As discussed in Section 2.2, we want to exploit the correlation between protons and lambda particles which originates from the strong interaction and is consequently limited in its range to a few fm [320].
As mentioned in Section 5.2.5, a significant amount of the selected primary proton candidates originates from feed-down of (dominantly) lambda particles. The decay Λ→p +π− has acτ of 7.89 cm [32], hence these daughter protons will not interact with other particles and therefore will not contribute on their own to a correlation.
A topic of current interest is residual correlations in correlation functions [321, 322]. The correlation of a pair of mother particles can survive their decay and thus feed-down to another system. The issue is discussed in many occasions for pp correlations in Au-Au collisions registered with STAR [323–326]. Also the measurement of pp correlations by ALICE [195]
makes an effort to take the remanent correlation from pΛ pairs into account. The effect was discussed already in 1999 by F. Wang in [327]. Fig. 6.6 (left) shows the correlation of pΛ pairs when the momentum difference of the primary proton and the proton of the lambda decay Λ→pdecπis studied. We see that the peak of the pΛ correlation at a momentum difference of the proton lambda pair q(pΛ)≈0 is still present in the ppdec system. However, it is shifted toq(ppdec)≈80 MeV/c. This shift is explained by two effects. The main reason for the peak displacement is the momentum released in the decay of the Λ of p = 101 MeV/c [32]. As pictured in Fig. 6.6 (right) this release directly translates itself into the ppdecrelative momentum.
Ignoring any other influences, we simply expect a peak in the ppdec correlation function at q = 101 MeV/c. The reason why the peak in Fig. 6.6 (left) appears at a lower momentum difference of aboutq(ppdec)≈80 MeV/ccan be understood by phase-space considerations, which give aq2 dependence for the number of pairs. The more abundant high-qpairs dilute the signal, which moves the peak to slightly lower momentum differences, ending up at q ≈80 MeV/c.
Using the same argumentation we can explain the second feature of Fig. 6.6 (left), namely the lowering of the maximum of the correlation function from the pΛ to the ppdec system by more than a factor of ten. The few ppdec pairs originating fromq(pΛ) equal to about 0 to 20 MeV/c, which show a value in C2 ≈1.7 in the example, are diluted by the increase in the number of pairs with q2, for which the pairs are less and less correlated.
A significant number of pΛ pairs stem from ΛΛ pairs, where one partner decayed via Λ→pdecπand the decay proton is selected, thus forming a pdecΛ pair. The measurement of ΛΛ pairs by STAR, presented at the Quark Matter Conference 2012 [328], shows no correlation.
Consequently, the feed-down from Λ hyperons was considered an uncorrelated background in the proton sample. The update from the STAR collaboration [329] and the preliminary measurement by ALICE [317], both appearing at the end of 2014, show a slight anti-correlation.
Keeping in mind the dilution by the decay momentum of about a factor of ten, the assumption
6.5 Purity Correction 117
pΛ
ppdec
q (MeV/c) C2
40 80 120 160 200
0
decay momentum p = 101 MeV/c
Λ
pdec p
q (pΛ) ≈ 0 MeV/c
π q (ppdec) ≈ 101 MeV/c
Figure 6.6: Left: Residual pΛ correlation in the pp system. Taken from [327]. Right:
Kinematics of the decay Λ→pdecπ with its consequence on the ppdec system.
of an uncorrelated background for protons from weak decays is a good approximation.
The interest in associated strangeness production near threshold yields a rich set of data for the reaction pp→pΛK+ and pp→pΣ0K+ [330–332]. Fig. 6.7 (left) shows the enhancement of the cross-section for the Λ production over the one for the Σ0. The clear increase of the ratio from 2.2 for an excess energy of more than 700 MeV towards small excess energies, where the ratio reaches a value of about 30 is interpreted as an absence of a considerable final-state interaction in the pΣ0 system. As shown on the right in Fig. 6.7, the absolute cross-section with the pΛ in the final state can not be described by pure phase-space effects, i. e. a quadratic dependence on the excess energyσ ∝a·2, whereais a constant. However, the pΣ0 system perfectly agrees with such a description. This implies that pΣ0 pairs will not be correlated.
It was elaborated in Section 5.2 that the contamination of the proton sample from misidenti-fied particles is negligible; in Section 5.3.1 it was discussed that and Section E.2 it was discussed that by looking up the differential single-particle purity and/or feed-down fraction, the pair purity can be obtained. While the value of the pair purity can be gained straight forward by dividing the purity-weighted pair distribution by the non-weighted counts, the calculation of the fluctuation of the pair purity is a little more involved. Calculating this fluctuation is desirable in order to give a good graphical representation of the pair purity, get a good understanding of the applied pair purity correction, and test possible systematic variations of the pair purity with, e. g., relative pair momentum. In Section F.2 it is shown, that within the root framework neither the standard error calculation, nor a binomial error, nor variations of it are applicable,
Figure 6.7: Left: Enhancement of σpp→pΛK+ overσpp→pΣ0K+ vs. the excess energy of the system. The ratio increases from about 2.2 at high excess energy towards about 30 at low excess energy. Right: Absolute cross section for the reaction pp→pΛK+ (closed symbols) and pp→pΣ0K+ (open symbols). Expectation from pure phase space
as dashed lines. Both taken from [332].
and only the central limit theorem appropriately describes the fluctuations. In order to apply the central limit theorem, see Eq. F.1, knowledge of the purity fluctuations is necessary. While the single-particle Λ purity varies by about 0.25 in the accepted region of phase space (see the right panel of Fig. 5.9), the fluctuations of the lambda pair-purity are reduced, since most Λ forming a pair stem from the high-purity region|η|<0.9. As shown in Fig. 6.8, the root mean square of the lambda pair-purity distribution is only 0.010. The effect is presented in a bit more detailed way but with slightly different particle selection criteria in [333].
purΛ
0.7 0.75 0.8 0.85 0.9 0.95 1
counts
0 10 20 30 40 50 60 70 80 90
103
×
This Thesis Pb-Pb, 2.76 TeV Centrality 0-10%
pairs pΛ
mT
all ) < 0.2 GeV/c (pΛ
q
= 0.93
Λ〉 pur
〈
) = 0.010 RMS(purΛ
Figure 6.8: Fluctuation of the lambda purity in 0–10% most central events for pairs of pΛ with any mT and q(pΛ)<0.2 GeV/c.
Fig. 6.9 shows the Λ pair-purity purΛ(q, mT) for the 0–10% most central events and
6.5 Purity Correction 119
mT≥1.9 GeV/c2. The bigger fluctuations with decreasing relative momentumq are described by the derived vertical error bars. The pair purity is very high, above 95%, and shows no or only a very weak dependence onq, note the scale of ±0.01 of they axis. Shown as a red line in the figure is a fit of the pair purity with a constant in the region q < 0.15 GeV/c. The good description of the data by the constant confirms that for lowq the Λ pair purity can be regarded as independent ofq. In the current implementation, the fit value is used to correct the data.
c) (GeV/
q
0 0.1 0.2 0.3 0.4 0.5
purityΛ
0.94 0.95
0.96 data
const. fit
= 2.76 TeV sNN
Pb-Pb,
centrality: 0−10%
pairs Λ p
c2
1.90 GeV/
T ≥ m ALICE preliminary
0.0001 fit value: 0.9510±
ALI−PREL−69404
Figure 6.9: Λ pair-purity for the 0–10% most central events and pairs withmT ≥ 1.9 GeV/c2. Obtained by looking up for each pair the single particle purity, see Fig. 5.9 (right).
Similar to the Λ pair-purity, the pair fraction of protons and Λ from feed-down is obtained.
The overall purity, which includes the Λ purity (purΛ) and the feed-down fraction for the sample of Λ (fΛ) and protons (fp), factorizes according to
D
pur(q, mT)E
pair =D
purΛ(q, mT)·(1−fp(q, mT))·(1−fΛ(q, mT))E
pair (6.23)
≈D
purΛ(q, mT)E
pair·D
(1−fp(q, mT))E
pair·D
(1−fΛ(q, mT))E
pair, (6.24) where the brackets denote averaging over pairs. The approximation of Eq. 6.24 is valid, since the variation of the purities and feed-down fractions in phase space, i. e. (y, pT), is small and only partially correlated. The numerical values for the variables of Eq. 6.24 are cataloged for all centrality classes andmT selections in Table 6.1. With this, the unique pΛ and pΛ correlation
functionC2corr. can be recovered via the correction C2corr.(q, mT) =
1
hpur(q, mT)ipair ·(C2raw(q, mT)−1)
+ 1. (6.25)
No difference in the corrected correlation functions between pairs of particles and pairs of anti-particles were found. Therefore, they were combined following the procedure by the PDG (see Eq. 6.19). The following discussion applies to the correlation functions merged over matter
and anti-matter.
centrality (%) mT (GeV/c2) pair hpurΛipair hfΛipair hfpipair hpuripair
0.0-10.0 1.0–1.4 pΛ 0.928 0.328 0.256 0.464
0.0-10.0 1.0–1.4 pΛ 0.955 0.320 0.156 0.548
0.0-10.0 1.4–1.6 pΛ 0.948 0.323 0.220 0.500
0.0-10.0 1.4–1.6 pΛ 0.944 0.318 0.147 0.549
0.0-10.0 1.6–1.9 pΛ 0.943 0.325 0.217 0.499
0.0-10.0 1.6–1.9 pΛ 0.939 0.321 0.149 0.542
0.0-10.0 1.9–∞ pΛ 0.951 0.331 0.214 0.500
0.0-10.0 1.9–∞ pΛ 0.947 0.327 0.151 0.542
10.0-30.0 1.0–1.6 pΛ 0.962 0.323 0.208 0.516
10.0-30.0 1.0–1.6 pΛ 0.970 0.317 0.117 0.585
10.0-30.0 1.6–∞ pΛ 0.964 0.323 0.186 0.531
10.0-30.0 1.6–∞ pΛ 0.960 0.318 0.116 0.579
30.0-50.0 1.0–∞ pΛ 0.979 0.316 0.149 0.570
30.0-50.0 1.0–∞ pΛ 0.981 0.310 0.081 0.621
Table 6.1: Purities used to correct the raw correlation function for uncorrelated contaminations. They were obtained from constant fits for low relative momenta q <0.15 GeV/c.