BLED WORKSHOPS
IN PHYSICS
VOL. 13, NO. 1
p. 42
Proceedings of the Mini-Workshop
Hadronic Resonances
Bled, Slovenia, July 1 - 8, 2012
The implications of the two solar mass neutron star
for the strong interactions ⋆
Vikram Sonia, Pawel Haenselb and Mitja Rosinac,d
aNational Physical Laboratory, K.S. Krishnan Marg, New Delhi 110012, India
bNicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18,
PL-00-716 Warszawa, Poland
cFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, P.O. Box
2964, 1001 Ljubljana, Slovenia
dJ. Stefan Institute, 1000 Ljubljana, Slovenia
Abstract. The existence of a star with such a large mass means that the equation of state
is stiff enough to provide a high enough pressure up to a fairly large density , about four
times the nuclear density.
1 Introduction
Equations of state (EOS) that involve nonrelativistic constituents counteract grav-
itational infall of matter through a fermi pressure that is proportional to the den-
sity to the (5/3) power, unlike fermi pressures of relativistic constituents that go
as density to the (4/3) power. Clearly the nonrelativistic nucleons are favoured
over quarks for stiffer EOS’s that can lead to larger mass for the stars.
However, a pure nonrelativistic fermi gas of neutrons is not sufficient to give
large masses for neutron stars. Such a non interacting gas can give stars of max-
imum mass 0.7 solar mass - this a general relativistic effect coming from the Op-
penheimer – Volkoff equation where the pressure needs to be proportional to
density to a power greater than (5/3) . On the other hand, for white dwarfs fermi
pressure of a nonrelativistic electron gas is all that is needed to counteract gravity
and have stable stars. This enhanced pressure is provided by nuclear interactions
like the hard core.
It is known that stars with soft, relativistic quark matter cores surrounded by
a nonrelativistic n+p+e plasma in beta equilibrium can give maximum mass for
neutron stars ∼ 1.6 solar mass [1, 2].
It is also known that there are many nucleon based neutron stars models
that have neutron stars with maximum mass above 2 solar masses, eg. the APR
98 EOS of Akmal, Pandharipande and Ravenhall [3].
If we can show that matter in neutron stars is entirely composed of nucleon
degrees of freedom then we can have a simple resolution of this problem. Can we?
⋆ Talk delivered by V. Soni
The implications of the two solar mass neutron star . . . 43
2 The Maxwell construction between nuclear matter and quark
matter
A simple way to look at whether nucleons can dissolve into quark matter is to
plot EB, the energy per baryon in the ground state of both phases versus 1/nB,
where nB is the baryon density. The slope of the common tangent between the
two phases then gives the pressure and the intercept the common baryon chemi-
cal potential. For the quark matter equation of state see Fig.1.
Fig. 1. The Maxwell construction: Energy per baryon plotted against the reciprocal of the
baryon number density for APR98 equation of state (dashed line) and the 3-flavour pion-
condensed phase (PC) for three different values of mσ (solid lines). A common tangent
between the PC phase and the APR98 phase in this diagram gives the phase transition
between them. The slope of a tangent gives the negative of the pressure at that point, and
its intercept gives the chemical potential. As this figure indicates, the transition pressure
moves up with increasing mσ, and at mσ below ∼750 MeV a common tangent between
these two phases cannot be obtained. (From Fig. 2 of Soni and Bhattacharya [2] or Fig. 3 of
the preprint [4])
This is based on an effective chiral symmetric theory that is QCD coupled
to a chiral sigma model. The theory thus preserves the symmetries of QCD. In
this effective theory chiral symmetry is spontaneously broken and the degrees of
freedom are constituent quarks which couple to colour singlet, sigma and pion
fields as well as gluons. The nucleon in such a theory is a colour singlet quark
soliton with three valence quark bound states [5]. The quark meson couplings are
set by matching mass of the nucleon to its experimental value and the meson self
coupling which sets the tree level sigma particle mass is set from pi-pi scattering
to be of order 800 MeV. Such an effective theory has a range of validity up to
44 Vikram Soni, Pawel Haensel and Mitja Rosina
centre of mass energies ( or quark chemical potentials) of ∼ 800 MeV. For details
we refer the reader to ref. [2].
This is the simplest effective chiral symmetric theory for the strong inter-
actions at intermediate scale and we use this consistently to describe, both, the
composite nucleon of quark boundstates and quark matter. We expect it to be
valid till the intermediate scales quoted above. Of course inclusion of the higher
mesonic degrees of freedom like the rho and A1 would make for a more complete
description. We work at the mean field level the gluon interactions are subsumed
in the colour singlet sigma and pion fields they generate. We could further add
perturbative gluon mediated corrections but they do not make an appreciable
difference.
As can be seen from Fig.1, it is the tree level value of the sigma mass that
determines the intersection of the two phases; the higher the mass the higher the
density at which the transition to quark matter will take place. In [2] it was found
that above,mσ ∼ 850MeV, stars with quark matter cores become unstable as their
mass goes up beyond the allowed maximummass. So, if we want purely nuclear
stars we should, in this model, work at,mσ ≥ 850 MeV [2].
From Fig. 1, for the tree level value of the sigma mass ∼850MeV, the common
tangent in the two phases starts at 1/nB ∼ 1.75 fm3 ( nB ∼ 0, 57/fm3) in the
nuclear phase of APR [A18 + dv +UIX] and ends up at 1/nB ∼ 1.25 fm3 (nB ∼
0.8/ fm3) in the quark matter phase.
At the above densities between the two phases there is a mixed phase at the
pressure given by the slope of the common tangent and the at a baryon chemical
potential given by the intercept of the common tangent on the vertical axis. If we
are to stay in the nuclear phase the best way is to look at the central density of
the nuclear (APR) stars and if it so happens that they are at lower density than
that at which the above phase transition begins the we can safely say that the star
remains in the nuclear phase.
Going Back to the APR phase in in fig 11 of APR [3] we find that for the APR
[A18 + dv +UIX] the central density of a star of 1.8 solar mass is nB ∼ 0.62 /fm3,
very close to the initial density at which the phase transition begins.
The reason we are taking a static star mass of 1.8 solar mass from APR [3] is
that for PSR-1614, the star is rotating fast at a period of 3 millisec and we expect a
∼ 15% diminution of the central density from the rotation [6]. Equivalently, since
the above paper reports results for static stars, the central density of a fast rotating
1.97 solar mass star ∼ the central density of a static 1.8 solar mass star.
Now we have found that in above scenario the central density is of the same
order as the density at which the above phase transition begins in the nuclear
phase. Ideally we would like the central density to be a little less than the initial
density at which the above phase transition begins in the nuclear phase.
3 Beyond the Maxwell tangent construction for the phase
transition
How dowe change the crossover andMaxwell tangent construction for the phase
transition? There are 2 ways of moving the crossover between the 2 phases and
The implications of the two solar mass neutron star . . . 45
also the initial density at which the above phase transition begins in the nuclear
phase to higher density.
(i) By increasing the tree level mass of the sigma we can move the quark
matter curve up (Fig. 1), thus moving the initial density at which the above phase
transition begins in the nuclear phase to higher density. However we have to be
careful. There is not much freedom here, as this is what also determines the π−π
scattering.
(ii) By softening the nuclear EOS at high density, e.g. by including hyperons
or pi condensates. But this will increase the central density of the star and also
reduce its maximum mass.
Of these the option (i) is a safer option as it does not disturb the central den-
sity or maximum mass of the nuclear star. However, the Maxwell construction is
not the final word on the phase transition. The exact nature of the transition is
not just given by the energy /baryon in the quark matter phase ( which depends
mainly onmσ) but will depend on the quark binding inside the nucleon ( which
depends mainly o the quark meson coupling ) and the nucleon nucleon repulsion
as we squeeze them. This is not captured by the Maxwell construction.
The nucleon binding in this model is very high (Fig. 2) [5]
Fig. 2. Dependence of the quark energy on the soliton size X in the quark soliton model
(From Fig. 2 of Kahana, Ripka and Soni [5])
The quark eigenfunctions are smaller than the radius of the nucleon; they
spread over about 0.5 fermi. This yields a quark wave function size of ∼1 fermi
or kinetic energy of about 200 MeV. The unbound mass of the quark is given by
gfπ ∼ 500 MeV and effectively they must contribute 313 MeV to the mass of the
nucleon , giving the quark binding energy of ∼ 400 MeV.
We can see that the quarks will become unbound ( go to the continuum)
when the energy eigenvalue is larger than the unbound mass of the quark which
46 Vikram Soni, Pawel Haensel and Mitja Rosina
is given bymfree = gfπ ∼ 500MeV. This happens when in the dimensionless units
used in Fig. 2 ǫ ≥ 1 at X= 3.12/1.94 = 1.6. This translates into R =(1.6/2.5) fm−1 ∼
0.6 fm−1 .This is the effective radius of the squeezed nucleon at which the bound
state quarks are liberated to the continuum. By inverting the volume occupied
by the nucleon and assuming hexagonal close packing, this translates to nucleon
density of 1/(6R3) ∼ 0.77 fm−3.
Thus the quark bound states in nucleon persist untill a much higher density
∼ 0.8/fm3. In other words, nucleons can survive well above the density at which
the Maxwell phase transition begins and appreciably above the central density of
the APR 2-solar-mass star.
Another feature is the the nucleon nucleon potential. It has been found for
skyrmions and such quark-quark solitons with skyrmion configurations that there
is a strong N-N repulsion that forces the lowest baryon number NB = 2 configu-
ration to become toroidal [7]. This is an indication that nucleon nucleon potential
becomes strongly repulsive.
It thus follows that the phase transition from nuclear to quark matter will
encounter a potential barrier before the quarks can go free. This effect cannot be
seen by the coarse Maxwell construction which does not track their transition.
This will modify the simple minded Maxwell construction which assumes only
the energy and pressure that exist independently in the 2 phases. Here is where
the internal structure of the nucleon will delay the transition.
All in all this produces a very plausible scenario of how the ∼2 solar mass
star can be achieved in a purely nuclear phase.
4 Consequences and discussion
A simple consequence of this unexpected scenario at high density is that the the
phase diagram of QCD which plots temperature versus baryon chemical poten-
tial, the quark matter transition for finite density ( in the range above) will be
lifted up along the temperature axis.
References
1. J. M. Lattimer and M. Prakash, Astrophysical J. 550 (2001) 426;
2. V. Soni and D. Bhattacharya, Phys. Lett. B 643 (2006) 158.
3. A. Akmal, V. R. Pandharipande and D. G. Ravenhall, Phys. Rev. C 58 (1998) 1804.
4. V. Soni and D. Bhattacharya, arXiv. Hep-ph/0504041 v2.
5. S. Kahana, G. Ripka and V. Soni, Nuclear Physics A 415 (1984) 351.
6. Haensel et al., New Astronomy Reviews 51 (2008) 785; arXiv:0805.1820;
arXiv:1109.1179 v2.
7. M. S. Sriram et al., Phys. Lett. B 342 (1995) 201.