Screened charge and phase transition in QCD quark matter

Screened Charge and Phase Transition in QCD Quark Matter

VIKRAM SONI

Service de Physique Th~,orique, CEA-SACLA Y, 91191 Gif-sur-Yvette, Cedex, France

INTRODUCTION

This lecture spans developments in the subject of quark matter (T=0) governed by Quantum Chromodynamics, The object is to examine phase transitions (possibly to nuclear matter) that occur in this system in the framework of many body perturbation theory. The subject is vast and often convoluted in formalism. This shall be treated in a simple way and not detailed, keeping the interests of the unsuspec- ting reader very much in mind. Obviously, the choice of the expansion parameter or effective coupling in renormalization group improved perturbation theory is impor- tant. A new effective coupling, the screened charge, the appropriate one for the case of matter shall be introduced in distinction to the usually employed vacuum OCD running coupling and the consequent differences in the physics and the fransi- tion parameters highlighted.

It is expected that nuclear matter transits into the quark phase when the nucleons overlap. This is crudely estimated to be between one to ten times, the nuclear density, nN, if we take the nucleon radius to be 0,9F to 0.SF. The anticipation is that such matter exists in the interior of neutron stars, in the early universe or can be produced in ion ion collisions,

For simplicit~ we will look at colour singlet, neutron matter at T=0 which goes into a quark gas at high enough density. The feature of asymptotic freedom in QCD informs us that at very high euclidean momentum transfer quarks become almost non interacting. Space like momentum transfer translates into inverse interparticle distance. This is the crucial feature which allows us to treat a very high density quark gas as virtually free (Collins and Perry, 1975 ; Kisslinger and Morley, ]976). The ground state of matter at T=0 and infinite density is then

i) Free fermi seas of quarks (multiplicity N c (No, of colours) x Nf (No, of fla- yours)) with an equal occupation of all colours.

ii) Since, T=0 and the effective coupling ~ 0~the theory is Abelian in this limit and there are no antiquarks and gluons.

As the density is reduced~we can use many body pertUrbation theory to treat the system, Also, one should recall the presence of matter will screen the interaction at distances greater than the interparticle distance.

585

586 Vikram Soni

In section l we demonstrate the construction of the appropriate effective coupling, the screened charge,through the renormalization group. Sec 2 begins with an introduction to the length scales relevant to the phase transition in a QCD quark gas followed by a discussion in 2.1, of an instability in its plasma spectrum and in 2.2 by the pressure zero phase transition in the context of improved many body perturbation theory.

I. RENORMALIZATION GROUP AND SCREENED CP~RGE

We know that in quantum field theory it is only in the lowest or tree approximation (Born approximation) that the coupling constant is a fixed number and the theory is simple. In higher order there are loops in the Feynman diagrams for any scattering process. These loops introduce divergences and a regularization or renormalization is required to extract sensible and finite results. This is accomplished by multiplying the Green's function by infinite renormalization constants such that the Green's function are normalized to the free ones on the mass shell. This introduces a mass scale in the renormalization constants;what is more, is that the renormalization group tells us that the normalization can be done off mass shell i.e. the scale introduced through the normalization is arbitrary,

The renormalization group states that the physics is invariant under a set of transformations by multiplicative constants on i) the fields ii) vertices and iii) couplings of the theory which satisfy certain relations amongst themselves. Set us illustrate these for Quantum Electrodynamics (QED)

A' ~ Z I/2 A , ~' ~I/2 ~, , ~ -I o ~ Lf rffy Zffy Fffy

,2 Z~2 z-l Z 2 D' ~ Z D G' ~ Zf G and e = ~v o ~' o ffy 2

e

(1)

where A~(D~v), ~(G)~Tff~ and e 2 are the photon field operator (Green's function), fermion field operator (Green's function), the vertex function and the square of the coupling (or charpe) respectively. The Z's are the multiplicative constant of the transformation. It can be trivially seen that Dyson's equation or the S-matrix elements are invariant under this transformation.

Note i) the transformation has, a priori, nothing to do with divergences.

ii) the multiplicative constants can be parametrized by a mass scale M, set by the normalization condition (we shall see this in the following considerations)

iii) the invariances of a theory, e,g. gauge invariance in OED, give constraints on these constants, namely, Zf = Zffy. These constraints are called the Ward Identities.

For QED, we then have the renormalization group relationship

D' = Z D and e '2 = Z "l e 2 (2) ~ o ~ o

An exactly similar relationship exists for OCD in the Coulomb gauge' (Taylor and Frenkel, 1976 ; Shuryak, ]980 ; Soni, 1983 a) that relates the Coulomb gluon propagator, D ab and the coupling g2

OO'

D 'ab = Z ab and g,2 z-l 2 oo o Doo = o g

There are problems in usln 8 covariant gauges in Quark Ratter due to the ~lack of Lorentz invariance (Soni, 1983a).

Screened Charge and Phase Transition in QCD Quark Matter 587

We must point out that we shall renormalize the quark mass, m, on shell, that is, as the pole of the quark propagator order by order in perturbation theory. The quark mass is thus excluded from the transformation above and independent of the normalization point.

The full Coulomb gluon propagator

D a b . + 2 2 2 2 . 2 2 ooab { ~ab( I ~I 00tP0,Po,l~i,m ,m ,g2(M)) = d(~ 2 ..,g (M)) =

, . oo o t i - 7 -7 - , j s

where d is the dimensionless function

d(;2 2 2 2 2 2 R 2 2 2 )o oo ,po,~i,m ,M ,g (M)) =I/{l - oo(~2,po,~i,m ,M2,g2(M)) D

h where (p,po) is the four momentum, ~i the i t flavour quark chemical potential, m, the common quark mass and M the normalization point Rwoo is the renormalized (see later) proper self energy which is composed of a vacuum part, Rw~o , and a matter part AWoo

R = R V + A~ OO OO OO

The unrenormalized vacuum part is divergent but the matter part is finite. The renormalized proper self energy follows from th~ removal of the infinity and fixing its value uniquely at a given ÷2 - Z momentum p - M (where M is the normalization point) through a normalization condition. We shall use the normalization condition

d(;2 = M2' Po =0' ~i' m2' M2' g2(M)) = ] (3)

+2 = Note~the normalization point is chosen as p = M2, Po 0. The details of the above procedure and the evaluation of R~oo are not described here but can be found in (Soni, 1983 I). The above normalization condition is identified as the screened charge scheme and as we shall see gives rise to a new effective coupling, the screened charge.

The usual scheme, which we shall dub the vacuum scheme, has a normalization condition which is independent of the matter oart A~ The condition is d( +2 = M 2,

^ ~ 2 . o o . . . . . P .

P° =does ~ ~ym he M ~r~a tio f~w;~rAa 1~nZ~h:m d 1~h it

scheme will satisfy it). More specifically, in the vacuum scheme

R oo(52,po=0,~i,M2,g2(M)) R V .-~2 0 M 2 2"M'" = ootP ,Po = , • ,g t ))

2 + A~oo(~2,po=0,~i,g 0 ! ) )

where the matter part does not have an explicit M dependence or

2 R~oo (;2 = M2) = A~oo (;2 = M2' ~i' g (M))

For a complete discussion on schemes see (Soni, 1983, b).

Now consider the set, therenormalized function d (instead of Dab) and ~2(M~ as specified above. Let as construct a renormallzatlon group trans~ormatmon on these that changes the normalization point from ~I to M' while preserving the condition (3). Further, we shall work with the restriction Po = 0 as before,

d,P~2 2 ÷2 i,M2,g2(M)) -I 2 kp ,ui,M' ,g2(M')) = Z o d(p ,~ and g'2(M') =Zp~ (M) (4)

588

+2 Putting p = M '2 and using (3)

Vikram Soni

-+2 2 g2 Z -- I/d(p =M' ~i' M2 (M))

O ' '

This ~ives the dimensionless constant of the transformation, which is parametrized the old and new normalization scales.

Clearly g2(M) cannot be a physical coupling since it can altered at will through the renormalization group. A physical coupling must be invariant under change of the normalization point. The quantity

2 2 2,g2(M)) g (M) (5) glNV(P ,~i ) = d(~2,~i,M

is by construction invariant under change of M or renormalization group. It is a function of the momentum transfer, ~2, and ~., the chemical notential. It is called the invariant or effective couplingland is c~nstructed via the renormalization group. Notice that g~NV = g2(M) at p2 = M 2. Let us briefly illustrate its meaning,

Consider the electron-electron scattering born graph

Fig. l. Electron electron born scattering

~2 The scattering amplitude, is a physical quantity and goes like e2(M)/lql , Clearly, the presence of eZ(M) is not physical. The renormalization group invariance of the physics on the so called improvement is recovered if we make the replacemeht

2 e2(M) ~ PiNy(q). What does this mean ?

2 + elNV(q) is renormalization group i~vari~nt. Let us choose the normalization point M=O (on shell). Then e~NV(~=0) = eg = e~(M=0) % 1/137, the fine structure constant. Now as ~2 ~ O.the scattering is well described by the Born approximation using the coupling, Co, as there are no radiative corrections. For 72 # O, however, loop or radiative corrections enter and change the scattering amplitude~ The summing of leadin~ corrections t~ all ordersis exactly what is accomplished by the use of e~NV(~ 2) instead of e~. This shows us the meaning of the invariant coupling, Also, what is measured in an experiment is clearly g~NV and not g2(M),

In QCD the only experimental information we have, is for the vacuum theory effective coupling at asymptotic momentum transfer from scattering experiments, Of course, we are interested in a theory of quark matter The only way to connect with experiment is to impose that (~i/IP+o]) << I where I~oi is a deep asymptotic momentum transfer at which we connect with vacuum OCD experiment. This is to ensure that th~ matter contribution is negligible at this point, We shall choose Ipol = 1000 GeV ~ - all reasonable densities of interest then satisfy the above restriction,


The invariant coupling or screened charge is given by

2 (->2. go Po ) 2 "-+2,~i,...) (6)

glNV ~p = l -

Po \IPl 2 IPol "

This is for quark mass equal to zero. Also, for simplicity some small terms have been left out from the matter (~.) dependent piece, For convenience we have introduced a factor, A, to distinguis~ between the screened charge (A=]) scheme and the vacuum scheme (A=0).

->2 The value of go(Po ) is fixed by the scale length A from vacuum OCD.

We list the main properties of the screened charge (A=I) is compared the vacuum QCD effective coupling (A=0)

i) It is the correct effective charge to use in quark matter. This is proved in (Soni, ]983, b)

ii) It merges with the Usual vacuum OCD effective coupling for deep asymptotic mo- Mentum transfer, ~2 + = (in view'of the constraint (~/r~ I) << ]).its deve-

->2 • % • • lopment as p is decreased ( ~i) is then dlfferent due ~o matter screenlng. This renders it infrared (~2 ~ O) finite (Fig.2) for all densities higher than a certain critical density (~). This is in distinction to the vacuum effective coupling which diverges as a simple pole at I~l = A for all densities,

T K .,4"

N . >

102.4

101.a

101.2

100. 6

0

160.6

1(} 1.2

lO a

lO'2.;

v/C 258 MeV 500 MeV 103 HeY 103 MeV

h = 500 MeV ~ = 2.115 m=O

101 102.1s 104.5 106.2S

p (MeV) _~

Fig.2. A plot of the screened change for different, ~, till it diverges for ~u = ~ = 258 MeV.

590 Vikram Soni

iii) It is always smaller than its vacuum counterpart due to screening extendin~ the validity of perturbative calculations to much lower density,

iv) The screened charge diverges at a critical ~i or density and at a critical momentum transfer pc.

The invariant coupling or screened charge can also be written in terms of g2(M) where M is an arbitrary normalization point

where

2 g2 (M)

' 7 - ,.n ..

(7)

g2o (~o) g2 (M) =

I + 8 g022 (~) ~ Ln<~ + A l . , o t Al. ~ ~2-- i~o-~ )j j,-

2. PHASE TRANSITIONS IN QUARK MATTER

We shall now look at the phase transition in quark matter. Let us begin by looking at the possible length scales in the theory. First consider QED. The only dimen- sional parameter in the theory is the electron mass, m, which is essentially kine- matic. The dimensionless parameter which carries the dynamics is the electron charge or coupling. The size of the bound state (positronium or hydrogen) follows from the observation that it should depend inversely on the strength of the charge. This givesa length scale like % I/me 2 (~ = c = l),which is the Bohr radius. Alternative- ly, consider an electron ~as in a neutralising background (say protons) which ~oes into Hydrogen atoms below a threshold density. Many body perturbation theory yields the ground state of the system in powers of ~, the fine structure constant. Retai- nin~ only the exchange term (order, ~ = 1/137) we find that the energy density of the spectrum has a minimum (pressure = O) when kF/m % ~ (I/137)~ The fact that ~ is small fixes the P = 0 transition to be non-relativistic no matter what the value of m.

This simple potential picture remains ~ood till ~ is small, l~en ~ is large the system becomes relativistic and the complications of the field theoretic vacuum enter. In quantum field theory, as we have seen, the observed charge (~) is not a constant but a function of momentum transfer. In Q.E.D., however, the charge is so small that all renormalization effects on it are negligible for all sensible length scales. Another remark is that the renormalization introduces an arbitrary mass scale into the theory - the normalization point. In Q.E.D~ it can be seen that this scale can be eliminated from the theory by going to the mass shell, e.g~ for the photon self energy,so no new length scale is introduced into the theory,

Contrast this with OCD. Consider the gluon propagator. The self energy has a pure gluonic loop : Whenever the vacuum bubble has only massless particles (unlike the e+e - loop in QED) on shell renormalization is not possible. This is not surprising as gluons are not seen. The scale introduced in renormalization cannot be eliminated. This is true, regardless of whether the quarks are massive or not. Thus QCD has a hidden length scale, apart from the quark mass, often associated with A, the scale length. Whether such a length scale becomes relevant should be determined by the value ~ .... We know this to be large even at length scales smaller than at nuclear densi~TUflfm).Only if ~QCD is small - at high k F (~ ] GeV) - and therefore


m > I GeV may the non-relativistic picture become possible in OCD ; otherwise the ~hase transition in quark matter is likely to be governed by a new dynamical length scale distinct from the Bohr radius.

2.1 The plasma spectrum insltabiilit ~

The free Coulomb propagator is just the static Fourier transform of the Coulomb potential. At one loop order dynamics enters through the frequency dependence of the proper self-energy. The plasma spectrum is given by the poles of the full (I loop) Coulomb propagator :

+2 R .+2 P - ~oo tp ' Po =~°' ~i' g2(M)' M2) 0

whore ~ooR is the renormalized proper polarization (self-ener~v).~.

(9)

For the non relativistic electron gas this ~ives the usual plasma spectrum, In the quark gas, whereas the matter polarization gives rise to screeninz (exactly as in the electron gas) the vacuum antiscreens. The competition can give rise to an instability, namely, a zero mode.

The renormalization group improved Coulomb gluon propagator is evaluated in (Soni, 1983 b ; Soni and Murugan, 1983). It has the property i) gauge invariance, as its renormalization is the inverse of the charge (gZ(M)) and the charFe renormalization is gauge invariant ii) though, obviously it is not renormalization group invariant, its poles are.

Thus the poles of the propagator given by (9) are physical.

On setting ~=0 we find a solution to the above equation. T~is signals a zero mode which occurs at a critical value of ~ic and a wavevector }p] = Pc, The instability is found to correspond to a density wave in the colour octet Coulomb gluon channel with a wavevector Pc" In general, this would break colour symmetry as it implies a vacuum expectation value for the colour non singlet coulomb gluon <~a>, However, since the threshold conditions are identical for all colours it is possible that the density wave will occur with <oa~ga > # 0 as a eolour singlet,

In the screened charge scheme the poles of the Coulomb propagator which are scheme independent coincide with divergence in the screened charge~ This would invalidate the assertions above. Even so, one can look at higher densities (g~NVI = I) where the poles are complex, The Fourier transform of the Coulomb oroDagator is the Coulomb potential. It is clear that the complex pole will give rise to a sinusoidal modulation from the real part of pc(=PR)an as a damped envelope from the imaFinary part of p (=pl). It is found that the damplng does not completely kill the periodic variations (or PR > Pl ) allowing for a local density wave even though there is no long range order.

Table I lists the critical density and wave ve?tor Pc for the density w~ve instabi- l~ty. This is for pure neutron matter (n u = ~ n= = n=) and with the assumption ~in ~ (~- m2). For a detailed discussion of theUresu~ts of Sec.2.1 and 21"2 see

and tlurugan, 1983).

2.2 Pressure Zero phase transition

We shall now use a different criterion for a phase transition in quark matter. This is to look for a minimum of the ground state energy density or the zero pressure

592

TABLE I

m(MeV)

Vikram Soni

~ c2 2an d • c A Listing of the Critical Value of m6 c'k~u=v~u-m. ~a~

which the Screened Charge Diverges for uifferent Quark Mass

m(A = 290 MeV and A = 500 MeV)

A =290 MeV(g~ = 1.869) A = 500 MeV (g~ =2.115)

~u c nB/nnuclea r k~%~ ~ nB/nnuclea r (MeV) (MeV) (MeV) (MeV)

0 150 150 0.2568 258 258 1.3065

50 150 158 0.2568 258 263 1.3065

IO0 ]40 172 0.2088 250 260 1.1887

300 I25 325 0.1486 234 380 .9748

500 120 523 0.1315 215 544 .7561

I000 ]15 1006 0.1157 200 ]019 .6086

1500 I12 1504 0.1069 198 1513 .5906

2000 110 2002 0.1013 196 2110 .58

Also listed nB/nnuc!ear, the ratio of the baryon density, nB,

corresponding to m6~ =k~u(n. = 1.2933 x 10 -2 × (k~ (MeV)) 3 3 " . ~ u baryons/fm and nnuclea r xs the nuclear densit~.~aken to be

0.17 baryons/fm 3 (assumption m=mu=md and ~d=(2~/36u corresponding to neutron matter)

condition in renormalization group improved many body perturbation theory (Baym and Chin, I976 ; Chapline and Nauenberg, 1977 ;F~eedman and M~Lerran, 1977). We shall outline the procedure and display and interpret the results. Further, we shall make a comparison of the results from the screened charge scheme and vacuum scheme.

i) Many body perturbation theory gives a Feynman graph expansion in orders of the coupling ~ = (g2(M)/4~) for the renoramlized thermodynamic potential, ~, or free energy density (T=0) (Kisslinger and Morley, |976, 1979 ; Freedman and McLerran, 1977 ; Baluni, 1978).

= ~o + ~ ~I + 2 ~2 = E - i ~ ~.n.x x (10)

where E is the ground state energy density, ~i and n. are the i th flavour quark • 2 I • chemical potential and number densxty, ~ = g (M)/4z, the QCD fxne structure cons-

tant defined earlier (8).

ii) ~(M) is nothing hut the renormalized coupling at a normalization point, M(8). As in the case of the Born scattering discussed earlier, the expression for ~ is manifestly not renormalization group invariant, The group imnrovement of ~ can be carriedoutbythe replacement of M by the average chemical potential M2=(~/nf)=~ 2 un the ight hand side of (I0). This gives rise to a ~i or density dependent invariant coupling we express as ~(~) and can be shown to correspond exactly to the renormalization group improvement of ~ (Kisslin~er and Morley, 1976, 1979; Freedman and McLerran, |977~ Ba.vm. 1977 )Mur " and Soni 1983)~', This ~ives ~This' is similar to the screened charge 'except that the role of I~J is taken over by ~ .


+ ~(~) ~ + 2(~) ~2 " ' " (II) = ~o o

iii) The quark density is given by

ni = - ~i

From the density constraints for neutron matter, n u = I/2 n d = n B (n B is the baryon density) we can relate the u and d quark chemical potentials.

iv) Finally the i~ condition is implemented to determine uniquely the critical u(d) quark chemical potential and density and also ac(~c)"

a) We shall first work with the expression for ~ to lowest order in the coupling,

~ = ~ + ~ 1 O

where ~o i s t h e f r e e q u a r k gas t e r m and a~ l i n c o r p o r a t e s i n t e r a c t i o n t o l o w e s t o r d e r .

It is customary in this order (ep,. electron gas), instead, to work with the energy ,density E = ~ + I ~/.n.

i z ~

E = E ° + e(M) E l (12)

where e(M) has been defined earlier. To This order we can make the substitution, ~2 = k 2. + m 2, above (kfi = i th cuark fermi momentum). E is then written as a fulnction I of kf: • The renormalization group im1~rovement follows on scaling kc. and _is achleved by settlng M 2 = (~ k2i/Nf) = k 2 in (12) above (instead of M = Dasin (II)).

Using the neutron matter constraints n B = n u = I/2 n D = kf3u/~ 2 = k3d/2n 2 we can convert from E(kfi) to E(nB). The pressure zero condition is

2 3 [K/nB] P = n B ~n B 0 (13)

I t i s u s e f u l t o n o t e t h a t ~ ( k f ) o r a ( n B) i s now a f u n c t i o n o f t h e b a r y o n d e n s i t y , rib, and t h i s g i v e s r i s e t o a t e r m i n t h e p r e s s u r e , P3

E 1 ~ (rib) P3 = n ~ ~n B ~ ~2(nB) (14)

n B

which is proportional to ~2(nB). This can be easily seen from the form of the QCD B function?B M = M(~/~M)~(M) = A ~2(M).

The results for the screened charge scheme, for different quark masses, are listed in Table 2. We briefly discuss them.

594 Vikram Soni

TABLE 2

A = 500 MeV E

m kf~ -n--B ~c(~£) n B MeV MeV MeV baryons/fm 3 nB/nnuclear

0 333 1288 2.19 ,4869 2.8649

2 333 ]288 2.1973 .4869 2.8646

5 332.5 ]287 2.2054 .4847 2.8511

50 315 II93.2 2.6702 .4122 2,4247

I00 265 964.5 6.2246 .2454 1.4435

No solution for I00 MeV <m <500 MeV,

500 220 1410,1 ,99065 .1404 ,8258

900 270 2673.1 .45356 .2595 1.5264

1000 275 2978.3 .39986 ,2742 1.6129

1500 300 4491.4 ~25792 ~356 2,094]

2000 310 5996.5 ,18877 .3928 2.3105

A = 290 MeV E

m kf~ u n-'~ ~ (~f) n B MeV MeV MeV c baryons/fm3 nB/nnuclear

0 192.5 743.22 2.2203 0.094 0.5533

50 163.25 607.09 4.0942 0.0573 0.3375

No solution for 50 MeV <m <300 MeV

300 129 851,98 0.93394 0.0283 0.1665

500 154 ]483.1 0.47463 0,0481 0.2833

900 174 2695.5 0.24753 0.0694 0.4086

I000 ]77 2996,7 0.22155 0.0731 0.4301

1500 182 4499.7 0.14206 0.9795 0.4676

2000 192 6001.5 0.1110 0.0933 P.549

There are two regions (i) small m, 0 < m < ]00 MeV and ii) large m > 500 MeV, where we find a solution to (]4) (A = 500 MeV).

i) for mass, m=0, or close to zero the term P3 is negative due to the negativity of the B function in QCD. Besides, it is the only term in the pressure which is negative. Since it goes as, e2, it makes for a fairly high value of ~(kf) for the pressure to go negative (ec(~f) = 2.]9). As has been observed (Baym and Chins1976& Chap]ine and Nauenberg)1977).this term mocks the ~IT bag pressure and can be idenr tilled with the bag pressure, B (coincidentally ~c(kf) is of the same order as the MIT bag coupling). This is a consequence of negative sign of the QCD 8 function : for QED, 8, is positive, and we could never form a self Bound (P,=O) state, Region

. . . . . . . . in ''~ " ~i) Is dls~inct, as the transltlon denslty goes down wlth increase m contrary


to the "metal insulator" transition characterized by the Bohr radius.

ii) For very large mass the value of the coupling at PrO, ec(kf), is small and thus the term P~ is not significant. Here, E, acts as the usual non relativistic exchange energy which goes negative when kf/m < 2.53 (see for example Baym, 1977) as in the electron gas. The phase transition is now similar to that for an electron gas and is not set by any dynamical sca~e linked to QCD, but by the quark mass. This is corroborated by the behaviour of the transition density which now increases with the mass.

For small quark mass the deciding length scale is not the quark mass but the dynamical of hidden length scale of QCD associated with the scale length. A,whereas for very large m the transition is characterized by the Bohr radius and small coupling as in Q~D. Large quark masses are unlikely to be relevant for the nuclear matter transition since we know the u and d quark are not heavy.

Unfortunately, there are problems with this lowest order calculation which render the results~at most,qualitative .

i) The expansion parameter for OCD (SU3 colour ) may be taken to be ~/2 (Kisslinger and Morely, 1977). The lowest value of ~ in region (i) is 2.17 which makes perturbation theory not valid.

ii) Again, in region (i) the crucial term for Pffi0 is P3 which is of order e2(~f). A consistent calculation wauld then demand that E be computed to ~2(~f).

To see if any sensible perturbation analysis is possible we must go to order ~2(kf) and see if the perturbation expansion parameter ~c(kf)/2 at transition is small compared to I.

Table 3 compares the results for the screened charge and vacuom schemes for mffi0 and fixed A. The transition density is down by a factor of six in the screened charge scheme, though ec(kf) is the same in both schemes (see Murngan and Soni, 1983). Clearly the choice of scheme has a significant influence one the numbers and the physics.

TABLE 3 Comparison of (prO) Critical Parameters (Order ~) for the Screened Charge and Vacuum Schemes

nB/nnuclea r ec(kf) E/nB(MeV) A(MeV)

Screened Charge Scheme l 2.19 940 360

Vaduum SCheme 6 2.19 1710

Next we work with the thermodynamic potential to order ~2 for the case m=O. Here, the self consistent calculation outlined at the begining of this section gives a reasonable value for the expansion parameter, Uc(kf)/2~l'/2, for the transition. Table IV compares the results for the screened charge and vacuum schemes. Clearly, again a much lower charge density is obtained for the screened charge scheme, In this order the credibility of the results is vastly improved.

We have seen that, though a density wave instability is suspect and anyway occurs at a density far lower than the P=O transition, a precursor local density local density wave is possible at higher density, The prO transition indicates that the

596 Vikram Soni

TABLE 4 Comparison of (P=0) Critical Parameters (order ~2) for the Screened Charge and Vacuum Schemes

nB/nnuclea r ~c(~) E/nB(MeV) A(MeV)

Screened Charge Scheme 3.8 ,816 1621 300

Vacuum Scheme 13 1.04 2270

energy density has a minimum at a certain inter particle distance. Since, for light quarks, quantum fluctuations and the absence of a hard core are unlikely to locali- ze the quarks a hydrogen like self bound state at this inter particle distance is more likely. The attractive velocity dependent forces that arise from the precursor local density wave could give rise to such a bound state.

We cannot make a definite statement about whether the transition is to nuclear matter or not. This depends on the transition density and energy per baryon being close to nuclear matter. There is always the possibility that there is an inter- mediate transition. ~at can be definitely rules out is a transition density less than the naive overlap density. This brings up the question of the correct value for A. The transition density is sensitive to A and goes down with decreasing A, A realistic transition density (greater than overlap) will, then, fix a minimum value of A.

REFERENCES

Collins, J.C. and Perry, M.J. (1975), Phys. Rev. Lett. 34, 1353 Kisslinger, M.B. and Morley, P.D. (1976) Phys. Rev. D3, 2765 Taylor, J.C. and Frenkel, J. (1976) Nucl. Phys. BI09, 439 Shuryak, E.V. (1980), Phys. Reports 61, 72 Soni, V. (1983) a : Nucl. Phys. B216, 244 Soni, V. (1983) b : Nucl. Phys. B216, 267 Freedman, B.A. and Melcrran, L.D. (1977), I,II,III. Phys. Rev. DI6, 1130, 1147, 1169 Baym G, and Chin, S.A. (1976), Phys. Lett. B62, 241 Chapline, G and Nauenberg, M. (1977) Phys. Rev. DI6, 450 Baluni, V (1978), Phys. Rev. D17, 2092 Kisslinger, M.B. and Morley, P.D. (1979). Phys. Reports 51, 64 Murugan, V and Soni, V (1983), preprint Baym, G. (1977) Neutron stars and properties of matter at high density, Nordita preprint Kisslinger M.B. and Morley P.D. (1977) Phys. Lett. 67B, 371

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Screened charge and phase transition in QCD quark matter