High density matter in the universe

NuckarPhysks A328 (1979) 320-351 ©North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

HIGH DENSITY MATTER IN THE UNIVERSE V. CANUTO*

Insdtrrte for Space Studies, Goddard Space Flight Center, NASA, New York, NY 10025, USA B . DATTA**

International Centre for Theonedcal Physics, Trieste, Italy. Received 23 February 1979

Abrtraet: Some aspects concerning the behaviour of matter at high densities are considered. First, it is

shown that high-energy proton-proton scattering can be used, based on the Landau hydrodynamical model, to obtain information about the equation of state . Next, we consider pulsar interiors that have baryonic liquids composed of nucleons and various hyperons . We present a new set of hyperonc potentials based on the one-boson-exchange model, and outline a multi-component, self-consistent many-body treatment for the baryonic liquid . Lastly, we emphasize the role of the spin-2 interaction, arising out of the nucleon-nucleon fmeson exchange, and present a relativistic model for dense matter that includes such interaction .

1. Introduction In this paper we shallpresent a review of the work done in the last fewyears on the subject of high density matter . Re-invigorated by the discovery of the long-sought neutron stars, the field of dense matter has received a great deal of attention reaching its peak in the years 1973-74 . Now that the gross properties of neutron stars are understood, the intensity of research has somewhat subsided and it therefore seems appropriate to take stock and thereupon decide which parts of the subject are not yet fully understood . The state of the art as of 1975 was reviewed by one of the authors in two papers 1.2), where the most reliable equations of state were presented and discussed. The conclusion was that no large disagreement existed among the different researchers, except perhaps on the question of the existence of a solid core. The possibility of a solid core in a neutron star, however, remains physically plausible even though the mechanism needed to realize it seems to be more complicated than a pure two-body repulsive potential. An exhaustive review of the most important physical parameters concerning neutron stars was then presented in 1975 at the International * Also with Dept . of Physics, City College of the City University of New York, New York, NY 10031, USA. ** Present address : Tata Institute of Fundamental Research, Bombay 400005, India. 320

HIGH DENSrrY MATTER IN UNIVERSE

321

School of Physics Enrico Fermi (Varenna), and the reader is referred to that paper 3) for a detailed presentation of the results implied by the equations of state discussed in refs . 1.z). In 1976 a paper by Canuto `) summarized the most recent work on dense matter with consequences for particle theory and cosmology. Finally, at the Eighth Texas Symposium on Relativistic Astrophysics an up-todate review of neutron stars waspresented andthe most reliable equations of state up to about 6 x 1015 g . cm3, and the corresponding limiting mass for a stable neutron star, were discussed 5). It was also the purpose of that paper to point out the discrepancy existing between the results of the relativistic and non-relativistic computations . Keeping in mind the above series of review articles, we decided to present here three topics which were only marginally discussed in those papers, and which constitute, in our opinion, three promising directions of research for the future years. The first concerns the high density proton-proton scattering, the second the physics of hyperons and the third the importance of spin-2 interaction at nuclear and higher than nuclear densities. 2. The pp scattering It is certainly true that neutron stars represent the densest stable objects in the skies, and as such constitute avery good probe to check our understanding of nuclear forces and many-body systems. However, one need not go as far as 3.5 Kpc to find a physical situation characterized by a very high density. In fact, the pp scattering processes, routinely studied in many high energy laboratories, can be viewed as mechanisms whereby a hot blob of dense matter is formed and subsequently dissipated. This picture goes under the name of hydrodynamic model of high energy diffusion, a mechanism first proposed by Fermi 6) to understand cosmic ray phenomena. Improved by Pomeranchuk 7) and later by Landau e), the hydrodynamic model has suffered periods of total neglect when it was erroneously believed that thepredictedmultiplicity of secondary particles versus energy wouldgo like too high a power law of the energies. The reggeon-exchange mechanism, yielding only a log E dependence, seemed to fit the data much better . We shall show that the hydrodynamic model can fit the existing data as well as any other model, with the added advantage that such a fit yields a value for the velocity of sound, and therefore an equation of state p = c;e. The physical mechanism underlying the hydrodynamic model is simple . The two incoming protons are considered not as point particles but as an extended system made of pions. As shown in fig. 1, the two extended systems are Lorentz contracted along the direction of motion by a factor Mc2/E, thus making them appear as two books facing each other. Once they have collided, they form a blob of high density matter, clearly non-degenerate . The reason for the high density is the following. Each proton has, at rest, a volume Vo ^. (fi/mc)3 . When in motion, the volume

322

V. CANUTO AND B. DATA

HIGH ENERGY P-P SCATTERING

p " -'Iff " J c+pr

" 1.5

10~

E L " 103 GeV ; p~ 1e T'u. " P Be +(P+ Π)

4 E L (GeV) (q/cm3) q/Cm3

up. uy

Fig. 1 . High-energy pp scattering.

becomes V

2 _h Mc 2 _4a _ fit 3 ~m nc mc E '

wherewe have contracted one of the lengths by the factor Mc2/E. At the moment of collision, the volume is therefore V= _41r~ fit 3 2Mc 2 3 mac) E ' and so the density p becomes _e_1_E

p c _ c V=1 .5x1014ELg .cm_

a

where the lab energy EL is in GeV. A density much larger than nuclear density is then easy to obtain for experiments on pp scattering can, in fact, easily reach EL 103 GeV. The hadronic matter condensed in volume (2) will subsequently expand under its own pressure. For a full hydrodynamic study see Canuto and Tsiang 9). Once the expansion has taken place, one must follow the time evolution of the relevant thermodynamic quantities like T, E, entropy, etc. This can be done by solving the relativistic Navier-Stokes equations

HIGH DENSTTY MATTER IN UNIVERSE

323

where the energy-momentum tensor T,w is expressed in standard fashion as

Tw = ps v + (P + e)u,~u v + Tp,v = Two,)

+T,v .

Two ingredients are now needed . An equation of state and an expression for the coefficient of viscosity q entering in the viscosity part T, Even if T, were known, it is clear that we would not be able to solve eq . (4). Landau s) gave an exact solution of eq . (4) for the case i .e., when there is no viscosity. This is a key part of the whole problem. In order to be able to use such a model and to compare the results with experimental data, we must be able to write the multiplicity and the transverse momentum (the quantities of experimental interest). When eq . (7) holds, that is when viscosity is absent, the entropy is constant and we can write for the multiplicity N-S=sV=7,V,

(8)

where S and s represent total and specific entropy. Landau's model, based on Fermi's idea, can therefore be applied only from the moment n = 0, when S becomes constant . In his original paper, Landau postulated that 71 = 0 from the very beginning of the expansion (something he half proved and half postulated) so that the multiplicity versus E can be computed at any time during the expansion, using eq. (8). We shall discuss the reliability of this assumption later. For the time being, however, let us remember that what we shall do next is valid only when q = 0, whenever that might occur during the expansion. For an adiabatic process, we have, from the first law of thermodynamics, With we have or, since E - T,

TdS=pdV+dU=O . E=EV

(9)

E =El V- V-('+' ;)

(10)

P=c ;E,

T - V-'2 ,

E - Tcl+`;1i`.

(11)

Substituting (11) into (8), we get

(12)

V . CANUTO AND B . DATTA

324

a formula of general validity, giving the multiplicity as a function of energy and volume . The second important parameter, the transverse momentum (PT), has been computed by Chaichian et a1. 1°). Their result is (PT)

~ 1E 1- '2.

(13)

If the entropy is constant from the very beginning of the expansion, then the volume appearing in eq. (12) can be identified with the volume atthe time of encounter of the two initial blobs, i.e., with eq . (2). We derive then

In table 1 below, we have written out explicitly the values for N and (pr) corresponding to different values of c;. The predictions of eqs. (14) and (15) should be compared with the data presented in figs . 1 and 2 (where the multiplicity N is called n,). Clearly, the first case (Pomeranchuk model) giving N -EL 2 is ruled out. So is the second case, c; = 3 (Fermi model). In fact, N- EL1/4 is still too high a power dependence . The conclusion seems to be that if n = 0 from the very beginning, one needs an equation of state which is stiffer than the free one: P P = se.

(16)

c; > 3.

(17)

that is, we must have the restriction How realistic is Landau's assumption that the viscosity is very small from the very beginning of the expansion? Let us state clearly that we do not possess a reliable value for n either theoretically or experimentally. We can therefore only speculate, indicating in each case the reliability of the computations. The fundamental dependence of 71 or T is unknown, although Feinberg 11) has given reasons to believe that 71 - T3. The constant of proportionality is, however, TABLE 1 Multiplicity and transverse momentum as functions of c ; n=0

E-E,.~-Ei

c:

N

(PT)

0

E E~ const .

con~t. Et const.

1

HIGH DENSITY MATTER IN UNIVERSE

32 5

soot

11 - 1 1 1 1 -1loz

103

104 E L(GeV)

IOS

E L (GeV)

EE cs

77

Ns

v3

0

E v4

Evn

I

0

IgE

IgE

us

Ts

E uz

to£

Ts

E us

E.1n

~P7>

Fig. 2. Transverse momentum and multiplicity versus lab energy for pp scattering.

unknown. The lack of knowledge of this primary parameter can be somewhat circumvented by the following trick . There is a point during the expansion when kT = m"c2, when the initial blob is believed to separate and split into subsystems. This must occur rather late during the expansion. (In fact, the initial temperature is much higher than m,rc 2 .) At that point q can be a function of h, c and m only. Since has dimensions of g - cm-' - sect, ,7 .., m3a C3/f2 . (18) What is the corresponding Reynold's number R ? As we know, R is defined as the ratio bf any of the components of Two) eq . (5), to any of the components of r,,,, If the ratio is large, the non-viscous part dominates . This is how Landau arrived at the conclusion that 71 must be small. We have R

=

EL/ r1c,

(19)

where L is a typical length of the system . Using E V

E i(L

(20)

32 6

V. CANUTO AND B. DATTA

R becomes

Z

E 1 R = _-.

(21)

Substituting for il from eq . (18), we derive R =E/mc2.

(22)

EZ - E~.m. = 2Mc2EL,

(23)

Using the relation we finally derive

/2EL _M

(24)

For EL _ 103 GeV, it then follows that

R -.. 300 .

(25)

We have thus obtained a value for R at one point in the evolution. This number is large, but not as large as we might have hoped for. (The larger the value of R, the better the assumption of Landau.) Besides, we must remember that it refers solely to the final stages, when T - mc 2. At earlier stages, when the temperature is higher, 71 is certainly larger than (18), and so R is smaller than (25) - perhaps by more than a factor of ten. Therefore, it is questionable that Landau's assumption, 71 = 0, is valid all throughout the expansion. 11'12), who has worked on and improved A remedy has been proposed by Feinberg the understanding of the hydrodynamic model. Using an empirical expression of the type +7 - Tk and eq. (11) to eliminate T, we get from eq . (19) R

_El a+cl-k>~ ;~ia+~ ;~ =L(VI

(26)

This expression is still general. Using the expression [see eq . (20)]: V = VOL,

(27)

we now obtain R - (Lll

with L1

*E-t1+(l-k)c ;}i+c ; .

(28) (29)

A large Reynold's number can be attained only if L > Ll. Ll defines the minimum distance at which we can start applying the concept of constant entropy. We have therefore solved the problem of knowing what volume to use in eqs. (12) and (13).

HIGH DENSITY MATTER IN UNIVERSE

327

We must use the general expression (27) with L = L1. This is quite differentfromLandau's approach. Landau took eq . (27) with L = E-' as from (2). Finally, we have N .r Eck-1>ik

(30) (31)

It is important to note that N is independent of the equation of state. Also the expression for N is valid for any value of c;. For this reason, we use eq . (30) to fix k and eq. (31) to fix c;. From the data in figs. 1 and 2, we derive 2k 1

<4,

or, k < 2,

(32)

c : > 3.

(33)

Therefore, To the best of our knowledge, this is the first time that such a clear prediction can be made using experimentally available data. Whether 71 is zero or not, we reach the conclusion that c; >3 [eqs. (17) and (33)]. The fact that this conclusion is independent of the major uncertainty parameter of the problem, namely q, makes the approach more credible . 3. Hyperons The possibility that heavy baryons such as hyperons (E, A) should be present in matter whose density is in excess of nuclear density has been suggested, for energetic -15 ). Heavy pulsars can then be expected to possess reasons, by several authors 13 hadronic liquids composed of neutrons, protons and hyperons . It is therefore imperative that the influence of hyperons on the equation of state for dense matter be investigated . The appearance of hyperons is expected to keep the energy of the system low enough so as to allow for a non-relativistic treatment. The derivation of the equilibrium properties of an assembly of baryons composed of different species requires the knowledge of all the two-body interactions . In addition, the multicomponent many-body aspect of the problem must be treated self-consistently. In the work that has been done so far, the above points have not been treated adequately. For a review, see ref. 2).

0

Va

0

0

0

(_) L+3 158.88f(r)e r f(r) -j +0 .425ß/r +0 .1813/r2

0

146 .98f(r)e r f(r)=I +0.2078/r +0.0432/r 1

-52 .45f(r)é r f(r)-i +0 .3481/r +0 .1212/r2

0

48 .99e r

(_)L+s52 .96r

0

0

-17.48~ r

0

-161 .39f(r)er

-32.99f(r)e r f(r)-0.0298/,.1 +0.0102/r 3

-3.2071f(r) °r

f(r)-0 .1194/r2 +0.0ß26/,-3

f(r)-0 .1727fr +0 .0298/r1

_s, -364.86f(r)e r

f(r)=0.3456/1 +0 .1194/,.1

0

f(r)-0.1727/r +V0.0596/r 2 +0.0102/r 3

r

a

f(r)-0 .1814/r +0.0329/r1 _(_) L+s 144.34 eL", xf(r) r f(r)=0.0329/r1 +0 .012/r3

_(_)L+S 1762.12 Cr"r xf(r)-

0

f(r)-0.1814/r +0 .0658/r1 +0 .012/r 3

L+i72 .17f(r) (_) -Jr" xer

-(-)L+32689.03 . e -Ar" x-

-483 .99~ r 16.5f(r)e

K' -5.5118fm-'

S - 5.7919fm -' _3,

K'(1100, T -1, S)

S(1141, T -0,S)

0

f(r)=0.3456/r +0.2398/r 1 +0.0826/r3

1.6f(r)r

-983.41 er

a,a2 .ß934fm-'

K -2.3485fm-'

Xm4.ß122fm-'

17=2.8731fm-'

Q(570, T=0, S)

K(495, T= }, PJ

X(948, T= 0, PJ

17(566, T -0,P,)

VIS

Vr

V

Vc

TABLE 2

AN potential : T-

f(r)-0 .2258/r +0.102/r +0 .023/r 3

(_)L+s[458.22-r", 144.03 f(r)] ~ r

-r"r (_f+s964 .31e r

K* - 4.4285fm_1

K*(890, T-+J, V)

f(r)-0 .0493/,1 +0.0218/r3

14 .94f(r)e r

f(r)=0.2221/r +0.0493/x1

+3 288 .06 ~r"r xj
(_f

f(r)-0.2258/r +0.051/r2

L'r 6 _(_)L+s697 .33f(r)r f(r)=j +0 .2258/r +0 .051/f L+3 1922 .92 _(_) -989.22f(r) er r"r xf(r)

f(r)-0 .2221/r +0.0986/rs +0 .0218/r3 ew -159.1f(r)r f(r) -f +0 .2221/r +0 .049311'

7.47f(r)] er

[106 .07-

1174.94er

d-4.5025fm_'

eo(887, T-0, V)

rJ

W N

HIGH DENSITY MATTER IN UNIVERSE

329

As a first step towards a realistic description of the many-body baryonic liquid, we present here a set of hyperon-nucleon and hyperon-hyperon two-body potentials. The many-body aspect of the problem is discussed later. The potential is based on a model that assumes that the baryon-baryon interaction is describable, in principle, by the exchange of mesons . The motivation for choosing the one-boson-exchange (OBE) model for the potential comes largely from the relative success such models have met in the nucleon-nucleon case . The lack of adequate experimental data on hyperon-nucleon and hyperon-hyperon scattering prevents the construction of good phenomenological potentials . The details of the OBE model can be found in the works of Brown et al. '6--"'), who presented the AN potential on the basis of fit to A p scattering data. Here we have extended the model to include all pairs of hyperons and nucleons . The mesons considered are octets of pseudo-scalar mesons (or, K, q), vector mesons (p, K*, ¢), scalar mesons (S, K', S), the unitary singlet meson (a-) and the unitary singlet vector meson (w). At very short distances, the interaction is represented by a hard core . The relevant details of the model are summarized in the appendix, where we also list the values ofthe parametersinvolved. For any particular pair of baryons, the two-body potential is written as a sum of central, spin, tensor, spin-orbit and quadratic spin-orbit contributions: V = Ve + VQQ1 -

Q2 + Vz,S12+

VLsL ' S+ VO W12-

We have listed all the two-body potentials in tables 2-9. For the baryon liquid, the determination of the concentration of each species of particles as a function of density requires that the Gibbs free energy be minimized subject to the constraints that the total number of baryons remain constant and the system be electrically neutral. This reduces the problem to one of solving a set of simultaneous equations involving the chemical potential of each species. For a system composed of e-, n, p, A, and E, the equations are Jun = Aé +Jup,

1uE- = $Zn+iaé, JUA = JLn.

uE° _ JUn iUE' = Wn - Ué

The chemical potential is given by p, =

(c 2p 2

+m 2c4)

1/2

+U(P),

where U(p) is the single-particle potential for a particle with ferrai momentum p. U(p) must include the interactions that a particle has with the rest of the particles of

VQ

Vi

Vo

Vc

r

0

0

f(r) °s +1 .4286/r +2 .0409Ir2

- 20.15f(r)-

-6 .72 e r

0

ar - 0.7 fm -'

A(138, T=1, P.)

0

0

0

0

139 .36f(r)e r f(r) = +0 .2078/r +0 .0432/r2

_, 49 .73f(r)e r f(r) = +0 .3481/r +0 .1212/r2

-'

46 .45r

0

X-4.8122 fm

X° (948, T= 0, P.)

16 .58r

0

- n - 2.8731 fm- '

q(566, T =0, Pa

0

0

f(r) =s +0 .4702/r +0 .2211/r2

+s 4 .54f(r)e r

_(_)L+s1 .51e r

0

K -2.1266 fm- '

K(495, T =}, Pà

TABLE 3 EN potential: T=

r

e-o*

-153 .38f(r) er f(r)=0.3456/r +0.1194/r2 - 2.8f(r) er f(r)=0.1194/r2 +0.0826/r3

0

f(r) = 0.3456/r +0 .2388/r2 +0 .0826/r3

1.4 f(r) e-~ r

-926.03

Q- 2.8934 fm -'

v(570, T = 0, S)

-ar

115.17 f(r) e ar r f(r)=0 .0416/r2 +0.017/r3

r f(r)-0.2039/r +0.0416/r2

2045 .36 f(r)e

0

f(r) = 0.2039/r +0 .0832/r2 +0 .017/r3

Car r f(,) Car -57.58 r 4230 .9

8 -4.9036fm-1

8(966, T=1, S)

<

W W O

r

f(r) = 0.1845/r +0.034/r2 -K 'r (_)L+s9.83f(r)

f(r) =0.1726/r +0 .0298/x2

f(r) = 0.0298/r2 +0 .0102/r3

-28.82 f(r) e r f(r) = 0.034/r2 +0.0126/r3

r'r (_)L+S 134.83 f(r)r

_gr -350 .59f(r)e r

-SI

0

f(r)=0.18451r +0.068/x 2 +0.0126/x3

0

f(r)- 0.1726/r +0 .0596/r2 +0 .0102/x3

14 .41 f(r) e

-493.2e_F r

K'- 5.421 fm-'

S -5.7919 fm-' r. . ( -)L+s231 .17 ér r K'r _(_)L+s4 .91 f(r)

K'(1100, T=J, S)

S(1141, T-0, S)

w

13 .05 f(r) e-

f(r) = 0.0493/r2 +0.0218/r3

f(r) = 0.2575/r +0.0663/rz eor -288 .92 f(r) f(r) = 0.0663/r2 +0.0342/r3

f(r) - 0.2221/r +0.0493/r2

f(r)=1 r +0 .2221/r +0 .0493/x2 -.,r -926 .87 f(r) er

- 148 .7f(r) e

783.04 f(r) e 7 r f(r) = I +0 .2575/r +0 .0663/r2 e -Pr 2567 .94 f(r) r

r

f(r)=0.2221/r +0 .0986/r2 +0 .0218/r3

6.53f(r)]

r

e=*"

f(r) = 0.2575/r +0 .1326/r2 +0 .0342/r3

144.46 f(r)]

[99.13-

-[522.03e-°r

1155 .08

m -4.5025 fm-'

p - 3.8832 f3n-' éw -1279.22r

m(887, T=0, V)

(cont.)

p(765, T=1, V)

TABLE 3

f(r) -j +0.2317/r +0.0537/r2 _(_)L+s 184 .58 ear*r x f(r) r f(r) = 0.2317/r +0.0537/r2 -(_ )L+s28 .0 K"r x f(r) r f(r)=0.0537/r2 +0 .0248/r3

f(r)=0 .2317/r +0.1074/r2 +0.0248/r3 (_)L+a 17 .56f(r)e

_K*r _ ( _ )L+s 106.83 e r _(_)c+s[11 .71r*r 14 .0f(r)le

K* - 4.315 fm_'

K*(890, T-J, V)

w w

0

0

VQ

0

49 .73 f(r) r !(r)=3 +0 .3481/r +0 .1212/r 2

10.08 f(r) r f(r)= +1 .4286/r +2.0409/r2

VT

16 .58r

VQ

0

C-11

.. 3 .36r

0

0

71-2.8731 fm-1

or - 0.7fm -1

Vc

n(566, T = 0, PJ

or(138, T =1, PJ X - 4 .8122fm- '

x o(948, T - 0, PJ

lC.

0

0

0

(_)L+s9.08 f(r) é r f(r)=i +0 .4702/r +0 .2221/r2

( - )`+s 3 .03 ~

0

K - 2.1266fm- '

K(495, T = J, P.)

0

139.36 f(r)r f(r) +0 .2078/r +0 .0432/r 2

C- .w 46 .45r

0

4

EN potential : T a

TABLE

r

r f(r)=0 .0416/,.2 +0 .017/,.3

-57.58 f(r) e

-2.8 f(r) e- ~ r f(r)=0 .1194/r 2 +0.0826/r3

-sr

- 1022 .68f(r)r f(r)=0 .2039/r +0 .0416/r2

0

f(r)=0 .2039/r +0.0832/r 2 +0.017/,.3

28.79f(r) e

~ a. -2115 .45r

8 - 4.9036fm - '

8(966, T -1, S)

- 153 .38f(r)-~ r f(r)=0 .3456/r +0.1194/r2

0

f(r)=0.3456/r +0 .2388/r2 +0.0826/r3

1 .4f(r) e " r

-926 .03r

v - 2 .8934fm- '

0(570, T = 0, S)

1

bd

W W N

_s, Car

r

f(r) - 0.0298/r2 +0.0102/x3

-28.82 f(r) e

f(r) =0.1726/r +0.0298/r 2

-350.59f(r)

0

f(r)-0.1726/r +0.0596/r 2 +0.0102/r3

14 .41f(r) esr

~rr f(r)f(r) - 0.1845/r +0.034/? 19 .66x er'r Ar) f(r) - 0.034/r2 +0.0126/r3

-(_)L+s 269.67x

0

f(r)=0 .1845/r +0.068/,.2 +0.0126/r 3

( _)L+s9.83 f(r)eeK~r

e_r" e-o r

_,

f(r) - 0.0537/r2 +0.0248/,.3 f(r)=0.0493/r2 +0.0218/r 3 f(r) = 0.0663/r2 +0.0342/r 3

f(r) - 0.2317/r +0.0537/x2 r"r (_)L+s56.0 f(r) Ç f(r) = 0.2221/r +0.(493/i 13.05 f(r) e-r

r

f(r) =0.2575/r +0.0663/,.2 ` 144.46f(r)ew

f(r)= +0.2317/r +0.0537/r2 (_) L+s 369.17f(r)

( -)L+s[23 .41e 29.01 f(r)] r f(r)=0 .2317/r +0.1074/? +0.0248/r 3 _(_)L+x35 .12 f(r)

-926.87f(r)e r

f(r)=3 r +0.2221/r +0.0493/r 2

-148.7 f(r)e

6.53 f(r)]2 r f(r)=0.2221/r +0.0986/r2 +0.0218/r3

[99.13-

1155 .08 e~

-1283 .97f(r)e r

f(r) -I r +0.2575/r +0.0663/,.2

-391 .52f(r) e

[261.01f(r)Je-pr 72.23 r f(r)-0.2575/r +0.1326/r2 +0.0342/r3

639 .61

K' -4.315 fin-1 , r"r (_)L+s213 .66 r

w-4 .5025 fm-1

p-3 .8832 fm-1

K'- 5.421 fm-1

S -5.7919 fm-1 _a -493.2e~ _( -)L+s462.34

K'(890, T=}V)

w(587, T-0, V)

p(765, T-1, V)

K'(1100, T = J, S)

(cont.)

S(1141, T=0, S)

TABLE 4

w w w

7d

xy

r

-r

0

0

Va

88.35 f(r) 'w r f(r)=3+0.3481/r +0 .1212/r2

29 .45

0

Vz .s

VT

Vo

Vc

T = 0, Pj

n - 2.8731 fm - 1

n (566, T = 0, PO

0

0

123 .78f(r) e-"' r f(r)=3+0 .2078/r +0 .0432/r 2

41 .26 en r

0

X - 4 .8122 fm-1

x o (948,

-o

f(r)=0.1194/x2 +0 .0826/r 3

- 134.7f(r)e r f(r)=0 .3456/r +0 .1194/r 2 -or -2 .27 f(r) e r

0

e

r

f(r)=0 .1727/r +0.0298/r2 -Sr -7 .8 f(r) e r f(r)=0 .0298/r2 +0 .0102/r 3

-103 .58f(r)

0 -Sr

3 .9f(r)er

1 .14f(r) e-r r f(r)=0 .1727/r +0 .0596/r2 +0 .0102/r3

~sr -171 .94r

~ur -997 .22r

f(r)=0.3456/r +0.2388/r2 +0.0826/r3

S - 5 .7919fm-1

S(1141, T =0, S)

0 - 2 .8934 fin'

0(570, T =0, S)

M potential : T=O

TABLE 5

e~

f(r) - }+0.2221/r +0.0493/x2 -825 .1f(r)r f(r)=0 .2221/r +0.0493/r 2 -... 10 .6 f(r) r 2 f(r)=0 .0493/r +0 .0218/r3

-133 .98 f(r) e~

f(r)=0.2221/r +0 .0986/x2 +0 .0218/x3

[89.22 - 5 .3 f(r)l

1120.17r

a - 4 .5025fm-1

m(887, T =0, V)

â

â

C

W W A

0

0

0

vu

77 .17f(r)e!~ r f(r)-}+0 .3481/r +0.1212/r 3

-12.51f(r)er f(r)- }+1 . 4286/r +2.0409/r3

0

25 .72-r

0

-4.17e r

0

17 m2.8731 fm-'

a" m0.7 fm-1

Vu

VT

Vc

rr(566,T-0,P.)

a"(138,T-1,P,)

0

0

108.13t(41 r f(r)-}+0.2078/r +0.0432/r3

36.04r

0

X-4 .8122fm-'

X(948,T-0,P.)

f(r)-0 .1194/r2 +0.0826/r3

f(r)-0.0416/r3 +0 .017/r3

2237.85f(r)f(r)=0.2039/r +0.0416/r -h 99.89f(r)~ r

-118.17f(r)-!_ f(r) - 0.3456/rr +0 .1194/r3 -T -1 .73f(r)é r

0

f(r)=0 .2039/r +0.08321r2 +0.017/r 3

r

0

f(r)=0.3456/r +0.2388/, 2 10 .0826/r3

-49 .44f(r)

-1005.75e7 r 0 .87f(r) er

-a 6266 .86~ r -a,

8-4.9036fm -'

8(966,T=1,S)

o-2.8934 fm-'

0(570,T=0,S)

TABLE 6 2S paendal: T=0

f(r)-0.1727/r +0.0298/r2 _~ -5 .95f(r)~ r f(r)=0 .0298/,3 +0 .0102/r3

-92.21 tir) -

0

f(r)-0.1727/r +0 .0596/r3 +O.0102/,3

-178 .56e _ry .98f(r) ~ r

-r.

Su5.7919 fm-'

S(1141,T-0,S)

f(r)-0.2575/r +0 .0663/Î3 -P - 133.7f(r)er f(r)-0 .0663/r3 +0.0342/x3

2098 .06f(r)-

571.3f(41 r f(r)-} +0.2575/r +0.0663/r3

f(r)-0.2575/r +0 .1326/rz +0 .0342/r3

66.83f(r)J~ e r

-[380.87-

-r -1758.78~ r

pm3.8832 fm -'

p(763,T-1,V)

8 .09f(r)° r f(r)-0 .0493/r3 +0.0218/r 3

f(r) -0.2221/rr +0.0493/r2

-718.43f(r)-

-117.04f(r)°r f(r)=1+0 .2221/r +0.0493/r3

f(r)-0.2221/r +0.986/x3 +0.0218/r3

4.04f(r)] er

[78.03-

1085 .03r

a -4.5025 Im -'

to(887,T-0,V)

w w

y Crl



0~

cö 0 1118 .92f(r)~ r f(r)-0 .2039/r +0.0416/r'

0 - 118 .17f(r) r f(r)-0 .3456/r +0 .1194/r2

f(r)-0.2039/r +0 .0832/r2 +0 .017/r'l

- 24.97f(r) 6

0.87f(r) e r f(r)-0.3456/r +0 .2388/r= +0 .0826/r3

3133.43r

-1005.75r

8-4.9036ßn-1

0

f(r)-0.1727/r +0.0596/r3 +0 .0102/r3

2 .98f(r) e r

-178.56r

S - 5 .7919[m-1

-I

f(r)-0 .2575/r +0 .1326/r+0 .0342/,3 0r 285 .65 f(r) r Ar) +0 .2575/r +0 .0663/t

33 .42f(r)]r

-[190 .43-

-879.39r

p - 3 .8832tm - '

p(765, T -1, V)

Vu

0

0

0

- 66 .85f(r)! r - 0.0663/r2 +0 .0342/r 3

f(r)-1 +0.2078/r +0.0432/r2

108. 13 f(r)

36.04r

0

c - 2.8934[m-1

S(1141, T - 0, S)

49.94f(r)e -1 .73f(r) e-5 .95f(r)e r r r f(r) 0 0 0 f(r)=0 .1194/r2 f(r) 0 .0416/rz - 0.0298/,3 va +0.0826/r 3 +0.017/r3 +0 .0102/r3

f(r) -j +0.3481/r +0.1212/r 2

77 .17 f(r)

25 .72r

0

X-4.8122tm-1

8(966, T -1, S)

1049 .03f(r)r f(r)-0 .2575/r +0 .0663/x'

-6 .25f(r)Q! ~ r f(r) -j +1 .4286/r +2.0409/.3

-2 .08r

0

,i - 2.8731tm-1

1r-0.7tm -1

X°(948, T = 0, P,) c(570, T - 0, S)

- 92 .21f(r)~ r f(r)=0.1727/r +0 .0298/r2

V.

Vc

n (566, T - 0, Pj

w(138, T -1, P,)

TwsIB 7 XX potential: T-1

8.09f(r) °r - 0.0493/r 2 +0.0218/r 3

-718 .43f(r)r f(r)=0.2221/r +0.0493/r2

f(r) -i +0.2221/r +0.0493/r

-117 .04f(P) e-

f(r)-0 .2221/r +0.0986/r +0.0218/r3

e 4.04f(r)]r

[78.03-

1085 .03r

w-4.5025tm-1

m(887, T - 0, V)

py G

W W O~

TABLE

8 T-2

Va

VT

0

0

0

0

t

f(r) -i +0.3481/r +0.1212/x3

77.17 f(r)

f(r) -k +1 .4286/r +2.0409/r 3

6.25 f(r) _W

-Ir

Vo

e

â.. 2.08r

é. 25.72r

0

0

V_

x.

C

0

0

f(r)-1 +0 .2078/r +0 .0432/r3

108 .13 f(r) -

é 36 .04x

0

C

0

j(r)=0.1727/r +00596/r3 . +0.0102/r3

s

f(r)=0.2039/r +0.0416/r 3 -49 .94f(r) a r f(r)-0 .0416/r 3 +0.017/

- 1 .73f(r)r j(r)-0.1194/r3 +0 .0826/x3

-5 .95f(r) a r j(r)=0.0298/r 3 +0.0102/x3

j(r) - 0.1727/r +0.0298/r3

é~ e8 -1118 .92 f(r)~ -92 .21j(r)

0

f(r)=0.2039/r +00832/r . 3 +0.017/r3

j(r) - 0.3456/r +0.1194/f

~ -118 .17f(r) e-

0

f(r)-0.3456/r +02388/r3 . +0.0826/r 3

-100575. a-e : 0.87f(r)r

S-5 .7919fm -1

S(1141, T=O, S)

t a -3133 43-178 56a -a r . t 24.97f(r)-2.98f(r)x x

d-4.9036fm-'

0-2.8934fm-'

+1-2.8731fm-'

,r-0.7fm-' X-4.8122fm-'

8(966, T-1 . S)

potentiel:

w(138, T-1 . PJ iy(566, T-0, P.) xo (948, T-0. PJ o(570, T-0, S)

F.E

C

66.85 f(r)r f(r)-0.0663/r3 +0.0342/x3

r

C~

j(r)=0.2575/r +0.0663/r2

-1049.03 f(r)

f(r) -i +0.2575/r +0.0663/x3

-285 .65 j(r) ~

f(r) - 0.2575/r +01326/3 . +0.0342/r3

t 33.42 j(r)] ~ r

I190 .43-

e 879 03-

p-3.8832fm- '

p(765, T-1, V)

e~

e

!(r)-0.2221/r +0.0493/r3 a 8.09f(r)r f(r)-0.0493/ r2 +0.0218/x3

-718.43f(r)

f(r)=I +0 .2221/r +0 .0493/r2

-117.04 f(r)

f(r)-0.2221/r r +00986/3 . +0.0218/r3

e1085 03r [78 .034 .04 j(r)]tr

u-4 .5025fm -1

m(887, T-0, V)

w w v

tri

z y

C7

x

71 & 2.8731 fm -' 0

-27.53!:! r

-92.57f(r)L7

f(r)=I +0.3481/r +0.1212/r 2

ir & 0.5744 fm -'

0

(-)L+s1 .ler

(-)L+53 .3 f(r)!_~

f(r) -i +1 .741/r +3.0311/r 2

0

0

Vc

V

VT

Vu

Va

0

0

17(566,T-0.P.)

r(138,T-1.P.)

0

0

115.69f(r) e ~ r f(r) -# +0 .2078/r +0 .0432/r2

38.56e r

0

x &4.8122 fm -'

X°(948,T-0.P.)

f(r)-0.1194/r' +0 .0826/r3

-1 .99f(r)!r

f(r) -0.3456/r +0 .1194/r2

-126.47f(P)e-r

0

f(r)=0.3456/r +0 .2388/r2 +0 .0826/

0.99f(r) !r

-1001.48l r

a&2.8934 fm-'

0(570,T-0,S) S &5 .7919fm''

S(1141,T-0.S)

_(_)L+53 .45 -a xf(r)e r f(r)-0.0419/P2 +0.0172/r3

_(_)L+572.38 -a xf(r)e r f(r) - 0.2046/r +0.0419/r2

0

f(r)-0 .2046/r +0.0838/r2 +0.0172/r3

f(r)-0.0298/r' +0 .0102/r3

f(r) - 0.1727/r +0 .0298/r' a 6.81 f(r)! r

_i, 98 .02f(r)e r

0

f(r)-0 .1727/r +0.0596/r' +0.0102/r 3

-(-) L+5 189.72e 175.22e r r L+a1 .73f(r)ear -3 (_) .41f(r) ! r r

8 &4.8864 fm -'

8(966,T-1 .S)

TABLE 9 AI potential : T- 3

0

0

0

0

0

p&3.8616 fm -'

p(765,T-1, V)

f(r) -0.0493/r' +0.0218/r3

9.26 f(r) !r

f(r)-0.2221/r +0 .0493/r2

f(r) -j +0 .2221/r +0 .0493/r2 C-770 .47 f(r)-

-125 .23 f(r) ! r

f(r)-0.2221/r +0 .0936/r2 +0 .0218/r3

4.63 f(r)3 !-

[83.49-

1102.28lr

&&4.5025 fm -'

&(887,T-0.V)

HIGH DENSITY MATTER IN UNIVERSE

339

the same species as well as of all the other species. It is given by U(P ')=(Z

L Jo ' P1 pi{K(P IPl) - eich .}+ '~t Jo ' PidPJ(PlIPl)]~ i

where K is the (Brueckner) reaction matrix and nk the concentration of particles of kth species. The presence of several different species of particles makes a self-consistent treatment of the problem rather difficult. In view of this, several different many-body techniques have been devised, and it is not clear which one is the most reliable approach . To further complicate the problem, it has been suggested 19) that the hyperons will be characterized by a shift in their rest masses because of the dense surrounding medium . None of the previous computations have included such 2) . dispersion effects. For a comprehensive review, see ref. To sum up, we have stressed the need for using more representative hyperonic interactions and a reliable many-body method to describe the hyperonic liquid inside neutron stars. The equilibrium calculations using the new hyperonic potentials presented earlier and employing a fully self-consistent theory is presently under investigation by us. 4. The spin-2 interaction When the density of matter exceeds nuclear density by an order of magnitude (or more), conventional non-relativistic descriptions that make use of the concept of a static potential are expected to be unreliable . For a more realistic description, one must employ a relativistic treatment, and, in addition, include short-range interactions characteristic of such high-density regions. It is known that at nuclear densities the, scalar and vector interactions play the dominant role . Based on such ideas, several relativistic models for dense matter have 2&--22) . been proposed in recent years Although these models, in principle, constitute an improvement over the conventional non-relativistic theories, their usefulness is restricted for the following reason . As one goes up the density scale and deviates considerably from the region of nuclear density, attractive interactions, arising from the exchange of spin-2 f° mesons (mass =1260 MeV) among nucleons, are expected to be important. In this section, we consider such short-range interactions, and present the results of a relativistic many-body calculation for high density neutron matter in which, for the first time, forces arising from the exchange of spin-2 mesons are included, in addition to the traditional scalar (spin-0) and vector (spin-1) 23,24 interactions. In what follows we shall present only y the main equations ). The equations of motion that determine the behaviour of a gas of fermions ('P) coupled via scalar (o-), vector (A) and spin-2 (gr..) mesons are (h =1= c) :

340

V. CANUTO AND B. DATTA

( -a2+m .,)a-=ga #», h "" +a"( -I---gF,,h A h o")= go~-g~e âya4, R,+mi (~g" '

16mf2

= m(t",. -ig"Yt~h°e) .

(35) (36) (37)

In writing these equations, we have used a formal analogy with gravitation (since the spin-2 field couples to external fields in the same way as gravitation does). The mass term in eq. (37) has been constructed in such a way that when sources are absent, the equation reduces to the Pauli-Fierz equation for a free massive spin-2 field 23) . The source term t" is the stress tensor corresponding to all the fields except the free spin-2 field, and is given by ~d,~.y

al

l M, - ig..A,)* +a"cra,a

- im,,g,,YAah"OAa+F,Fp h'e-ig,~h"ha°FA,,

(38)

The quantities ea and d,,a are vierbein fields, given by (N,, a = 0, 1, 2, 3) e,,We Yb

nab = e"aeb = h V

d,db,Îab = d' Ld,,.

"Y

=

_

g /L 1Y,

g,,,,

where 71. b = diag . (-1,1,1,1). In order to determine the equation of state, we shall evaluate the conserved total stress tensor of the system, given by T', = 0», +-J-g h"ßtgY

(39)

(9v is the stress tensor of the free spin-2 field) . Eq. (34) and its hermitian conjugate give (40)

We shall then identify the average value of the quantity ~g¢eâya~/r with the fermion number density n . The averaging is done by expanding the fermion field in terms of creation and annihilation operators and using the prescription of normal ordering. In principle, one can determine the functional dependence of the meson fields on 41, and obtain T; as a function of n. Since the diagonal components of T" Y give the energy density (e) and pressure (P) of the system (assuming the system to be a perfect

HIGH DENSITY MATTER IN UNIVERSE

341

fluid), this will eventually yield an equation of state in parameec form

e =s(n),

(41)

P = P(n ).

(42)

However, the nature of the coupled equations makes them very difficult to handle . On the other hand, one can make considerable progress by resorting to the Hartree approximation. This consists in assuming that at high densities, the fermions "see" only an average, effective cloud of mesons . In other words, when nucleons are very dense, the source terms in the meson field equations will be large, and therefore, the quantum fluctuations about the expectation values of the meson fields can be expected to be small. Thus, in accordance with this approximation, we shall take the meson fields to be c -numbers, and of the following form Q

o A" -> &>o,

wo,

1+A

g""- (

1+A

(43) 1+A

where (ro, Ao, X and A are space-time dependent. With this assumption, we can neglect all terms that involve derivatives of the meson fields. Eqs. (34)-(37) then reduce to a set of coupled algebraic (instead of partial differential) equations. These are solved (numerically), taking the number density n as the independent parameter of the theory. Once this is done, it is straightforward to evaluate the equation of state. The expression for pressure and energy density in the Hartree approximation are P

32[MNF(X,

s=-

2 32f

A) - z~H(X, A)~ô+2

A)+4~Gi(X_

z mN F(X, A)+

48,~

2 2 m (~ 1+X),

Gz(X, A)+im~(X, A)~ô+2

m(

2 z

).

(45)

Here y stands for the spin degeneracy factor for the fermions . The other quantities are as follows:

hT(X, A) _ (1 +X)1(1 + AA 0 Gz(X,A) = 1 J~d3k(k*z+mr*,z)~, 0

342

V. CANUTO AND B . DATTA kF

= (6WZn/y)Y,

(1+X)k*2=(1+A)k2,

mN2 =(1+A)1(MN-g~o)2 .

(1+,y)3

For purpose of actual computation of the equation of state, we shall identify the scalar, vector and spin-2 mesons with (r (700 MeV), m (784 MeV) and f° (1260 MeV) 26). The coupling constants for scalar and vector mesons are 2'): g!'/4v =13 .9 and g2./41r =10.0. For f°-meson coupling constant, three experimen2') tal values are cited in ref. : f2 =2.91, 6.55 and 7.44. The pressure (P) and the energy per particle (e/n) of neutron matter as functions of neutron number density are presented in figs . 3 and 4. The most noticeable feature of this equation of stateis that beyond a critical density (which depends on the values of the coupling constants and masses of the mesons), the pressure becomes a monotonically decreasing function of the neutron number density. A comparison with other relevant equations of state is presented in fig. 5. In table 10 we have given the numerical values of P, e/n and p = e/c2 versus n for two values of the spin-2 coupling constant . Turning next to astrophysical application, we show, in fig. 6, the results for the gravitational mass of a stable neutron star that the various equations of state yield.

24.0 20.0

v

N+Q+w

16.0 12 .0 8.0

(P/mw) X 10 3

/ \\

\ /

(b)

f2 -2 .91

4.0

~r

0.0

c

-4.0

f2 " 10.0

-8.0 -12 .0

W~

f2-7.44,i~~~

n(f,n 3)

1 \

1

1

~~~~

\ \\

\ \

\

1 10 3

Fig. 3 . Pressure of neutron matter (in dimensionless units) versus neutron number density. The solid curve corresponds to the case where spin-2 mesons are absent .

343

HIGH DENSITY MATTER IN UNIVERSE 3.0 2.8 2.6 2.4 2 .2 2 .0 1 .8

_fz=2 .91

1.6 1 .4

'

i

(C) f2=6 .55 ' __ ____--_-__,____ ____fz=7aa z_==_________________-__ = - f2 =10

1 .2 1 .0 0.8 0.6 0.4

0.2

10 -2

10 3

10",

n(fm -3 )

Fig . 4 . Energy per particle (in units of MNC 2 ) versus neutron number density . The solid curve corresponds to the case where spin-2 mesons are absent .

Equation of state for neutron matter f2

n

P

f= 6.55

= 2 .91 s/n

(fm-3)

(g . Cm 3 )

(GeV)

7.295 6.590 5 .931 5 .318 4 .749 4 .222 3 .735 3 .287 2 .877 2 .502 2 .161 1 .853 1 .576 1 .327 1 .107 0.912 0 .741

1 .837 (16) 1 .647 (16) 1 .471 (16) 1 .307 (16) 1 .155 (16) 1 .014 (16) 8 .849 (15) 7 .663 (15) 6 .583 (15) 5 .604 (15) 4 .724 (15) 3 .937 (15) 3 .242 (15) 2 .635 (15) 2 .111(15) 1 .667 (15) 1 .297 (15)

1 .412 1 .402 1 .391 1 .378 1 .364 1 .348 1 .329 1 .308 1 .283 1 .256 1 .226 1 .192 1 .154 1 .113 1 .070 1 .025 0 .981

P

(dyn .

Cm 2 )

1 .076 (36) 1 .068 (36) 1 .057 (36) 1 .040 (36) 1 .017 (36) 9.890 (35) 9.545 (35) 9.132 (35) 8.648 (35) 8 .089 (35) 7 .452 (35) 6 .738 (35) 5 .949 (35) 5 .095 (35) 4 .193 (35) 3 .271 (35) 2 .367 (35)

' Entries in parentheses represent powers of ten .

(g . Cm 3)

P

e/n (GeV)

P (dyn . Cm 2)

1 .317 (16) 1 .187 (16) 1 .065 (16) 9 .519 (15) 8 .466 (15) 7.492 (15) 6.592 (15) 5.764 (15) 5 .006 (15) 4.314 (15) 3.688 (15) 3.124 (15) 2 .620 (15) 2 .172 (15) 1 .780 (15) 1 .441 (15) 1 .150 (15)

1 .012 1 .010 1 .007 1 .004 1 .000 0 .995 0 .990 0 .983 0 .976 0 .967 0.957 0 .946 0.932 0.918 0.902 0.886 0.870

2 .442 (35) 2.572 (35) 2 .679 (35) 2 .762 (35) 2 .817 (35) 2 .845 (35) 2 .844 (35) 2 .810 (35) 2 .742 (35) 2 .638 (35) 2 .495 (35) 2 .312 (35) 2 .088 (35) 1 .822 (35) 1 .519 (35) 1 .186 (35) 8.367 (34)

log p(g cnn3) Fig . S . Pressure versus mass density for neutron matter. For curves A-O, see ref . s) . Curves (b) and (c) as from the present theory.

The two insets on the right hand side of fig. 6 represent observational results due to Joss and Rappaport 28) and Avni Z9) ; the ranges of masses given by these authors are respectively (1.4-1 .84) ME) and (1-2.3) Me, where Me is the solar mass. The curves A-G refer to non-relativistic computations [for details see ref. s )] . Curve L is due to Pandharipande and Smith 3°) who included pion tensor interaction in a nonrelativistic way. Finally, curves O and N are results corresponding to equations of state of Bowers et al. 22) and Walecka st) . A general review of these computations can be found in ref. s). In all these computations, non-relativistic as well as relativistic, it is assumed that the high density regime is dominated by the exchange of vector

345

HIGH DENSITY MATTER IN UNIVERSE Y Avni Vela X-1

SMC X-1

f- 2.7

C on X-3 Her X-1

-4-24

Cygnus X-1 I

2

3

4

5

Joss - Rappoport - Vela X-1

1.

SMC X-1

K:f

Cen X-3 Her X-[

-

I Both@-Johnson Pandhoripaudo Moszkosrski Arponen Canuto-Chltre

PSR 1913+16 1 .8

"?-

2

3

4

5

C,D A,8 (Y) E F G Present Work Case ( b )

Fig. 6 . Neutron star gravitational mass (in units of solar mass) versus central density . For curves A-0, see ref. s) . The horizontal scales in the two insets refer to masses of the pulsars in units of the solar mass .

mesons, which produce repulsion among nucleons. Fig. 6 indicates that when (attractive) spin-2 interactions are included, the maximum mass for a neutron star turns out to be 1 .75 MO - a value significantly lower than the predictions of the previous relativistic calculations . We also see that our result bunches together with the results of the models A-G. This indicates that our model, by including spin-2 interactions in a relativistic way, tends to narrow the divergence between the sets of results A-G and L, N, O. This closeness, however, does not necessarilyimply that the non-relativistic descriptions are adequate . What is happening is the result of competition between two important physical aspects of our theory, namely (i) relativistic effects and (ii) strong, attractive NN interaction, arising from the exchange of f° mesons .

346

V. CANUTO AND H . DATTA

S. Disc"on In this paper we have dealt with three promising directions of research concerning high density matter, namely (i) the high-energy pp scattering, (ii) the physics of hyperons in pulsars and (iii) the importance of short-range, attractive spin-2 interactions. Regarding (i), we have indicated that high-energy.phenomena characterizing the pp system can be meaningfully used to study the behaviour of high-density matter, and in particular the equation of state P = c;e. The analysis however is hamperedby a lack of a satisfactory knowledge of the viscosity of hadronic matter, although dimensional arguments for the viscosity seem to suggest that 71 ^-T'l c " . So far as the problem of the hyperonic liquid is concerned, the present status can be summarized by saying that all the computations published so far indicate that the equation of state is not radically altered from the one corresponding to a pure neutron gas. This, however, does not close the issue, since our knowledge of the hyperonic interactions is far from perfect. Should future experiments on the hyperonic interactions reveal 'unexpected features, the equation of state could be significantly changed. Lastly, we have emphasized the role of the spin-2 interaction atvery high densities . We have seen that the inclusion of such interaction brings important and so far unpredictable new features into the behaviour of the equation of state. The spin-2 interaction dominatesover the spin-1 and spin-0 interactions at high densities, where it makes the equation of state rather soft. When neutron matter gets so dense that the individual neutrons tend to overlap, it is conceivable that a phase transition to quark matter will take place. Such a possibility has been investigated by several authors 3"') . However, inadequacies in describing quark matter as also dense neutron matter make the estimates for the critical transition density somewhat uncertain. We 42) have recently examined this problem in the context of our relativistic model for dense neutron matter, presented in sect . 4. For quark matter, we have adopted the MIT bag model 43) . Our investigation indicates that for the three experimentally suggested values of the spin-2 coupling constant f2 (see sect . 4), there is no first-order phase transition to quark matter . Furthermore, by taking f2 as a free parameter, we find that thevalue of f2 at which neutron matter - quark matter phase transition first takes place is 0.33 - ten times smaller than the smallest value suggested by experiments. In fig. 7, we have indicated the phase transition (by dashed lines) as it appears on the equation of state (corresponding tof2 = 0.33) . The critical transition density is 3.9 fm -3 , which corresponds to a matter density of 1 .7 x 10' 6 g . cm3. If there is a clear basis for realising a transition to quark matter at high densities (as the success of the quark model for hadronic structures would indicate), then the above results imply a density dispersion of the coupling constant. In particular, the coupling constant should decrease with increasing density. Such a possibility is currently under investigation by us.

HIGH DENSITY MATTER IN UNTVERSE

34 7

EV

N N C T L

a

n (fm-) Fig . 7 . Equation of state (pressure versus number density). The phase transition is denoted by dashed lines. P, and n, are the critical pressure and density (see text).

One of the authors (B.D.) wishes to thank Professor A. Salam for generous hospitality at the International Centre for Theoretical Physics, Trieste, Italy. Appendix We give here a brief summary of the OBE model and list the values of the parameters involved. The conventions of Bjorken and Drell 44) are used to write Feynman graphs . Charge independence and unitary symmetry are used to reduce the number of free parameters. For part (b) of the table below, we have used the notations and conventions of Gasiorowicz 43) . The choice of the mixing parameter a follows Gell-Mann The coupling constants are the rationalized ones. w and kQ are the mass and momentum of the exchanged meson.

n.

A. INTERACTION LAGRANGIAN AND VERTEX FACTORS FOR MESON-BARYON INTERACTIONS

(a)

Space-time dependence

scalar

Interaction Langrangian g.:

Vertex factor ig.

348

V. CANUTO AND B. DATTA

- igvs . 1~ys 00 :

pseudoscalar vector

- {gv: ~rY°00.

:-2NL

:

gpys

- a.0a):} - ilgvy a- ~[y °,y a lkal

(b) Isotopic spin dependence unitary singlet

g[1VN + SH + S " X +M l

unitary octet

49W " ir+(1-2a)frH " zr-2i(1-a) .9xXV " zr +2a~ " A

T

.'jr

+11X .Ir+

(3-4a) MV71 ,/3

(3-2a) 2a 2a S~ +T~ .~~ -TAA~ J3 -(.TK°-X+2-K°'TES) +

3-4a ^, (,~AK`+K°t,!H) d3

-(1-2a)(NTK " X+.X " K+TN) 3-2a --~-3(NAK+KtXN)} . B. THE OBE POTENTIAL

For the approximations used to simplify the field-theoretic OBE graphs, see refs . 16,17). At one baryon-baryon-meson vertex, baryon A-i-baryon B with coupling constant gAB; at the other vertex, baryon C-> baryon D with coupling constant gcD. j, is defined in table 3 (g) and eqs. (6)--(8) of ref. l'). For direct elastic graphs g = is . I.F. stands for isospin factor and x = N,r. (a)

Pseudoscalar exchange

V(r) = (I.F.) it 3 9AC9BD r_1 Ql , Q2+(1+1+ ' S12 e x, 64fmAcmBDL3 3 x x) x mu -Mdw,1(M1+M,),

S12 = -I(ff1 ' r)(o2 ' r)-Ql " Q2.

349

HIGH DENSITY MATTER IN UNIVERSE

(b) Scalar exchange V(r) =

84sn h

x

~Cl

8~z LMAMs +M~J

2 2 1 r + +-l O3 ' Pz + 64 MAMBMc D~ 64MwMBMcMD lz x xJ A4 1 _1 1 _ r_1 _z~ 1 ( +x)L S + w x VtifwMB 16MAMBMc~41~ L4 + Mc D~ 4 1l 2 X L)(QZ L)+(az L)(~i ' L)}} . + 116MAMBMcMiD [P+ J) (°i " (c) Vector exchange BD 2 V(r) =(I .F.) AgA~ Xz[(A IB1+N, A3B3)-2h z{A4B3+A3B4 %1 1 2_ + } \x x)1L . S+L 3~z{A4B4 2 +24 z (A5B4+A4Bs)+4w A5Bs}- â{AzBz

-j(AlB2+A2B,)

1 2 2 -2(A 5B4 +A4Bs)-ja 2A5Bs} ~1 +x + x)] ff 1

* ffz

-A{A4B4+za z (ASB4+A4Bs) +494A5Bs} x (3 + x + z)S12+ -A2A5B5

}

3{AzBz-2(AiB4+A4Bs)

(X x

A i =1+ Az__

_

N,z +fwBiiz (MA+MB) 8MAMB 4gABNMAMB '

1 _ fAB(MA+MB) wMB 4M 2gABgMAMB'

_ Mw +MB a2AB A3 _ 4MAMB + 8,ugwsMAMB' A4 =A3+ fwB .

w

AS=-

fAB 4WgABMAMH '

V. CANUIb AND B. DATTA

350

The B, are obtained from the A, by the substitution A-" C, B -" D everywhere . C. THE VALUE OF THE PARAMETERS

The quantities with asterisk denote search parameters, adjusted to fit Ap scattering data. See ref. ts) . (a) Masses (Me V): 8 = 966 K'= 1100 S=1141 u = 570* N=939 (b)

ir =138 K=495 n=566 Xo = 948 A=1115

p = 765 K* = 890 ¢=930 w = 887 E=1193

Hard-core radius: 0.46 fm

(c) Coupling constants :

scalar octet g, =11 .0* scalar singlet gQ = 8.245* pseudoscalar octet gv, =13.5 pseudoscalar singlet gr = 6.61 vector octet gv = 3.07 ft =1 .5 g vector singlet gd = 7.35 fm=0

a, = 0.30 a = 0.60 a,=0 at = 0.75

References 1) V. Canuto, Ann. Rev. Astr. Ap . 12 (1974) 167 2) V. Canuto, Ann. Rev. Astr. Ap . 13 (1975) 335 3) V. Canuto, Proc . Enrieo Fermi Summer School on Physics and Astrophysics of Neutron Stair and Black Holes, Varenna, Italy, 1975, in press 4) V. Canuto, Proc . NATO Summer School on many degrees of freedom in particle theory, 1976 (Plenum Press, 1978) 5) V. Canuto, Ann. N.Y. Academy of Sciences 302 (1977) 514 6) E. Fermi, Prog . Timor. Phys . 5 (1950) 570; Phys. Rev. 81 (1951) 683 7) Y. Pomeranchuk, Doklady Akad. Nauk SSSR 78 (1951) 889 8) L. Landau, Izv. Akad . Nauk SSSR (ser . fiz.) 17 (1953) 51 9) V. Canuto and E. Tsiang, Ap . J. 213 (1977) 27 10) M. Chaichian, H. Satz and E. Suhonen, Phys. Lett . SOB (1974) 362 11) E. L. Feinberg, Phys. Lett. 52B (1974) 203 12) E. L. Feinberg, Proc. P. N. Lebedev Physics Institute, ed. D. V. Skobel'tsyn (Consultants Bureau, New York, 1967) vol. 29, p. 151 13) A. G. W. Cameron, Ap . J. 130 (1959) 884

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