Medium-mass hypernuclei and the nucleon-isospin dependence of the three-body hyperon-nucleon-nucleon force
EEPJ manuscript No. (will be inserted by the editor)
Medium-mass hypernuclei and the nucleon-isospindependence of the three-body hyperon-nucleon-nucleon force
Diego Lonardoni and Francesco Pederiva National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Physics Department, University of Trento, via Sommarive 14, I-38123 Trento, Italy INFN-TIFPA, Trento Institute for Fundamental Physics and Application, Trento, ItalyReceived: XXX / Revised version: XXX
Abstract
We report quantum Monte Carlo calculations of single- Λ hypernuclei for A < based on phe-nomenological two- and three-body hyperon-nucleon forces. We present results for the Λ separation energyin different hyperon orbits, showing that the accuracy of theoretical predictions exceeds that of currentlyavailable experimental data, especially for medium-mass hypernuclei. We show the results of a sensitiv-ity study that indicates the possibility to investigate the nucleon-isospin dependence of the three-bodyhyperon-nucleon-nucleon force in the medium-mass region of the hypernuclear chart, where new spec-troscopy studies are currently planned. The importance of such a dependence for the description of thephysics of hypernuclei, and the consequences for the prediction of neutron star properties are discussed. The recent observation of gravitational waves from themerger of two neutron stars (NSs) [1,2] provides new con-straints on the equation of state (EoS) of high densitymatter. An important example is the tidal deformabilityparameter, which is connected to the EoS of matter inthe NS core [3]. This matter is characterized by a densityseveral times larger than the typical density of a nucleus,and by an extreme nucleon asymmetry, i.e., the numberof neutrons largely exceeds that of protons.While it would be desirable to have a theory of NSstructure directly based on quantum chromodynamics, atpresent we still need to rely on effective theories usingnucleons (and possibly mesons) as active degrees of free-dom [4–14]. These have proven to be remarkably suc-cessful in the description of low-energy properties of nu-clei [15–19], and they have also been employed for theprediction of NS structure [17, 20, 21].In the inner core of a NS, Pauli blocking might fa-vor the appearance of heavier baryons, such as hyperons ( Y ) [22]. Hyperons are expected to make the EoS softer,reducing the maximum mass that a NS can stably support.However, different models predict very different results (1 . M (cid:12) (cid:46) M max (cid:46) M (cid:12) ) [23–31], and there is no clearconsistency with the observation of heavy NSs [32, 33].This puzzle can only be solved by performing accu-rate many-body calculations based on a well constrainedtheory of the hyperon-nucleon ( Y N ) and hyperon-hyperon ( Y Y ) interactions. Unfortunately, both Y N and
Y Y inter-actions are not well constrained by current experiments.Direct
Y N scattering is technically difficult to perform, and few data are available [34]. Some information is avail-able for
Y Y scattering, like the derivation of the ΛΛ scat-tering length from heavy-ion collisions [35], and the Na-gara event [36], but it only allows for constraining at mosta central contact interaction. Moreover, as in the non-strange sector, many-body hypernuclear forces ( Y N N , Y Y N , . . . ) are expected to be relevant, and need to beconstrained. Experimental information on hypernuclei—bound states of nucleons and hyperons—is thus the onlybasis for the construction of a realistic hyperon-nucleonpotential. Several attempts at using a theory that includesthe whole baryon octet, and accounts for the so called ΛN - ΣN coupling, have been made [37–41]. However, theseschemes need a relatively large number of parameters thatare presently difficult to access experimentally and requireextra theoretical constraints.In Refs. [42,43], a phenomenological model for the ΛN and ΛN N forces [44, 45], inspired by the phenomenologi-cal
N N [4, 46] and
N N N [47–49] potentials, was derivedusing quantum Monte Carlo algorithms. This interactionwas introduced with the precise aim of building a realisticmodel that i) is tightly connected to the experimentallyavailable information and ii) contains the least number ofparameters possible. Its success was shown by the capa-bility of accurately describing the properties of Λ hyper-nuclei in a wide mass range [42, 43]. In order to make afirst connection to NS properties, the same interaction wassuccessively employed to study hyper-neutron matter [30].In this paper, we address an additional crucial point. Ina hypernucleus, contributions to the binding energy from Y N N interactions, as they are introduced in our scheme, a r X i v : . [ nu c l - t h ] A ug D. Lonardoni and F. Pederiva: Medium-mass hypernuclei and the isospin dependence of the three-body
ΛNN force ⇤⇤ ⌃ NN ⇡⇡ (a) ⇤⇤ ⌃ ⇡⇡ NNNN (b) ⇤⇤ ⇡⇡ NNNN (c) ⇤⇤ ⌃ NN ⇡⇡ N (d) Figure 1.
Hyperon-nucleon two-pion exchange diagrams. Panel (a) represents the two-body ΛN channel. Panels (b)-(d) arethe three-body ΛNN channels. might be dominated by contributions of
Λpn triplets, mainlydue to the Pauli principle suppressing
Λnn ( Λpp ) contribu-tions. However,
Λnn terms are expected to be largely dom-inant in the NS core. By means of the auxiliary field dif-fusion Monte Carlo (AFDMC) method [50], we try to as-sess the extent of the nucleon-isospin dependence of thesethree-body contributions by studying the ground state ofmedium-mass hypernuclei. In particular, we provide pre-dictions for A = 40 , potassium hypernuclei, the study ofwhich is the goal of the current hypernuclear experimentalprogram at the Thomas Jefferson National Accelerator Fa-cility (JLab)—approved experiment E12-15-008 [51]. Thisis an important step toward a coherent description of hy-pernuclear experiments and the phenomenology related toNSs, eventually including the information obtained fromgravitational waves observations. The Hamiltonian for single- Λ hypernuclei AΛ Z , with A thetotal number of baryons, is of the form: H = − (cid:126) m N (cid:88) i ∇ i + (cid:88) i ΛNN force include in the AFDMC propagation. The spin/isospin-dependent components of Eq. (3) can be recast as a sumof quadratic operators, involving only spin and isospin oftwo particles at a time. This allows us to fully include thepotentials described above in the imaginary time propa-gation of the trial state of Eq. (4). The sampling of spa-tial coordinates and auxiliary fields for the spinor rota-tions is done as in Ref. [17] via a “plus-minus” sampling.This allows one to sensibly lower the variance comparedto the simpler sampling scheme employed in Ref. [43]. TheFermion sign problem [17] is controlled by constraining theevolved configurations to have positive real overlap withthe trial function (constrained-path approximation) [68].Expectation values of the quantities of interest are evalu-ated as described in Refs. [17, 43]. The AFDMC binding energy for selected nuclei are re-ported in Tab. 1. Although the employed two- and three-body nucleon interaction model is quite simple (AV4’ +UIX c ), AFDMC predicts binding energies of nuclei for ( A − < within ≈ of the experimental val-ues. This agreement can be further improved by employ-ing more realistic two- and three-body forces, and at thesame time improving the wave functions and propagationtechniques, as done, for instance, in Refs. [19,69,70]. How-ever, this is not relevant for the current work where theobservable of interest is an energy difference ( B Λ ) , not sen-sitive to the details of the employed nuclear force [42]. It ishowever interesting to note that the employed simplifiednuclear potential retains the basic structure of a pion-lesseffective interaction at leading order, and it is capable of Table 1. Binding energies (in MeV) for selected nuclei with ≤ ( A − ≤ . The experimental total angular momentum,parity, and isospin ( J π , T ) are also shown. Results are calcu-lated by employing the two-nucleon AV4’ potential plus thecentral component of the three-nucleon UIX force, see text fordetails. Experimental energies are shown for comparison. A − Z AV4’+UIX c Exp H (cid:0) + , (cid:1) − . − . H (cid:16) 12 + , (cid:17) − . − . He (cid:0) + , (cid:1) − . − . O (cid:16) − , (cid:17) − . − . O (cid:0) + , (cid:1) − . − . K (cid:16) 32 + , (cid:17) − . − . Ca (cid:0) + , (cid:1) − . − . Ca (cid:0) + , (cid:1) − . − . K (cid:16) 12 + , (cid:17) − . − . Ca (cid:0) + , (cid:1) − . − . spdfg B L ( M e V ) A −2/3 emulsion (K − , p − ) ( p + ,K + ) (e,e’K + ) AFDMC 0510152025300.0 0.1 0.2 0.3 0.4 0.5 Figure 2. Λ separation energy for different hyperon orbits.Solid symbols are the updated experimental results from dif-ferent production mechanisms [34]. Empty red circles are theresults of this work. Dashed lines are just a guide to the eye. providing a reasonable description of the ground-state en-ergy of nuclei up to calcium.In Fig. 2 we present the Λ separation energy of hy-pernuclei from A = 3 to A = 49 calculated with AFDMCfor different hyperon orbits. Theoretical results are consis-tent with available experimental data over the entire massrange analyzed. Remarkably, even though the three-body ΛN N force was fit to the Λ separation energy in the s -orbit only, the agreement between theory and experimentfor B Λ in different hyperon orbits is also very good.We report in Tabs. 2 and 3 details of the AFDMCcalculations for Λ hypernuclei. In the former, the Λ sep-aration energy for s - and p -shell hypernuclei is shown. Inthe latter, the B Λ values of the expected dominant peaksin s -, p -, and d -orbits are reported for hypernuclei with ≤ A ≤ , in comparison with the limited available Table 2. Λ separation energy (in MeV) for A ≤ in s - and p -orbits. For each hyperon orbit, AFDMC results (first column)are compared to available experimental data [34] (second col-umn). The quoted experimental value for Λ O in s -orbit is thesemiempirical result of Ref. [52]. For each system, the total an-gular momentum, parity, and isospin ( J π , T ) of the dominantpeaks (measured or expected) are also shown. AΛ Z s -orbit p -orbit Λ H (cid:16) 12 + , (cid:17) . . Λ H (cid:0) + , (cid:1) . . Λ He (cid:0) + , (cid:1) . . Λ He (cid:16) 12 + , (cid:17) . . Λ O (cid:0) − , (cid:1) . . (cid:0) + , (cid:1) . . Λ O (cid:16) 12 + , (cid:17) . . ∗ (cid:16) − , (cid:17) . —. Lonardoni and F. Pederiva: Medium-mass hypernuclei and the isospin dependence of the three-body ΛNN force 5 Table 3. Λ separation energy (in MeV) for A ≥ . AFDMCresults at the top, experimental data at the bottom. Data for Λ Ca and Λ V are taken form Ref. [34]. The s -orbit B Λ for Λ Ca and Λ V are from Ref. [71] and Ref. [72], respectively. Foreach system, the total angular momentum, parity, and isospin ( J π , T ) of the dominant peaks (measured or expected) are alsoshown. AΛ Z s -orbit p -orbit d -orbit Λ K (cid:0) + , (cid:1) . (cid:0) − , (cid:1) . (cid:0) + , (cid:1) . Λ Ca (cid:16) 12 + , (cid:17) . (cid:16) − , (cid:17) . (cid:16) 52 + , (cid:17) . Λ Ca (cid:16) 12 + , (cid:17) . (cid:16) − , (cid:17) . (cid:16) 52 + , (cid:17) . Λ K (cid:0) + , (cid:1) . (cid:0) − , (cid:1) . (cid:0) + , (cid:1) . Λ Ca (cid:16) 12 + , (cid:17) . (cid:16) − , (cid:17) . (cid:16) 52 + , (cid:17) . Λ Ca (cid:0) + , (cid:1) . . (cid:0) − , (cid:1) . (cid:0) + , (cid:1) . Λ V (? , 2) 21 . , 2) 13 . , 2) 5 . Λ V (cid:0) − , (cid:1) . — — (cid:0) − , (cid:1) . — — experimental information for the neighboring hypernuclei Λ Ca , Λ V , and Λ V . In the light sector, AFDMC resultsare in general compatible with observations. The Λ sep-aration energy of Λ H is slightly lower than the currentexperimental value. However, it is consistent with the oldemulsion data of . , to which the employed spin-independent CSB potential was originally fit [54]. In themedium-mass region for A ≥ , results for different hy-peron orbits show a consistent pattern, and they are com-patible with the experimental data for the nearest hyper-nucleus. Exception is B Λ in the d -orbit for A ∼ , forwhich the only available experimental value indicates amuch smaller Λ separation energy. However, this result isfrom the ( K − , π − ) exchange reaction on Ca performedat Saclay in 1979 [73] and never confirmed in followingexperiments. Theoretical calculations of the photoproduc-tion cross section of Λ K [74] indicate a value for B Λ in the d -orbit much closer to the one of this work. This, in addi-tion to the very low statistics for the ( π + , K + ) productionof Λ Ca [71], emphasizes the need of accurate experimentalresearch in the medium-mass region of the hypernuclearchart.In view of the hypernuclear experimental program atJLab, aimed at the high precision spectroscopy of Λ hyper-nuclei for A = 40 , with an electron beam, the AFDMCpredictions for Λ K and Λ K are of particular interest. Sta-tistical uncertainties due to the use of a Monte Carloprocedure for A = 40 , are smaller than those of cur-rent nearby experimental data. Quantum Monte Carlocould then be used to extract accurate information onthe hyperon-nucleon force, if supported by more accuratemeasurements of the Λ separation energy. In this partic-ular regime, it is possible to start addressing the funda-mental problem of the nucleon-isospin dependence of thethree-body ΛN N interaction, which is of great importance in extrapolating the experimental information on hyper-nuclei to the study of NS matter.The current version of the ΛN N potential does not de-pend on whether the two nucleons are in a isospin singletor a isospin triplet state (see Fig. 3). For symmetric hyper-nuclei the Pauli principle suppresses any strong contribu-tion from the Λnn or Λpp channels. On the other hand, inhyper-neutron matter or in matter at β -equilibrium, thecontribution of the nucleon-isospin triplet channel mightbecome relevant. By looking at the nucleon-isospin asym-metry, defined as δ = ( N − Z ) /A with N the number ofneutrons and Z that of protons, Λ K and Λ K are charac-terized by having the smallest and largest nucleon-isospinasymmetry along the hypernuclear chart. Moreover, theyare large enough to manifest central densities comparableto saturation density, but they are still accessible usingab-initio techniques such as AFDMC. A = 40 , Λ hy-pernuclei are thus excellent candidates to be used to ex-tract information on the hypernuclear force relevant forthe description of strange neutron-rich nuclei and for theprediction of NS properties.In order to test the sensitivity of our AFDMC predic-tions to a possible nucleon-isospin dependence of the hy-pernuclear force, we performed a proof-of-principle study.The first two terms of Eq. (3) depend upon the nucleon-isospin via the operator τ i · τ j . This can be written in termsof the projectors on the singlet and triplet nucleon-isospinchannels, τ i · τ j = − P T N =0 ij + P T N =1 ij . The nucleon-isospindependent terms of Eq. (3) can then be expressed as: v Tλij τ i · τ j = − v Tλij P T N =0 ij + (1 + C T ) v Tλij P T N =1 ij , (7)where the additional parameter C T has been introduced inorder to single out the nucleon-isospin triplet contributioncompared to the original formulation of Eq. (3).By (artificially) varying the C T parameter we can testwhether our AFDMC predictions for B Λ are affected byan unbalance in the strength of the nucleon-isospin tripletcomponent compared to the nucleon-isospin singlet one. Inother words, the aim of this study is to answer the follow-ing question: can AFDMC calculations be used to resolve,at least at this elementary level, the nucleon-isospin de-pendence of the ΛN N force starting from experimentalhypernuclear data only? Isospin-triplet components haveto play a crucial role in strange neutron-rich systems, suchas A = 48 hypernuclei. What is not obvious is whetherAFDMC calculations for such large systems have a chanceto appreciate this effect (considering Monte Carlo statis-tical errors). We point out once more, that, besides the n p ⇤ n n ⇤ p p ⇤ T N = 0 , T N = 1 Figure 3. Nucleon-isospin T N = 0 and T N = 1 configurationsin ΛNN triplets. D. Lonardoni and F. Pederiva: Medium-mass hypernuclei and the isospin dependence of the three-body ΛNN force L O L Ca L Ca B L ( M e V ) C T = −2.0 C T = −1.0 C T = 0.0 C T = 0.5 1215182124 L O L Ca L Ca Figure 4. Λ separation energy in s -orbit for medium-masshypernuclei. Empty symbols are the AFDMC results for dif-ferent choices of the parameter C T . Solid symbols are avail-able experimental data (same color scheme of Fig. 2): Λ Ofrom ( π + , K + ) [34] and ( K − , π − ) [75]; Λ N from ( e, e (cid:48) K + ) [76]; Λ Ca from ( π + , K + ) [71]; Λ V from ( π + , K + ) [34]; Λ V from ( e, e (cid:48) K + ) [72]. Shaded areas are current experimental con-straints in the regions A ∼ , , . implications in hypernuclear physics, resolving this de-pendence has a potential strong impact on the study ofneutron star matter.In Fig. 4 we show the results of such a sensitivitystudy. The Λ separation energy for Λ O , Λ Ca , and Λ Ca is reported for different choices of the C T parameter. For C T = 0 the original parametrization of the three-bodyforce is recovered, and the results are the same as in Fig. 2.For C T = − the nucleon-isospin triplet component ofEq. (7) is turned off, for C T = 0 . it is enhanced, andfor C T = − it changes sign. Notice that for C T (cid:54) = 0 Monte Carlo errors are larger. This is not a consequenceof the introduction of the C T parameter in Eq. (7), rathera matter of statistics. Simulations for C T (cid:54) = 0 have beenperformed sampling a reduced number of configurationscompared to those for C T = 0 . Considering the proof-of-principle nature of this study and the size of experimentaluncertainties on B Λ , this does not affect the conclusionsthat one can extract from Fig. 4.As one could have expected, for symmetric hypernucleithe variation of B Λ due to the change of C T is not large, inparticular for the lightest system. All results obtained fordifferent values of C T fall within the uncertainty band de-termined by the available experimental data for neighbor-ing hypernuclei, despite the large variation of the controlparameter ( C T ∈ [ − . , . . The case of Λ Ca differs;one finds a large nucleon-isospin asymmetry δ ≈ . ,for which the variation of the control parameter greatlyaffects the prediction of B Λ , in particular for C T < .Even though the range of C T considered here is unrealis-tically large, the results for A = 49 suggest that AFDMCcalculations for the employed hypernuclear potentials are sensitive to a possible nucleon-isospin dependence of thethree-body hyperon-nucleon force for A (cid:38) and δ (cid:54) = 0 .However, the current experimental information in theregion of interest (40 (cid:46) A (cid:46) is not sufficiently accu-rate, as clearly visible from the gray bands in Fig. 4. Thehigh precision spectroscopy of Λ K and Λ K that will beperformed in the coming years at the Thomas JeffersonLaboratory will provide a new and valuable set of con-straints to be used in the construction of realistic hyperon-nucleon interactions. The accuracy of the expected exper-imental data should be sufficient to allow one to re-fit the W D and C T parameters of Eqs. (3) and (7) in order tosimultaneously reproduce the Λ separation energy of Λ K and Λ K with AFDMC calculations. This will provide aunique opportunity to investigate how the nucleon-isospindependence of a many-body hyperon-nucleon force affectsthe hypernuclear spectrum, in particular for neutron-richsystems. It could elucidate the role of three-body forces inthe determination of the unusually large spin-dependenthypernuclear CSB in light nuclei [60–62], and it could shedsome light on the implications for the EoS of hypernuclearmatter and for the prediction of NS properties. We have performed quantum Monte Carlo calculations ofsingle- Λ hypernuclei in the mass range ≤ A ≤ us-ing phenomenological two- and three-body hypernuclearinteractions. The accuracy of these calculations allows forthe prediction of Λ separation energies with statistical er-rors comparable or even smaller than current experimen-tal uncertainties, especially for medium-mass hypernuclei.From a sensitivity study we can learn that quantum MonteCarlo methods can be used to resolve a possible nucleon-isospin dependence of the three-body ΛN N force start-ing from hypernuclear data. We conclude that future hy-pernuclear experiments on medium-mass targets will pro-vide an opportunity to develop more accurate many-bodyhyperon-nucleon interactions, crucial for the descriptionof the physics of hypernuclei and even more for the pre-diction of neutron star properties, for which new and morestrict constraints have become available [3, 32, 33].Preliminary results for the nucleon-isospin dependence ofthree-body hyperon-nucleon-nucleon force were obtainedby F. Catalano during the initial phase of this work. Wethank S. Reddy, S. Gandolfi, A. Lovato, L. Contessi, A. 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