7 Relativistic Nucleus-Nucleus Collisions and the QCD Matter Phase Diagram
419
To proceed one employs a Gaussian ansatz on either side of Eq. (7.83). The
experimental data are parametrized by source “radius” parameters R ij (K),
C(q, K) = 1 + λ(K) exp [−
ij
R
2
ij (K)q i q j ]
(7.84)
employing λ(K) essentially as a fudge factor (related nominally to possible
coherence of emission effects in the source, which would dilute or remove the
Bose-Einstein statistics effect of C → 2 for q → 0 but have never been seen in
nuclear collisions). In Eq. (7.84) the sum runs over three of the four components of
q, due again to the on-shell requirements [287]. For the emission function S(x, K)
a Gaussian profile is assumed about an “effective source center” x(K), thus
S(x, K) → S(x(K), K) × G
(7.85)
where G is a Gaussian in coordinates ˜
x μ (K) relative to the center coordinates
x
μ (K). Inserting Eq. (7.85) into (7.83) one finally obtains
C(q, K) = 1 + λ(K) exp [−q μ q v
˜
x
μ
˜
x
ν
]
(7.86)
where ˜
x μ ˜
x ν are the elements of the space-time variance of the correlation function, which re-interpret the “radii” R 2
ij in Eq. (7.84). Assuming azimuthal symmetry
(central collisions), cartesian parametrizations of the pair relative momentum q
coordinates (corresponding to fixation of the space-time variance in Eq. (7.84)) have
been introduced by Yano, Koonin and Podgoretskii [289], and, alternatively, by Pratt
[290]. The latter, out-side-longitudinal coordinate system has the “long” direction
along the beam axis. In the transverse plane, the “out” direction is chosen parallel
to K T = (p 1T + p 2T )/2, the transverse component of the pair momentum K. The
“side” direction is then orthogonal to the out- and long-direction but, moreover, it
has the simplest geometrical interpretation (see ref. [287] for detail), to essentially
reflect the transverse system size [288]. The parameters of Eq. (7.86) are thus
defined; as an example we quote, from identification of Eq. (7.84) with (7.86), the
resulting geometrical definition of the “side” radius,
R
2
side (K) =
˜
y(K)
2
.
(7.87)
Overall, this model of Fourier related correlators, C(q, K) the experimentally
accessible quantity (see Eq. (7.82)), and S(x, K) the to-be-inferred spatial freezeout fireball configuration, leads to the Gaussian ansatz [287]
C(q, K) = 1 + λ(K) exp[−R
2
out (K)q
2
out − R
2
side (K)q
2
side −
R
2
long (K)q
2
long + cross terms]
(7.88)
419
To proceed one employs a Gaussian ansatz on either side of Eq. (7.83). The
experimental data are parametrized by source “radius” parameters R ij (K),
C(q, K) = 1 + λ(K) exp [−
ij
R
2
ij (K)q i q j ]
(7.84)
employing λ(K) essentially as a fudge factor (related nominally to possible
coherence of emission effects in the source, which would dilute or remove the
Bose-Einstein statistics effect of C → 2 for q → 0 but have never been seen in
nuclear collisions). In Eq. (7.84) the sum runs over three of the four components of
q, due again to the on-shell requirements [287]. For the emission function S(x, K)
a Gaussian profile is assumed about an “effective source center” x(K), thus
S(x, K) → S(x(K), K) × G
(7.85)
where G is a Gaussian in coordinates ˜
x μ (K) relative to the center coordinates
x
μ (K). Inserting Eq. (7.85) into (7.83) one finally obtains
C(q, K) = 1 + λ(K) exp [−q μ q v
˜
x
μ
˜
x
ν
]
(7.86)
where ˜
x μ ˜
x ν are the elements of the space-time variance of the correlation function, which re-interpret the “radii” R 2
ij in Eq. (7.84). Assuming azimuthal symmetry
(central collisions), cartesian parametrizations of the pair relative momentum q
coordinates (corresponding to fixation of the space-time variance in Eq. (7.84)) have
been introduced by Yano, Koonin and Podgoretskii [289], and, alternatively, by Pratt
[290]. The latter, out-side-longitudinal coordinate system has the “long” direction
along the beam axis. In the transverse plane, the “out” direction is chosen parallel
to K T = (p 1T + p 2T )/2, the transverse component of the pair momentum K. The
“side” direction is then orthogonal to the out- and long-direction but, moreover, it
has the simplest geometrical interpretation (see ref. [287] for detail), to essentially
reflect the transverse system size [288]. The parameters of Eq. (7.86) are thus
defined; as an example we quote, from identification of Eq. (7.84) with (7.86), the
resulting geometrical definition of the “side” radius,
R
2
side (K) =
˜
y(K)
2
.
(7.87)
Overall, this model of Fourier related correlators, C(q, K) the experimentally
accessible quantity (see Eq. (7.82)), and S(x, K) the to-be-inferred spatial freezeout fireball configuration, leads to the Gaussian ansatz [287]
C(q, K) = 1 + λ(K) exp[−R
2
out (K)q
2
out − R
2
side (K)q
2
side −
R
2
long (K)q
2
long + cross terms]
(7.88)