From Hadrons to Quarks

Abstract

In this chapter, we look at some simple models to illustrate how a phase transition from hadronic matter to a quark-gluon plasma can occur. First we determine when hadrons start forming clusters which can be considered as quark matter. After considering the thermodynamics of an ideal hadron gas and of an ideal quark-gluon medium, we introduce bag pressure and baryon repulsion as interaction features to specify under which conditions strongly interacting matter prefers to consist of hadrons and when it wants to turn into a plasma of unbound quarks and gluons. Finally we show that also a simple string model yields localized hadrons at low density, while at high density color charges can move around freely by changing partners. In all cases, very basic physical notions are found to lead to a two-phase structure of matter.

The time has come, the walrus said, to talk of many things: of shoes, and ships, and sealing wax, of cabbages, and kings, and why the sea is boiling hot, and whether pigs have wings. Lewis Carrol, Through the Looking Glass

Appendix: Bose and Fermi Gas Partition Functions

The partition function for an ideal gas of particles of mass m at temperature T in a volume V is given by [17]

$$\displaystyle \begin{aligned} \ln Z_s(T,V) = -s {V \over 2\pi^2} \int_0^{\infty} dp~p^2~ \ln\left[1 - s \exp (-\sqrt{p^2 + m^2}/T)\right], {} \end{aligned} $$

(4.62)

with s = +1 for bosons, s = −1 for fermions. For simplicity we assume in both cases no internal degrees of freedom. Expanding the logarithm as a power series, we obtain

$$\displaystyle \begin{aligned} \ln Z_s(T,V) = s{V \over 2\pi^2} \sum_{n=1}^{\infty} {1\over n} \int_0^{\infty} dp~p^2 \left(s~\exp(-\sqrt{p^2 + m^2}/T) \right)^n; {} \end{aligned} $$

(4.63)

the first term in this sum constitutes the Boltzmann limit. The momentum integration can be carried out to give

(4.64)

where K ₂(x) is the modified Hankel function of second order. In the limit x → 0, it becomes

$$\displaystyle \begin{aligned} K_2(x) = {2 \over x^2} \left[ 1 + {x^2 \over 4} + \mathrm{O}(x^4)\right]. {} \end{aligned} $$

(4.65)

Inserting this into Eq. (4.64), we get

(4.66)

The sums in Eq. (4.66) are Riemann zeta functions [18]:

$$\displaystyle \begin{aligned} \zeta(r) = \sum_{n=1}^{\infty} {1 \over n^r} {} \end{aligned} $$

(4.67)

and

$$\displaystyle \begin{aligned} (1-2^{(1-r)}) \zeta(r) = \sum_{n=1}^{\infty} {(-1)^{n+1} \over n^r} {} \end{aligned} $$

(4.68)

In particular, we have

$$\displaystyle \begin{aligned} \sum_{n=1}^{\infty} {1 \over n^4} = \zeta(4) = {\pi^4 \over 90}, \end{aligned} $$

(4.69)

$$\displaystyle \begin{aligned} \sum_{n=1}^{\infty} {(-1)^{n+1} \over n^4} = (1-2^{-3})\zeta(4) = {7 \over 8}~ {\pi^4 \over 90}, \end{aligned} $$

(4.70)

$$\displaystyle \begin{aligned} \sum_{n=1}^{\infty} {1 \over n^2} = \zeta(2) = {\pi^2 \over 6}, \end{aligned} $$

(4.71)

$$\displaystyle \begin{aligned} \sum_{n=1}^{\infty} {(-1)^{n+1} \over n^2} = (1-2^{-1})\zeta(2) = {1 \over 2}~ {\pi^2 \over 6}. \end{aligned} $$

(4.72)

Inserting this into Eq. (4.66) gives as leading terms

$$\displaystyle \begin{aligned} \ln Z_B(T,V) = {\pi^2 \over 90} V T^3\left[ 1 - {15 \over 4 \pi^2} \left({m\over T}\right)^2 + \mathrm{O}((m/T)^4)\right] {} \end{aligned} $$

(4.73)

for the partition function for bosons and

$$\displaystyle \begin{aligned} \ln Z_F(T,V) ={7\over 8} {\pi^2 \over 90} V T^3\left[ 1 - {30 \over 7 \pi^2} \left({m\over T}\right)^2 + \mathrm{O}((m/T)^4)\right]. {} \end{aligned} $$

(4.74)

for fermions. In both cases, the first term is the Stefan-Boltzmann limit. Comparing Eqs. (4.66) with (4.73), we see that the numerical coefficient of the bosonic partition function in the limit m → 0 is π ²/90 ≃ 0.1097 for correct Bose-Einstein statistics, while it becomes 1/π ² ≃ 0.1013 in the Stefan-Boltzmann limit. This provides the justification for using the latter form whenever m ≪ T.