13 Elliptic Flow in Relativistic Heavy-Ion Collisions
165
(Fig. 13.3 (center)). So, it is believed that the elliptic flow helps in understanding the
interactions between the constituents at an early time in the evolution of the produced
system and hence sensitive to the equation of state of the system.
The invariant triple differential distribution of particles emitted in the collision is
given by the following Fourier expansion [21]:
E
d
3 N
dp 3 =
1
2π
d
2 N
p t dp t dy
1 +
∞
n=1
2v n cos(n(φ − R P ))
(13.1)
where p is the momentum of the particle, p t is the transverse momentum, E is the
energy, y is the rapidity, φ is the azimuthal angle and ψ R P is the reaction plane
angle (Fig. 13.3 (right)). Sine terms vanish in the above Fourier expansion due to the
reflection symmetry with respect to the reaction plane and are not included in the
(13.1). The v 2 for n = 2 represents the elliptic flow and is obtained as
v 2 = =cos(2(φ − R P ))
(13.2)
here, angular brackets represent an average over the particles, summed over all events
in a given sample.
13.2.1 Elliptic Flow Methods
The elliptic flow is found to be very useful for understanding relativistic nuclear
collisions but its value spreads over a range of 20% determined by different analysis methods. It is well known that non-flow correlations not related to the reaction
plane and fluctuations affect the measured v 2 values. We will discuss the standard
event plane method [22, 23], Q-cumulants [24] and probability p(v 2 ) [25]. A higher
accuracy is needed in determining v 2 to compare with the relativistic viscous hydrodynamic calculations to extract the ratio of the shear viscosity to entropy.
13.2.1.1 Event Plane Method
The reaction plane ( R P ) cannot be determined experimentally for an event. So one
uses the event plane which is good approximation to the reaction plane. The nth
order event plane is defined as
E P
n
= tan
−1
(Y n / X n )/n
(13.3)
165
(Fig. 13.3 (center)). So, it is believed that the elliptic flow helps in understanding the
interactions between the constituents at an early time in the evolution of the produced
system and hence sensitive to the equation of state of the system.
The invariant triple differential distribution of particles emitted in the collision is
given by the following Fourier expansion [21]:
E
d
3 N
dp 3 =
1
2π
d
2 N
p t dp t dy
1 +
∞
n=1
2v n cos(n(φ − R P ))
(13.1)
where p is the momentum of the particle, p t is the transverse momentum, E is the
energy, y is the rapidity, φ is the azimuthal angle and ψ R P is the reaction plane
angle (Fig. 13.3 (right)). Sine terms vanish in the above Fourier expansion due to the
reflection symmetry with respect to the reaction plane and are not included in the
(13.1). The v 2 for n = 2 represents the elliptic flow and is obtained as
v 2 = =cos(2(φ − R P ))
(13.2)
here, angular brackets represent an average over the particles, summed over all events
in a given sample.
13.2.1 Elliptic Flow Methods
The elliptic flow is found to be very useful for understanding relativistic nuclear
collisions but its value spreads over a range of 20% determined by different analysis methods. It is well known that non-flow correlations not related to the reaction
plane and fluctuations affect the measured v 2 values. We will discuss the standard
event plane method [22, 23], Q-cumulants [24] and probability p(v 2 ) [25]. A higher
accuracy is needed in determining v 2 to compare with the relativistic viscous hydrodynamic calculations to extract the ratio of the shear viscosity to entropy.
13.2.1.1 Event Plane Method
The reaction plane ( R P ) cannot be determined experimentally for an event. So one
uses the event plane which is good approximation to the reaction plane. The nth
order event plane is defined as
E P
n
= tan
−1
(Y n / X n )/n
(13.3)