The Fireball Paradigm

Abstract

In this chapter we show how multihadron production can be related to thermodynamical considerations. Following a general introduction to the topic, we discuss the statistical hadronization model, in which each species is produced according to its phase space weight, and show that this leads to a universal hadronization temperature found in e ⁺ e ⁻ annihilation as well as in hadron-hadron and nucleus-nucleus collisions. In the final part we address deviations from a universal statistical hadronization description.

Es wurde schon vor längerer Zeit geschlossen, daß bei einem sehr energiereichen Stoß eines Kernteilchens auf ein anderes viele Mesonen mit einem Schlag erzeugt werden können Werner Heisenberg, Zeitschrift für Physik 126 (1949) 569 (It was concluded quite some time ago that in an energetic collision of one nuclear particle with another, many mesons could be created with one bang) (Werner Heisenberg)

Appendices

Appendix 1: Scattering Matrix and Phase Space

The probability for the production of N particles in the collision of two incident hadrons of four-momenta q ₁ and q ₂ is determined by

(11.25)

where W ² = (q ₁ + q ₂)² is the (squared) center-of-mass collision energy and p ₁, …, p _N are the four-momenta of the final hadrons. For simplicity, we assume here for the time being only one species of identical scalar hadrons of mass m ₀ and neglect any effects of quantum statistics. The unitary scattering matrix S maps the initial state onto the final state. The Fermi model [69] is obtained by assuming that the squared S-matrix element is proportional to the probability of finding the N final hadrons as free particles inside a spatial volume V ,

$$\displaystyle \begin{aligned} |\langle p_1\ldots p_N|S|q_1 q_2\rangle|^2 \sim \int_V d^3x_1\ldots d^3x_N \prod_1^N |\Phi_{\mathbf{p}_i}(\mathbf{x}_i)|^2. {} \end{aligned} $$

(11.26)

With plane wave states

$$\displaystyle \begin{aligned} \Phi_{\mathbf{p}_i}(\mathbf{x}_i) = \sqrt{2p_{i0} \over (2\pi)^3} \exp\{i \mathbf{p}_i \mathbf{x}_i \} {} \end{aligned} $$

(11.27)

for the final hadrons we obtain the standard form for the relativistic N-particle phase space volume,

(11.28)

with W ² = E ² −P ² for the invariant center of mass energy of the system. The delta functions project the N-particle state onto fixed total energy E and momentum P. The production probability can then be expressed in terms of the phase space volume Q _N (W) as

$$\displaystyle \begin{aligned} P_N(W,V) = {Q_N(W,V) \over Q(W,V)}, {}\end{aligned} $$

(11.29)

where the normalization Q(W) is given by the sum over all states at fixed energy and volume

$$\displaystyle \begin{aligned} Q(W,V) = \sum {1\over N!} Q_N(W,V), {}\end{aligned} $$

(11.30)

assuming, as mentioned, that all particles are identical.

The Laplace transforms of Q _N (W) and Q(W) are just the canonical and grand canonical partition functions of an ideal gas in the Boltzmann limit,

$$\displaystyle \begin{aligned} {\mathcal{Z}}_N(T,V) = \int d^3 P dW ~Q_N(W,V) \exp\{-W/T\} \simeq \left[ V z(T) \right]^N {}\end{aligned} $$

(11.31)

and

$$\displaystyle \begin{aligned} {\mathcal{Z}}(T,V) = \int d^3 P dW ~Q(W,V) \exp\{-W/T\} = \sum_N {1 \over N!} {\mathcal{Z}}_N(T,V) \simeq \exp\{V z(T)\}, {} \end{aligned} $$

(11.32)

with

(11.33)

here K ₂(x) is the Hankel function of purely imaginary argument. For small x, K ₂(x) ≃ (2/x ²), so that for high temperatures

(11.34)

This can be used to invert the Laplace transform (11.32), leading to [70]

(11.35)

Hence the average multiplicity

(11.36)

increases as W ^3/4, so that the average energy per secondary grows as W ^1/4.

Appendix 2: Exact Charge Conservation

The phase space integral (11.28) specifies the totality of allowed states consisting of a fixed number N of constituents, subject to the conservation of overall energy and momentum and contained in a volume V . The corresponding partition function (11.30) then provides the grand canonical sum over all N-body states, with the temperature T determining the average overall energy; the effect of momentum conservation actually becomes asymptotically negligible [71]. Quantum-mechanically, we write the grand canonical partition function as

$$\displaystyle \begin{aligned} Z(T,V) = \mathrm{Tr}\left[\exp\{-{\mathcal{H}}/T\}\right] = \sum_N \mathrm{Tr}_N\left[\exp\{-{\mathcal{H}}/T\}\right], {} \end{aligned} $$

(11.37)

where the trace runs over all possible N-particle states and ${\mathcal {H}}$ denotes the Hamiltonian. In the grand canonical framework, the conservation of a discrete charge Q (for simplicity, we restrict ourselves to only one additive conserved charge) is taken into account through fugacities $\exp \{-\mu {\mathcal {Q}}\}$, with ${\mathcal {Q}}$ for the charge operator and μ for the corresponding chemical potential,

$$\displaystyle \begin{aligned} Z(T,V,\mu) = \mathrm{Tr}\left[\exp\{-({\mathcal{H}} -\mu {\mathcal{Q}})/T\}\right]. {} \end{aligned} $$

(11.38)

For illustration, consider a system containing particles of charge zero and mass m ₀, together with particles of charge ± 1 and a common mass m ₁. The grand canonical partition function then becomes

$$\displaystyle \begin{aligned} \ln Z(T,V,\mu) = V\left[ z_0 + z_1(e^{\mu/T} + e^{-\mu/T})\right] = V \left[z_0 + 2 z_1 \cosh(\mu/T)\right], {} \end{aligned} $$

(11.39)

and the average charge density q = 〈Q〉/V  is given by

$$\displaystyle \begin{aligned} \langle q \rangle = T {\partial \ln Z(T,V,\mu) \over \partial\mu} =2 z_1\sinh(\mu/T), {} \end{aligned} $$

(11.40)

with

$$\displaystyle \begin{aligned} z_i =d_i {m_i^2 T \over 2 \pi^2} K_2(m_i/T), ~i=0,1. {} \end{aligned} $$

(11.41)

for the weight factor of a single particle of mass m _i and degeneracy d _i , in the Boltzmann limit. For a system of vanishing average charge density, Eq. (11.40) thus implies μ = 0, and from Eq. (11.39) we see that the system contains an equal number of positive and negative particles, in addition to the neutral ones. We thus have

$$\displaystyle \begin{aligned} \ln Z(T,V,\mu=0) = V \left[z_0 + 2 z_1 \right] {} \end{aligned} $$

(11.42)

for the corresponding grand canonical partition function.

Such a grand canonical formulation provides a correct description of the system only if it contains sufficiently many charged particles, so that fluctuations can be neglected. If there are only very few such particles, charge conservation has to be implemented exactly, not on the average. One thus has to “project” out the section of phase space associated to a fixed quantum number. We define the partition function at fixed Q as

$$\displaystyle \begin{aligned} Z_Q(T,V) = \mathrm{Tr}_Q\left[\exp\{-({\mathcal{H}}/T)\}\right] = \mathrm{Tr}\left[\exp\{-({\mathcal{H}}/T)\} {\mathcal{P}}_Q \right], {} \end{aligned} $$

(11.43)

where ${\mathcal {P}}_Q$ is the corresponding projection operator onto a fixed charge Q [72,73,74,75]. The partition function ${\mathcal {Z}}_Q(T,V)$ is thus a sum over all possible N = 2, 3, … particle clusters of fixed total charge Q. Using it, the grand canonical partition function (11.30) can be “re-organized”, i.e., rewritten as a sum over Q instead of as sum over N,

$$\displaystyle \begin{aligned} Z(T,V,\mu) = \sum_{Q=-\infty}^{Q=+\infty} Z_Q(T,V) \lambda^Q, {}\end{aligned} $$

(11.44)

where

$$\displaystyle \begin{aligned} \lambda = \exp\{\mu/T\} {} \end{aligned} $$

(11.45)

is the corresponding fugacity. Each term in the sum (11.44) is grand-canonical in terms of particle number, but canonical in charge. The series (11.44) can be inverted to obtain the partition function Z _Q (T, V ) at fixed charge Q, using the Cauchy formula. The result is

(11.46)

with the function ${\tilde {Z}}(T,V,\phi )$ obtained from the grand canonical partition function Z(T, V, μ) by a Wick rotation μ _s /T → iϕ, so that $\lambda \to \exp \{i \phi \}$.

For our system containing charges 0, ±1, we thus obtain

(11.47)

so that the “charge-canonical” partition function for exactly vanishing overall charge Q = 0 becomes

(11.48)

where I ₀(x) is the Bessel function of purely imaginary argument. From this we get

$$\displaystyle \begin{aligned} \ln Z_0(T,V) = V \left[ z_0 + {1\over V} \ln I_0(2V z_1) \right]. {} \end{aligned} $$

(11.49)

Comparing Eq. (11.45) to the grand canonical form (11.39), we see that the exact conservation of charge leads to the replacement

$$\displaystyle \begin{aligned} z_1 ~\to~{1\over V} \ln I_0(2V z_1). {} \end{aligned} $$

(11.50)

Since for large argument

$$\displaystyle \begin{aligned} I_0(x) = {e^x \over \sqrt{2\pi x}} \left[1 + O(1/x)\right], {} \end{aligned} $$

(11.51)

the two forms become identical in the large volume limit, as expected. For small V , corresponding to small particle numbers, the contribution of the charged particles is suppressed relative to that of the neutral particles, if the overall charge is fixed to zero. It can be shown, in fact, that the canonical (c) and grand canonical (gc) densities n _Q of particles of charge Q are related by [75]

$$\displaystyle \begin{aligned} n_Q^c \simeq n_Q^{gc}\left\{ {I_Q(2 V z_Q) \over I_0(2 V z_Q)}\right\}. {} \end{aligned} $$

(11.52)

This relation is approximate, since it neglects the possibility of multiply-charged states; the role of these is addressed in [75]. Since I ₁(x)/I ₀(x) → 1 for x →∞, exact charge conservation leads to an effective charge suppression by a factor approaching unity in the large volume limit. Since the volume V  is determined by the overall multiplicity, which in turn grows with the collision energy, the suppression of the form (11.52) disappears with increasing $\sqrt {s}$.

The formalism just sketched for the case of an Abelian (additive) charge 0, ±1 can be extended to several Abelian as well as non-Abelian charges [72,73,74,75]; for details, we refer to the cited works. We note here only that while exact baryon number conservation does lead to the correct non-strange abundances in elementary hadron-hadron collisions and in e ⁺ e ⁻ annihilation, the strangeness reduction obtained through exact strangeness conservation is not sufficient to account for the reduced strange particle production observed in these reactions. This requires an additional suppression, as given, e.g., by the suppression factor γ _s applied per strange quark in the hadron in question, as discussed above.