Nuclear Equation of State for Compact Stars and Supernovae

Isaac Vidaña CFC, University of Coimbra

Pulsar and their Environments May 11th – 13th 2015

University of Orléans (France)

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Ø  This is not a conventional talk. I will not present the results of a particular work. The scope of this talk is to initiate & motivate the discussion on the topic of the nuclear EoS

Ø  Some of the people already know this talk because they were in Lyon last November, so I would like to apologize if they find it repetitive

The Four Topics of the WG2 as defined in the MOU of NewCompstar

q  Nuclear EoS for Compact Stars & Supernovae

q  Low-energy QCD & Super-dense matter

q  Superfluidity & Superconductivity in Compact Stars

q  Transport phenomena & Reaction rates for Compact Stars & Supernovae

Nuclear EoS for Compact Stars & Supernovae One of the tasks of WG2 is:

Validate available EoS across different experiments & observations. The WG will define a protocol where the EoS, and the underlying interactions, are consistently checked and compared with nuclear experiments & observations of compact stars

To such end we need to: v  Define where we are today (not an easy task)

v  Define where we want to go tomorrow (even more difficult)

•  search, compile & classify existing EoS

•  Perhaps perform a collective effort to construct reliable EoSs to be used in astrophysical applications ? Partly done with CompOSE

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What do we know to build the Nuclear EoS ?

v  Masses, radii & other properties of ~ 3200 isotopes

v  Around ρ0 the nuclear EoS can be characterized by few isoscalar & isovector parameters

x = ρ − ρ03ρ0

, β =ρn − ρp

ρn + ρp

EAρ,β( ) = E0 +

12K0x

2 +16Q0x

3 + Esym + Lx +12Ksymx

2 +16Qsymx

3!

"#

$

%&β 2 + ⋅ ⋅ ⋅

J. Erler et al., Nature 486, 509 (2012)

v  Isoscalar parameters

v  Isovector parameters

Less certain. Large variation of the prediction of the different models

K0 = 9ρ0

2 ∂2EIS (ρ)∂ρ2

ρ=ρ0

≈ 240± 20 MeV , Q0 = 27ρ03 ∂

3EIS (ρ)∂ρ3

ρ=ρ0

≈ −500÷300 MeVE0 ≈ −16 MeV ,

L = 3ρ0∂EIV

∂ρ ρ=ρ0

Ksym = 9ρ02 ∂

2EIV

∂ρ2ρ=ρ0

, Qsym = 27ρ03 ∂

3EIV

∂ρ3ρ=ρ0

Esym =12∂ 2E / A∂β 2

β=0

,

J. M. Lattimer & A. W. Steiner, EPJA 50, 40 (2014)

ü  Neutron skin thickness in heavy nuclei ü  Giant & pygmy resonances in neutron-rich nuclei ü  n/p & t/3He ratios in nuclear reactions ü  Isospin fragmentation & isospin scaling in nuclear multi-fragmentation ü  Neutron-proton correlation functions at low relative momenta ü  Isospin diffusion/transport in heavy ion collisions ü  Neutron-proton differential flow

Symmetry Energy Sensitive Observables

§  Sub-saturation densities

§  Supra-saturation densities

ü  π-/π+ & K-/K+ ratios in heavy ion collisions ü  Neutron-proton differential transverse flow ü  n/p ratio of squeezed out nucleons perpendicular to the reaction plane ü  Nucleon elliptic flow at high transverse momenta

Constraints of the Nuclear EoS from HIC

1

10

100

1 1.5 2 2.5 3 3.5 4 4.5 5

neutron matter

Akmalav14uvIINL3DDFermi GasExp.+Asy_softExp.+Asy_stiff

P (M

eV/fm

3)

ρ/ρ0

v  Collective flow constraints confirms the softening of the EoS at high densities

v  Constraints from kaon production are consistent with the flow constraints and bridge gap to GMR constraints

v  Symmetry energy dominates the uncertainty in the neutron matter EoS

P. Danielewicz et al., Science 298, 1592 (2002)

Astrophysical determination of the Nuclear EoS

F. Ozel & D. Psaltis, PRD 80, 103003 (2009) F. Ozel, G. Baym & T. Guver, PRD 82, 101301(R) (2010)

²  Piecewise polytropic EoS above ρ0 from mass-radius relation of 3 type-I X-ray bursts

ρi−1 < ρ ≤ ρi, ε =αiρ +βiρΓi , P = Γi −1( )βiρΓi

v  SLy below ρ0

v  Piecewise poytropic above ρ0

Astrophysical determination of the Nuclear EoS

A. W. Steiner, J. M. Lattimer & E. F. Brown, ApJ 722, 33 (2010)

²  Nuclear parameters determined in a Bayesian data analysis of:

v  3 type-I X-ray burst

v  3 transient low mass X-ray binaries

v  Cooling of 1 isolated NS, RX J1856-3754

Parameters in the range expected from nuclear systematics & lab. experiments

Approaches to the Nuclear EoS

Phenomenological approaches Microscopic ab-initio approaches

Based on effective density-dependent interactions with p a r a m e t e r s a d j u s t e d t o reproduce nuclear observables and compact star properties

Based on two- & three-body realistic interactions. The EoS is obtained by “solving” the compl i ca t ed many-body problem

v  Variational: APS, CBF, FHNC, LOVC

v  Monte-Carlo: VMC, DMC, GFMC, AFDMC

v  Diagrammatic: BBG (BHF), SCGF

v  RG methods: Vlow k & SRG from χEFT potentials

v  DBHF

v  Liquid drop type: BPS, BBP, LS, OFN

v  Thomas-Fermi: Shen

v  HF: NV, Sk, BSk, PAL, RMF, RHF, QMC d

v  Statistical models: HWN, RG, HS

I apologize for all those approaches

I have missed

A glance on the phenomenological models

Proliferation of phenomenological models predicting different SM & NM EoS Skyrme RMF

J. R. Stone et al., PRC 68, 034324 (2003) T. Klan et al., PRC 74, 035802 (2006)

Recently M. Dutra et al., (arXiv:1405.3633) have analyzed 263 parametrizations of 7 different types of RMF imposing constraints from SM, PNM & Symmetry Energy and its derivatives. Similar analysis was done for 240 Skyrme forces by M. Dutra et al., (PRC 85, 035201 (2012))

Difficulties of microscopic approaches

v  Different NN potentials in the market

v  Short range repulsion makes any perturbation expansion in terms of V meaningless. Different ways of treating SRC

v  C o m p l i c a t e d c h a n n e l & operatorial structure (central, spin-spin, spin-isospin, tensor, spin-orbit, …)

A comparison of some microscopic approaches

Compare different many-body techniques using the same NN interaction to find the sources of discrepancies & ultimately determine “systematic error” associated to the nuclear EoS predicted predicted by many-body theory

The NN: Av18 & simpler ones: Av4’, Av6’ & Av8’ (3BF are not included)

€

Vij = Vp (rij )Oijp

p=1,18∑

Oijp=1,14 = 1,

σ i ⋅σ j( ),Sij,

L ⋅S,L2,L2

σ i ⋅σ j( ),

L ⋅S( )

2"#$

%&'

⊗ 1, τ i ⋅τ j( )"# %&

€

Oijp=15,18 = Tij ,

σ i ⋅ σ j( )Tij ,SijTij , τ zi + τ zj( )[ ]

M. Baldo, A. Polls, A. Rios, H.-J. Schulze & I.Vidaña, PRC 86, 064001 (2012)

A comparison of some microscopic approaches

M. Baldo, A. Polls, A. Rios, H.-J. Schulze & I.Vidaña, PRC 86, 064001 (2012)

Tensor & spin-orbit and their in-medium treatment are at the heart of most of the observed discrepancies

Symmetric nuclear matter Pure neutron matter

Hyperons: Exotic Components of Neutron

Stars Interiors

What do we know to include hyperons in the EoS ?

Something but not much to put stringent constraints on the YN & YY interactions

Ø  Data from about 40 single Λ and 3 double Λ hypenuclei

Ø  Very few YN scattering data due to

ü  difficulties of preparation of hyperon beams.

ü  no hyperon targets available.

(e,e’K+)

€

Λ16N

O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006)

Λ Hypernuclear Chart

Present status of Λ Hypernuclear Spectroscopy

The 3D Nuclear Chart

Double Λ-hypernuclei (S=-2) studied with:

),( +− KK

Single Λ-hypernuclei (S=-1) studied with:

€

(π +,K +),(K−,π−),(e,e'K +)

Ordinary nuclei (S=0)

N

S Z

Total cross YN sections Differential YN cross sections

§  YN: Only about 35 data points, all from the 1960´s §  10 new data points, from KEK-PS E251 collaboration (2000)

(cf. > 4000 NN data for Elab < 350 MeV) §  YY: No scattering data at all

Hyperons in NS interiors have been considered by many authors since the pioneering work of Ambartsumyan & Saakyan (1960). They are expected to appear in the core of neutron stars at ρ ~ (2-3)ρ0 when µN is large enough to make the conversion of N into Y energetically favorable.

€

n + n→ n + Λ

p + e− → Λ + ν e −

n + n→ p + Σ−

n + e− → Σ− + ν e −

€

µΣ−

= µn + µe − −µνe−

µΛ = µn

Effect of Hyperons in the EoS and Mass of Neutron Stars

“stiff” EoS

“stiff” EoS

“soft” EoS

“soft” EoS

Relieve of Fermi pressure due to the appearance of hyperons è

EoS softer è reduction of the mass

Measured Neutron Star Masses (2015)

Observation of ~ 2 Msun neutron stars

Dense matter EoS stiff enough is required such that

Mmax EoS[ ] > 2M¤

Can strangeness (hyperons, kaons, quarks) still be present in the interior of neutron stars in view of this constraint ?

updated from Lattimer 2013

Exotic Matter: The Hyperon Puzzle

“Hyperons è “soft (or too soft) EoS” not compatible (mainly in microscopic approaches) with measured (high) masses. However, the presence of hyperons in the NS interior seems to be unavoidable.”

ü  can YN & YY interactions still solve it ?

ü  or perhaps hyperonic three-body forces ?

ü  what about quark matter ?

Solution I: YY vector meson repulsion

General Feature: Exchange of scalar mesons generates attraction (softening), but the exchange of vector mesons generates repulsion (stiffening)

Add vector mesons with hidden strangeness (φ) coupled to hyperons yielding a strong repulsive contribution at

high densities

Dexhamer & Schramm (2008), Bednarek et al, (2012), Weissenborn et al., (2012)

(explored in the context of RMF models)

ü  σ2, σ3, σ4 terms ü  ρ2, ω2, ω4 terms ü  “hidden strangeness” mesons: σ*, φ

  (σ*2, φ2)

ü  gYV couplings: from SU(6) to SU(3) vary z=g8/g1 & α=F/(F+D) ü  gYS couplings adjusted by fitting UB

(N)

(UΛ

(N)=-30, UΣ(N)=+30, UΞ

(N)=-28 MeV)

ü  σ2, σ3, σ4 terms ü  ρ2, ρ4, ω2, ω4 terms ü  “hidden strangeness” mesons: σ*, φ

(σ*2, φ2, φ4)

ü  cross terms: ω2ρ2, φ2ω2, φ2ρ2

ü  gYV couplings fixed by SU(6) ü  gYS couplings adjusted by fitting UB

(B) (UΛ

(N)=-28, UΣ(N)=+30, UΞ

(N)=-18 MeV) (UΞ

(Ξ)=UΛ(Ξ)=2UΛ

(Λ), UΛ(Λ)=-5 MeV)

Weissenborn et al. (2012) Bednared et al. (2012)

Mmax > 2M¤

Mmax compatible with 1.97M¤

See also the extensive parameter study of Oertel et al. (2014)

Solution II: can Hyperonic TBF solve this puzzle ?

NNN Force

Natural solution based on: Importance of NNN force in Nuclear Physics (Considered by several authors: Chalk, Gal, Usmani, Bodmer, Takatsuka, Loiseau, Nogami, Bahaduri, IV)

NNY, NYY & YYY Forces

Energy density

Pres

sure

NN, NY & YY

NN, NY, YY NNN, NNY, NYY & YYY

Can hyperonic TBF provide enough repulsion at high densities to reach 2M ?

¤

?

The results are contradictory

I. V. et al. (2011) Yamamoto et al. (2015)

BHF with NN+YN+phenomenological YTBF. Different strength of YTBF includin the case of universal TBF

BHF wi th NN+YN+universa l repulsive TBF (mult ipomeron exchange mecanism)

1.27 <Mmax <1.6M¤ Mmax > 2M

¤

It should be mentioned also the recent DBHF calculation of hyperonic matter by Katayama & Saito (2014)

Ø  DBHF includes some TBF effects in a natural way

Ø  Mmax compatible with 2M Ø  But the construction of YN is a

bit obscure in this work

¤

And the recent Quantum Monte Carlo calculation by Lonardoni et al. (2015)

v  First Quantum Monte Carlo calculation on neutron+Λ matter

v  Strong dependence of Λ onset on Λnn force

v  Some of the paramametrizations of the Λnn force give maximum masses compatible with 2M

¤

Solution III: Quark Matter Core

Ozel et al., (2010), Weissenborn et al., (2011), Klaehn et al., (2011), Bonano & Sedrakian (2012), Lastowiecki et al., (2012), Zdunik & Haensel (2012)

To yield Quark Matter should have: Mmax > 2M¤

§  significant overall quark repulsion stiff EoS

§  strong attraction in a channel strong color superconductivity

General Feature:

Some authors have suggested that the hyperon core in neutron stars could be replaced by a cores of uds quark mater. Massive neutron stars could actually be hybrid stars with a stiff quark matter core

§  BQ transition must be rather low ~ 2ρ0-3ρ0 §  Density jumps λi should not bee too large (< 1.2) §  vsound ~ 0.8-0.9 if is observed

requires: Mmax > 2M¤

§  Stiffening EoS µb increases §  If instability (reconfinement) µb

Q (P)> µbH (P)

Zdnunik & Haensel (2012)

2.2− 2.4M¤

Stability of QM core and Mmax

Is there also a Δ isobar puzzle ? The recent work by Drago et al. (2014) calculation have studied the role of the Δ isobar in neutron star matter

v  Constraints from L indicate an early appearance of Δ isobars in neutron stars matter at ~ 2-3 ρ0 (same range as hyperons)

v  Appearance of Δ isobars modify the composition & structure of hadronic stars

v  Mmax is dramatically afected by the presence of Δ isobars

If Δ potential is close to that indicated by π-, e-nucleus or photoabsortion nuclear reactions then EoS is too soft Δ puzzle simlar to the hyperon one

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A First Attempt of Classification

Approach Type β-eq. T=0 Finite T Inhom. Matt.

Hom. Matt. CompOSE Form

LDM Phen X X X X X X Table

TF Phen X X X Table

HF Phen X X X Table

SM Phen X X X X X X Table

Var Micr X X X Analytic & Table

MC Micr X X X Analytic & Table

Diagr Micr X X X X Table

RG Micr X X Table

DBHF Micr X X X Table

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Open Questions

v  Why is the isovector part of the EoS & particularly the density dependence of the nuclear symmetry energy so uncertain ?

v  What is the role of many-body cluster contributions ?

v  Is there a solution for the hyperon puzzle ? . If yes, what it is ?

v  Is there also a Δ puzzle ?. If yes, what is the solution ?

v  And many, many others …

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Incomplete List of References

Phenomenological approaches

[BPS] G. Baym, C. Pethick & P. Sutherland, Astrophys. J 170, 299 (1971) [BBP] G. Baym, H. A. Bethe & C. Pethick, Nucl. Phys. A 175, 225 (1971) [LS] J. M. Lattimer & F. D. Swesty, Nucl. Phys. A 535, 331 (1991) [OFN] M. Oertel, A. Fantina & J. Novak, Phys. Rev. C 85, 055806 (2012)

v  Liquid drop type

v  Thomas-Fermi [Shen] H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, Nucl. Phys. A 637, 435 (1998)

v  Hartree-Fock [Sk] D. Vautherin & D. M. Brink, Phys. Rev. C 3, 626 (1972) [NV] J. N. Negele & D. Vautherin, Nucl. Phys. A 207, 298 (1973) [RMF] B. D. Serot & J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986) [RHF] A. Boussy et al., Phys. Rev. Lett. 55, 1731 (1985); Phys. Rev. C 36, 380 (1987) [PAL] M. Prakash, T. L. Ainsworth & J. M. Lattimer, Phys. Rev. Lett. 61, 2518 (1988) [QMC] P. A. M. Guichon, A. W. Thomas & K. Tsushima, Nucl. Phys. A 814, 66 (2008) [BSk] S. Goriely. N. Chamel & J. M. Pearson, Phys. Rev. C 82, 035804 (2010)

v  Statistical models [HWN] W. Hillebrandt, R. Wolf & K. Nomoto, Astron. & Astrophys. 133, 175 (1984) [RG] A. Raduta & F. Gulminelli, Phys. Rev. C 82, 065801 (2010) [HS] M. Hempel & J. Schaffner-Bielich, Nucl. Phys. A 837, 210 (2010)

Microscopic approaches

[FHNC] S. Fantoni & S. Rosati, Nuov. Cim. A 20, 179 (1974) [LOCV] J. C. Owen, R. F. Bishop & J. M. Irvine, Nucl. Phys. A 277, 45 (1978) [CBF] A. Fabrocini & S. Fantoni, Phys. Lett. B 298, 263 (1993) [APR] A. Akmal, V. R. Pandharipande & D. G. Ravenhall, Phys. Rev. C 58, 1804 (1998)

v  Variational

v  Monte Carlo

v  Diagrammatic [SCGF] L. P. Kadanoff & G. Baym, Quantum Statistical Mechanics (Benjamin, NY., 1962) [BBG] B. D. Day, Rev. Mod. Phys. 39, 719 (1967)

v  Rernomalization Group methods [Vlow k] S. K. Bogner, T. T. S. Kuo & A. Schwenk, Phys. Rep. 286, 1 (2003) [SRG ] S. K. Bogner, R. J. Furnstahl & R. J. Perry, Phys. Rev. C 75, 061001(R) (2007)

v  DBHF B. Ter Haar & R. Malfiet, Phys, Rep. 149, 207 (1987); Phys. Rev. C 36, 1611 (1987) R. Brockmann & R. Machleidt , Phys. Rev. C 42, 1965 (1990) L.Shen et al.,, Phys. Rev. C 56, 216 (1997); C. Fuchs et al., Phys. Rev. C 58, 2022 (1998)

[VMC] R. B. Wiringa, Phys. Rev. C 43, 1585 (1991) [AFDMC] K. E. Schmidt & S. Fantoni, Phys. Lett. B 446, 99 (1999) [GFMC] J. Carlson et al., Phys. Rev C 68, 025802 (2003)

The discussion is served