Abnormal phase in dense neutron star matter

Nuclear Physics A290 (1977) 493-500 ; © North-XollatriPt~lirhiny Co., Atrtttudmtt Not to be reproduced by photoprhtt or microfilm without written permiraion tirom the pub]bhar

ABNORMAL PHASE IN DENSE NEUTRON STAR MATTER EBBE M. NYMAN t Department ojPhysics, Strate University ojNew York at Stony Brook, Stony Brook, New York 11794, USA and ~b~ Akademi, 20500 .~bo S0, Finland and MANNQUE RHO Service de Physique Théorique, Centre d'Études Nucléaires de Saclay, BP no. 2, 911910 Gif- sur- Yvette, Fronce Received 11 February 1977 Abstract : The possibility that neutron stars may possess an "abnormal" central region of the Lee-Wick type is discusved. It is found that when the abnormal equation of state includes quantum corrections and short-range repulsion as required in normal nuclear matter, the gravitational pressure in stable neutron stars is insufficient to induce a phase transition to abnormal matter.

1 . Introduction Recently, Lee and Wick t) suggested, on the basis of renormaliTable Lagrangians involving scalar mesons and nucleons, that at sufficiently high baryon densities matter would undergo a phase transition into an "abnormal" phase with properties very different from the normal one. Using the Q-model Lagrangian in the semiclassical approximation, Lee 2) predicted that abnormal nuclear matter would be bound, and thus that small "lumps" ofit might occur or could be produced in nature . Subsequently, quantum corrections were studied by Lee and Margulies j): their result was that for certain values of the parameters in the Lagrangian, quantum corrections can be adjusted to be quite small. However in this version, the tr-model is associated with a three-body force which in normal nuclear matter would have a strength that cannot be easily accomodated in traditional nuclear physics 4). It was observed by Nyman and Rho (IVR) s) that in the presence of quantum corrections the only parameter in the Q-model - the mass of the scalar meson - may instead be chosen such as to eliminate any embarrassingly large three-body force and thus to render the theory compatible with the standard lore of nuclear physics. A further, albeit qualitative, study of quantum corrections e) has indicated that when a strong three-body force is constrained to be absent, quantum corrections to the energy of the abnormal state are e,~ectively repulsive. As a consequence, in the version of NR, t Supported in part by USERDA Contract No. E(11-1}3001 .

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E . M . NYMAN AND M . RHO

no abnormal state of nuclear matter is expected to be bound at zero pressure . Nevertheless, given enough external pressure (e.g. in heavy-ion collisions) an abnormal state can still be lower in energy than the normal one, and hence relevant to physics under certain circumstances. It must be stressed that no claim can be rigorously made as to which of the two alternatives (and other possibilities that can be thought of) must be realized in nature . This is because we do not know how to obtain a fully satisfactory solution to the fieldtheoretic many-body problem even within the Q-model, let alone the validity thereof. In this situation, it seems best to let experiments ultimately settle the issue ; in the absence thereof, however, we shall be guided by what is established in conventional nuclear physics and assume the absence of large three-body forces in normal nuclear matter to be a necessary condition. The situation in neutron matter dißers from the above in two important aspects : firstly, normal neutron matter is in any case unbound, and secondly, in the central region of a massive neutron star, the matter is indeed subject to a very high pressure from gravitational forces. Thus, an interesting possibility to explore is whether the core of neutron stars might not be in the form of abnormal neutron matter, stable agaïnst gravitational collapse into a black hole. . The possibility of abnormal neutron stars was previously discussed by Kàllman'~ who also used the v-model Lagrangian and the semi-classical approximation. In this approximation, however, neutron matter is self-bound (or nearly so) even without gravitational attraction . This led him to predict a new family of stable neutron stars, made entirely of matter in the abnormal phase. It is the aim of this paper to reexamine this issue in light of the important role of the quantum corrections discussed above which were not considered in the work of Kälhnan. This will be done in an approximate way by taking into consideration only those aspects of vacuum fluctuations that we consider to be qualitatively important. In sect. 2, we obtain simplified forms of the equation of state for both normal and abnormal neutron matter, the latter along the lines discussed extensively in refs . s" s). This section is not seV-contained and should be read in conjunction with these references. The normal-abnormal neutron matter transition is discussed in sectRi, along with possible consequences on the structure of massive neutron stars. Sect. 4 contains a brief discussion of .other possible phases at high density, in particular, pion condensation and quark matter, and some questions raised concerning them. 2. Equadon of state for normal and abnormal neutron matter We shall first determine the equation of state of abnormal neutron matter. This requires some minor modifications in the nuclear matter results of ref. e~ i.e., leaving out the protons ûom the Fermi sea. (Protons will still contribute, however, whenever they occur as proton-antiproton pairs.) In this note, we do not wish to repeat any ofthe rather involved considerations of irreducible diagrams, vacuum propagators,

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49 5

renormalization counter terms, etc, which are, in fact, necessary to go beyond the semi-classical approximation in the abnormal case. Rather, we formulate our results in the simplified language described in ref.'). In the quasi-classical (tree) approximation, the total energy perparticle in abnormal neutron (or nuclear) matter takes the simple form where n is the fermion number density (the symbol p being reserved for the energy density and c the speed of light. The first term is the contribution from the energy density U,~ of the abnormal vacuum and the second term, where kF. is the Fermi momentum, is the average kinetic energy which takes this simple form because in the abnormal case nucleons have no rest mass . The net effect of the quantum corrections calculated in ref. a) or ref. e) is to add more terms, in general density-dependent, to eq. (1). However, as discussed in ref. s), at high nuclear densities one may take this effect into account by simply modifying the value of U~,~ appropriately. [In the simplifying approximations used in ref. 6~ this is, in fact, an exact statement above a certain Fermi momentum.] This procedure is invalidated partly by the Fock energy, which has a stronger density dependence, but this is small numerically, so we shall ignore it here . It is difficult to estimate Uo~ accurately with the inclusion of quantum corrections because of the lack of practicable methods to calculate systematically higher loop graphs in the presence of nuclear or neutron matter . In ref. 6), we made some plausible approximations to simplify the set of equations and arrived at a certain range of possible values for the energy of nuclear matter. This is summarized in fig. 9 of ref. 6). This constitutes an inherent uncertainty in our calculation, but fortunately it is not too serious for the qualitative effect that we are interested in. To be specific, we shall choose a typical set of parameters leading to the curve labelled "a in fig. 9 of ref. e~ which in terms of eq. (1) corresponds to U~,~

500 MeV/frn3.

In the absence of quantum corrections, we would have U,~ = 180 MeV/fm3 corresponding to the result of Lee 2). In the present application, the protons are missing and this modifies U,~ slightly. The corresponding neutron-matter value comes out to be We shall use this as a canonical value in this note. To have an idea how sensitive our results are to this quantity, smaller values (i.e. less repulsion from vacuum fluctuation) will also be considered . An important feature which is missing in eq. (1), and which must be taken into account, is the short-range repulsion between neutrons. We shall use the excludedvolume approximation e), which amounts to evaluating the kinetic term in eq. (1)

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E. M. NYMAN AND M. RHO

using an elevated Fermi momentum, where d is the radius ofthe excluded (hard-core) region and a is a phenomenologically determined paraméter. The interparticle distance is denoted r, such that n -1 = ~;. Clearly, only the combination ad will enter the calculation. It is therefore convenient to rewrite eq. (3) as KF = ~/(1-~/q~ where q now is a phenomenological parameter. The equation of state is obtained from eqs. (1) and (3) by evaluating the pressure

where p = ElV = n(ElA) is the energy density of the matter. Clearly our calculation would make sense only when ~ < q. The equation of state will, however, be unphysical already at densities lower than the value corresponding to kF. = q, since in this simple model the speed ofsound may exceed the speed of light, or equivalently the pressure (P) may exceed the energy density p. The same problem arises in nonrelativistic treatments of normal neutron or nuclear matter . It turns out, however, that the densities at which our equation of state will actually be used are safely below the regiôn where this unphysical behaviour occurs t. Therefore, causality is never violated, although to what extent one can have confidence in such an equation of state near phase transition points is of course hard to assess. Although it is not our primary concern, an equation of state of normal neutron matter is nevertheless needed for our purpose because it enters into the determination ofthe densities between which the normal-abnormal phase transition occurs. We have decided to use the equation of state corresponding to the Reid potential 9), as given by Pandharipande, Fines and Smith t°). Since it is possible that this equation is unrealistically soft, we shall also consider a very hard one, namely the mean-field result of ref. t°). It appears unlikely that the true equation of state is outside of these two extreme cases. The next step is to describe the repulsion in the normal statein terms ofthe excluded volume effects and then determine the repulsive core parameter q therefrom, a more important reason of considering specifically normal equations of state. An essential assumption that we make is that the same value of q determined from a fit to the normal state can be used to describe repulsion in the abnormal phase. To proceed, we also

need an attractive term in the expression for the energy per particle ofnormal neutron matter . We shall simply take it tobe proportional to the fermion number density. This leads to the following expression : E/A = mc s- ~+ ôKF/~

f For instance, for We vacuum energy of eq. (2), causality is violated when kF > 5.6 fm'', while the phase transition occurs at kF x 4.2 fm'' for the Reid equation of state.

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where m is the rest mass ofthe neutron. Note that the core parameter q enters through expression (4) for the enhanced Fermi momentum KF in the kinetic energy term. There.are a number of ways in which the coe6cient D could be determined from nuclear theories or data, one of the simplest being to compute it from the Skyrme interaction (see, e.g., Vautherin t t)). Depending on which one of thé versions used, one would then get a range of values for D. Here we wish to obtain a'quantitative fit to the Reid equation of state and shall therefore search for the value of D within such a range t. At the same time, the value of q is varied such as to reproduce the equation of state in the relevant region of densities and pressures.

Fig. 1 . Equations of state (pressure P versus mass density p/c=) for neutron matter . R refers to the Reid . potential and MF to the mean-field model of ref. '~. Shown are also the fits actually used in this paper, as well as the line P = p below which curves must lie .

As seen in fig. 1, the simple formula (6) reproduces the Reid equation of state with remarkable accuracy if . This gives us some confidence in the manner in which the short-range repulsion is treated. The Reid fit shown in fig. 1 corresponds to D = 250 MeV ~ fm3,

q=6.9fm - ',

(7a)

whereas the mean-field fit uses D = 320 MeV ~ fm3,

q = 4.S fm- t.

t Note that when the kinetic~nergy term is expanded in powers of kF, there will be a repulsive kF term, part of which should be wmbined with the attractive term -Dn before the comparison with other theories is made . f ~ The fit is, in fact, better than one is entitled to expect, as seen from comparison with exact results for the hard-sphere gas at low densities. At higher densities, which are of interest here, there seems to be no problem .

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E . M . NYMAN AND M . RHO

The value q = 6.9 fm t corresponds to ad = 0.28 fm in eq . (3), which is quite consistent with the radius of the strongly repulsive region in the Reid potential. The fit to the mean-field model is less impressive and somewhat unsatisfactory at high densities, for the reason that while the mean-field model automatically satisfies the constraint P < p at high densities the function by which the curve is fitted does not. As a further test of our fits to the equation of state, we have computed the mass versus central-density curves for neutron stars by integrating the TolmanOppenheimer-Volkov equation (see, e.g., ref. ts)). Again, the Reid fit gives excellent agreement and the mean-field fit somewhat less good, but still in reasonable accord with the results of ref. to). 3. Plisse traction in dense neutron star matter To determine the densities at which phase transitions occur, we perform the standard double-tangent construction equivalent to the Maxwell construction of thermodynamics . It is convenient to display the results in a coordinate system where constant pressure corresponds to straight lines. The phase transition takes place along the lowest possible straight line connecting the two phases . As a function of the pressure, the volume per particle thus changes discontinuously, signalling a first-order phase transition.

2000

leoo ~ 12ao

W 0

0

1A 12 1 .4 WA (fm 3 ) Fig. 2. Energy per particle;of normal and abnormal neutron matter versus volume per partick . Shown are the normal and abnormal states for the Reid and the mean-field fits. 1'he dashed lines connecting the curves are the double tangents corresponding to coexistence of the two phases. 0 .2

Ox

OB

00

The results of such a construction are given in fig. 2. With our fit to the Reid potential, the phase transition takes place from a normal fermion number density n = 2.5 particles/fin3 to an abnormal one of 2.9 particles/fm a. This density is some 15 times the normal nuclear matter density. The corresponding mass density p is 7.4x lo ts gJcm 3 which exceeds by a significant amount the maximum possible

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49 9

central density in a stable neutron star [4.1 x lOls g/cm3 for the Reid equation of state 1 °) or slightly higher with our fit to it]. The conclusion remains essentially unmodified also for the stiffequation of state (the mean field fit). Here the transition goes from n = 1.0 fm -3 (p = 3.2 x lOl s g/cm 3) to n = 1.2 fm -3, the p for the maximum stable neutron star in this case being 1 .6 x 101 s 8/cm' . Let us now comment on the sensitivity of this conclusion to the values of the parameters of the theory . As mentioned before, the precise value of U~ is not known, though we do not expect it to be too different from (2). A smaller%value of U,~ would lowerthe transition density and a higher value would increase it. It turns out, however, that the effect is not too drastic for a not too violent variation of U,~ around the canonical value (2): for instance, lowering U,~ to 300 MeV/fm3 merely lowers the _ normal transition density from 2.5 fm s to 2.1 fm a with the Reid fit. Another important parameter of the theory is the "repulsion-core parameter q, which of course may not (and need not) be the same in two phases. We can see roughly how different the parameter can be in different phases by considering the possibility of an abnormal central region in a massive neutron star, say, Her X-1 with its mass 1.3 times the solar mass 1°). In the case of the Reid equation of state, such a star is already quite close to gravitational collapse . Any phase transition necessarily softens further the equation of state and therefore it is unlikely that a significant fraction of the mass is found in the abnormal state. A small abnormal part can, however, be obtained by softening the hard core or raising q in the abnormal state to x 9 fm -1. On the other hand, it would be possible to obtain a significant central region of abnormal neutron matter if the q in the abnormal state were considerably larger, say, q ~ 10 fm -1 with the mean-field equation of state; but it is difficult to imagine that the core parameter can differ in two phases by a factor of two (compare eq. (7b)). Thus most probably neutron stars do not possess a significant portion of the central region consisting of abnormal matter alone. 4. Disarssioo So far, pions have been ignored in the consideration of phase transitions in dense neutron matter . This is not to mean that they play no role at all, but that their influence, expected to be less important than scalar mesons as far as the abnormal state of the Lee-Wick type is concerned, is much harder to take into account systematically in a calculation of this sort. In fact, it is currently believed 13 .1x) that pious will condense (macroscopically) at a density of2 or 3 times that of normal nuclear matter : thus this is perhaps more relevant to neutron stars. However as recent studies show la), the effect of the pion condensation on equation of state is marginal and perhaps even insignificant (this may not be true, however, in the cooling of neutron stars 1 s)). Current ideas on the structure of hadrons suggest that nuclear or neutron matter shôuld, at ultrahigh densities, consist not of elementary nucleons, but of quarks

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E. M. NYMAN AND M. RHO

which ultimately behave as free particles t6) . Asymptotically the equation of state would then be P = 3p. One specific realiTation of these ideas is the MIT bag model ") which enables one to estimate explicitly at what densities the transition to free quark states occurs. Calculations's) indicate that this process may occur at matter densities that are also deep in the region of gravitational collapse. In view ofinherent uncertainties in the models ofhadron structtue, we cannot be too sure that a transition does not take place already at densities commensurate with the central region of massive neutron stars. This intriguing possibility is bound to attract a lot of attention from various domains of physics. An interesting question that arises is : If the transitions to an abnormal state and a quark-bag state do take place, which one comes first? In our picture, should one identify the neutron bag with the excluded volume, then matter would become abnormal before becoming a giant quark bag. Hut the transition points are not too far apart. All this would be an academic issue, were the phase transitions to take place deep in the region ofgravitational collapse. But suppose that this is not so. Then we are faced with this problem: how good is the Q-model at ultrahigh densities? The normal-abnormal phasè transition corresponds to the restoration of chiral _ symmetry spontaneously broken in the normal phase' a . s ) . It may be that this phase transition is symptomatic of unrealistic features of the model at high densities rather than real ones: in fact it is claimed t9) that in gauge models with a neutral current, symmetry breaking increases rather than decreases as baryon density is raised . Even so it is not quite clear to us how relevant this observation is to the issue as to whether or not the Q-model makes sense at high densities. These questions await both experimental and theoretical clarifications . One of us (E.M.N .) is grateful to the Theory Group ofCEN Saclay for the hospitality extended to him while part of this work was being done. References

1) T. D. Lee and G. C. Wick, Phys. Rev. D9 (1974) 2291 2) T. D. Lee,. Rev. Mod. Phys. 47 (1975) 267 3) T. D. Lee and M. Margulies, Phys. Rev. Dll (1975) 1591 4) S. Barshay and G. E. Brown, Phys . Rev. Lett . 34 (1975) 1106 5) E. M. Nyman and M. Rho, Phys . Lett. 60B (1976) 134 ô) E. M. Nyman and M. Rho, Nucl . Phys. A268 (1976) 408 7) C. G. Kitllman, Phys . L;ett . SSI~ .(1975) 178 8) A. Bohr and B. R. Mottelson, Nuclear structure, vol. 1 (Benjamin, NY, 1969) 9) R. V. Reid, Ann. of Phys . SO (1968) 411 10) V. R. Pandharipande, D. Pines and R. A. Smith, Ap . J. 208 (1976) 550 11) D. Vautherin, in Heavy-ion, high-spin states and nuclear structure (IAEA, Vienna 1975) pp. 159-177 12) C . Misner, K. Tltorne and J. Wheeler, Gravitation (Freeman, San Francisco, 1974) 13) A. B. Migdal, Pion field in nuclear medium, to be published 14) G. E. Brown and W. Weise, Phys. Reports 27C (1976) 1 15) J. N. Bahcall and R. A. Wolf, Phys. Rev. 140B (1965) 1445, 1452 ; D. K. Campbell, R. F. Dachen and J. T. Manassah, Phys. Rev. D12 (1975) 979 l6) J. C. Collins and M. J. Perry, Phys. Rev. Lett . 34 (1975) 1353 ; J. R. Ipser, M. H. Kislinger and P. D. Morley, unpublished 17) A. Chador, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys . Rev. D9 (1974) 3471 18) G. Baym and S. A. Chin, Phys. Lett . 62B (1976) 241 ; G. Chapline and M. Nauenberg, Nature 264 (1976) 235 19) D. A. Kinhnits and A. D. Linde, Ann. of Phys . 101 (1976) 195