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PHYSICAL REVIEW C SEPTEMBER 1997VOLUME 56, NUMBER 3
Constraints on the v- and s-meson coupling constants with dibaryons
Amand Faessler,1,* A. J. Buchmann,1,† and M. I. Krivoruchenko1,2,‡
1Institut fur Theoretische Physik, Universita¨t Tubingen, Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany2Institute for Theoretical and Experimental Physics, B.Cheremushkinskaya 25, 117259 Moscow, Russia
~Received 21 February 1997!
The effect of narrow dibaryon resonances on basic nuclear matter properties and on the structure of neutronstars is investigated in mean-field theory and in relativistic Hartree approximation. The existence of massiveneutron stars imposes constraints on the coupling constants of thev and s mesons with dibaryons. In theallowed region of the parameter space of the coupling constants, a Bose condensate of the light dibaryoncandidatesd1(1920) andd8(2060) is stable against compression. This proves the stability of the ground stateof heterophase nuclear matter with a Bose condensate of light dibaryons.@S0556-2813~97!00209-4#
PACS number~s!: 14.20.Pt, 21.65.1f, 26.60.1c
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The prospect of observing the long-livedH particle pre-dicted in 1977 by Jaffe@1# stimulated considerable activity iexperimental dibaryon searches. It was proposed to invegateH-particle production in different reactions@2#. So far,experiments@3# have not provided an unambiguous signatufor theH particle. The existence of theH particle remains anopen question, which must eventually be settled by furtexperiments. Nonstrange dibaryons with exotic quantnumbers, which have a small width due to zero couplingthe NN channel, are promising candidates for experimensearches@4#. The data on pionic double charge exchanreactions on nuclei@5# exhibit a peculiar energy dependenat an incident total pion energy of 190 MeV, which caninterpreted@6# as evidence for the existence of a narrowd8dibaryon with quantum numbersT50, Jp502 and a totalresonance energy of 2063 MeV. Recent experiments at TUMF ~Vancouver! and CELSIUS~Uppsala! seem to supporthe existence of thed8 dibaryon@7#. The properties of thed8dibaryon were recently analyzed in the constituent qumodel @8#. A method for searching narrow, exotic dibaryoresonances in thepp→ppgg reaction is discussed in Re@9#. Recently, some indications for ad1(1920) dibaryon havebeen found in this reaction@10#.
When the density of nuclear matter is increased beyoncritical value, production of dibaryons becomes energeticfavorable. Dibaryons are Bose particles. Therefore, they cdense in the ground state and form a Bose conden@11,12#. An exactly solvable model for a one-dimensionFermi system in which the fermions interact through a ptential that leads to a resonance in the two-fermion channanalyzed in Ref.@13#. When the density is increased beyoa critical value, the behavior of the system can be interprein terms of a Bose condensation of two-fermion resonanIn the limit of vanishing decay width, a dibaryon can bapproximately described as an elementary field. The effecnarrow dibaryon resonances on nuclear matter has recebeen analyzed in mean-field theory~MFT! @14,15#.
Although a dibaryon Bose condensate does not exisordinary nuclei, dibaryons affect the properties of nuclei a
*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]
560556-2813/97/56~3!/1576~6!/$10.00
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nuclear matter through a Casimir effect: The presence ofbackgrounds-meson mean field inside the nucleus modifithe nucleon and dibaryon masses. This in turn modifiesvacuum fluctuations of the nucleon and dibaryon fields athus affects the energy density and pressure of the sysThis effect, which is well known@16# for nucleons, can beevaluated within the relativistic Hartree approximatio~RHA!. We recall that in a loop expansion of quantuhadrodynamics~QHD!, MFT corresponds to the lowest approximation ~no loops!, while the RHA corresponds to thone-loop approximation in the calculation of the equationstate for nuclear matter.
At zero temperature, a uniformly distributed systembosons with an attractive potential is energetically unstaagainst compression and collapses@17#. In such a case, thelong wave excitations~sound in the medium! have an imagi-nary dispersion law: The square of the sound velocitynegativeas
2,0. The amplitude of these excitations increaswith time, resulting in the instability of the system. Itnecessary to analyze dispersion laws of other elementarycitations also. We shall see, however, that in MFT andRHA only sound waves can generate an instability. Tground state of nuclear matter with a Bose condensatedibaryons is either stable or unstable against small pertutions depending on whether the repulsivev-meson exchangeor the attractives-meson exchange interaction betwedibaryons dominates.
In this paper, we investigate the possibility that dibarymatter is unstable against compression. If dibaryon matteunstable, the formation of dibaryons in nuclear matter woprovide a possible mechanism for a phase transition to qumatter. If the central density of a massive neutron starceeds a critical value for the formation of dibaryons, tneutron star could convert into a quark star, a strange staa black hole. Some of the observed pulsars are identiquite reliably with ordinary neutron stars@18#. From the re-quirement that dibaryon formation be not energeticallyvored at densities lower than the central density of neutstars with a mass of 1.3M ( , we derive constraints on thcoupling constants of thev ands mesons with the possibled1(1920) andd8(2060) dibaryons. We conclude that narrodibaryons in this mass range can only form a Bose condsate that is stable against perturbations. The effect ofdibaryons on the stability and structure of neutron stars
1576 © 1997 The American Physical Society
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56 1577CONSTRAINTS ON THEv- AND s-MESON COUPLING . . .
different phenomenological models is analyzed in Refs.@12,19#. Constraints on the binding energy of strange matter@20#derived from the existence of massive neutron stars arecussed in Ref.@21#.
The dibaryonic extension of the Walecka model@16# isobtained by including dibaryons in the Lagrangian dens@14,15#
L5C~ i ]mgm2mN2gss2gvvmgm!C11
2~]ms!2
21
2ms
2s221
4Fmn
2 11
2mv
2 vm2 1~]m2 ihvvm!w*
3~]m1 ihvvm!w2~mD1hss!2w* w. ~1!
Here,C is the nucleon field,vm ands are fields of thev ands mesons,Fmn5]nvm2]mvn is the field-strength tensorand w is the dibaryon isoscalar-scalar~or isoscalar-pseudoscalar! field. Here, mv and ms are the v- ands-meson masses andgv ,gs ,hv ,hs are the coupling con-stants of thev ands mesons with nucleons (g) and dibary-ons (h).
The s-meson mean fieldsc determines the effectivenucleon and dibaryon masses in the medium:
mN* 5mN1gssc , ~2!
mD* 5mD1hssc . ~3!
The nucleon scalar density in the RHA is defined byexpression@16#
rNS5^C~0!C~0!&5gE dp
~2p!3
mN*
E* ~p!u~pF2upu!
24mN3 z~mN* /mN!, ~4!
where
4p2z~x!5x3lnx112x2 52 ~12x!21 11
2 ~12x!3
and where the statistical factorg is g52 for neutron matterandg54 for nuclear matter. The last term in Eq.~4! is dueto the Dirac sea. This vacuum contribution renormalizesscalar density.
Here, we investigate the properties of nuclear matterlow the critical density for formation of dibaryons: i.e., thdibaryon condensate is zero,^w(0)&50. The vacuum contri-bution to the scalar density of dibaryons can be found to
2mD* rDS52mD* ^w~0!* w~0!&5mD3 z~mD* /mD!. ~5!
It differs from the corresponding vacuum contribution to tnucleon scalar density in sign and statistical factor@oneshould replace 4(52S32I)→1#. The self-consistency condition for the nucleon effective mass has the form
mN* 5mN2gs
ms2 ~gsrNS1hs2mD* rDS!. ~6!
The renormalized vacuum contribution to the nucleenergy-momentum tensor has the form@16#
^TmnN ~0!&vac524gmnmN
4 h~mN* /mN!, ~7!
is-
y
e
e
e-
e
where
16p2h~x!5x4lnx112x2 72 ~12x!2
1 133 ~12x!32 25
12 ~12x!4.
For dibaryons, we get the expression
^TmnD ~0!&vac5gmnmD
4 h~mD* /mD!. ~8!
The elementary excitations in nuclear matter with a Bocondensate of dibaryons correspond to nucleons andnucleons,v mesons,s mesons, and dibaryons and andibaryons. The dispersion laws for these quasiparticlesbe found in Ref.@15#. The nucleon and antinucleon dispesion laws have the same form as in the vacuum withreplacementmN→mN* . Therefore, they cannot generateinstability of the system. The dispersion laws ofv mesons,smesons, and antidibaryons turn out to be real, too. The opossible source of instability is the dibaryon quasipartiexcitations. The latter are responsible for long wavelenperturbations of the system and are connected with the etence of sound in the medium.
The square of the sound velocity has the form@15#
as25
a
11a, ~9!
where
a52rDS
ms2
ms2 S hv
2
mv2 2
hs2
ms2 D ~10!
andms25ms
212hs2rDS . We see thatas
2.0 for
hv2
mv2 .
hs2
ms2 . ~11!
The validity of inequality~11! is a sufficient condition forthe stability of the ground state of nuclear matter with a Bocondensate of dibaryons. Later, we show that a violationthe inequality~11! for light dibaryons is in contradiction withthe existence of massive neutron stars.
The physical meaning of the inequality~11! can be clari-fied by considering the interaction energy of uniformly dtributed dibaryon matterrDV(x)5rDV5const:
W51
2 E dx1dx2rDV~x1!rDV~x2!V~ ux12x2u!. ~12!
The Yukawa potentialV(r ) for two dibaryons has the form
V~r !5hv
2
4p
e2mvr
r2
hs2
4p
e2msr
r. ~13!
Integration gives
W51
2NDrDVS hv
2
mv2 2
hs2
ms2 D , ~14!
whereND is the number of dibaryons. When the dibaryodensityrDV increases, the energy~for as
2.0! increases alsothe pressure is positive, and therefore the system is stab
At present, the coupling constants of the mesons wdibaryons are not known with sufficient precision to drawdefinite conclusion concerning the stability of dibaryon m
nd depthe dibaryonssity is
1578 56FAESSLER, BUCHMANN, AND KRIVORUCHENKO
FIG. 1. The effective nucleon mass~a!, asymmetry coefficient~b!, incompressibility of nuclear matter at saturation density~c!, and thecoupling constantsCs
25gs2(mN /ms)
2 andCv2 5gv
2 (mN /mv)2 in the RHA, versus the ratiohs /(2gs) of s-dibaryon (hs) ands-nucleon (gs)coupling constants. The MFT results are shown for comparison. The coupling constants are fixed by fitting the minimum position aof the energy per baryon number at the saturation density of nuclear matter. The solid, dashed, and dotted curves correspond to thH~2220!, d8~2060!, andd1~1920!, respectively. A satisfactory description of the properties of nuclear matter at the saturation denpossible forhs /(2gs),0.8.
ae
i-
rvWa
nf
Iaan
eth
no
anfr
ti
e-HDe
sion
at
ses
. 2.
a-er,
OS-t
ut
ter. Given that the ratiohs /(2gs) between thes-meson cou-plings with dibaryons and nucleons is fixed, one can extrthe v- and s-meson couplings with nucleons by fitting thnuclear matter binding energyE/A2mN5215.75 MeV atthe empirical equilibrium densityr050.148 fm23. The em-pirical equilibrium densityr0 , which is determined from thedensity in the interior of208Pb @16#, corresponds to the equlibrium Fermi wave numberkF51.30 fm21.
In Fig. 1, we show how relevant nuclear matter obseables are influenced by the presence of dibaryons in thelecka model. The dependence of the effective nucleon mon the ratiohs /(2gs) at the equilibrium density is shown iFig. 1~a!. In Figs. 1~b! and 1~c! we show the dependence othe incompressibilityK59r0(]2«/]r2)ur5r0
and the asym-
metry coefficienta4 on the ratiohs /(2gs). In Fig. 1~d!, thevaluesCs
25gs2(mN /ms)2 and Cv
2 5gv2 (mN /mv)2 are plot-
ted. Note thaths /(2gs)50 is equivalent to the RHA with-out dibaryons. For comparison, we give the MFT results.MFT, dibaryons do not influence the properties of nuclematter below the critical density for the formation ofdibaryon Bose condensate. The RHA results do not setively depend on the mass of the dibaryon.
When the ratiohs /(2gs) approaches the value 0.8, thsystem of equations does not yield physical solutions andempirical equilibrium properties of nuclear matter canlonger be reproduced. Forx→1, z(x)5O„(12x)4
…, and thezero-point contributions to the scalar density of nucleonsdibaryons, which have opposite signs, are comparable4gs
4/mN'hs4/mD . Dibaryon effects become large fo
hs /(2gs)'0.5(4mD /mN)1/4'0.84. The greater thedibaryon mass, the greater the upper limit of the critical rahs /(2gs). This effect is seen in Fig. 1.
ct
-a-ss
nr
si-
e
dor
o
In this work, we study the physical implications of theffective Lagrangian~1! describing nucleon and dibaryon degrees of freedom. Other baryons can be included in the Qframework in a similar way. Their effect is quite small. Whave checked that the Casimir effect caused by the incluof other octet baryons~238516 degrees of freedom! shiftsthe critical valuehs only by about 25% if one assumes ththe s meson is an SU~3!f singlet. The inclusion of octet anddecuplet baryons~23814310556 degrees of freedom!with a universal sigma-meson coupling constant increathe critical value ofhs /(2gs) to 1.2.
The saturation curve for nuclear matter is shown in FigThe equation of state~EOS! in the RHA is softer than inMFT. The contributions of the vacuum zero-point fluctutions of nucleons and dibaryons partially cancel each othand therefore the inclusion of dibaryons makes the Estiffer. This is shown in Fig. 2, where the RHA with dibaryons ~dashed curve! lies above the RHA calculation withou
FIG. 2. Saturation curve for nuclear matter in the RHA: withodibaryons ~solid line! and with the inclusion ofH dibaryons~dashed line! for hs /(2gs)50.6.
ea
e-tiathol
n
fo
e
ohein
ry-heed
ro-ta-
toarkaxi-
s andestses
a-m-
-
ilib-
don-foral-
tu
tclu
56 1579CONSTRAINTS ON THEv- AND s-MESON COUPLING . . .
dibaryons~solid line!. The same effect is seen in Fig. 3, thdependence of the effective nucleon mass on the Fermi wnumberkF being shown for the case of theH particle withhs /(2gs)50.6. In MFT, the effective nucleon mass dcreases faster than in the RHA. As a result of the parcancellation of the nucleon and dibaryon contributions tovacuum scalar density, the dashed lines lie below the slines.
In Fig. 4 we show the critical densities for the formatioof dibaryonsH~2220!, d8~2060!, and d1~1920! in nuclearand neutron matter versus the ratiohs /(2gs), both in MFTand the RHA. For each dibaryon, three different valuesthe v-dibaryon coupling are assumed, namely,hv /hv
max51,0.8, and 0.6. Here,hv
max5hsmv /ms is the maximum value
FIG. 3. The effective nucleon mass versus the Fermi momenof nucleons in nuclear matter~upper curves! and in neutron matter~lower curves!. The solid lines correspond to the RHA withoudibaryons, and the dashed lines correspond to the RHA with insion of H dibaryons.
ve
leid
r
for the v-dibaryon coupling constant for which which thinequality ~11! is violated. Note thathv /hv
max51 corre-sponds toas
250 and hv /hvmax50.8 and 0.6 correspond t
as2,0. In the RHA, dibaryons occur at higher densities. T
coupling constanthv determines the energy of dibaryonsthe positivev-meson mean field. The greaterhv , the greaterthe density that is required to make the production of dibaons energetically favorable. This effect is seen in Fig. 4: Tsolid lineshv /hv
max51 lie above the long-dashed and dashlines hv /hv
max50.8 and 0.6, respectively.If dibaryon matter is unstable against compression, p
duction of dibaryons with increasing density results in insbility of neutron stars with a subsequent phase transitionquark matter and the conversion of neutron stars to qustars, strange stars, or black holes. In such a case, the mmum masses of neutron stars are determined by the masthe coupling constants of the mesons with the lightdibaryon. In Fig. 5 we show the minimal neutron star masfor which dibaryons can occur.
The MFT and RHA EOS for neutron matter at suprnuclear densities are matched smoothly with the BauBethe-Pethick EOS@22# at densitiesrdrip,r,0.8r0 whererdrip54.331011 g/cm3 and then with the Baum-PethickSutherland EOS@23# at densitiesr,rdrip . The maximumneutron star masses are sensitive to the value of the equrium Fermi wave number. If we chosekF51.42 fm21 in-stead of kF51.30 fm21, the maximum masses in MFT~without dibaryons! are reduced from 3M ( down to 2.6M (
@16#. The choicekF51.30 fm21 provides less stringent antherefore more conservative constraints for the mesdibaryon coupling constants. We do not show the resultsthe d1~1920! dibaryon, because its condensation starts
m
-
n
t
d
-s-
FIG. 4. Critical densities forthe formation of H~2220!,d8~2060!, andd1~1920! dibaryonsin nuclear and neutron matter iboth MFT and the RHA, versusthe s-dibaryon coupling constanfor hv /hv
max51, 0.8, and 0.6~thesolid, longed-dashed, and dashelines, respectively!. Here,hv
max5hsmv /ms is the maximumvalue of thev-dibaryon couplingconstant for which dibaryon matter is unstable against compresion ~see text!.
f
in
iomao
-tato
neemf
er
athe
lin
le
tck
able
v-
heh
esti-
ed
eleeancem
cespec-
a
t-
r
onds
to
eson
is
pec-osser-
1580 56FAESSLER, BUCHMANN, AND KRIVORUCHENKO
ready at a densityr'r0 and results in the conversion oneutron stars with very low massesM,0.2M ( into quarkstars.
In Fig. 6 we show the parameter space for the couplconstants of dibaryons withs andv mesons. As mentionedin the beginning, our discussion is restricted to the regas
2,0 for which dibaryon matter is unstable against copression. The requirement of stability of normal nuclear mter at the saturation density allows us to get constraintsthe coupling constants. The corresponding curves~straightlines in the MFT case! marked by arrows with white arrowheads restrict the parameter space of the coupling consfrom below. The dotted line in the MFT case, which refersthe d1~1920! dibaryon, is very close to the dash-dotted lias
250. For low valueshs , the dotted line lies above the linas
250. In this case, dibaryon matter unstable against copression cannot exist. The window in parameter spaceunstabled1~1920!-dibaryon matter is, however, much largin the case of the RHA.
With the conservative assumption that pulsars with a m1.3M ( are ordinary neutron stars, the constraints onmeson-dibaryon coupling constants can be further improvThe corresponding curves~straight lines in the MFT case!are shown in Fig. 6. We see that the dotted and dashedlie above or very close to the lineas
250. This means that ford1~1920! and d8~2060! dibaryons, dibaryon matter unstabagainst compression cannot exist@owing to the very smallwindow for d8~2060! at higher values ofhs#. For the Hparticle, there is a window in parameter space betweenline as
250 and the solid curves marked by arrows with bla
FIG. 5. Lowest neutron star masses, for which dibaryon formtion becomes energetically favorable, versus thes-dibaryon cou-pling constant forhv /hv
max51, 0.8, and 0.6~the solid, longed-dashed, and dashed lines, respectively!. Here,hv
max is the maximumvalue of thev-dibaryon coupling constant for which dibaryon mater is unstable against compression (as
2,0). The results are shownfor both MFT and the RHA and forH~2220! andd8~2060! dibary-ons. In the case of thed1~1920! dibaryon, the resulting neutron stamasses are very small (,0.2M ().
g
n-t-n
nts
-or
ssed.
es
he
arrowheads that corresponds to dibaryon matter unstagainst compression.
The H-particle interaction was studied in the nonrelatiistic quark cluster model@24,25#, which simultaneously de-scribes theNN-phase shifts. The coupling constants of tmesons with theH particle can be fixed by fitting the deptand the position of the minimum of theHH-adiabatic poten-tial @26# to give hv /(2gv)50.89 and hs /(2gs)50.80.These values are marked on Fig. 6 with a cross. Thesemates are used in the MFT calculations@14,15#. They corre-spond to unstable dibaryon matter and are in the allowregion of the parameter space for theH particle. The ener-getically favorable compression ofH matter can lead to theformation of absolutely stable strange matter@20# and theconversion of neutron stars to strange stars@27#.
The MFT and RHA EOS of the extended Walecka modare both very stiff. In softNN interaction models such as thReid potential~for a review of nuclear matter models se@18#!, the central density of neutron stars is much larger thin stiff models. Therefore, the conditions for the occurrenof new forms of nuclear matter are more favorable. FroFigs. 5 and 6, we see that the softer RHA EOS produlower upper limits on the neutron star masses and, res
- FIG. 6. Parameter space for thes- and v-dibaryon couplingconstants in MFT and the RHA. The dash-dotted lineas
250 dividesthe parameter space into two parts. The upper left part correspto dibaryon matter stable against compression~square of the soundvelocity is positiveas
2.0!, and the lower right part correspondsdibaryon matter unstable against compression (as
2,0). The solid,dashed, and dotted curves restrict the parameter space for the mcoupling constants with theH particle and thed8~2060! andd1~1920! dibaryons. In the left upper region, dibaryon formationenergetically unfavorable in ordinary nuclei~curves marked by ar-rows with white arrowheads! and in neutron stars of a mass 1.3M (
~curves marked by arrows with black arrowheads!. In the lowerright parts of the parameter space, dibaryons could appear, restively, in ordinary nuclei and in massive neutron stars. The crrefers to theH-particle coupling constants with the mesons, detmined from the adiabaticH-H potential@26#.
odeofonerhe
intu
isethn
ryonlude-
e
is-
oftFG
56 1581CONSTRAINTS ON THEv- AND s-MESON COUPLING . . .
tively, more stringent constraints on the meson-dibarycoupling constants as compared to the stiffer MFT EOS,spite the fact that in the RHA dibaryons occur at highdensities~see Fig. 4!. One can assume that this effect isgeneral validity and that softer EOS give more stringent cstraints on the meson-dibaryon coupling constants. Thfore, we consider the constraints given in Fig. 6 as ratconservative.
In conclusion, we have shown that the assumption ofstability of d1~1920!- andd8~2060!-dibaryon matter againscompression is in contradiction with the hypothesis that psars of a mass of 1.3M ( are ordinary neutron stars. Thconclusion is valid for all narrow dibaryons with the samquantum numbers in the same mass range. On the ohand, theH particle is sufficiently heavy, its condensatio
.
ky
p.
, J.
a
nd
ko
B
B
ne-r
-e-r
-
l-
er
starts at higher densities, and constraints on meson-dibacoupling constants are not as stringent. One cannot excthe possibility thatH particles form in nuclear matter a condensate that is unstable against compression. Finally, thv-ands-meson coupling constants withd1~1920! andd8~2060!dibaryons should obey the inequality~11!. The correspond-ing coupling constants with theH particle should lie abovethe solid curves in Fig. 6.
The authors are grateful to B. V. Martemyanov for dcussions of the results. One of the authors~M.I.K.! acknowl-edges hospitality of the Institute for Theoretical Physicsthe University of Tu¨bingen, the Alexander von HumboldStiftung for support with a Forschungsstipendium, and Dand RFBR for Grant No. Fa67/20-1.
cts
1-
na-
J.
.
-
;
ki,
me
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