Nucleon correlations and
neutron star physics
T.Takatsuka (Iwate) and R. Tamagaki (Kyoto)
KEK Ｗｏｒｋｓｈｏｐ on
「Short-range correlations and tensor structure
at J-PARC」 2009. 9.25
Contents
(1) Equation of state of neutron star matter
(2) Relevance of s.r.c. and tensor-coupling to the
neutron (3P2+3F2) superfluidity
(3) Unique structure caused by the OPE- tensor
correlation : ALS structure~π０ condensation
(Ｃ１) Effects of proton high-momentum
components on NS cooling due to the nucleon
direct URCA process
(Ｃ２) Origin of universal s.r. repulsion in the baryon
system from the confinement
(1) Equation of State(EOS) of
neutron star(NS) matter
At first, we see the features of the short-range
correlations (s.r.c.) in EOS of neutron matter (the
main component of NSs), from the viewpoint that
nuclear force has strong state-dependence.
・State dependence of T=1 int. and nn correlation
・Density dependence of E/N (EOS) and partial-wave
contribution to E/N
・Effects of s.r.c. on M (mass) and R( radius) of NSs
References prior to 1993: Kunihiro, Muto, Takatsuka, Tamagaki and
Tatsumi, Prog. Theor. Phys. Supplment 112(1993).
Strong state- dependence of
nn correlations in neutron matter
• Each partial wave gives largely different
contribution to E/N, but the <G>/N (total
interaction energy) is close to <G(1S0)>/N.
• This comes from the cancellation in the
three 3P-wave sum and also between the
repulsive 3P-waves (in average) and
attractive 1D2-wave.
• The s.r.c. due to the repulsive core plays a
substantial role in the EOS of NS matter.
• We note the competition between
repulsive core and outside attraction in
1S0 and 3P2 .
(2) Relevance of s.r.c. and tensor-coupling
to nucleon superfluidity (SF)
•
At low density in the crust of NS, neutrons dripped from n-rich nuclei turn
into the 1S0-type SF. This SF appears in the region of ~ (10 3  1 / 2)  
•
The 1S0 gap equation is of the well-known BCS type:
(k )  
1


2
1
dk
'
k
'

k
'
|
V
(
S0 ) | k 

0
(k ' )
,
E (k ' )
E ( k )  {(  ( k )   ) 2   ( k ) 2 }1 / 2
•
Δ/E describes the pairing correlation near the Fermi surface and also
the s.r.c. far from the Fermi surface, which should be solved with full V.
•
At moderate density in the fluid core of NS in (~0.7→ several)
neutrons turn into the (3P2+3F2)-type SF .
•
At moderate density, protons as small component turn into the 1S0-type SF.
0 ,
Here focus on the (3P2+3F2) SF coupled
by the tensor force; Vcouple= (6 6 / 5)VT
although main attraction for pairing comes from the LS int.
(3) Unique structure caused by the OPEtensor interaction (ALS ~π0 condensation)
Variational calculations have shown the new phase
regarded as neutral pion condensation
A. Akmal, V.R. Pandaripande and D.G. Ravenhall,
Phys. Rev. 58C (1998), 1804.
Growth of the long-range
tensor correlation plays
a key role in the phase
transition from the low density
phase to high density phase.
In neutron matter, the transition
occurs at ρ=0.2fm -3= 1.25ρ0 .
・ The long range (OPE-dominated) part of
tensor correlation is taken into account
in the ALS structure, equivalent to the
π0 condensation.
・ In the ALS structure, the Fermi surface
becomes cylindrical ; the axis is along kc
(condensed momentum) and the side consists
of the two dimensional Fermi circle.
・ After taking a new model state, there still
remains tensor correlation of short range.
（Ｃ１）Ｅｆｆｅｃｔ ｏｆ proton ｈｉｇｈ-ｍｏｍｅｎｔｕｍ
components ｏｎ ＮＳ ｃｏｏｌｉｎｇ
ｄｕｅ ｔｏ ｎｕｃｌｅｏｎ ｄｉｒｅｃｔ ＵＲＣＡ
• Recent progress: The s.r.c. provides the high
momentum components of n & p well above the
Fermi surface. Especially for protons, the np-tensor
correlation plays an important role.
e.g., M. Alvioli, C. Ciofi degli Atti and H. Morita,.Phys. Rev. Letters, !00
(2008),162503
• Question arises as to at what extent such highmomentum components influence NS phenomena.
• Recently it has been suggested that high momentum
p components enhance the neutrino emissivity of
NSs due to the nucleon direct URCA process.
Neutrino emissivity by Direct URCA
• Nucleon direct URCA (NDU) is the most efficient process
as neutrino cooling of NS (among N,π-cond.,K-cond.,Y-mix.,q)
n→p＋e- +  e , p+e-→n+ e (μ possible for μe>mμ).
“Direct” : without by-stander nucleon, the momentum
conservation holds among three Fermi momenta of the
degenerate fermions (kn=kp+ke ) , within the allowance of
small neutrino’s energy ckν~kBT=(0.01-0.1MeV) .
• This becomes possible when proton mixing x=Z/A amounts
to >~10% , depending sensitively on symmetry energy,
at density higher than several  0 . (now still open)
problem)
• At moderate densities (1~3)  0 , because of x<~5%,
normally NDU is forbidden, when we take the sharp Fermi
surfaces slightly diffused due to finite temperature.
But, in such situation, NDU becomes
possible, if some high momentum
components above the Fermi
surfaces exist , e.g. for protons
due to the np tensor correlation.
This point has been noted recently,
e.g.L.Frankfurt, M. Sargsian
and M.Strikman, Int.J.Mod.Phys.,
Vol.23, no.20(2008),2991.
They give an estimate of enhancement factor
R, where Pnp is the probability for a proton
to have momentum k>kp, taking density
~nuclear density , Z/N=0.1 and Pnp=0.1.
At the internal temperature of NS as
kBT=(0.1-0.01) MeV, R becomes of
the order of (0.5~16).
This gives enormously large neutrino
emissivity, and provideｓ a new
problem in NS cooling.
.
 NDU  c(k T )  (k
6
B
p
 ke  k n ) R
R  0.16 Pnp (MeV / k BT )
3/ 2
For the direct URC A ( B1  B2  l  l plus its inverse),
neutrino emmisivity is given as
2
   2  E  ( E f  Ei )n( p1 )1  n( p2 )1  n( pl )
pi 1~ 4
  |  B2 , l , l | H  | B1  |2 ,
spins
Usually th e Fermi - Dirac distributi on function is used :
1
n( pi ) 
1  exp(( Ei  i ) / k BT )
• What is the problem?
• Usually a large Δp to
suppress large NDU
emissivity is used,
to avoid too cooled
NSs which cannot
be observed.
• For the nucleons
well above the Fermi
surface, suppression
of SF does not work.
The curves and marks taken
from S.Tsuruta,Proc. of
IAU 2003 Symp.
(C2) Origin of universal s.r. repulsion in the
baryon system from the confinement
R. Tamagaki, Prog. Theor. Phys. 119 (2008), 965 and
arXiv:0801.2289.
R. Tamagaki, Prog. Theor. Phys. Suppl. 174 (2008), 233.
Two motivations
(1) Necessity of universal repulsion of
3-body int. (3BI)
to avoid the dramatic softening in EOS of NS matter
due to the hyperon-mixing,
S.Nishizaki, Y.Yamamoto and T. Takatsuka, Prog. Theor. phys.
108(2002),703.
(2) String-junction structure of the baryon shown
by recent lattice QCD calculations
T. Takahashi and H. Suganuma, Phys. Rev. D70 (2004), 074506.
Regarding this universal nature as
originating from the color degrees of
freedom, we study the origin of the
universal 3BI repulsion from the viewpoint
of the confinement mechanism in QCD,
adopting the string-junction model (SJM).
M. Imachi, S. Otsuki and F. Toyoda, Prog. Theor.Phys. 54 (1975),
280; 55 (1976), 551; 57 (1979)17.
Mass of
neutron star (NS)
with Y- mixed core
versus central
density/  0 ,
with use of the
universal repulsion
of 3BI, derived
in the string-junction
model (SJM)
• This is an extension of the previous
approach to understand the origin of
repulsive core in baryon-baryon
interaction, based on the stringjunction model.
R. Tamagaki, Bulletin of the Institute for Chemical Research,
Kyoto Univ. 60, No.2 (1982),190.