NNoname manuscript No. (will be inserted by the editor)
Topics in Low-Energy QCD with Strange Quarks
Wolfram Weise
Received: date / Accepted: date
Abstract
Recent developments and pending issues in low-energy strong inter-action physics with strangeness are summarized. Chiral SU(3) effective fieldtheory has progressed as the appropriate theoretical framework applied toantikaon- and hyperon-nuclear systems. Topics include antikaon-nucleon in-teractions and the Λ (1405), ¯ KN N systems, recent developments in hyperon-nucleon interactions and strangeness in dense baryonic matter with emphasison new constraints from neutron star observations.
Keywords
Chiral effective field theories · strangeness · antikaon-nucleoninteractons · hyperon-nucleon interactions · neutron stars PACS · · Within the hierarchy of quark masses in QCD, strange quarks are special. Withtheir mass m s ∼ . s -quarks are not extremely light (unlike the u and d quarks with their masses m u,d of only a few MeV), but at the same time stillwell separated on the mass scale from the heavy ( c , b and t ) quarks. Systemswith strange quarks are thus of interest as test areas for the interplay betweenspontaneous and explicit chiral symmetry breaking in low-energy QCD.High-precision antikaon-nucleon threshold physics is one such testing ground.The driving attractive s-wave K − N interactions at threshold are determinedentirely by the kaon energy and decay constant. The leading-order thresh-old amplitudes are T ( K − p ) | thr = 2 T ( K − n ) | thr = m K /f . The appearance W. WeisePhysik-Department, Technische Universit¨at M¨unchen, D-85747 Garching, GermanyandECT*, Villa Tambosi, I-38123 Villazzano (TN), ItalyE-mail: [email protected] a r X i v : . [ nu c l - t h ] J a n Wolfram Weise of the pseudoscalar decay constant f ∼ . f is an orderparameter . The kaon mass m K (actually the kaon energy at zero momentum)reflects explicit breaking of chiral symmetry by the non-vanishing (strange)quark mass, m K being proportional to m s + m u . In the chiral limit of van-ishing quark masses the corresponding threshold amplitude for the interactionof a pseudoscalar Nambu-Goldstone boson with a nucleon would vanish inaccordance with Goldstone’s theorem.Confinement implies that QCD in the low-energy limit is realized as atheory of hadronic rather than quark-gluon degrees of freedom. Spontaneouschiral symmetry breaking implies further that the appropriate framework isChiral Effective Field Theory (ChEFT), a systematic approach describing theinteractions of the pseudoscalar Nambu-Goldstone bosons amongst each otherand with “heavy” sources such as baryons. For the two-flavor case with almostmassless u, d quarks and spontaneously broken SU (2) L × SU (2) R symmetry,chiral perturbation theory (ChPT) has been applied successfully to ππ , πN and N N scattering. Moreover, in-medium ChPT has been developed as anappropriate tool for treating the nuclear many-body problem at moderatedensities (for recent reviews see [2,3] and refs. therein).Three-flavor QCD at low energy is represented by SU (3) L × SU (3) R ChEFTwhere now the kaon mass appears prominently in the symmetry-breakingmass term. This theory involves the pseudoscalar meson octet coupled tothe baryon octet. However, perturbative (ChPT) methods are in general notapplicable in the sector with strangeness. This is partly a consequence ofthe stronger explicit chiral symmetry breaking by the strange quark masswhich implies, in particular, significantly stronger interactions of antikaonsas compared to those of pions near their respective thresholds. For exam-ple, the existence of the Λ (1405) just 27 MeV below ¯ KN threshold rulesout a perturbative ChPT treatment as an expansion in powers of “small”momenta. Non-perturbative strategies, based on the chiral SU (3) EFT effec-tive Lagrangian at next-to-leading order as input but solving coupled-channelsLippmann-Schwinger equations to all orders, have been well established as themethod of choice to deal with these problems [4]. While sacrificing the rigor-ous power-counting scheme of ChPT, the gain in physics insights using thisnon-perturbative approach emerges as a major benefit.In the following, selected topics of recent interest in this field are discussed,including low-energy ¯ KN and ¯ KN N interactions, progress in understand-ing the nature and structure of the Λ (1405), new developments concerninghyperon-nucleon interactions derived from ChEFT, and the possible role ofstrangeness in dense baryonic matter in view of the established existence oftwo-solar-mass neutron stars. In the chiral limit, f (cid:39)
86 MeV. Explicit chiral symmetry breaking shifts this value tothe empirical f π = 92 . ± . f K = 110 . ± . -1-0.5 0 0.5 1 1.5 1340 1360 1380 1400 1420 1440 R e [ T K - p - K - p ] ( f m ) E c.m. (MeV) Re f ( K − p → K − p )[ fm ] √ s [ MeV ] I m [ T K - p - K - p ] ( f m ) E c.m. (MeV) [ fm ] Im f ( K − p → K − p ) √ s [ MeV ] Λ ( ) !" (cid:82) !" !" Pole structure in the complex energy plane
Resonance state ~ pole of the scattering amplitude ∼ T ij ( √ s ) ∼ g i g j √ s − M R + i Γ R / D. Jido, J.A. Oller, E. Oset, A. Ramos, U.G. Meissner, Nucl. Phys. A 723, 205 (2003) Λ (1405) in meson-baryon scattering T. Hyodo, D. Jido, arXiv:1104.4474 d o m i n a n t l y π Σ dominantly ¯KN Fig. 1
Lower panel: Real and imaginary parts of the K − p scattering amplitude from chiralSU(3) coupled-channels dynamics [6]. The threshold points indicating the real and imaginaryparts of the corresponding scattering length are constrained by the SIDDHARTA kaonichydrogen measurement [5]. Upper panel: two-poles scenario of coupled ¯ KN and πΣ channels[7]. K -nucleon interaction and structure of the Λ (1405) Applications of chiral SU(3) coupled-channels dynamics to threshold and sub-sthreshold ¯ KN interactions have focused prominently on the K − p system. Itsdynamics includes the K − p ↔ ¯ K n charge exchange channel and the strongcoupling to the πΣ continuum. Goldstone’s theorem implies that the drivingattractive s-wave interactions are proportional to the energies of the partici-pating pseudoscalar mesons. With input constrained by the strong interactionenergy shift and width deduced from the SIDDHARTA kaonic hydrogen mea-surement [5], calculated real and imaginary parts of the K − p amplitude [6]are shown in Fig. 1. The resulting complex K − p scattering length (includingCoulomb corrections) is [6]:Re a ( K − p ) = ( − . ± .
10) fm , Im a ( K − p ) = (0 . ± .
15) fm . (1)The uncertainties in a ( K − p ) derive primarily from those of the kaonic hy-drogen data. Further extensions of such calculations based on chiral SU (3)effective field theory have recently been reported in [8,9].The SU (3) ChEFT coupled-channels approach also predicts K − n ampli-tudes. While the weaker attraction in this I = 1 channel does not produce a Wolfram Weise quasibound state or resonance, the NLO calculations still suggest a sizeable K − n scattering length [6]:Re a ( K − n ) = (0 . +0 . − . ) fm , Im a ( K − n ) = (0 . +0 . − . ) fm . (2)It is important to provide empirical constraints for this quantity in order toarrive at complete information for both isospin I = 0 and I = 1 channels ofthe ¯ KN system. This can be achieved by accurate antikaon-deuteron thresholdmeasurements such as the planned SIDDHARTA-2 kaonic deuterium experi-ment and a related proposal at J-PARC.The Λ (1405) with isospin I = 0 emerges from coupled-channels dynamicsas a quasibound ¯ KN state embedded in the πΣ continuum. A characteristicfeature of the chiral SU (3) coupled-channels approach is the appearance of twopoles in the T matrix (see [7] for a review and refs. therein). The calculationthat leads to the amplitude in Fig. 1 produces a pole located at ( E , Γ /
2) =(1424 ± , ±
5) MeV and a second one at ( E , Γ /
2) = (1381 ± , ± KN boundstate component. This pole is quite well determined, with its imaginary partrepresenting the decay width into the open πΣ phase space driven by the¯ KN → πΣ interaction. The location of the second pole is more ambiguous,less well determined by the fits to existing scattering data and kaonic hydrogenmeasurements. Its dominant component reflects a broad resonance structure inthe πΣ channel. The coexistence of these two coupled modes implies that the Λ (1405) is not described by a single, unique spectral function but by an entan-glement of the two modes. The πΣ spectra observed in different photon- andhadron-induced reactions are expected to differ in their shape and location oftheir maximum, depending on which of the ¯ KN or πΣ components of the cou-pled modes are more strongly involved in the particular reaction mechanismat work. For example, the maximum of the π − Σ + invariant mass spectrumproduced in γp → K + π − Σ + photoproduction at JLab [10] is observed around1420 MeV, as expected for a process driven by a primary γ → K + K − vertexand followed by the t-channel exchange of the K − that is absorbed by the pro-ton and then converted to the observed π − Σ + final state. On the other hand,the more complex hadron-induced reactions tend to emphasize more stronglythe primary πΣ channels and produce a maximum in the πΣ spectrum aroundthe nominal 1405 MeV.Hence the Λ (1405) is obviously not a standard quark model baryon. Con-sider a Fock space expansion | Λ ∗ (cid:105) = a | uds (cid:105) + b | ( udu )(¯ us ) (cid:105) + c | ( uus )(¯ ud ) (cid:105) + . . . , (3)where the first term on the r.h.s. corresponds to a “bare” three-quark statewhile the subsequent terms schematically describe N ¯ K and Σπ quasimolec-ular configurations. Actually any baryon state has an expansion of this kind.The issue is how the coefficients of this Fock expansion arrange themselves,governed by the strong interactions of the constituents. For a baryonic state inthe proximity of a meson-baryon threshold, a quasi-molecular structure tendsto be favored. opics in Low-Energy QCD with Strange Quarks 5 m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| Λ ( ) : RECENT NEWS (contd.) Λ ( ) Structure of from Lattice QCD | Λ ∗ ! = a | uds ! + b | ( udu )( ¯us ) ! + . . . ! (cid:83) " $ % & ' ( ( (((((( )* + , - ( . / *01 ( , + ( (cid:135) ! " % & ' ( ) *+, ' ( - " . " / " "+ ' ( ( < ((( / - + : ; (- < + - /0 ( ( (cid:47) = A ( . - - ( < B ( + ( , + < - ( + + (( ( ( C - < + D E = F A ( : ; (- < + - /0 ( ( %? $ @ $ G ( F ( H I + - , - ' ( J + + K + I - ' ( L C M@ = %??% A ( ?>>??@ N ( D O I / ' ( L C G@ = %? A ( ?F>?? N ( : + - , < ' ( D - ' ( P - , ' ( ; H @Q> = A ( F%@ N ( R , ' ( L + / , ' ( ; H MF@ = A ( QQ N ( S B / / ' ( ! - / ' ( ; MT = %? A @@ N ( quasimolecular constituent quarkdominated m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| m π [ MeV ]
296 570 ¯KN ¯KN π Σ π Σqqq qqq |! Λ ∗ | n "| “light” quarks structure ¯KN “heavy” quarks m π ! . m π ! . J.M.M. Hall et al. (Adelaide group 2014)
Fig. 2
Leading components of the Λ (1405) deduced from a lattice QCD computation [11]for different light-quark masses indicated in terms of the pion mass. In this context a recent lattice QCD study [11] of the Λ (1405) comes to in-teresting conclusions (see Fig.2). When all quarks are kept at a relatively largemass scale characteristic of the strange quark mass, the Λ (1405) behaves morelike a three-quark system reminiscent of the naive quark model. As the lightquarks approach their physical masses, the Λ (1405) emerges as a state withdominant quasi-molecular ¯ KN structure. This is where lattice QCD meetschiral SU (3) coupled-channels dynamics.The Λ (1405) thus figures as prototype of a non-conventional hadronicspecies displaying a quasi-molecular weak-binding structure close to a thresh-old (in this case the ¯ KN threshold). Several cases of analogous phenomena areobserved and discussed in the physics with charmed quarks close to thresholdsinvolving D or D ∗ mesons. KN N three-body systems
Low-energy antikaon-neutron interactions are accessible through the investi-gation of the ¯ K -deuteron three-body system. Examples of calculations of the K − deuteron scattering length using data-constrained K − p input togetherwith predicted K − n amplitudes are presented in Fig. 3. Different approaches(non-relativistic EFT [12], Faddeev [13] and three-body multiple scatteringcalculations [14]) are seen to arrive at reasonably consistent results. Startingfrom the K − p and K − n amplitudes derived in [6] gives [14]: a ( K − d ) = ( − .
55 + i 1 .
66) fm , (4)with 10-20 % uncertainties [15] resulting from the “fixed scatterer” (no-recoil)treatment of the nucleons, uncertainties in the K − n amplitude and neglectof K − d → Y N absorption. Note that while the real parts of the individ-ual scattering lengths (1) and (2) would tend to cancel in leading order, the
Wolfram Weise
Fig. 4. Uncertainty of the boundary of allowed values for A Kd . The central value(solid line) corresponds to the average of a p from scattering data and the SID-DHARTA value; uncertainty (shaded area) from the combined errors of these twosources. In summary, we have reanalysed the predictions for the kaon-deuteronscattering length in view of the new kaonic hydrogen experiment from SID-DHARTA. Based on consistent solutions for input values of the K − p scatter-ing length, we have explored the allowed ranges for the isoscalar and isovectorkaon-nucleon scattering lengths and explored the range of the complex-valuedkaon-deuteron scattering length that is consistent with these values. In partic-ular, the new SIDDHARTA measurement is shown to resolve inconsistenciesfor a , a , and A Kd as they arose from the DEAR data. A precise measure-ment of the K − d scattering length from kaonic deuterium would thereforeserve as a stringent test of our understanding of the chiral QCD dynamicsand is urgently called for. Acknowledgments
We thank Akaki Rusetsky for stimulating discussions. Partial financial supportfrom the EU Integrated Infrastructure Initiative HadronPhysics2 (contractnumber 227431) and DFG (SFB/TR 16, “Subnuclear Structure of Matter”)is gratefully acknowledged.
References [1] M. Bazzi, G. Beer, L. Bombelli, A. M. Bragadireanu, M. Cargnelli, G. Corradi,C. Curceanu, A. d’Uffizi et al. , [arXiv:1105.3090 [nucl-ex]]. Re a ( K − d ) [ fm ] Im a ( K − d ) [ fm ] excludedexcluded ( a )( b )( c ) Fig. 3
Real and imaginary parts of the K − deuteron scattering length subject to empiricalconstraints from K − p data. (a): non-relativistic EFT calculation [12]; (b): Faddeev calcu-lation using separable amplitudes [13]; (c) three-body multiple scattering calculation [14]using K − p and K − n amplitudes from [6]. K − pn ↔ ¯ K nn charge exchange process is important in determining the result(4).Predictions for a ( K − d ) can in principle be tested by measuring the strong-interaction energy shift ∆E and width Γ of 1s kaonic deuterium. This requiresa non-trivial three-body calculation of the K − deuteron atomic system in thepresence of strong interactions, but a good estimate can already be obtainedusing the improved Deser formula [16]: ∆E − i Γ − µ α a ( K − d )1 − µα (1 − ln α ) a ( K − d ) , (5)where µ is the kaon-deuteron reduced mass and α is the fine structure constant.With a ( K − d ) taken from (4) this gives ∆E = 0 .
87 keV and Γ = 1 .
19 keV,again with an estimated uncertainty of 10 - 20 %.A great amount of activities has been focused in recent times on the possi-ble existence of quasibound antikaon-nuclear clusters, inspired by earlier phe-nomenological considerations [17]. A prototype example is the K − pp system.Here we briefly report on theoretical expectations using interactions that arebased on chiral SU (3) dynamics with coupled channels. The present situationis summarized in Table 1 which collects results from three different varia-tional and Faddeev three-body calculations. While these computations differin details, there is now a high degree of at least qualitative consistency inthe predictions that a possible K − pp cluster is expected to be weakly bound(with a binding energy not exceeding about 15-20 MeV), but short-lived witha width around 40 MeV or larger. Such a large width would make experimentalsearches for a quasibound K − pp state not an easy task.The weak-binding scenario resulting from chiral SU (3) dynamics withenergy-dependent interactions and strong ¯ KN ↔ πΣ channel coupling dif-fers conceptually and quantitatively from the earlier phenomenological picture opics in Low-Energy QCD with Strange Quarks 7 Table 1
Binding energies and widths of the K − pp system from variational [18,20] andFaddeev [19] calculations using energy-dependent chiral SU(3) based interactions as input.Variational [18] Faddeev [19] Variational [20]B [MeV] 17-23 9-16 16 Γ [MeV] 40-70 34-46 41 [17] that assumes an energy-independent local ¯ KN potential and generates the Λ (1405) as a single pole in the ¯ KN amplitude. While both approaches (chi-ral SU (3) dynamics vs. phenomenology) reproduce kaonic hydrogen and theexisting (admittedly not very accurate) K − p scattering data, they differ sig-nificantly in their extensions below ¯ KN threshold. The energy-independentpotential model would suggest a much more strongly bound K − pp state thanthe approach based on chiral SU (3) dynamics.A possible way of distinguishing empirically between chiral SU (3) two-pole dynamics and single-pole phenomenolgy is offered by the K − d → nπΣ reaction with an incident K − momentum around 1 GeV/c and detection ofthe outgoing neutron, as discussed in [21]. The neutron energy spectrum re-flects the πΣ mass distribution. Below K − p threshold the spectra of the threecharge combinations, π − Σ + , π Σ and π + Σ − , are expected to feature charac-teristic differences between the energy-dependent two-poles approach and theenergy-independent single-pole phenomenology. A corresponding experimentalproposal (E31) is on the way at J-PARC.Several dedicated experiments are now being pursued in a focused searchfor a possible K − pp quasibound state. The HADES collaboration [22] hasperformed a detailed partial wave analysis of the pp → K + Λp reaction with3.5 GeV incident protons. A possible quasibound K − pp cluster should showup via its decay into Λp . No significant signal was found, and an upper limitfor the cluster production cross section of about 1 - 4 µ b has been deduceddepending on the assumed width of the hypothetical K − pp quasibound state.Further recent examples of dedicated activities are the He( K − , n ) X (E15) [23]and d ( π + , K + ) X (E27) [24,25] experiments at J-PARC. The first set of E15data [23] shows no structure in the deeply-bound region but a small excessbelow K − pp threshold in the energy range expected for a possible weakly-bound cluster. On the other hand, E27 observes a very broad structure in arange 60 - 140 MeV below K − pp threshold [25]. While definitive conclusioncannot be drawn so far, the more focused exclusive measurements and analysisof the final state X continue. Along with successful applications to kaon- and antikaon-baryon systems, chi-ral SU (3) effective field theory provides also the appropriate framework for atheory of hyperon-nucleon interactions. Recent progress has been made [26] indeveloping this ChEFT framework at next-to-leading order (NLO) such that Wolfram Weise note: moderate attraction at low momenta strong repulsion at higher momenta
Author's personal copy J. Haidenbauer et al. / Nuclear Physics A 915 (2013) 24–58
Fig. 2. “Total” cross section σ (as defined in Eq. (24)) as a function of p lab . The experimental cross sections are takenfrom Refs. [54] (filled circles), [55] (open squares), [69] (open circles), and [70] (filled squares) ( Λ p → Λ p ), from [56]( Σ − p → Λ n , Σ − p → Σ n ) and from [57] ( Σ − p → Σ − p , Σ + p → Σ + p ). The red/dark band shows the chiral EFTresults to NLO for variations of the cutoff in the range Λ = ,...,
650 MeV, while the green/light band are results toLO for Λ = ,...,
700 MeV. The dashed curve is the result of the Jülich ’04 meson-exchange potential [37]. also for Λ p the NLO results are now well in line with the data even up to the Σ N threshold.Furthermore, one can see that the dependence on the cutoff mass is strongly reduced in the NLOcase. We also note that in some cases the LO and the NLO bands do not overlap. This is partlydue to the fact that the description at LO is not as precise as at NLO (cf. the total χ values inTable 5). Also, the error bands are just given by the cutoff variation and thus can be consideredas lower limits.A quantitative comparison with the experiments is provided in Table 5. There we list theobtained overall χ but also separate values for each data set that was included in the fittingprocedure. Obviously the best results are achieved in the range Λ = χ exhibits also a fairly weak cutoff dependence so that one can really speak ofa plateau region. For larger cutoff values the χ increases smoothly while it grows dramatically Author's personal copy
J. Haidenbauer et al. / Nuclear Physics A 915 (2013) 24–58 Λ p S and P phase shifts δ as a function of p lab . The red/dark band shows the chiral EFT results toNLO for variations of the cutoff in the range Λ = ,...,
650 MeV, while the green/light band shows results to LO for Λ = ,...,
700 MeV. The dashed curve is the result of the Jülich ’04 meson-exchange potential [37].Fig. 7. The Λ p phase shifts for the coupled S – D partial wave as a function of p lab . Same description of curves asin Fig. 6. state in the Σ N system. It should be said, however, that the majority of the meson-exchangepotentials [36,38,39] produce an unstable bound state, similar to our NLO interaction. The onlycharacteristic difference of the chiral EFT interactions to the meson-exchange potentials mightbe the mixing parameter $ which is fairly large in the former case and close to 45 ◦ at the Σ N threshold, see Fig. 7. It is a manifestation of the fact that the pertinent Λ p T -matrices (for the S → S , D → D , and S ↔ D transitions) are all of the same magnitude.The strong variation of the S – D amplitudes around the Σ N threshold is reflected inan impressive increase in the Λ p cross section at the corresponding energy, as seen in Fig. 2. LO NLO LONLO repulsion phase shift
Hyperon - Nucleon Interaction (contd.)
J. Haidenbauer, S. Petschauer, N. Kaiser,U.-G. Meißner, A. Nogga, W. W.Nucl. Phys. A 915 (2013) 2422 k F (1/fm) -60-45-30-150 U ! ( M e V ) k F (1/fm) -20-15-10-50510 S ! ( M e V f m ) Figure 2: The Λ s.p. potential U Λ ( p Λ =
0) (left) and the Scheerbaum factor S Λ (right), as a function of the Fermi momentum k F . The red / darkband shows the chiral EFT results to NLO for variations of the cuto ff in the range Λ = . .. ,650 MeV, while the green / light band are results toLO for Λ = . . . ,700 MeV. The hatched band is the NLO fit from [4] without antisymmetric spin-orbit force. The dashed curve is the result ofthe J¨ulich ’04 meson-exchange potential [43]. also in Table 3 and denoted by NLO † (650). Clearly in this case S Λ is significantly larger. A closer inspection of therespective partial-wave contributions reveals what one expects anyway, namely that the main di ff erence is due to theantisymmetric spin-orbit force whose contribution is given in the column labelled with P ↔ P . It is zero for the LOinteraction (because there is no antisymmetric spin-orbit force at that order) and also for the original NLO interaction,where it has been assumed to be zero. With the corresponding contact term included its strength can be used tocounterbalance the sizeable spin-orbit force generated by the basic interaction so that the small S Λ is then achievedby a cancellation between the spin-orbit and antisymmetric spin-orbit components of the G -matrix interaction. Note,however, that the results in Table 3 indicate that such a cancellation is not the only mechanism that can providea small spin-orbit force. For example, let us look at the predictions for the interaction (denoted by) NLO (600 ∗ ),where all two-meson-exchange contributions involving the η - and / or K meson have been omitted. Here, a very small S Λ is achieved without any antisymmetric spin-orbit force. It is simply due to overall more repulsive contributions,notably in the P and P partial waves. We want to emphasize that the interaction NLO (600 ∗ ) reproduces all Λ N and Σ N scattering data with the same high quality as the other EFT interactions at NLO, see the results in Ref. [4].Finally, it is worthwhile to mention that also the J¨ulich meson-exchange potentials predict rather small values for theScheerbaum factor. For both interactions there is a substantial contribution from the antisymmetric spin-orbit forcewhich is primarily due to vector-meson ( ω ) exchange [32]. The magnitude is comparable to the one of the chiral EFTinteraction at NLO.The dependence of S Λ on k F can be seen in Fig. 2. Obviously, except for the LO interaction, the density de-pendence is fairly weak. A weak density dependence of S Λ was also observed in Ref. [20] in a calculaton of theScheerbaum factor for a YN interaction based on the quark model.Let us now come to the Σ hyperon. In the course of constructing the NLO EFT interaction [4] it turned outthat the available YN scattering data can be fitted equally well with an attractive or a repulsive interaction in the S partial wave of the I = / Σ N channel. Indeed the underlying SU(3) structure as given in Table 1 of Ref. [4]suggests that there should be some freedom in choosing the interaction in this particular partial wave. From that tableone can see that the S partial wave in the I = / Σ N channel belongs to the “isolated” 10 representation so that9 LONLO Λp S Fig. 4
Left: Singlet s-wave Λp phase shift derived from the haperon-nucleon interactionbased on chiral SU (3) EFT [26] in leading (LO) and next-to-leading order (NLO). Uncer-tainty bands refer to variations within a range of momentum space cutoffs 0.5 - 0.7 GeV. Thedashed curve is the result obtained with the phenomenological J¨ulich04 potential. Right: Sin-gle particle potential U Λ of a Λ hyperon in nuclear matter as function of Fermi momentum[27]. Notations as in the left figure. the relevant one- and two-meson exchange mechanisms involving the completepseudoscalar meson octet are properly incorporated. The important two-pionexchange ΛN interaction with ΣN intermediate states emerges naturally inthis approach.Fig. 4 (left) shows as an example the momentum dependence of the S phase shift for Λp scattering calculated in such a framework. While the leading-order (LO) result displays attraction over the whole range of momenta, theNLO calculation shows a turnover from attraction at low momenta to repulsionat high momenta. This behavior is qualitatively consistent with characteristicfeatures of hyperon-nucleon potentials deduced from lattice QCD [28] whereit is found that intermediate-range attraction turns to strong short-range re-pulsion at distances r (cid:39) . Λ single particle potential U Λ in nuclearmatter derived from a Brueckner G-matrix calculation using the ChEFT in-teraction (with inclusion of the dominant S - D channel) is shown in Fig. 4(right). The dependence of U Λ on the nuclear Fermi momentum k F demon-strates the stabilization of the NLO potential at values consistent with proper-ties of Λ hypernuclei. Repulsive effects are expected to act more prominently athigher densities, several times the density of normal nuclear matter ( (cid:37) (cid:39) . − ).Such repulsive interactions of the Λ with nucleons in dense baryonic matterare presently under active discussion in view of the new constraints impliedby the existence of massive neutron stars. The recent observation of two neu-tron stars with M (cid:39) M (cid:12) sets strong constraints on the required stiffness ofthe equation of state (EoS) in order to support such objects against gravi- opics in Low-Energy QCD with Strange Quarks 9 ρ th Λ towards a densityregion where the contribution coming from the hyperon-nucleon potential cannot be compensated by the gain inkinetic energy. It has to be stressed that (I) and (II) givequalitatively similar results for hypernuclei. This clearlyshows that an EoS constrained on the available bindingenergies of light hypernuclei is not sufficient to draw anydefinite conclusion about the composition of the neutronstar core.The mass-radius relations for PNM and HNM obtainedby solving the Tolman-Oppenheimer-Volkoff (TOV)equations [47] with the EoS of Fig. 1 are shown in Fig. 2.The onset of Λ particles in neutron matter sizably reducesthe predicted maximum mass with respect to the PNMcase. The attractive feature of the two-body Λ N interac-tion leads to the very low maximum mass of . M ⊙ ,while the repulsive Λ NN potential increases the pre-dicted maximum mass to . M ⊙ . The latter resultis compatible with Hartree-Fock and Brueckner-Hartree-Fock calculations (see for instance Refs. [2–5]). M [ M ] R [km]PNM (cid:82) N (cid:82) N + (cid:82)
NN (I) (cid:82)
N + (cid:82)
NN (II)0.00.40.81.21.62.02.42.8 11 12 13 14 15
PSR J1614-2230PSR J0348+0432
Figure 2. (Color online) Mass-radius relations. The key isthe same of Fig. 1. Full dots represent the predicted max-imum masses. Horizontal bands at ∼ M ⊙ are the ob-served masses of the heavy pulsars PSR J1614-2230 [18] andPSR J0348+0432 [19]. The grey shaded region is the excludedpart of the plot due to causality. The repulsion introduced by the three-body force playsa crucial role, substantially increasing the value of the Λ threshold density. In particular, when model (II) forthe Λ NN force is used, the energy balance never favorsthe onset of hyperons within the the density domain thathas been studied in the present work ( ρ ≤ . fm − ).It is interesting to observe that the mass-radius relationfor PNM up to ρ = 3 . ρ already predicts a NS massof . M ⊙ (black dot-dashed curve in Fig. 2). Evenif Λ particles would appear at higher baryon densities,the predicted maximum mass is consistent with present astrophysical observations.In this Letter we have reported on the first QuantumMonte Carlo calculations for hyperneutron matter, in-cluding neutrons and Λ particles. As already verifiedin hypernuclei, we found that the three-body hyperon-nucleon interaction dramatically affects the onset of hy-perons in neutron matter. When using a three-body Λ NN force that overbinds hypernuclei, hyperons appeararound twice saturation density and the predicted max-imum mass is . M ⊙ . By employing a hyperon-nucleon-nucleon interaction that better reproduces theexperimental separation energies of medium-light hyper-nuclei, the presence of hyperons is disfavored in the neu-tron bulk at least until ρ = 0 . fm − and the lowerlimit for the predicted maximum mass is . M ⊙ .Therefore, within the Λ N model that we have consid-ered, the presence of hyperons in the core of the neutronstars cannot be satisfactory established and thus there isno clear incompatibility with astrophysical observationswhen lambdas are included. We conclude that in order todiscuss the role of hyperons - at least lambdas - in neu-tron stars, the Λ NN interaction cannot be completelydetermined by fitting the available experimental energiesin Λ hypernuclei. In other words, the Λ -neutron-neutroncomponent of the Λ NN will need additional theoret-ical investigation and a substantial additional amountof experimental data. In particular, there are somefeatures of the hyperon-nucleon interaction ( Λ -neutron-neutron channels, spin-orbit contributions) which mightbe efficiently constrained only by experiments involvinghighly asymmetric hypernuclei and/or excitation of thehyperon. We believe that our conclusions will not changequalitatively if other hyperons and/or a v ΛΛ are includedin the calculation.We would like to thank J. Carlson, S. C. Pieper, S.Reddy, A. W. Steiner, and R. B. Wiringa for stimulatingdiscussions. This research used resources of the NationalEnergy Research Scientific Computing Center (NERSC),which is supported by the Office of Science of the U.S.Department of Energy under Contract No. DE-AC02-05CH11231. The work of D.L. and S.G was supported bythe U.S. Department of Energy, Office of Nuclear Physics,under the NUCLEI SciDAC grant and A.L. by the De-partment of Energy, Office of Nuclear Physics, under con-tract No. DE-AC02-06CH11357. The work of S.G. wasalso supported by a Los Alamos LDRD grant. F.P. isalso member of LISC, the Interdisciplinary Laboratoryof Computational Science, a joint venture of the Univer-sity of Trento and the Bruno Kessler Foundation. [1] V. A. Ambartsumyan and G. S. Saakyan, Sov. Astro. AJ , 187 (1960).[2] E. Massot, J. Margueron, and G. Chanfray, EuroPhys. n − matter ΛN ΛN ΛN ΛNN ( ) ΛNN ( ) ++ ChEFT QMC R [ km ] MM O . Fig. 5
Mass-radius relations for neutron stars. Solid black curve: ChEFT result (nucleon +pion degrees of freedom) taken from [29]; colored curves: QMC computations [30] including Λ hyperons with phenomenological ΛN forces and two versions of repulsive ΛNN three-bodyinteractions. Version
ΛNN (2) reproduces the systematics of hypernuclear binding energies. tational collapse. An EoS based on ChEFT with “conventional” nucleon andpion degrees of freedom can produce sufficient pressure at high density, gen-erated by repusive three-body forces and the impact of the Pauli principle onthe in-medium nucleon-nucleon effective interaction [29] (see Fig. 5). However,neutrons in the core of the star tend to be replaced by Λ hyperons at densities(typically around 2-3 (cid:37) ) where this becomes energetically favorable. Thenthe EoS would soften too much so that maximum neutron star masses of 2 M (cid:12) cannot be sustained any more.A recent advanced quantum Monte Carlo (QMC) computation of neutronstar matter, with hyperons added [30], emphasizes this issue. While this cal-culation still uses phenomenological ΛN input interactions, the conclusionsare nonetheless instructive. When parametrized repulsive ΛN N three-bodyforces are added subject to the condition that the systematics of hypernuclearbinding energies be reproduced, the admixture of Λ ’s in neutron star mat-ter gets strongly reduced such that the pressure to support a 2 M (cid:12) star canbe maintained as demonstrated in Fig. 5. The pending question is whetherthe necessary repulsive effect can be entirely relegated to a hypothetical ΛN N three-body force, or whether at least a large part of it comes from momentum-dependent ΛN two-body interactions as they appear in the SU (3) ChEFTtreatment [26] at next-to-leading order. Progress has been made in establishing chiral SU (3) effective field theory asthe adequate realization of low-energy QCD with strange quarks. It defines aconsistent and well organized coupled-channels framework for kaon-, antikaon- and hyperon-nuclear interactions. The investigation of strangeness S = − B = 1 , Λ (1405) as a weakly bound (quasi-molecular) ¯ KN state imbedded in thestrongly coupled πΣ continuum. Threshold and subthreshold ¯ KN N physicsis proceeding towards a focused experimental program. Concerning the roleof strangeness in dense baryonic matter, new constraints imposed by the exis-tence of two-solar-mass neutron stars and the required stiffness of the equationof state imply a quest for strong short-distance repulsion in hypernuclear two-and three-body interactions. Lattice QCD studies are on their way to providefurther basic information on these issues.
Acknowledgements
Collaborations with Thomas Hell, Tetsuo Hyodo, Yoichi Ikeda andShota Ohnishi on topics reported in this paper are gratefully acknowledged. Useful commentsand suggestions by Avraham Gal are much appreciated. I also thank Daniel Gazda, NorbertKaiser, Maxim Mai and Stefan Petschauer for discussions. This work was supported in partby BMBF and DFG (CRC 110 “Symmetries and Emergence of Structure in QCD”).
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