DENSE NEUTRON MATTER

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DENSE NEUTRON MATTER

Mannque Rho

To cite this version:

Mannque Rho. DENSE NEUTRON MATTER. Journal de Physique Colloques, 1980, 41 (C2), pp.C2-

1-C2-8. �10.1051/jphyscol:1980201�. �jpa-00219793�

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DENSE NEUTRON MATTER *

Abstract.- Neutron star matter with densities exceeding that of the normal nuclear matter (n ^ 0.17/fm3) is discussed in light of the recent developments in under-

standing of the structure of the nucleon in terms of its basic constituents : quarks and gluons. Two different scenarios of phase transition-pion condensation followed by quark matter or percolation followed by quark plasma-are possible depending upon different confinement sizes : the "Little Bag" or the "Big Bag" (M.I.T. b a g ) . Conse- quences on neutron star properties are briefly discussed.

1. Introduction.- The dense neutron matter to be found in neutron stars is a meeting ground for the two different fields of physics : astrophysics and nuclear physics. On the one hand, it plays a crucial role in the theory of collapse of large stars and the possible formation of stable neutron stars, and on the other hand, it provides valuable information on the structure of the nucleon in terms of quarks and gluons. Just what happens when nuclear matter is highly compressed beyond the normal number density n = 0.17/fm (or the matter density p =* 2.8 x

14 3

10 g/cm ) has always been an intriguing question in both nuclear physics and astrophysics, but the issue acquires greater ur- gency as one realizes its profound implica- tion on the fundamental structure of hadrons, some aspects of which are uniquely probed in the unusual conditions that are provided by higher densities, than at higher energies.

Recent studies on the collapse of large stars (say M ~ 5 - 10 M0) strongly suggest that the implosion preceding an explosion proceeds in a surprisingly ordered manner (thus producing low entropy) so that nucleons stay in heavy nuclei all the way to the point at which collapse is halted. The stop-

ping density turns out to depend upon many factors : equation of state, general rela- tivity effects etc. A rough estimate indi- cates that a bounce must occur at a density 3 to 6 times that of nuclear matter . This is clearly an ideal low-temperature dense matter heavy-ion physicists would lo- ve to produce in their laboratories. What happens after the bounce-how the supernovae explosion proceeds-is a subject of in- tensive investigation on which I have little to say ; however it seems reasonable to accept that if neutron stars are formed after explosion, they will have a mass of order of 1 solar mass (M„) and an initial temperature of about 30 MeV (=s3 x i 0l l oK ) which will subsequently get lowered by neu- trino emission. The cooling scenario is found to depend sensitively on wether or not pion condensates are formed and ulti- mately on the basic structure of the nucleon.

In this talk, I would like to discuss the physics of nuclear or neutron matter at densities exceeding that of nuclear matter.

I shall not touch at all on subnuclear densities. It is true that despite the enor- mous effort that has so far been spent, our knowledge of nuclear matter at the normal density is still incomplete : No one 2 has sueeessfully calculated the binding<e-

3E

Invited talk given at the International Colloquium on the Physics of Dense Matter, Paris, France (September 17-22, 1979)

JOURNAL DE PHYSIQUE Colloque C2, supplément au n° 3, Tome 41, mars 1980, page C2-1

Mannque Rho

CEN - SACLAI - Boîte Postale n° 2 - 91190 GIF sur YVETTE - FRANCE

Résumé.- La matière des étoiles à neutrons dont les densités dépassent celle de la matière nucléaire normale (n ^0,l7/fm3) est étudiée à la lumière des développements récents permettant de comprendre la structure du nucléon à partir de ses constituants fondamentaux : quarks et gluons„ Deux scénarios différents : condensation de pions puis formation d'un gaz de quarks ou percolation et formation d'un plasma de quarks sont possibles selon l'importance du confinement : "petit sac" ou "grand sac" (M.I.

T. bag). On discute brièvement certaines conséquences sur les propriétés des étoiles à neutrons.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980201

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c2-2 JOURNAL DE PHYSIQUE

nergy', the saturation density and the com- pressibility modulus of nuclear matter starting from a nucleon-nucleon potential. It may therefore sound absurd to talk on matter densities that are much higher and about which uncertainties are surely greater. One main justification for this-and the hope-is that dramatically new phenomena will take place at higher densities making qualitative- ly significant impact on the structure of

neutron star matter. Furthermore it is not completely absurd to entertain the hope that by understanding higher density regime, one may get to gain an additional insight into what happens at or below nuclear matter density.

There have been much discussion on the observational aspects of the dense neutron matter3 ; some will also be dispussed by M.

Soyeur later in this meeting4. The present situation, both in nuclear matter and in neutron star matter,, may be safely summarized by saying that there are no unequivocal evi- dences for or against those startling new phenomena predicted theoretically and that the issue is wide open.

As the matter is compressed beyond the nor- ma1 density no

"

^0,17/fm3

,

one expects several or many changes to occur in the properties of nuclear or neutron matter. They may be phase transitions of abrupt nature or a system may evolve into another in a smooth and undramatic manner. Both are con- jectured theoretically. The character of transitions is highly model-dependent (not at all surprising in view of the fact that we totally lack a praticable theory of the strong interactions at non-asymptotic regime) and awaifs to be determined by experiments.

what I shali do here is to give may personal view of the present status of and the future directions for the following phenomena :

1) Pion condensation ; 2) Lee-Wick matter ;

3) quark percolation : and 4) quark plasma, Approximate (albert very c0nservative)ranges of densities involved are summarized in Fig.

1. _1.

-

ⁿ

_I-

^Oumk ^Matter

-

5 10

y5

nln,

c Perc - --A Lee

-

^Weh

-

2. Pion condensation5.- Wether or not pion condensation takes place at high densities is expected to depend upon the size of confinement region (e.g. bag) in which quarks are confined. If one takes the M.I.T. view 6 that the bag radius can be as large as l f m or greater, pion condensation at any densi-

-

ty is inconceivable. Rather the scenario proposed by Baym 7

,

i.e., a quark percolation followed by a uniform quark matter, could be much more likely, although it cannot be ruled out that there may be some o- verlap between various outwardly different pictures. On the other hand the alternative view that quarks are confined in a much smaller bag8 could accomodate the phenomenon of pion condensation ; in what follows we shall assume this to be the case.

consider a neutron Fermi sea filled up to

un.

In neutron stars, there are some pro- tons around whose charge is neutralized by electrons ; let the proton chemical potential be

u

Then neutron decay

P

-

can occur spontaneously if the following is satisfied :

un - up

2 ma (2

Since pion is a boson, it will occupy a single mode in a manner familiar from boson condensation. It is a phase transition, most likely of a second order (though there may be one or several first orders lur- king around), the order parameter being the ground state exceptation value of the charged pion field (IT-) or equivalently of a particle-hole in the pion channel. The order parameter implies a spontaneous breakdown of isospin invariance ; thus pion condensation corresponds to an isospin oscillation becoming soft 9

.

In neutron star matter in beta equilibrium, 11,

- ^up

is not far from mT at the nuclear matter density, Thus it might be thought that a higher density (i.e. pn) would suf- fice to induce the condensation. A caveat

~ig.1.- Ranges of critical densities invylved for pion condensation (n) , Lee-tl7ick matter (Lee-wick) , quark percolation (perc) and quark matter.

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to this was quickly realized : as the density is increased, the repulsion in the s-wave n-N interaction operative in the isovector channel increases linearly (at low density) compensating the increase in pn. However as first pointed out by ~ i ~ d a l " and by Sawyer and ~cala~ino", the attractive p-ware a-N interaction also grows as the density increases in such a way as to overcome the net repulsion that is brought in by nuclear interactions. It is instructive to see what this means in conventional nuclear physics.

The attraction at the intermediate range in nucleon-nucleon interaction may be interpre- ted in terms of the diagram 2a ;

Figure 2 : (a) A mechanism for intermediate attrac- tion ; (b) Three-body force which descri- bes Pauli effect of (a) (among others) ;

(c) effect of three-body force in the ground state leading to pion condensation at a higher density.

Now this interaction, if it occurs inside neutron matter, is inhibited for those intermediate states n that are occupied. The effect of this Pauli exclusion principle, which gives rise to a repulsion, can be e- quivalently described by a three-body force Fig.2b. However the latter is more general ;

when fully generalized, it can lead to a , term contributing to the ground state cor- relation energy that represents the zero point fluctuation of the ground state with the quantum numbers of pion. The condensation corresponds to this fluctuation develo- ping instability. Thus as long as the pion degrees of freedom are' relevant, the instability in the pion channel is inevitable.

Theoretical efforts seem to converge to a relatively narrow range of density at which the phase transition is expected :

2no 5 ^nc

I

4n0

where no is the normal matter number density

-

^0,17/fm3.

The observational aspects of pion condensation have been discussed extensively by many people 5r13114. What one expects .?!is that the

equation of state would be softened, the maximum stable neutron star mass would be lowered and the radius reduces, thus rai- sing the central density. Some typical ef- fects5 are shown in Fig. 3.

10

sotmn

Figure 3 : Effect of pion condensation on the pres- sure P(a), and on the maximum stable mass M of a neutron star (b,c) ₀

.

Probaly the most spectacular effect might be seen in the cooling of hot neutron stars. Pion condensates would provide an efficient refrigeration by the process :

- -

n

+

"n-"+ n

+

^e

+

^ve,lowering the temperature by many order of magnitude at the initial stage. It has been recently sugges- ted14 that the failure to observe X-ray emission from "neutron stars" in the young supernovae SN1006, Tycho and Cas A by the Einstein observatory15 may be accounted for by rapid cooling through pion condensates. This is an interesting possibility if one can be sure that neutron stars are indeed present.

A convenient field theory lor studying equations of state is the linear a-model 16

in which fermions (nucleons), a triplet of pions and a scalar meson ( 0 ) are coupled in a chirally symmetric way ; a small chiral symmetry breaking term is added to account for the pion mass m 2.140 MeV. The

71

normal nuclear matter in this Lagrangian is described by

where fT is the pion decay constant, while

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C2-4 JOURNAL DE PHYSlQUE

the pion condensed state is signalled by a non-zero value of a + Now the empirical fact

c'

that chiral symmetry is realized in nature in the Narnbu-Goldstone mode is equivalent to stating that

Thus pion condensation corresponds to making a chiral rotation on a circle with the radius = fv, the rotation angle 0 or rather sin 0 representing the strength of condensates. In as much as the chiral radius is cons- trained to a fixed constant f the possibi-

a'

lity of having a Lee-Wick abnormal state (which has a small or vanishing chiral ra-

2 +2

dius, or oc

+

^ac

--

⁰⁾ is by fiat precluded, This probaly is a right thing to do if the confinement bag turns out to be as small as in the Little Bag (described below).

3. Lee-Wick abnormal matter17.- If the a-model is taken seriously, then in the mean- field approximation, the strongly attractive a-N coupling starts eating up the nucleon mass as the density is increased, and trig- gers a first order phase transition to an abnormal state in which nucleons become ef- fectively massless. The effective mass of the nucleon is essentially proportional to

+2 1/2

( 5 :

+

xc)

,

^i.e., the radius of the chiral

in consequence the results obtained so far cannot be taken at their face values. Howe- ver one can gain a rough, qualitative understanding of the phenomenon by recogni- zing1' that the cause for the precocious onset of the phase transition is an intri- sically large three-body attraction (Fig.

Figure 4 : Three-body a t t r a c t i o n due t o G-exchange whichplays a c r u c i a l r o l e f o r t h e phase

t r a n s i t i o n .

and that such a large many-body attraction is incompatible with the present, rather precise knowldege of the experimental binding energy of the nuclear matter at the normal density. This defect has been reme- died in an approximate way by taking into account quantum f luctuations18 ; the relevant density is now foulid to be strongly pushed up. The results of Ref. 18 are given in Fig. 5 (for two different equations of state at the normal density).

circle and play the role of order parameter

-

^1600-

in this phase transition : meff is disconti-

- 5

^1x10:

nuous in going from meff

= %

^Y940 MeV to (Y

m eff = ^0. In terms of chiral symmetry, the BOO

phenomenon corresponds to the restoration of ^{$ LOO-}

W

chiral symmetry (Wigner-Weyl mode) which is 0

0 0.2 0.4 0.6 0.8 1.0 22 1.4 spontaneously broken in the normal matter _V/A(f_m3

1

(Nambu-Goldstone mode). A simple calculation with the linear a-model with a suitable ad-

justment to take into account repulsive N-N inizeractions shows that the abnormal state can be formed at a density as low as 1.5 times the matter density in symmetric nuclear matter and somewhat higher in neutron star matter (this because of an agditional repulsion possible between pairs of nucleons with T = 1).

The question as to how trustworthy such a simple-minded treatment can be in real systems has been examined18. Unfortunately this cannot be done in fully satisfactory way and

Figure 5 : Energy per p a r t i c l e of normal and abnormal neutron matter versus volume per par- t i c l e . Shown a r e t h e normal and abnormal s t a t e s f o r t h e Reid and t h e mean-field f i t s . The dashed l i n e s connecting t h e curves a r e t h e double tangents corresponding t o coexistence of t h e two phases.

The density at which an abnormal neutron matter is formed ranges roughly from n l/fm3 to n = 3/fm3, the corresponding mass density mostly exceeding the maximum possible density of a stable neutron star.

In view of the recent developments in the renormalizable field theory of the strong interactions (namely quantum chromodyna-

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mics), a serious doubt is cast on the vali- dity of consideration based on the.0-model or any other effective Lagrangians involving nucleons and mesons under conditions extra- polated far from the normal nuclear region.

For instance, in the M.I.T. bag picture 6

,

nucleons and mesons would lose their identi- ties rapidly as the matter is compressed and their degrees of freedom could be more or less meaningless at the density range involved. In the "Little Bag" picture, nucleons and mesons could survive to high densities until the quark phase sets in ; however the likely phenomenon at intermediate densities would be pion condensation rather an abnormal Lee-Wick matter.

Despite the arguments against it, it would be premature to rule out the possibility of a Lee-Wick type phase transition. It is true that no laboratory experiments, e.g. relati- vistic heavy ion collision, have uncovered anything unusual and spectacular at high densities, but one should not forget that those experiments are nct done satisfying a 2 2 the conditions required for producing such a phenomenon. It is also possible that a Lee-Wick type phase transition does take place in QCD ; the o-model may be a phenome- nological approximation of the truth. The transition Goldstone mode

-

Wigner mode is believed to take place in both cases ; only the connections (if any) are not transparent.

In as much as it is a symmetry that is involved, the physics may not be entirely dif- f erent

.

4. Quark percolation.- A. Big Bag ⁶

If quarks are confined in a spherical bag of radius-lfm as the M.I.T. picture implies, then the bags would already be close-packed at the normal nuclear matter density. ~aym' has suggested that at such a density, the bags may have formed an infinite cluster through which quarks can percolate, in a manner analogous to water percolating through coffee beans ; quarks would be deconfined though still localized, forming a color sin- glet. As n exceeds no, quarks would become delocalized making a Mott-like transition and the system would then become a uniform quark matter. One may consider the color electric conductivity to be an order parameter-zero

below and non-zero above a critical percolation density nperc

-

in analogy to the familiar (dc) electrical conductivity 7

.

^If

color is absolutely confined as many theo- rists believe, the order parameter cannot be a measurable quantity so we would never be able to "see" this transition directly even if it took place in nature.

The percolation transition is a universal phenomenon, pervading not only physical systems but also biological systems2'. Gi- ven a chance, things will percolate !

-

and roughly independently of dynamics. If quarks do percolate in nuclear matter, many puzzles are posed : Why is a nucleus what it is ? ; What is the nuclear force

in terms of a network of connected bags with deconfined but localized quarks ? ; what is the exchange of mesons that has been found so successful in describing the nucleon-nucleon potential ?

B.. LittZe Bag ⁸

To reconcile the old (successful) meson- exchanqe picture with the modern view of the strong interactions (QCD), a sugges- tion has been put forward that quarks in the nucleon are confined in a much smaller bag than that of the M.I.T. ; i.e., with a radius roughly one-third fm 8

.

The reaso- ning that goes into this picture is as follows : If chixal SU(2) x SU(2) symmetry

(arising from massless up and down quarks) which is badly broken in the M.I.T. model

is to be restored, one is naturally led to a Goldstone excitation of the vacuum with the pion quantum number (which could be a collective quark-antiquark state). The pion then couples to the quarks on the surface of the bag. The bag is impermeable to the Goldstone excitation but pDrous to the pion quantum number : The Goldstone excitation stops at the surface but the quantum number is carried inside by quark-antiquark pairs.. This surface coupling generates a strong pressure inward, thus squeezing the bag. An effort to derive a Little Bag in a fully consistent way has met with only a partial success so far2' but a workis in progress to improve on this aspect of the problem, We may note that hadrons not coupled to Goldstone excitations e.g. charmed

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C2-6 JOURNAL DE PHYSIQUE

particles are not influenced by this additional pressure ; hence they will be stabili- zed at a large radius.

Consequences of the Little Bag are many :

The meson exchange picture is naturally ac- comodated ; pion emission or absorption can be given a simple description ; and it lea- ves room for pion condensation, Because of the small confinement size, the percolation density will be pushed up to more than 10 no. In a naive interpretation, the "meson cloud" around the bag might pose serious problems in high-energy processes, but we believe that the concept of "pion cloud" is a lot more delicate matter. A model calculation suggests that a compact bag surrounded by a pion cloud (for .the nucleon) can be ef- fectively described as consisting of three

"quasi-quarks" behaving non-relativistically and obeying SU(4) [su(~) if strange quark is considered symmetry]; the quasi-quark car- ries an effective mass and resembles the quasi-particle in Landau's theory of Fermi liquids.

5.. Quark ~ a t t e r ~ ~ . - Whatever may be the scenario at intermdiate densities, there is .

little doubt in the framework of QCD that at asympotic densities, a nuclear or neutron matter will necessarily go over to a quark plasma, a uniform degenerate matter of un- confined quarks with 6 x nf(nf = number of flavors) Fermi seas. This follows from the density dependence of the gluon coupling a c

-

^g2/4s^;

from QCD Lagrangian.. The only thing that can be done at the moment is to calculate the low and high density regimes separate- ly, the former in terms of the nucleon degrees of freedom and the latter in terns of the quark degrees of freedom, extrapola- te them toward each other and then assu- ming a first-order phase transition do a Maxwell construction to find the transition density.

Several people have done this calculation 24'25 evaluating the QCD diagrams to order a

: (e.g. Fig. 6) plus an RPA bubble.

quark

Figure 6 : QCD diagrams t o order a2 f o r t h e quark phase.

2 5 An exemplary result is shown in Fig. 7

.

In the M.I.T. bag model

Figure 7 : Energy per p a r t i c l e vs. baryon d e n s i t y

reflecting the celebrated asymptotic freedom. f o r neutrons with Mean F i e l d (a,) and

unforqunately the precise knowledge of what Reid p o t e n t i a l (b) and f o r quark matter i n QCD ( c ) , Upper and lower curves a r e

happen's at the asymptotic regime is little r e s p e c t i v e l y without and with s t r a n g e

consolation to the woes that one has if one ^quarks^;^A^{= .4} ^GeV

wants to calculate what happens in the real which has the unrealistic feature of a fi- system of neutron star matter, since the xed gluon coupling constant at all densities coupling constant is never so small over the the neutron matter-quark matter transition relevant range of densities and hence pertur-density ranaes from l/fm3 to

--

^{5/fm de-}3 bative caJculations

-

the only method of cal-pending upon the equation of state of nor- culation known.-break down. ~ o b o d y knows how mal density neutron matter. The results in to calculate with any confidence &he nuclear the QCD calculation [with the running cou- equation of state at, say, n , 2no starting pling constant ac(kF)] are quite similar,

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the range of density being roughly the same.

The transition density involved, as in the case of the Lee-Wick matter, appears to be too large to be relevant for a stable quark star ; it exceeds the maximum stable density, One cannot, however, rule out the possibility that the transition is a smooth one

(without going through a discontinuity of some sort) and occurs at a much lower density. Kislinger and ~ o r l e y ~ ~ argue that this is indeed the case with the consequence that the equation of state is much stiffer than imagined just above the normal matter density and no softening occurs until hyperonic matter (e.g. strange quark) appears at about

p c 7 x 10 g/cm3 14 ; the maximum stable neutron star mass would be --, 2.3M and the ra-

8

dius

-

15km both of which are considerably larger than the pion condensation scenario would predict.

6. Conclusion.- We are rapidly realizing that in order to have a realistic picture of what happens in neutron stars, we have to understand how what might turn out to be a correct strong-interaction theory, QCD, works in detail in the regime where strong- coupling and weak-coupling meet. High-ener-

will be different from the standard scenario (including pion condensation).. There is no reason to believe that cooling could be faster. It is thus tantalizing that the surface temperature of neutron stars can say something about the structure of the nucleon.

References.

/1/ H.A. Bethe, G.E. Brown, J. Applegate and J.M. Lattimer, Nucl. Phys. A=, 221 (1979) and references given the- rein

/2/ Any recent reviews on nuclear matter /3/ D. Pines, Invited talk, this Conferen-

ce

/4/ M. Soyeur, Invited talk in this Confe- rence

/5/ For review, see S.O. BBckman and W.

Weise, G. Baym and D. Campbell, and A.

B. Migdal, in M e s o n s in N u c l e i , edited by M. Rho and D.H. Wilkinson (North- Holland, Amsterdam, 1979)

/6/ A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Phys. Rev. 10,2599 (1974);

T. De Grand, R.L. Jaffe, K. Johson and J. Kiskis, Phys.Rev.

12,

2060 (1975) /7/ G. Baym, Physica, Schwinger Festsch-

rift, to be published

gy experiments are silent on this area. Thus /8/ G.E. Brown and M. Rho, Phys. Lett.

82B

the dense neutron matter has a rather unique 177 (1979) ; G.E. Brown, M. Rho and V.

Vento, Phys. Lett.

E,

383 (1979) role in uncovering the contents of QCD that

are unreachable otherwise. Much depend on /9/ G. Baym, ,Phys. Rev. Lett.

30,

1340 (1973)

the size of the confinement bag and how the

bag boundaries disolve when they overlap. ^/lo/ A.B. Migdal, JETP 34, 1.184 (1972) Although there are some compelling arguments /11/ R.F. Sawyer, Phys. Rev. Lett.

29,

³⁸²

27 from high energy data fdr a small bag si- (1972) ; D.J. Scalapino, Phys. Rev.

Lett.

2,

386 (1972) ze of R G 0.3

-

0.4 fm to which I subscribe,

/12/ A recent review is given by G.E. Brown, the issues are highly controversial and the The Nucleon-Nucleon Interaction and interpretation much disputed. the ~uclear ~ a n ~ - ~ o d ~ Problem, NORDITA

-78/4, 1978 The neutron star may provide an answer to

this question : If the Little Bag is a cor- /13/ 0. Maxwell, G.E. Brown, D.K. campbell, R.F. Dashen and J.T. Manassah, Ap. J.

rect picture, the likely scenario would be 216, 77 (1977) pion condensation at n

--

2

-

4 no followed

/14/ 0. Maxwell and M. Soyeur, Pion Conden- at higher density (- 10nJ by quark matter. sation as a possible Explanation for If on the other hand, the bag is large as the absence of Hot stark in Young SU-

pernovae Remrnants, M.I.T. prepint 1979 in the M.I.T. picture, the scenario would be

a percolation which starts at or below nu- /15/ R. Novick e t aZ, ~ull. AM. st. Soc, 11 445 (1979) ; S. Murray e t a l , Bull.

clear matter density and which at higher

%I

Ast. Soc.

11,

462 (1979) densities will go over to a quark plasma.

/16/ See G. Baym and D. Campbell, Ref. 5, We know very little of what is to be obser- for an elementary introduction to the ved of neutron stars in this case, but we subject and a complete list of refe-

rences.

conjecture that the neutron star cooling

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C2-8 JOURNAL DE PHYSIQUE

/17/ T.D. Lee and G.C. Wick, Phys. Rev.

2,

2291 (1974)

/18/ E.M. Nyman and M. Rho, Nucl. Phys.

w,

493 (1977)

/19/ S. Barshay and G.E. Brown, Phys. Rev.

Lett.

34,

1106 (1975)

/20/ P.G. de Gennes, La Recherche

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919 (1976)

/21/ G.E. Brown, J, Delorme, M. Rho and V.

Vento, to be published

/22/ J.H. Jun, E.M. Nyman, M. Rho and V. Ven- to, unpublished

/23/ J.C. Collins and M.J. Perry, Phys. Rev.

Lett. 34, 1353 (1975) ; G. Baym and S.

Chin, P h y s . Lett.

el

241 (1976) ; G.

Chapline and M. Nauenberg, Nature 259, 377 (1976); B. Keister and L. Kisslinger Phys. Lett.

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Rho and G. Ripka (North-Holland, Amster- dam, 1978)

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