The appearance of the true vacuum state is closely related to the adhesive-adhesive interaction. In the above discussion, quark confinement is a natural feature of the hypothetical structure of the real vacuum. It is a straightforward exercise, performed at the beginning of the next section, to reduce the large partition functionZ to an expression in terms of the mass spectrum.m2;b/.
The ideas of the statistical bootstrap have found a very successful application in the description of hadronic reactions [11] during the last decade. We see that the theoretical description of the two hadronic phases—the individual hadron gas and the quark-gluon plasma—is consistent with observations and with current knowledge of elementary particles. Furthermore, we note that the bootstrap equation (27.17) uses virtually all the same approximations as our description of the level density in Eq.
The Hot Hadronic Gas
The last of the important thermodynamic quantities is entropyS. It is a function of the chemical potential, i.e. We note that [see Eq. Therefore, at 'D'0 we find a singularity in the point particle quantities "pt,pt andPpt. We can easily verify that this is correct by establishing the average number of clusters present in the hadronic gas.
This is achieved by introducing the artificial fugitive Nin Eq. 27.5) in the sum over N, where N is the number of clusters. We note the striking fact that the hadronic gas phase obeys the "ideal" gas equation, although of course hNi is not constant as in a real ideal gas, but is a function of thermodynamic variables. Its position, of course, depends on the actually given form'.ˇ; /, i.e. on the set of "input" particles fmb;gbgas and the value of the constant H in Eq.
In the case of three elementary pions C, 0 and and four elementary nucleons (spin ˝ isospin) and four antinucleons, we have from Eq. Beyond it, the ordinary hadronic world ceases to exist. In the shaded region, our theory is not valid because we have neglected Bose-Einstein and Fermi-Dirac statistics. For D0, the curve ends at TDT0, where T0, the 'limiting temperature of hadronic matter', is the same as that which appears in the mass spectrum.
This apparently large value of T0 seemed necessary to give an average maximum decay temperature of the order of 145 MeV, as required by [18]. Indeed, since the energy density along the critical curve is constant (D 4B), the critical curve can be reached and, if the energy density becomes > 4B, we enter a region that cannot be described without making assumptions about the structure the interior and dynamics of 'elementary particles'fmb;gbg—here pions and nucleons—enter the input function'.ˇ;.
The Quark–Gluon Phase
HeregD.2sC1/.2IC1/CD12 counts the number of components in the quark gas, and q is the fugacity related to the quark number. Since each quark has baryon number 1/3, we find where, as before, there is the possibility of preserving the baryon number. Finally, let us introduce the vacuum concept, which accounts for the fact that the perturbing vacuum is an excited state of the 'true' vacuum which has been renormalized to have a vanishing thermodynamic potential,˝ D ˇ1lnZ. Therefore in the disturbing vacuum.
27.57), it follows that, when the pressure disappears, the energy density is 4B, independent of the values of enT that fix the line PDO. Thus, above the critical curve of the T-plane, we exposed the quark-gluon plasma to an external force. To get an idea of the shape of the T-plane PD0 critical curve for the quark-gluon plasma, we rewrite Eq.
It is quite likely that with proper handling of the adhesive field and plasma corrections and with a larger B1=4 190MeV, the desired Tq D T0 value corresponding to the choice of the statistical bootstrap will follow. Since the quark plasma is the phase into which individual hadrons dissolve, it is sufficient if the pressure of the quark plasma vanishes within the limit set for the non-expanding positive pressure of the hadron gas. Figure 27.1b shows the pressure as a function of volume (aP;Vdiagram) along the dashed straight line at constant temperature.
Between volumes V1 and V2, matter coexists in the two phases, the relative fractions being determined by the size of the actual volume. This is precisely the domain where our description of the hadron gas is currently failing, while the quark-gluon plasma is also starting to suffer from infrared problems.
Nuclear Collisions and Inclusive Particle Spectra
Furthermore, we see that, along the critical curve of the hadronic gas, the baryon density decreases as the temperature increases. Before discussing this point further, we note that the hadronic gas branches of the curves in Figure 27.3 show very similar behavior to that shown at constant temperature in Figure 27.1b. However, what we cannot see from looking at Figure 27-3 is that there will be a discontinuity in the variables and T at this point, unless the parameters are chosen so that the critical curves of the two phases coincide.
A further aspect of the equations of state for the hadrone gas is also illustrated in Figure 27.3a. As shown in Figure 27.3b, entropy initially increases by as much as 50-100% in the dense phase of matter due to pion production and resonance decay. Reviewing Figure 27-2, it seems that a possible test of the equations of state for the hadronic gas would be to measure the temperature in the hot fireball zone as a function of the nuclear collision energy.
Their kinetic energy resembles the temperature in each phase of expansion. Therefore, it may be more practical to calculate the average transverse amount of emitted particles. The threshold value thus obtained is the observable 'average temperature' of the interaction residues, while the initial temperature Tcr at a given Ek;lab (solid line in Fig. 27.4) is difficult to observe.
Figure 27.4 shows the average pion and nucleon temperatures as a function of the kinetic energy of the heavy ions. Figure 27.5 shows the dependence of the average transverse moments of pions and nucleons on the kinetic energy of heavy ion projectiles.
Strangeness in Heavy Ion Collisions
Our choice of the freeze-out condition was made in such a way that the nucleon temperature at Ek;lab=AD1:8GeV is about 120 MeV. A third effect has so far been omitted - the emission of pions from the two-body decay of long-lived resonances [1] would lead to an effective temperature which is lower in nuclear collisions. Thus, if we assume equilibrium in the quark plasma, we find the density of the strange quarks to be (two spins and three colors).
Since the mass ms of the strange quarks in the perturbing vacuum are assumed to be in the order of 280–300 MeV, the equilibrium assumption for ms=T 2 may indeed be correct. 27.71), we could use the Boltzmann distribution again, because the density of strangeness is relatively low. When the quark matter dissociates into hadrons, some of the numerous, instead of being bound in a qskaon, may end up in aq q santibaryon and, in particular2, aΛorΣ0. The probability for this process appears to be similar to that for antinucleon production by the antiquarks present in the plasma.
Therefore, we would like to argue that the study of Λ and Σ0 in nuclear collisions for 2 Total strangeness yield is not indicative of the phase transition to quark plasma, since the increase (p . q=D 1:25) in yield can be reinterpreted as due to a change in hadronic volume. Our conclusions about the importance of Λ as an indicator of the phase transition to quark plasma remain valid since the production of Λ in the hadronic gas phase will only be possible in the very first stages of the nuclear collisions, if sufficient center-of-mass energy is available. In particular, in the case of hadronic gas, we have completely abandoned a more conventional Lagrangian approach in favor of a semi-phenomenological statistical bootstrap model of hadronic matter that incorporates those properties of hadronic interaction that we believe are most important. in nuclear collisions. In particular, the attractive interactions are encompassed by the rich, exponentially growing hadronic mass spectrum.m2;b/, while the introduction of the finite volume of each hadron accounts for an effective short-range repulsion. Our approach leads us to the equations of state of hadronic matter that reflect what we have included in our considerations. This work has just begun and it is too early to say whether the features of strong interactions that we have chosen to include in our considerations are the most relevant. Perhaps the most interesting aspect of our work is the realization that the transition to quark matter will occur at much lower baryon density for highly excited hadronic matter than for matter in the ground state (TD 0). The exact baryon density of the phase transition depends somewhat on the pocket constant, but we estimate it to be about 2–40atT D150MeV. The detailed study of the various aspects of this phase transition, as well as of possible characteristic signatures of quark matter, remains to be performed. We have given here only a very preliminary report on the status of our current understanding. We believe that the occurrence of the quark plasma phase is observable, and we have therefore proposed a measurement of the relative yield of Λ=N p between 2 and 10 GeV/N kinetic energies. In the quark plasma phase, we expect a significant enhancement of ΛN production, which will most likely be visible in Λ=N p relative velocity. Open Access This book is distributed under the terms of the Creative Commons Attribution Non-commercial License, which permits any non-commercial use, distribution, and reproduction in any medium, provided the original author(s) and sources are credited.Summary
