The effects are studied of H-dibaryons on the structure of neutron stars. It was found that H particles could be present in neutron stars for a wide range of dibaryon masses. The appearance of dibaryons softens the equations of state, lowers the maximum neutron star mass, and affects the transport properties of dense matter. The parameter space is constrained for dibaryons by requiring that a 1.44 solar mass neutron star be gravitationally stable

Frieman, Joshua A.

Haensel, Pawel

Olinto, Angela V.

English

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Fermi National Accelerator Laboratory
FERMILAB -Pub-91/176- A
June, 1991
DIBAI:LYONS IN NEUTRON STARS
Angela V. Olinto
Department of Astronomy and Astrophysics
Enrico Fermi Institute,
University of Chicago, Chicago, IL 60637
Pawel Haensel
N. Copernicus Astronomical Center,
Polish Academy of Sciences,
Bartycka 18, PL-00716 Warszawa, Poland
/ •
r
and
Joshua A. Frieman
NASA/Fermilab Astrophysics Center,
Fermi National Accelerator Laboratory,
P.O.Bo= 500, Batavia, 11 60510
/
ABSTRACT
We investigate the effects of H-dibaryons on the structure of neutron stars. We find
that H-particles could be present in neutron stars for a wide range of dibaryon masses. The
appearance of dibaryons softens the equation of state, lowers the maximum neutron star
mass, and affects the transport properties of dense matter. We-constrain the parameter
space for dibaryons by requiring that a 1.44 Me neutron star be gravitationally stable.
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The existence of long-llved (or even stable) six-quark states with baryon number 2 was
first proposed by Jaffe 1 on the basis of bag model considerations; experimental searches
for them are being actively pursued, 2 with one recent report of a positive detection, a Of
the possible dibaryon states, the H-particle, with the quark content of two A's (uuddgs)
in a flavor-singlet spin-zero state (jr = 0+), is the best candidate for metastability. Bag,
Skyrme, quark duster, and lattice QCD model calculations predict a mass for the H-
dibaryon in the range ms "" 2.1 - 2.24 GeV, with most models indicating a value 20 - 80
MeV below that of two A's (2mA = 2.231 GeV). 4 In this case, the H-particle would be
stable with respect to the strong force, decaying only via weak interactions.
Dibaryons wiU be present in neutron stars whenever the chemical potential of neutron
star matter is greater than the effective mass of the dibaryon. Since they are bosons, hke
pions and kaons, dibaryons can, in principle, condense in the cores of neutron stars, but
with one important difference: dibaryons carry baryon number. If dibaryon interactions
are neglected, a Bose condensate of dibaryons would cause the collapse of neutron stars
with central densities higher than the minimum density for the appearance of dibaryons, s
At some level, however, such interactions must be present, and such an instability will
not be realized. At very short distances, the interactions among the underlying quark
components of the dibaryons will give rise to an effective repulsive interaction between
dibaryons. Even a very small self-coupling can stabilize the system against collapse. If
the interactions are relatively weak, however, the equation of state of dense matter will
be softened by the presence of dibaryons.
At a critical baryon number density, nB> r_, dibaryons begin to appear inside neu-
tron stars. We study this transition by considering a mixture of neutrons and dibaryons in
chemical equilibrium. (For self-consistency, we should also include protons, As, and other
light hyperons, but this does not qualitatively change the results for the parameter range
of interest, s) In equilibrium, the chemical potentials of dibaxyons (/_s) and neutrons (#,)
are related by
= (1)
Of the many candidate equations of state for neutron matter, we choose the medium-
stiffnessBethe-Johnson z (BJ1) model as a representative and analyticallytractable ex-
ample. (Clearly,itwould be of interestto make a systematic study with differentnuclear
equations of state;here, we have chosen a familiar model to qualitativelyillustratethe
effectsof dibaryons.) In itspolytroplcincarnation, we can write the energy per neutron
as
e,, = P--_-_=mn + 236n_ MeV, (2)
nn
where the neutron number densityr_ is given in fin-s, ra,,isthe neutron mass in MeV,
and a = 1.54. The neutron pressure and chemical potentialcan be written as:
,n,2 d£n
Pn = _"-_ = 363.44n,_ +*MeV fm-S, (3)
/_ = mr + 599.44n,_ MeV. (4)
The adiabatic index is then r,, = a + i = 2.54.
We shall assume that the H-dibaryon is spatially compact, with a size comparable to
that of nucleons. This assumption is borne out by bag model estimates 4 and corroborated
by a recent study e of the quark structure of the H-particle. Then, at the nucleon level,
in nuclear mean field theory, the H-partlcle can be modeUed as a complex scalar field, _b,
with effective Lasrangian°:
L,= -lg 0. o  t - _ _ - L,_a, (5)
where n __ 3 describes higher order multl-body interactions, and L_ includes the coupling
to other hadrons. In mean field theory, the primary effect of the H interaction with
nucleons is to induce a shift ms _ rn_ in the effective H-mass, due to the mean field
of the neutrons. Although densitydependent, to firstapproximation we can absorb this
effectin a mean renormallzed mass for H; for simplicity,we willdenote this effective
mass by ms hereafter. As a first approach to modelling the dibaryons that is simple but
qualitatively reliable, we will neglect the higher order self-interactions between dibaryons.
(These terms may be important at high densities, but in that regime the description of
the EOS in terms of nucleons is suspect at any rate.)
We can estimate the dibsryon self-coupling _ by using a variant of the P-matrix
formalism of Jaffe mid Low, TM applied to diquarks by Donoghue and SateeshJ t If we
consider two dibaryons brought together, the energy of the resulting configuration can be
written as E2 -- 2ms+AE(r,A), where r is their separation. As r _ as, E2 --_ M4, where
M4 and/74 are the mass and radius of the 12-quark baryon number-4 state. The mass
M4 can be estimated from the bag model, 4 M4 -_ 4.7 GeV. Calculating the interaction
energy for the two-dibaryon state con_ned inside a cavity of radius R, we find that tt
EI,_(R,)_) = 0.34)_/4E2R s, where E 2 = p_+ rn_ and the momentum p = lr/R from the
vanishing of _ at the cavity boundary. Taking the cavity to be the size of the baryon
number-4 state, R = R4, and equating AE(R4,,_) = EI_(R4,,_), we obtain
(-')=   .sRI(M, - 2m,,) • (6/
For the range/_ = 1.1 - 1.4 fm and mR = 2 - 2.2 GeV, we estimate J_ _ (3.4 - 13) x 10 _.
Given the crudeness of the approximations involved, this estimate should be considered
a guide to the expected order of magnitude for the coupling.
Since the self-coupllng A lies comfortably above a critical value A" -- 41rm_/m_v_
" 4 x 10 -37, we can use the equation of state for self-interacting bosons derived by
Colpi, Shapiro, and Wasserman 12 to excellent approximation (corrections are of order
0(_'/,_) ._ 10-4°). For a spherically symmetric scalar field with time-independent energy-
momentum tensor, the lowest energy solution is of the form
(7)
where q?(r) is a real function of the radial coordinate. In the limit ,_ >> )_*, the solution
3
to the scalar field equation of motion to lowest order in A*/A is just
@=A-i_ _-_-1 for r<R, (8)
where R denotes the radius of the configuration. 13 The dibaryon equation of state is
obtained by substituting thissolution into the scalarfieldenergy-momentum tensor,
and the H-particle current
Defining the dimensionless parameter = = _=]rn 2, the dibaryon energy density and pres-
sure are then given by:
Ps = po(z- 1)(3z+ 1), (11)
pH= p0Cz-1)', (12)
where
Po m_/4A MeV( m_z )4(_) (13)= --523--_- 2000 MeV "
The parameter z isrelated to the baryon number density of dibaryons by:
ns = 2ns = 8P°(z- 1)'V_. (14)
7rtH
(Thus, z --41 ms the dibaryon density goes to zero.)This specifiesthe dibaryon equation
of state in parametrized form. (Alternatively, eliminating z, the equation of state can
be written t_ PH = (4po/9)[(1 + 3pH/4po) I/2 -- 112.) We can then calculate the dlbaryon
chemical potential:
+ (is)PH = =
_tH
and the adiabaticindex
FH=p_+pR ( dpH _ _ 4z
PH _ d--_H/ 3z- 1 (16)
4
At low density (as --* 0), Fu --* 2, and at high density (p D, Po) rs --, 4/3; the latter
reflects the fact that the dibaryon EOS reduces to that of a free relativistic gas in the
high density limit. Since FH < F,_, dibaryons generally soften the equation of state.
Imposing chemical equilibrium between neutrons and dibaryons, we find that the
threshold baryon number density for the appearance of dibaryons is:
( mR - 2rn,_ _ l/"f.m_ a fm_a ( mu - 2m_, ) l/°n_ = \l198.88MeV) = 0.45 _ -- 2--ms (17)
This threshold is independent of the self-coupling ,_. For m_ __ 2rnA, the threshold baryon
density for the appearance of H-particles is about three times nuclear matter density.
For densities above n_, the equation of state will be that of a neutron-dibaryon mix-
ture:
nB = nn _- 2ntl,
P-- pn W PJ_,
p = + pB =
= (p+ p)/ B = (18)
Imposing chemical equilibrium, the system can be described with one parameter, for
example the neutron density n,,.
We have constructed models of neutron stars by integrating the Oppenheimer-Volkoff
equation of hydrostatic equilibrium using the above equation of state. The effect of the
appearance of dibaryons can be seen in Fig. 1, which shows the neutron star mass M vs.
radius R for models with (solid curve) and without dibaryons (dashed curve). For this
choice of parameters, mar = 2200 MeV and _ = 8000, the maximum neutron star mass is
reduced when dlbaryons are included, a consequence of the softening of the equation of
state.
In Fig. 2, we plot the neutron star mass threshold for dibaryons to be present: a
neutron star with B J1 equation of state and a mass above the threshold will have an
5
admixture of dibaryons in its core. Note that dibaryons are present in stars of mass
1.44M e for a wide range of H-partlcle masses. In particular, even if the H is unstable
(rnH _ 2rnA), it seems to be welcome.
The dependence of the neutron star maximum mass on the dibaryon mass and coupling
constant is shown in Fig. 3. The constraint that the maximum neutron star mass exceed
1.44M®, the mass of the binary pulsar 14 PSR1913+16, implies a lower bound on the
strength of dibaryon interactions (,_) for a given d/baryon mass. In Fig. 4, we show this
constraint on the dibaryon parameter space for the neutron-dibaryon model. Interestingly,
this lower bound on A as a function of mu is close to the bag model estimate of A, Eqn.(6).
We also note that if mB < 2m,_ = 1879 MeV, then 'Mibaryon matter" would be absolutely
stable at zero pressure; in this case, d/baryons would form 'boson stars '12 (perhaps with
a small crust of nuclear matter) with m_mum mass M,,,. = 0.22(Ap,')'/'.,_l,_s =
1.44MoC ,/2.s× 10')'/'(2,,,,,/,,,H)'.
Aside from their effects on neutron star structure, d/baryons may also have impor-
tant consequences for the transport and cooling properties of neutron stars. Since they
are a non-ideal Bose gas, the low-lying excitations will have a phonon spectrum, and
the d/baryons will presumably act as a superfluid. In fact, one can demonstrate this di-
rectly by considering the spectrum of small fluctuations Is about the solution (7). Also,
since they are not Pauli-blocked, neutrino emission from H-H and H-n scattering may be
considerably enhanced over the modified URCA process, thereby accelerating the eaxly
stages of neutron star cooling. We conclude that dibaryons could play an important role
in neutron star physics.
Part of this work was completed while two of us (J.F. and A.O.) were visitors at
the Copernicus Astronomical Center in May 1990; we thank the Center for its warm
hospitality. J.F. and A.O. acknowledge support from the DOE and NASA (through grant
NAGW-1340) at Fermilab, and A.O. from NSF (Grant AST-22595) and DOE (grant
DE-FG02-90ER40606) at the University of Chicago.
References
1. R. L. Ja_e, Phys. Reg. Left. 38, 195 (1977).
2. K. Imal, "Nucleas and Quark Matter with Strangeness," in Proc. of the Twelfth
International Conference on Particlesand Nuclei (PANIC XII), (June 1990). S.
Aoki, et al, Phys. Reg. Lett. 65, 1729 (1990).
3. B. A. Shahbazian, et al.,Phys. Left. 235B, 208 (1990).
4. For a discussionof ba8 model calculationswith referencesto the literature,see E.
Faxhi and R. L. Jaffc,Phys. Rev. D30, 2379 (1984). Skyrme model treatments
include: A. P. Balachandran, et al.,Phys. Rev. Lett. 52, 887 (1984); A. P. Bal-
achandran, F. Lizzi,and V. G. J. Rodgers, Nucl. Phys. B258, 525 (1985); S. A.
Yost and C. R. Nappi, Phys. Re_. D32, 816 (1985); R. L. Jaffe and C. L. Ko-
rpa, NucL Phys. B258, 468 (1985). Lattice calculationsof the H-mass arc: P.
B. Mackenzie and H. B. Thacker, Phys. Reg. Left. 55, 2539 (1985); Y. Iw_aki,
T. Yosie, and Y. Tsuboi, Phys. Reg. Left. 80, 1371 (1988); T. Yosie,Y. Iwasaki,
and S. Sakai, Nucl. Phys. B (Proc. Suppl.) 17, 413 (1990). Quark clustermodel
calculationsinclude: M. Oka, K. Shimizu, and K. Yazaki, Phys. Left. 180B, 365
(1983); iVucl.Phys. A464, 700 (1987); U. Straub, el al.,Phys. Lett. 200B, 241
(1988); Y. Koike, K. Shimizu, and K. Yazaki, Nucl. Phys. A513, 653 (1990); S.
Takeuchi and M. Oka, Phys. Reg. Left.66, 1271 (1991).
5. M. I. Krivoruchenko, P_'ma Zh. Eksp. Teor. Fir. 46, No. 1,5 (1987).
8. In the Bethe-Johnson model, inclusionof the lighthyperons generallyleads to only
a slightsoftening of the equation of state from the pure neutron case, due to the
increased phase space. For other equations of state,however, thismay not be true;
see,e.g.,J. Ellis,J. Kapusts, and K. Olive,Nucl. Phys. B348, 345 (1991).
7. H. Bethe and K. Johnson, Nucl. Phys. A230, 1 (1974).
8. R. Tamagaki, Kyoto University preprint (1990).
9. The metric signature is (-+++).
10. R. L. Jaffe and F. E. Low, Phys. Rev. D19, 2105 (1979).
11. J. F. Donoghue and K. S. Sateesh, Phys. Rev. D38, 360 (1988). Note that our
definitions of )_ differ, our _ = 4)_DS.
12. M. Colpi, S. Shapiro, and I. Wasserman, Phys. Rev. Left. 57, 2485 (1986).
13. Here and below we have neglected gravitational corrections to the scalar field so-
lution, i.e., we are treating _b as a field in fiat spacetime; this has no effect on the
derived equation of state.
14. J. Weisberg and J. Taylor, Phys. Rev. Left. 52, 1348 (1984).
15. R. Friedberg and T. D. Lee, preprint (1988); R. Davis and E. P. S. Shellard, Phys.
Rev. Left. 63, 2021 (1989); J. Frieman and B. Lynn, Nucl. Phys. B329, 1 (1990).
FIGURE CAPTIONS
Figure 1. Gravitational mass M versus stellar radius R for B J1 equation of state
(solid curve) and for B J1 with cUbaryons of mass mR - 2200MeV and self-coupling
- 8000 (dashed curve).
Figure 2. Threshold neutron star mass above which dibaryons appear, as a function
of dibaryon mass.
Figure 8. Maximum neutron star mass as a function of _ for mH = 2000, 2100, 2200,
2300 MeV. The horizontal line at 1.44M o denotes the mass of the binary pulsar.
Figure 4. Minimum value of )_ required for stability of a 1.44Mo neutron star,
)_,_i,J1000_ as a function of H-mass mH.
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Dibaryons in neutron stars

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