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we are closing in on neutron stars both observationally and theoretically .
observationally more and better masses and radii are determined by a number of different methods .
the  large " masses implied in few cases ( as in the vela x-1 pulsar where a mass of @xmath2 has been suggested ; @xcite ) poses a real challenge to models of neutron stars built using the so - called modern equation of state ( eos ) where the uncertainties are reduced by improved two- and three - body forces , relativistic effects and many - body calculations @xcite .
even the stiffest eos so far developed seem to be facing difficulty in accounting for the extreme values ( up to @xmath3 ; sect .
4 ) . in @xcite ,
hereafter ob , as an alternative , we constructed an eos of dense matter based on the skyrme model for strong interactions which represents baryons as solitons of classic pionic fields .
the resulting compact objects we named _ skyrmion stars _ ( sss ) @xcite are intrinsically heavier ( due to the stiffness of the skyrmion fluid ; hereafter sf ) than any other type of compact stars computed using modern eos .
sss can be as massive as @xmath4 leading us to speculate ( given the above mentioned observations ) that these might exist in nature .
we expect only older sss that have accreted enough mass to reach these extreme masses . ] .
the paper is presented as follows : sect .
2 is devoted to the discussion of the skyrme model and its link to quantum - chromodynamics ( qcd ) .
the problem of the _ missing attraction _ in the skyrme model is described . in sect .
3 , we discuss the role the dilaton ( the glueball potential in qcd ) could play in curing such a problem by binding skyrmions together to form the sf .
we end sect . 3 by reminding the reader of the basic properties of the sf and the resulting eos used to construct models of sss .
the astrophysical implications follow in sect . 4 where we compare sss to stars constructed using modern eos of dense matter .
we conclude in sect .
we first give a brief overview of the skyrme model .
the fundamental principles are discussed at a basic level .
the interested reader is referred to ref .
@xcite for a thorough introduction to the topic .
skyrme constructed a model of pion interactions consisting of a conventional model of weak meson interactions plus an additional ( higher - order ) term thought to take into account indirect effects of heavier mesons like the @xmath5-meson .
the now well - known skyrme lagrangian density is usually written as @xcite , @xmath6 where , @xmath7 is the skyrme term ( @xmath8 is the chiral field and @xmath9 is interpreted as the pion decay constant ) , and @xmath10 [ u^{+}\partial^{\mu}u , u^{+}\partial^{\nu}u ] ) \label{three}\ ] ] is the quadratic term introduced by skyrme to keep the skyrmion stable against the derrick instability @xcite ( @xmath11 is the skyrme parameter ) .
skyrme found that his model contained ` topologically nontrivial ' configurations ( extended objects ) of the meson fields , namely topological solitons , which he identified as baryons . for twenty years
the skyrme model was overshadowed by the tremendous success of qcd and only in the early 1980 s after the establishment of its link to low energy qcd that the model was revived .
the success of qcd is limited to the high energy regime , while at low energy it remains virtually intractable .
the reason for this is that qcd has a running coupling constant @xmath12 ; it is a function of momentum transfer , or distance . at short distances of the order of 0.1 fm or less ( high energy and momentum transfer of several gev ) qcd
is characterized by a small enough @xmath12 that it is treated perturbatively .
all of the results obtained in this regime are consistent with experimental data .
this is the phase in which the relevant degrees of freedom are quarks and gluons and it is called the asymptotic freedom phase . at large distances of the order of 1 fm or more ( low energy and momentum transfer of 1 gev or less ) @xmath12 is of the order of unity and qcd is a nonperturbative theory .
this is the confinement phase in which quarks are confined inside hadrons and the hadronic degrees of freedom are more relevant .
this phase , which is the most practical , is the most mathematically complex .
it should provide all properties of hadrons such as masses , sizes , magnetic moments , lifetimes , scattering properties and , in principle , all nuclear phenomena .
the first major step to overcome this problem was taken by @xmath13thooft @xcite .
he found that in the limit of a large number of colors ( large @xmath14 ) , @xmath15 could be used as an expansion parameter . in this limit
, qcd simplifies a great deal and @xmath13thooft went on to show that at large @xmath14 , qcd is equivalent to a local field theory of mesons and ` glueballs ' ( bound states of gluons , without quarks ) , with an effective interaction between them of order @xmath15 .
the second step was taken by witten @xcite .
assuming confinement , he showed that baryons in large @xmath14 qcd behave much like solitons in a weakly coupled local field theory of mesons . in this limit ,
baryon masses scale as @xmath16 , where @xmath17 is the strength of the meson coupling , while baryon sizes are of order 1 .
solitons in weakly coupled theories have masses that scale as @xmath18 and sizes that tend to constants as @xmath17 tends to zero .
even though the mesons are weakly interacting , the solitons interact strongly as do baryons in qcd .
the next natural step , it seems , is to derive the effective meson lagrangian from the fundamental qcd lagrangian .
this task , as it turned out , is immensely difficult .
its achievement is equivalent to the solution of the intractable original problem of low energy nonperturbative qcd in the confinement phase .
nevertheless , the form of the resulting effective lagrangian is being narrowed down under reasonable assumptions .
assuming chiral symmetry is spontaneously broken in qcd ( with the physical pions as the resulting goldstone bosons , and taking the low energy limit in which one expects the golstone bosons to dominate the dynamics ) it has been shown that the first term in the resulting low energy effective lagrangian is @xcite : @xmath19 which is astonishingly similar to the skyrme term ( eq .  [ two ] ) ! this picture , which emerged from large-@xmath20 qcd , is precisely what skyrme had in mind long before qcd .
further work showed that the similarities between the skyrme model and to what is described as the nature of mesons and baryons in large @xmath21 qcd is simply intriguing .
the skyrme model , however , as it is build was known to predict an isospin independent spin - orbit force with the _ wrong _ sign .
that is , it predicts a _ repulsive _ interaction .
the product ansatz for the two - baryon system as suggested in ref .
@xcite ( eq .  [ two ] ) , beyond its relative simplicity as compared to other two - baryon field configurations which can be found in the literature @xcite , becomes exact for large @xmath22 separation .
unfortunately , it is not the case for the isoscalar component of the spin - orbit force since the standard skyrme model predicts an isospin independent spin - orbit force with the _ wrong _ sign .
namely , it predicts a _ repulsive _ interaction while the phenomenological bonn potential @xcite as the paris potential @xcite gives an _ attractive _ one .
this came to be known as the problem of the _ missing attraction_. extensions of the skyrme model consisted on including higher - order terms in powers of the derivatives of the pion field @xcite .
expressed in terms of an @xmath23 matrix @xmath8 which , as we have said characterizes the pion field , a six - order term corresponding to @xmath24-meson exchange @xcite , @xmath25 where @xmath26 is the baryon current , @xmath27 the @xmath24-meson mass and @xmath28 a dimensionless parameter related to the @xmath29 width , might be a good candidate to solve the problem of the @xmath22 isoscalar spin - orbit force . while it was believed that the inclusion of such a term leads to the correct sign ( _ attractive _ interaction ) for the isoscalar spin - orbit potential @xcite , recent calculations ( see ref .
@xcite for e.g. ) proved that by taking into account the second part of of the sixth - order term the anomaly of the skyrme model ( _ repulsive _ force instead of _ attractive _ ) remains .
the treatment of the spin - orbit part of the two - pion exchange potential within the skyrme model needed to be improved in order to correct the anomaly of that sign .
below we explain how and why the dilaton field was suggested as a plausible cure .
in ref . @xcite and @xcite the authors explored the idea of coupling the skyrmion to the dilaton field .
this idea to _ account for a scalar field _
confines the skyrmion and provides the attractive term missing in the original skyrme formalism ( see ob for more on this ) .
we start by writing the energy of @xmath30 skyrmions per unit volume ( parameterized by the density , @xmath31 ) at finite temperature . in the mean field approximation
it is given by ( where we adopt natural units with @xmath32 ) , @xmath33 here , @xmath34 is the strength of the coupling of the @xmath35-meson ( of mass @xmath36 ; @xmath37 is the mean - field value ) to skyrmions while @xmath38 represents the isospin degrees of freedom ( @xmath39 for neutron matter and @xmath40 for symmetric nuclear matter ) ) , the @xmath5 meson coupling has been omitted which reduces the analysis to symmetric nuclear matter only . as such in ob the isospin degrees of freedom ( @xmath41 ) was taken as a free parameter as to allow for the two regimes  pure neutron matter and symmetric matter  to be explored . ] .
@xmath42 is the contribution to energy of a single skyrmion while @xmath43 is the skyrmion s momentum . @xmath44 and @xmath45 parameterize the single particle and anti - particle distribution functions .
the pressure of such an ensemble at @xmath46 is then simply given by @xmath47 .
one finds that the contribution of the vector meson field ( @xmath35 ) to the pressure grows with density ( @xmath48 ) , and is positive ; the dilaton potential @xmath49 ( where @xmath50 is the mean - field value ) gives a negative contribution to the pressure , acting to bind the system into a fluid ; the sf .
once the eos describing such a fluid is derived ( ob ) the corresponding compact objects ( sss ) are then computed @xcite .
sss in our picture are not boson / soliton stars where the soliton is a global structure over the scale of the star but rather form their constituent baryons as topological solitons using pions fields .
this is fundamentally different from other 
exotic " stars which also follow from solutions to an effective non - linear field theory of strong forces ( see also @xcite ; @xcite and references therein ) . in what follows , we conclude this letter by describing astrophysical cases plausibly supporting the existence of sss in nature .
in fig .  [ fig.1 ] we compare the @xmath51 relation for skyrmion stars ( ob ) to the theoretical @xmath51 curve obtained using six recent realistic models for the eos ( uu , bbb1 , bbb2 , bpal12 , hyp , and k@xmath52 ) . the solid curves labeled ss1 and ss2 are for strange stars ( the data was kindly provided to us by the authors of ref .
the triangle depicts the mass - radius constraint from fits to x - ray bursts in 4u 1636 - 53 . inside the triangle
is the allowed range of @xmath53 and @xmath54 which satisfies the compactness constraints as modeled in ref .
@xcite ( see their figure 4 ) , and clearly favoring stiffer eoss .
our modeled stars ( ob ) cross the triangle suggestive of 4u 1636 - 53 as a plausible ss candidate .
_ qpos _ are neutron stars emitting x - rays at frequencies of the orbiting accreting matter
. such _ quasi - periodic oscillations _ ( qpo ) have been found in 12 binaries of neutron stars with low mass companions .
if the qpo originate from the innermost stable orbit @xcite ) of the accreting matter , their observed values imply that the accreting neutron star has a mass of @xmath55 in the case of 4u 1820 - 30 ; this would rule out most modern eoss allowing only the stiffest ones .
sss is one possibility given that the gravitational mass of the maximum stable non - rotating ss has a value of @xmath4 ( ob ) . for completeness
, one should note that even by making the modern / recent eos stiffer at high densities in a smooth way , the maximum mass can never exceed @xmath56 due to the causality condition @xcite .
we gave a brief historical overview of the skyrme model , its predictions of hadronic interactions and its interesting connection to qcd .
here we showed how the repulsive term in the skyrme model can be removed by coupling the skyrmion to the dilaton field .
this lead to the concept of the skyrmion fluid and the related hypothetical stars we called skyrmion stars .
the stiffness of the skyrmion fluid allows for sss to be as massive as @xmath57 .
the sss show unique features ; for a given mass their radii are in general larger than those of neutron stars constructed using modern eoss .
we discussed examples in astrophysics where sss might constitute plausible candidates .
future observations constraining the mass - radius plane of compact stars would most likely prove or rule out the existence of sss in nature .
* acknowledgements .
* i am grateful to s. morsink , g. klbermann and r. bhaduri for encouraging help and valuable discussions .
the research of r.o . is supported by grants from the natural science and engineering council of canada ( nserc ) .
we discuss the skyrme model for strong interactions and the concept of a skyrmion fluid . the corresponding compact objects , namely skyrmion stars , constructed from the equation of state describing such a fluid are presented and compared to models of neutron stars based on modern equation of states . 
 we suggest plausible skyrmion star candidates in the * 4u 1636@xmath053 * and * 4u 1820@xmath030 * low mass x - ray binary systems where the suggested masses of the accreting compact companion ( @xmath1 ) remain a challenge for neutron star models .     
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