342
R. Stock
occurs an idealized t = z = 0 interaction “point”. Toward positive t the light cone
proper time profiles of progressing parton-hadron matter evolution are illustrated.
The first profile illustrated here corresponds to the end of formation time τ 0 . From
our above discussion of the e + e − annihilation process one obtains a first estimate,
τ 0 ≥ 0.25 fm/c (including interpenetration time of 0.15 fm/c at RHIC) which
refers to processes of very high Q 2 ≥ 10 3 GeV 2 , far above the saturation scale
Q 2
s discussed in the previous section. The latter scale has to be taken into account
for low p T hadron production.
It is the specific resolution scale Q 2 of a QCD sub-process, as enveloped in
the overall collision dynamics of two slabs of given transverse partonic structure
function density, that determines which fraction of the constituent partons enters
interaction. In the simple case of extremely high Q 2 processes the answer is that
all constituents are resolved. However, at modest Q 2 (dominating bulk hadron
production) the characteristic QCD saturation scale Q 2
s (x) gains prominence,
defined such that processes with Q 2 < Q 2
s do not exploit the initial transverse
parton densities at the level of independent single constituent color field sources
(see Eq. (7.11)). For such processes the proper formation time scale, τ 0 , is of order
of the inverse saturation momentum [61], 1/Q s ∼ 0.2 fm/c at
√
s = 200 GeV. The
first profile of the time evolution, sketched in Fig. 7.18, should correspond to proper
time t = τ 0 = 0.25 fm/c at RHIC energy. At top SPS energy,
√
s = 17.3 GeV, we
cannot refer to such detailed QCD considerations. A pragmatic approach suggests
to take the interpenetration time, at γ ≈ 8.5, for guidance concerning the formation
time, which thus results as τ 0 ≈ 1.5 fm/c.
In summary of the above considerations we assume that the initial partonic
color sources, as contained in the structure functions (Fig. 7.14), are spread out in
longitudinal phase space after light cone proper time t = τ 0 ≈ 0.2 fm/c, at top
RHIC energy, and after τ 0 ≈ 1.5 fm/c at top SPS energy. No significant transverse
expansion has occurred at this early stage, in a central collision of A ≈ 200
nuclei with transverse diameter of about 12 fm. The Bjorken estimate [45] of initial
energy density (Eq. (7.1)) refers to exactly this condition, after formation time
τ 0 . In order to account for the finite longitudinal source size and interpenetration
time, at RHIC, we finally put the average τ 0 ≈ 0.3 fm, at
√
s = 200 GeV,
indicating the “initialization time” after which all partons that have been resolved
from the structure functions are engaged in shower multiplication. As is apparent
from Fig. 7.18, this time scale is Lorentz dilated for partons with a large longitudinal
momentum, or rapidity. This means that the slow particles are produced first toward
the center of the collision region, and the fast (large rapidity) particles are produced
later, away from the collision region. This Bjorken “inside-out” correlation [45]
between coordinate- and momentum-space is similar to the Hubble expansion
pattern in cosmology: more distant galaxies have higher outward velocities. This
means that the matter created in A+A collisions at high
√
s is also born expanding,
however with the difference that the Hubble flow is initially one dimensional
along the collision axis. This pattern will continue, at
√
s = 200 GeV, until the
system begins to feel the effects of finite size in the transverse direction which
will occur at some time t 0 in the vicinity of 1 fm/c. However, the tight correlation
R. Stock
occurs an idealized t = z = 0 interaction “point”. Toward positive t the light cone
proper time profiles of progressing parton-hadron matter evolution are illustrated.
The first profile illustrated here corresponds to the end of formation time τ 0 . From
our above discussion of the e + e − annihilation process one obtains a first estimate,
τ 0 ≥ 0.25 fm/c (including interpenetration time of 0.15 fm/c at RHIC) which
refers to processes of very high Q 2 ≥ 10 3 GeV 2 , far above the saturation scale
Q 2
s discussed in the previous section. The latter scale has to be taken into account
for low p T hadron production.
It is the specific resolution scale Q 2 of a QCD sub-process, as enveloped in
the overall collision dynamics of two slabs of given transverse partonic structure
function density, that determines which fraction of the constituent partons enters
interaction. In the simple case of extremely high Q 2 processes the answer is that
all constituents are resolved. However, at modest Q 2 (dominating bulk hadron
production) the characteristic QCD saturation scale Q 2
s (x) gains prominence,
defined such that processes with Q 2 < Q 2
s do not exploit the initial transverse
parton densities at the level of independent single constituent color field sources
(see Eq. (7.11)). For such processes the proper formation time scale, τ 0 , is of order
of the inverse saturation momentum [61], 1/Q s ∼ 0.2 fm/c at
√
s = 200 GeV. The
first profile of the time evolution, sketched in Fig. 7.18, should correspond to proper
time t = τ 0 = 0.25 fm/c at RHIC energy. At top SPS energy,
√
s = 17.3 GeV, we
cannot refer to such detailed QCD considerations. A pragmatic approach suggests
to take the interpenetration time, at γ ≈ 8.5, for guidance concerning the formation
time, which thus results as τ 0 ≈ 1.5 fm/c.
In summary of the above considerations we assume that the initial partonic
color sources, as contained in the structure functions (Fig. 7.14), are spread out in
longitudinal phase space after light cone proper time t = τ 0 ≈ 0.2 fm/c, at top
RHIC energy, and after τ 0 ≈ 1.5 fm/c at top SPS energy. No significant transverse
expansion has occurred at this early stage, in a central collision of A ≈ 200
nuclei with transverse diameter of about 12 fm. The Bjorken estimate [45] of initial
energy density (Eq. (7.1)) refers to exactly this condition, after formation time
τ 0 . In order to account for the finite longitudinal source size and interpenetration
time, at RHIC, we finally put the average τ 0 ≈ 0.3 fm, at
√
s = 200 GeV,
indicating the “initialization time” after which all partons that have been resolved
from the structure functions are engaged in shower multiplication. As is apparent
from Fig. 7.18, this time scale is Lorentz dilated for partons with a large longitudinal
momentum, or rapidity. This means that the slow particles are produced first toward
the center of the collision region, and the fast (large rapidity) particles are produced
later, away from the collision region. This Bjorken “inside-out” correlation [45]
between coordinate- and momentum-space is similar to the Hubble expansion
pattern in cosmology: more distant galaxies have higher outward velocities. This
means that the matter created in A+A collisions at high
√
s is also born expanding,
however with the difference that the Hubble flow is initially one dimensional
along the collision axis. This pattern will continue, at
√
s = 200 GeV, until the
system begins to feel the effects of finite size in the transverse direction which
will occur at some time t 0 in the vicinity of 1 fm/c. However, the tight correlation