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Microscopic Equations of State

Microscopic Equations of State. M. Baldo. Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Catania, Italy. ECT* 2017. Outlook. 1. Microscopic many -body theories . 2. Choice of the force. 3. Comparison of the results . 4. The saturation point and around .

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Microscopic Equations of State

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  1. Microscopic Equations of State M.Baldo Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Catania, Italy ECT* 2017

  2. Outlook • 1. Microscopicmany-body theories. • 2. Choice of the force. • 3. Comparison of the results. • 4. The saturationpoint and around. • 5. Higherdensity • 6. Where do we stand ?

  3. Experimental and • observationalconstraints • 1. Nuclearstructure • 2. Heavyions • 3. Neutron Stars • 4. Gravitationalwaves

  4. What can weget from thisoverall set of constraints ? • 1. The constraintsrestrict the properties of the EOS • butsurelythey do notfixit. An ample family of • EOS can be compatible with the phenomenological • bounds. • 2. The constraints are obtainedgenerallythrough the • use of phenomenological Energy DensityFunctionals, • which can generate spuriouscorrelationsamong • physicalquantities. • One can follow a differentapproach : • Develop a microscopicmany-body theory of the EOS and compare • with the phenomenologicalconstraints. Thenonegets : • Selection of the EOS • 2. Hints on the structure of nuclearmatter

  5. The constraints are coming from quitedifferentphysicalconditions. • H.I. : small asymmetry, high temperature. • SN : high asymmetry and high temperature. • NS : high asymmetry and low temperature. • GW : very high density, asymmetry and temperature (NS mergers). • A microscopictheory must be able to treatallthesephysical • situations.

  6. There are two main basic elements in the microscopic approach. • The microscopicmany-body approach • 1. The Brueckner-Bethe-Goldstoneexpansion (BBG) • and Coupled Cluster (CC) expansion • 2. Self-consistentGreen’sfunction. • 3. The variationalmethod • 4. The relativisticDirac-Bruecknerapproach • 5. The renormalizationgroup • 6. Quantum Monte Carlo • The choice of the bare nucleonic force • 1. Mesonexchangemodels • 2. Chiralapproach • 3. Quark models

  7. Two-body forces Meson exchange models Three-body forces

  8. BBG Two-body force only Two and three body correlations One can get an estimate of the theoretical uncertainy by comparing the EOS with different choices of the auxiliary single particle potential.

  9. Three-body forces are necessary to get the correct saturation point. However their contribution is much smaller than the two-body one (around saturation) Phenomenological three-body forces BHF (M.B. et al. PRC87, 064305 (2013) ) Variational (Akmal et al. PRC58, 1804 (1998) )

  10. Dirac – Brueckner. Two-body forces only NN interactions Bonn A, B, C T. Gross-Boelting et al. NPA 648, 105 (1999)

  11. Chiral expansion approach, from QCD symmetry Two-body forces Pion exchange + point interactions Three-body forces Systematic hierarchy of the relevance of the forces The quark degrees of freedom do not appear explicitly

  12. Chiral force + RG, perturbative calculation Hebeler et al. PRC 83, 031301 (2011) No saturation with only two-body forces Three-body forces essential and large even at saturation

  13. Confirmation of the perturbative approach for the evolved chiral forces A. Carbone, A. Polls and A. Rios, PRC 88, 044302 (2013)

  14. Difficulty in fitting both few-body and Nuclear Matter saturation point Hagen et al., PRC 89, 014319 (2014) Coupled Cluster calculations up to selected triples, chiral forces. Situation similar to the one for meson exchange models.

  15. D. Logoteta, I. Bombaci and A. Kievsky, PRC 94, 064001 (2016) Strength of the TBF reduced

  16. Bound and scattering three-body system

  17. Including three-body correlations in Nuclear Matter

  18. Relevance of three-body correlations for the QM interaction At saturation n = 0.157 fm-3 K = 219 MeV E/A = -16.3 MeV This is at variance with respect to the other NN interactions that need three-boy forces (non relativistic) M.B. andK.Fukukawa, PRL113, 242501 (2015)

  19. Comparing two-body and three-body CORRELATIONS

  20. Comparing the QM EOS with other models

  21. Comparison with other non relativistic models for pure Neutron Matter

  22. Let us consider a brief survey of the comparison with the phenomenological constraints. Only the EOS that give the correct saturation point will be included Nuclear structure : Symmetry energy Heavy ions

  23. Overall comparison of the symmetry energy below saturation. IAS + neutron skin data Fair agreement among different EOS Some discrepancy close to saturation From : M.B. & G.F. Burgio, Prog. Part. Nucl. Phys. 2016

  24. Symmetry energy above saturation Substantial disagreement among the different EOS No relevant constraints from Heavy Ion data (up to now) Higher density constraints would be quite selective (CBM experiment at FAIR)

  25. EoS for NS matter i.e. beta-stable nuclear matter with components :

  26. The constraint from the observed maximum mass. Hatched area : Bayesiananalysis by Lattimer & Steiner, EPJA50, 40 (2014) Differentfunctionals, includingSkyrme. Crustincluded. Sharma et al., A&A 584, A103 (2015)

  27. Neutron Star mass as a function of radius and central density QM force Other (microscopic) EOS. Dramatic effect of the hyperon component

  28. Other hyperon-nucleon and hyperon-hyperon interaction models

  29. Possible solution Introducing multi-body forces for hyperons Multi-pomeron exchange potential (MPP) Only nucleons With hyperons Universal repulsive force for all baryon sectors, including hyperons Yamamoto et al., PRC90, 045805 (2014)

  30. Universal repulsive baryon-baryon interaction related to three anf four-body forces. • One can cocludethat : • Extra repulsionisneeded. • The multi-body forces in the hyperonic • sector must be atleastas strong as in the • nucleonicsector. Similar conclusion in D. Lonardoni et al., PRL114, 092301 (2015).

  31. A similar conclusion is obtained also in DBHF, assuming SU(6), which is equivalent to take the same TBF in the nucleonic and hyperonic sector Katayama & Saito, PLB 747, 43 (2015)

  32. Can the solution come from the quark degrees of freedom ? Shaded area : mixed phase QP : pure quark matter Introducing the quark degrees of freedom Bag model with density dependent bag constant Hyperons mainly disappear and the maximum mass is determined by the quark EOS, but it is still below the observational limit G.F. Burgio et al., PLB 526, 19 (2002)

  33. With respect to the MIT bag model there is need of additional repusion at high density. This problem has been approached within several schemes 1. Color dielectric model 2. Nambu – Jona Lasinio model + additional interactions 3. Dyson – Schwinger equation 4. Field correlator method 5. Freedman & McLerran model of QCD With a suitable choice of the parameters they are able to reach the two solar mass limit (but one must check that hyperons are prevented to appear or they have little effect )

  34. The quark matter EOS can be as stiff as the nucleonic EOS at high density T. Koyo et al., PRD91, 045003 (2015), extended NJL model Vector + diquark interaction

  35. CONCLUDING REMARKS • Thereis a set of microscopicnucleonic EOS • that are compatible with the phenomenological • constraints. More constraints are expected • from GW, heayions and astrophysical data • Theysubstantiallyagree up to density just above • saturation, in particular on the symmetryenergy • Disagreementsappearathigherdensity, whichmeans • thatconstraints in thisdensityregionwould be • veryeffective in selecting the microscopic EOS • Ifhyperonic and quark degrees of freedom are • introduced, the observedmasses of NS require • a substantialadditionalrepulsion with respect to • the simplestmodels, either to stiffen the EOS • or to hinder the appearence of these ‘exotic’ • components. A sound QCD theoreticalbasis • for thisrepulsionisstilllacking

  36. A systematicapplication of thesemicroscopic • many-body theories to the calculations of other • NS properties (e.g. MURCA, transport, GW, Moment • of inertia) are still scarse. Hopefullythiscouldprovide • furtherselection. Major uncertainties : • Three-body forcesunknownat high density. • Theirrelevanceis model dependent. • If quark degrees of freedom are introducedtheir • relevanceseems to be reduced to a minimum. • The effect of ‘exotic’ components (mainlyhyperon and • quark) hasnot a sound theoreticalframework

  37. The microscopic EOS can be compatible with most of the constraints. The latter can give, especially the astrophysical ones, hints on the direction where to move, e.g. the additional repulsion at high density in the ‘ exotic ‘ sector. The two solar mass constraint seems not so selective for the pure nucleonic EOS. The symmetry energy above saturation looks very selective for the nucleonic EOS ( -- heavy ion ). It is indeed the mutual interaction between phenomenology and theory that can support additional progresses in the field MANY THANKS !

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