1.6 fm-1, is much larger than that between7ΛHe and
7ΛLi*at the same cutoff. In the second column in Tab.4.13we also listed the difference between the Σ±-probabilities,δΣ =PΣ+ −PΣ−, for7ΛHe and7ΛBe. It is further noticed that the kinetic contribution
∆CSBT has the same sign as the CSB splitting due toΣ+andΣ−rest mass difference, which is given by [145]
∆TCSBM
R =δΣ(mΣ− −mΣ+), mΣ−−mΣ+ =8.1 MeV.
Chapter 4 Results forA=4−7Hypernuclei
λYN δΣ ∆CSBT ∆CSBNN,nucl ∆CSBNN,C ∆CSBY N,nucl ∆CSBY N,C ∆CSB ∆exactCSB
[fm-1] [%] [keV] [keV] [keV] [keV] [keV] [keV] [keV]
7ΛHe−7ΛLi*
0.836 -0.34 -16.8 -2.0 15.0 -52.1 1.0 -54.9 -53.3 1.0 -0.14 -6.8 -8.0 34.0 -24.4 -0.0 -5.2 -5.4 1.60 -0.18 -9.8 -13.0 58.0 -16.7 1.0 19.5 19.4 2.0 -0.26 -12.8 -13.0 59.0 -14.1 2.0 21.1 19.9 2.6 -0.36 -19.8 -12.0 53.0 -9.8 2.0 13.4 14.3 3.0 -0.41 -21.8 -10.0 46.0 -7.7 3.0 9.5 8.5 4.0 -0.41 -21.8 -7.0 26.0 -4.4 2.0 -5.2 -9.7
7ΛHe−7ΛLi*[146] 20(230)
7ΛBe−7ΛLi*
0.836 0.35 16.7 -5.0 -19.0 21.5 -5 9.2 8.0
1.0 0.15 7.7 -12.0 -49.0 19.8 -2.0 -35.5 -37.1 1.6 0.21 8.7 -20.0 -83.0 16.3 -3.0 -81.0 -81.1 2.0 0.29 13.7 -21.0 -83.0 13.8 -4.0 -80.5 -82.4 2.6 0.41 17.7 -19.0 -74.0 9.8 -6.0 -71.5 -71.8 3.0 0.47 21.7 -16.0 -64.0 7.5 -7.0 -57.8 -59.6 4.0 0.46 20.7 -11.0 -35.0 4.5 -6.0 -26.8 -34.0
7ΛBe−7ΛLi*[113] 0.12 3.0 -70.0 50.0 -17.0
7ΛBe−7ΛLi*[137] −100(90)
Table 4.13: Perturbative calculation of the CSB splitting in the A=7 isotriplet for model spaceN = 10.
Same NN and YN potentials as in Table4.12. The second columns is the difference between probabilities of findingΣ+andΣ−for7ΛHe and7ΛBe. Note that in Gal calculations [113] (second last row), 0.12 % is the totalΣprobability but not the difference betweenPΣ+andPΣ−. This extremely smallΣ-admixture leads to a negligibly weak∆CSBT contribution. The negative 70 keV contribution attributed to the Coulomb forces is in fact a net effect of the nuclear-core Coulomb energy modification∆CSBNN,C and theΣpCoulomb interaction
∆CSBY N,nucl.
However, its size is noticeably smaller,∆CSBT ≈ 2
3∆TMCSB
R at most. Additionally, we observe a strong influence the of SRG-YN evolution on∆CSBY N,nucl, the value of which however vanishes as the YN potential being less and less evolved. This is not a surprise. Let us remind that there is no explicit CSB terms in the employed version of the YN interaction model. The only source of the∆CSBY N,nuclis then the physical masses of nucleons andΣ’s that were used in the generator of the flow equation Eq. (2.5). Furthermore, the CSB effect induced by the YN Coulomb interactions,∆CSBY N,C, is rather small but stable, its size is of order of 6 keV in7ΛBe -7ΛLi*.
Let us finally summarize that, without explicit CSB terms in YN interactions, our results for CSB in the A=7 isotriplet are rather small but in a good agreement with the experimental values,
4.7 A=7 isotriplet and CSB splittings
λYN δΣ ∆CSBT ∆CSBNN,nucl ∆CSBNN,C ∆CSBY N,nucl ∆CSBY N,C ∆CSB ∆exactCSB
[fm-1] [%] [keV] [keV] [keV] [keV] [keV] [keV] [keV]
7ΛHe−7ΛLi*
0.836 -0.38 -18.3 0.0 9.0 -58.0 1.0 -66.3 -64.2 1.0 -0.14 -7.3 -6.0 27.0 -25.5 -0.0 -11.8 -12.0 1.6 -0.18 -8.3 -11.0 48.0 -17.0 1.0 12.7 11.7 2.0 -0.25 -12.3 -11.0 49.0 -13.9 1.0 12.8 12.2
2.6 -0.35 -17.3 -9.0 41.0 -9.4 2.0 7.3 7.0
3.0 -0.40 -20.3 -8.0 34.0 -7.2 2.0 0.5 1.3
4.0 -0.41 -20.3 -3.0 14.0 -4.3 2.0 -11.6 -11.4
7ΛBe−7ΛLi*
0.836 0.39 18.4 -2.0 -10.0 23.6 -5.0 25.0 23.3 1.0 0.16 7.4 -9.0 -38.0 20.6 -2.0 -21.0 -22.9 1.6 0.21 9.4 -17.0 -69.0 16.5 -3.0 -63.1 -64.5 2.0 0.29 13.4 -18.0 -68.0 13.8 -4.0 -62.8 -65.2 2.6 0.39 18.4 -15.0 -58.0 9.4 -6.0 -51.2 -52.6 3.0 0.45 19.4 -13.0 -46.0 7.2 -6.0 -38.4 -39.2 4.0 0.46 20.4 -5.0 -20.0 4.3 -7.0 -7.3 -8.0
Table 4.14: Perturbative calculation of the CSB splitting in the A=7 isotriplet for model spaceN =8. The second columns is the difference between probabilities of findingΣ+andΣ−for7ΛHe and7ΛBe. Calculations based on the same interactions as in Table4.12.
furthermore, they are are dominantly given by the Coulomb NN interaction. In order to verify the sufficiency of studying CSB splittings for the largest model spaceNmax =10, we additionally show the CSB splitting calculated with the model space sizeN =8 in Table4.14. Comparing the two tables, one notices that all the different contributions to CSB except∆CSBNN,C converge very quickly (within several keV at most). The slight discrepancies of about 7 keV for∆CSB(7ΛHe−7ΛLi) and 15 keV for∆CSB(7ΛBe−7ΛLi) between the two model space sizes, occurring at large flow parameters, are primarily due to the slow convergence of∆CSBNN,C. We therefore conclude that the model space N =10 is sufficient to understand the essence of CSB in the A=7 isotriplet.
Chapter 4 Results forA=4−7Hypernuclei