Volume 98B, number 1,2 PHYSICS LETTERS i January 1981

PHASE TRANSITION TO QUARK MATTER IN NEUTRON STARS

Enrique ALVAREZ Departamento de Ffsiea Tedrica, Universidad Autdnoma de Madrid, Canto Blanco, Madrid 34, Spain

Received 24 July 1980 Revised manuscript received 30 October 1980

A calculation is made of the critical density Pt at which it is more favorable for the matter to be in the quark phase than in the nuclear one. The equation of state of the quark phase is obtained assuming that the QCD effective lagrangian satisfactorily describes the interaction at all energies, and by making a relativistic Hartree-Fock approximation. The main result we get is that Pt > Pro, the maximum central density of stable neutron stars.

1. I n t r o d u c t i o n . In the core of a neutron star the energy density can be greater than that of close pack- ing, so that the possibility is open for the appearance of quark matter [1]. As the neutron stars are stable only when the central density is lower than a certain value, Pro, a most important problem is a precise de- termination of the critical density Pt'

This calculation has in fact been done perturbative- ly by a number of authors [2]. While these results are certainly valid in the asymptotic region, where the cor- responding chemical potential,/aa, is much greater than a typical hadronic mass, m h: ,u a >),/1 h [because then the strong running coupling constant is very small, as(/aa) "~ 1] some doubts can be raised in the region when the energy per particle ,1 (,on-1)NM in the nu- clear phase (NM) and in quark matter are of the same order, owing to the fact that then as(~t ) ~ 1.

The aim of this paper is to expose the results of a complementary, non-perturbative calculation, i.e. the quantum chromodynamical theory of interactions of quarks and gluons, QCD, in the Har t ree-Fock approxi- mation. This kind of cluster approximations is typically good for great space-t ime separations between parti- cles (i.e. n - 1 / 3 >> XD, where ~D is the effective range of the interaction - the generalized Debye length).

+1 Throughout this paper we shall use natural units, so that the pressure p, and the energy density p, shall both be ex- pressed in g cm -3. n is the baryonic density, expressed in cm -3, so that pn -1 is the energy per baryon.

General ly, ;k D ~ n - a , a > 1/3 (for example, for an ordinary electromagnetic plasma a = 1/2), so that the quantity ~D n 1/3 goes to zero as the density grows, and the criterium is well satisfied. To be specific, in QCD it can be shown that k D ~ [gs(/a)/a]-I ~ n - l ~ 3

X (log n) -1/2. Other related non-perturbative approaches in relativistic many-body theory are those of Kalman [3], Walecka [4], Chin [5] (who studied a system of fermions with scalar-vector interactions), Baym and Chin [6] (who studied quark matter in the MIT bag model) and Bowers et al. [7]. The calculations of K~ill- man [8], Shuryak [9] and Abrikosov [10] on the ef- fects o f instantons on the equation o f state o f quark matter are also worth mentioning.

We shall assume that quark matter with QCD inter- actions will reach, after a sufficiently long time, an equilibrium state, with a chemical potential/a associated to the conserved baryon current

ju - t r t~ 7u~ . (1.1)

The equilibrium thermodynamical problem is com- pletely solved once the baryon density

n = fdo,,i# = f i o d3x (1,2) ~: t---const.

is known, because then the thermodynamic potential (at T = 0) can be deduced from

n --- - ~ n ] ~ # , (1.3)

and the pressure and energy density of the system will

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Volume 98B, number 1,2 PHYSICS LETTERS 1 January 1981

be respectively given by:

p = - ~ 2 , O =~2+/dn. (1.4)

2. Relativistic Hartree-Fock approximation. In order to calculate the baryon density, one needs the quark propagator, G(p). It is easy to get an equation relating it with the quark-quark-gluon vertex ,2, r~(p, q) -((4, ~A~)+):

( ~ b + ~ - m q ) G ( p ) - g X A ~ G P ~ l ( p , ~ ) = i . (2.1)

The vertex is given as a combination of the two-body Green function for quarks, G(p, q), and the quark- quark-three-gluon vertex, ~EFG, t ,~p tp, q) ----

- E F G • ( ( ~ A a A ~ A o ) + ) .

rA(p, q) = A~(q)

× [ - g tr xAToa(P, q) + g2fDALfDEFGEF~(p,q)], (2.2)

where A~ is the free gluon propagator, and fABC are SU~ structure constants. The Hartree approximation is made now, which in position space means

It It ? t! G(x,x )G(x ,x ), G(x,x ,x ,x ) ~ G ( x , x ) G ( x " , x ' ) - ' . . . . (2.3)

i.e. the correlations between quarks are neglected, and the Green functions with gluons are replaced by the free values (so that the last term of eq. (2.2) is also neglected, because it contains an odd number of glu. ons).

The Hartree ("tadpole") part of the approximation [the one corresponding to the second term in eq. (2.3)] is identically zero, because it is proportional to tr XA G(p), and the quark Green function at equilibrium must be proportional to the unit in color space, 8p,q,. Incidentally, were it not for this circumstance, serious infrared divergences would be encountered due to the presence in this term of a factor A~(0) in momentum space, which diverges badly because of the massless- ness of gluons.

It is then concluded that only the exchange inter- action contributes, so that the Hartree-Fock approx- imation to the quark propagator obeys

*2 We shall use the notation

th roughou t ,

(~b +/~ - mq) G(p)

+ ~g2vp~x~(g) a(g + p) "to a(p) = i. (2.4)

This equation can be formally solved by decomposi- tion in the Dirac algebra:

a(p) =- C l (p ) + a2(P) . ~ + a 3(p) . ~ , (2.S)

~g(~) a(~ + p)

-=g~r[Al(P ) + A2(P) • Zb + A3(P) • /~] , (2.6)

getting

[~ + ~de(P) ~ -- me(P)] G(p) = iZ, (2.7)

where

Z - l = l + ( e - 2 ) A 2 ,

and the (momentum-dependent) effective mass and chemical potential are given by the equations

Pc(P) =-- Z[/d + (e - 2) A31 , (2.8)

me(P) = Z[m + (e - 4) A1] . (2.9)

The trivial solution of eq. (2.7) is, of course

GHF = Z . SF(/de(p), me(P) ) . (2.10)

Even if the Hartree-Fock problem has been (formally) completely solved, the calculation of finite values for /de(P) and me(P ) from eqs. (2.8) and (2.9) is a difficult task, because these are, in fact [due to the implicit de- pendence of A i on/de(P) and me(P)] non-linear integral equations.

Leaving aside, for the moment, the general problem, we shall systematically use in the sequel the crudest ap- proximations in order to get a feeling of some general aspects of the result, which can be of astrophysical in- terest. In practice, then, the determination of the effec- tive mass and chemical potential as functions o fp will be an iterative process: using first (2.10) with/de(O) -=/d and me(0) = m constants, the quantities Gi(o) , Ai(0), i = 1,2, 3 can be determined and used in eqs. (2.8) and (2.9) to calculate/de(1)(P) and me(1)(p ). Returning then to (2.10), the loop is closed.

It is to be stressed that the constants A i admit a Laurent expansion in terms of e:

a;=- Z; (2.11) a=--I

It can be shown that (A~-I))(O) = (2x(3-1))(o) = 0, so

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that eq. (2.8) for the effective chemical potential has no poles in it in the first iteration. Eq. (2.9) for the effective mass can be written as

-1 me(i)Z(i_l ) = m - 4(A(-1))( i_l ) /e + (A~-I))(i_I)

- 4 ( A ] 0 ) ) ( i _ l ) + O (e ) , ( 2 . 1 2 )

so that a mass renormalization is needed, rn ~ m + 6m, with 6m - 4(A~-l))(i_l)/e_ + mf(i), where mf(i) is a

finite arbitrariness.

3. Results. When calculating the total baryonic charge of a physical system, it is necessary to take away the unphysical contribution of the vacuum (this is equivalent to a certain normal ordering of the inter- action).

From (1.2), it is easy to see that

/U(x) = tr 7 u Gt(p) ,

so that

n = tr fdau "yP[Gt(ff) -- Gtac (p)]

- ~ ] t r3 ,0AG t , (3.1) flavors

with

ACt = ZS~(/ae(~) , me(P))

t - - ZvacSv(/lvae, mvac) •

It can be shown that the dominant term, when # is large, is given by the single integral

n = n -2 ~ [ Z ( E ' P ) ( E + t l e - I ~ ) flavors ./ E

× O(l%(p) -- E(p)) p2 rip. (3.2)

This integral can be approximated to give the result

O(f) 8 ~ [ I ( p + ) - I ( p _ ) ] , (3.3) n - k 1 + 87r 2 flavors

with

f(la, rn0) --- g + 2 - 2(1 + p + m0)1/2 ,

k = m0[(1 + m2) 1/2 - 1]/2(1 + 8n2)(mo - ~ ) ,

and

p± _=_ ½[p + (p2 _ 4m0)1/2 ]

and

I (p) -- ~p5

+ l~-~-6e-X/P (-- 24p 5 + 69~p 4 2X2p 3 + k3p 2 -- k4p)

- ~-~;k5 E i ( - k / p ) .

It is perhaps worth remarking that formula (3.3) gives zero unless/a >/acrit - 1 + 2x/1 + m 0. This critical value of the chemical potential then plays the same role in this theory as (in the free case) the minimal value of the chemical potential (/a = m), the one which is ob- tained when the Fermi momentum is zero, i.e. the would-be "physical mass" of the quarks. Thus this quantity,/acrit, is (in the precise statistical sense speci- fied above) the analogue of the "free" quark mass.

4. Conclusions. Numerical computations show that quark matter begins to exist only at # > P c n t ~ 4 GeV, which corresponds to a baryon density of n ~, 8.58 × 1039 cm -3 and a pressure and energy density of (p, p) = (1.53 X 1016 , 4.60 X 1016) g cm -3 . As can be seen in fig. 1, the equation of state always remains very close to the line p = ~-p.

In order to see at what point it is energetically more favorable for the matter to be in the quark phase than in the nuclear one, it is necessary to compare the ener-

IOgp (gr/cm~

p.p"

p'l/3P

ER

16 17 18 t9

Fig. 1. Equation of state for quark matter on a logarithmic scale. For the sake of comparison, the linesp = p andp = ~p are also represented.

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Volume 98B, number 1,2 PHYSICS LETTERS 1 January 1981

T~ 2'

pr; I (9r l

-MATT r~ ~

tc,,P)

Fig. 2. Energy per particle in the nuclear phase and in the quark one. In the WNM situation, Maxwell's double tangent is indicated by the thin line.

gy per particle in the two phases. We shall do that with two different models for nuclear matter (cf. Alvarez [11 ], where a complete specification is given): one called BCZ, obtained by matching Canuto's preferred model [12] with one of the models o f Bowers et al. [7] and which is a soft model in the sense that it asymp- totically tends tq p = ½p; and the other, called WNM, which is obtained by matching Canuto's model with the relativistic scalar-vector model of Walecka [4] ; this is a hard model in the sense that it asymptotically tends to p = p.

When carrying out that comparison, it is seen (cf. fig. 2) that in a soft situation (BCZ) at the point where quark matter (QM) begins to exist, the energy per par- ticle in the quark phase is lower than that in the nu- clear phase: (,on-1)Q M <~ (pn-1)BCZ , so that it is favor- able for the system to be in the quark phase from the moment this is possible. This means that a phase tran- sition cannot occur, and the critical density is Pt ~ 5 X 1016 g c m -3 .

As can also be seen in the same figure, in a hard (WNM) situation, a first order phase transition is pos- sible corresponding (via the realization of Maxwell's double tangent construction) to Pt ~ 1.3 X 1016 g cm -3 and a baryon density o f n --~ 5 X 1039 cm -3 . In fig. 3 the matched equation o f state is represented in both cases,

Owing to the fact that Pt > P m in both hard and

Io¢ p

17

16

P'P p , I / 3 P o / ( I )

Iogp ( g ~ )

15 16 17 18

Fig. 3. Matched equation of state. In the WNM case, the double tangent construction implies a line of constant pressure (II), in contrast with the BCZ situation (I), in which no Maxwell con- struction is possible.

soft situations, it seems reasonable to conclude that the same result will remain true for the real equation of state for nuclear matter, which is undoubtedly some- where between BCZ and WNM. Also, due to the ex- treme simplicity of the result, it is probable that its main characteristics will remain true in a refined cal- culation o f the self-consistent parameters of the Har- t r ee -Fock approximation.

In conclusion, there is no quark matter in neutron stars in this approach, but the possibility is open for the existence of a third family of quark stars at much greater central densities [13]. More details on this work will be given elsewhere.

It is a pleasure to acknowledge very stimulating dis- cussions with Drs. M.B. Gavela and R. Hakim.

References

[ 1 ] G. Baym, Neutron stars and the physics of matter at high density, Lecture notes prepared for the 1977 Les Houches Summer School in Nuclear physics (1977).

[2] M.B. Kislinger and P.D. Morley, Phys. Rev. D13 (1976) 2765, 2771; Phys. Lett. 67B (1977) 371; 69B (1977) 257; Astrophys. J. 219 (1978) 1017; B.A. Freedman and L.D. McLerran, Phys. Rev. D16 (1977) 1130, 1147, 1169;D17 (1978) 1109; V. Baluni, Phys. Rev. D17 (1978) 2092; Phys. Lett. 72B (1978) 381.

[3] G. Kalman, Phys. Rev. D9 (1974) i656.

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[4] J.D. Walecka, Ann. Phys. 83 (1974) 491;Phys. Lett. 59B (1975) 109.

[5] S.A. Chin, Ann. Phys. 108 (1977) 301. [6] G. Baym and S.A. Chin, Phys. Lett. 62B (1976) 241. [7] R.L. Bowers, J.A. Campbell and R.L. Zimmermann,

Phys. Rev. D7 (1973) 2278, 2289. [8] C.G. K~illman, Phys. Lett. 85B (1979) 392.

[9] E.V. Shuryak, Phys. Lelt. 81B (1979) 65. [10] A.A. Abrikosov Jr., Phys. Lett. 90B (1980) 415. [11] E. Alvarez, Astron. Astrophys. 84 (1980) 9. [12] V. Canuto, Annu. Rev. Astron. Astrophys. 12 (1974)

167; 13 (1975) 335. [13] R.L. Bowers, A.M. Gleeson and R.D. Pedigo, Astrophys.

J. 213 (1977) 840.

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