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The quark-hadron phase transition in the early universe
Volume 119B, number 1,2,3
PHYSICS LETTERS
16 December 1982
THE QUARK-HADRON PHASE TRANSITION IN THE EARLY UNIVERSE E. SUHONEN NORDITA, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark and Department of Theoretical Physics 1, University of Oulu, Oulu, Finland Received 19 July 1982
By using perturbative QCD with bag corrections for quark-gluon plasma and a resonance gas model for hadronic matter, we study the effect of the phase transition on the cooling rate of the early universe.
With increasing energy density, a hadronic matter is expected to undergo a phase transition to a q u a r k gluon plasma, There has been much speculation that both plasma formation and the subsequent transition may take place in ultra-relativistic heavy ion collisions [ 1 ]. The most obvious situation for such a transition is however in the early universe. The standard cosmological model [2] allows one to extrapolate back to times when the universe probably was in a plasma state. Due to the expansion the energy density decreased and there was an era when the transition from quarks and gluons to hadrons occurred. The Yang-Mills theory on the lattice is the only presently available description of a strongly interacting matter for the whole temperature range. It indeed predicts two phases and a transition [3]. Above the critical temperature, the results obtained in the lattice evaluation are in full agreement [4] with those provided by a bag model approach. To describe hadronic matter one may use Walecka's relativistic mean field theory [5], resonance gas model [6] or potential models [7]. With a suitable two-phase input the equilibrium between both phases can be determined [8-11 ] from the crossover of the pressure according to Gibbs' rules. The standard cosmological model is based on the Einstein field equations which in the RobertsonWalker metric are [12]
1 Permanent address. 0 031-9163/8210000--0000/$02.75 © 1982 North-Holland
R -2 [(dR/dr) 2 + k] - ~ r O e = O,
(1)
R 3 dP/dt = (d/dt)[R3(e + P)] = 0 .
(2)
Here R is the cosmic scale factor, t is the proper time measured by a comoving observer, k has the value +1, - 1 , 0 for a closed, open or flat universe, respectively, G is the gravitational constant and e and P denote the total energy density and the isotropic pressure. Eq. (2) which results from the energy conservation can be written equivalently, e -1 de/dt + 3 (1 + P / e ) R - 1 dR/dt = 0 .
(2a)
When supplemented by the equation of state, eqs. (1) and (2) are sufficient to describe the evolution of R ( t ) in the early universe. To get the equation of state for a strongly interacting matter we assume that the quark-gluon phase is described by perturbative QCD up to the first order in the running coupling constant, together with a bag term to account for the nonperturbative influence of the physical vacuum. For the hadronic phase we consider an ideal gas of ground state hadrons and all possible resonance excitations. The resulting energy density in such a two-phase picture is not unlike [10] that found in the lattice evaluation. At a high temperature T the model universe conrains bosons and fermions with masses m i ~ T and multiplicity factors gB and gF respectively, at thermal equilibrium. Since the asymmetry between baryons and antibaryons is exceedingly small we can limit ourselves to the case of a vanishing chemical potential. 81
Volume lI9B, number 1,2,3
PHYSICS LETTERS
the radiation and lepton terms added, we obtain with the choice of a = 4 from eqs. (8) and (9)
The pressure Per and energy density eer of such an extra relativistic gas are given by l
1
1 _2,7, 4
Per =~eer = ~ g ~
i
,
g=gB +~gF •
/
(3.4)
The particles which contribute to g at temperatures of interest, T = O(100 MeV), are photons 7 and leptons e ±, #±, ve, De, v~, Du, vr and Dr in both phases and in addition gluons and two quark flavours in the plasma phase. These yield the values 51.25 and 14.25 for the total multiplicity factors of extra relativistic partides in the plasma and hadronic phase, respectively. By taking into account also the contributions of strange quarks and antiquarks, and interactions up to the order % = g2/4zr, together with the bag term B, we have Pp = (51.25/90)~r2T 4 + !~-aszrT4 +Ps - B,
(5)
ep=(51.25/30)zt2T 4 +Tas~rT 11 4 +-~-asT a19 2 4 + e s + B , (6) for the energy density in the plasma phase. For the pressure Ps and energy density e s of strange quarks and antiquarks we use the integral expressions given by Kapusta [13] with the value 280 MeV for the mass of the strange quark. In the case of three quark flavours the running coupling constant is
a s = 21r/91n(4T/A),
(7)
where A is the scale-fixing parameter. The partition function of a resonance gas is given by lnZ(T,V)-
V
16 December 1982
; dm r(m)
(2~r) 3 o
Ph = (TE/2~r2)
,c f
2m~
(3m2K2(m/T)
am
+ (14.25/90),r2T 4 for the pressure in the hadronic phase. The corresponding energy density becomes e h = 3P h + (T/2n2)(3ma~KI(m/T)
+c ; dmm-lexp(m/Te)Kl(m/T)) 2mn + (14.25/30),r2T 4 .
(8)
where the mass spectrum r(m) has from hadron dynamics [14] the expression r(m) = 38(m - m~) + cO(m - 2mn)m-ae bm ,
(9)
with constant a, b and c and the pion mass mzr. It is well known [15] that the spectral form (9) leads to critical behaviour with Tc = lib as the critical temperature. The order of the associated transition is determined by the parameter a. For 7/2 < a ~< 9/2 we have a finite energy density and diverging specific heat, consistent with lattice calculations [3,16]. With 82
(11)
The equation of state is given by eqs. (5) and (6) for T ~ Tc and by eqs. (10) and (11) for T < Tc. In the equations of state we have four as yet unfixed parameters. For the scale parameter A we take the usual value 100 MeV. As the critical temperature we use the values 180 MeV and 190 MeV, in accord with phenomenological considerations [17] as well as with lattice calculations [ 16]. As the bag constant we use the value B 1/4 = 235 MeV, as suggested by spectroscopic studies [18]. This leaves c the only undetermined parameter which we F~ by normalizing the hadronic pressure to plasma value at the critical temperature. At high temperatures the term k in the dynamical equation (1) is negligible and eqs. (1) and (2b) yield
e-lde/dt + 3(1 +P/e)(~nGe) 1/2 = 0 . X f d3k exp[--(k 2 + m2)l/2/T] ,
(10)
(12)
If one would assume the quantity 1 + PIe to be a constant, one would get an approximative solution t = [2/3(1 +P/e)] (3/8nGe) 1/2 .
(13)
Due to the interactions the quantity 1 + Pie varies slowly with the temperature, and we integrate eq. (12) numerically. An integration of eq. (12) yields the time-temperature relation for the early universe. The resulting solution is applicable both to the plasma and hadronic phase but not at the time when the temperature has the critical value. During a first order phase transition the pressure
Volume 119B, number 1,2,3
PHYSICS LETTERS
temperature is 180 MeV the resonances play a minor role and it takes 4.72/as to complete the transition. We remark that the values of the pressure and energy density and therefore the time-temperature relation are strongly dependent [10] on the value of the bag constant. For a comparison we have done the calculations also by using hard-core repulsive potentials [7,11 ] for hadronic interactions. In the case of T c = 180 MeV the results are almost the same as the ones provided by a resonance gas. In the case of T c = 190 MeV, however, there is no crossover o f the plasma pressure with the hadronic one in the uncertainty ranges o f rrrt- and NN.potentials given in ref. [ 11 ].
T[MeV] 280
240
200
16¢
\
12(
,
, , , J,t 5 10
,
, \
20
16 December 1982
,~.,
40
Fig. 1. The time-temperature relation in the quark-gluon plasma and hadronic phase, for the critical temperature Tc = 190 MeV (solid line) and 180 MeV (dashed line).
I am greatly indebted to Professor L. van Hove for the suggestion of this investigation and to Dr. V.V. Dixit and Dr. J. Lodenquai for pointing ou t a mistake in eq. (16) of the original manuscript. The kind hospitality at NORDITA is also acknowledged.
remains constant but the energy density decreases from the critical plasma value ep(Tc) to the critical hadronic one eh(Tc). In the case of a constant pressure eq. (2) can be reformulated into
References
t[#s]
dR3/R 3 + d(e +P)c/(e +P)c = O,
(14)
from which (e +P)c ~ R - 3
,
(15)
where the lower index c refers to the critical values at T = T c. From eq. (1), with e given by (15), we obtain
th(T~) - t p ( T ) = [1/6zre(Tc) ] 1/2 ~ (16) X a r c s i n / Px/-P-~c)[X/ep(Tc)- X/eh(Tc)l ~x,/[ep(Tc) + P(Tc) ] [ e h ( r c) +P(rc)l 1 for the duration of the phase transition. The model universe is in the plasma phase until tp(Tc), undergoes a transition during th(Tc) - tp(Tc) and is in the hadronic phase since th(Tc). The time-temperature relation as obtained from eqs. (5), (6) and (12) for the plasma, from eq. (16) for the phase transition and from eqs. (10), (11) and (12) for the hadronic state is shown in the figure. If the bag value is B = (235 MeV) 4 and the phase transition occurs at T c = 190 MeV, about a half of the critical energy density eh(Tc) is shared among resonances and the duration of the phase transition is 1.04 #s. If the critical
[1] See e.g. Proc. 5th High energy heavy ion study Berkeley, California 1981) LBL-12652 (Berkeley, 1981). [2] G. Gamov, Phys. Rev. 70 (1946) 572. [3] J. Engels, F. Karsch, I. Montvay and H. Satz, Nucl. Phys. B205 [FS5] (1982) 545. [4] H. Satz, Phys. Lett. 113B (1982) 245. [5] J.D. Walecka, Phys. Lett. 59B (1975) 109; Ann. Phys. 83 (1974) 491; R.A. Freedman, Phys. Lett. 71B (1977) 369. [6] R. Hagedorn, Suppl. Nuovo Cimento III (1965) 147. [7] S. Weinherg, Phys. Rev. 166 (1968) 1568; G.E. Brown and A.D. Jacson, The nucleon-nucleon interaction (North-Holland, Amsterdam, 1976). [8] S.A. Chin, Phys. Lett. 78B (1978) 552. [9] J. Kuti, B. LukAcs,J. Pol6nyl and K. Szlach~nyi, Phys. Lett. 95B (1980) 75. [10] V.V. Dixit, H. Satz and E. Suhonen, Critical behaviour in a quark plasma, Bielefeld preprint BI-TP 82/06 (April 1982) Z. Phys. C, to be published. [11] K.A. Olive, Nucl. Phys. B190 [FS3] (1981) 483; B198 (1982) 461. [12] See e.g.: S. Weinberg, Gravitation and cosmology, (Wiley, New York, 1972). [131 J.I. Kapusta, Nucl. Phys. B148 (1979) 461. (141 R. Hagedorn, in: Carg6se Lectures in Physics, Vol. 6 ed. E. Schatzrnan (Gordon and Breach, New York, 1973);
83
Volume 119B, number 1,2, 3
PHYSICS LETTERS
M. Jacob, ed., Dual theory (North-Holland, Amsterdam, 1974); A. Chodos, R.L. Yaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471. [15] N. Cabibbo and G. Parisi, Phys. Lett. 59B (1975) 67. [16] L.D. McLerran and B. Svetitsky, Phys. Lett. 98B (1981) 195; J. Kuti, J. Pol6nyi and K. Szhch~nyi, Phys. Lett. 98B (1981) 199;
84
16 December 1982
J. Engels, F. Karsch, I. Montvay and H. Satz, Phys. Lett. 101B (1981) 89; K. Kajantie, C. Montonen and E. Pietarinen, Z. Phys. C9 (1981) 253. [ 17] See e.g.: N. Cabibbo and L. Sertorio eds., Hadronic matter at extreme energy density, Plenium, New York, 1980). [18] P. Hasenfratz, R.R. Horgan, J. Kuti and J.M. Richard, Phys. Lett. 95B (1980) 299.
PHYSICS LETTERS
16 December 1982
THE QUARK-HADRON PHASE TRANSITION IN THE EARLY UNIVERSE E. SUHONEN NORDITA, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark and Department of Theoretical Physics 1, University of Oulu, Oulu, Finland Received 19 July 1982
By using perturbative QCD with bag corrections for quark-gluon plasma and a resonance gas model for hadronic matter, we study the effect of the phase transition on the cooling rate of the early universe.
With increasing energy density, a hadronic matter is expected to undergo a phase transition to a q u a r k gluon plasma, There has been much speculation that both plasma formation and the subsequent transition may take place in ultra-relativistic heavy ion collisions [ 1 ]. The most obvious situation for such a transition is however in the early universe. The standard cosmological model [2] allows one to extrapolate back to times when the universe probably was in a plasma state. Due to the expansion the energy density decreased and there was an era when the transition from quarks and gluons to hadrons occurred. The Yang-Mills theory on the lattice is the only presently available description of a strongly interacting matter for the whole temperature range. It indeed predicts two phases and a transition [3]. Above the critical temperature, the results obtained in the lattice evaluation are in full agreement [4] with those provided by a bag model approach. To describe hadronic matter one may use Walecka's relativistic mean field theory [5], resonance gas model [6] or potential models [7]. With a suitable two-phase input the equilibrium between both phases can be determined [8-11 ] from the crossover of the pressure according to Gibbs' rules. The standard cosmological model is based on the Einstein field equations which in the RobertsonWalker metric are [12]
1 Permanent address. 0 031-9163/8210000--0000/$02.75 © 1982 North-Holland
R -2 [(dR/dr) 2 + k] - ~ r O e = O,
(1)
R 3 dP/dt = (d/dt)[R3(e + P)] = 0 .
(2)
Here R is the cosmic scale factor, t is the proper time measured by a comoving observer, k has the value +1, - 1 , 0 for a closed, open or flat universe, respectively, G is the gravitational constant and e and P denote the total energy density and the isotropic pressure. Eq. (2) which results from the energy conservation can be written equivalently, e -1 de/dt + 3 (1 + P / e ) R - 1 dR/dt = 0 .
(2a)
When supplemented by the equation of state, eqs. (1) and (2) are sufficient to describe the evolution of R ( t ) in the early universe. To get the equation of state for a strongly interacting matter we assume that the quark-gluon phase is described by perturbative QCD up to the first order in the running coupling constant, together with a bag term to account for the nonperturbative influence of the physical vacuum. For the hadronic phase we consider an ideal gas of ground state hadrons and all possible resonance excitations. The resulting energy density in such a two-phase picture is not unlike [10] that found in the lattice evaluation. At a high temperature T the model universe conrains bosons and fermions with masses m i ~ T and multiplicity factors gB and gF respectively, at thermal equilibrium. Since the asymmetry between baryons and antibaryons is exceedingly small we can limit ourselves to the case of a vanishing chemical potential. 81
Volume lI9B, number 1,2,3
PHYSICS LETTERS
the radiation and lepton terms added, we obtain with the choice of a = 4 from eqs. (8) and (9)
The pressure Per and energy density eer of such an extra relativistic gas are given by l
1
1 _2,7, 4
Per =~eer = ~ g ~
i
,
g=gB +~gF •
/
(3.4)
The particles which contribute to g at temperatures of interest, T = O(100 MeV), are photons 7 and leptons e ±, #±, ve, De, v~, Du, vr and Dr in both phases and in addition gluons and two quark flavours in the plasma phase. These yield the values 51.25 and 14.25 for the total multiplicity factors of extra relativistic partides in the plasma and hadronic phase, respectively. By taking into account also the contributions of strange quarks and antiquarks, and interactions up to the order % = g2/4zr, together with the bag term B, we have Pp = (51.25/90)~r2T 4 + !~-aszrT4 +Ps - B,
(5)
ep=(51.25/30)zt2T 4 +Tas~rT 11 4 +-~-asT a19 2 4 + e s + B , (6) for the energy density in the plasma phase. For the pressure Ps and energy density e s of strange quarks and antiquarks we use the integral expressions given by Kapusta [13] with the value 280 MeV for the mass of the strange quark. In the case of three quark flavours the running coupling constant is
a s = 21r/91n(4T/A),
(7)
where A is the scale-fixing parameter. The partition function of a resonance gas is given by lnZ(T,V)-
V
16 December 1982
; dm r(m)
(2~r) 3 o
Ph = (TE/2~r2)
,c f
2m~
(3m2K2(m/T)
am
+ (14.25/90),r2T 4 for the pressure in the hadronic phase. The corresponding energy density becomes e h = 3P h + (T/2n2)(3ma~KI(m/T)
+c ; dmm-lexp(m/Te)Kl(m/T)) 2mn + (14.25/30),r2T 4 .
(8)
where the mass spectrum r(m) has from hadron dynamics [14] the expression r(m) = 38(m - m~) + cO(m - 2mn)m-ae bm ,
(9)
with constant a, b and c and the pion mass mzr. It is well known [15] that the spectral form (9) leads to critical behaviour with Tc = lib as the critical temperature. The order of the associated transition is determined by the parameter a. For 7/2 < a ~< 9/2 we have a finite energy density and diverging specific heat, consistent with lattice calculations [3,16]. With 82
(11)
The equation of state is given by eqs. (5) and (6) for T ~ Tc and by eqs. (10) and (11) for T < Tc. In the equations of state we have four as yet unfixed parameters. For the scale parameter A we take the usual value 100 MeV. As the critical temperature we use the values 180 MeV and 190 MeV, in accord with phenomenological considerations [17] as well as with lattice calculations [ 16]. As the bag constant we use the value B 1/4 = 235 MeV, as suggested by spectroscopic studies [18]. This leaves c the only undetermined parameter which we F~ by normalizing the hadronic pressure to plasma value at the critical temperature. At high temperatures the term k in the dynamical equation (1) is negligible and eqs. (1) and (2b) yield
e-lde/dt + 3(1 +P/e)(~nGe) 1/2 = 0 . X f d3k exp[--(k 2 + m2)l/2/T] ,
(10)
(12)
If one would assume the quantity 1 + PIe to be a constant, one would get an approximative solution t = [2/3(1 +P/e)] (3/8nGe) 1/2 .
(13)
Due to the interactions the quantity 1 + Pie varies slowly with the temperature, and we integrate eq. (12) numerically. An integration of eq. (12) yields the time-temperature relation for the early universe. The resulting solution is applicable both to the plasma and hadronic phase but not at the time when the temperature has the critical value. During a first order phase transition the pressure
Volume 119B, number 1,2,3
PHYSICS LETTERS
temperature is 180 MeV the resonances play a minor role and it takes 4.72/as to complete the transition. We remark that the values of the pressure and energy density and therefore the time-temperature relation are strongly dependent [10] on the value of the bag constant. For a comparison we have done the calculations also by using hard-core repulsive potentials [7,11 ] for hadronic interactions. In the case of T c = 180 MeV the results are almost the same as the ones provided by a resonance gas. In the case of T c = 190 MeV, however, there is no crossover o f the plasma pressure with the hadronic one in the uncertainty ranges o f rrrt- and NN.potentials given in ref. [ 11 ].
T[MeV] 280
240
200
16¢
\
12(
,
, , , J,t 5 10
,
, \
20
16 December 1982
,~.,
40
Fig. 1. The time-temperature relation in the quark-gluon plasma and hadronic phase, for the critical temperature Tc = 190 MeV (solid line) and 180 MeV (dashed line).
I am greatly indebted to Professor L. van Hove for the suggestion of this investigation and to Dr. V.V. Dixit and Dr. J. Lodenquai for pointing ou t a mistake in eq. (16) of the original manuscript. The kind hospitality at NORDITA is also acknowledged.
remains constant but the energy density decreases from the critical plasma value ep(Tc) to the critical hadronic one eh(Tc). In the case of a constant pressure eq. (2) can be reformulated into
References
t[#s]
dR3/R 3 + d(e +P)c/(e +P)c = O,
(14)
from which (e +P)c ~ R - 3
,
(15)
where the lower index c refers to the critical values at T = T c. From eq. (1), with e given by (15), we obtain
th(T~) - t p ( T ) = [1/6zre(Tc) ] 1/2 ~ (16) X a r c s i n / Px/-P-~c)[X/ep(Tc)- X/eh(Tc)l ~x,/[ep(Tc) + P(Tc) ] [ e h ( r c) +P(rc)l 1 for the duration of the phase transition. The model universe is in the plasma phase until tp(Tc), undergoes a transition during th(Tc) - tp(Tc) and is in the hadronic phase since th(Tc). The time-temperature relation as obtained from eqs. (5), (6) and (12) for the plasma, from eq. (16) for the phase transition and from eqs. (10), (11) and (12) for the hadronic state is shown in the figure. If the bag value is B = (235 MeV) 4 and the phase transition occurs at T c = 190 MeV, about a half of the critical energy density eh(Tc) is shared among resonances and the duration of the phase transition is 1.04 #s. If the critical
[1] See e.g. Proc. 5th High energy heavy ion study Berkeley, California 1981) LBL-12652 (Berkeley, 1981). [2] G. Gamov, Phys. Rev. 70 (1946) 572. [3] J. Engels, F. Karsch, I. Montvay and H. Satz, Nucl. Phys. B205 [FS5] (1982) 545. [4] H. Satz, Phys. Lett. 113B (1982) 245. [5] J.D. Walecka, Phys. Lett. 59B (1975) 109; Ann. Phys. 83 (1974) 491; R.A. Freedman, Phys. Lett. 71B (1977) 369. [6] R. Hagedorn, Suppl. Nuovo Cimento III (1965) 147. [7] S. Weinherg, Phys. Rev. 166 (1968) 1568; G.E. Brown and A.D. Jacson, The nucleon-nucleon interaction (North-Holland, Amsterdam, 1976). [8] S.A. Chin, Phys. Lett. 78B (1978) 552. [9] J. Kuti, B. LukAcs,J. Pol6nyl and K. Szlach~nyi, Phys. Lett. 95B (1980) 75. [10] V.V. Dixit, H. Satz and E. Suhonen, Critical behaviour in a quark plasma, Bielefeld preprint BI-TP 82/06 (April 1982) Z. Phys. C, to be published. [11] K.A. Olive, Nucl. Phys. B190 [FS3] (1981) 483; B198 (1982) 461. [12] See e.g.: S. Weinberg, Gravitation and cosmology, (Wiley, New York, 1972). [131 J.I. Kapusta, Nucl. Phys. B148 (1979) 461. (141 R. Hagedorn, in: Carg6se Lectures in Physics, Vol. 6 ed. E. Schatzrnan (Gordon and Breach, New York, 1973);
83
Volume 119B, number 1,2, 3
PHYSICS LETTERS
M. Jacob, ed., Dual theory (North-Holland, Amsterdam, 1974); A. Chodos, R.L. Yaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471. [15] N. Cabibbo and G. Parisi, Phys. Lett. 59B (1975) 67. [16] L.D. McLerran and B. Svetitsky, Phys. Lett. 98B (1981) 195; J. Kuti, J. Pol6nyi and K. Szhch~nyi, Phys. Lett. 98B (1981) 199;
84
16 December 1982
J. Engels, F. Karsch, I. Montvay and H. Satz, Phys. Lett. 101B (1981) 89; K. Kajantie, C. Montonen and E. Pietarinen, Z. Phys. C9 (1981) 253. [ 17] See e.g.: N. Cabibbo and L. Sertorio eds., Hadronic matter at extreme energy density, Plenium, New York, 1980). [18] P. Hasenfratz, R.R. Horgan, J. Kuti and J.M. Richard, Phys. Lett. 95B (1980) 299.