104 CHAPTER 6. DISCUSSION
0.5 1
3Dλ
0 0.2 0.4 0.60-10%
0.5 1
[fm] sR
0 2 4
6 π+π-+
π π
-K
-+K
-K+
K+
0.5 1
[fm] oR
0 2 4 6
0.5 1
[fm] lR
0 2 4 6
0.5 1
3Dλ
0 0.2 0.4 0.610-20%
0.5 1
[fm] sR
0 2 4 6
0.5 1
[fm] oR
0 2 4 6
0.5 1
[fm] lR
0 2 4 6
0.5 1
3Dλ
0 0.2 0.4 0.620-40%
0.5 1
[fm] sR
0 2 4 6
0.5 1
[fm] oR
0 2 4 6
0.5 1
[fm] lR
0 2 4 6
[GeV/c]
mT
0.5 1
3Dλ
0 0.2 0.4 0.640-70%
[GeV/c]
mT
0.5 1
[fm] sR
0 2 4 6
[GeV/c]
mT
0.5 1
[fm] oR
0 2 4 6
[GeV/c]
mT
0.5 1
[fm] lR
0 2 4 6
Figure 6.1: Comparison ofmT dependence of HBT radii between charged pions and kaons for four centrality bins.
6.2. FINAL SOURCE ECCENTRICITY 105
[GeV/c]
mT
0 0.5 1
s/RoR
0 0.5 1
1.5 0-10%
π+
π+
π
-π
-K
-+K
-K+
K+
[GeV/c]
mT
0 0.5 1
0 0.5 1
1.5 10-20%
[GeV/c]
mT
0 0.5 1
0 0.5 1
1.5 20-40%
[GeV/c]
mT
0 0.5 1
0 0.5 1
1.5 40-70%
Figure 6.2: Comparison of Ro/Rs between charged pions and kaons.
6.2.1 Initial Eccentricity vs Final Eccentricity
Figure 6.3 shows final eccentricity of charged pions and charged kaons as a function of initial eccentricity. Result for charged pions in this analysis is consistent with the result from the STAR [64]
experiment within systematic uncertainties. In this analysis the pion pairs in 0.2 < kT < 2.0 GeV/c (⟨kT⟩≈0.54 GeV/c) are analyzed to get more statistics. In the analysis of [64], the final eccentricity was determined as the average of three kT bins (0.15 < kT <0.25, 0.25< kT <0.35, 0.35 < kT < 0.60 GeV/c). The consistency with the STAR experiment would indicate that our result is dominated by the pairs with lower kT (∼0.54 GeV/c) and the approximation of final eccentricity is held in our analysis. Final eccentricity of charged pions increases with increasing initial eccentricity, that is, with centrality going from central collision to peripheral one, while the final eccentricity is about half of initial eccentricity. It indicates that the source expands to in-plane direction with the large elliptic flow, while it still has elliptical shape at freeze-out because the final eccentricity has finite value. In other words, it indicates that the expansion time is not long enough to vanish the initial eccentricity. On the other hand, final eccentricity of charged kaons is larger than that of charged pions and close to initial eccentricity.
Here we need to note that the averagekT of kaons is larger than that of pions. The HBT radius has kT dependence as shown in the previous section and the emission region of pairs with higher kT corresponds to the region closer to the surface of the source under the picture of the radial flow. Therefore the εf inal also depends on the kT. To make it clear whether the difference is due to the different average kT between both particle species, we discuss the difference by comparing theεf inal for both species at the samemT in the next section.
Figure 6.4 shows the oscillation amplitudes for several combinations of HBT radii for charged pions and kaons following the similar definitions of Eq. (6.1) as a function of the initial eccentricity.
Left top panel is the same as Fig. 6.3. These oscillation amplitudes other than 2R2s,2/Rs,0would also reflect the spatial eccentricity at final state, and would additionally contain the temporal informa-tion and more sensitive contribuinforma-tion from the collective flow compared toRs (e.g. see Eq. (2.37)).
There is no significant difference between both species except Rs and Rl. The amplitudes includ-ing the temporal information, such as Ro and Ros, show the similar trends that the values in all central regions are larger than the values ofRs for pions. Especially in most central collisions, the amplitudes of Ro has finite value with although the εf inal is almost close to zero from Rs. Under the assumption that Ro and Rs have the same spatial information, that result may indicate that spatial eccentricity is quite small, but the emission duration of particles has azimuthal dependence or the flow anisotropy makes the oscillation only for Ro and Ros because Ro explicitly has those
106 CHAPTER 6. DISCUSSION
initial
0 0.2 0.4 ε
final ε
0 0.2 0.4
K
-+K
-K+
K+
π
-π -++
+π π
Sys. error from BW STAR (PRL93 12301)
final
= ε
initial
ε
Au+Au 200GeV
Figure 6.3: Initial eccentricity vs final eccentricity for charged pions and kaons. ThekT ranges are 0.2 < kT < 2.0 GeV/c for pions and 0.3 < kT <2.0 GeV/c for kaons. The initial eccentricity is calculated by a Monte-Carlo Glauber simulation, where color bands shows the systematic uncer-tainties. A band consisting of two solid lines represents the 30% systematic uncertainties derived from the definition of the final eccentricity [67]. Result for pions measured by STAR [64] experiment is also shown. Dashed line showsεinitial=εf inal.
dependence as shown in Eq. (2.20) and Eq. (2.37).
6.2.2 mT Dependence of Final Eccentricity
As shown in the previous section, it has been found that the εf inal of kaons is larger than that of pions and almost the same as the initial eccentricity. The HBT radii does not represent the whole source size, but the size of emission region for a expanding source. Under the picture of the collective radial flow, all particles have a common transverse velocity at the same spatial position and it is expected that particles being closer to the source surface get larger velocity. Therefore the measured length decreases with increasing the transverse momentum of pairs (kT). At the presence of such a radial flow effect, the mT scaling needs to be tested for the comparison of both species.
Figure 6.5 shows the relative amplitude of the azimuthal HBT radii for charged pions and kaons as a function ofmT in two centrality bins, where the centrality regions of pions and kaons are set to be the same. The left top panel corresponds to the εf inal. The relative amplitudes of Rs and Ro show monotonic increases withmT, except for Ro in 0-20%. These increases will be basically due to the variation of the emission region and qualitatively agree with the results of an ideal hydrodynamic model [26] shown in Fig. 6.6, which are calculated for non-central Au+Au collisions at RHIC energy. In Fig. 6.6, the slope of kT dependence of the relative aptitudes of Ro is steeper than that of Rs above ∼0.15 GeV/c, which is due to the effect of flow. On the other hand, our results does not show such strong kT(mT) dependence in Ro. We note that that model does not
6.2. FINAL SOURCE ECCENTRICITY 107
initial
0 0.2 ε0.4
0 0.2 0.4
s,0
/ R2 s,2
2R2
initial
0 0.2 ε0.4
0 0.2 0.4
o,0
/ R2 o,2
-2R2
initial 0 0.1 0.2 0.3 ε0.4 0
0.2 0.4
l,0
/ R2 l,2
-2R2
initial
0 0.2 ε0.4
0 0.2 0.4
s,0
/ R2 o,2
-2R2
initial
0 0.2 ε0.4
0 0.2 0.4
s,0
/ R2 os,2
2R2
K
-+K
-K+
K+
π
-π -++
+π π
final
= ε
initial
ε
Au+Au 200GeV
Figure 6.4: Relative amplitudes of azimuthal HBT radii for charged pions and kaons with respect to 2nd-order event plane, where color bands shows the systematic uncertainties. Dashed lines show the line of x-axis=y-axis.
incorporate the recent theoretical improvements described in Sec. 1.3.5, such as prethermal flow, stiffer equation of state, and viscosity.
The−2R2o,2/R2s,0 and 2R2os/Rs,0 show similar trends to the −2R2o,2/Ro,02 , which values in 0-10%
seem to be independent of mT. The Ro and Ros contain temporal information in addition to the geometrical information and may be different sensitivity to flow anisotropy due to the definition of the outward direction.
When we compare the results of charged pions and kaons, in 0-20% centrality both species are consistent at the same mT, while in non-central collisions the relative amplitude ofR2s of kaons is still larger than that of pions even at the same mT. At higher mT, the relative amplitude of R2s does not represent the whole source eccentricity, but would reflect the eccentricity. The relative amplitudes calculated by other HBT parameters do not show any significant difference between pions and kaons. On the other hand, for the average radii, theRs shows a good agreement between pions and kaons, and theRoandRlshow the difference in central and mid-cental collisions. It may be difficult to intuitively understand these data, and the average radii and the relative amplitudes representing the variation of the emission region between in-plane and out-of-plane directions would be at least different physics quantities. Further discussion is done in the context of the Blast-wave study in the following section.
108 CHAPTER 6. DISCUSSION
[GeV/c]
mT
0 0.5 1
0 0.2
0.4 s,0
/ R2 s,2
2R2
[GeV/c]
mT
0 0.5 1
0 0.2
0.4 o,0
/ R2 o,2
-2R2
[GeV/c]
mT
0 0.5 1
0 0.2
0.4 l,0
/ R2 l,2
-2R2
[GeV/c]
mT
0 0.5 1
0 0.2
0.4 s,0
/ R2 o,2
-2R2
[GeV/c]
mT
0 0.5 1
0 0.2
0.4 s,0
/ R2 os,2
2R2 Au+Au 200GeV
0-20%
π
-π -++ π π+
20-60%
π
-π -++ π π+
0-20%
K
-+K
-K+
K+
20-60%
K
-+K
-K+
K+
Figure 6.5: Relative amplitude of azimuthal HBT radii for charged pions and kaons with respect to 2nd-order event plane as a function of mT in two centrality bins in Au+Au 200 GeV collisions, where color bands shows the systematic uncertainties. The relative ratio of Rs shown in the left top panel corresponds to the εf inal.
[degree]
Ψ2
- φ
0 50 100 150
]2[fm 2 sR
0 5 10
[GeV/c]
kT
0.00 0.15 0.30 0.45
[degree]
Ψ2
- φ
0 50 100 150
]2[fm 2 oR
0 10 20 30 40
[GeV/c]
kT
0 0.2 0.4
Relative amplitude
0 0.1 0.2
Rs Ro
Figure 6.6: Azimuthal oscillations and relative amplitudes of the HBT radii calculated by an ideal hydrodynamic model [26] with b=7 fm Au+Au collisions at√
sNN=130 GeV, for 4kT values. Data points are read from the figures in [26].