7 Relativistic Nucleus-Nucleus Collisions and the QCD Matter Phase Diagram
347
between coordinate and momentum space that prevails at the onset of the dynamical
evolution in A+A collisions at high
√
s. As all hadronic transverse momentum
spectra initially follow an approximately exponential fall-off (see below) the bulk
hadronic output is represented by thermal transverse spectra at p T ≤ 2 GeV/c. We
shall turn to high p T information in later sections.
Furthermore we shall focus here on mid-rapidity production in near central
A+A collisions, because hydrodynamic models refer to an initialization period
characterized by Bjorken-type longitudinal boost invariance, which we have seen
in Figs. 7.7 and 7.9 to be restricted to a relatively narrow interval centered at
mid-rapidity. Central collisions are selected to exploit the azimuthal symmetry
of emission, in an ideal impact parameter b → 0 geometry. We thus select the
predominant, relevant hydrodynamic “radial flow” expansion mode, from among
other, azimuthally oriented (directed) flow patterns that arise once this cylindrical
symmetry (with respect to the beam direction) is broken in finite impact parameter
geometries.
In order to define, quantitatively, the flow phenomena mentioned above, we
rewrite the invariant cross section for production of hadron species i in terms of
transverse momentum, rapidity, impact parameter b and azimuthal emission angle
ϕ p (relative to the reaction plane),
dN i (b)
p T dp T dy dϕ p
=
1
2 π
dN i (b)
p T dp T dy
1 + 2v i
1 (p T , b) cos ϕ p + 2v i
2 (p T , b) cos(2ϕ p ) + . . .
(7.20)
where we have expanded the dependence on ϕ p into a Fourier series. Due to reflection symmetry with respect to the reaction plane in collisions of identical nuclei,
only cosine terms appear. Restricting to mid-rapidity production all odd harmonics
vanish, in particular the “directed flow” coefficient v
i
1 , and we have dropped the ydependence in the flow coefficients v i
1 and v i
2 . The latter quantifies the amount of
“elliptic flow”, to which we turn in Sect. 7.4. In the following, we will restrict to
central collisions which we shall idealize as near-zero impact parameter processes
governed by cylinder symmetry, whence all azimuthal dependence (expressed by the
v i
1 , v i
2 , . . . terms) vanishes, and the invariant cross section reduces to the first term
in Eq. (7.20), which by definition also corresponds to all measurements in which the
orientation of the reaction plane is not observed.
Typical transverse momentum spectra of the latter type are shown in Fig. 7.20,
for charged hadron production in Au+Au collisions at
√
s = 200 GeV, exhibiting
mid-rapidity data at various collision centralities [97]. We observe a clear-cut
transition, from bulk hadron emission at p T ≤ 2 GeV/c featuring a near-exponential
cross section (i.e. a thermal spectrum), to a high p T power-law spectral pattern.
Within the context of our previous discussion (Sect. 7.2.4) we tentatively identify
the low p T region with the QCD physics near saturation scale. Hadron production
at p T → 10 GeV/c should, on the other hand, be the consequence of primordial
leading parton fragmentation originating from “hard”, high Q 2 perturbative QCD
processes.
347
between coordinate and momentum space that prevails at the onset of the dynamical
evolution in A+A collisions at high
√
s. As all hadronic transverse momentum
spectra initially follow an approximately exponential fall-off (see below) the bulk
hadronic output is represented by thermal transverse spectra at p T ≤ 2 GeV/c. We
shall turn to high p T information in later sections.
Furthermore we shall focus here on mid-rapidity production in near central
A+A collisions, because hydrodynamic models refer to an initialization period
characterized by Bjorken-type longitudinal boost invariance, which we have seen
in Figs. 7.7 and 7.9 to be restricted to a relatively narrow interval centered at
mid-rapidity. Central collisions are selected to exploit the azimuthal symmetry
of emission, in an ideal impact parameter b → 0 geometry. We thus select the
predominant, relevant hydrodynamic “radial flow” expansion mode, from among
other, azimuthally oriented (directed) flow patterns that arise once this cylindrical
symmetry (with respect to the beam direction) is broken in finite impact parameter
geometries.
In order to define, quantitatively, the flow phenomena mentioned above, we
rewrite the invariant cross section for production of hadron species i in terms of
transverse momentum, rapidity, impact parameter b and azimuthal emission angle
ϕ p (relative to the reaction plane),
dN i (b)
p T dp T dy dϕ p
=
1
2 π
dN i (b)
p T dp T dy
1 + 2v i
1 (p T , b) cos ϕ p + 2v i
2 (p T , b) cos(2ϕ p ) + . . .
(7.20)
where we have expanded the dependence on ϕ p into a Fourier series. Due to reflection symmetry with respect to the reaction plane in collisions of identical nuclei,
only cosine terms appear. Restricting to mid-rapidity production all odd harmonics
vanish, in particular the “directed flow” coefficient v
i
1 , and we have dropped the ydependence in the flow coefficients v i
1 and v i
2 . The latter quantifies the amount of
“elliptic flow”, to which we turn in Sect. 7.4. In the following, we will restrict to
central collisions which we shall idealize as near-zero impact parameter processes
governed by cylinder symmetry, whence all azimuthal dependence (expressed by the
v i
1 , v i
2 , . . . terms) vanishes, and the invariant cross section reduces to the first term
in Eq. (7.20), which by definition also corresponds to all measurements in which the
orientation of the reaction plane is not observed.
Typical transverse momentum spectra of the latter type are shown in Fig. 7.20,
for charged hadron production in Au+Au collisions at
√
s = 200 GeV, exhibiting
mid-rapidity data at various collision centralities [97]. We observe a clear-cut
transition, from bulk hadron emission at p T ≤ 2 GeV/c featuring a near-exponential
cross section (i.e. a thermal spectrum), to a high p T power-law spectral pattern.
Within the context of our previous discussion (Sect. 7.2.4) we tentatively identify
the low p T region with the QCD physics near saturation scale. Hadron production
at p T → 10 GeV/c should, on the other hand, be the consequence of primordial
leading parton fragmentation originating from “hard”, high Q 2 perturbative QCD
processes.