- interaction, hypernuclei & hyperonic matter

Isaac Vidaña Universidade de Coimbra

“Interaction forte dans la matière nucléaire: nouvelles tendaces”

Ecole Internationale Joliot-Curie, 27 Sept.- 3 Oct. 2009, Lacanau (France) Plan of the Lecture

Introduction & Historical Overview

Birth, Life & Death of Hypernuclei

Hyperon-Nucleon Interaction

Hypernuclear Matter & Star Properties What is a hyperon ?

A hyperon is a made of one , two or three strange

Hyperon Quarks I(JP) Mass (MeV) Λ uds 0(1/2+) 1115 Σ+ uus 1(1/2+) 1189 Σ0 uds 1(1/2+) 1193 Σ- dds 1(1/2+) 1197 Ξ0 uss 1/2(1/2+) 1315 Ξ- dss 1/2(1/2+) 1321 Ω- sss 0(3/2+) 1672 What is a ?

A hypernucleus is a bound system of with one or more strange (Λ,Σ,Ξ,Ω- ).

H. Bando, 1, 54 (1986)

In a simple single- model: , and hyperons are considered distinguishable placed in independent effective potential wells in which 12 Pauli exclusion principle is applied. Simple s.p. model of " C

! Since hyperons are distinguishable from nucleons, they are privileged probes to explore states deep inside the nucleus, extending our knowledge of conventional to flavored .

ΛΛ

Hyperons can change the nuclear nuclear structure. For instance the glue-like role of the Λ hyperon can facilitate the existance of neutron-rich hypernuclei, being a more suitable framework to study matter with extreme n/p ratios as compared to ordinary nuclei.

A hypernucleus is a “laboratory” to study hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions. A simple model of hypernuclei: Hyperon-Nucleus effective potential

Hypernucleus = Ordinary Nuclear Core + Hyperon in a hyperon-nucleus effective potential

eff r r r r + r r % V"N (r) = V0 (r) + VS (r) S N # S " + VT (r)S12 + Vls(r) L $ S + Vals(r) L $ S ( ) ( ) ( )

! tensor antisymmetric central -orbit spin-spin symmetric (zero in NN spin-orbit due to Pauli principle) Present status of Λ Hypernuclear Spectroscopy

Λ Hypernuclear Chart

16 " N

!

(e,e’K+)

O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006) The 3D nuclear chart

Double Λ-hypernuclei (S=-2) studied with: (K ! , K + )

Single Λ-hypernuclei (S=-1) studied with: ( +,K + ),(K , ),(e,e'K + )

S Z Ordinary nuclei (S=0) N Brief History of Hypernuclear Physics First hypernuclear event observed in a nuclear emulsion by Marian Danysz and Jerzy Pniewski in 1952

Incoming high energy cosmic ray

Collision with the nucleus

Nuclear fragments that eventually stop in the emulssion

One fragment containing a hyperon disintegrates weakly To commemorate the discovery of Danysz and Pniewski a postcard was issued by the Polish Post in May 1993

(200.000 postcards, postcard price 2000 zl, stamp 1500 zl) A few years earlier, in 1989, the postmask designed on the basis of the first hypernucleus observation was used for the 20th International Physics Olympiad at the Warsaw post office number 64 Historical Overview

1953 → 1970 : hypernuclear identification with visualizing techniques emulsions, bubble chambers

1962 : first double Λ hypernucleus discovered in a nuclear emulsion irradiated by a beam of K- at CERN 1970 → Now : Spectrometers at accelerators: CERN (up to 1980) BNL : (K-, π-) and (π+, K+) production methods KEK : (K-, π-) and (π+, K+) production methods

Φ - π- After 2000 : Stopped at DA NE (FINUDA) : (K stop, ) The new electromagnetic way : HYPERNUCLEAR production with BEAM (e,e’K+) at JLAB &MAMI-C International Hypernuclear Network

PANDA at FAIR • 2012~ SPHERE at JINR • Anti- beam • Heavy ion beams • Double Λ-hypernuclei • Single Λ-hypernuclei • γ-ray spectroscopy HypHI at GSI/FAIR MAMI C • Heavy ion beams • Single -hypernuclei at • 2007~ Λ • Electro-production extreme isospins • Magnetic moments JLab • Single Λ-hypernuclei • 2000~ • Λ-wavefunction • Electro-production • Single Λ-hypernuclei FINUDA at DAΦNE + - J-PARC • Λ-wavefunction • e e • Stopped-K- reaction • 2009~ - • Single Λ-hypernuclei • Intense K beam • γ-ray spectroscopy • Single and double Λ-hypernuclei (2012~) • γ-ray spectroscopy for single Λ

Basic map from Saito, HYP06 Hypernuclei: from the cradle to the grave Production of Λ hypernuclei can occur by …

Strangeness exchange: (BNL, KEK, JPARC) (replace a u or d with an s quark)

A A K + Z Z +

Where the K- in-flight or stopped

Associated production: (BNL, KEK, GSI) (produces an ss pair) + A A + " + Z#$ Z + K

! Electroproduction: (JLAB, MAMI-C) e AZ e K + A Z 1 + + + ( )

A A Z(e,e’K) (Z-1)Λ elementary process e’ + p + K + e + e beam K p ~4 GeV Λ

Hypernucleus New Production kinematics - π- Λ High momentum transfer n(K , )  Low momentum transfer  hyperon has large probability of being bound.

 - π- Low momentum transfer Attenuation of (K , ) reaction in matter (resonance states). Interacion with outer shell neutrons replacing it with a Λ in the same shell.

n(π+,K+)Λ , p(γ,K+)Λ

 High momentum transfer  hyperon has large probability of escaping the nucleus.

 Longer π+ and K+ mean free path  interaction with interior nucleons, significant angular momentum transfer. Measurement of hypernuclear masses

M A # M = B # B A + M" # MN " Z A Z A Z " Z

Stopped K- reaction In-flight reactions - π - π π+ + ! (K stopped, -) (K in-flight, -) ( ,K )

A A Kin flight + Z Z + A A Kstopped + Z Z + + A A + " + Z#$ Z + K

2 2 2 r r 2 M A E M M A p M A E E M A p p Z = ( # $ K $ Z ) $ # Z = ( # $ K $ Z ) $ ( # $ K ) " ! "

Need only π- outgoing momentum Need incident & outgoing momenta  One Spectrometer  Two Spectrometers ! ! Example: spectrum for a (π+,K+) on a heavy target

+ 139 139 + " + La# $ La + K

 Energy resolution: 2.5 MeV !  Clear shell structure

 Obtained with a typical magnetic spectrometer for the detection of K+

T. Hasegawa et al., Phys. Rev. C 53, 1210 (1996) The FINUDA experiment @ DAΦNE(Frascati)

DAΦNE: Double Annular e+e- Φ-factory for Nice Experiments

e+e- collider dedicated to the production of Φ resonance

FINUDA: FIsica NUcleare at DAΦNE

produce hypernuclei by stopping negative originating from Φ decay in nuclear target

e+ + e" # $ # K + + K " " A A " Kstopped + Z#$ Z + %

!

! 12 FINUDA results on ΛC Very good agreement between FINUDA results & E368 @ KEK ones

FINUDA @ DAΦNE E369 @ KEK + + (Kstopped , ) (" ,K )

!

M. Agnello et al., Phys. Lett. B 622, 35 (2005) H. Hotchi et al., Phys. Rev. C 64, 044302 (2001) The (e,e’K+) reaction Relatively new (JLAB, MAMI-C). Excellent energy resolution of aaenergy spectrum. Although the cross section is 10-2 aasmaller than that of (π+,K+) this is aacompensated by larger beam aaintensity.

The experimental geometry requires two spectrometers to detect:

 the scattered which aadefines the virtual  the kaons Hypernuclear spectrum from the (e,e’K+) reaction

12 + 12 C(e,e’K ) ΛB spectrum Energy left inside the nucleus (related to the excitation energy)

E x = E e E e' E K +

Incoming electron Outgoing kaon

Outgoing electron

V. Rodrigues, PhD Thesis, University of Houston (2006) In summary …

The Λ single particle Hypernuclear binding states energies

Hypernuclear binding energies show saturation as ordinary nuclei Production of Σ hypernuclei Λ Production mechanisms similar to the ones considered± for hypernuclei like, e.g., exchange (K-,π )

4 ΣHe Σ hypernuclear states in E ~ 4 MeV, Γ ~ 7 MeV p-shell hypernuclei

T. Nagae et al., Phys. Rev. Lett. 80, 1605 (1998) S. Bart et al., Phys. Rev. Lett. 83, 5238 (19989) What do we know about double Λ hypernuclei ?

Not so much ….

Nagara event

A A A#1 B"" ("" Z) = B" ("" Z) + B" ( " Z)

A A A$1 "B## (## Z) = B# (## Z) $ B# ( # Z)

!

! The production of double Λ hypernuclei

Ξ- conversion in two Λ’s:

"# + p $ % + % + 28.5 MeV

Ξ- production:

!  (K-,K+) reaction (BNL, KEK)

K " + p # $" + K +

production (PANDA@FAIR)

$ + ! p + p " # + #

! Hypernuclear γ-ray spectroscopy

Produced hypernuclei can be aain an . Energy released by emission aaof neutrons or protons or γ- aarays when hyperon moves to aalower states.

Excellent resolution with Ge (NaI) aadetectors. Λ depth potential in nucleus ~ 30 MeV aa observation of γ-rays limited to low aaexcitation region. γ-ray transition measures only energy Hyperball aadifference between two states. Hypernuclear fine structure and the spin-dependent ΛN interaction 16 γ-ray spectrum of ΛO

Observed twin peaks aademonstrate hypernuclear 16 - aafine structure for ΛO (1  aa1-,0-) transitions.

Small spacing in twin peaks aacaused by spin-dependent aaΛN interaction.

Recent analysis revealed aaanother transition at 6758 keV 16 - - aacorresponding to ΛO (2 0 ).

M. Ukai et al., Phys. Rev. C 77, 05315 (2008) The Weak Decay of Λ hypernuclei Decay observables

free 9 %1 " ~ "# = 3.8 $10 s

Hypernuclear lifetimes-Decay Rates BNL 91 ! KEK 95, 98 Jülich 93,97, 98

" = "M + "NM + "2N

= "# 0 + "# $ + "n + "p + "np !

"N # NNN

pN ~ 340 MeV ! " # N$ pN ~ 100 MeV "N # NN

pN ~ 420! MeV W. M. Alberico et al., Phys. Rev. C 61, 044314 (2000) (well reproduced by theoretical models) ! ! Building the Hyperon-Nucleon Interaction The Hyperon-Nucleon interaction …

Study of the role of strangeness in low and medium energy nuclear physics.

Test of SU(3)flavor symmetry. Input for Hypernuclear Physics & (Neutron Stars).

But due to:  difficulties of preparation of aahyperon beams.  no hyperon targets available. 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) YN meson-exchange models

Strategy: start from a NN model & impose SU(3)flavor constraints

σ δ 1 scalar: , "s = L = g " # # $ M M ( B B ) M " = i# 5 pseudocalar: π,K,η ps ! " = # µ , " = $ µ% vector: ρ,K,ω, φ v T ! ! ! " " ' (1) (1) PM ' (2) (2) p1 p2 VM p1 p2 = u (p1)gM #M u(p1) u (p2 )gM #M u(p2 ) " 2 2 (p1 $ p1) $ mM mM (1)Γ (1) (2)Γ (2)  gM M gM M r r r r + r r % ! V(r) = V0 (r) + VS (r)(" 1 #" 2 ) + VT (r)S12 + Vls(r) L $ S + Vals(r) L $ S ( ) ( )

! The Nijmegen & Jülich models Nijmegen Jülich (Nagels, Rijken, de Swart, Maessen) (Holzenkamp, Reube, Holinde, Speth, Haidenbauer, Meissner, Melnitchouck) Based on Nijmegen NN potential. Based on Bonn NN potential. Momemtum & Configuration Momemtum Space & Full Space. energy-dependence & non- locality structure.

exchange of nonets of pseudo- higher-order processes involving scalar, vector and scalar. π- and ρ-exchange (correlated 2π- exchange) besides single exchange. Strange vertices related by SU(3) Strange vertices related by SU(6) symmetry with NN vertices. symmetry with NN vertices. Gaussian Form Factors: Dipolar Form Factors:

2 k 2 2 2 " 2 $ " # m ' 2 F (k ) M M 2 2# M M = & 2 2 ) FM (k ) = e % "M # k (

! ! Scattering amplitudes

Scattering amplitudes describing the hyperon-nucleon scattering are obtained by solving the Lipmann-Schwinger equation

T = V + V 1/E T

1 T = V + V T E

! Chiral Effective Field Theory for YN

Strategy: start from a chiral effective lagrangian in a way similar to the NN case

Leading order (LO) Contact terms

1 L1 = Ci Ba Bb ("iB)b ("iB)a 2 L2 = Ci Ba ("iB)a Bb ("iB)b 3 L3 = Ci Ba ("iB)a Bb ("iB)b  V B1B2 "B3B4 = C B1B2 "B3B4 + C B1B2 "B3B4 #r $#r S T 1 2 ! One-pseudoscalar meson exchange ! D F L = iB " µ D B # M B B + B " µ" u ,B + B " µ" u ,B µ 0 2 5{ µ } 2 5 [ µ ]  r r r r $ 1 % k $ 2 % k V B1B2 "B3B4 f f ( )( ) = # B1B2M B3B4 M 2 2 k + mM Polinder, Haidenbauer & Meissner, NPA 779, 244 (2006)!

! Lippmann-Schwinger equation cut-off with the regularized

( p 4 + p' 4 ) # F(p, p',") = e "4

! Cutoff dependence: Λ=550-700 MeV

YN data rather well described Total cross YN sections Differential YN cross sections

Solid: NSC97f NPA 779, 224 (2006) Green band: EFT Dashed: Jülich04 Light hypernuclei properties

3 ( HΛ ) binding energy cutoff independent

Λ=550 Λ=600 Λ=650 Λ=700 Jülich04 NSC97f Expt. -2.35 -2.34 -2.34 -2.36 -2.27 -2.30 -2.354(50)

Deuteron B( 2H): -2.224 MeV

4 4 A=4 doublet: HΛ - HeΛ

4 HΛ Λ=550 Λ=600 Λ=650 Λ=700 Jülich04 NSC97f Expt. + Esep(0 ) 2.63 2.46 2.36 2.38 1.87 1.60 2.04

+ Esep(1 ) 1.85 1.51 1.23 1.04 2.34 0.54 1.00 Δ Esep 0.78 0.95 1.13 1.34 -0.48 0.99 1.04

CSB-0+ 0.01 0.02 0.02 0.03 -0.01 0.10 0.35 CSB-1+ -0.01 -0.01 -0.01 -0.01 -0.01 0.24

(All units are given in MeV) Low-momentum YN interaction

Idea: start from a realistic YN interaction & integrate out the high- momentum components in the same way as as been done for NN.  Lippmann-Schwinger Equation

$ Vlow k 2 2 1 T(k",k;E k ) = Vlowk (k",k) + P % dqq Vlowk (k",q) T(q,k;E k ) # 0 E k & H0 (q)  phase shift equivalent ! Conditions  energy independent dT" = 0; V = " " # $ :Vlowk = Vbare d" lowk  softer (no hard core) ! !  hermitian !R enormalization Group Flow Equation

2 d 2 Vlowk (k#,")T(",k;" ) Vlowk (k#,k) = $ d" % E k $ H0 (") B. -J. Schaefer et al., Phys. Rev. C 73, 011001 (2006) !

!

! 1 S0 (I=3/2)matrix elements and phase-shift for ΣNΣN Λ=500 MeV

1 S0 (I=1/2)matrix elements and phase-shift for ΛNΛN Λ=500 MeV

B. -J. Schaefer et al., Phys. Rev. C 73, 011001 (2006) Cut-off dependence

1 Λ Λ S0 (I=1/2)matrix elements for N N (NSC97f)

B. -J. Schaefer et al., Phys. Rev. C 73, 011001 (2006) Hypernuclear Matter & Neutron Star Properties Well known facts about Neutron Stars

Formed from the collapse remnant of a massive star after a Type II, Ib or Ic supernova.

~ 57 Baryonic number: Nb 10 (“giant nuclei”)

Mass: M ~ 1-2 M ± MPSR1913+16 = (1.4411 0.0035)M

Radius: R ~ 10-12 km

Density: ρ ~ 1015 g/cm3

ρ ~ -30 3 universe 10 g/cm ρ ~ 3 sun 1.4 g/cm ρ ~ 3 earth 5.5 g/cm Magnetic field: B ~ 108…16 G

Electric field: E ~ 1018V/cm

Temperature: T ~ 106…11 K

Shortest rotational period: PB1937+2 = 1.58 ms

Latest discovery: PSR in Terzan 5: PJ1748-244ad = 1.39 ms

Accretion rates: 10-10 to 10-8 M/year Let’s have a look into the neutron star interior

In a traditional and conservative picture the internal composition of a neutron star has been modelled by a uniform fluid of neutron rich nuclear matter in equilibrium with respect to weak interactions

# n " p + e + $ e # µp = µn " µ " + µ# # e e" p + e " n + $ e #

! ! But because of ….

ρ ~ ρ The value of the central density is high: c (4-8) 0 ρ -3 14 3 ( 0 = 0.17 fm = 2.8 x 10 g/cm )

The rapid increase of the nucleon chemical potential with density

More exotic degrees of freedom are expected in the neutron star interior, in particular hyperons. Hyperons are expected to appear in the core of ρ ~ ρ neutron stars at (2-3) 0

µ # = µn + µ # # µ$ " e e#

µ% = µn

!

n + n " n + # $ p + e " # + % e $ n + n " p + &$ $ $ n + e " & + % e $

! Hyperons in Neutron Stars

Since the pioneering work of Ambartsumyan & Saakyan (1960) …

Relativistic Mean Field Models: Glendenning, 1985; Knorren, Prakash & Ellis, 1995; Shaffner-Bielich & Mishustin, 1996

Non-realtivistic potential model: Balberg & Gal, 1997

Quark-meson coupling model: Pal et al., 1999

Brueckner-Hartree-Fock theory: Baldo, Burgio & Schulze, 2000; Engvik, Hjorth-Jensen, Polls, Ramos & Vidaña, 2000

Chiral Effective Lagrangians: Hanauske et al., 2000

Density dependent field models: Hofmann, Keil & Lenske, 2001 Effect of Hyperons in the EoS and Mass of Neutron Stars

Hyperons make the EoS softer  reduction of the mass β-stable Neutron Star Matter

The equilibrium composition of the neutron star material is determined by the requirement of:

Charge neutrality

Equilibrium with respect to weak interacting processes

b1 " b2 + l + # l b2 + l " b1 + # l

! Relativistic Mean Field approach of hyperonic matter

Using the Lagrangian density …

+ µ µ 1 r r µ . L = 1" B - i# µ$ % mB + g&B& % g'B# µ' % g(B# µ) * ( 0" B B , 2 / 1 1 1 + $ &$ µ& % m2& 2 % ' ' µ2 + m2' ' µ 2 ( µ & ) 4 µ2 2 ' µ 1 1 1 1 % (r * (r µ2 + m2 (r * (r µ % bm (g &)3 % c(g &)4 4 µ2 2 ( µ 3 N &N 4 &N µ + 1" 3 (i# µ$ % m3 )"3 3 " = $ " %$ " ; &r = $ &r %$ &r µ# µ # # µ µ# µ # # µ B = n, p,",#$,#0,#+,%$,%0; & = e$,µ$ !

! ! one arrives at the hyperonic EoS in the mean field approximation

1 3 1 4 1 2 2 1 2 2 1 2 2 " = bmN (g#N#) + c(g#N#) + m# # + m$$0 + m% %03 3 4 2 2 2 k k FB F) 2J +1 2 1 B k 2 m g k 2dk k 2 m2 k 2dk + ( 2 ' + ( B + #B#) + ( 2 ' + ) B 2& 0 ) & 0

! 1 3 1 4 1 2 2 1 2 2 1 2 2 p = " bmN (g#N#) " c(g#N#) " m# # + m$$0 + m% %03 3 4 2 2 2 k k FB 4 F) 4 1 2JB +1 k dk 1 1 k dk + 2 + 2 ( ' 2 2 ( ' 2 2 3 B 2& 0 3 & 0 k m k + (mB + g#B#) ) + )

! Coupling Constants

, , , The nucleon coupling constants gσN gωN gρN b and c are constrained ρ by the empirical values of density 0, energy per particle E/A, compression modulus K, symmetry energy asym and effective mass m* at nuclear saturation.

, The hyperon coupling constants gσY gωY and gρY are constrained by: the binding of Λ hyperon in nuclear matter, hypernuclear levels and neutron star masses.

Assuming that all hyperons in the octet have the same coupling, the hyperon couplings are expressed as a ratio to the above mentioned nucleon couplings

g"Y g#Y g$Y x" = ; x# = ; x$ = ; g"N g#N g$N

! Two astrophysical constraints to the hyperon couplings

Red shift of EXO0748-676, z ~ 0.35 Mass of Ter 5 I M ~ 1.68 M

Below dark: EoS compatible with red shift Above light: EoS compatible with mass

B. D. Lackey, M. Nayyar & B. J. Owen, PRC 73, 024021 (2006) Brueckner-Hartree-Fock approach of hyperonic matter

Bethe-Goldstone Equation

Q G " = V + V B5B6 G " ( )B B ;B B B1B2 ;B3B4 % B1B2 ;B5B6 ( )B B ;B B 1 2 3 4 " # E # E + i$ 5 6 3 4 B5B6 B5 B6

Single particle energy & single particle potential ! 2k 2 E (k) = M c 2 + h + Re U (k) Bi Bi 2M 2 [ Bi ] Bi

r r r r U (k) = k k G " = E + E k k Bi $ $ i j ( Bi B j ) i j B k#k ! j FB j

! Note that the Bethe-Goldstone equation

Q G = V + V G " # H0 + i$ is formally identical to the Lippman-Schwinger equation for the scattering of two particles in the vaccum. ! 1 T = V + V T " # K + i$

In fact the G-matrix can be considered as a generalization of the T-matrix to the medium, when one takes into account the presences o!f other particles. Medium effects are taken into account through …

Pauli blocking of the intermediate states

The Pauli operator Q prevents the scattering to any occupied state, limiting the phase space of the intermediate states.

Dressing of the intermediate particles

The modification of the single-particle spectrum due to the inclusion of the averaged potential U “felt” by a particle due to its interaction with the others must be taken into account in the propagator. Enery density & Pressure of β-stable Hyperonic Matter

k FBi d 3k $ 2k 2 1 1 ' " = 2 & M c 2 + h + Re[U N ]+ Re[UY ]) + * 3 Bi 2M 2 Bi 2 Bi Bi 0 (2#) %& Bi () k 1 F, k 2 m2 k 2dk ++ 2 * + , , # 0

#$ ! p = " %$ #"

! and Strangeness channels

S = 0 S = -1 S = -2 S = -3 S = -4

& "" # "" "" # $N "" # %%) ( + I = 0 (NN " NN) ( $N # "" $N # $N $N # %%+ "" # "" ( + ( ) ' %% # "" %% # $N %% # %%*

% "N # "N "N # $N( & "# $ "# "# $ %#) I = !1 /2 ' ! * ( ! + & $N # "N $N # $N) ' %# $ "# %# $ %#*

& N N N N ) " # " " # $% " !# %% I = 1 NN! " NN ( + ( ) ( $% # "N $% # $% $% # %%+ "" # "" ( + ( ) ' %% # "N %% # $% %% # %%*

I = !3 /2 ("N # "N) ("# $ "#) ! !

I = 2 ("" # "") ! !

! Neutron Star Matter Composition

RMFT BHF

Free hyperons

NSC89 YN

N.K. Glendenning, ApJ 293, 470 (1985) M. Baldo et.al., Phys. Rev. C 61, 055801 (2000) Neutron Star Matter EoS (I)

RMFT BHF

M. Baldo et.al., Phys. Rev. C 61, 055801 (2000)

No YY interaction ! Neutron Star Matter EoS (II)

Additional softening from YY interaction

BHF

I. V. et.al., Phys. Rev. C 62, 035801 (2000) Structure equations for Neutron Stars: TOV Equations

Since neutron stars have masses M ~ 1-2 M, and radii R ~ 10-20 km, the value of the gravitational potential on the neutron star surface is of the order 1 GM ~ 1 c 2R

with escape velocities of the order of c/2. Therefore, general relativistic effects become very important and thus the structure equations read ! 1 dp m(r)#(r)$ p(r) '$ 4*r3 p(r)m(r)'$ Gm(r)'" = "G &1 + )& 1 + )& 1 " ) dr r2 % c 2#(r)(% c 2 (% c 2r ( dm = 4*r2#(r) dr

m(r = 0) = 0 with the boundary conditions p(r = R) = psurf !

! Neutron Star Structure

RMFT BHF

hyperons

N.K. Glendenning, ApJ 293, 470 (1985) H.-J- Schulze, I.V., A. Polls & A. Ramos Phys. Rev. C 73, 08801 (2006) Implications for Neutron Star Structure

The presence of hyperons reduces the maximum mass of Neutron Stars by an amount ΔMmax~ (0.5-0.8)M

Microscopic EoS “very soft EoS” not compatible with measured masses of NS

Need for extra pressure at high densities Two-body forces: Improved YN and YY Three-body forces: NNY, NYY and YYY