356
R. Stock
emitting source. Given such conditions, the total multiplicity per collision event
(the invariant yield divided by the total overall inelastic cross section) should be
employed in the SHM analysis, as is exemplified in Fig. 7.25.
7.3.1 Hadronic Freeze-Out from Expansion Flow
The hadronic multiplicities result from integration of the invariant triple differential
cross section over p T and y. Instrumental, experiment-specific conditions tend to
result in incomplete p T and/or y acceptances. It is important to ascertain that the
effects of hydrodynamic transverse and longitudinal flow do not blast a significant
part of the total hadron yield to outside the acceptance, and that they, more generally,
do not change the relative hadron yield composition, thus basically affecting the
SHM analysis. To see that hadronization incorporates only the internal energy in the
co-moving frame [110], we first assume that hadrochemical freeze-out occurs on a
sharp hypersurface , and write the total yield of particle species i as
N i =
d 3 p
E
p
μ d
3 σ μ (x) f i (x, p) =
d
3 σ μ (x)j
μ
i (x)
(7.26)
where d 3 σ is the outward normal vector on the surface, and
j
μ
i (x) = g i
d
4 p 2(p
0 )δ(p
2
− m
2
i ) p
μ (exp [p · u(x) − μ i ]/T ± 1)
−1
(7.27)
is the grand canonical number current density of species i, μ i the chemical potential,
u(x) the local flow velocity, and g i the degeneracy factor. In thermal equilibrium it
is given by
j
μ
i (x) = ρ i (x)u
μ (x) with
ρ i (x) = u μ (x)j
μ
i (x) =
d
4 p 2(p
0 )δ(p
2
− m
2
i ) p · u(x) f i (p · u(x); T ; μ i )
=
d
3 p
f i (E p ; T , μ i ) = ρ i (T , μ i ).
(7.28)
Here E p is the energy in the local rest frame at point x. The total particle yield of
species i is therefore
N i = ρ i (T , μ i )
d
3 σ μ (x)u
μ (x) = ρ i (T , μ i ) V (u
μ )
(7.29)
where only the total comoving volume V of the freeze-out hypersurface depends
on the flow profile u μ . V is thus a common total volume factor at hadronization (to
R. Stock
emitting source. Given such conditions, the total multiplicity per collision event
(the invariant yield divided by the total overall inelastic cross section) should be
employed in the SHM analysis, as is exemplified in Fig. 7.25.
7.3.1 Hadronic Freeze-Out from Expansion Flow
The hadronic multiplicities result from integration of the invariant triple differential
cross section over p T and y. Instrumental, experiment-specific conditions tend to
result in incomplete p T and/or y acceptances. It is important to ascertain that the
effects of hydrodynamic transverse and longitudinal flow do not blast a significant
part of the total hadron yield to outside the acceptance, and that they, more generally,
do not change the relative hadron yield composition, thus basically affecting the
SHM analysis. To see that hadronization incorporates only the internal energy in the
co-moving frame [110], we first assume that hadrochemical freeze-out occurs on a
sharp hypersurface , and write the total yield of particle species i as
N i =
d 3 p
E
p
μ d
3 σ μ (x) f i (x, p) =
d
3 σ μ (x)j
μ
i (x)
(7.26)
where d 3 σ is the outward normal vector on the surface, and
j
μ
i (x) = g i
d
4 p 2(p
0 )δ(p
2
− m
2
i ) p
μ (exp [p · u(x) − μ i ]/T ± 1)
−1
(7.27)
is the grand canonical number current density of species i, μ i the chemical potential,
u(x) the local flow velocity, and g i the degeneracy factor. In thermal equilibrium it
is given by
j
μ
i (x) = ρ i (x)u
μ (x) with
ρ i (x) = u μ (x)j
μ
i (x) =
d
4 p 2(p
0 )δ(p
2
− m
2
i ) p · u(x) f i (p · u(x); T ; μ i )
=
d
3 p
f i (E p ; T , μ i ) = ρ i (T , μ i ).
(7.28)
Here E p is the energy in the local rest frame at point x. The total particle yield of
species i is therefore
N i = ρ i (T , μ i )
d
3 σ μ (x)u
μ (x) = ρ i (T , μ i ) V (u
μ )
(7.29)
where only the total comoving volume V of the freeze-out hypersurface depends
on the flow profile u μ . V is thus a common total volume factor at hadronization (to