Achievements and Perspectives in Low-Energy QCD
with strangeness
ECT*, Trento (Italy), 27 – 31 October 2014
Ignazio Bombaci
Dipartimento di Fisica “E. Fermi”, Università di Pisa
INFN Sezione di Pisa
Role of strangeness for the physics of Neutron Stars
Strangeness in Neutron Stars
Confined within hadrons (hyperons, strange mesons)
Deconfined (Strange Quark Matter)
“Neutron Stars”
Nucleon Stars
Hyperon Stars
Hybrid Stars
Strange Stars
I. Bombaci, A. Drago, INFN Notizie, n. 13, 15 (2003)
Relativistic equations for stellar structure
Static and sphericaly symmetric self-gravitating mass distribution
 
2 ( r ) 2 2
2 ( r ) 2
2
2
2
2
2
ds  g  dx dx  e
c dt  e
dr  r ( d  sin  d )
 = ( r), = ( r) metric functions
e
(r)

2G m ( r ) 
 1 

c 2r


1 / 2
for the present case the Einstein’s field equations take the form called the
Tolman – Oppenheimer – Volkov equations (TOV)
dP
 G
m(r)  (r)
dr
dm
r
2


1  P(r) 


2
 c (r) 
2
 4 r  ( r )
dr
d
dr

1
 (r )c2
dP 
1  P(r)
dr 
  ( r )c2




1

 1  4


3
r P(r)
m(r )
c
2




 2 Gm ( r ) 
1 

2


c
r 
One needs the
equation of state (EOS) of
dense matter, P = P(ρ),
up to very high densities
1
The Oppenheimer-Volkoff maximum mass
There is a maximum value for the gravitational mass of a Neutron Star that
a given EOS can support. This mass is called the Oppenheimer-Volkoff mass
Pressure
Mmax = (1.4 – 2.5) M
M
“stiff” EOS
“soft”
“stiff” EOS
“soft”
density
The OV maximum mass represent the key physical quantity to separate (and distinguish)
Neutrons Stars from Black Holes.
Mmax(EOS)  all measured neutron star masses
R
Measured Neutron Star masses in Relativistic binary systems
Measuring post-Keplerian parameters:
* very accurate NS mass measurements
* model independent measuremets within GR
PSR B1913+16 NS (radio PSR) + NS (“silent”) (Hulse and Taylor 1974)
PPSR = 59 ms, Pb= 7 h 45 min
  4.220 / yr
Mp = 1.4408 ± 0.0003 M
(Mercury:   43 arcsec/ 100 yr )
Mc = 1.3873 ± 0.0003 M
Orbital period decay in agreement with GR predictions over about 40 yr
→ indirect evidence for gravitational waves emission
PSR J0737-3039 NS(PSR) + NS(PSR)
P1 = 22.7 ms, P2 = 2.77 s
Pb= 2 h 24 min
M1 = 1.34 M
(Burgay, et al 2003)
  16.88o / yr
M2 = 1.25 M
Two “heavy” Neutron Stars
MNS = 1.97 ± 0.04 M
PSR J1614–2230
NS – WD
binary system (He WD)
MWD = 0.5 M (companion mass)
Pb = 8.69 hr
(orbital period)
i = 89.17  0.02
P = 3.15 ms
(PSR spin period)
(inclination angle)
P. Demorest et al., Nature 467 (2010) 1081
PSR J0348+0432
NS – WD
MNS = 2.01 ± 0.04 M
binary system
MWD = 0.172  0.003 M
(companion mass)
Pb = 2.46 hr (orbital period)
i = 40.2  0.6
P = 39.12 ms (PSR spin period)
(inclination angle)
Antoniadis et al., Science 340 (2013) 448
Measured Neutron Star Masses
Mmax  Mmeasured
Mmax  2 M
PSR J0737-3039
PSR J0737-3039 comp
PSR J1614-2230
PSR J0348+0432
Very stringent
constraint on the
EOS
Neutron star physics in a nutshell
1) Gravity compresses matter at very high density
2) Pauli priciple
Stellar constituents are different species of identical fermions (n, p,….,e-, μ-)
antisymmetric wave function for particle exchange
Pauli principle
Chemical potentials
 n ,  p , e
rapidly increasing functions of density
3) Weak interactions change the isospin and strangeness content of
dense matter to minimize energy
Cold catalyzed matter
(Harrison, Wakano, Wheeler, 1958)
The ground state (minimum energy per baryon) of a system of hadrons and
leptons with respect to their mutual strong and weak interactions at a
given total baryon density n and temperature T = 0.
swiss cheese lasagne spaghetti meet-balls
Nucleon Stars
outer crust
nuclei, e-
-stable nuclear matter
drip = 4.3 1011 g/cm3
p  e  n   e
inner crust
nuclei, n, e-
n  p  e  e
~1.5 1014 g/cm3

Nuclear matter
core
 e  m
n, p, e- , -
e      e  
 Equilibrium with
n   p  e
respect to the weak
interaction processes
  e
 Charge neutrality
neutrino-free matter
    0
np  ne  n
To be solved for any given value of the total baryon number density nB
Proton fraction in -stable nuclear matter and role of
the nuclear symmetry energy
ˆ   n   p  
 ( E / A)
 ( E / A)
2
x

n
 = (nn – np )/n = 1 – 2x
n = nn + np
n
x = np /n
Energy per nucleon for asymmetric nuclear matter
Symmetry energy
1  2 ( E / A)
E sym ( n ) 
2  2
 0
(MeV)
E ( n,  ) E ( n, 0 )

 E sym ( n )  2  S 4 ( n )  4  ....
A
A
proton fraction
E ( n ,  ) E ( n, 0 )

A
A
The “parabolic approximation” (*)
E ( n,  )
E ( n, 0 )

 E sym ( n )  2
A
A
β2
(*) Bombaci, Lombardo, Phys. Rev: C44 (1991)
Proton fraction in -stable nuclear matter and role of
the nuclear symmetry energy
In the “parabolic approximation”:
E sym ( n ) 
ˆ  4 Esym(n) 1 2x
E ( n,   1 )
E ( n,   0 )

A
A
 = 0 symm nucl matter
 = 1 pure neutron matter
Chemical equil.+charge neutrality
(no muons)
if
3 2 ( c ) 3 n x ( n)  4 E sym ( n) 1  2 x ( n)   0
3
x<<1/2
Symmetry en.
1 1  4 E sym ( n ) 


x eq ( n ) 
2


c
3 n 

proton fraction
3
The composition of -stable nuclear
matter is strongly dependent on the
nuclear symmetry energy.
M. Baldo, I. Bombaci, G. Burgio, Astr. & Astrophys. 328 (1997)
Microscopic approach to nuclear matter EOS
input
Two-body nuclear interactions: VNN
“realistic” interactions: e.g. Argonne, Bonn, Nijmegen interactions.
Parameters fitted to NN scatering data with χ2/datum ~1
Three-body nuclear interactions: VNNN
semi-phenomenological. Parameters fitted to
• binding energy of A = 3, 4 nuclei
or
• empirical saturation point of symmetric nuclear matter: n0= 0.16 fm-3 , E/A = -16 MeV
AV18
AV18/UIX
Exp.
Nuclear Matter at n = 0.16 fm-3
Epot(2BF)/A ~ -40 MeV
B(3H)
7.624
8.479
8.482
B(3He)
6.925
7.750
7.718
B(4He)
24.21
28.46
Epot(3BF)/A ~ - 1 MeV
28.30
Values in MeV
A. Kievsky, S. Rosati, M.Viviani, L.E. Marcucci, L. Girlanda, Jour. Phys.G 35 (2008) 063101
A. Kievsky, M.Viviani, L. Girlanda, L.E. Marcucci, Phys. Rev. C 81 (2010) 044003
Z.H. Li, U. Lombardo, H.-J. Schulze, W. Zuo, Phys. Rev. C 77 (2008) 034316
VNN + VNNN
Quantum
Many-Body Theory
EOS
β-stable matter
TOV
Neutron Star
properties
observational data
(measured NS properties)
e.g.
Brueckner-Hartree-Fock
VNN
GNN
Microscopic EOS for nuclear matter:
Brueckner-Bethe-Goldstone theory
G  (  )  V  V
k a k b Q  k a k b
  e
ka kb

G  (  )
( k a )  e  ( k b )
2k 2
e ( k ) 
 U ( k )
2m
U ( k )   kk' | G  ( e

k'
( k )  e  ( k ' ))

| kk'
A
Energy per baryon in the Brueckner-Hartree-Fock (BHF) approximation
E 1
2k 2
1

~
E ( nn , n p )    
 U τ (k) 
A A τ k 2M
2 τ k

1
1   n
2
1
n p  1    n
2
nn 
Mass-Radius relation for Nucleon Stars
Maximum mass configuration
for Nucleon Stars
PSR J1614-2230
EOS
MG/M
R(km)
nc / n0
BBB1
1.79
9.66
8.53
BBB2
1.92
9.49
8.45
WFF
2.13
9.40
7.81
APR
2.20
10.0
7.25
BPAL32
1.95
10.54
7.58
KS
2.24
10.79
6.30
WFF: Wiringa-Ficks-Fabrocini, 1988.
BPAL: Bombaci, 1995.
BBB: Baldo-Bombaci-Burgio, 1997.
APR: Akmal-Pandharipande-Ravenhall, 1988.
KS: Krastev-Sammarruca, 2006
Mmax = (1.8
2.3) M
Z.H. Li, H.-J. Schulze,
PSR J0348+0432
V18: Argonne V18 + mTBF
BOB: Bonn B + mTBF
N93: Nijmegen 93 + mTBF
UIX: Argonne V18 + Urbana IX
Message taken from Nucleon
Stars
(i.e. Neutron Stars with a pure nuclear matter core)
NN interactions essential to have “large” stellar mass
For a free neutron gas Mmax = 0.71 M
(Oppenheimer and Volkoff, 1939)
NNN interactions essential
(i) to reproduce the correct empirical saturation point of
nuclear matter
(ii) to reproduce measured neutron star masses, i.e. to have
Mmax > 2 M
models of Nucleon
Stars
(i.e. Neutron Stars with a pure nuclear matter core)
are able to explain
measured Neutron Star masses
as those of
PSR J1614-2230 and PSR J0348+0432
MNS ≈ 2 M
Happy?
Not the end of the story!
Hyperon Stars
Why is it very likely to have hyperons in the core of a Neutron Star?
Pauli principle. Neutrons (protons) are identical Fermions, thus
their chemical potentials (Fermi energies) increase very rapidly
as a function of density.
The central density of a Neutron Star is “high”: nc  (6 – 9) n0
(n0 = 0.16 fm-3)
above a threshold density, ncr  (2 – 3) n0 , weak interactions
in dense matter can produce strange baryons (hyperons)
n + e-   +  e
p + e-   +  e
etc.
A.V. Ambarsumyan, G.S. Saakyan, (1960)
G.S. Saakyan, Y.L. Vartanian (1963)
V.R. Pandharipande (1971)
In Greek mythology Hyperion (Ὑπερίων)
was one of the twelve Titan son of Gaia and
Uranus
Hyperons appear in the stellar core above
a threshold density cr  (2 – 3) 0
n + e-   + e
p + e-   +  e
etc.
I. Vidaña, Ph.D. Thesis (2001)
 p  n  e  

n      
0
0
n  e    


  e
n p  n   ne  n  n   n 
Av18+TNF+NSC97e
UΣ-(k=0, n0) = – 25 MeV
m   1115.7 Me V
m    1197.5 Me V
n  
 n  e  

Av18+TNF+ESC08b
D. Logoteta, I. Bombaci (2014)
TNF: Z H.. Li, U. Lombardo, H.-J. Schulze. W. Zuo, Phys. Rev. C 77 (2008)
Microscopic approach to hyperonic matter EOS
input
2BF: nucleon-nucleon (NN), nucleon-hyperon (NY), hyperon-hyperon (YY)
e.g. Nijmegen, Julich models
3BF: NNN, NNY, NYY, YYY
Hyperonic sector: experimental data
1.
YN scattering (very few data)
2.
Hypernuclei
Hypernuclear experiments
FINUDA (LNF-INFN),
PANDA and HypHI (FAIR/GSI),
Jeff. Lab, J-PARC
C. Curceanu, talk at INFN 2014, Padova 2014
Microscopic EOS for hyperonic matter: extended Brueckner theory
G( )
B1B2 B3 B4


V
B1B2 B3 B4
QB5 B6
 V
B5 B6
B1B2 B5 B6
G( )
  eB  eB
5
B5 B6 B3 B4
6
2 2

k
eB i (k )  M Bi c 2 
 U Bi (k )
2M Bi
U Bi (k )  


 kk ' | GB B B B (  eBi  eB j ) | kk ' 
B j k 'k F B
i j
i
j
j
V is the baryon -baryon interaction for the baryon octet
-
-
( n, p, ,  , 0, +,  ,  0 )

Energy per baryon in the BHF approximation
E /N B  2  
Bi
k F [ Bi ]
0
d 3k
(2 )3


 2k 2 1 N
1 Y


2
M
c


U
(
k
)

U
(
k
)
 Bi

Bi
Bi
2
M
2
2


Bi


Baldo, Burgio, Schulze, Phys.Rev. C61 (2000) 055801;
Vidaña, Polls, Ramos, Engvik, Hjorth-Jensen, Phys.Rev. C62 (2000) 035801;
Vidaña, Bombaci, Polls, Ramos, Astron. Astrophys. 399, (2003) 687.
The Equation of State of Hyperonic Matter
Av18+TNF+ESC08b
D. Logoteta, I. Bombaci (2014)
Av18+TNF+ESC08b
Av18+TNF
Composition of hyperonic beta-stable matter
Baryon number density b [fm-3]
Av18+TNF+NSC97e
Hyperonic Star
MB = 1.34 M
Radial coordinate [km ]
I. Vidaña, I. Bombaci,
A. Polls, A. Ramos,
Astron. and Astrophys.
399 (2003) 687
Composition of hyperonic beta-stable matter
Baryon number density b [fm-3]
Av18+TNF+NSC97e
Hyperonic Star
MB = 1.34 M
Hyperonic core
NM
shell
Radial coordinate [km ]
I. Vidaña, I. Bombaci,
A. Polls, A. Ramos,
Astron. and Astrophys.
399 (2003) 687
Composition of hyperonic beta-stable matter
Av18+TNF+ESC08b
D. Logoteta, I. Bombaci (2014)
The stellar mass-radius relation
Z.H. Li, H.-J. Schulze,
interaction: NN + NY + YY + NNN
PSR J0348+0432
PSR B1913+16
NY,YY: Nijmegen NSC89 potential (Maessen et al, Phys. Rev. C 40 (1989)
The stellar mass-radius relation
D. Logoteta, I. Bombaci (2014)
Av18+TNF+ESC08b
see also: H.-J. Schulze, T. Rijken, Phys. Rev. C 84 (2011) 035801
Hyperons in Neutron Stars: implications for the stellar structure
The presence of hyperons reduces the maximum mass of neutron stars:
Mmax  (0.5 – 1.2) M
Therefore, to neglect hyperons always leads to an overstimate of the
maximum mass of neutron stars
Microscopic EOS for
hyperonic matter: ”very soft”
non compatible with measured
NS masses
Need for
extra pressure
at high density
Improved NY, YY
two-body interaction
Three-body forces*:
NNY, NYY, YYY
(*) A preliminary study: I. Vidana, D. Logoteta, C. Providencia, A. Polls, I. Bombaci, EPL 94 (2011) 11002
Hyperons in Neutron Stars: implications for the stellar structure
The presence of hyperons reduces the maximum mass of neutron stars:
Mmax  (0.5 – 1.2) M
Therefore, to neglect hyperons always leads to an overstimate of the
maximum mass of neutron stars
Microscopic EOS for
hyperonic matter: ”very soft”
non compatible with measured
NS masses
Need for
extra pressure
at high density
Improved NY, YY
two-body interaction
Three-body forces*:
NNY, NYY, YYY
More experimental
data from
hypernuclear physics
(*) A preliminary study: I. Vidana, D. Logoteta, C. Providencia, A. Polls, I. Bombaci, EPL 94 (2011) 11002
Hyperons in Neutron Stars: implications for the stellar structure
The presence of hyperons reduces the maximum mass of neutron stars:
Mmax  (0.5 – 1.2) M
Therefore, to neglect hyperons always leads to an overstimate of the
maximum mass of neutron stars
Microscopic EOS for
hyperonic matter: ”very soft”
non compatible with measured
NS masses
Need for
extra pressure
at high density
Theory: baryonic forces from
SU(3) chiral effective theory
(Petschauer’s talk, yesterday)
Improved NY, YY
two-body interaction
Three-body forces*:
NNY, NYY, YYY
More experimental
data from
hypernuclear physics
(*) A preliminary study: I. Vidana, D. Logoteta, C. Providencia, A. Polls, I. Bombaci, EPL 94 (2011) 11002
Estimation of the effect of hyperonic TBF
on the maximum mass of neutron stars
BHF calculations: NN (Av18) + NY (NSC89)
TBF: phenomenological density dependent contact terms
 3  a NN n N2  b NN n N
NN
1
n N  n 
 a N n N n   b N  n N n 
n N  n
N
n N  n
 a N n N n   b N  n N n 
n N  n
N
N
N
Energy density form inspired by S. Balberg, A. Gal, Nucl Phys. A 625, (1977) 435
I.Vidaña, D. Logoteta, C. Providencia, A. Polls, I. Bombaci, EPL 94 (2011) 11002
we assume:
a N  a N
 N   N
bN  bN
a NY b NY

x
a NN b NN
empirical saturation point of symmetric NM
n0  0.16 fm -3
~
E 0  16 MeV
a NN , bNN ,  NN
K 0  210 285 MeV
Binding energy of Λ in NM
B   28 MeV  U  ( k  0 )  a NY n0  b NY n0
NY
 NY
effect of hyperonic TBF on the maximum mass of neutron stars
I.Vidaña, D. Logoteta, C. Providencia, A. Polls, I. Bombaci, EPL 94 (2011) 11002
Neutron Stars in the QCD phase diagram
Lattice QCD
at μb=0 and finite T
► The transition to Quark
Gluon Plasma is a crossover
Aoki et ,al., Nature, 443 (2006)
675
► Deconfinement transition
. temperature Tc
HotQCD Collaboration
Tc= 154 ± 9 MeV
Bazarov et al., Phys.Rev. D85
(2012) 054503
Wuppertal-Budapest Collab.
Tc= 147 ± 5 MeV
Borsanyi et al., J.H.E.P. 09
(2010) 073
Cristalline Color
superconductor
Neutron Stars:
high μb and low T
Quark deconfinement transition
expected of the first order
Z. Fodor, S.D. Katz, Prog. Theor Suppl. 153
(2004) 86
Lattice QCD calculations are
presently not possible
1st order phase transitions are triggered by the nucleation of a
critical size drop of the new (stable) phase in a metastable mother phase
H

Q
Virtual drops of the stable phase are
created by small localized fluctuations
in the state variables of the
metastable phase
P0
H = Q  0
pressure
H 
TH = TQ  T
P(H) = P(Q)  P(0)  P0
Q 
 H  PH  sH T
nb, H
 Q  PQ  sQT
nb,Q
1st order phase transitions are triggered by the nucleation of a
critical size drop of the new (stable) phase in a metastable mother phase
H

Q
Virtual drops of the stable phase are
created by small localized fluctuations
in the state variables of the
metastable phase
P0
pressure
Astrophysical consequences of the nucleation process of quark matter (QM)
in the core of massive pure hadronic compact stars (“Hadronic Stars”, HS).
Berezhiani, Bombaci, Drago, Frontera, Lavagno, Astrophys. Jour. 586 (2003) 1250
I. Bombaci, I. Parenti, I. Vidaña, Astrophys. Jour. 614 (2004) 314
I. Bombaci, G. Lugones, I. Vidaña, Astron. &Astrophys. 462 (2007) 1017
Metastability of Hadronic Stars
M
Hadronic Stars
(no quark matter)
Quark Stars
Mmax(HS)
(Oppenheimer-Volkoff mass)
Mcr critical mass
Metastable
hadronic stars
Mthr( = )
Hadronic Stars above
a threshold value of
their gravitational
mass are metastable
to the conversion to
Quark Stars (QS)
(hybrid stars or
strange stars)
stable HSs
R
Berezhiani, Bombaci, Drago, Frontera, Lavagno, Astrophys. Jour. 586 (2003) 1250
I. Bombaci, I. Parenti, I. Vidaña, Astrophys. Jour. 614 (2004) 314
I. Bombaci, G. Lugones, I. Vidaña, Astron. &Astrophys. 462 (2007) 1017
Metastability of Hadronic Stars
M
Hadronic Stars
(no quark matter)
Quark Stars
Mcr , critical mass
Mmax(HS)
of hadronic stars.
.
(Oppenheimer-Volkoff mass)
. Two branches of
compact stars
Mcr critical mass
. stellar conversion
Metastable
HSQS
hadronic stars
Mthr( = )
Econv  1053 erg
(possible
energy source for some GRBs)
stable HSs
extension of the concept of
R limiting mass of compact stars
Berezhiani, Bombaci, Drago, Frontera, Lavagno, Astrophys.
Jour. 586
(2003)
1250 one
with respect
to the
classical
I. Bombaci, I. Parenti, I. Vidaña, Astrophys. Jour. 614 (2004) 314 given by
I. Bombaci, G. Lugones, I. Vidaña, Astron. &Astrophys. 462
(2007) 1017and Volkoff
Oppenheimer
Quantum nucleation theory
I.M. Lifshitz and Y. Kagan, 1972; K. Iida and K. Sato, 1998
Quantum fluctuation of a virtual drop of QM in HM
QM drop
R
U(R) = (4/3) R3 nQ* (Q* - H ) + 4 R2

av R3
+ as R2
Hadronic
Matter
I. Bombaci, I. Parenti, I. Vidaña, Astrophys. Jour. 614 (2004) 314
Hadronic Stars: nucleons + hyperons
Bombaci, Parenti, Vidaña, Astrophys. Jour. 614 (2004) 314
D. Logoteta, I. B. (2014)
SQM EOS: Alford et al. Astrophys. J. 629 (2005); Fraga et al., Phys. Rev. D 63 (2001)
Conclusions
The presence of hyperons reduces the maximum mass of neutron stars,
thus, to neglect hyperons always leads to an overstimate of the maximum
mass of neutron stars.
“Hyperon puzzle” in Neutron star physics
Mmax < 2 M
quest for extra pressure at high densities
(i)
► strong short-range repulsion in NY, YY interactions
► repulsive NNY, NYY, YYY 3-baryon interactions
(ii) or, the transition to Strange Quark Matter produce a stiffening
of the EOS due to e.g. non-perturbative quark interactions
NS → Quark Stars (hybrid or strange stars)