13 Elliptic Flow in Relativistic Heavy-Ion Collisions
177
Fig. 13.14 The eccentricity scaled v 2 {4} i.e.,v 2 {4}/{4}, versus d N ch /dη with eccentricity ({4})
taken from the MCG-N (left), MCG-Q (middle), or fKLN-CGC (right) model at
√
s NN = 200 and
62.4 GeV for Au+Au and Cu+Cu collisions as indicated in the figure
Assuming v 2 ∝ , then v 2 / in the reaction plane is given by v 2 {4}//{4} [56].
The mean multiplicity versus v 2 {4}//{4} displayed in Fig. 13.14 for Au+Au and
Cu+Cu collisions at 200 and 62.4 GeV for above three models. The data points
for different system and energies seem to fall on top of each other as expected if
v 2 ∝ . The MCG-N exhibits the sharp increase of v 2 {2}// violating the approximation v 2 ∝ whereas the fKLN-CGC shows a saturation with v 2 ∝ . The MCG-Q
displays intermediate trend between the sharp rise and the saturation. The v 2 saturation or nearly saturation trend exhibited in central Au+Au collisions at 200 GeV is
consistent with a nearly perfect liquid behaviour.
New Q-cumulants method used to determine experimentally flow fluctuations
does not require the knowledge of either reaction plane or participant plane. The
estimation of v 2 {2} can be biased by correlations such as resonance decay, jets, etc.,
which are not related to the reaction or participant plane and denoted as δ. Voloshin et
al., [57], have shown that v 2 {2} and v 2 {4} for the Gaussian form of flow fluctuations
in the participant plane are correlated as
v 2 {2}
2
= =v
2
+ σ v
2
+ δ
(13.38)
v 2 {4}
2
∼ ∼v
2
− σ v
2
(13.39)
v 2 {2}
2
− v 2 {4}
2
∼ δ + 2σ v
2
(13.40)
where v 2 is the elliptic flow, δ is the non-flow contribution and σ v represent flow
fluctuations. This is because fluctuations increase v 2 {2} and decrease v 2 {4} and the
approximations are valid for σ v 2 /v 2 1. It is not possible to determine the value
of σ v 2 from the measurements of v 2 {2} and v 2 {4}. However, we can find the upper
limit on the flow fluctuations taking δ = 0 in [(13.38)–(13.40)], i.e.,
R v(2−4) = σ v 2 /v 2 =
(v 2 {2} 2 − v 2 {4} 2 )
(v 2 {2} 2 + v 2 {4} 2 )
(13.41)
177
Fig. 13.14 The eccentricity scaled v 2 {4} i.e.,v 2 {4}/{4}, versus d N ch /dη with eccentricity ({4})
taken from the MCG-N (left), MCG-Q (middle), or fKLN-CGC (right) model at
√
s NN = 200 and
62.4 GeV for Au+Au and Cu+Cu collisions as indicated in the figure
Assuming v 2 ∝ , then v 2 / in the reaction plane is given by v 2 {4}//{4} [56].
The mean multiplicity versus v 2 {4}//{4} displayed in Fig. 13.14 for Au+Au and
Cu+Cu collisions at 200 and 62.4 GeV for above three models. The data points
for different system and energies seem to fall on top of each other as expected if
v 2 ∝ . The MCG-N exhibits the sharp increase of v 2 {2}// violating the approximation v 2 ∝ whereas the fKLN-CGC shows a saturation with v 2 ∝ . The MCG-Q
displays intermediate trend between the sharp rise and the saturation. The v 2 saturation or nearly saturation trend exhibited in central Au+Au collisions at 200 GeV is
consistent with a nearly perfect liquid behaviour.
New Q-cumulants method used to determine experimentally flow fluctuations
does not require the knowledge of either reaction plane or participant plane. The
estimation of v 2 {2} can be biased by correlations such as resonance decay, jets, etc.,
which are not related to the reaction or participant plane and denoted as δ. Voloshin et
al., [57], have shown that v 2 {2} and v 2 {4} for the Gaussian form of flow fluctuations
in the participant plane are correlated as
v 2 {2}
2
= =v
2
+ σ v
2
+ δ
(13.38)
v 2 {4}
2
∼ ∼v
2
− σ v
2
(13.39)
v 2 {2}
2
− v 2 {4}
2
∼ δ + 2σ v
2
(13.40)
where v 2 is the elliptic flow, δ is the non-flow contribution and σ v represent flow
fluctuations. This is because fluctuations increase v 2 {2} and decrease v 2 {4} and the
approximations are valid for σ v 2 /v 2 1. It is not possible to determine the value
of σ v 2 from the measurements of v 2 {2} and v 2 {4}. However, we can find the upper
limit on the flow fluctuations taking δ = 0 in [(13.38)–(13.40)], i.e.,
R v(2−4) = σ v 2 /v 2 =
(v 2 {2} 2 − v 2 {4} 2 )
(v 2 {2} 2 + v 2 {4} 2 )
(13.41)