Thermodynamics of massless quarks and gluons Finite-size hadron formation and phase transition to the quark-gluon plasma state

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T H E R M O D Y N A M I C S OF MASSLESS QUARKS AND GLUONS. FINITE-SIZE H A D R O N F O R M A T I O N AND PHASE T R A N S I T I O N TO THE QUARK-GLUON PLASMA STATE Jishnu DEY Hooghly Mohsin College, Chmsurah, WestBengal, India

Mira DEY Department of Physlcs, Maulana Azad College, Calcutta 700 013, India

and Partha GHOSE Brittsh Deputy High Commission, British CounctlDivision 5, ShakespeareSaranL Calcutta 700 071, India

Received 31 August 1988;revised manuscript received 31 January 1989

The free energyF of massless quarks and gluons confined in a spherical bag has been calculated by computingthe discrete allowed states and taking appropriately weighted sums. The behaviour of F as a function of the bag radius R shows a unique solution R upto a transition temperature T~beyondwhich the model shows a first-order phase transition to QGP (R=~). We also find a limiting temperature To> Tsbeyondwhichhadrons cannot exist.

In their original paper introducing the bag model, Chodos et al. [1 ] considered a relativistic gas of quarks confined in a bag of finite size. In the thermodynamic limit ( V ~ o o ) a Hagedorn-like behaviour for the mass spectrum was predicted, and for bags of finite size certain corrections to these were indicated. The precise form of this mass spectrum in the thermodynamic limit was Obtained by Kapusta [ 2 ]. The finite-size corrections were first calculated by Jennings and Bhaduri for gluons in a cubic bag and by Bhaduri et al. for quarks in a spherical bag [ 3 ]. In these papers the effective smooth single-particle densities g(e) for gluons and quarks were calculated to include finite-size effects. This was done by approximating the calculated discrete sums over states in a finite bag by integrals over energy. Such approximations are valid for large values of TR where R is the radius of the bag at temperature T. For small R such approximations give reliable results for large enough T. However, hadronic bags are small (R = 0.5 fro) and the critical temperature for the phase transition to the 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

quark-gluon plasma phase (QGP) is around 200 MeV as indicated by lattice gauge calculations. So we expect T R < 1. An expansion in inverse powers of TR for a realistic bag is therefore highly questionable. In this paper we calculate the free energies and hence all thermodynamic properties of small bags directly from the computed discrete allowed states of quarks and gluons in such bags without either approximating the sum over discrete states by integrals or resorting to an expansion in inverse powers of TR. These discrete states have been obtained by solving the equations of motion with linearised boundary conditions. The extrema of the free energy lead to a set of solutions for the bag radius R (T). For T less than a transition temperature Ts, there is only one solution for R. For T > Ts there are two solutions for R. The smaller of the two corresponds to a minimum of F and therefore to a hadron, whereas the larger solution corresponds to a maximum of F and is thermodynamically unstable. There is also a critical temperature To> T, beyond which there are no extrema o f F 161

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and hence no stable solutions or hadrons. Consider massless quarks and gluons confined in a bag of radius R. In the limit of SU (3) flavour symmetry we can work with massless quarks. We set the chemical potential/ae=0, and take only a mesonic system (n +, n-; k +, k - etc.) with an equal number of quarks and antiquarks. This corresponds to considering the central region of a heavy-ion collision which has the quantum numbers of the QCD vacuum. Throughout this paper we shall consider such a hadronic system only. The kinetic energy of quarks and gluons enclosed in a cavity gives rise to an effective temperature T. Unless probed by colourless objects like pions, the system remains isolated at a fixed temperature. Our aim is to place hadrons in heat baths with which they can exchange energy, and study their properties as the temperature is varied. These changes are studied in the framework of the conventional bag model with linear boundary conditions. The price to pay for this linearisation is the coupling of the quarks to an "induced" axial gauge field [ 4 ]. The energetic collision of hadrons (as, for example, in heavy-ion collisions) can produce a hot gas of pions [ 5 ] which can couple to the quarks and provide the heat bath. The discrete energy levels q, of confined massless quarks and gluons can be found by solving the equations of motion with appropriate boundary conditions. The energy eigenvalues for quarks with p = (1, 2) for J = ( L - ½, L + ½) are given by JL (qP,L R ) = ( -- )PJL+ ~(qPLR) •

( 1)

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dc q, EG = -~- +8 ~ e x p ( q J T ) - 1 - - d o / R + EG( T) ,

(3b)

where dQ= 0.03 and d o = 0.37 are coefficients of the zero-point energies for quarks [ 6 ] and gluons [ 7-10 ]. We take dG from Brevik et al. [ 10]. The factor 36 is the degeneracy for an equal number of q and ~t (/tr= 0) of three flavours and colours; the factor 16 likewise is the degeneracy of gluons. Therefore, the total bag energy is given by Eb,g =EQ +EG + B V ,

(3c)

where B is the bag parameter. The single-particle levels are calculated from eqs. ( 1 ) and (2). Eq. (3) give us the temperature-dependent part of the energy E ( T ) for a given R. The sum over the states in eq. (3) is truncated at an energy of 4 5 / R MeV. A truncation at 2 0 / R is sufficient for resuits good upto the fourth significance figure. This means, for example, summing over 86 levels for gluons and 95 for quarks. The free energies of quarks and gluons are given by FQ= -g dQ - 1 8 T ~

ln[l+exp(-q,/T)]

(4a)

Fo= ~d~ - +8T~ ln[1-exp(-q,/T)],

(4b)

and

i

respectively. The total free energy of the bag is then

For gluons we have the TM mode fb,g =FQ + f ~ + B V . (q, LR)jL_~ (q,LR) = L j L ( q , L R ) .

(2a)

In addition the gluons have another mode (TE) given by jL(q,,LR) = 0 .

(2b)

The energy of quarks and gluons are given, respectively, by EQ=~

Once the free energies are obtained one can evaluate the pressure from the relation P = - ( OF / O V ) r = - B + Pg,s,

where it is trivial to check that Pus V - 1Egas = ~ (EQ +EG) •

P=0 (3a)

and so Pgas = B •

(5b)

(6)

This leads to Ebag=4BV.

162

(5a)

For hadrons we must have

q, +18 ~ e x p ( q i / T ) + l

=_do/R+EQ(T ) ,

(4c)

(7)

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and B 1/4= 200 MeV. It is clear from these plots that at low temperatures below T = 150 meV F has a minimum. Between 130-150 MeV there are two extrema of F. The minimum point occurs at a lower value of R and corresponds to a possible hadron, whereas the maximum occurs at a higher value fo R and corresponds to an unstable state because (OP/OV)T>O here. At around T = 150 MeV the two extrema coalesce into a degenerate critical point where the second derivative of F vanishes, that is,

At T = 0 we get the QCD vacuum condition B = d/4~R 4

27 April 1989

(8)

from eq. ( 5 ). For 31/4 __200 MeV we get an equivalent R-- 0.42 fm. This shows that the bag pressure is related to the Casimir energy. Since do >> dQ, it follows that this Casimir energy is predominantly due to gluons. This is consistent with the qualitative result obtained by Shifman et al. [ 11 ] from QCD sumrules which lead to a vacuum energy density l el = (240 MeV)* generated by gluon condensates

(0P/0V)~=0.

< GG>.

(9)

This and the fact that F h a s no extrema above T = 150 MeV tells us that this is the critical isotherm for deconfinement. This is also clear from fig. 2, which is a plot of the hadron radius against temperature the curve turns at T = To---150 MeV. It is remarkable that the simple thermodynamics of the bag is able to give us a Tc quite consistent with lattice gauge theories. (This does not however mean that the model is able to reproduce all results oflanice gauge theories. ) Also, we find that around T = 132 MeV the maximum in F shifts to a very large radius. For such large bags the

Our aim is to study how this vacuum behaves as the temperature is raised. This is expected to happen in the central region of heavy ion collisions. Eqs. (3) and (4) indicate that the quark and gluon levels will be populated as the temperature rises forming hadronic systems whenever the equilibrium condition given by eq. (6) holds. We shall presently see that there is a critical temperature To beyond which no such hadronic solutions exist. Our results are displayed in fig. 1, which is a plot o f F as a function of R for three typical temperatures 350

&

300

)

A

LL 250

200 -

0.4

I O.S

I 0.6

I 0.7 R (fro)

I 0-8

I 0-9

--~

Fig. 1. Free energy F as a function of R for three different temperatures. The arrows denote the extrema in F.

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This essentially completes the derivation of om main results. Having already computed F and E, we found it instructive to fit them to polynomials in R and T. This enables us to establish contact with previous work [3,12-15] and obtain analytic expressions for the critical constants. We are interested in a radius o f about 0.5 fm and in a temperature range o f 100-150 MeV, that is, TR < 1. In this range the dominant quark energy fits very easily to a form (see table 1 )

.90

.80 B ~/4=200MeV ~.70 E

EQ( T) = ½aQR3T4 q- bQ T - C Q R T 2 .

(12a)

n,,,

whereas for gluons a similar fit .60

E~( T) = ] a c R 3T 4 + bc T - C o R T 2

150 le

.40 "T~h ,

j

110

i

120

,Ts ,

130

1

140

"-~ 150

T (MeV)

is not as good. This is not surprising. Balian and Duplantier [ 9 ] had also shown that the low-temperature expansion o f the gluon free energy is limited to a very narrow range of TR. Since real gluons contribute very little at this low temperature, a fit o f the form (12b) is adequate for our purpose. The total energy E of the bag with pressure B is then given by

Fig. 2. The bag radius as a function of temperature for extrema of free energy Fcorresponding to P= 0. The lower curve for T~to T~corresponds to minima in F and the upper one to maxima in F.

Ebag ~ l a R 3 T 4 + b T - c R T Z + d / R + B V ,

thermodynamic limit is valid and in such a limit

b = bc + bQ = 0.965 + 0 . 8 5 5 6 ,

F= - l aR3T4+BV,

c=Cc+CQ=5.166+5.79,

(10)

21_3 a = a G + a Q = ~3n2 _ 3 - --r~-n = -~-3

d = d ~ + dQ=0.37 +O.03 .

a=~Tt 3 .

The entropy S is given by

Ts = ( 12nB/a)l/a=o.66B~/4 .

Notice that unlike in refs. [ 12-15 ] this can be clearly seen as a transition o f a small bag (Rs~ 0.43 fm) to a large unstable bag and therefore to a Q G P state. Notice that To> T~ and corresponds to deconfinement. When placed in a heat bath at T = Ts a hadron can absorb sufficient latent heat and make a transition from R = R ~ to R = c e . It is a first-order phase transition. On the other hand, if a quark-gluon soup at a temperature higher than T¢ is cooled it may form hadrons with radius R~ at T~. The region T~ to T~ is metastable and corresponds to supercooling (superheating). 164

,

S = 4aR 3 T 3 - 2 c R T + b In R T + b' , ( 11 )

(13a)

with

where

This yields a transition temperature [ 12-15 ]

(12b)

(13b)

(14)

where the integration constant has been written as b In R + b'. The free energy fit is then F~E-

TS

= B V - ~aR 3T4 + b T( 1 - l n R T ) -b'T+cRTa+d/R,

(15)

where b' = b~ + b~ = 3.48 + 3.2735. To show the relative importance of the various temperature-dependent finite-size terms we have listed them in table 2 for a typical case. These numbers clearly show the importance o f the finite-size terms -

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Table 1 Temperature-dependent part of the calculated and fitted energyE(T) and E' (T) and free energyF(T) and F' (T) for quark and gluon states in a spherical cavity of radius R = 0.5 fro. The zero-point energyd/R = 157.86 MeV. T (MeV)

EQ Eb FQ

F~ Ea E~, F~ F~

110

120

130

140

150

20.05 20.00 - - 1.80 - 1.79 1.44 0.28 -0.14 -0.16

37.78 37.84 --5.47 -5.46 3.34 1.68 -0.36 -0.24

65.05 65.00 - 10. l 1 - 10.10 6.84 6.68 -0.80 -0.57

104.5 103.5 - 17.28 - 17.24 12.73 16.33 - 1.59 - 1.46

158.7 155.7 -27.77 -27.60 21.96 31.82 -2.90 -3.23

Table 2 Contributions to energyand free energy from various terms in the polynomialfit at R=0.5 fm, T= 125 MeV.

EQ FQ Eo Fo

Volume term

b term

b' term

c term

Total

172.40 -- 57.47 87.57 -29.19

106.95 229.91 120.62 259.31

-- 409.19 -435.00

- 229.23 229.23 -204.53 204.53

50.12 -- 7.51 3.67 -0.35

the usual volume term alone is totally inadequate to account for the computed free energy (energy). The pressure is given by

P= - (OF /OV) r= - B + Pga~

t

= - Bert(T) + Pga~,

(16a)

(16b)

TsRs= b+

Beff( T ) = B - aT4/12n

( 17 )

decreases with temperature [ 12-15 ]. Such a n effective B is also expected qualitatively from Q C D sumrules [ 16 ]. As before for hadrons we get

Eb~g=4BV.

(18)

We get from eq. (9) -

3b+x/9bZ+32dc 4c

b2c2 ~

=0.29.

(21)

(16c)

where

R~T~ =

(20)

one can also calculate TsRs i n d e p e n d e n t of the bag pressure since at this temperature (defined by Be~(Ts) = 0 ) the finite-size corrections cancel a m o n g themselves to give

=-B+aT4/12n + b T / 4 n R 3 - c T 2 / 4 n R 2 +d/4nR 4

Tc=O,7456B 1/4

=0.422.

(19)

This is i n d e p e n d e n t of the bag pressure B. This along with eq. (16) leads to

At Ts= 132.4 MeV, the transition from R s = 0.434 fm to an infinite radius involves an absorption of latent heat L = 7 2 5 . 6 M e V / f m 3. At To= 149 MeV the transition from 0.6 fm to Q G P involves a change in energy density of 170 M e V / f m 3. The effect of perturbative one-gluon exchange between quark a n d gluon pairs can be estimated as shown by Kapusta [ 17 ] and used by Bhaduri et at. [ 3 ] a n d recently by G a g n o n a n d Marleau [ 15 ]. We follow ref. [ 15 ] a n d use Amom= 100 MeV. The temperatures T~ a n d Tc change only by a few per cent to 137 a n d 155 MeV, respectively, with the corresponding radii 0.424 a n d 0.532 fm. Some of these numerical results are qualitatively similar to those obtained from Q C D lattice calculations [ 18 ]. 165

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We have not projected out good colour singlet states for the quarks a n d gluons which m a y be justified only for the large-volume t h e r m o d y n a m i c limit as discussed by Anishetty [ 19 ]. It is comforting however to note that the real gluons due to finite t e m p e r a t u r e are u n i m p o r t a n t . As discussed b y K a p u s t a a n d m o r e recently by G a g n o n a n d M a r l e a u [ 15 ], the p r o p e r colour projection will reduce the allowed n u m b e r o f states and decrease the c o n t r i b u t i o n further. In case o f the mass spectrum this affects only the pre-exponential factor, whereas finite-size terms are m o r e imp o r t a n t and change the exponential. We have p a i r e d quarks o f each flavour with antiquarks o f the same flavour in the same spirit as Bhaduri, D e y a n d Srivastava [ 3 ]. They o b t a i n a reasonable fit to the observed meson spectrum without projecting colour singlet states. We thus hope that colour projection m a y not destroy the qualitative picture o f the phase transition that we obtain. Recent calculations o f Skagerstam [20] seem to support this. In s u m m a r y , we generate the energy (free energy) for discrete states o f massless quarks a n d gluons for a spherical cavity o f radius R at a given t e m p e r a t u r e T. The a d d i t i o n o f the suitable r e n o r m a l i z e d zero-point energies o f these particles ( p r e d o m i n a n t l y the gluon C a s i m i r energy) gives us the pressure equation. This yields the radii a n d t e m p e r a t u r e s o f stable h a d r o n s as well as the transition points o f h e a t e d h a d r o n s to QGP. The inclusion o f chemical potentials is u n d e r consideration. This work is s u p p o r t e d in part by a U G C grant F10-45/87 ( S R - I I I ) . It is a pleasure to acknowledge useful correspondence with R.K. Bhaduri. We are also grateful to R.L. Jaffe for a useful discussion during the I C P A - Q G P conference in B o m b a y (1988). P.G. would like to thank the British Council for their encouragement a n d support. J.D. a n d M.D. wish to t h a n k Professor R. Vinh M a u a n d the m e m b e r s o f the theory division, IPN, for hospitality at Orsay for a month. M.D. wishes to acknowledge a week's visit

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to Saclay a n d useful discussions with Professor M. R h o a n d Professor J.P. Blaizot, a n d especially with Professor R. Balain.

Note added. It has been p o i n t e d out b y Schrieffer et al. [21] a n d Lee [22] that models like the Q C D bag could be relevant in u n d e r s t a n d i n g p h e n o m e n o l ogy o f high To superconductors. The general idea o f working out bag t h e r m o d y n a m i c s , possibly in lower dimensions, could therefore be relevant for such studies. References [ 1] A. Chodos et al., Phys. Rev. D 9 (1974) 3471. [2] J.I. Kapusta, Phys. Rev. D 23 ( 1981 ) 2444. [3] B. Jennings and R.K. Bhaduri, Phys. Rev. D 26 (1982) 1750; R.K. Bhaduri, J. Dey and M.K. Srivastava, Phys. Rev. D 31 (1985) 1965. [4] M. Rho, Topological objects in hadron physics, Saclay preprint. [ 5 ] J.I. Kapusta and K.A. Olive, Nucl. Phys. A 208 ( 1983 ) 478. [6] K. Milton, Phys. Rev. D 27 (1983) 439. [7] T. Boyer, Ann. Phys. 59 (1970) 474. [8] J. Schwinger, L. de Raad and K. Milton, Ann. Phys. 115 (1979) 1; J. Baacke and Y. Igarashi, Phys. Rev. D 27 (1987) 460; J. Baacke and A. Schenk, Z. Phys. C 37 ( 1988 ). [ 9 ] R. Balian and D. Duplantier, Ann. Phys. 112 ( 1978 ) 165. [ 10 ] I. Brevik and Kolbenstvedt, Ann. Phys. 143 (1982) 179. [ 11 ] M.A. Schifman et al., Nucl. Phys. B 147 ( 1979 ) 505. [ 12 ] R.D. Pisaraki, Phys. Lett. B 110 ( t 982) 155. [13]F. Takagi, Phys. Rev. D 35 (1987) 2226; Z. Phys. C 37 (1988) 259. [ 14] S.A. Chin, Phys. Lett. B 78 (1978) 552. [15] R. Gagnon and L. Marleau, Phys. Rev. D 36 (1987) 910. [ 16 ] J. Dey and M. Dey, Phys. Lett. B 176 (1986) 469. [ 17] J. Kapusta, Nucl. Phys. B 196 (1982) 1. [ 18] E.V. Shuryak, Phys. Rep. 115 (1984) 269. [ 19] R. Anishetty, J. Phys. G 10 (1984) 419; see also G. Auberson, L. Epele and G. Mahoux, Phys. Lett. B 153 ( 1985 ) 303; Nuovo Cimento 94A ( 1986 ) 1; G. Auberson et al., Saclay preprint/85/166. [20] B.S. Skagerstam, Z. Phys. C 24 (1984) 97. [ 21 ] J.R. Sehrieffer et al., Phys. Rev. Lett. 60 (1988) 944. [22] T.D. Lee, Nature 330 (1987) 460.