Critical behaviour in baryonic matter

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Z. Phys. C - Particles and Fields 31, 167-174 (1986)

and Finds

@ Springer-Verlag 1986

Critical Behaviour in Baryonic Matter

B. Berg 1, J. Engels 2, E. Kehl 2, B. Waltl 2, H. Satz 3

1 Department of Physics, Florida State University, Tallahassee, FL 32306, USA 2 Fakult~it ftir Physik, Universitiit, D-4800 Bielefeld, Federal Republic of Germany 3 Fakult~it ftir Physik, Universit~it, D-4800 Bielefeld, Federal Republic of Germany, and

Department of Physics Brookhaven National Laboratory, Upton, NY 11973, USA Received 19 February 1986

Abstract. First we consider the phenomenology of deconfinement and chiral symmetry restoration for strongly interacting matter at non-vanishing baryon number density. Subsequently, we present numerical results obtained by a Monte Carlo evaluation of statistical QCD on an 83 x3 lattice, using Wilson fermions with N I = 2 , in fourth order hopping parameter expansion, and suppressing the imaginary part of the fermion action. We consider baryonic chemical potentials up to /~a=0.6

(I~/AL~--200);

ⁱⁿ

this range, the critical parameters for deconfinement and chiral symmetry restoration are found to coin- cide.

I. Introduction

The prediction of the phase structure of strongly interacting matter is one of the most challenging problems in statistical QCD. We expect that with increasing density, hadronic matter will be trans- formed into a plasma of coloured, massless quarks and gluons: it should undergo deconfinement and chiral symmetry restoration. The increase in density can be achieved either by heating or by compression, and hence the phase of the system will depend on both temperature and baryon number density.

In the case of vanishing baryon number density, deconfinement and chiral symmetry restoration have been investigated in a variety of lattice evaluation schemes, and the thermodynamics of such "mesonic"

matter is slowly emerging [1]. Quantitative studies of the baryon number dependence, however, have been initiated only quite recently [2, 3]; and this topic will form the main subject of our paper.

To clarify the phenomena which we want to in- vestigate, it seems helpful to first consider a simple schematic model, stripped of all but the essential dynamics. This will be the topic of the first section.

Following it, we will turn to lattice QCD at non- vanishing baryonic chemical potential p. Treating the quarks (we will consider two flavours) as Wilson fermions in low order hopping parameter expansion, we will calculate the basic thermodynamic observables and study the pattern of deconfinement and chiral symmetry restoration in baryonic matter.

II. Basic Phenomenology

Let us first look at hadron and quark systems of vanishing baryon number density. Consider an ideal gas of massless pions. Its pressure is

7~2 ~2

P~=90 x 3 x

T4=~oT4, ⁽¹⁾

taking into account the three possible charge states.

For an ideal plasma of massless quarks, antiquarks and gluons, the pressure becomes

Pq=90 • 90 T4, (2)

including two flavours (u and d), two spin orientations and three colours for quarks and antiquarks, eight colours and two spin orientations for gluons.

Comparing the two states, we note that the pressure of the plasma - with more degrees of freedom always exceeds that of the pion gas. Matter in equilibrium would thus always be in the plasma

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-B

Tc I 4

. T 4

Fig. 1. Ideal pion gas (P~) vs. ideal quark-gluon plasma with bag pressure (Pq')

P

#4

Fig. 2. Ideal Fermi gas of quarks (Pq) vs. ideal Fermi gas of massless nucleons with hard core repulsion (P~)

phase. The essential dynamical input to change this is the (non-perturbative) bag pressure B, which re- duces Pq:

Pq' = Pq-B. (3)

The result is shown in Fig. 1; there now is a cross- over, which determines

/ 90 \~/4

To= [3~2 B)

-~0.72B 1/4 (4t

as transition temperature. Including the pion mass and/or further resonant states, such as p and co, does not lead to any qualitative change of this picture;

neither do perturbative corrections to the quark- gluon plasma.

For baryonic matter, we will now consider the other extreme: T = 0 at nonzero baryonic chemical potential g. A perfect Fermi gas of massless protons and neutrons has the pressure

1 • 4 - ~4

= 24n~ 6 n2" (5)

The ideal quark plasma, with coloured u and d quarks, gives

4 @ 4 p q4

~ = • 2 9 (6)

At equilibrium, pq = p/3, and hence 1 #4

P q - 2 7 6n 2" (7)

Here we find that the nuclear matter phase domi- nates at all g. To change this, we have to take into account the repulsion between nucleons, which puts a bound on the compression of nuclear matter. F o r nucleons with hard cores of volume

VN,

the nuclear pressure becomes

& 9 (8)

/'/-- 1 + n vN' here

= 2 3 (9)

n 3~2/1

is the density of a perfect Fermi gas of nucleons.

Hence we find

~4

(I0)

P N - - 6 n 2 + 4 # 3 VN'

The behaviour of P~ vs. Pq is shown in Fig. 2: the hard core repulsion provides a much weaker growth of P~ at large/~ and thus leads to a cross-over, with

(39 n2~ 1/3 ~7.27 V N a/3 (11)

U c = \ VN !

as the critical chemical potential. - Here also a more realistic picture, with massive nucleons and bag pressure, does not lead to qualitative changes*.

We thus note that on a purely phenomenological level, deconfinement is at g = 0 basically determined by the bag pressure, while at T = 0 , it is the nucleon repulsion which is crucial.

To compare deconfinement and chiral symmetry restoration, we must allow a third phase: constituent quark matter. Consider a non-interacting gas of coloured constituent quarks of mass

mo~-~m N,

together with massless pions as Goldstone bosons; relative to the physical vacuum, there will be a confining bag pressure B e, with IBel <lBI. F o r this phase, we have a t / ~ = 0

* The nucleon mass m N does, however, determine a lower bound for the bag pressure, if there is to be a cross-over [4]: from Pq (p

= m N ) - - B > 0 we obtain B1/4>(162n2) 1/4 mN_~0.158mN_~148 MeV

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P

- B Q 84

-B

P~

c TCH

T ~

Fig. 3. Constituent quark matter (P~), compared to ideal pion gas (P~) and ideal quark-gluon plasma (Pq')

-BQ 1 -8

P

Fig. 4. Constituent quark matter (P6), compared to ideal Fermi gas of hardcore nucleons (P~) and ideal Fermi gas of massless quarks (P~')

, 7zz[21P(mo, T)

3 I T 4

PQ=90 P(O, T) + - B e '

(12)

where

P(mQ, T)

is the pressure of an ideal gas of massive fermions with one intrinsic degree of freedom. In Fig. 3, we compare (12) to the ideal pion pressure (1) and that of the chirally symmetric plasma of massless quarks. There are now in general two transitions: deconfinement at

T~=f(BQ, mo)

and chiral symmetry restoration at

TcH=f(BQ, B, me).

Whether the intermediate constituent quark phase actually occurs, or whether it leads to a pressure below P~ in the hadronic regime and below Pq' once the plasma pressure crosses P~ - the behaviour in- dicated by the dashed curve in Fig. 3 - depends on the actual values of the parameters involved. Lattice calculations [5-8] have led to T~_ Tcn and would thus support the latter case. On a phenomenological level, the question is studied in detail in [9].

At T = 0 , we must add to the properties of constituent quark matter a hard core baryonic repulsion between the quarks, characterized by an intrinsic constituent quark volume VQ(<-~VN). The resulting pressure then is

P~= Pc(me' g) Be; (I3)

l + no V~2

here

13.

m ~ (

# [ , 2 \1/2 / /./2 5~

3 i~ e

+sln [&+

⁽¹⁴⁾

denotes the pressure of an ideal gas of massive fermions [4] with two spin, two flavour and three colour degrees of freedom. Similarly,

/qQ = ~ 2 (./2 2 -- 9 rn~2 ) 3/2 (15)

is the baryon number density for the constituent quark system; here, as above, # is the baryonic chemical potential. In Fig. 4, we compare the pressure of constituent quark matter, (13), with that of the plasma of massless quarks and with that of nuclear matter. F o r the sake of consistency, we have now included the nucleon mass in calculating PN, and the bag pressure B in Pq. Again we obtain in general two transitions: deconfinement at #c and chiral symmetry restoration at /~cn. The crucial question for tattice studies thus is whether these phenomena are indeed distinct, or if they occur at the same value of the baryonic chemical potential.

III. Lattice Q C D at p ~ 0

The starting point for statistical Q C D is the partition function

Z(T, #) = Tr { e - ( n - urn/r}, (16) where H is the Hamiltonian, p the baryonic chemical potential, and N the net baryon number of the system. On an asymmetric but isotropic Euclidean lattice with N.(N~) spatial (temporal) lattice sites, the partition function becomes

Z(N~, N~; g2;fl)= f I~ d U e -s~

{det Q} Ns. (17) links

Here

SG(U) : g6~ ~ ( 1 --89 Re

TrUUUU)

(18) gives the gauge field action as plaquette sum, with U denoting the gauge group elements and g: the bare

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coupling on the lattice. The fermion determinant, det Q, results from the integration of the quark fields, with

It gives in this approximation the main quark con- tribution, since the second term amounts only to the shift

s v = ~ ~ fQ~P f (19)

f

as quark action; we consider here N I massless quark species. In Wilson's formulation [10], Q has the form

3

Q = 1 - t o ~ M ~ - 1 -~cM (20)

v = 0

with

] +(1 +7~) Um+,6, m+~, V=1,2,3 (21a) (M*)"m - J ( 1 -7o)/-7.,. b . . . . 9e""

/

+(1 +2o) U~+, c5, m+o e-u", ^, v = - 0 . (21 b) Here U,,, is the group variable associated to the link between two adjacent sites n and m; n + 9 denotes the site obtained by a unit shift in the v direction.

The strength of the quark interaction is characterized by the hopping parameter ~c(g2); a(g 2) denotes the lattice spacing. For the sake of simplicity, we shall here write all formulae for equal couplings and equal lattice spacings in space and temperature di- rections. In actual calculations, they are of course set equal only after all operations (differentiation) are carried out.

The introduction of the chemical potential in (21) follows the prescription of [2] and [11]; a more general form is discussed in [12]. A common feature of all forms is that with /~q=0, the U and U + terms in (21b) are no longer hermitean conjugates, and as a consequence, detQ becomes complex. Note, however, that Z remains real, since both ~d U and ~d U + cover the complete group space.

In the hopping parameter expansion, we obtain for the quark action

S v - N i l n d e t ( 1 - ~cM) = - N f T r _~= M', (22) which gives for N~ < 4 as leading terms

SF = --2N'(2~C) N~ 2 {L~ e~'+L*e-e"}

s i t e s x

- 16 Ns. K4 ~ R e T r U U U U + O(KS), (23) with fl=-N~a for the inverse temperature and

N ~

L ~ = T r [ I Ux;~,~+ 1 (24)

t = l

for the thermal Wilson loop at spatial site x; the sum in the first term of (23) runs over all such sites.

6/g z -+ (6/g 2 + 48 N I ~c 4) (25)

in the gauge field action. Writing the first term of the quark action as

SF(L ) = -- 4 Nj.(2 ~c) N~ ~ {ReL~ cosh fl#

x

+ i I m L x sinh fl/~} (26)

we see explicitly that it is complex for # + 0. F r o m I m Z = ~[I d U e-S~ -ResF

9 sin [ - 4 N I (2 to) s~ ~ Im Lx sinh fl #] (27)

x

we also have explicitly, by changing variables U-+U +, that I m Z = 0 and hence Z real. In this order of the hopping parameter expansion we therefore obtain for the partition function

Z(N,, N "r, g

2,

,) = ~ H d g e - Sb-ReSr(L) cos[imSF(L)] '

(28)

where S~, denotes the gauge action with the shift

(25)

and SF(L ) is given by (26).

Using the form (28), we now define the thermodynamic average of an quantity f ( U ) in the usual way

( f ) - - ~H d U e- S'~-ReSF(L) f cos(ImSr(L))/

f I~d U e- *''~- ReS~,(L) cos(Im SF(L)). (29) F r o m this, we see immediately that

( I t a L y ) = 0 Vx, (30)

(ImSF(L)) ~ ( ~ I t a L y ) =0, (31)

x

since ImL~ changes sign under the transformation U-+U +. Consider now ( I m L ~ ) ; the integration over U, according to (29), gives us the average over configurations. On the other hand, ~ I m L ~ is the lattice

x

average for a given configuration. For large enough lattices and sufficiently many configurations we expect these two averages to agree, if we are not at a critical point: we can then imagine the large lattice to be obtained by combining sufficiently many equilibrium configurations on a smaller lattice.

These arguments have led to the approximation of

"partial quenching", in which we set I m S v = 0 every- where [3]. In this case, the usual Monte Carlo evaluation can be carried out with e x p { - S ' ~ - R e S v ( L ) } as weight, and all results to be presented here are

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obtained in this way. Thus now, with partial quenching,

( f ) R = S H d U e- s ; - ReSF(L)f/S H d U e- s'~- R e S F ( L ) (32) defines the t h e r m o d y n a m i c average.

We had initially considered it possible to test this partial quenching by calculating

( f ) = ( f cos(ImSv(L)))R/(cos(ImSF(L)))i~, (33) i.e., by including cos(ImSF(L)) as part of the observ- able whose average is to be calculated. It turns out that this procedure is not feasible, for the following reason. On an 8 3 x 3 lattice, we find even after 30,000 lattice sweeps, that ( ~ I m L x ) is of order

x

unity, rather than zero, as required in (31). Hence a Monte Carlo evaluation of (33) without importance sampling to assure ( ~ I m L x ) = 0 cannot be expected

x

to give reasonable results, since the values of cos(ImSF(L)) obtained with e x p { - S ' - R e S F ( L ) } as G weight for the Metropolis algorithm fluctuate wild- ly.

If we ignore these difficulties and just calculate e.g. ( R e L ) according to (33) with e x p { - S ' G

- R e S F ( L ) } as M o n t e Carlo weight, we obtain ( R e L ) -~ ( R e L ) R , together with (cos(ImS~.(L))) ~-0:

a smoothly varying function, such as ReL, becomes uncorrelated from the randomly fluctuating cos(ImSr(L)), and hence can in good a p p r o x i m a t i o n be taken outside of the integral. Thus the agreement between ( R e L ) and ( R e L ) R is here simply a consequence of the fluctuations of cos(ImSr(L)). In this situation, partial quenching appears to be the most reasonable procedure to follow. As all averages from here on are defined by (32), we shall now drop the subscript R on the averages ( )R.

To evaluate t h e r m o d y n a m i c observables as func- tions of p and T, we need explicit expressions for

~c(g 2) and a(g2). F o r the hopping parameter, we shall use the weak coupling form E13]

~:(g2) = ~ [1 + O. 11 g2 + 0 (g4)], (34) and for the lattice spacing the renormalization group relation with N I = 2

~ _ 4 n 2 (g6) 345 [8n 2 ( ; ) ] } a ( g 2 ) A L = e x p ( 29 + ~ l o g L 29

(35) In b o t h cases we expect some deviations at the g2 values actually used; but these relations should suf- fice to give us at least a reasonable qualitative im- pression of the resulting critical behaviour.

IV. Numerical Results

Our evaluation was performed on an 83x 3 lattice with Nj.=2. We have included terms up to ~c 4 in the hopping p a r a m e t e r expansion (23). F o r each g2 value, we carried out about 3,00(~4,000 lattice sweeps.

Using these results, we have studied the T and # behaviour of the thermal Wilson loop ( R e L ) as deconfinement measure, of ( ~ 0 ) as chiral symmetry measure, and of the overall energy density e.

In our study, we have considered that baryonic chemical potential in the range 0<pa__<0.6, i.e., up to about # ~ 3 0 0 - 4 0 0 MeV. The reason for stopping here is given by the truncation of the hopping parameter expansion: increasing p has a similar effect as increasing ~c and hence necessitates the inclusion of more terms in (22). To obtain some idea of the error made by including only terms up to order ~c 4, we have calculated the energy density of an ideal Fermi gas on an 83x 3 lattice for various # values and c o m p a r e d the results with those given by the hopping p a r a m e t e r expansion up to ~c 4 for this quantity [14]. The ratio

K4[~3 )r163 3)/sfuu(83 X 3) (36)

8 F ~,v

varies from 0.96 to 1.09 as # a is increased from 0 to 0.6; in the # range considered, the truncation error thus is 10% or less for an ideal Fermi gas. F o r larger #, the error increases; we have therefore only gone up to # a = 0 . 6 . The full result obtained on an 83 x 3 lattice is of course not identical with the ideal gas value in the continuum [15]. However, up to pa

=0.6 the difference between the continuum value and the lattice results with N~= 3 is essentially inde- pendent of #.

Let us note at this point one of the disadvan- tages of the hopping p a r a m e t e r a p p r o a c h : the truncation error is evidently N~ dependent, and hence this a p p r o a c h is not very suitable for studying the scaling behaviour of t h e r m o d y n a m i c observables.

In Fig. 5, we now show the deconfinement measure ( R e L ) as function of 6/g 2 for different # a values. There is a clear shift of the deconfinement point to lower 6/g 2, i.e., to lower temperatures, as the baryonic chemical potential increases. At the same time, the change in regimes becomes less ab- rupt with growing pa. - In Fig. 6, we show the onset of deconfinement obtained at fixed t e m p e r a t u r e by increasing #a. It is seen that a variation of # results in a behaviour quite similar to that obtained for a variation in T, so that deconfinement can indeed be induced either way.

The behaviour of the overall energy density, e==-{T2(~lnZ/c?T)u,v+#T(OlnZ/c?#)T,v}/V (37)

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<ReL>

1.5 -- #o ~ 0.600

0488

/ ~ ' 0,400

/ / ~ 0.a30

1.0 ~ 06~00

0.5

0 I I I I I I I I

5.0 5.2 5.4 5.6

6 / g 2

Fig. 5. ( R e L ) vs. 6/g 2 for different # a . C u r v e s are o n l y to g u i d e the eye; n a i v e s t a t i s t i c a l e r r o r s are s m a l l e r t h a n t h e d a t a p o i n t s

<ReL>

1.5

1.0

0.5

I I I I I I I I

0.0 O. 2 0.4 0.6 0.8

# o

Fig. 6. < R e L ) vs. # a at 6 / g 2 = 5 . 2 ; the c u r v e is o n l y to g u i d e the eye

calculated for different # a values as function of coupling 6/g 2 a n d t e m p e r a t u r e T/AL, is shown in Fig. 7.

It is n o r m a l i z e d here to the value esB(8~4 3 x 3) for an ideal gas of q u a r k s and gluons, also calculated on an 8 3 x 3 lattice in 4th order h o p p i n g p a r a m e t e r expansion. T h e lattice evaluation p r o c e d u r e of b o t h e and

~4 (~3 3) is described in [16]. We note in Fig. 7 SSB~U X

again a rapid d e c o n f i n e m e n t transition, b e c o m i n g slightly "softer" with increasing #a. In Fig. 8, we show the c o r r e s p o n d i n g b e h a v i o u r as function of # at a fixed value of T.

W e can n o w use either the e results or those for ( R e L ) to determine the transition parameters. At

T c, Cv~(C%/OT ) s h o u l d b e c o m e singular; < R e L ) should b e c o m e exponentially small there. In Fig. 9,

o.1 9 cO v ,.T on

~ 0

IO

4.8 Î 5.0 Î Î 5.2 Î Î 5.4 Î Î 5,6 Î

6 / g 2

v

O

b _I _I _I

100 150 200

TIAL

Fig. 7 a , b. E n e r g y d e n s i t y e,, n o r m a l i z e d to the c o r r e s p o n d i n g ~c 4 ideal gas v a l u e o n t h e s a m e lattice, vs. 6/g 2 a a n d vs. t e m p e r a t u r e T/A L b

A .'h"

~O v .., m

cO

Ru'l 1.C

0.5

I

j

I I I I I

0.2 0.4 0.6 ^I 0.8 ^I

# a

1.C

0.5

j

I I I I I I

0 IO0 200 30O

#IAL

Fig. 8 a , b. E n e r g y d e n s i t y 8, n o r m a l i z e d to the c o r r e s p o n d i n g ~:4 ideal g a s v a l u e o n t h e s a m e lattice, vs. # a a a n d vs. #/A L b

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5C

40

3C

20 I I

5.0 5.2

t i T 4 ~ ( r 4 )

60

5.4 I 6/g 2

O(6/g 2)

~00

150

100

50

Fig. 9. Energy density and its derivative (~specific heat) vs. 6/g 2 at/~a =0.488

T IA L l

150~L

125 -

t O 0 - I"

0

X 9 9

X

tt

X

I I I

100 200 ^I ^I

lz /AL

Fig. 11. Phase diagram for deconfinement ( x ) and chiral symmetry restoration (e)

ReL > a (ReL)

1.5 8 0 (61gZ)

1.0 6 2

~

5.0 5.2 5.4

o

6 / g z

Fig. 10. ( R e L ) and its derivative vs. 6/g 2 at/za=0.488

3.3

3.2E

3.2

:o ;oO0,O ~

a t l I I I I

5.0 5.2 5.4 5.6

6 / g 2 Fig. 12. ( ~ / ) vs. 6/g 2 at different #a

Table 1. Critical parameters for deconfinement and chiral symmetry restoration, as obtained on an 8 3 x 3 lattice

l J a lz/Ar 6/g~ 2 TJA L 6/g~ TJA L 6/g~H TcH/AL

(from e) (from ~) (from ReL) (from ReL) (from ~ ) (from ~ff)

0.1 45 5.299 151 5.288 149 5.290 150

0.2 89 5.282 148 5.294 150 5.292 150

0.33 139 5.239 140 5.239 140 5.243 141

0.4 161 5.194 132 5.214 136 5.218 136

0.488 184 5.t51 125 5.158 126 5.156 t26

0.6 203 5.063 112 5.077 114 5.081 114

w e s h o w as a n e x a m p l e t h e b e h a v i o u r o f e / T 4 vs. 6/g z t o g e t h e r w i t h its d e r i v a t i v e . T h e c r i t i c a l c o u p l i n g s t h u s o b t a i n e d , t o g e t h e r w i t h t h e r e s u l t i n g c r i t i c a l t e m p e r a t u r e s , a r e l i s t e d in T a b l e 1. A l s o l i s t e d t h e r e a r e t h e c o r r e s p o n d i n g p o i n t s o f m a x i m u m v a r i a t i o n o f ( R e L ) , a n d t h e t e m p e r a t u r e o b t a i n e d

f r o m t h e m ; a n i l l u s t r a t i o n o f t h i s f u n c t i o n a l b e - h a v i o u r is s h o w n in F i g . 10. It is s e e n t h a t t h e t w o d e t e r m i n a t i o n s o f T~(#) a g r e e q u i t e well, l e a d i n g t o t h e v a r i a t i o n o f T~ w i t h p as s h o w n in F i g . 11. A t t h e h i g h e s t v a l u e o f t h e b a r y o n i c c h e m i c a l p o t e n t i a l s t u d i e d h e r e ( p / T = l . 8 o r # / A L ~ _ 2 0 0 ), t h e c r i t i c a l

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t e m p e r a t u r e has d r o p p e d by a b o u t 25 ~o:

Tr (# = O)/T~ ( # / A L ~- 200) ~ 0.74. (3 8) F i n a l l y , we w a n t to c o n s i d e r chiral s y m m e t r y resto- r a t i o n a n d its r e l a t i o n to d e c o n f i n e m e n t . As is k n o w n , the W i l s o n f e r m i o n f o r m u l a t i o n is n o t ideal for this purpose, since the chiral s y m m e t r y m e a s u r e ( ~ b ) never vanishes o n a finite lattice. Nevertheless, for # = 0 it is f o u n d to show a rapid v a r i a t i o n pre- s u m a b l y related to the onset of chiral s y m m e t r y r e s t o r a t i o n 1-16], a n d it therefore a p p e a r s m e a n i n g f u l to study the # - d e p e n d e n c e of this variation. T h e results are s h o w n in Fig. 12; we see again a clear shift to lower t u r n i n g p o i n t values of 6/g 2 with in- c r e a s i n g / ~ a . D e f i n i n g a g a i n the critical p a r a m e t e r as that given by the p o i n t of m a x i m u m v a r i a t i o n , we o b t a i n for ( ~ b ) the values s h o w n in T a b l e 1. T h e y are seen to agree very well with those o b t a i n e d for the d e c o n f i n e m e n t point. W e therefore c o n c l u d e that o u r results p r o v i d e u p to # a = 0 . 6 a c o m m o n p o i n t of d e c o n f i n e m e n t a n d chiral s y m m e t r y restoration.

I n Fig. 11, we have the resulting phase d i a g r a m for b o t h d e c o n f i n e m e n t a n d chiral s y m m e t r y restoration, as it emerges form the t r a n s i t i o n p a r a m e t e r s listed in T a b l e 1.

Acknowledgement. We thank the Bochum Computer Center and SCRI, Tallahassee, for providing us with facilities (CYBER 205) and computer time.

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