(slide 1)
(soft) physics from particle spectra 
Roppon Picha 
UCD Nuclear Physics Group 
9 Feb 2005 
outline: 
1. heavy ion collsion 
2. chemical freeze-out 
3. statistical model 
4. kinetic freeze-out 
5. blast wave model 
6. conclusions 

(slide 2)
heavy ion collisions 

- ultimate goal of HIC = study properties of hot and dense quark-gluon 
plasma 
- QGP = thermalized system of free quarks and gluons 
- elementary collisions -> dilute 
- HIC -> lots of particles -> final-state thermalization 
- thermalization -> collectivity 

time before \tau_0 is pre-equilibrium 
\tau_0 = formation/thermalization time 
(Bjorken, Phys. Rev. D27, 140 (1983)) 

(slide 3)
freeze-out evolution 

- chemical and kinetic freeze-outs are based on similar idea: 
expansion rate > collision rate 
- both chem. and kin. equilibria require thermalization, but at 
different degrees 
- T_{ch} -> inelastic collisions stop -> chemical equilibrium 
- T_{kin} -> elastic collisions stop -> kinetic equilibrium 
- T_{crit} -> around 175 MeV 

(slide 4)
particle spectra 

- what do they tell us? 
momentum and energy distribution 
- hadron multiplicities -> production at chemical freeze-out 
- shape contributions: 
a thermal source with temperature T -> statistical, 
Boltzmann-like, e^{-E/T}, same slopes for all particles 
boosted -> different shapes for different masses 

(slide 5)
statistical model of chemical equilibrium

- a tool to tell where the system is on the phase diagram 
- basic ideas:
thermally equilibrated (constant temp.)
chemically equilibrated (constant densities (n))

grand canonical ensemble 
Z = \sum_i \exp(- \frac{E_i - \mu N_i}{T})

Braun-Munzinger et al, nucl-th/0311005 
Braun-Munzinger et al, nucl-th/0304013 
Cleymans et al, J. Phys. G25, 281 (1999) 

(slide 6)
statistical model

- model's parameters: chemical freeze-out temperature (T_{ch}), chemical 
potentials (\mu), and strangeness saturation factor (\gamma_s) 
number density of particle i: 

\rho_i = \frac{g_i}{2\pi^2} \int_0^{\infty} \frac{p^2 \,dp}{\exp((E_i -
\mu)/T) \pm 1}

\rho_i = \gamma_{s}^{\langle s + \bar{s} \rangle} \frac{g_i}{2\pi^2} 
m^2_i T_{ch} K_2 \left( \frac{m_i}{T_{ch}} \right) \lambda_q^{Q_i} 
\lambda_s^{s_i}

\lambda_q \equiv \exp(\mu_q/T_{ch})

\lambda_s \equiv \exp(\mu_s/T_{ch})

Q_i = \langle u + d - \bar{u} - \bar{d} \rangle_i

s_i = \langle s - \bar{s} \rangle_i

\gamma_s \equiv {s density}{equilibrium density}

Rafelski, Phys. Lett. B262, 333 (1991)
Sollfrank, J. Phys. G23, 1903 (1997)
Sollfrank et al, Phys. Rev. C59, 1637 (1999)

(slide 7)
particle ratios

- model input is a set of particle ratios 

(slide 8)
statistical model fits 

Kaneta and Xu, QM04 nucl-th/0405068Braun-Munzinger et al, PLB518, 41 (2001)

130 GeV Au+Au
200 GeV Au+Au
20 GeV Au+Au

(slide 9)
chemical freeze-out 

- result is surprisingly consistent with other heavy ion experiments 
- inelastic collisions stop when energy per hadron is about 1 GeV 

Karsch, hep-lat/0401031 
Cleymans and Redlich, PRL81, 5284 (1998) 

(slide 10)
spectra shape

- system of particles freezes out kinetically when density and
temperature drop at a point where the particles no longer scatter

mean free path \approx system size

time between collisions \approx Hubble time (1/H)

- natural observable to study transverse flow -> p_T or m_T spectra 

m_T \equiv \sqrt{p^2_T + m^2_0}

Schnedermann and Heinz, PRC50, 1675 (1994)
Kolb, nucl-th/0304036

(slide 11)
more on spectra shape


NA44, PRL 78, 2080 (1997)

- previously (SPS): obtain T for each 
particle, plot T vs m, then ->

T_{slope} = T_{kin} + m \langle \beta_T \rangle^2 for p_T \leq m

T_{slope} = T_{kin} + \sqrt{\frac{1 + \langle \beta_T \rangle}{1 - 
\langle \beta_T \rangle}} for p_T \gg (blueshift)

problem: the value of T depends on fit range

current: hydrodynamics-based blast wave model -> simultaneous fit to all 
particles

(slide 12)
relativistic hydrodynamics

energy momentum tensor for a fluid cell: energy density velocity

T^{\mu \nu} (x) = (e(x) + p(x))u^{\mu} (x) u^{\nu} (x) - g^{\mu \nu} 
p(x)

x = (t, \vec{x})
e(x) = energy density
p(x) = pressure 
u(x) = velocity

``charge'' current at x: 

j^{\mu}_i (x) = n_i(x) u^{\mu} (x)

T^{\mu \nu} \equiv flow of p^{\mu} in the \nu-direction

the tensor tells us about energy and momentum at every point in 4-d
space-time

``charges'' = net baryon, net strangeness, net electric charge, ... etc

Kolb and Heinz nucl-th/0305084

(slide 13)
motion

fluid motion is determined from

- conservation of energy and momentum: \partial_{\mu} T^{\mu \nu} = 0

- conservation of ``charges'': \partial_{\mu} j^{\mu} = 0

- equation of state (EoS) = pressure as a function of energy, and charge 
densities: p(\varepsilon, n_i)

\mu = 0,1,2,3 = t,x,y,z

(slide 14)
more on eqn of state

Typical EOS used with RHIC data: 

- EOS plasma = ideal gas of massless q, qbar, g with a bag 
constant. stiff. (EOS I)

- EOS hadrons = gas of hadrons and resonances. soft. (EOS H)

- EOS quark-hadron transition (EOS Q)

Kolb and Heinz, nucl-th/0305084 

(slide 15)
intro to blast wave

- blast wave is a parametrization of hydro

- describes final freeze-out condition, but not how the system evolves

- basic ideas: 

rescattering of produced particles -> fluid-like flow

assume boost invariant (true for small region at midrapidity) 

(slide 16)
blast-wave model

Schnedermann et al, PRC48, 2462 (1993)

parameters: kinetic freeze-out temperature (Tkin), flow velocity 
(\beta), and flow profile parameter (n)

\frac{dN}{m_T \dmT} \propto \int_0^R r \,dr\, m_T I_0 \left( \frac{p_T
\sinh \rho(r)}{T_{kin}}\right) K_1 \left( \frac{m_T \cosh
\rho(r)}{T_{kin}} \right)

(integrated over \phi)

\rho(r)=\tanh^{-1} \beta_r
transverse rapidity

\beta_r = \beta_{surf} (r/R)^n
flow profile -> Hubble-like 
(n = 2 best matches hydro, but isn't important)

(slide 17)
blast wave fits 

global (T, beta) to fit 6 particles all at once. 

Au+Au @ 19.6 GeV 

(slide 18)
kinetic freeze-out and collective flow

- saturation of temperature around SPS, or even AGS, energies

- strong collective flow indicates dense system increases with energy,
RHIC flow about 3 times AGS flow

- necessary condition for QGP (thermalization), but not direct evidence

(slide 19)
conclusions

- particle spectra provide a mean to uncover rich information about the
heavy ion collisions

- chemical freeze-out: ratios consistent with statistical model 

- vanishing baryon density at increasing energy 

- kinetic freeze-out: blast wave model - globally explain light
particles well

- stronger collective flow with increasing energy 

- saturated Tkin < Tch 

- the results support thermalization, but are not direct evidences

- to do next: understand the theories/models involved better